Black-Scholes formula

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A Primer for the Mathematics of Financial Engineering by Dan Stefanica

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asset allocation, Black-Scholes formula, capital asset pricing model, constrained optimization, delta neutral, discrete time, Emanuel Derman, implied volatility, law of one price, margin call, quantitative trading / quantitative finance, Sharpe ratio, short selling, time value of money, transaction costs, volatility smile, yield curve, zero-coupon bond

From definition (3.6), we know that P(a :'S X :'S b) = Y~J-L P(a~X~b) Note that the constant random variable X = IL is a degenerate normal variable with mean IL and standard deviation O. h(x) = 93 FINANCIAL APPLICATIONS The Black-Scholes formula. CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORMULA. 94 Assume that the price of the underlying asset has lognormal distribution and volatility a, that the asset pays dividends continuously at the rate q, and that the risk-free interest rate is constant and equal to r. Let C (S, t) be the value at time t of a call option with strike K and maturity T, and let P(S, t) be the value at time t of a put option with strike K and maturity T. Then, Implied volatility. The concept of hedging. ~-hedging and r-hedging for options. Implementation of the Black-Scholes formula. 0(8, t) The Black-Scholes formula P(8, t) The Black-Scholes formulas give the price of plain vanilla European call and 4 put options, under the assumption that the price of the underlying asset ~as lognormal distribution.

We call this routine cum_disLnormal(tt), and include its pseudocode in Table 3.1. The pseudocode for implementing the Black-Scholes formulas using the routine cum_disLnormal(t) for approximating N(t), is given in Table 3.2. d l = (In(*)+(r q+~2)(T-t))/((J"VT-t);d2=dl-(J"VT-t C = Se-q(T-t)cum_dist-llormal( dd - K e-r(T-t)cum_dist-llormal( d 2 ) P = Ke-r(T-t)cum_disLnormal( -d2 ) - Se-q(T-t)cum_dist-llormal( -dl ) Example: Use the Black-Scholes formula to price a six months European call option with strike 40, on an underlying asset with spot price 42 and volatility 30%, which pays dividends continuously, with dividend rate 3%. Assume that interest rates are constant at 5%. CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORMULA. 110 Price a six months European put option with strike 40 on the same asset, using the Black-Scholes formula. Check whether the Put-Call parity is satisfied.

RISK-NEUTRAL DERIVATION OF BLACK-SCHOLES (4.47) While risk-neutral pricing does not hold for path-dependent options or American options, it can be used to price plain vanilla European Call and Put options, and is one way to derive the Black-Scholes formulas. CHAPTER 4. LOGNORMAL VARIABLES. RN PRICING. 134 Applying (4.45) to plain vanilla European options, we find that C(O) = e- rT ERN[max(8(T) - K,O)]; P(O) = e- rT ERN[max(K - 8(T), 0)], (4.48) (4.49) where the expected value is computed with respect to 8(T) given by (4.47). We derive the Black-Scholes formula for call options using (4.48). The Black-Scholes formula for put options can be obtained similarly from (4.49). By definition, max(8(T) _ K,O) _ - {8(T) - K, if 8(T) 2 K; 0, otherwise. d1 = d2 + ~VT = In ( S~) ) + (r _q + ~2) vfr ~ T T . (4.53) Note that the term d 1 from (4.53) is the same as the term d 1 from the BlackScholes formula given by (3.55) if t = o. Formula (4.52) becomes 1 = 8(0)e- qT . rn= v 27r 1 00 -d1 e- Ty2 dy - 1 00 Ke- rT 1 rn= V 27r 2 e- Tx dx


pages: 447 words: 104,258

Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues by Alain Ruttiens

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algorithmic trading, asset allocation, asset-backed security, backtesting, banking crisis, Black Swan, Black-Scholes formula, Brownian motion, capital asset pricing model, collateralized debt obligation, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, discounted cash flows, discrete time, diversification, fixed income, implied volatility, interest rate derivative, interest rate swap, margin call, market microstructure, martingale, p-value, passive investing, quantitative trading / quantitative finance, random walk, risk/return, Satyajit Das, Sharpe ratio, short selling, statistical model, stochastic process, stochastic volatility, time value of money, transaction costs, value at risk, volatility smile, Wiener process, yield curve, zero-coupon bond

Here, with a 3M = 90 days option, we have: and similarly: moneyness: strike: −3 60.819 −2 68.058 −1 75.296 —0 82.535 = F —1 89.774 —2 97.012 —3 104.251 Figure 10.5 Determination of a moneyness scale These values are shown in Figure 10.5 together with two strikes, K′′ and K′, respectively at: K′ = S − €10 = 73.06 corresponding to a moneyness of −1.31 K′′ = S + €10 = 93.06 corresponding to a moneyness of +1.45. The moneyness measure is mainly used with respect to the option smile, as developed in Chapter 12, Section 12.1.3. 10.2.5 Beyond the Black–Scholes formula The Black–Scholes formula is an answer to the diffusion equation (cf. Eq. 10.6, for call options) leading to an option valuation subject to the very specific assumptions as set in Section 10.2.1. This formula, and its variants, is called an “analytical” solution to option pricing, since if suffices to replace the variables of the formula by their values relating to the option to be priced. Moreover, the fact remains that the analytic – also called “close form” or “closed-form” – Black–Scholes formula presents the advantage of allowing a straightforward calculation of options sensitivities (cf. Section 10.5). However, in many instances, some of the Black–Scholes assumptions must be relaxed or modified, for example in the case of: incorporating dividend payments (options on equities); American options; options on interest rates, volatility, or other underlyings, that do not fit with the geometric general Wiener process (see Chapters 11 and 12); second generation options (cf.

They all pursue the same objective of modeling in a more or less realistic way how the underlying spot price will move over time, to compute what should be the option fair/theoretical price accordingly. 10.2 THE BLACK–SCHOLES FORMULA 10.2.1 Introduction F. Black and M. Scholes were the first to publish, 2 in 1973, a well-grounded formula for computing call and put options prices. The way their formula is established is useful to better understand the underlyings of option pricing. This formula is subject to rather restrictive hypotheses, which may be questioned in some circumstances but at least it constitutes a robust pricing tool, not necessarily the case for further, more complex, pricing models (cf. Chapter 15, Section 15.1), whose sophistication is also synonymous of real difficulties to properly assess correct values to their ingredients. The Black–Scholes formula is applicable to European options only, and provided the underlying financial instrument offers no return during the lifetime of the option: for example, a stock delivering no dividend during such period, or any non-financial commodity.

Let us compute a European call option maturing in 90 days (or = 90/365 = 0.2466 year) on L'OREAL stock quoting EUR 64.5 (data as of Jan 06), with an ATMS strike price of EUR 64.5; the risk-free interest rate is 2.514% p.a. and the stock volatility is 11.9% p.a. Equations 10.7–10.9 give: because ln(S/K) = ln(64.5/64.5) = ln1 = 0, d1 = (0.02514 + 0.1192/2) × 0.2466/0.119√0.246 = 0.134457 d2 = d1 − 0.119√0.2466 = 0.075363 hence, using the cumulative normal distribution N(0, 1), N(d1) = 0.553479, and N(d2) = 0.530037 → C = 64.5 × 0.553479 − 64.5 × e−0.02514×0.2466 × 0.530037 = €1.72 (rounded). 10.2.2 Variants of the Black–Scholes formula The Black–Scholes formula can be extended to European options on any kind of underlying offering a return ≠ 0, provided that its process can be reasonably modeled by a geometric Wiener process. This extension is valid if the underlying return can be considered as continuous in time.3 This will be the case of a LIBOR rate of return, for example. Let precise the above r return, by calling it rm, as the market rate of return, and calling ru the return of the underlying, both up to the maturity T of the option.

Monte Carlo Simulation and Finance by Don L. McLeish

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Black-Scholes formula, Brownian motion, capital asset pricing model, compound rate of return, discrete time, distributed generation, finite state, frictionless, frictionless market, implied volatility, incomplete markets, invention of the printing press, martingale, p-value, random walk, Sharpe ratio, short selling, stochastic process, stochastic volatility, survivorship bias, the market place, transaction costs, value at risk, Wiener process, zero-coupon bond, zero-sum game

/(dif+BLSPRICE(So,strike,r,T,sigma,0))]; plot(strike/So,re) xlabel(’Strike price/initial price’) ylabel(’relative error in Black Scholes formula’) The relative error in the Black-Scholes formula obtained from a simulation of 100,000 is graphed in Figure 3.14. The logistic distribution differs only slightly GENERATING RANDOM NUMBERS FROM NON-UNIFORM CONTINUOUS DISTRIBUTIONS151 Figure 3.14: Relative Error in Black-Scholes price when asset prices are loglogistic, σ = .4, T = .75, r = .05 from the standard normal, and the primary difference is in the larger kurtosis or weight in the tails. Indeed virtually any large financial data set will differ from the normal in this fashion; there may be some skewness in the distribution but there is often substantial kurtosis. How much difference does this slightly increased weight in the tails make in the price of an option? Note that the Black-Scholes formula overprices all of the options considered by up to around 3%.

Assume that we simulated the asset prices under this model and then valued a call option, say. Then since ln(ZT /Z0 ) has a N ((c − σ2 )T, σ2 T ) distribution 2 we could use the Black-Scholes formula to determine the conditional expected value 268 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS Z E0 [exp{− T rt dt}(ZT − K)+ |rs , 0 < s < T ] (5.18) 0 = EE0 [(S0 e(c−r)T eW − e−rT K)+ |rs , 0 < s < T ], where W has a N (−σ2 T /2, σ 2 T ) (c−r)T = E[BS(S0 e 1 , K, r, T, σ)], with r = T Z 0 Here, r is the average interest rate over the period and the function BS is the Black-Scholes formula (5.2). In other words by replacing the interest rate by its average over the period and the initial value of the stock by S0 e(c−r)T , the Black-Scholes formula provides the value for an option on an asset driven by (5.17) conditional on the value of r. The constant interest rate model is a useful control variate for the more general model (5.16).

It follows that the conditional distribution is lognormal with mean η = Xt q(t, T ) and volatility qR T 2 σ (u)du. parameter t We now derive the well-known Black-Scholes formula as a special case of 2.53. For a call option with exercise price E, the payoff function is V0 (ST ) = max(ST − E, 0). Now it is helpful to use the fact that for a standard normal random variable Z and arbitrary σ > 0, −∞ < µ < ∞ we have the expected value of max(eσZ+µ , 0) is eµ+σ 2 /2 Φ( µ µ + σ) − Φ( ) σ σ (2.57) where Φ(.) denotes the standard normal cumulative distribution function. As a result, in the special case that r and σ are constants, (2.53) results in the famous Black-Scholes formula which can be written in the form V (S, t) = SΦ(d1 ) − Ee−r(T −t) Φ(d2 ) where d1 = √ log(S/E) + (r + σ 2 /2)(T − t) √ , d2 = d1 − σ T − t σ T −t (2.58) 90 CHAPTER 2.

The Concepts and Practice of Mathematical Finance by Mark S. Joshi

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Black-Scholes formula, Brownian motion, correlation coefficient, Credit Default Swap, delta neutral, discrete time, Emanuel Derman, fixed income, implied volatility, incomplete markets, interest rate derivative, interest rate swap, London Interbank Offered Rate, martingale, millennium bug, quantitative trading / quantitative finance, short selling, stochastic process, stochastic volatility, the market place, time value of money, transaction costs, value at risk, volatility smile, yield curve, zero-coupon bond

We then see how this discrete model can be used as an approximation to a continuous model, and we deduce the Black-Scholes formula for the price of a call option via a limiting argument. Having developed the Black-Scholes formula, we then discuss in Chapter 4 its flaws and how these flaws affect its use in practice. This chapter is very much a foretaste for chapters near the end of the book where we study alternative models of price evolution which try to compensate for the shortcomings of the BlackScholes model. In Chapter 5, we step up a mathematical gear and introduce the Ito calculus. With this calculus we introduce the geometric Brownian motion model of stock price evolution and deduce the Black-Scholes equation. We then show how the BlackScholes equation can be reduced to the heat equation. This yields a derivation of the Black-Scholes formula. In Chapter 6, we step up another mathematical gear and this is the most mathematically demanding chapter.

(6.113) The final expectation is just the probability that ST is greater than K in the S-numeraire martingale measure. Using the solution of the SDE for a Brownian motion with drift, this is equal to the probability that Soe(' +a212)T+a,ITN(o,1) > K. (6.114) A straightforward computation gives us the first term in the Black-Scholes formula. Risk neutrality and martingale measures 170 To get the second term in the Black-Scholes formula, it is easier to use B as numeraire. Note that this neatly explains the division of the Black-Scholes formula into two terms with coefficients So and e-',T K. Note also that the computation of the first term is made substantially easier by the use of the correct numeraire. O For each complete market, we now have multiple martingales measure each one associated to a choice of numeraire. How do they relate to each other?

Index N, 65 N(0, 1), 57 N(µ, a2), 98 or-field, 458 accreting notional, 429 admissible exercise strategy, see exercise strategy, admissible almost, 257 almost surely, 99 American, 10 American option, see option, American amortising notional, 429 annualized rates, 302 annuity, 308 .anti-thetic sampling, 192 arbitrage, 19-20, 27-29,429 and bounding option prices, 29-39 arbitrage-free price, 45, 46 arbitrageur, 12-13, 18 Arrow-Debreu security, 152 at-the-forward, 31 at-the-money, 30, 66 auto cap, 429 bank, 12 barrier option, see option, barrier basis point, 429 basket option, 261 Bermudan option, see option, Bermudan Bermudan swaption, see swaption, Bermudan BGM, 429 implementation of, 450-453 BGM model, 322-355 automatic calibration to co-terminal swaptions, 342 long steps, 337 running a simulation, 337-342 BGM/J, 429 BGM/J model, see BGM model bid-offer spread, 21 Black formula, 173, 310-311 approximate linearity, 356 approximation for swaption pricing under BGM model, 341 Black-Scholes formula, see option, call, Black-Scholes formula for Black-Scholes density, 188 Black-Scholes equation, 69, 160, 161 for options on dividend-paying assets, 123 higher-dimensional, 271 informal derivation of, 114-116 rigorous derivation, 116-119 solution of, 119-121 with time-dependent parameters, 164 Black-Scholes formula, 65 Black-Scholes model, 74, 113, 430 Black-Scholes price, 19 Black-Scholes model, 76 bond, 4 6, 430 callable, 301 convertible, 7, 430 corporate, 7 government, 1 premium, 2 riskless, 5, 7 zero-coupon, 5, 24-26, 28, 302, 433 Brownian bridge, 230 Brownian motion, 97-100, 101, 107, 142, 260, 430 correlated, 263 higher-dimensional, 261-263 Buffett, Warren, 2 bushy tree, see tree, non-recombining calibration to vanilla options using jump-diffusion, 377 call, 301 call option, see option, call callable bond, see bond, callable cap, 309, 430 caplet, 309-311, 430 strike of, 309 caption, 326, 430 cash bond, 26, 430 Central Limit theorem, 56, 60 Central Limit Theorem, 64 Central Limit theorem, 278,463 central method, 238 533 Index 534 CEV, see constant elasticity of variance chain rule, 106 for stochastic calculus, 109 characteristic function, 408 Cholesky decomposition, 227 cliquet, 425, 430 call, 425 optional, 426 put, 425 CMS, see swap, constant maturity co-initial, 317, 340 co-terminal, 317, 340 commodities, 123 complete market, 152, 430 compound optionality, 426 conditional probability, 460 consol, 430 constant elasticity of variance, 113 constant elasticity of variance process, 355 constant maturity swap, see swap, constant maturity contingent claim, 152, 430 continuously compounding rate, 25, 26 control variate and pricing of Bermudan swaptions, 351 on a tree, 288 convenience yield, 123 convexity, 35-37, 81 as a function of spot price in a log-type model, 383 correlation, 466 between forward rates, 321, 335 correlation matrix, 268, 466 cost of carry, 123 coupon, 4, 301, 430 covariance, 466 covariance matrix, 466 and implementing BGM, 343 crash, 10, 86 credit default swap, 23 credit rating, 316, 430 cumulative distribution function, 461 cumulative normal function, 65, 435, 437 default, 1 deflated, 168 Delta, 76, 80,430 and static replication, 246, 248 Black-Scholes formula for call option, 80 integral expression for, 189 Delta hedging, see hedging, Delta dependent, 461 derivative, 10, 430 credit, 11 weather, 11 Derman-Kani implied tree, 381 deterministic future smile, 244, 426 digital, 430 digital option, see option, digital dimensionality, 224, 438 dimensionality reduction, 229 discount curve, 431 discretely compounding money market account, 324 displaced diffusion model, 355 distribution log-normal, see log-normal distribution diversifiable risk, 431 diversification, 8 dividend, 7, 431 scrip, 25 dividend rate, 25 dividends and the Black-Scholes equation, 121-123 drift, 60, 111 of a forward rate under BGM, 330 real-world, 64 Dupire model, 381 dynamic replication, see replication, dynamic early exercise, 68 equivalent martingale measure for a tree with jumps, 363 equivalent probability measures, see probability measure, equivalence European, 10 European contingent claim, 116 exercise, 10 exercise boundary, 289 exercise region, 289 exercise strategy admissible, 286 expectation, 431, 462 conditional, 155 fat tails, 85, 431, 464 Feynman-Kac theorem, 161 fickle, 377 filtration, 143, 154, 162 first variation, see variation, first fixed leg, 306 fixed rate, 431 floating, 300 floating leg, 306 floating rate, 431 floating smile, see smile, floating floor, 309,431 floorlet, 309, 431 floortion, 326, 431 forward contract, 9, 22, 181, 431 and risk-neutrality, 137 value of, 26 forward price, 26, 31 forward rates, 303-305 forward-rate agreement, 23, 304, 431 Fourier transform, 395, 408 FRA, see forward-rate agreement free boundary value problem, 290 Gamma, 77, 80, 431 and static replication, 246, 248 Black-Scholes formula for a call option, 80 non-negativity of, 384 Index Gamma distribution, 402 Gamma function, 402 incomplete, 405 Gaussian distribution, 57, 103 Gaussian random variable synthesis of, 191 gearing, 300 geometric Brownian motion, 111, 114 gilt, 314 Girsanov transformation, 214 Girsanov's theorem, 158, 166, 210-213, 368, 390, 431 higher-dimensional, 267-271 Greeks, 77-83, 431 and static replication, 246, 248 computation of on a tree, 186 of multi-look options, 236-238 heat equation, 119, 120-121 Heath, Jarrow & Morton, 322 hedger, 12-13, 18 hedging, 4, 8, 11, 67-68, 431, 441 and martingale pricing, 162-164 Delta, 18-19, 68, 73, 76, 115, 118, 162 exotic option under jump-diffusion, 535 Ito's Lemma, 106-110 application of, 111-114 multi-dimensional, 264 joint density function, 464 joint law of minimum and terminal value of a Brownian motion with drift, 213 without drift, 208 jump-diffusion model, 87, 364-381 and deterministic future smiles, 244 and replication of American options, 293 price of vanilla options as a function of jump intensity, 374 pricing by risk-neutral evaluation, 364-367 jump-diffusion process, 361 jumps, 86-88 jumps on a tree, 362 Kappa, 79 knock in, 202 knock out, 202 knock-in option, see option, barrier knock-out option, see option, barrier kurtosis, 85, 432, 464 375 Gamma, 77 in a one-step tree, 44-45 in a three-state model, 49 in a two-step model, 51 of exotic options, 424 vanilla options in a jump-diffusion world, 372 Vega, 79 hedging strategy, 17-18, 44, 76 stop-loss, 143 hedging, discrete, 76 HJM model, 322 homogeneity, 274, 281, 383 implied volatility, see volatility, implied importance sampling, 193 in-the-money, 30 incomplete, 431 incomplete market, 50, 361, 367-375, 389, 390 incomplete model, 89 incremental path generation, see path generation, incremental independent, 461 information, 2, 4, 113, 140-145, 162, 401 conditioning on, 145 insider trading, 3 insurance, 12 inverse cumulative normal function, 192, 435, 436 inverse floater, 359 Ito, 97 Ito calculus higher-dimensional, 261, 263-266 Ito process, 106, 154 law of large numbers, 69, 191, 462 law of the minimum of a Brownian motion drift, 215, 216 law of the unconscious statistician, 463 Leibniz rule, 110 leveraging, 300 LIBID, 432 LIBOR, 302, 315, 432 LIBOR market model, 322 LIBOR-in-arrears, 312-313 LIBOR-in-arrears caplet pricing by BGM, 326 LIBOR-in-arrears FRA pricing by BGM, 326 likelihood ratio, 195, 237 liquidity, 21 Lloyds, 6 log-normal distribution, 61 log-normal model, 58 approximation by a tree, see tree, approximating a log-normal model for stock price movements, 112 log-type model, 382-385 long, 21, 432 low-discrepancy numbers, 193 the pricing of exotic options, 445-447 lucky paths, 369 marginal distribution, 465 Margrabe option, see option, Margrabe market efficiency, 2-4 weak, 3, 4, 99 market maker, 74 market model, 432 market price of risk, 89, 112 Index 536 Markov property, 3, 98, 99 strong, 210 martingale, 129, 145, 432 and no arbitrage, 146 continuous, 154-160 discrete, 146 higher-dimensional, 267 martingale measure, 148 choice of, 376 uniqueness, 150 martingale pricing and time-dependent parameters, 164-165 based on the forward, 172-175 continuous, 157-160 discrete, 145-154 equivalence to PDE method, 161-162 with dividend-paying assets, 171 martingale representation theorem, 162 maturity, 5 maximal foresight, 296 mean-reverting process, 390 measure change, 368 model risk, 244 moment, 432 moment matching, 193 and pricing of Asian options, 231-233 money-market account, 26, 114, 430 moneyness, 385 monotonicity theorem, 27 Monte Carlo simulation, 69, 462 and price of exotic options using a jump-diffusion model, 379 and pricing of European options, 191 computation of Greeks, 194-195 variance reduction, 192 Moro, 435 mortgage, 301 multi-look option, see option, multi-look Name, 6 natural payoff, 330 NFLWVR, 132, 135 no free lunch principle, 19 no free lunch with vanishing risk, see NFLWVR no-arbitrage, 45 non-recombining tree, see tree, non-recombining normal distribution, see Gaussian distribution, 461 notional, 304 numeral e, 168, 174, 310, 312, 314, 324 change of, 167 numerical integration and pricing of European options, 187-190 option, 9-12 American, 68, 144, 284, 429 boundary conditions for PDE, 290 lower bounds by Monte Carlo, 293-295 PDE pricing, 289-291 pricing on a tree, 287-289 replication of, 291-293 seller's price, 297 theoretical price of, 287 upper bounds by Monte Carlo, 295-297 American digital, 219 American put, 219 Asian, 222, 429 pricing by PDE or tree, 233-234 static replication of, 249-251 barrier, 69, 429 definition, 202-204 price of down-and-out call, 217, 218 basket, 261, 429 Bermudan, 284,429 binary, 429 call, 10, 181, 430 American, 32 Black-Scholes formula for, 65, 160 down-and-in, 202 down-and-out, 202 formula for price in jump-diffusion model, 366, 367 pay-off, 29 perpetual American, 299 pricing under Black-Scholes, 114 chooser, 294 continuous barrier expectation pricing of, 207-208, 216-219 PDE pricing of, 205-207 static replication of, 244-247, 252-256 static replication of down-and-out put, 244-246 continuous double barrier static replication, 246-247 digital, 83, 257 call, 83 put, 83 digital call, 181 Black-Scholes formula for price of, 183 digital put, 181 Black-Scholes formula for price of, 183 discrete barrier, 222 static replication of, 247-249 double digital, 130 European, 431 exotic, 10, 87 Monte Carlo, 444445 pricing under jump-diffusion, 379-381 knock-in, 431 knock-out, 69, 432 Margrabe, 260, 273-275 model-independent bounds on price, 29-39 multi-look, 223 Parisian, 432 path-dependent, 223 and risk-neutral pricing, 223-225 static replication of, 249-251 power call, 182 put, 10, 181,432 Black-Scholes formula for, 65 pay-off, 30 Index quanto, 260, 275-280 static replication of up-and-in put with barrier at strike, 251-252 trigger, 433 vanilla, 10 with multiple exercise dates, 284 out-of-the-money, 30 path dependence weak, 225 path generation, 226-230 incremental, 228 using spectral theory, 228 path-dependent exotic option, see option, path-dependent pathwise method, 195, 236 PDE methods and the pricing of European options, 195-196 Poisson process, 364 positive semi-definite, 467 positivity, 7, 28 predictable, 162 predictor-corrector, 340 present valuing, 302 previsible, 370 pricing arbitrage-free, 22 principal, 5, 301 probability risk-neutral, see risk-neutral probability probability density function, 461 probability measure, 458 equivalence, 147 product rule for Ito processes, 110 pseudo-square root, 468 put option, see option, put put-call parity, 30, 65, 67 put-call symmetry, 252-256 quadratic variation, 100, see variation, quadratic quanto call, 277 quanto drift, 276 quanto forward, 277 quanto option, see option, quanto quasi Monte Carlo, 194 Radon-Nikodym, 214 Radon-Nikodym derivative, 213 random time, 88, 143 random variable, 459 real-world drift, see drift, real-world recombining trees implementing, 443 reflection principle, 208-210 replication, 23, 116 and dividends, 122 and the pricing of European options, 196-198 classification of methods, 257 dynamic, 198, 257 in a one-step tree, 48-49 537 in a three-state model, 50 semi-static and jump-diffusion models, 381 static, 198 feeble, 257 mezzo, 257 strong, 243, 257 weak, 243, 257 repo, 315 restricted stochastic-volatility model, see Dupire model reverse option, 320 reversing pair, 319 Rho, 79 rho, 432 risk, 1-2, 8, 9 diversifiable, 8-9 purity of, 9 risk neutral, 19 risk premium, 46, 60, 64, 111, 119,432 risk-neutral distribution, 64 risk-neutral density as second derivative of call price, 137 in Black-Scholes world, 139 risk-neutral expectation, 64 risk-neutral measure, 148, 432 completeness, 166 existence of, 129 uniqueness, 166 risk-neutral pricing, 64 65, 140 higher-dimensional, 267-271 risk-neutral probability, 47, 52, 54, 59, 128 risk-neutral valuation, 59 in a one-step tree, 45-48 in a three-state model, 50 in jump models, 86 two-step model, 52 riskless, 1 riskless asset, 28 Rogers method for upper bounds by Monte Carlo, 295, 350 sample space, 458 self-financing portfolio, 28, 116-117, 128, 163, 369 dynamic, 28 share, 6-7, 432 share split, 57 short, 432 short rate, 25, 433 short selling, 21 simplex method, 295 skew, 433, 464 smile, 74-77 displaced-diffusion, 356,420 equity, 421 floating, 88, 385, 407, 413-414 foreign exchange, 413 FX, 424 interest-rate, 355-357, 424 jump-diffusion, 378, 415 sticky, 88, 413-414 sticky-delta, 413 538 smile (cont.) stochastic volatility, 398, 416 time dependence, 414-415 Variance Gamma, 406,417 smile dynamics Deiman-Kani, 420 displaced-diffusion, 420 Dupire model, 420 equity, 421 FX, 424 interest-rate, 424 jump-diffusion, 415 market, 413-415 model, 415-421 stochastic volatility, 416 Variance Gamma, 417 smoothing operator, 120 spectral theory, 228 speculator, 12, 18 split share, see share split spot price, 31 square root of a matrix, 467 standard error, 191 standard deviation, 463 static replication, see replication, static stepping methods for Monte Carlo, 439 stochastic, 433 stochastic calculus, 97 stochastic differential equation, 105 for square of Brownian motion, 107 stochastic process, 102-106, 141 stochastic volatility, 88, 389 and risk-neutral pricing, 390-393 implied, 400 pricing by Monte Carlo, 391-394 pricing by PDE and transform methods, 395-398 stochastic volatility smiles, see smile, stochastic volatility stock, 6-7, 433 stop loss hedging strategy, 18 stopping time, 143, 286, 346 straddle, 182, 257 Index lower bound via local optimization, 347 lower bounds by BGM, 345-349 pricing by BGM, 325 upper bounds by BGM, 349-352 cash-settled, 327 European, 310 price of, 313-314 payer's, 309,432 pricing by BGM, 323 receiver's, 309, 432 swaptions rapid approximation to price in a BGM model, 340 Taylor's theorem, 67, 80, 108 term structure of implied volatilities, 334 terminal decorrelation, 339, 352 Theta, 79, 433 and static replication, 246, 248 time homogeneity, 33, 333 time value of money, 24-26 time-dependent volatility and pricing of multi-look options, 235 Tower Law, 155 trading volatility, see volatility, trading of trading volume, 401 transaction costs, 21, 76, 90, 91 trapezium method, 188 tree with multiple time steps, 50-55 and pricing of European options, 183-186 and time-dependent volatility, 184 approximating a log-normal model, 60-68 approximating a normal model, 55-58 higher-dimensional, 277-280 non-recombining, 184 one-step, 44-50 risk-neutral behaviour, 61 trinomial, 184 with interest rates, 58-59 trigger FRA, 318 trigger swap, 325 pricing by BGM, 325 trinomial tree, see tree, trinomial strike, 10, 433 strong static replication, see replication, static, strong sub-replication, 369-375 super-replication, 369-375 swap, 300, 305-309, 433 constant maturity, 328 payer's, 306, 432 pricing by BGM, 323 receiver's, 306, 432 value of, 308 swap rate, 433 swap-rate market model, 340 swaption, 301, 309, 433 Bermudan, 301, 310, 342 and factor reduction, 352-355 lower bound via global optimization, 347 underlying, 10 uniform distribution, 461 valuation risk-neutral, see risk-neutral valuation value at risk, 433 Vanna, 433 VAR, 433 variance, 433, 463 Variance Gamma mean rate, 402 variance rate, 403 Variance Gamma density, 408 Variance Gamma model, 88, 404-407 and deterministic future smiles, 244 Variance Gamma process, 401-403 Index variation, 157, 367 first, 99, 367, 409 quadratic, 368 second, see variation, quadratic Vega, 79, 82, 433 integral expression for, 189 Vega hedging, see hedging, Vega volatility, 60, 65, 66, 73-74, 111 Black-Scholes formula as linear function of, 66 forward, 426 implied, 73, 197 instantaneous curve, 320, 333 539 root-mean-square, 320 time-dependence and tree-pricing, 294 trading of, 73 volatility surface, 363 weak static replication, see replication, static, weak Wiener measure, 141, 142 yield, 5, 24, 433 annualized, 25 yield curve, 319, 433 zero-coupon bond, see bond, zero-coupon

Mathematical Finance: Core Theory, Problems and Statistical Algorithms by Nikolai Dokuchaev

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Black-Scholes formula, Brownian motion, buy low sell high, discrete time, fixed income, implied volatility, incomplete markets, martingale, random walk, short selling, stochastic process, stochastic volatility, transaction costs, volatility smile, Wiener process, zero-coupon bond

Then HBS,c(x, K, σ, T, r)=xΦ(d+)−Ke−rTΦ(d−), HBS,p(x, K, σ, T, r)=HBS,c(x, K, σ, T, r)−x+Ke−rT, (5.18) where and where (5.19) This is the celebrated Black-Scholes formula. Note that the formula for put follows from the formula for call from the put-call parity (Corollary 5.51). Numerical calculation via the Black-Scholes formula MATLAB code for Φ(·) function[f]=Phi(x) N=400; eps=abs(x+4)/N; f=0; pi=3.1415; for k=1:N; y=x-eps*(k-1); f=f+eps/sqrt(2*pi)*exp(-y^2/2); end; Here N=400 is the number of steps of integration that defines preciseness. One can try different N=10, 20, 100,…. (See also the MATLAB erf function.) © 2007 Nikolai Dokuchaev Mathematical Finance 96 MATLAB code for Black-Scholes formula (call) function[x]=call(x, K, v, T, r) x=max(0, s-K); if T>0.001 d=(log(s/K)+T*(r+v^2/2))/v/sqrt(T); d1=d-v*sqrt(T); x=s*Phi(d)-K*exp(-r*T)*Phi(dl); end; MATLAB code for Black-Scholes formula (put) function[x]=put(x,K,v,T,r) x=call(x,K,v,T,r)-s+K*exp(-r*T); end; Problem 5.57 Assume that r=0.05, σ=0.07, S(0)=1.

Here E* is the expectation defined by the risk-neutral probability measure (this measure gives the same probability distribution of P(·) as the original measure for the case when r=a). Find an analogue of the Black-Scholes formula for the call option. (Hint: (1) find the expectation via calculation of an integral with a certain (known) probability density; (2) for simplicity, you may take first r=0.) Black-Scholes formula Problem 5.79 Let HBS,c(s, K, r, T, σ) and HBS,p(s, K, r, T, σ) be the Black-Scholes prices for call and put options respectively. Here is the volatility, r≥0 is the bank interest rate, s=S(0) is the initial stock price, K is the strike price. (i) Are these functions increasing (decreasing) in s? Prove. (Hint: use the basic riskneutral valuation rule.) (ii) Find the limits for these functions as: (a) T→+∞; (b) σ→+∞; (c) T→+0; (d) σ→+0. (Hint: use the Black-Scholes formula.) Challenging problems Problem 5.80 Let ã(t)≡â≠0 not depend on time, and let a self-financing strategy be defined in closed-loop form such that where is the corresponding discounted wealth.

MATLAB code for the price of an option with payoff F(x)=1+cos(x) function [f]=option(s,r,T,v) N=800; eps=0.01; f=0; pi=3.1415; for k=1:800; x=-4+eps*(k-1); f=f+eps/sqrt(2*pi*T) *exp(-x^2/(2*T))*(1+cos(s*exp((r-v^2/2)*T+v*x))); end; f=exp(-r*T)*f; © 2007 Nikolai Dokuchaev Continuous Time Market Models 95 Problem 5.56 (i) Write your own code for calculation of the fair price for payoff F(S(T)) where F(x)=|sin(4x)|ex. (ii) Let (S(0), T, σ, r)=(2, 1, 0.2, 0.07). Find the option price with payoff F(S(T)). 5.9.5 Black-Scholes formula We saw already that the fair option price (Black-Scholes price) can be calculated explicitly for some cases. The corresponding explicit formula for the price of European put and call options is called the Black—Scholes formula. Let K>0, σ>0, r≥0, and T>0 be given. We shall consider two types of options: call and put, with payoff function ψ where ψ=(S(T)−K)+ or ψ=(K−S(T))+, respectively. Here K is the strike price. Let HBS,c(x, K, σ, T, r) and HBS,p(x, K, σ, T, r) denote the fair prices at time t=0 for call and put options with the payoff functions F(S(T)) described above given (K, σ, T, r) and under the assumption that S(0)=x.

Mathematics for Finance: An Introduction to Financial Engineering by Marek Capinski, Tomasz Zastawniak

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Black-Scholes formula, Brownian motion, capital asset pricing model, cellular automata, delta neutral, discounted cash flows, discrete time, diversified portfolio, fixed income, interest rate derivative, interest rate swap, locking in a profit, London Interbank Offered Rate, margin call, martingale, quantitative trading / quantitative finance, random walk, short selling, stochastic process, time value of money, transaction costs, value at risk, Wiener process, zero-coupon bond

What we have just derived is the celebrated Black–Scholes formula for European call options. The choice of time 0 to compute the price of the option is arbitrary. In general, the option price can be computed at any time t < T , in which case the time remaining before the option is exercised will be T − t. Substituting t for 0 and T − t for T in the above formulae, we thus obtain the following result. Theorem 8.6 (Black–Scholes Formula) The time t price of a European call with strike price X and exercise time T , where t < T , is given by C E (t) = S(t)N (d1 ) − Xe−r(T −t) N (d2 ) 8. Option Pricing 189 with 1 2 ln S(0) (T − t) X + r + 2σ √ d1 = , σ T −t 1 2 ln S(0) (T − t) X + r − 2σ √ d2 = . (8.11) σ T −t Exercise 8.15 Derive the Black–Scholes formula P E (t) = Xe−r(T −t) N (−d2 ) − S(t)N (−d1 ), with d1 and d2 given by (8.11), for the price of a European put with strike X and exercise time T .

The precise relationship comes from a version of the Central Limit Theorem: It can be shown that the option price given by the Cox–Ross– Rubinstein formula tends to that in the Black–Scholes formula in the continuous time limit described in Chapter 3. Figure 8.2 Option price C E in continuous and discrete time models as a function of time T remaining before the option is exercised 190 Mathematics for Finance Rather than looking at the details of this limit, we just refer to Figure 8.2 for illustration. It shows the price C E of a European call with strike X = 100 on a stock with S(0) = 100, σ = 0.3 and m = 0.2. (Though m is irrelevant for the Black–Scholes formula, it still features in the discrete time approximation.) The continuous compounding interest rate is taken to be r = 0.2. The option price is computed in two ways, as a function of the time T remaining before the option is exercised: a) (solid line) from the Black–Scholes formula for T between 0 and 1; b) (dots) using the Cox–Ross–Rubinstein formula with T increasing from 0 to 1 in N = 10 steps of duration τ = 0.1 each; the discrete growth rates for each step are computed using formulae (3.7).

As a result, the price of the American call is higher than that of the European call. Exercise 8.12 Compute the prices of European and American puts with exercise and strike price X = 14 dollars expiring at time 2 on a stock with S(0) = 12 dollars in a binomial model with u = 0.1, d = −0.05 and r = 0.02, assuming that a dividend of 2 dollars is paid at time 1. 8.3 Black–Scholes Formula We shall present an outline of the main results for European call and put options in continuous time, culminating in the famous Black–Scholes formula. Our treatment of continuous time is a compromise lacking full mathematical rigour, which would require a systematic study of Stochastic Calculus, a topic 186 Mathematics for Finance treated in detail in more advanced texts. In place of this, we shall exploit an analogy with the discrete time case. As a starting point we take the continuous time model of stock prices developed in Chapter 3 as a limit of suitably scaled binomial models with time steps going to zero.


pages: 313 words: 34,042

Tools for Computational Finance by Rüdiger Seydel

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bioinformatics, Black-Scholes formula, Brownian motion, commoditize, continuous integration, discrete time, implied volatility, incomplete markets, interest rate swap, linear programming, London Interbank Offered Rate, mandelbrot fractal, martingale, random walk, stochastic process, stochastic volatility, transaction costs, value at risk, volatility smile, Wiener process, zero-coupon bond

These may be elementary evaluations of functions like the logarithm or the square root such as in the Black-Scholes formula, or may consist of a subalgorithm like Newton’s method for zero finding. There is hardly a purely analytic method. The finite-difference approach, which approximates the surface V (S, t), requires intermediate values for 0 < t < T for the purpose of approximating V (S, 0). In the financial practice one is basically interested in V (S, 0) only, intermediate values are rarely asked for. So the only temporal input parameter is the time to maturity T − t (or T in case the current time is set to zero, t = 0). Recall that also in the Black-Scholes formula, time only enters in the form T − t (−→ Appendix A4). So it makes sense to write the formula in terms of the time to maturity τ , τ := T − t , which leads to the compact version of the Black-Scholes formulas (A4.10), 4.8 Analytic Methods 1 S √ + r+ log K σ τ 1 S + r− d2 (S, τ ; K, r, σ) := √ log K σ τ d1 (S, τ ; K, r, σ) := # σ2 τ 2 # σ2 τ 2 167 (4.40) VPeur (S, τ ; K, r, σ) = −SF (−d1 ) + Ke−rτ F (−d2 ) VCeur (S, τ ; K, r, σ) = SF (d1 ) − Ke−rτ F (d2 ) (dividend-free case).

An immediate candidate for the lower bound V low is the value VPeur provided by the Black-Scholes formula. Thus, VPeur (S, τ ; K) ≤ VPam (S, τ ; K) ≤ VPam (S, τ ; Kerτ ) The latter of the inequalities is the monotonicity with respect to the strike, see Appendix D1. Following [Mar78], an American put with strike Kerτ rising exponentially with τ at the risk-free rate is not exercised, so VPam (S, τ ; Kerτ ) = VPeur (S, τ ; Kerτ ) , 168 Chapter 4 Standard Methods for Standard Options which serves as upper bound. This allows to apply the Black-Scholes formula (4.40) to the European option and provides the upper bound to VPam (S, t; K). The resulting approximation formula is V := αVPeur (S, τ ; Kerτ ) + (1 − α)VPeur (S, τ ; K) . (4.41) The parameter α will depend on S, τ, K, r, σ, so does V . Actually, the BlackScholes formula (4.40) suggests that α and V only depend on the three parameters S/K, rτ , and σ 2 τ .

To match the extreme cases, γ should vanish for τ = 0, and γ ≈ 1 for large values of τ . [Joh83] suggests σ2 τ , b0 + b1 b0 = 1.04083 , b1 = 0.00963 . γ := σ2 τ (4.44) The constants b0 and b1 are again obtained by a regression analysis. The analytic expressions of (4.43), (4.44) provide an approximation of Sf , and then by (4.42), (4.41) an approximation of VPam for S > Sf , based on the Black-Scholes formulas (4.40) for VPeur . 4.8 Analytic Methods Algorithm 4.16 169 (interpolation) For given S, τ, K, r, σ evaluate γ, S f , β, α . Evaluate the Black-Scholes formula for VPeur for the arguments in (4.41). Then V from (4.41) is an approximation to VPam . Numerical experiments show that the approximaton quality of S f is poor. But for S not too close to S f the approximation quality of V is quite good. As reported in [Joh83], the error is small for rτ ≤ 0.125, which is satisfied for average values of the risk-free rate r and time to maturity τ .


pages: 206 words: 70,924

The Rise of the Quants: Marschak, Sharpe, Black, Scholes and Merton by Colin Read

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Albert Einstein, Bayesian statistics, Black-Scholes formula, Bretton Woods, Brownian motion, capital asset pricing model, collateralized debt obligation, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, David Ricardo: comparative advantage, discovery of penicillin, discrete time, Emanuel Derman, en.wikipedia.org, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, floating exchange rates, full employment, Henri Poincaré, implied volatility, index fund, Isaac Newton, John Meriwether, John von Neumann, Joseph Schumpeter, Kenneth Arrow, Long Term Capital Management, Louis Bachelier, margin call, market clearing, martingale, means of production, moral hazard, Myron Scholes, naked short selling, Paul Samuelson, price stability, principal–agent problem, quantitative trading / quantitative finance, RAND corporation, random walk, risk tolerance, risk/return, Ronald Reagan, shareholder value, Sharpe ratio, short selling, stochastic process, The Chicago School, the scientific method, too big to fail, transaction costs, tulip mania, Works Progress Administration, yield curve

Through his clever mathematical construct, Merton was able to relax some of the strongest assumptions that offended those most critical of the Black-Scholes formula. Consequently, he was able to increase the generality of the Black-Scholes formula. He demonstrated that dividends and early calls could also be accommodated by his enveloping portfolio. He managed to show that this enveloping portfolio was smooth and hence the differential equation for the enveloping portfolio could be solved. In fact, this enveloping portfolio of an option for an underlying stock that may exhibit jumps and dividends nonetheless follows the same differential equation defined by Black. Merton demonstrated that the Black-Scholes formula is accurate, even if the option may need to be continuously rebalanced at each point in time to accommodate jumps or ex-dividend repricing. 156 The Rise of the Quants We might imagine that there could be great profits to be had if Black, Scholes, and Merton kept their equation secret and started their own investment firm.

At the time, financial theorists, especially Merton, were increasingly skeptical of the static and backward-looking characteristics of the CAPM, and were seeking to create dynamic extensions of it. Merton was convinced that the Black-Scholes formula, which was a special case of Spreckle’s derivation, must be a further special case of a more general and dynamic CAPM. Black had already realized that it might be safer to interpret the formula as Scholes had done while wearing his Chicago arbitrage and efficient market lenses. The intuition Meanwhile, Merton managed to overcome his skepticism of any approach based on the static CAPM model.1 He reasoned that, even if he looked from the dynamic CAPM perspective at a portfolio that is readjusted at each period in time, he could mimic the option returns specified by the Black-Scholes formula by combining positions on the underlying stock with borrowing at the risk-free interest rate.

They were left with the Black-Scholes equation: C(S,t) = SN(d1 ) − Ke− r (t ∗ −t) N(d 2 ) where K is the strike price, d1 ⫽ (ln(S/C) ⫹ (t*⫺t)(r ⫹ v2/2))/(r(t*⫺t)1/2), d2 ⫽ d1 ⫺ v(t*⫺t)1/2, t* is the expiration date, and the optimal hedge ␦C/␦S is simply N(d1). As in Spreckle’s solution and, for that matter, Bachelier’s derivation 70 years earlier, N(d) is the normalized cumulative probability distribution function. Alternative derivations of the Black-Scholes formula Black and Scholes had made a few implicit assumptions in their analysis. First, they assumed that the number of shares outstanding does not change before the settlement date. If so, this would dilute the price of the stock and affect the option price. Similarly, they assumed that no dividends are paid and that the stock evolution follows a log-normal random walk with a constant drift and volatility.


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The Misbehavior of Markets by Benoit Mandelbrot

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Albert Einstein, asset allocation, Augustin-Louis Cauchy, Benoit Mandelbrot, Big bang: deregulation of the City of London, Black-Scholes formula, British Empire, Brownian motion, buy low sell high, capital asset pricing model, carbon-based life, discounted cash flows, diversification, double helix, Edward Lorenz: Chaos theory, Elliott wave, equity premium, Eugene Fama: efficient market hypothesis, Fellow of the Royal Society, full employment, Georg Cantor, Henri Poincaré, implied volatility, index fund, informal economy, invisible hand, John Meriwether, John von Neumann, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market microstructure, Myron Scholes, new economy, paper trading, passive investing, Paul Lévy, Paul Samuelson, Plutocrats, plutocrats, price mechanism, quantitative trading / quantitative finance, Ralph Nelson Elliott, RAND corporation, random walk, risk tolerance, Robert Shiller, Robert Shiller, short selling, statistical arbitrage, statistical model, Steve Ballmer, stochastic volatility, transfer pricing, value at risk, Vilfredo Pareto, volatility smile

He spends much of each day studying the fast-changing “volatility surface” of the options market—an imaginary 3-D graph of how price fluctuations widen and narrow as the terms of each option contract vary. By the Black-Scholes formula, there should be nothing of interest in such a surface; it should be flat as a pancake. In fact it is a wild, complex shape. Tracking it and predicting its next changes are fundamental ways in which Citigroup’s options traders make money. About 10 percent of the world FX options market is of a class called exotic. It has mind-numbing combinations of precise options terms tailor-made to pay off only under certain circumstances. These combinations are obscure to most people, but perhaps just what the CFO of GM needs to guard against one particular risk that worries him in his company’s yen-based cash-flow. None of this would exist if the original Black-Scholes formula were accurate. Of course, the formula remains important; it is the benchmark to which everyone in the market refers, much the way, say, people talk about the temperature in winter even though whether they actually feel cold also depends on the wind, the snow, the clouds, their clothing, and their health.

In the options industry, where mistakes can cost millions, that is exactly what they have received. Hundreds of scholarly papers, several textbooks, and scores of financial conferences have been devoted to studying the errors. A wide error range. This diagram, from Schoutens 2003, plots the volatility that the standard Black-Scholes formula would infer from the market prices for one family of options. All the curves here show the same type of option, but with different times, T, to maturity. The “strike” price at which each contract can be exercised is on the bottom scale; the volatility that the Black-Scholes formula infers from the data is on the vertical scale, in standard deviations. If the formula were right, there would be nothing much to see: just one flat line. Improving or replacing Black-Scholes is one of the liveliest subdisciplines in mathematical finance.

In hindsight, the newspaper appears to have underestimated, and thus under-played, the importance of the exchange’s opening. 72 “The answer came…” From the eulogy of Black, Scholes 1995. The account of their discovery given here is based on the published recollections of the participants, including Black 1989, Scholes 2001, and the autobiographical essays Merton and Scholes 1997. 73 “The Black-Scholes formula permitted…” The Black-Scholes formula looks complex, but working with it is a simple matter of plugging numbers into their proper places in a spreadsheet or calculator. The price of a call option to buy a stock at a specific price and time is: Here, C0 is the price of the call option; S0 is the current stock price; X is exercise price at which the option contract allows you to buy the stock; r is the risk-free interest rate; and T is the time to maturity.


pages: 345 words: 86,394

Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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Albert Einstein, asset allocation, beat the dealer, Black-Scholes formula, Brownian motion, butterfly effect, capital asset pricing model, collateralized debt obligation, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, discrete time, diversified portfolio, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, fudge factor, implied volatility, incomplete markets, interest rate derivative, interest rate swap, iterative process, London Interbank Offered Rate, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, market bubble, martingale, Myron Scholes, Norbert Wiener, Paul Samuelson, quantitative trading / quantitative finance, random walk, regulatory arbitrage, risk/return, Sharpe ratio, statistical arbitrage, statistical model, stochastic process, stochastic volatility, transaction costs, urban planning, value at risk, volatility arbitrage, volatility smile, Wiener process, yield curve, zero-coupon bond

First let me reassure you that you won’t theoretically lose money in either case (or even if you hedge using a volatility somewhere in the 20 to 40 range) as long as you are right about the 40% and you hedge continuously. There will however be a big impact on your P&L depending on which volatility you input. If you use the actual volatility of 40% then you are guaranteed to make a profit that is the difference between the Black-Scholes formula using 40% and the Black- Scholes formula using 20%. V(S,t;σ) - V(S,t;σ̃), where V(S,t;σ) is the Black-Scholes formula for the call option and σ denotes actual volatility and σ̃ is implied volatility. That profit is realized in a stochastic manner, so that on a marked-to-market basis your profit will be random each day. This is not immediately obvious, nevertheless it is the case that each day you make a random profit or loss, both equally likely, but by expiration your total profit is a guaranteed number that was known at the outset.

The expectation being over the risk-neutral measure for the diffusion but the real measure for the jumps. There is a simple solution of this equation in the special case that the logarithm of J is Normally distributed. If the logarithm of J is Normally distributed with standard deviation σ ′ and if we writek = E [J − 1] then the price of a European non-path-dependent option can be written as In the above and and VBS is the Black-Scholes formula for the option value in the absence of jumps. This formula can be interpreted as the sum of individual Black-Scholes values each of which assumes that there have been n jumps, and they are weighted according to the probability that there will have been n jumps before expiry. Jump-diffusion models can do a good job of capturing steepness in volatility skews and smiles for short-dated option, something that other models, such as stochastic volatility, have difficulties in doing.

Because that is a statistical measure, necessarily backward looking, and because volatility seems to vary, and we want to know what it will be in the future, and because people have different views on what volatility will be in the future, things are not that simple. Example Actual volatility is the σ that goes into the Black-Scholes partial differential equation. Implied volatility is the number in the Black-Scholes formula that makes a theoretical price match a market price. Long Answer Actual volatility is a measure of the amount of randomness in a financial quantity at any point in time. It’s what Desmond Fitzgerald calls the ‘bouncy, bouncy.’ It’s difficult to measure, and even harder to forecast but it’s one of the main inputs into option pricing models. It’s difficult to measure since it is defined mathematically via standard deviations which requires historical data to calculate.


pages: 350 words: 103,270

The Devil's Derivatives: The Untold Story of the Slick Traders and Hapless Regulators Who Almost Blew Up Wall Street . . . And Are Ready to Do It Again by Nicholas Dunbar

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asset-backed security, bank run, banking crisis, Basel III, Black Swan, Black-Scholes formula, bonus culture, break the buck, capital asset pricing model, Carmen Reinhart, Cass Sunstein, collateralized debt obligation, commoditize, Credit Default Swap, credit default swaps / collateralized debt obligations, delayed gratification, diversification, Edmond Halley, facts on the ground, financial innovation, fixed income, George Akerlof, implied volatility, index fund, interest rate derivative, interest rate swap, Isaac Newton, John Meriwether, Kenneth Rogoff, Long Term Capital Management, margin call, market bubble, money market fund, Myron Scholes, Nick Leeson, Northern Rock, offshore financial centre, Paul Samuelson, price mechanism, regulatory arbitrage, rent-seeking, Richard Thaler, risk tolerance, risk/return, Ronald Reagan, shareholder value, short selling, statistical model, The Chicago School, Thomas Bayes, time value of money, too big to fail, transaction costs, value at risk, Vanguard fund, yield curve, zero-sum game

Imagine that the option factory is up and running and selling its products in the market. By assuming that smart, aggressive traders like Meriwether would snap up any mispriced options and build their own factory to pick them apart again using the mathematical recipe, Black, Scholes, and Merton followed in Miller’s footsteps with a no-arbitrage rule. In other words, you’d better believe the math because, otherwise, traders will use it against you. That was how the famous Black-Scholes formula entered finance. When the formula was first published in the Journal of Political Economy in 1973, it was far from obvious that anyone would actually try to use its hedging recipe to extract money from arbitrage, although the Chicago Board Options Exchange (CBOE) did start offering equity option contracts that year. However, there was now an added incentive to play the arbitrage game because Black, Scholes, and Merton had shown that (subject to some assumptions) their formula exorcised the uncertainty in the returns on underlying assets.

He set up his own highly lucrative hedge fund, LTCM, which made $5 billion from 1994 to 1997, earning annual returns of over 40 percent. By April 1998, Merton and Scholes were partners at LTCM and making millions of dollars per year, a nice bump from a professor’s salary. By the late 1990s, investment banks were supplanting exchanges as the favored market-making institution for options and other derivatives, but LTCM worked with both. The original mathematics behind the Black-Scholes formula had gone through several generations of upgrades and refinements since 1973 and was gathering acolytes daily. According to Black-Scholes, the cost of manufacturing options increased with market volatility. Traders learned to use the option price as a kind of “fear gauge,” measuring what the market expected future volatility to be. (In 2005 the CBOE would adopt this fear gauge in the form of a new index called the VIX.)

How could arbitrage trades that were immunized from swings in fundamental markets such as equities or interest rates lose $4 billion in a matter of months? How come VAR, the tool that LTCM and all other big trading banks used to control their exposures, broke down, when it had worked like a dream in 1994? These trades were supposedly safe bets because of the no-arbitrage principle. For example, the Black-Scholes formula suggested that buyers of options were being overcharged compared with the replication cost over time (which tracked underlying market volatility). So LTCM sold options and paid the replication costs, earning a profit as the option price converged on the replication cost, as the quants’ calculations said it would. Because of smart organizations like LTCM (and the fund attracted lots of imitators) the “ought” would become the “is.”


pages: 819 words: 181,185

Derivatives Markets by David Goldenberg

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Black-Scholes formula, Brownian motion, capital asset pricing model, commodity trading advisor, compound rate of return, conceptual framework, Credit Default Swap, discounted cash flows, discrete time, diversification, diversified portfolio, en.wikipedia.org, financial innovation, fudge factor, implied volatility, incomplete markets, interest rate derivative, interest rate swap, law of one price, locking in a profit, London Interbank Offered Rate, Louis Bachelier, margin call, market microstructure, martingale, Myron Scholes, Norbert Wiener, Paul Samuelson, price mechanism, random walk, reserve currency, risk/return, riskless arbitrage, Sharpe ratio, short selling, stochastic process, stochastic volatility, time value of money, transaction costs, volatility smile, Wiener process, Y2K, yield curve, zero-coupon bond, zero-sum game

The idea behind implied volatility is that the Black–Scholes formula embodies an implicit volatility estimator. If we compare market option prices to Black–Scholes model option prices, we can extract the Black–Scholes implicit volatility estimator. Since option prices incorporate a wide variety of forward views of volatility, implied volatility could be a better estimator of unknown volatility than the historical estimator, which is a backward looking estimator. OPTION PRICING IN CONTINUOUS TIME 585 B. The Implied Volatility Estimator Method Volatility is one of the key parameters in the Black–Scholes formula, but it is unobservable. Why not let the model generate estimates of that are consistent with the assumption that the market prices options using the Black–Scholes formula? This is a good idea. In order to implement it, all we have to do is plug all the parameters, except , into the Black–Scholes formula.

This is interesting, because it is an economic rationale for level-dependent volatility, and not just a statistical, or purely mathematical generalization of constant . 16.8.3 Modeling Changing Volatility, the Deterministic Volatility Model There are three initial ways to alter , the instantaneous volatility of percentage returns, that appears in the Black–Scholes formula. 1. =(t)=t , meaning that is not constant but is a deterministic function of time. Black–Scholes can be easily modified to accommodate this case simply by averaging t . A modification of the Black–Scholes formula holds even if the instantaneous variance of percentage returns, t , depends on time. One obtains it by substituting ∫ T t s2ds for 2 (T − t ) in the Black–Scholes formula. OPTION PRICING IN CONTINUOUS TIME 587 Of course, this assumes that the functional dependence of on time is known, or can be estimated. 2. =(St ), meaning that depends upon the current stock price.

OPTION PRICING IN CONTINUOUS TIME 581 Therefore (GBM 16) is equal to, e rT Y0 ∫ e ∞ 1 - (z ′ )2 2 z ′<−(z (K )− T ) 2 ( dz′ ) = erT Y0 N ⎡⎣− z(K ) − T ⎤⎦ (GBM 17) Now all we have to do is to calculate, ( ) − z(K ) − T = − ⎡ ⎛ K ⎞ ⎛ 2 ⎞ ⎤ ⎢ln ⎜ ⎟ − ⎜r − ⎟T ⎥ 2 ⎠ ⎦⎥ ⎢⎣ ⎝ Y0 ⎠ ⎝ + T 2T ⎡ ⎛Y ⎞ ⎛ 2 ⎞ ⎤ ⎢ln ⎜ 0 ⎟ + ⎜r − ⎟T ⎥ 2 ⎠ ⎦ ⎣ ⎝K ⎠ ⎝ = + T T ⎡ ⎛Y ⎞ ⎛ 2 ⎞ ⎤ ⎢ln ⎜ 0 ⎟ + ⎜r − ⎟T + 2T ⎥ 2 ⎠ ⎣ ⎝K ⎠ ⎝ ⎦ = T = ⎡ ⎛Y ⎞ ⎛ 2 ⎞ ⎤ ⎢ ln ⎜ 0 ⎟ + ⎜ r + ⎟T ⎥ 2 ⎠ ⎦ ⎣ ⎝K ⎠ ⎝ T ≡ d1 This is the definition of d1 in the Black–Scholes formula, where we again used the fact that, ⎛K ⎞ ⎛Y ⎞ − ln ⎜ ⎟ = ln ⎜ 0 ⎟ ⎝K ⎠ ⎝ Y0 ⎠ Hence, ∞ e rT Y0 ∫ z′<−(z(K )− = erT Y0 N (d1 ) ⎡ ⎛ 1 2⎞ ⎢exp ⎜ − ( z′) ⎟ T) ⎠ ⎣ ⎝ 2 ⎤ 2 ⎥dz′ ⎦ This completes the derivation of the integral in GBM (8a). (GBM 18) 582 OPTIONS To get the full European call option price, we have to remember to discount by B(0,T). When we do so, we obtain that our entire European call option formula reduces to, C (Y0 ,T ;K ) = e −rT ⎡⎣e rT Y0 N (d1 ) − K N (d2 )⎤⎦ = Y0 N (d1 ) − e−rT K N (d2 ) Therefore, the Black–Scholes formula is given by, C (Y0 ,T ;K ) = Y0 N (d1 ) − e −rT K N (d 2 ) d1 = ⎡ ⎛Y ⎞ ⎛ 2 ⎞ ⎤ ⎢ln ⎜ 0 ⎟ + ⎜r + ⎟T ⎥ 2 ⎠ ⎦ ⎣ ⎝K ⎠ ⎝ T (Black–Scholes, [0,T]) and, d2 = d1 − T N(di ) is the cumulative normal distribution up to di , i=1,2.


pages: 317 words: 84,400

Automate This: How Algorithms Came to Rule Our World by Christopher Steiner

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23andMe, Ada Lovelace, airport security, Al Roth, algorithmic trading, backtesting, big-box store, Black-Scholes formula, call centre, cloud computing, collateralized debt obligation, commoditize, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, Donald Trump, Douglas Hofstadter, dumpster diving, Flash crash, Gödel, Escher, Bach, High speed trading, Howard Rheingold, index fund, Isaac Newton, John Markoff, John Maynard Keynes: technological unemployment, knowledge economy, late fees, Marc Andreessen, Mark Zuckerberg, market bubble, medical residency, money market fund, Myron Scholes, Narrative Science, PageRank, pattern recognition, Paul Graham, Pierre-Simon Laplace, prediction markets, quantitative hedge fund, Renaissance Technologies, ride hailing / ride sharing, risk tolerance, Sergey Aleynikov, side project, Silicon Valley, Skype, speech recognition, Spread Networks laid a new fibre optics cable between New York and Chicago, transaction costs, upwardly mobile, Watson beat the top human players on Jeopardy!, Y Combinator

About a year after the men had put their algorithm to work, a thunderclap sounded above Wall Street. In 1973 Fischer Black and Myron Scholes, both professors at the University of Chicago, published a paper that included what would become known as the Black-Scholes formula, which told its users exactly how much an option was worth. Algorithms based on Black-Scholes would over the course of decades reshape Wall Street and bring a flock of like-minded men—mathematicians and engineers—to the front lines of the financial world. The Black-Scholes solution, quite similar to Peterffy’s, earned Myron Scholes a Nobel Prize in 1997 (Black had died in 1995). Change didn’t happen overnight. The Black-Scholes formula, a partial differential equation, was brilliant. But most traders didn’t peruse academic journals. Even if they did, employing the formula wasn’t simple; it took significant math skills to wield.

“You know, you still have our Nobel Prize,” Jarecki said to Scholes. The remark elicited a dry grimace. “He was not amused,” Jarecki says. For traders who understood it, Black-Scholes gave them a way to calculate the exact price at which options should be traded. It was like having a cheat sheet for the market. There was money to be made by anybody who could accurately calculate each factor within the Black-Scholes formula and apply it to options prices in real time. Traders using the formula would sell options that were priced higher than the formula stipulated and buy ones that were priced lower than their fair price. Do this enough times with enough securities and a healthy profit was virtually guaranteed. TO BE A WALL STREET HACKER IN 1980: PERFECT PLACE, PERFECT TIME The late 1970s marked the faint dawn of the hacker era on Wall Street, when algorithms began to step in front of humans, a trend that has come to dominate all financial markets in every corner of the world.

“I instantly calculated how much money I could save in twenty years by no longer smoking,” he explains. “I needed everything.” Peterffy returned to the pits with a renewed focus. He stuck to his sheets, as always, but with DuPont haunting him, he didn’t make what he called “cowboy bets.” He slowly rebuilt his capital, one grinding day at a time. Sticking to his algorithmic system, he rarely experienced days with substantial losses. Even though the Black-Scholes formula had been published seven years before, it wasn’t moving the markets enough to bother Peterffy or others who were cashing in on its genius. As effective as his algorithms and sheets were, Peterffy was only one man. He needed more people in the pits. So he slowly hired more traders. To prevent losses and keep control of how his traders operated, he trained them to bid and offer only off of values on his sheets, which he would update with fresh numbers from his algorithm every night.


pages: 313 words: 101,403

My Life as a Quant: Reflections on Physics and Finance by Emanuel Derman

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Berlin Wall, bioinformatics, Black-Scholes formula, Brownian motion, capital asset pricing model, Claude Shannon: information theory, Donald Knuth, Emanuel Derman, fixed income, Gödel, Escher, Bach, haute couture, hiring and firing, implied volatility, interest rate derivative, Jeff Bezos, John Meriwether, John von Neumann, law of one price, linked data, Long Term Capital Management, moral hazard, Murray Gell-Mann, Myron Scholes, Paul Samuelson, pre–internet, publish or perish, quantitative trading / quantitative finance, Richard Feynman, Sharpe ratio, statistical arbitrage, statistical model, Stephen Hawking, Steve Jobs, stochastic volatility, technology bubble, the new new thing, transaction costs, value at risk, volatility smile, Y2K, yield curve, zero-coupon bond, zero-sum game

In a lesser but similar way, I wanted to find a set of consistent rules that would let you convince a poor trader that an option formula was right without resorting to the advanced mathematics behind dynamic options replication. I thought about what the Black-Scholes formula really tells you. In principle, you can derive the formula from the Merton strategy of dynamic replication; from this point of view, the formula dictates in exquisite detail exactly how to synthesize a stock option out of a chang ing mixture of stock and riskless bonds. But looked at more naively, the formula gives you the fair price of the option in terms of the current price of the stock and the current price of a riskless bond. Its key insight is that the option is a mixture. Like the ancient Greeks' mythological centaur, part horse and part man, a call option is a hybrid, too-part stock and part bond. From this point of view, I came to regard the BlackScholes formula as a simple and sensible way of interpolating from the known market prices of a stock and a bond to the fair value of the hybrid.

In fruit salad terms, you might start with 50 percent apples and 50 percent oranges, and then, as apples increase in price, move to 40 percent apples and 60 percent oranges; a similar decrease in the price of apples might dictate a move to 70 percent apples and 30 percent oranges. In a sense, you are always trying to keep the price of the mixture constant as the ingredients' prices change and time passes.The exact recipe you need to follow is generated by the Black-Scholes equation. Its solution, the Black-Scholes formula, tells you the cost of following the recipe. Before Black and Scholes, no one even guessed that you could manufacture an option out of simpler ingredients, and so there was no way to figure out its fair price. This discovery revolutionized modern finance. With their insight, Black and Scholes made formerly gourmet options into standard fare. Dealers could now manufacture and sell options on all sorts of underlying securities, creating the precise riskiness clients wanted without taking on the risk themselves.

To our amazement, we discovered that even for 10,000 rehedgings on a one-year option-that is, for more than thirty rebalancings in a day-we still couldn't obtain the exact Black-Scholes value. There was always a residual discrepancy This seemed wrong, so I wrote my own version of the program and found the same small but significant discrepancy. This was very puzzling; it suggested that the Black-Scholes formula was less applicable to the conditions of actual markets than we had expected. I was perturbed enough to want to speak to Fischer about this, and went over to his office in another building on Goldman's growing campus. When I explain what I had found, he briefly became quite excited at the apparent inability of Merton's replication method to produce the exact Black-Scholes value, and said something like, "You know, I always thought there was something wrong with the replication method."


pages: 374 words: 114,600

The Quants by Scott Patterson

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Albert Einstein, asset allocation, automated trading system, beat the dealer, Benoit Mandelbrot, Bernie Madoff, Bernie Sanders, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Brownian motion, buttonwood tree, buy low sell high, capital asset pricing model, centralized clearinghouse, Claude Shannon: information theory, cloud computing, collapse of Lehman Brothers, collateralized debt obligation, commoditize, computerized trading, Credit Default Swap, credit default swaps / collateralized debt obligations, diversification, Donald Trump, Doomsday Clock, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, Gordon Gekko, greed is good, Haight Ashbury, I will remember that I didn’t make the world, and it doesn’t satisfy my equations, index fund, invention of the telegraph, invisible hand, Isaac Newton, job automation, John Meriwether, John Nash: game theory, law of one price, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, merger arbitrage, money market fund, Myron Scholes, NetJets, new economy, offshore financial centre, old-boy network, Paul Lévy, Paul Samuelson, Ponzi scheme, quantitative hedge fund, quantitative trading / quantitative finance, race to the bottom, random walk, Renaissance Technologies, risk-adjusted returns, Rod Stewart played at Stephen Schwarzman birthday party, Ronald Reagan, Sergey Aleynikov, short selling, South Sea Bubble, speech recognition, statistical arbitrage, The Chicago School, The Great Moderation, The Predators' Ball, too big to fail, transaction costs, value at risk, volatility smile, yield curve, éminence grise

He programmed the formula into his HP computer, and it quickly produced a graph showing the price of a stock option that closely matched the price spat out by his own formula. The Black-Scholes formula was destined to revolutionize Wall Street and usher in a wave of quants who would change the way the financial system worked forever. Just as Einstein’s discovery of relativity theory in 1905 would lead to a new way of understanding the universe, as well as the creation of the atomic bomb, the Black-Scholes formula dramatically altered the way people would view the vast world of money and investing. It would also give birth to its own destructive forces and pave the way to a series of financial catastrophes, culminating in an earthshaking collapse that erupted in August 2007. Like Thorp’s methodology for pricing warrants, an essential component of the Black-Scholes formula was the assumption that stocks moved in a random walk.

“I realized that the existence of the smile was completely at odds with Black and Scholes’s 20-year-old foundation of options theory,” wrote Emanuel Derman, a longtime financial engineer who worked alongside Fischer Black at Goldman Sachs, in his book My Life as a Quant. “And, if the Black-Scholes formula was wrong, then so was the predicted sensitivity of an option’s price to movements in its underlying index. … The smile, therefore, poked a small hole deep into the dike of theory that sheltered options trading.” Black Monday did more than that. It poked a hole not only in the Black-Scholes formula but in the foundations underlying the quantitative revolution itself. Stocks didn’t move in the tiny incremental ticks predicted by Brownian motion and the random walk theory. They leapt around like Mexican jumping beans. Investors weren’t rational, as quant theory assumed they were; they panicked like rats on a sinking ship.

A group of economists at the University of Chicago, led by free market guru Milton Friedman, were trying to establish an options exchange in the city. The breakthrough formula for pricing options spurred on their plans. On April 26, 1973, one month before the Black-Scholes paper appeared in print, the Chicago Board Options Exchange opened for business. And soon after, Texas Instruments introduced a handheld calculator that could price options using the Black-Scholes formula. With the creation and rapid adoption of the formula on Wall Street, the quant revolution had officially begun. Years later, Scholes and Robert Merton, an MIT professor whose ingenious use of stochastic calculus had further validated the Black-Scholes model, would win the Nobel Prize for their work on option pricing. (Black had passed away a few years before, excluding him from Nobel consideration.)

Analysis of Financial Time Series by Ruey S. Tsay

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Asian financial crisis, asset allocation, Bayesian statistics, Black-Scholes formula, Brownian motion, capital asset pricing model, compound rate of return, correlation coefficient, data acquisition, discrete time, frictionless, frictionless market, implied volatility, index arbitrage, Long Term Capital Management, market microstructure, martingale, p-value, pattern recognition, random walk, risk tolerance, short selling, statistical model, stochastic process, stochastic volatility, telemarketer, transaction costs, value at risk, volatility smile, Wiener process, yield curve

The unobservability of volatility makes it difficult to evaluate the forecasting performance of conditional heteroscedastic models. We discuss this issue later. In options markets, if one accepts the idea that the prices are governed by an econometric model such as the Black–Scholes formula, then one can use the price to obtain the “implied” volatility. Yet this approach is often criticized for using a specific model, which is based on some assumptions that might not hold in practice. For instance, from the observed prices of a European call option, one can use the Black–Scholes formula in Eq. (3.1) to deduce the conditional standard deviation σt . The resulting value of σt2 is called the implied volatility of the underlying stock. However, this implied volatility is derived under the log normal assumption for the return series.

Continuous-Time Models and Their Applications 6.1 6.2 6.3 6.4 6.5 Options, 222 Some Continuous-Time Stochastic Processes, 222 Ito’s Lemma, 226 Distributions of Stock Prices and Log Returns, 231 Derivation of Black–Scholes Differential Equation, 232 221 ix CONTENTS 6.6 Black–Scholes Pricing Formulas, 234 6.7 An Extension of Ito’s Lemma, 240 6.8 Stochastic Integral, 242 6.9 Jump Diffusion Models, 244 6.10 Estimation of Continuous-Time Models, 251 Appendix A. Integration of Black–Scholes Formula, 251 Appendix B. Approximation to Standard Normal Probability, 253 7. Extreme Values, Quantile Estimation, and Value at Risk 7.1 7.2 7.3 7.4 7.5 7.6 7.7 8. 256 Value at Risk, 256 RiskMetrics, 259 An Econometric Approach to VaR Calculation, 262 Quantile Estimation, 267 Extreme Value Theory, 270 An Extreme Value Approach to VaR, 279 A New Approach Based on the Extreme Value Theory, 284 Multivariate Time Series Analysis and Its Applications 299 8.1 Weak Stationarity and Cross-Correlation Matrixes, 300 8.2 Vector Autoregressive Models, 309 8.3 Vector Moving-Average Models, 318 8.4 Vector ARMA Models, 322 8.5 Unit-Root Nonstationarity and Co-Integration, 328 8.6 Threshold Co-Integration and Arbitrage, 332 8.7 Principal Component Analysis, 335 8.8 Factor Analysis, 341 Appendix A.

We begin the chapter with some terminologies of stock options used in the chapter. In Section 6.2, we provide a brief introduction of Brownian motion, which is also known as a Wiener process. We then discuss some diffusion equations and stochastic calculus, including the well-known Ito’s lemma. Most option pricing formulas are derived under the assumption that the 221 222 CONTINUOUS - TIME MODELS price of an asset follows a diffusion equation. We use the Black–Scholes formula to demonstrate the derivation. Finally, to handle the price variations caused by rare events (e.g., a profit warning), we also study some simple diffusion models with jumps. If the price of an asset follows a diffusion equation, then the price of an option contingent to the asset can be derived by using hedging methods. However, with jumps the market becomes incomplete and there is no perfect hedging of options.


pages: 389 words: 109,207

Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone

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Albert Einstein, anti-communist, asset allocation, beat the dealer, Benoit Mandelbrot, Black-Scholes formula, Brownian motion, buy low sell high, capital asset pricing model, Claude Shannon: information theory, computer age, correlation coefficient, diversified portfolio, Edward Thorp, en.wikipedia.org, Eugene Fama: efficient market hypothesis, high net worth, index fund, interest rate swap, Isaac Newton, Johann Wolfgang von Goethe, John Meriwether, John von Neumann, Kenneth Arrow, Long Term Capital Management, Louis Bachelier, margin call, market bubble, market fundamentalism, Marshall McLuhan, Myron Scholes, New Journalism, Norbert Wiener, offshore financial centre, Paul Samuelson, publish or perish, quantitative trading / quantitative finance, random walk, risk tolerance, risk-adjusted returns, Robert Shiller, Robert Shiller, Ronald Reagan, Rubik’s Cube, short selling, speech recognition, statistical arbitrage, The Predators' Ball, The Wealth of Nations by Adam Smith, transaction costs, traveling salesman, value at risk, zero-coupon bond, zero-sum game

They were the same except for an exponential factor incorporating the risk-free interest rate. Thorp had not included this because the over-the-counter options he traded did not credit the trader with the short-sale proceeds. The rules were changed when options began trading on the Chicago Board of Exchange. Black and Scholes accounted for this. Otherwise, the formulas were equivalent. The Black-Scholes formula, as it was quickly christened, was published in 1973. That name deprived both Merton and Thorp of credit. In Merton’s case, it was a matter of courtesy. Because he had built on Black and Scholes’s work, he delayed publishing his derivation until their article appeared. Merton published his paper in a new journal that was being started by AT&T, the Bell Journal of Economics and Management Science.

Thorp considers the Merton paper “a masterpiece.” “I never thought about credit, actually,” Thorp said, “and the reason is that I came from outside the economics and finance profession. The great importance that was attached to this problem wasn’t part of my thinking. What I saw was a way to make a lot of money.” Man vs. Machine FEW THEORETICAL FINDINGS changed finance so greatly as the Black-Scholes formula. Texas Instruments soon offered a handheld calculator with the formula programmed in. The market in options, warrants, and convertible bonds became more efficient. This made it harder for people like Thorp to find arbitrage opportunities. Of necessity, Thorp was constantly moving from one type of trade to another. In 1974 Thorp and Regan changed the name of their fund to Princeton-Newport Partners, a name steeped in the Ivy League and East Coast old money.

In some cases, the funds’ trading is dictated completely by computer printouts, which not only suggest the proper position but also estimate its probable annual return. “The more we can run the money by remote control the better,” Mr. Thorp declares. The Journal linked Thorp’s operation to “an incipient but growing switch in money management to a quantitative, mechanistic approach.” It mentioned that the Black-Scholes formula was being used by at least two big Wall Street houses (Goldman Sachs and Donaldson, Lufkin & Jenrette). The latter’s Mike Gladstein offered the defensive comment that the brainy formula was “just one of many tools” they used. “The whole computer-model bit is ridiculous because the real investment world is too complicated to be reduced to a model,” an unnamed mutual fund manager was quoted as saying.


pages: 289 words: 113,211

A Demon of Our Own Design: Markets, Hedge Funds, and the Perils of Financial Innovation by Richard Bookstaber

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affirmative action, Albert Einstein, asset allocation, backtesting, beat the dealer, Black Swan, Black-Scholes formula, Bonfire of the Vanities, butterfly effect, commoditize, commodity trading advisor, computer age, computerized trading, disintermediation, diversification, double entry bookkeeping, Edward Lorenz: Chaos theory, Edward Thorp, family office, financial innovation, fixed income, frictionless, frictionless market, George Akerlof, implied volatility, index arbitrage, intangible asset, Jeff Bezos, John Meriwether, London Interbank Offered Rate, Long Term Capital Management, loose coupling, margin call, market bubble, market design, merger arbitrage, Mexican peso crisis / tequila crisis, moral hazard, Myron Scholes, new economy, Nick Leeson, oil shock, Paul Samuelson, Pierre-Simon Laplace, quantitative trading / quantitative finance, random walk, Renaissance Technologies, risk tolerance, risk/return, Robert Shiller, Robert Shiller, rolodex, Saturday Night Live, selection bias, shareholder value, short selling, Silicon Valley, statistical arbitrage, The Market for Lemons, time value of money, too big to fail, transaction costs, tulip mania, uranium enrichment, William Langewiesche, yield curve, zero-coupon bond, zero-sum game

This is a point made by John Danaher in the introduction to Brazilian JiuJitsu: Theory and Technique, by Renzo Gracie and Royler Gracie with Kid Peligro and John Danaher (Montpelier, VT: Invisible Cities Press, 2001). 270 bindex.qxd 7/13/07 2:44 PM Page 271 INDEX Accidents/organizations, 159–161 Accountants, failure (reasons), 135 Accounting conventions, problems, 138 Accounting orientation, 137–138 Adaptation, best measure, 232–233 Adverse selection, 191–192 American depositary receipts (ADRs), 68 America Online (AOL), 139 Amex Major Market Index (XMI) futures, 12 Analytically driven funds, 248 Analytical Proprietary Trading (APT), 44–45 initiation, 189 remnant, form, 190 A Programming Language (APL), 43–47 asset, problem, 45 Armstrong, Michael, 130 Arthur Andersen, failure, 135 Artificial markets, 229 Asia Crisis (1997), 3, 115 Asian currency crisis, 114 Asian economies, 118 Asia-Pacific Economic Cooperation (APEC), 63 Assets class, hedge fund classification, 245 direction, hedge fund classification, 246 Asynchronous pricing, 225 AT&T Wireless Services IPO, SSB underwriting, 130 Back-office functions, 39 Bacon, Louis, 165 Bamberger, Gerry, 185–187, 251 Bankers Trust lawsuit, 38 purchase announcement, 75 Bank exposure, 146–147 Bank failures, 146 Bank of Japan, objectives/strategies, 166 Baptist Foundation, restatements/liability, 135 Barings (bank) bankruptcy, 39 clerical trading error, 38–39 derivatives cross-trading, 143 Beard, Anson, 13 Beder, Tanya, 204 Behavior, economic theory, 231 Berens, Rod, 73 Bernard, Lewis, 42, 52 Biggs, Barton, 11 Black, Fischer, 9 Black Monday (1929), 17 Black-Scholes formula, 9, 252 Block desk, 184–185 trading positions, 186 Bond positions, hedging, 30 Booth, David, 29 Breakdowns, explanation, 5–6 Broker-dealer block-trading desk, usage, 184 price setting role, 213–214 Bucket shop era, 177 Buffett, Warren, 62, 99, 181, 198 arb unit closure, 87–88 Bushnell, Dave, 129–131 Butterfly effect, essence, 227 Capital cushions, 106 Capitalism, 250 Cash futures, 251 arbitrageurs, 19, 23 spread, 19 trade, 19 Cerullo, Ed, 41 Cheapest-to-deliver bond, 251 Chicago Board Options Exchange (CBOE), 252 Black-Scholes formula, impact, 9–10 Citigroup Associates First Capital Corporation, 128 consolidation, impact, 132–134 Japanese private banking arm, 133 management change, Fed reaction, 133 organizational complexity/structural uncertainty, 126 Citron, Robert, 38 Coarse behavior benefits, 232–233 consistency, 236–237 271 bindex.qxd 7/13/07 2:44 PM Page 272 INDEX Coarse behavior (Continued) decision rules, 233 in humans, 235–237 measurement of, 238–239 response based on, 236 rules, optimality, 238 Cockroach example, 232–233, 235 Collateral, usage, 218 Collateralized mortgage obligations (CMOs), 71–75, 250 Commercial Credit, Primerica purchase, 126 Competitive prices, 36 Complexity by-product, 143 implications, 156 importance, 144–146 Consumer lending violations, Federal Reserve fine, 132 Control-oriented risk management, 200 Convergence Capital, 80 Convergence trades, 122 Convertible bond (CB) strategy, 57–58 Cooke, Bill, 185–187 Corporate defaults, possibility, 29–30 Corporate political risk, 140 Corrigan, Gerald, 196–198 Countervailing trades, 213 Credit Suisse First Boston, 72–73 Crises, causes, 240 da Vinci, Leonardo, 136 Denham, Bob, 62–63, 99, 195 Derivatives customization, 143 trading strategy, 30 Deterministic nonperiodic flow, 228 Detroit Edison, Fermi-1 experimental breeder reactor, 161–164 Deutsche Bank, investment banking (problems), 72–73 Dimon, Jamie, 77–78, 91, 97–98, 126 Distressed debt, event risk, 248–249 Dow Jones Industrial Average (DJIA), 2, 12 Dynamic hedge, 12, 161 Dynamic system, 228–229 Ebbers, Bernard, 70 Economic catastrophe, 257 Efficient markets hypothesis, 211 Einstein, Albert, 224–226 Emerging market bonds, 71 Enron restatements/liability, 135 U.S.

If the portfolio declines in value, the hedge is increased, so that finally, if the portfolio value falls well below the floor price, the portfolio is completely hedged. Thus the portfolio is hedged when it needs it and is free to take market exposure when there is a buffer between its value and the floor value. Because the hedge increases and decreases over time, it is called a dynamic hedge. The hedging method of portfolio insurance is based on the theoretical work of Fischer Black, Robert Merton, and Myron Scholes. Their work is encapsulated in the Black-Scholes formula, which makes it possible to set a price on an option. No other formula in economics has had as much impact on the world of finance. Merton and Scholes both received the Nobel Prize for it. (Fischer Black had died a few years before the award was made.) The theory and mathematics behind it were readily embraced by the academic community. Adopted from the mathematics of the heat transfer differential equation of physics and employing the new tools of stochastic calculus, it appealed to an academic core that seemed to derive a twisted pleasure from the mathematically arcane.

Adopted from the mathematics of the heat transfer differential equation of physics and employing the new tools of stochastic calculus, it appealed to an academic core that seemed to derive a twisted pleasure from the mathematically arcane. Despite its esoteric derivation, the formula was timely and—a rarity for work on the mathematical edge of economics—was immediately applicable. First, there was a ready market that required such a pricing tool: the Chicago Board Options Exchange (CBOE) opened for business in 1973, the same year both the paper presenting the Black-Scholes formula and a 9 ccc_demon_007-032_ch02.qxd 2/13/07 A DEMON 1:44 PM OF Page 10 OUR OWN DESIGN more complete exposition on option pricing by Merton were published.1 Second, although the formula required advanced mathematics and computing power, it really worked, and it worked in a mechanistic way. The formula gave rise to portfolio insurance through the work of two University of California at Berkeley finance professors, Hayne Leland and Mark Rubinstein.2 With John O’Brien, their marketing partner, they founded a management company, Leland O’Brien Rubinstein Associates (LOR), in 1981 to sell their technique.


pages: 425 words: 122,223

Capital Ideas: The Improbable Origins of Modern Wall Street by Peter L. Bernstein

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Albert Einstein, asset allocation, backtesting, Benoit Mandelbrot, Black-Scholes formula, Bonfire of the Vanities, Brownian motion, buy low sell high, capital asset pricing model, corporate raider, debt deflation, diversified portfolio, Eugene Fama: efficient market hypothesis, financial innovation, financial intermediation, fixed income, full employment, implied volatility, index arbitrage, index fund, interest rate swap, invisible hand, John von Neumann, Joseph Schumpeter, Kenneth Arrow, law of one price, linear programming, Louis Bachelier, mandelbrot fractal, martingale, means of production, money market fund, Myron Scholes, new economy, New Journalism, Paul Samuelson, profit maximization, Ralph Nader, RAND corporation, random walk, Richard Thaler, risk/return, Robert Shiller, Robert Shiller, Ronald Reagan, stochastic process, the market place, The Predators' Ball, the scientific method, The Wealth of Nations by Adam Smith, Thorstein Veblen, transaction costs, transfer pricing, zero-coupon bond, zero-sum game

See also Diversification computer-based dart-board efficiency of market mean-variance analysis of optimal options risk trading variance volatility Portfolio insurance Black Monday and Portfolio management by bank trust departments economic policy and equity management and interior decorator approach to intertemporal capital asset pricing model for Ito’s lemma and Liquidity Preference theory of risk calculations “Portfolio Selection” (Markowitz) Portfolio Selection: Efficient Diversification of Investment (Markowitz) Positive sum theory Predictions of stock movement: see Forecasting, Market theories (general discussion) Price(s). See also Capital Asset Pricing Model; Random price fluctuations; specific types of securities arbitrage Black/Scholes formula of: see Black/Scholes formula earnings ratio efficient markets and future of growth stocks information and interest rates and intrinsic value and manipulation risk and security analysis and shadow transfer trends value differentiation zero downside limit on “Price Movements in Speculative Markets: Trends or Random Walks” (Alexander) “Pricing of Options and Corporate Liabilities, The” (Black/Scholes) Probability theory Procter & Gamble Profit maximization Program trading Prospective yield “Proposal for a Smog Tax, A” (Sharpe) Puts: see Options Railroads RAND Random Character of Stock Prices, The (Cootner) “Random Difference Series for Use in the Analysis of Time Series, A” (Working) Random price fluctuations/random walks selection of securities and “Random Walks in Stock Market Prices” (Fama) Rational Expectations Hypothesis “Rational Theory of Warrant Pricing” (Samuelson) Regulation of markets Return analysis: see Risk/return ratios Review of Economics and Statistics Review of Economic Studies, The “RHM Warrant and Low-Price Stock Survey, The” Risk arbitrage calculations diversification and dominant expected return and minimalization portfolio premium return ratios Rosenberg’s model stock prices and of stocks vs. bonds systematic (beta) trade-offs valuation of assets and “Risk and the Evaluation of Pension Fund Performance” (Fama) Risk-free assets Rosenberg Institutional Equity Management (RIEM) “Safety First and the Holding of Assets” (Roy) Samsonite Savings rates Scott Paper Securities analysis Securities and Exchange Commission Security Analysis (Graham/Dodd) Security selection Separation Theorem Shadow prices “Simplified Model for Portfolio Analysis, A” (Sharpe) Singer Manufacturing Company Single-index model Sloan School of Management Standard & Poor’s 500 index “State of the Art in Our Profession, The” (Vertin) Stock(s) cash ratios common expected return on growth income international legal restrictions on market value variance volatility Stock market (general discussion) Black Monday (October, 1987, crash) “Stock Market ‘Patterns’ and Financial Analysis” (Roberts) Supply and demand theory Swaps Tactical asset allocation theory Tampax Taxes.

Fischer Black had been stuck on the options problem for “many, many months” when he started working with Myron Scholes. Scholes collaborated with Black to unlock the puzzles of option pricing. Their article on the subject was rejected at first as excessively specialized, but thanks to Merton Miller’s intervention it finally appeared just as the Chicago Board Options Exchange opened for business in 1973. The Black-Scholes formula was soon in general use there and has subsequently formed the basis for many investment, trading, and corporate finance strategies. (©1990 photography by Andy Feldman) In 1968, MIT was the only graduate school that would accept Robert Merton, now of Harvard Business School, when he decided to abandon math for economics. Paul Samuelson immediately selected Merton as an assistant and collaborator and stimulated his interest in option pricing.

If the olive harvest is about the same every year, there is little risk that press capacity and olive production will be badly matched. If the harvest is unexpectedly large, the olive grower will want to hedge against the possibility that he will have no access to the presses when his crop comes in. In light of all these considerations, how does an investor determine whether an option is cheap, expensive, or priced about right? The answer is to use the Black-Scholes formula. The investor knows the current prices of the stock and the option, the price at which the option can be exercised, the time to expiration, and the going rate of interest. With this information, the model will provide an estimate of the stock’s volatility that is implied in the price of the option. Then it is up to the investor to judge whether the market’s expectations about volatility look too low, too high, or about right.


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Models. Behaving. Badly.: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life by Emanuel Derman

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Albert Einstein, Asian financial crisis, Augustin-Louis Cauchy, Black-Scholes formula, British Empire, Brownian motion, capital asset pricing model, Cepheid variable, creative destruction, crony capitalism, diversified portfolio, Douglas Hofstadter, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, Henri Poincaré, I will remember that I didn’t make the world, and it doesn’t satisfy my equations, Isaac Newton, law of one price, Mikhail Gorbachev, Myron Scholes, quantitative trading / quantitative finance, random walk, Richard Feynman, Richard Feynman, riskless arbitrage, savings glut, Schrödinger's Cat, Sharpe ratio, stochastic volatility, the scientific method, washing machines reduced drudgery, yield curve

Black-Scholes provides a recipe for manufacturing a call by borrowing money to buy shares of the stock. The model tells you exactly how many shares to buy initially and then, at every future instant of time and at every future stock price, how much additional stock to buy or sell so that the stock you own will replicate the payoff of the option contract. The value of the option is the total cost of its manufacture, the cost of all the required trading with borrowed money. The Black-Scholes formula explains how the option value—the estimated cost of trading— depends on the stock price, the interest charged for borrowing, and the riskiness of the stock itself. Just as a weather model makes assumptions about how fluids flow and how heat undergoes convection, just as a soufflé recipe makes assumptions about what happens when you whip egg whites, so the Black-Scholes Model makes assumptions about the riskiness of stock prices, that is, about how stock prices fluctuate.

It is better to ensure that one owns a portfolio that will not suffer too badly under disastrous scenarios than it is to try to estimate the probability of destruction. So die the dreams of financial theories. Only imperfect models remain. The movements of stock prices are more like the movements of humans than of molecules. It is irresponsible to pretend otherwise. aMyron Scholes together with Robert C. Merton, who derived a different proof of the Black-Scholes formula and developed much of the elegant mathematics associated with options pricing, received the 1997 Nobel Memorial Prize in Economics for their work on the model. Fischer Black died two years earlier. Chapter 6 Breaking The Cycle Caught in a fiendish cage • Avoiding pragmamorphism • The great financial crisis and the abandonment of principle • The point of financial models • Be sophisticatedly vulgar • Let the dirt be visible • Beware of idolatry • The modelers’ Hippocratic oath • We need free markets, but we need them to be principled • Once in a blue moon, people stop behaving mechanically “Alas” said the mouse, “the world is growing smaller every day.

Markets are by definition vulgar, and the most useful models are vulgar too, using variables (such as price per square foot) that crowds use to describe the value of the assets they trade. One should build vulgar models in a sophisticated way. Some of the best and most practical models involve interpolation, not in prices but rather in the intuitive variables sophisticated users employ to estimate value, for example, volatility. Of course over time crowds and markets get smarter, and yesterday’s High Dutch becomes tomorrow’s patois. The Black-Scholes formula, which translate estimates of volatility into option prices, seemed so arcane when it burst upon the world that Black and Scholes had great difficulty getting their paper accepted for publication. Then, as users of the model grew more experienced, volatility became common currency. Nowadays traders and quants have grown so sophisticated that they talk fluently about models with stochastic volatility, a volatility that is itself volatile.


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I.O.U.: Why Everyone Owes Everyone and No One Can Pay by John Lanchester

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asset-backed security, bank run, banking crisis, Berlin Wall, Bernie Madoff, Big bang: deregulation of the City of London, Black-Scholes formula, Celtic Tiger, collateralized debt obligation, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, Daniel Kahneman / Amos Tversky, diversified portfolio, double entry bookkeeping, Exxon Valdez, Fall of the Berlin Wall, financial deregulation, financial innovation, fixed income, George Akerlof, greed is good, hindsight bias, housing crisis, Hyman Minsky, intangible asset, interest rate swap, invisible hand, Jane Jacobs, John Maynard Keynes: Economic Possibilities for our Grandchildren, John Meriwether, laissez-faire capitalism, light touch regulation, liquidity trap, Long Term Capital Management, loss aversion, Martin Wolf, money market fund, mortgage debt, mortgage tax deduction, mutually assured destruction, Myron Scholes, negative equity, new economy, Nick Leeson, Norman Mailer, Northern Rock, Own Your Own Home, Ponzi scheme, quantitative easing, reserve currency, Right to Buy, risk-adjusted returns, Robert Shiller, Robert Shiller, Ronald Reagan, shareholder value, South Sea Bubble, statistical model, The Great Moderation, the payments system, too big to fail, tulip mania, value at risk

The interacting factors of time, risk, interest rates, and price volatility were so complex that they defeated mathematicians until Fischer Black and Myron Scholes published their paper in 1973, one month after the Chicago Board Options Exchange had opened for business. The revolutionary aspect of Black and Scholes’s paper was an equation enabling people to calculate the price of financial derivatives based on the value of the underlying assets. The Black-Scholes formula opened up a whole new area of derivatives trading. It was a defining moment in the mathematization of the market. Within months, traders were using equations and vocabulary straight out of Black-Scholes (as it is now universally known) and the worldwide derivatives business took off like a rocket. The total market in derivative products around the world is today counted in the hundreds of trillions of dollars.

., 36–37, 39–40, 43, 63–71, 73, 75, 77–78, 84, 116, 120–21, 127, 150, 152, 163, 183, 185, 190, 195, 204, 211–12, 219–20, 225, 227–28 wish list of, 186–87, 195 zombie, 43, 229 banking-and-credit crisis, 192–96, 215–21, 225–28, 231 aftermath of, 215–17 bases of, 201–2 causes of, 182–83, 186, 196, 205–7, 217 economists on, 192–94 failure in forecasting of, 193–94, 211 journalists on, 192–93 profits in, 78, 227–28 and regulation, 182–83, 194–96, 202, 205, 211, 225–26 and risk, 192–95, 202, 205–7, 216 Bank of America, 39 Bank of England, 36, 52, 102, 167, 177–78, 206 and banking-and-credit crisis, 194–95 and interest rates, 178, 180 and regulation, 180–81, 195 Barclays, Barclays Bank, 11, 35–36, 77, 146, 227 Baring, Peter, 52 Barings Bank, 51–52, 54, 180 Barofsky, Neil, 219 Basel rules, 154, 208 derivatives and, 67–68, 120, 183 Bear Stearns, 39, 190 Belair-Edison Community Association, 127 Belgium, 40 Bell, Madison Smartt, 89 bell curve, 154–56, 160 Berlin Wall, fall of, 12, 16, 18, 23 Bernstein, Peter, 149 “Big Bang,” 22, 195–96, 200–201 Bitner, Richard, 124–27, 131 Black, Conrad, 59 Black, Fischer, 45, 47–48, 147 Black-Scholes formula, 48, 54, 116–17, 151 Blank, Victor, 40 BNP Paribas, 36, 77 bond market, bonds, 20, 22–23, 58–59, 73, 107–12 Broad Index Secured Trust Offering (BISTRO), 70–71, 121 corporate, 154, 210 derivatives and, 58, 63–67, 112, 114, 118–19, 210–11 of governments, 29–30, 61–62, 103, 109, 118, 144, 176–77, 208 incentives and, 209–11 investing and, 62–63, 102–3, 107–8, 111, 208–9 investment grade, 62 junk, 42, 62, 208 prices and, 61, 63, 102–3, 108–10, 144 in raising capital, 59, 61–63, 102–3 ratings of, 61–63, 114, 118–19, 208–11 risk and, 61–63, 103, 118, 144, 154, 208 Russia’s default and, 55–56, 162, 164–65 bonuses, 19, 37, 76–78, 207, 218, 224, 228 Bradford & Bingley, 40 Bragason, Valgarður, 10–11 British Airways, 199 Brown, Gordon, 12, 33, 88, 178 Buffett, Warren, 150 credit rating of, 123, 125 derivatives and, 56–57, 78 Bush, George W., 2, 78, 99, 142, 203, 219 regulation and, 19–20, 191, 195 businesses, 15, 58–63, 105–6, 187, 198–99, 221 balance sheets of, 29–34, 37, 106 banks and, 195, 229 bonds and, 59, 61–63, 102–3, 154, 208, 210 derivatives and, 112, 114, 153 lending to, 41–42, 60, 108 offshore, 70, 72 regulation and, 183, 195 risk and, 37, 145, 150–51, 153–54, 195 Canada, banks of, 116, 211–12 capitalism, 12–19, 116 banks and, 19, 25, 182–83, 202, 218, 228, 231 communism vs., 12, 16–17 failure of, 228, 230 free-market, 13–19, 21, 23–24, 96, 105n, 143, 173–75, 184, 192, 196, 202–4, 230–32 in Hong Kong, 13–14 laissez-faire, 142–43, 173, 182–83, 189, 191, 195–96, 202, 211–12 Marxist analysis of, 15–16 regulation and, 182, 192 as secular religion, 202–4 success and spread of, 14–15, 18–19, 21, 23–24 Carville, James, 22–23 cash ratios, 25 Cassano, Joseph, 201 check-clearing systems, 33 Chicago Board Options Exchange, 48 Chicago Mercantile Exchange, 47 China, People’s Republic of, 115, 124 economic boom in, 3–4, 14, 108–9 Hong Kong and, 13–14 U.S. investment of, 109, 176–77 Cisneros, Henry, 99 Citigroup, 120, 163, 219–20, 227 Citron, Robert, 51 City of London, 32, 195–97, 199–202, 217–18 and banking-and-credit crisis, 205–6 and Big Bang, 195–96, 200–201 derivatives and, 56–57, 79, 201 and financial vs. industrial interests, 197, 199 ideological hegemony of, 21–23 Wimbledonization of, 195–96 Civil Justice Network, 85, 128–29, 131 Cleveland, Ohio, 83 Clinton, Bill, 22, 43, 107 housing and, 99–100 regulation and, 19–20 Coggan, Philip, 25 cognitive illusions, 141–42 Cold War, 201–2 end of, 16, 18, 21, 164 collateralized debt obligations (CDOs), 183, 201, 210–12 bonds and, 112, 114, 118–19, 210–11 of CDOs, 119, 206 Gaussian copula function and, 116–17, 157–60, 163 mathematics and, 115–16 mortgages and, 75–76, 112–22, 132, 157, 159–60, 172, 210 risk and, 114–15, 117–22, 158–60, 163, 167, 212 securitization and, 113–14, 11719, 122 shortage of borrowers for, 121–22 tranching and, 117–18, 122 commodities, 227 derivatives and, 47, 49n, 51–52, 184 prices of, 3–4, 107–8, 148–49 Commodity Futures Modernization Act, 184 communism, 12, 16–18, 23 competition, 58, 96, 105n, 203, 226–27 regulation and, 187–88, 226 Confessions of a Subprime Lender (Bitner), 124, 127 Congress, U.S., 77, 100, 204 regulation and, 184–86 risk and, 142–43, 164–66 conservatism, housing and, 98 correlation, correlations: CDOs and, 115–16, 158, 167 risk and, 74, 148–49, 158–59, 165, 167 credit, 8, 169–73 banks and, 24–26, 37, 41, 43, 209, 211 bubbles in, 42, 60, 109, 170, 176, 216–17, 221, 223 CDOs and, 114–15, 119–20, 172 crunch in, 37, 41, 43, 54n, 77, 84–86, 92–93, 94n, 136, 163–64, 169, 171–73, 182, 193, 201–2, 215–16, 218–19 histories and ratings on, 85, 100, 123–26, 158, 163, 165, 208–11 housing and, 84–86, 92–93, 94n, 100, 109, 112, 125, 129–30, 132, 163–64 Iceland’s economic crisis and, 10–12 interest rates and, 172–73, 175, 209 risk and, 136, 158, 165 see also banking-and-credit crisis Crédit Agricole, 36 credit cards, 27, 217 credit ratings and, 123–24 Iceland’s economic crisis and, 9, 11–12 risk and, 158–59, 163 credit default swaps (CDSs), 20, 63, 65–80, 117, 158–59, 183–86 AIG and, 75–78, 201 attractive aspects of, 72–74 examples of, 57–58 Exxon deal and, 67–70, 121 over-the-counter trading of, 184–85, 201 regulation and, 68, 70, 73, 184–86 risk and, 58, 66–70, 72–75, 78–80, 212 securitized bundles of, 69–70, 74 streamlining and industrializing of, 68–69 unfortunate side effect of, 74–75 Credit Suisse, 36, 227 Cuomo, Andrew, 99 Cutter family, 126–27 Darling, Alistair, 172, 220 debt, debts, 27–29, 34, 59–63, 118, 172n, 179, 216, 229 in balance sheets, 27–28, 30–31 benefits of, 59–61 bonds and, 59, 61–63, 208, 210 credit and, 123–26, 221 derivatives and, 52, 67, 69–72 housing and, 93, 100, 132, 176 paying the bill and, 220–22 personal, 221–22 regulation and, 181, 190 Russian default on, 55–56, 162, 164–65 see also collateralized debt obligations default, defaults, default rates, 162–65 CDOs and, 114–15 on mortgages, 159–60, 163, 165, 229 risk and, 154, 159–60, 163 of Russia, 55–56, 162, 164–65 see also credit default swaps Demchak, William, 69 democracy, democracies, 15–18, 108–9, 179, 213 free-market capitalism and, 15, 17, 23 housing and, 87, 98 DePastina, Anthony, 85 Depository Institutions Deregulation and Monetary Control Act (DIDMCA), 100 deregulation, see regulation, deregulation derivatives, 45–58, 63–80, 86, 210–12 in balance sheets, 30–31, 70 banks and, 20, 51–54, 57–58, 63–71, 74–75, 77, 79, 115–17, 120–21, 132, 183–84, 200, 205–6, 211 Black-Scholes formula and, 48, 54, 116–17, 151 bonds and, 58, 63–67, 112, 114, 118–19, 210–11 Buffett and, 56–57, 78 and City of London, 56–57, 79, 201 complexity of, 52–54, 56–57 Enron and, 56, 105–6, 185 futures and, 46–47, 49n, 51–52, 54, 184 Greenspan on, 166, 183–84 in history, 45–48, 147 mathematics and, 47–48, 52–54, 115–17, 166 offshore companies and, 70, 72 options and, 46–47, 50–52, 151, 174, 184 over-the-counter trading of, 184–85, 201, 205–6 prices and, 38, 46–52, 54, 56, 75, 158–59, 166 regulation and, 68, 70, 73, 153, 183–86, 200–201 risk and, 46–47, 49–52, 54–55, 57–58, 66–75, 78–80, 114–15, 117–22, 151, 153, 158–60, 163, 166–67, 184–85, 205, 212 size of market in, 48, 56, 80, 117, 201 see also collateralized debt obligations; credit default swaps Detroit, Mich., 81–82 Deutsche Bank, 36, 77, 83, 227 diversification, 146–48, 177 dividends, 101, 147–48 Doctorow, E.

., 77, 100, 204 regulation and, 184–86 risk and, 142–43, 164–66 conservatism, housing and, 98 correlation, correlations: CDOs and, 115–16, 158, 167 risk and, 74, 148–49, 158–59, 165, 167 credit, 8, 169–73 banks and, 24–26, 37, 41, 43, 209, 211 bubbles in, 42, 60, 109, 170, 176, 216–17, 221, 223 CDOs and, 114–15, 119–20, 172 crunch in, 37, 41, 43, 54n, 77, 84–86, 92–93, 94n, 136, 163–64, 169, 171–73, 182, 193, 201–2, 215–16, 218–19 histories and ratings on, 85, 100, 123–26, 158, 163, 165, 208–11 housing and, 84–86, 92–93, 94n, 100, 109, 112, 125, 129–30, 132, 163–64 Iceland’s economic crisis and, 10–12 interest rates and, 172–73, 175, 209 risk and, 136, 158, 165 see also banking-and-credit crisis Crédit Agricole, 36 credit cards, 27, 217 credit ratings and, 123–24 Iceland’s economic crisis and, 9, 11–12 risk and, 158–59, 163 credit default swaps (CDSs), 20, 63, 65–80, 117, 158–59, 183–86 AIG and, 75–78, 201 attractive aspects of, 72–74 examples of, 57–58 Exxon deal and, 67–70, 121 over-the-counter trading of, 184–85, 201 regulation and, 68, 70, 73, 184–86 risk and, 58, 66–70, 72–75, 78–80, 212 securitized bundles of, 69–70, 74 streamlining and industrializing of, 68–69 unfortunate side effect of, 74–75 Credit Suisse, 36, 227 Cuomo, Andrew, 99 Cutter family, 126–27 Darling, Alistair, 172, 220 debt, debts, 27–29, 34, 59–63, 118, 172n, 179, 216, 229 in balance sheets, 27–28, 30–31 benefits of, 59–61 bonds and, 59, 61–63, 208, 210 credit and, 123–26, 221 derivatives and, 52, 67, 69–72 housing and, 93, 100, 132, 176 paying the bill and, 220–22 personal, 221–22 regulation and, 181, 190 Russian default on, 55–56, 162, 164–65 see also collateralized debt obligations default, defaults, default rates, 162–65 CDOs and, 114–15 on mortgages, 159–60, 163, 165, 229 risk and, 154, 159–60, 163 of Russia, 55–56, 162, 164–65 see also credit default swaps Demchak, William, 69 democracy, democracies, 15–18, 108–9, 179, 213 free-market capitalism and, 15, 17, 23 housing and, 87, 98 DePastina, Anthony, 85 Depository Institutions Deregulation and Monetary Control Act (DIDMCA), 100 deregulation, see regulation, deregulation derivatives, 45–58, 63–80, 86, 210–12 in balance sheets, 30–31, 70 banks and, 20, 51–54, 57–58, 63–71, 74–75, 77, 79, 115–17, 120–21, 132, 183–84, 200, 205–6, 211 Black-Scholes formula and, 48, 54, 116–17, 151 bonds and, 58, 63–67, 112, 114, 118–19, 210–11 Buffett and, 56–57, 78 and City of London, 56–57, 79, 201 complexity of, 52–54, 56–57 Enron and, 56, 105–6, 185 futures and, 46–47, 49n, 51–52, 54, 184 Greenspan on, 166, 183–84 in history, 45–48, 147 mathematics and, 47–48, 52–54, 115–17, 166 offshore companies and, 70, 72 options and, 46–47, 50–52, 151, 174, 184 over-the-counter trading of, 184–85, 201, 205–6 prices and, 38, 46–52, 54, 56, 75, 158–59, 166 regulation and, 68, 70, 73, 153, 183–86, 200–201 risk and, 46–47, 49–52, 54–55, 57–58, 66–75, 78–80, 114–15, 117–22, 151, 153, 158–60, 163, 166–67, 184–85, 205, 212 size of market in, 48, 56, 80, 117, 201 see also collateralized debt obligations; credit default swaps Detroit, Mich., 81–82 Deutsche Bank, 36, 77, 83, 227 diversification, 146–48, 177 dividends, 101, 147–48 Doctorow, E.


pages: 471 words: 124,585

The Ascent of Money: A Financial History of the World by Niall Ferguson

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Admiral Zheng, Andrei Shleifer, Asian financial crisis, asset allocation, asset-backed security, Atahualpa, bank run, banking crisis, banks create money, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, BRICs, British Empire, capital asset pricing model, capital controls, Carmen Reinhart, Cass Sunstein, central bank independence, collateralized debt obligation, colonial exploitation, commoditize, Corn Laws, corporate governance, creative destruction, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, currency manipulation / currency intervention, currency peg, Daniel Kahneman / Amos Tversky, deglobalization, diversification, diversified portfolio, double entry bookkeeping, Edmond Halley, Edward Glaeser, Edward Lloyd's coffeehouse, financial innovation, financial intermediation, fixed income, floating exchange rates, Fractional reserve banking, Francisco Pizarro, full employment, German hyperinflation, Hernando de Soto, high net worth, hindsight bias, Home mortgage interest deduction, Hyman Minsky, income inequality, information asymmetry, interest rate swap, Intergovernmental Panel on Climate Change (IPCC), Isaac Newton, iterative process, John Meriwether, joint-stock company, joint-stock limited liability company, Joseph Schumpeter, Kenneth Arrow, Kenneth Rogoff, knowledge economy, labour mobility, Landlord’s Game, liberal capitalism, London Interbank Offered Rate, Long Term Capital Management, market bubble, market fundamentalism, means of production, Mikhail Gorbachev, money market fund, money: store of value / unit of account / medium of exchange, moral hazard, mortgage debt, mortgage tax deduction, Myron Scholes, Naomi Klein, negative equity, Nick Leeson, Northern Rock, Parag Khanna, pension reform, price anchoring, price stability, principal–agent problem, probability theory / Blaise Pascal / Pierre de Fermat, profit motive, quantitative hedge fund, RAND corporation, random walk, rent control, rent-seeking, reserve currency, Richard Thaler, Robert Shiller, Robert Shiller, Ronald Reagan, savings glut, seigniorage, short selling, Silicon Valley, South Sea Bubble, sovereign wealth fund, spice trade, structural adjustment programs, technology bubble, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, Thomas Bayes, Thomas Malthus, Thorstein Veblen, too big to fail, transaction costs, value at risk, Washington Consensus, Yom Kippur War

Markowitz, a Chicago-trained economist at the Rand Corporation, in the early 1950s, and further developed in William Sharpe’s Capital Asset Pricing Model (CAPM).83 Long-Term made money by exploiting price discrepancies in multiple markets: in the fixed-rate residential mortgage market; in the US, Japanese and European government bond markets; in the more complex market for interest rate swapsbf - anywhere, in fact, where their models spotted a pricing anomaly, whereby two fundamentally identical assets or options had fractionally different prices. But the biggest bet the firm put on, and the one most obviously based on the Black-Scholes formula, was selling long-dated options on American and European stock markets; in other words giving other people options which they would exercise if there were big future stock price movements. The prices these options were fetching in 1998 implied, according to the Black-Scholes formula, an abnormally high future volatility of around 22 per cent per year. In the belief that volatility would actually move towards its recent average of 10-13 per cent, Long-Term piled these options high and sold them cheap. Banks wanting to protect themselves against higher volatility - for example, another 1987-style stock market sell-off - were happy buyers.

The sixteen partners were left with $30 million between them, a fraction of the fortune they had anticipated. What had happened? Why was Soros so right and the giant brains at Long-Term so wrong? Part of the problem was precisely that LTCM’s extraterrestrial founders had come back down to Planet Earth with a bang. Remember the assumptions underlying the Black-Scholes formula? Markets are efficient, meaning that the movement of stock prices cannot be predicted; they are continuous, frictionless and completely liquid; and returns on stocks follow the normal, bell-curve distribution. Arguably, the more traders learned to employ the Black-Scholes formula, the more efficient financial markets would become.97 But, as John Maynard Keynes once observed, in a crisis ‘markets can remain irrational longer than you can remain solvent’. In the long term, it might be true that the world would become more like Planet Finance, always coolly logical.


pages: 443 words: 51,804

Handbook of Modeling High-Frequency Data in Finance by Frederi G. Viens, Maria C. Mariani, Ionut Florescu

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algorithmic trading, asset allocation, automated trading system, backtesting, Black-Scholes formula, Brownian motion, business process, continuous integration, corporate governance, discrete time, distributed generation, fixed income, Flash crash, housing crisis, implied volatility, incomplete markets, linear programming, mandelbrot fractal, market friction, market microstructure, martingale, Menlo Park, p-value, pattern recognition, performance metric, principal–agent problem, random walk, risk tolerance, risk/return, short selling, statistical model, stochastic process, stochastic volatility, transaction costs, value at risk, volatility smile, Wiener process

References 115 There are four steps in the construction of this VIX as follows: • Compute the implied volatilities of entire option chain on SP500 and construct an estimate for the distribution of current market volatility. The implied volatility is calculated by applying Black–Scholes formula. • Use this estimated distribution as input to the quadrinomial tree method. Obtain the price of an at-the-money synthetic option with exactly 30-day maturity. • Compute the implied volatility of the synthetic option based on Black–Scholes formula once the 30-day synthetic option is priced. • Obtain the estimated VIX by multiplying the implied volatility of the synthetic option by 100. Please note that the most important step in the estimation is the choice of proxy for the current stochastic volatility distribution.

He assumes that the portfolio is rebalanced at discrete time δt fixed and transaction costs are proportional to the value of the underlying; that is, the costs incurred at each step is κ|ν|S, where ν is the number of shares of the underlying bought (ν > 0) or sold (ν < 0) at price S and κ is a constant depending on individual investors. Leland derived an option price formula which is the same as the Black–Scholes formula for European calls and puts with an adjusted volatility σ̂ = σ 1 + 2 κ √ π σ δt 1/2 . Following Leland’s idea, Hoggard et al. [3] derive a nonlinear PDE (partial differential equation) for the option price value in the presence of transaction costs. We outline the steps used in the next section. 14.1.1 OPTION PRICE VALUATION IN THE GEOMETRIC BROWNIAN MOTION CASE WITH TRANSACTION COSTS Let C(S, t) be the value of the option and be the value of the hedge portfolio.

See also Maximum likelihood estimation (MLE) Index Bernoulli(p) distribution, 190 Bessel function, 9, 376 Bessel function of the third kind, modified, 166 Best practices, 51 Bias, 253–254 estimated, 258, 259 of the Fourier covariance estimator, 264–266 Bias-corrected estimator, 261 Bid/ask orders, 29 Bid-ask price behavior, 236 Bid-ask spreads, 228, 229, 236, 238–239, 240 Big values, asymptotic behavior for, 338 Binary prediction problems, 48 Black–Litterman model, 68 Black–Scholes analysis, 383–384 Black–Scholes equation, 352, 400 Black–Scholes formula, 114, 115 Black–Scholes model(s), 4, 6–7, 334 boundary condition for, 354–355 in financial mathematics, 352 with jumps, 375 option prices under, 219 volatility and, 400 Black–Scholes PDE, 348. See also Partial differential equation (PDE) methods Board balanced scorecards (BSCs), 51–52, 59. See also Balanced scorecards (BSCs) designing, 59 Board performance, quantifying, 52 Board strategy map, 59–60 Boosting, 47–74 adapting to finance problems, 68 applications of, 68–69 combining with decision tree learning, 49 as an interpretive tool, 67 Boundary value problem, 319, 320 Bounded parabolic domain, 352, 368 Bozdog, Dragos, xiii, 27, 97 Brownian motion, 78, 120, 220 BSC indicators, 52, 53.


pages: 461 words: 128,421

The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street by Justin Fox

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activist fund / activist shareholder / activist investor, Albert Einstein, Andrei Shleifer, asset allocation, asset-backed security, bank run, beat the dealer, Benoit Mandelbrot, Black-Scholes formula, Bretton Woods, Brownian motion, capital asset pricing model, card file, Cass Sunstein, collateralized debt obligation, complexity theory, corporate governance, corporate raider, Credit Default Swap, credit default swaps / collateralized debt obligations, Daniel Kahneman / Amos Tversky, David Ricardo: comparative advantage, discovery of the americas, diversification, diversified portfolio, Edward Glaeser, Edward Thorp, endowment effect, Eugene Fama: efficient market hypothesis, experimental economics, financial innovation, Financial Instability Hypothesis, fixed income, floating exchange rates, George Akerlof, Henri Poincaré, Hyman Minsky, implied volatility, impulse control, index arbitrage, index card, index fund, information asymmetry, invisible hand, Isaac Newton, John Meriwether, John Nash: game theory, John von Neumann, joint-stock company, Joseph Schumpeter, Kenneth Arrow, libertarian paternalism, linear programming, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market design, Myron Scholes, New Journalism, Nikolai Kondratiev, Paul Lévy, Paul Samuelson, pension reform, performance metric, Ponzi scheme, prediction markets, pushing on a string, quantitative trading / quantitative finance, Ralph Nader, RAND corporation, random walk, Richard Thaler, risk/return, road to serfdom, Robert Bork, Robert Shiller, Robert Shiller, rolodex, Ronald Reagan, shareholder value, Sharpe ratio, short selling, side project, Silicon Valley, South Sea Bubble, statistical model, The Chicago School, The Myth of the Rational Market, The Predators' Ball, the scientific method, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, Thomas Kuhn: the structure of scientific revolutions, Thomas L Friedman, Thorstein Veblen, Tobin tax, transaction costs, tulip mania, value at risk, Vanguard fund, Vilfredo Pareto, volatility smile, Yogi Berra

As the market sputtered, and hedge fund after hedge fund closed, Princeton-Newport generated positive, usually double-digit gains every year. After several years of this, Thorp got the notice in the mail that his secret formula was about to become public. It was a preprint of the Black-Scholes article, sent by Fischer Black, who professed in an introductory letter to be a “great admirer” of Thorp’s work. After some initial puzzlement, Thorp realized that the Black-Scholes formula was the same as his. Not long after that, the easy options money had mostly disappeared. But Thorp displayed an uncanny ability to keep finding new sources of profit—and get out of them before they stopped working. He was also willing to discuss his trades, at least after he’d made his money on them—something few of his black-box imitators have done since. This trade-and-tell act started with a Wall Street Journal front-page story in 1974, in which Thorp laid out in detail how he’d made an 8.5 percent profit in three weeks on underpriced Upjohn Co. options.

After the 1987 crash, put options that were well out of the money (the stock was at $40, say, and the put allowed one to sell it for $10) traded at prices that, according to Black-Scholes, implied a similar crash every few years. Other options on the same stocks, though, continued to trade at prices that implied less extreme volatility. That was the smile—flat in the middle, rising at the edge. The Black-Scholes formula assumed that volatility would be constant, consistent, and normally distributed. That clearly wasn’t the case, and the search for better models of volatility was now on in earnest. One starting point was the statistical framework assembled twenty-five years before by Benoit Mandelbrot. Mandelbrot hadn’t predicted black Monday. He hadn’t written anything about finance in years. But anyone who had studied his market writings from the 1960s was far less surprised by events on Wall Street than those who had restricted their reading to standard finance textbooks.

., 154, 160, 165–66, 274, 352n. 3 Bernanke, Ben, xiii, 183 Bernoulli, Daniel, 51 beta, 87, 122–23, 126–27, 139, 152, 205, 207–8, 248–49, 345–46n. 30 Black, Fischer and asset pricing, 141, 149, 248, 322, 346–47n. 30 death, 235 and efficiency, 224 and Goldman Sachs, 224 and “joint hypothesis,” 105 and market crashes, 231 and market volatility, 138 and options, 144–47, 149, 277–78 and Ross, 150 and Shefrin and Statman, 358n. 25 and Summers, 199–201 and Thorp, 219 and Wells Fargo, 127 Black-Scholes formula, 147, 218–19, 233–34,237, 278–79, 280, 320 Bogle, John C., 112–13, 115, 122, 128–30, 306, 322 Bok, Derek, 169 bonds and bond markets, xi–xii, 13–14, 16, 21–22, 29, 38–39, 140–41, 167–68 book-to-price ratio, 208–9 Booth, David, 225 Bork, Robert, 158 Born, Brooksley, 244 Bossaerts, Peter, 297 Brealey, Richard, 355n. 38 Brennan, Michael, 284 Bretton Woods system, 92 Brinegar, Claude, 43–44 Brooks, John, 68, 118 Brown, Kathleen, 278 Brownian motion, 7, 13, 41, 65–69, 73 Bryan, William Jennings, 11 “Bubble Logic: Or, How I Learn to Stop Worrying and Love the Bull” (Asness), 261 Buchanan, James M., 159 Buffet, Warren, 118, 211–14, 214–16, 221–23, 229, 260, 271, 278–79, 323, 366n. 29 bull markets, 18, 61–62, 255, 279, 291 Burns, Arthur, 76, 217, 258 Bush, George W., 295 business cycle, 19–20, 28, 81, 309–10 Business Cycle Institute, 41 Business Week, 97–98 California Institute of Technology, 147 California Public Employees Retirement System (Calpers), 272, 273–74 Cambridge, University, 64 Camerer, Colin, 188 Cameron, David, 295 Campaign GM, 159 Campbell, John, 257 capital asset pricing model (CAPM).


pages: 505 words: 142,118

A Man for All Markets by Edward O. Thorp

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3Com Palm IPO, Albert Einstein, asset allocation, beat the dealer, Bernie Madoff, Black Swan, Black-Scholes formula, Brownian motion, buy low sell high, carried interest, Chuck Templeton: OpenTable, Claude Shannon: information theory, cognitive dissonance, collateralized debt obligation, compound rate of return, Credit Default Swap, credit default swaps / collateralized debt obligations, diversification, Edward Thorp, Erdős number, Eugene Fama: efficient market hypothesis, financial innovation, George Santayana, German hyperinflation, Henri Poincaré, high net worth, High speed trading, index arbitrage, index fund, interest rate swap, invisible hand, Jarndyce and Jarndyce, Jeff Bezos, John Meriwether, John Nash: game theory, Kenneth Arrow, Livingstone, I presume, Long Term Capital Management, Louis Bachelier, margin call, Mason jar, merger arbitrage, Murray Gell-Mann, Myron Scholes, NetJets, Norbert Wiener, passive investing, Paul Erdős, Paul Samuelson, Pluto: dwarf planet, Ponzi scheme, price anchoring, publish or perish, quantitative trading / quantitative finance, race to the bottom, random walk, Renaissance Technologies, RFID, Richard Feynman, Richard Feynman, risk-adjusted returns, Robert Shiller, Robert Shiller, rolodex, Sharpe ratio, short selling, Silicon Valley, statistical arbitrage, stem cell, survivorship bias, The Myth of the Rational Market, The Predators' Ball, the rule of 72, The Wisdom of Crowds, too big to fail, Upton Sinclair, value at risk, Vanguard fund, Vilfredo Pareto, Works Progress Administration

Then the CBOE announced it would start trading put options sometime in the following year, 1974. These options, like the call options we were already trading, were called American options, as distinguished from European options. European options can be exercised only during a short settlement period just prior to expiration, whereas American options can be exercised anytime during their life. If the underlying stock pays no dividends, the Black-Scholes formula, which is for the European call option, turns out to coincide with the formula for the American call option, which is the type that trades on the CBOE. A formula for the European put option can be obtained using the formula for the European call option. But the math for American put options differs from that for European put options, and—even now—no general formula has ever been found. I realized that I could use a computer and my undisclosed “integral method” for valuing options to get numerical results to any desired degree of accuracy for this as-yet-unsolved “American put problem.”

For most academic theorists, this was as close to impossible as anything can be. It was as though the sun suddenly winked out or the earth stopped spinning. They described stock prices using a distribution of probabilities with the esoteric name lognormal. This did a good job of fitting historical price changes that ranged from small to rather large, but greatly underestimated the likelihood of very large changes. Financial models like the Black-Scholes formula for option prices were built using the lognormal. Aware of this limitation in academia’s model of stock prices, as part of the indicators project we had found a much better fit to the historical stock price data, especially for the relatively rare large changes in price. So even though I was surprised by the giant drop, I wasn’t nearly as shocked as most. Though there was no major outside event to explain this one-day collapse, when I thought it through that evening I asked myself, Why did this happen?

Follow logic and analysis rather than sales pitches, whims, or emotion. Assume you may have an edge only when you can make a rational affirmative case that withstands your attempts to tear it down. Don’t gamble unless you are highly confident you have the edge. As Buffett says, “Only swing at the fat pitches.” 3. Find a superior method of analysis. Ones that you have seen pay off for me include statistical arbitrage, convertible hedging, the Black-Scholes formula, and card counting at blackjack. Other winning strategies include superior security analysis by the gifted few and the methods of the better hedge funds. 4. When securities are known to be mispriced and people take advantage of this, their trading tends to eliminate the mispricing. This means the earliest traders gain the most and their continued trading tends to reduce or eliminate the mispricing.


pages: 695 words: 194,693

Money Changes Everything: How Finance Made Civilization Possible by William N. Goetzmann

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Albert Einstein, Andrei Shleifer, asset allocation, asset-backed security, banking crisis, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bretton Woods, Brownian motion, capital asset pricing model, Cass Sunstein, collective bargaining, colonial exploitation, compound rate of return, conceptual framework, corporate governance, Credit Default Swap, David Ricardo: comparative advantage, debt deflation, delayed gratification, Detroit bankruptcy, disintermediation, diversified portfolio, double entry bookkeeping, Edmond Halley, en.wikipedia.org, equity premium, financial independence, financial innovation, financial intermediation, fixed income, frictionless, frictionless market, full employment, high net worth, income inequality, index fund, invention of the steam engine, invention of writing, invisible hand, James Watt: steam engine, joint-stock company, joint-stock limited liability company, laissez-faire capitalism, Louis Bachelier, mandelbrot fractal, market bubble, means of production, money market fund, money: store of value / unit of account / medium of exchange, moral hazard, Myron Scholes, new economy, passive investing, Paul Lévy, Ponzi scheme, price stability, principal–agent problem, profit maximization, profit motive, quantitative trading / quantitative finance, random walk, Richard Thaler, Robert Shiller, Robert Shiller, shareholder value, short selling, South Sea Bubble, sovereign wealth fund, spice trade, stochastic process, the scientific method, The Wealth of Nations by Adam Smith, Thomas Malthus, time value of money, too big to fail, trade liberalization, trade route, transatlantic slave trade, transatlantic slave trade, tulip mania, wage slave

Time only goes in one direction, and with it, the universe tends toward less organization, not more. The option pricing model is based on the principle of forecasting the range of future outcomes of the stock price by assuming it will follow a random walk that conforms to Regnault’s square-root of time insight. However, the Black-Scholes formula gives a solution to the option price today by mathematically rolling time backward. It reverses entropy. In this, it echoes the most basic trait of finance—it uses mathematics to transcend time. THERMODYNAMICS The Black-Scholes formula was published in 1973, just around the time that the Chicago Board Option Exchange began to trade standardized option contracts. Like Bachelier’s thesis, the path-breaking paper was not at first well received. The Journal of Political Economy, where it ultimately was published, needed serious urging from Chicago Professor Merton Miller to be convinced of its contribution.

Evidently none knew of Bachelier, and thus they had to retrace the mathematical logic of fair prices and random walks when they began work on the problem of option pricing in the late 1960s. Like Bachelier, they relied on a model of variation in prices—Brownian motion—although unlike Bachelier, they chose one that did not allow prices to become negative—a limitation of Bachelier’s work. The Black-Scholes formula, as it is now referred to, was mathematically sophisticated, but at its heart it contained a novel economic—as opposed to mathematical—insight. They discovered that the invisible hand setting option prices was risk-neutral. Option payoffs could be replicated risklessly, provided one could trade in an ideal, frictionless market in which stocks behaved according to Brownian motion. Later researchers4 developed a simple framework called a “binomial model” that was able to match the payoff of a put or a call by trading just the stock and a bond through time.

See also stock market, US Nicholas, Tom, 482–83 Nicholas de Anglia, 236 Nicholson, John, 394, 396–97, 399 Nippur, 65, 67 non-normality of security prices, 286 Northwest passage to Cathay, 311–12, 313–14 Norwegian Pension Fund Global, 512, 513, 515 number system: first evidence for, 28. See also Arabic numerals Objectivism, of Ayn Rand, 452 Ohio Company, 388–89 oil income, government investment of, 512 O’Keeffe, Georgia, 468, 475–76, 481 Old Babylonian period, 46, 49, 55–57, 65 Onslow’s Insurance, 370 operations research, 504, 507 opium trade, Chinese, 423, 425–26, 427, 441 Opium Wars, 425–26, 437, 441 option pricing: Bachelier on, 282–83; Black-Scholes formula for, 283–84; Brownian motion and, 276; fractal-based, 287; Lefèvre on, 279–82 options: defined, 280; on Law’s Mississippi Company shares, 357; in seventeenth-century Amsterdam stock market, 317; stock options as compensation, 171 oracle bones, Chinese, 146–47, 271 Ott, Julia, 469–70 owl coins, Athenian, 96–98, 101 Pacioli, Lucca, 246–47 paghe, 291–92 paper instruments, Chinese, 174–75; pawn tickets, 178–79; of Song dynasty, 186–89, 199 paper making and printing, 181–82 paper money: in American colonies, 386–88, 390, 400; Chinese invention of, 139, 168, 174–75, 183–84, 201–2, 400; Chinese nationalization of printing of, 185–86; eighteenth-century comeback of, 382, 399–400; of French revolutionary government, 391–92; of Law’s proposed land bank, 352–53 (see also land banks); Marco Polo’s account of, 191–93; Song dynasty collateral problem with, 387–88; Song dynasty color printing of, 182; of Song dynasty in military crisis, 199.


pages: 280 words: 79,029

Smart Money: How High-Stakes Financial Innovation Is Reshaping Our WorldÑFor the Better by Andrew Palmer

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Affordable Care Act / Obamacare, algorithmic trading, Andrei Shleifer, asset-backed security, availability heuristic, bank run, banking crisis, Black-Scholes formula, bonus culture, break the buck, Bretton Woods, call centre, Carmen Reinhart, cloud computing, collapse of Lehman Brothers, collateralized debt obligation, computerized trading, corporate governance, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, Daniel Kahneman / Amos Tversky, David Graeber, diversification, diversified portfolio, Edmond Halley, Edward Glaeser, endogenous growth, Eugene Fama: efficient market hypothesis, eurozone crisis, family office, financial deregulation, financial innovation, fixed income, Flash crash, Google Glasses, Gordon Gekko, high net worth, housing crisis, Hyman Minsky, implied volatility, income inequality, index fund, information asymmetry, Innovator's Dilemma, interest rate swap, Kenneth Rogoff, Kickstarter, late fees, London Interbank Offered Rate, Long Term Capital Management, loss aversion, margin call, Mark Zuckerberg, McMansion, money market fund, mortgage debt, mortgage tax deduction, Myron Scholes, negative equity, Network effects, Northern Rock, obamacare, payday loans, peer-to-peer lending, Peter Thiel, principal–agent problem, profit maximization, quantitative trading / quantitative finance, railway mania, randomized controlled trial, Richard Feynman, Richard Feynman, Richard Thaler, risk tolerance, risk-adjusted returns, Robert Shiller, Robert Shiller, short selling, Silicon Valley, Silicon Valley startup, Skype, South Sea Bubble, sovereign wealth fund, statistical model, transaction costs, Tunguska event, unbanked and underbanked, underbanked, Vanguard fund, web application

In 1973 a trio of American academics—Fischer Black, Myron Scholes, and Robert Merton—cracked the problem of what to pay for an option. The answer they came up with, expressed as what is now known as the Black-Scholes equation, was based on a simple idea: two things that had identical outcomes ought to cost the same. The price of the option ought to be the same as whatever it cost to construct an investment portfolio that achieved the same end. The Black-Scholes formula enabled the rapid pricing of options and paved the way for explosive growth in derivatives markets. Greek academics have even used it to work out what Thales should have paid for his olive-oil option more than fifteen hundred years ago.25 The third driver was technology. We have seen how a new technology like the railways required finance to adapt in order to provide appropriate financing and screening mechanisms.

Together the three men cracked the problem of how to price an option, a financial instrument that gives the buyer the right, but not the obligation, to buy or sell an underlying asset. The question of what price to pay for an option was one to which there was no rigorous answer until Black, Scholes, and Merton came along. The answer they came up with, expressed as what is now known as the Black-Scholes equation, was based on the idea that the price of the option ought to be the same as the cost of constructing a perfect hedge for the underlying asset. The Black-Scholes formula, which coincided with the computerization of trading, enabled the rapid pricing of options and paved the way for huge growth in derivatives markets.7 At a time when financial innovation and derivatives have become dirty words, Merton has become practiced at answering the criticisms thrown their way. “When you get asked, ‘What is it like to be an ax murderer?’ you tend to question the premise” is how he puts it.

How I Became a Quant: Insights From 25 of Wall Street's Elite by Richard R. Lindsey, Barry Schachter

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Albert Einstein, algorithmic trading, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, asset allocation, asset-backed security, backtesting, bank run, banking crisis, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, business process, buy low sell high, capital asset pricing model, centre right, collateralized debt obligation, commoditize, computerized markets, corporate governance, correlation coefficient, creative destruction, Credit Default Swap, credit default swaps / collateralized debt obligations, currency manipulation / currency intervention, discounted cash flows, disintermediation, diversification, Donald Knuth, Edward Thorp, Emanuel Derman, en.wikipedia.org, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, full employment, George Akerlof, Gordon Gekko, hiring and firing, implied volatility, index fund, interest rate derivative, interest rate swap, John von Neumann, linear programming, Loma Prieta earthquake, Long Term Capital Management, margin call, market friction, market microstructure, martingale, merger arbitrage, Myron Scholes, Nick Leeson, P = NP, pattern recognition, Paul Samuelson, pensions crisis, performance metric, prediction markets, profit maximization, purchasing power parity, quantitative trading / quantitative finance, QWERTY keyboard, RAND corporation, random walk, Ray Kurzweil, Richard Feynman, Richard Feynman, Richard Stallman, risk-adjusted returns, risk/return, shareholder value, Sharpe ratio, short selling, Silicon Valley, six sigma, sorting algorithm, statistical arbitrage, statistical model, stem cell, Steven Levy, stochastic process, systematic trading, technology bubble, The Great Moderation, the scientific method, too big to fail, trade route, transaction costs, transfer pricing, value at risk, volatility smile, Wiener process, yield curve, young professional

In the midst of reading about Black-Scholes, I was also deeply involved with writing the book with Victor Ginzburg from the University of Chicago. Ginzburg was a first-rate mathematician and a terrific writer. He was also a perfectionist to the highest degree. We would write and rewrite chapters endlessly; and each new theorem we added seemed to inspire the need for new chapters. The book went from a proposed hundred-page set of lecture notes into a long, involved project.15 As I was learning about the Black-Scholes formula I was growing increasingly frustrated with the Ginzburg book project. I needed relief. I decided to take everything I was learning about options pricing and write a book on the subject. Now why would I do that? By working with Ginzburg, I had learned to take extremely complicated ideas and explain them clearly and concisely. Also, since I was learning BlackScholes from scratch, I thought I could bring a fresh perspective to it.

In Toronto, I continued to work on the book with Ginzburg—which we had named Representation Theory and Complex Geometry—and work on mathematics research, but I also began to write a paper on options pricing. Thus, my career as a quant slowly began in Toronto in 1994. I never published that first paper, but I did post it on the Social Sciences Research Network (SSRN.com). It was called “An Options Pricing Formula with Volume as a Variable.” The idea was that the Black-Scholes formula relies on perfect dynamic replication of an option with a portfolio of the underlying stock and a riskless security. My idea was to ask, what if instead of perfect replication you can only replicate with a certain probability? What I did was show that if you could replicate a security with another with an arbitrarily high degree of probability, then you could obtain pricing formulas that had all the good properties associated with perfect replication.

When I discovered that the counterparty for most of his trades was a subsidiary of Goldman Sachs (such was the trader’s faith in his own models he neither knew nor cared), I put my foot down and got trading halted. The problem was that in order to derive closed-form solutions, one generally has to work in the Black-Scholes framework. Everyone knows that, due to the fatter than lognormal tails in most asset returns, farfrom-the-money options should generally be priced significantly above the Black-Scholes formula price. But these barrier and double barrier options were close to the money, so this problem doesn’t apply, right? Wrong; if a distribution has fat tails then it must have a taller thinner peak to compensate. Thus a formula derived in the Black-Scholes framework must price near-the-money barrier options too expensively. This is the crucial intuition that the trader had missed; departures from log-normality are important for pricing barrier options, but he had left them out because of his devotion to closed-form formulae and the assumptions that they entail.

Stocks for the Long Run, 4th Edition: The Definitive Guide to Financial Market Returns & Long Term Investment Strategies by Jeremy J. Siegel

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asset allocation, backtesting, Black-Scholes formula, Bretton Woods, buy low sell high, California gold rush, capital asset pricing model, cognitive dissonance, compound rate of return, correlation coefficient, Daniel Kahneman / Amos Tversky, diversification, diversified portfolio, dividend-yielding stocks, equity premium, Eugene Fama: efficient market hypothesis, fixed income, German hyperinflation, implied volatility, index arbitrage, index fund, Isaac Newton, joint-stock company, Long Term Capital Management, loss aversion, market bubble, mental accounting, Myron Scholes, new economy, oil shock, passive investing, Paul Samuelson, popular capitalism, prediction markets, price anchoring, price stability, purchasing power parity, random walk, Richard Thaler, risk tolerance, risk/return, Robert Shiller, Robert Shiller, Ronald Reagan, shareholder value, short selling, South Sea Bubble, survivorship bias, technology bubble, The Great Moderation, The Wisdom of Crowds, transaction costs, tulip mania, Vanguard fund

But the theory of options pricing was given a big boost in the 1970s when two academic economists, Fischer Black and Myron Scholes, developed the first mathematical for12 Chapter 16 will discuss a valuable index of option volatility called VIX. 266 PART 4 Stock Fluctuations in the Short Run mula to price options. The Black-Scholes formula was an instant success. It gave traders a benchmark for valuation where previously they used only their intuition. The formula was programmed on traders’ handheld calculators and PCs around the world. Although there are conditions when the formula must be modified, empirical research has shown that the Black-Scholes formula closely approximates the price of traded options. Myron Scholes won the Nobel Prize in Economics in 1997 for his discovery.13 Buying Index Options Options are actually more basic instruments than futures or ETFs. You can replicate any future or ETF with options, but the reverse is not true.

.): during 1966-1982, 85 Beating the Dow (O’Higgins and Downes), 147 Becker Securities Corporation, 344 Behavioral finance, 322–337 contrarian investing and investor sentiment and, 333–334, 335i excessive trading, overconfidence, and representative bias and, 325–328 fads, social dynamics, and stock bubbles and, 322–325 myopic loss aversion, portfolio monitoring, and equity risk premium and, 332–333 out-of-favor stocks and Dow 10 strategy and, 335–336 prospect theory, loss aversion, and holding on to losing trades and, 328–330 rules for avoiding behavioral traps and, 331 Bell, Heather, 351n Benchmarks, for fund performance, 342 Berkshire Hathaway, 43, 101n, 176i, 346 Bernanke, Ben, 246 Bernartzi, Shlomo, 332–333 Bernstein, Peter, 363 Best Food, 47 Bestfoods, 47 Beta, 140n Bethlehem Steel, 56i, 57 Bias: forward-looking, 330 367 Copyright © 2008, 2002, 1998, 1994 by Jeremy J. Siegel. Click here for terms of use. 368 Bias (Cont.): representative, 326 return, in stock indexes, 46–47 self-attribution, 326 survivorship, 18, 343, 343i Big Board (see New York Stock Exchange [NYSE]) Bikhchandani, S. D., 325n Bill and Melinda Gates Foundation, 7n Bills, interest rates on, 7–9, 9i Birrell, Lowell, 38–39 Black, Fischer, 265, 266n Black-Scholes formula, 266 Blake, Christopher R., 348 Blitzer, David, 107, 108n, 353n Blodget, S., Jr., 7n Blue Chip Economic Indicators, 217, 218 Bodie, Zvi, 35n Bogle, John C., 343n, 348 Bonds: current yield of, 111 equity premium and, 16–18, 17i government, interest rate on, above dividend yield on common stocks, 95–97 inflation-indexed, 35 long-term performance of, 7–9, 9i real returns on, 14–15, 15i return on, correlation with stock returns, 30–32, 31i Russian default on, 88 standard deviation of returns for, 30 stocks’ outperformance of, 26 yields on, stock yields related to, 95–97 (See also Fixed-income assets) Book value, 117 Bos, Roger, J., 353n Bosland, Chelcie C., 82 Boyd, John, 241n BP (British Petroleum), 177, 183 Index BP Amoco, 55 Brealey, Richard A., 171n Bristol Myers, 59n British American Tobacco, 63, 177 Brock, William, 295n, 304n Brown, Stephen J., 18n Browne & Co., 21n Bubbles: stock (see Stock bubbles) technology, 167 Buffett, Warren, 7n, 61, 104, 107, 187q, 268, 359q Bull markets: beginning of, 85–86 from 1982-1999, 14 Bureau of Labor Statistics (BLS), 241 Burns, Arthur, 210n Bush, George W., 69, 75 Business cycle, 207–219, 285 dating of, 208–211 definition of, 209–210 prediction of, 216–219 timing of, gains through, 214–216, 215i turning points of, stock returns around, 211–214, 212i–214i Buy-and-hold returns, 215 Buy and write strategy, 267 Buy programs, 258 Buybacks, 98 CAC index, 238 Calendar anomalies, 305–318 day-of-the-week effects, 316–318, 317i investing strategies for, 318 seasonal, 306–316 California Packing Co., 60i, 62 Calls, 264 Campbell, John Y., 35n, 87, 158 Capital asset pricing model (CAPM), 140, 141 Capital gains taxes: benefits of deferring, 69–70 failure of stocks as long-term inflation hedge and, 204 Capital gains taxes (Cont.): historical, 66, 67i increasingly favorable tax factors for equities and, 72–73 inflation and, 70–72, 71i total after-tax returns index and, 66, 68–69, 68i, 69i Capitalization-weighted indexing, 351–352, 352i fundamentally weighted indexation versus, 353–355 Carnegie, Andrew, 57 Carvell, Tim, 107n Cash flows, from stocks, valuation of, 97–98 Cash market, 257 Cash-settled futures contracts, 257 CBOE Volatility Index, 281–282, 282i Celanese Corp., 60i, 64 Center for Research in Security Prices (CRSP) index, 45, 46i, 141 Central bank policy, 247 (See also Federal Reserve System [Fed]) Chamberlain, Lawrence, 82 Chamberlain, Neville, 78 Channels, 40 technical analysis and, 294 Chartists (see Technical analysis) Chevron, 176i, 177 ChevronTexaco, 55 Chicago Board of Trade (CBOT): closure due to Chicago River leak, 253, 254i, 255 stock market crash of 1987 and, 273 Chicago Board Options Exchange (CBOE), 264–265 Volatility Index of, 281–282, 282i Chicago Gas, 47 in DJIA, 39i, 48 Chicago Mercantile Exchange, stopping of trading on, 276 Index Chicago Purchasing Managers, 244 China: global market share of, 178, 179i, 180, 180i sector allocation and, 177 China Construction Bank, 175 China Mobile, 177, 183 China National Petroleum Corporation, 182 Chrysler, 64 Chunghwa Telecom, 177 Cipsco (Central Illinois Public Service Co.), 48 Circuit breakers, 276–277 Cisco Systems, 38, 57n, 89, 104, 155, 157, 176i on Nasdaq, 44 Citigroup, 144, 175, 176i Clinton, Bill, 75, 227, 238 Clough, Charles, 86 CNBC, 48, 88 Coca-Cola Co., 59i, 61, 64 Cognitive dissonance, 328 Colby, Robert W., 295–296 Colgate-Palmolive, 59i Colombia Acorn Fund, 346 Comcast, 176 Common stock theory of investment, 82 Common Stocks as Long-Term Investments (Smith), 79, 83, 201 Communications technology, bull market and, 88 Compagnie Française des Pétroles (CFP), 184 Conference Board, 244 Conoco (Continental Oil Co.), 57 ConocoPhillips, 176i, 177, 183 Consensus estimate, 239 Consumer choice, rational theory of, 322 Consumer discretionary sector: in GICS, 53 global shares in, 175i, 176 Consumer Price Index (CPI), 245 369 Consumer staples sector: in GICS, 53 global shares in, 175i, 177 Consumer Value Store, 61 Contrarian investing, 333–334 Core earnings, 107–108 Core inflation, 245–246 Corn Products International, 47 Corn Products Refining, 47 Corporate earnings taxes, failure of stocks as long-term inflation hedge and, 202–203 Correlation coefficient, 168 Corvis Corporation, 156–157 Costs: agency, 100 effects on returns, 350 employment, 246 interest, inflationary biases in, failure of stocks as longterm inflation hedge and, 203–204 pension, controversies in accounting for, 105–107 Cowles, Alfred, 42, 83 Cowles Commission for Economic Research, 42, 83 CPC International, 47 Crane, Richard, 61 Crane Co., 59i, 60i, 61 Cream of Wheat, 62 Creation units, 252 Crowther, Samuel, 3 Cubes (ETFs), 252 Currency hedging, 173 Current yield of bonds, 111 Cutler, David M., 224n CVS Corporation, 61 Cyclical stocks, 144 DaimlerChrysler, 176 Daniel, Kent, 326n Dart Industries, 62 Dash, Srikant, 353n Data mining, 326–327 David, Joseph, 21 DAX index, 238 Day-of-the-week effects, 316–318, 317i Day trading, futures contracts and, 261 Dean Witter, 286 De Bondt, Werner, 302–303, 335 Defined benefit plans, 106–107 Defined contribution plans, 105–106 Delaware and Hudson Canal, 22 Deleveraging, 120 Del Monte Foods, 62 Department of Commerce, 203 Depreciation, failure of stocks as long-term inflation hedge and, 203 Deutsch, Morton, 324n Deutsche Post, 177 Deutsche Telekom, 177 Dexter Corp., 21n Diamonds (ETFs), 252 Dilution of earnings, 104 Dimensional Fund Advisors (DFA) Small Company fund, 142n Dimson, Elroy, 18, 19n, 20 Discounts, futures contracts and, 258 Distiller’s Securities Corp., 48 Distilling and Cattle Feeding, 47 in DJIA, 39i, 48 Diversifiable risk, 140 Diversification in world markets, 168–178 currency hedging and, 173 efficient portfolios and, 168–172, 169i–171i private and public capital and, 177–178 sector diversification and, 173–177, 174i The Dividend Investor (Knowles and Petty), 147 Dividend payout ratio, 101 Dividend policy, value of stock as related to, 100–102 370 Dividend yields, 145–149, 146i–149i interest rate on government bonds above, 95–97 ratio of market value to, 120, 120i Dodd, David, 77q, 83, 95q, 139q, 141, 145n, 150, 152, 289q, 304n, 334n Dogs of the Dow strategy, 147–149, 148i, 149i, 336 Dollar cost averaging, 84 Domino Foods, Inc., 47 Dorfman, John R., 147n Double witching, 260–261 Douvogiannis, Martha, 113n Dow, Charles, 38, 290–291 Dow Chemical, 58 Dow Jones & Co., 38 Dow Jones averages, computation of, 39–40 Dow Jones Industrial Average (DJIA), 37, 47 breaks 2000, 85 breaks 3000, 85 breaks 8000, 87 crash of 1929 and, 4 creation of, 38 fall in 1998, 88 firms in, 38–39, 39i following Iraq’s defeat in Gulf War, 85 long-term trends in, 40–41, 41i Nasdaq stocks in, 38 during 1922–1932, 269, 270i during 1980–1990, 269, 270i original firms in, 47–49 original members of, 22 predicting future returns using trend lines and, 41–42 as price-weighted index, 40 Dow Jones Wilshire 5000 Index, 45 Dow 10 strategy, 147–149, 148i, 149i, 336 Dow Theory (Rhea), 290 Index Dow 36,000 (Hassett), 88 Downes, John, 147 Dr.


pages: 272 words: 19,172

Hedge Fund Market Wizards by Jack D. Schwager

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asset-backed security, backtesting, banking crisis, barriers to entry, beat the dealer, Bernie Madoff, Black-Scholes formula, British Empire, Claude Shannon: information theory, cloud computing, collateralized debt obligation, commodity trading advisor, computerized trading, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, diversification, diversified portfolio, Edward Thorp, family office, financial independence, fixed income, Flash crash, hindsight bias, implied volatility, index fund, intangible asset, James Dyson, Long Term Capital Management, margin call, market bubble, market fundamentalism, merger arbitrage, money market fund, oil shock, pattern recognition, pets.com, Ponzi scheme, private sector deleveraging, quantitative easing, quantitative trading / quantitative finance, Right to Buy, risk tolerance, risk-adjusted returns, risk/return, riskless arbitrage, Rubik’s Cube, Sharpe ratio, short selling, statistical arbitrage, Steve Jobs, systematic trading, technology bubble, transaction costs, value at risk, yield curve

In 1967, I took some of the ideas about how to price warrants in the Random Character of Stock Prices by Paul Cootner and thought I could derive a formula if I made the simplifying assumption that all investments grew at the risk-free rate. Since the purchase or sale of warrants combined with delta neutral hedging led to a portfolio with very little risk, it seemed very plausible to me that the risk-free assumption would lead to the correct formula. The result was an equation that was equivalent to the future Black-Scholes formula. I started using this formula in 1967. Did you apply your formula (that is, the future Black-Scholes formula) to identify overpriced warrants and then delta hedge those positions? I didn’t have enough money to have a diversified warrant portfolio and to also place the hedge, since each side of a hedged position required separate margin. I used the formula to identify the most extremely overpriced warrants. I found warrants that were selling at two or three times what my formula said they should be priced at.

Small cap stocks were up 84 percent in 1967 and 36 percent in 1968. It was a terrible time to have net short exposure. However, the formula was good enough and the warrants were so overpriced that I still broke even on the naked short positions. The formula really proved itself under the most adverse circumstances. As far as I know, the short warrant positions I implemented during 1967 to 1968 were the first actual application of the Black-Scholes formula in the markets. When did Black-Scholes publish their formula? I believe they discovered it in 1969 and published it 1972 or 1973. Did you consider publishing your formula? The option-pricing formula seemed to me to be a big edge on everybody else. So I was happy just to use it. By 1969, I had started my first hedge fund, Princeton Newport Partners, and I thought that if I published the formula, I would lose the edge that was helping my investors.


pages: 701 words: 199,010

The Crisis of Crowding: Quant Copycats, Ugly Models, and the New Crash Normal by Ludwig B. Chincarini

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affirmative action, asset-backed security, automated trading system, bank run, banking crisis, Basel III, Bernie Madoff, Black-Scholes formula, buttonwood tree, Carmen Reinhart, central bank independence, collapse of Lehman Brothers, collateralized debt obligation, collective bargaining, corporate governance, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, discounted cash flows, diversification, diversified portfolio, family office, financial innovation, financial intermediation, fixed income, Flash crash, full employment, Gini coefficient, high net worth, hindsight bias, housing crisis, implied volatility, income inequality, interest rate derivative, interest rate swap, John Meriwether, labour mobility, liquidity trap, London Interbank Offered Rate, Long Term Capital Management, low skilled workers, margin call, market design, market fundamentalism, merger arbitrage, Mexican peso crisis / tequila crisis, money market fund, moral hazard, mortgage debt, Myron Scholes, negative equity, Northern Rock, Occupy movement, oil shock, price stability, quantitative easing, quantitative hedge fund, quantitative trading / quantitative finance, Ralph Waldo Emerson, regulatory arbitrage, Renaissance Technologies, risk tolerance, risk-adjusted returns, Robert Shiller, Robert Shiller, Ronald Reagan, Sharpe ratio, short selling, sovereign wealth fund, speech recognition, statistical arbitrage, statistical model, survivorship bias, systematic trading, The Great Moderation, too big to fail, transaction costs, value at risk, yield curve, zero-coupon bond

A call and put option give the holder the right to buy or sell a security at a given price, so the higher the security’s volatility, the greater the chance that the security’s price may move above or below the strike price, letting the investor make a profit. That’s why higher volatility means a higher option price. With a formula that relates an option’s price to the underlying security’s volatility, a trader could convert the option’s price into a volatility consistent with that price. This is called implied volatility. The Black-Scholes formula, discovered in 1973, is most commonly used for this purpose. It is named after one of LTCM’s principals, Myron Scholes, and the late Goldman Sachs partner Fischer Black. LTCM made volatility trades in both fixed income and equities. In the fixed-income arena, they noticed in 1998 that the implied volatility of 5-year options (i.e., options with five years to maturity) on German-denominated swaps was trading much lower than actual realized volatility.

See also Investment banks; specific banks bailouts of basic operations of fees charged by Greek debt exposure as hedge funds housing bubble and leverage and Maughan on provision of emergency credit by runs on trust between trust in Barclays Barclays Global Investors (BGI) Basel Committee: Basel I document Basel II document financial crisis and guidelines of overview of Bear Stearns: bank run on collapse of failure of hedge funds of history and reputation of J.P. Morgan and leverage of LTCM and near-collapse of repo system and window dressing by Begleiter, Steve Benn, Orson Berkshire Hathaway. See also Buffett, Warren Bernanke, Ben Black, Fischer Black-Scholes formula Blankfein, Lloyd Blasnik, Steve Bond arbitrage Born, Brooksley Box trade Brady Plan Brazilian C bonds Brendsel, Leland Broker-dealers Buffett, Warren Buoni del Tesoro Poliennali Buoni Ordinari del Tesoro Bush, George Butler, Angus Butterfly yield curve trades Callan, Erin Capital, contingency Capital adequacy ratio (CAR) Capital markets Capital ratio and leverage Capital-to-asset ratio Carhart, Mark Cash business Cassano, Joseph Caxton macro hedge fund Cayne, James E.

See also Risk management Basel I and credit risk failure of models of to account for crowding and interconnectedness Freddie, Fannie, and liquidity risk market risk measuring of mortgages reduction of prior to quarterly reports systemic risk tail risk Risk arbitrage trades Risk management: at Bear Stearns at JWMP and PGAM at JWM Partners at Lehman Brothers lessons from financial crisis of 2008 Risk management at LTCM: broad outlines as cause of failure diversification mathematics of framework for operations raw evidence Robertson, Julian Rosenblum, Ira Rosenfeld, Eric Rosengren, Eric Royal Dutch-Shell trade Rubin, Robert Russian government, default on debt by Russian markets Salomon Brothers: arbitrage trading group bond-trading group shutdown copycat positions Traveler’s Group purchase of Schapiro, Mary Scholes, Myron: Banco Nazionale del Lavoro project and Black-Scholes formula career of on diversification on economic system choices on financial models on insurance on Lehman failure LTCM and PGAM and on spreads on VaR Schwartz, Alan SEC (Securities and Exchange Commission) Securities: Agency asset-backed CMBS corporate debt high-yield illiquid liquid mortgage-backed municipal and tax-exempt Securitization Security price volatility Self help status Shadow banking system Sharpe, William Sharpe ratio Shed show Short swap spread trade Shustak, Robert Simon, James Size of firms and “too big to fail,” Slap hands Sloan, Bob Smith Breeden Mortgage Partners Snow, John Solender, Michael Solomon, David Soros, George Sowood Capital Spector, Warren Spoofing Standard & Poor’s Standard quant factors, erratic behavior of State Street Statistical arbitrage (stat arb) funds Steel, Robert Stock price manipulation Stress tests Stub quotes Subprime mortgage market collapse Subprime mortgages Sun, Tong-Sheng Swap business Swap spreads: behavior of longer-term computing returns from zero-coupon returns derivation of approximate returns historical average Italian Japanese mechanics of U.S.


pages: 349 words: 134,041

Traders, Guns & Money: Knowns and Unknowns in the Dazzling World of Derivatives by Satyajit Das

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accounting loophole / creative accounting, Albert Einstein, Asian financial crisis, asset-backed security, beat the dealer, Black Swan, Black-Scholes formula, Bretton Woods, BRICs, Brownian motion, business process, buy low sell high, call centre, capital asset pricing model, collateralized debt obligation, commoditize, complexity theory, computerized trading, corporate governance, corporate raider, Credit Default Swap, credit default swaps / collateralized debt obligations, cuban missile crisis, currency peg, disintermediation, diversification, diversified portfolio, Edward Thorp, Eugene Fama: efficient market hypothesis, Everything should be made as simple as possible, financial innovation, fixed income, Haight Ashbury, high net worth, implied volatility, index arbitrage, index card, index fund, interest rate derivative, interest rate swap, Isaac Newton, job satisfaction, John Meriwether, locking in a profit, Long Term Capital Management, mandelbrot fractal, margin call, market bubble, Marshall McLuhan, mass affluent, mega-rich, merger arbitrage, Mexican peso crisis / tequila crisis, money market fund, moral hazard, mutually assured destruction, Myron Scholes, new economy, New Journalism, Nick Leeson, offshore financial centre, oil shock, Parkinson's law, placebo effect, Ponzi scheme, purchasing power parity, quantitative trading / quantitative finance, random walk, regulatory arbitrage, Right to Buy, risk-adjusted returns, risk/return, Satyajit Das, shareholder value, short selling, South Sea Bubble, statistical model, technology bubble, the medium is the message, the new new thing, time value of money, too big to fail, transaction costs, value at risk, Vanguard fund, volatility smile, yield curve, Yogi Berra, zero-coupon bond

DAS_C07.QXP 8/7/06 4:45 PM Page 195 6 N Super models – derivative algorithms 195 The price of a European put option can be derived as follows: Ppe = K e-Rf.T . N(– d2) – S. N(– d1) Where Ppe = price of European put option Despite the formidable appearance of the equation, you need only high school maths to derive the option values. The papers would have remained obscure had it not been for the confluence of events. In 1973, the Chicago Board Options Exchange started trading options on leading stocks and the Black–Scholes formula quickly became the market standard for pricing and trading options. HewlettPackard calculators with preprogrammed Black–Scholes option pricing model became available – the age of the super model had arrived. In 1997, Scholes and Merton received the Nobel Prize for economics in recognition of their work. Black did not receive the award, having died of throat cancer in 1995. The Nobel Prize rewards longevity as much as intellectual achievement.

However, the text is different. 6 ‘What Worries Warren’ (3 March 2003) Fortune. 13_INDEX.QXD 17/2/06 4:44 pm Page 325 Index accounting rules 139, 221, 228, 257 Accounting Standards Board 33 accrual accounting 139 active fund management 111 actuaries 107–10, 205, 289 Advance Corporation Tax 242 agency business 123–4, 129 agency theory 117 airline profits 140–1 Alaska 319 Allen, Woody 20 Allied Irish Bank 143 Allied Lyons 98 alternative investment strategies 112, 308 American Express 291 analysts, role of 62–4 anchor effect 136 Anderson, Rolf 92–4 annuities 204–5 ANZ Bank 277 Aquinas, Thomas 137 arbitrage 33, 38–40, 99, 114, 137–8, 171–2, 245–8, 253–5, 290, 293–6 arbitration 307 Argentina 45 arithmophobia 177 ‘armpit theory’ 303 Armstrong World Industries 274 arrears assets 225 Ashanti Goldfields 97–8, 114 Asian financial crisis (1997) 4, 9, 44–5, 115, 144, 166, 172, 207, 235, 245, 252, 310, 319 asset consultants 115–17, 281 ‘asset growth’ strategy 255 asset swaps 230–2 assets under management (AUM) 113–4, 117 assignment of loans 267–8 AT&T 275 attribution of earnings 148 auditors 144 Australia 222–4, 254–5, 261–2 back office functions 65–6 back-to-back loans 35, 40 backwardation 96 Banca Popolare di Intra 298 Bank of America 298, 303 Bank of International Settlements 50–1, 281 Bank of Japan 220 Bankers’ Trust (BT) 59, 72, 101–2, 149, 217–18, 232, 268–71, 298, 301, 319 banking regulations 155, 159, 162, 164, 281, 286, 288 banking services 34; see also commercial banks; investment banks bankruptcy 276–7 Banque Paribas 37–8, 232 Barclays Bank 121–2, 297–8 13_INDEX.QXD 17/2/06 326 4:44 pm Page 326 Index Baring, Peter 151 Baring Brothers 51, 143, 151–2, 155 ‘Basel 2’ proposal 159 basis risk 28, 42, 274 Bear Stearns 173 bearer eurodollar collateralized securities (BECS) 231–3 ‘behavioural finance’ 136 Berkshire Hathaway 19 Bermudan options 205, 227 Bernstein, Peter 167 binomial option pricing model 196 Bismarck, Otto von 108 Black, Fischer 22, 42, 160, 185, 189–90, 193, 195, 197, 209, 215 Black–Scholes formula for option pricing 22, 185, 194–5 Black–Scholes–Merton model 160, 189–93, 196–7 ‘black swan’ hypothesis 130 Blair, Tony 223 Bogle, John 116 Bohr, Niels 122 Bond, Sir John 148 ‘bond floor’ concept 251–4 bonding 75–6, 168, 181 bonuses 146–51, 244, 262, 284–5 Brady Commission 203 brand awareness and brand equity 124, 236 Brazil 302 Bretton Woods system 33 bribery 80, 303 British Sky Broadcasting (BSB) 247–8 Brittain, Alfred 72 broad index secured trust offerings (BISTROs) 284–5 brokers 69, 309 Brown, Robert 161 bubbles 210, 310, 319 Buconero 299 Buffet, Warren 12, 19–20, 50, 110–11, 136, 173, 246, 316 business process reorganization 72 business risk 159 Business Week 130 buy-backs 249 ‘call’ options 25, 90, 99, 101, 131, 190, 196 callable bonds 227–9, 256 capital asset pricing model (CAPM) 111 capital flow 30 capital guarantees 257–8 capital structure arbitrage 296 Capote, Truman 87 carbon trading 320 ‘carry cost’ model 188 ‘carry’ trades 131–3, 171 cash accounting 139 catastrophe bonds 212, 320 caveat emptor principle 27, 272 Cayman Islands 233–4 Cazenove (company) 152 CDO2 292 Cemex 249–50 chaos theory 209, 312 Chase Manhattan Bank 143, 299 Chicago Board Options Exchange 195 Chicago Board of Trade (CBOT) 25–6, 34 chief risk officers 177 China 23–5, 276, 302–4 China Club, Hong Kong 318 Chinese walls 249, 261, 280 chrematophobia 177 Citibank and Citigroup 37–8, 43, 71, 79, 94, 134–5, 149, 174, 238–9 Citron, Robert 124–5, 212–17 client relationships 58–9 Clinton, Bill 223 Coats, Craig 168–9 collateral requirements 215–16 collateralized bond obligations (CBOs) 282 collateralized debt obligations (CDOs) 45, 282–99 13_INDEX.QXD 17/2/06 4:44 pm Page 327 Index collateralized fund obligations (CFOs) 292 collateralized loan obligations (CLOs) 283–5, 288 commercial banks 265–7 commoditization 236 commodity collateralized obligations (CCOs) 292 commodity prices 304 Commonwealth Bank of Australia 255 compliance officers 65 computer systems 54, 155, 197–8 concentration risk 271, 287 conferences with clients 59 confidence levels 164 confidentiality 226 Conseco 279–80 contagion crises 291 contango 96 contingent conversion convertibles (co-cos) 257 contingent payment convertibles (co-pays) 257 Continental Illinois 34 ‘convergence’ trading 170 convertible bonds 250–60 correlations 163–6, 294–5; see also default correlations corruption 303 CORVUS 297 Cox, John 196–7 credit cycle 291 credit default swaps (CDSs) 271–84, 293, 299 credit derivatives 129, 150, 265–72, 282, 295, 299–300 Credit Derivatives Market Practices Committee 273, 275, 280–1 credit models 294, 296 credit ratings 256–7, 270, 287–8, 297–8, 304 credit reserves 140 credit risk 158, 265–74, 281–95, 299 327 credit spreads 114, 172–5, 296 Credit Suisse 70, 106, 167 credit trading 293–5 CRH Capital 309 critical events 164–6 Croesus 137 cross-ruffing 142 cubic splines 189 currency options 98, 218, 319 custom repackaged asset vehicles (CRAVEs) 233 daily earning at risk (DEAR) concept 160 Daiwa Bank 142 Daiwa Europe 277 Danish Oil and Natural Gas 296 data scrubbing 142 dealers, work of 87–8, 124–8, 133, 167, 206, 229–37, 262, 295–6; see also traders ‘death swap’ strategy 110 decentralization 72 decision-making, scientific 182 default correlations 270–1 defaults 277–9, 287, 291, 293, 296, 299 DEFCON scale 156–7 ‘Delta 1’ options 243 delta hedging 42, 200 Deming, W.E. 98, 101 Denmark 38 deregulation, financial 34 derivatives trading 5–6, 12–14, 18–72, 79, 88–9, 99–115, 123–31, 139–41, 150, 153, 155, 175, 184–9, 206–8, 211–14, 217–19, 230, 233, 257, 262–3, 307, 316, 319–20; see also equity derivatives Derman, Emmanuel 185, 198–9 Deutsche Bank 70, 104, 150, 247–8, 274, 277 devaluations 80–1, 89, 203–4, 319 13_INDEX.QXD 17/2/06 4:44 pm Page 328 328 Index dilution of share capital 241 DINKs 313 Disney Corporation 91–8 diversification 72, 110–11, 166, 299 dividend yield 243 ‘Dr Evil’ trade 135 dollar premium 35 downsizing 73 Drexel Burnham Lambert (DBL) 282 dual currency bonds 220–3; see also reverse dual currency bonds earthquakes, bonds linked to 212 efficient markets hypothesis 22, 31, 111, 203 electronic trading 126–30, 134 ‘embeddos’ 218 emerging markets 3–4, 44, 115, 132–3, 142, 212, 226, 297 Enron 54, 142, 250, 298 enterprise risk management (ERM) 176 equity capital management 249 equity collateralized obligations (ECOs) 292 equity derivatives 241–2, 246–9, 257–62 equity index 137–8 equity investment, retail market in 258–9 equity investors’ risk 286–8 equity options 253–4 equity swaps 247–8 euro currency 171, 206, 237 European Bank for Reconstruction and Development 297 European currency units 93 European Union 247–8 Exchange Rate Mechanism, European 204 exchangeable bonds 260 expatriate postings 81–2 expert witnesses 310–12 extrapolation 189, 205 extreme value theory 166 fads of management science 72–4 ‘fairway bonds’ 225 Fama, Eugene 22, 111, 194 ‘fat tail’ events 163–4 Federal Accounting Standards Board 266 Federal Home Loans Bank 213 Federal National Mortgage Association 213 Federal Reserve Bank 20, 173 Federal Reserve Board 132 ‘Ferraris’ 232 financial engineering 228, 230, 233, 249–50, 262, 269 Financial Services Authority (FSA), Japan 106, 238 Financial Services Authority (FSA), UK 15, 135 firewalls 235–6 firing of staff 84–5 First Interstate Ltd 34–5 ‘flat’ organizations 72 ‘flat’ positions 159 floaters 231–2; see also inverse floaters ‘flow’ trading 60–1, 129 Ford Motors 282, 296 forecasting 135–6, 190 forward contracts 24–33, 90, 97, 124, 131, 188 fugu fish 239 fund management 109–17, 286, 300 futures see forward contracts Galbraith, John Kenneth 121 gamma risk 200–2, 294 Gauss, Carl Friedrich 160–2 General Motors 279, 296 General Reinsurance 20 geometric Brownian motion (GBM) 161 Ghana 98 Gibson Greeting Cards 44 Glass-Steagall Act 34 gold borrowings 132 13_INDEX.QXD 17/2/06 4:44 pm Page 329 Index gold sales 97, 137 Goldman Sachs 34, 71, 93, 150, 173, 185 ‘golfing holiday bonds’ 224 Greenspan, Alan 6, 9, 19–21, 29, 43, 47, 50, 53, 62, 132, 159, 170, 215, 223, 308 Greenwich NatWest 298 Gross, Bill 19 Guangdong International Trust and Investment Corporation (GITIC) 276–7 guaranteed annuity option (GAO) contracts 204–5 Gutenfreund, John 168–9 gyosei shido 106 Haghani, Victor 168 Hamanaka, Yasuo 142 Hamburgische Landesbank 297 Hammersmith and Fulham, London Borough of 66–7 ‘hara-kiri’ swaps 39 Hartley, L.P. 163 Hawkins, Greg 168 ‘heaven and hell’ bonds 218 hedge funds 44, 88–9, 113–14, 167, 170–5, 200–2, 206, 253–4, 262–3, 282, 292, 296, 300, 308–9 hedge ratio 264 hedging 24–8, 31, 38–42, 60, 87–100, 184, 195–200, 205–7, 214, 221, 229, 252, 269, 281, 293–4, 310 Heisenberg, Werner 122 ‘hell bonds’ 218 Herman, Clement (‘Crem’) 45–9, 77, 84, 309 Herodotus 137, 178 high net worth individuals (HNWIs) 237–8, 286 Hilibrand, Lawrence 168 Hill Samuel 231–2 329 The Hitchhiker’s Guide to the Galaxy 189 Homer, Sidney 184 Hong Kong 9, 303–4 ‘hot tubbing’ 311–12 HSBC Bank 148 HSH Nordbank 297–8 Hudson, Kevin 102 Hufschmid, Hans 77–8 IBM 36, 218, 260 ICI 34 Iguchi, Toshihude 142 incubators 309 independent valuation 142 indexed currency option notes (ICONs) 218 India 302 Indonesia 5, 9, 19, 26, 55, 80–2, 105, 146, 219–20, 252, 305 initial public offerings 33, 64, 261 inside information and insider trading 133, 241, 248–9 insurance companies 107–10, 117, 119, 150, 192–3, 204–5, 221, 223, 282, 286, 300; see also reinsurance companies insurance law 272 Intel 260 intellectual property in financial products 226 Intercontinental Hotels Group (IHG) 285–6 International Accounting Standards 33 International Securities Market Association 106 International Swap Dealers Association (ISDA) 273, 275, 279, 281 Internet stock and the Internet boom 64, 112, 259, 261, 310, 319 interpolation of interest rates 141–2, 189 inverse floaters 46–51, 213–16, 225, 232–3 13_INDEX.QXD 17/2/06 4:44 pm Page 330 330 Index investment banks 34–8, 62, 64, 67, 71, 127–8, 172, 198, 206, 216–17, 234, 265–7, 298, 309 investment managers 43–4 investment styles 111–14 irrational decisions 136 Italy 106–7 Ito’s Lemma 194 Japan 39, 43, 82–3, 92, 94, 98–9, 101, 106, 132, 142, 145–6, 157, 212, 217–25, 228, 269–70 Jensen, Michael 117 Jett, Joseph 143 JP Morgan (company) 72, 150, 152, 160, 162, 249–50, 268–9, 284–5, 299; see also Morgan Guaranty junk bonds 231, 279, 282, 291, 296–7 JWM Associates 175 Kahneman, Daniel 136 Kaplanis, Costas 174 Kassouf, Sheen 253 Kaufman, Henry 62 Kerkorian, Kirk 296 Keynes, J.M. 167, 175, 198 Keynesianism 5 Kidder Peabody 143 Kleinwort Benson 40 Korea 9, 226, 278 Kozeny, Viktor 121 Krasker, William 168 Kreiger, Andy 319 Kyoto Protocol 320 Lavin, Jack 102 law of large numbers 192 Leeson, Nick 51, 131, 143, 151 legal opinions 47, 219–20, 235, 273–4 Leibowitz, Martin 184 Leland, Hayne 42, 202 Lend Lease Corporation 261–2 leptokurtic conditions 163 leverage 31–2, 48–50, 54, 99, 102–3, 114, 131–2, 171–5, 213–14, 247, 270–3, 291, 295, 305, 308 Lewis, Kenneth 303 Lewis, Michael 77–8 life insurance 204–5 Lintner, John 111 liquidity options 175 liquidity risk 158, 173 litigation 297–8 Ljunggren, Bernt 38–40 London Inter-Bank Offered Rate (LIBOR) 6, 37 ‘long first coupon’ strategy 39 Long Term Capital Management (LTCM) 44, 51, 62, 77–8, 84, 114, 166–75, 187, 206, 210, 215–18, 263–4, 309–10 Long Term Credit Bank of Japan 94 LOR (company) 202 Louisiana Purchase 319 low exercise price options (LEPOs) 261 Maastricht Treaty and criteria 106–7 McLuhan, Marshall 134 McNamara, Robert 182 macro-economic indicators, derivatives linked to 319 Mahathir Mohammed 31 Malaysia 9 management consultants 72–3 Manchester United 152 mandatory convertibles 255 Marakanond, Rerngchai 302 margin calls 97–8, 175 ‘market neutral’ investment strategy 114 market risk 158, 173, 265 marketable eurodollar collateralized securities (MECS) 232 Markowitz, Harry 110 mark-to-market accounting 10, 100, 139–41, 145, 150, 174, 215–16, 228, 244, 266, 292, 295, 298 Marx, Groucho 24, 57, 67, 117, 308 13_INDEX.QXD 17/2/06 4:44 pm Page 331 Index mathematics applied to financial instruments 209–10; see also ‘quants’ matrix structures 72 Meckling, Herbert 117 Melamed, Leo 34, 211 merchant banks 38 Meriwether, John 167–9, 172–5 Merrill Lynch 124, 150, 217, 232 Merton, Robert 22, 42, 168–70, 175, 185, 189–90, 193–7, 210 Messier, Marie 247 Metallgesellschaft 95–7 Mexico 44 mezzanine finance 285–8, 291–7 MG Refining and Marketing 95–8, 114 Microsoft 53 Mill, Stuart 130 Miller, Merton 22, 101, 194 Milliken, Michael 282 Ministry of Finance, Japan 222 misogyny 75–7 mis-selling 238, 297–8 Mitchell, Edison 70 Mitchell & Butler 275–6 models financial 42–3, 141–2, 163–4, 173–5, 181–4, 189, 198–9, 205–10 of business processes 73–5 see also credit models Modest, David 168 momentum investment 111 monetization 260–1 monopolies in financial trading 124 moral hazard 151, 280, 291 Morgan Guaranty 37–8, 221, 232 Morgan Stanley 76, 150 mortgage-backed securities (MBSs) 282–3 Moscow, City of 277 moves of staff between firms 150, 244 Mozer, Paul 169 Mullins, David 168–70 multi-skilling 73 331 Mumbai 3 Murdoch, Rupert 247 Nabisco 220 Napoleon 113 NASDAQ index 64, 112 Nash, Ogden 306 National Australia Bank 144, 178 National Rifle Association 29 NatWest Bank 144–5, 198 Niederhoffer, Victor 130 ‘Nero’ 7, 31, 45–9, 60, 77, 82–3, 88–9, 110, 118–19, 125, 128, 292 NERVA 297 New Zealand 319 Newman, Frank 104 news, financial 133–4 News Corporation 247 Newton, Isaac 162, 210 Nippon Credit Bank 106, 271 Nixon, Richard 33 Nomura Securities 218 normal distribution 160–3, 193, 199 Northern Electric 248 O’Brien, John 202 Occam, William 188 off-balance sheet transactions 32–3, 99, 234, 273, 282 ‘offsites’ 74–5 oil prices 30, 33, 89–90, 95–7 ‘omitted variable’ bias 209–10 operational risk 158, 176 opinion shopping 47 options 9, 21–2, 25–6, 32, 42, 90, 98, 124, 197, 229 pricing 185, 189–98, 202 Orange County 16, 44, 50, 124–57, 212–17, 232–3 orphan subsidiaries 234 over-the-counter (OTC) market 26, 34, 53, 95, 124, 126 overvaluation 64 13_INDEX.QXD 17/2/06 4:44 pm Page 332 332 Index ‘overwhelming force’ strategy 134–5 Owen, Martin 145 ownership, ‘legal’ and ‘economic’ 247 parallel loans 35 pari-mutuel auction system 319 Parkinson’s Law 136 Parmalat 250, 298–9 Partnoy, Frank 87 pension funds 43, 108–10, 115, 204–5, 255 People’s Bank of China (PBOC) 276–7 Peters’ Principle 71 petrodollars 71 Pétrus (restaurant) 121 Philippines, the 9 phobophobia 177 Piga, Gustavo 106 PIMCO 19 Plaza Accord 38, 94, 99, 220 plutophobia 177 pollution quotas 320 ‘portable alpha’ strategy 115 portfolio insurance 112, 202–3, 294 power reverse dual currency (PRDC) bonds 226–30 PowerPoint 75 preferred exchangeable resettable listed shares (PERLS) 255 presentations of business models 75 to clients 57, 185 prime brokerage 309 Prince, Charles 238 privatization 205 privity of contract 273 Proctor & Gamble (P&G) 44, 101–4, 155, 298, 301 product disclosure statements (PDSs) 48–9 profit smoothing 140 ‘programme’ issuers 234–5 proprietary (‘prop’) trading 60, 62, 64, 130, 174, 254 publicly available information (PAI) 277 ‘puff’ effect 148 purchasing power parity theory 92 ‘put’ options 90, 131, 256 ‘quants’ 183–9, 198, 208, 294 Raabe, Matthew 217 Ramsay, Gordon 121 range notes 225 real estate 91, 219 regulatory arbitrage 33 reinsurance companies 288–9 ‘relative value’ trading 131, 170–1, 310 Reliance Insurance 91–2 repackaging (‘repack’) business 230–6, 282, 290 replication in option pricing 195–9, 202 dynamic 200 research provided to clients 58, 62–4, 184 reserves, use of 140 reset preference shares 254–7 restructuring of loans 279–81 retail equity products 258–9 reverse convertibles 258–9 reverse dual currency bonds 223–30 ‘revolver’ loans 284–5 risk, financial, types of 158 risk adjusted return on capital (RAROC) 268, 290 risk conservation principle 229–30 risk management 65, 153–79, 184, 187, 201, 267 risk models 163–4, 173–5 riskless portfolios 196–7 RJ Reynolds (company) 220–1 rogue traders 176, 313–16 Rosenfield, Eric 168 Ross, Stephen 196–7, 202 Roth, Don 38 Rothschild, Mayer Amshel 267 Royal Bank of Scotland 298 Rubinstein, Mark 42, 196–7 13_INDEX.QXD 17/2/06 4:44 pm Page 333 Index Rumsfeld, Donald 12, 134, 306 Rusnak, John 143 Russia 45, 80, 166, 172–3, 274, 302 sales staff 55–60, 64–5, 125, 129, 217 Salomon Brothers 20, 36, 54, 62, 167–9, 174, 184 Sandor, Richard 34 Sanford, Charles 72, 269 Sanford, Eugene 269 Schieffelin, Allison 76 Scholes, Myron 22, 42, 168–71, 175, 185, 189–90, 193–7, 263–4 Seagram Group 247 Securities and Exchange Commission, US 64, 304 Securities and Futures Authority, UK 249 securitization 282–90 ‘security design’ 254–7 self-regulation 155 sex discrimination 76 share options 250–1 Sharpe, William 111 short selling 30–1, 114 Singapore 9 single-tranche CDOs 293–4, 299 ‘Sisters of Perpetual Ecstasy’ 234 SITCOMs 313 Six Continents (6C) 275–6 ‘smile’ effect 145 ‘snake’ currency system 203 ‘softing’ arrangements 117 Solon 137 Soros, George 44, 130, 253, 318–19 South Sea Bubble 210 special purpose asset repackaging companies (SPARCs) 233 special purpose vehicles (SPVs) 231–4, 282–6, 290, 293 speculation 29–31, 42, 67, 87, 108, 130 ‘spinning’ 64 333 Spitzer, Eliot 64 spread 41, 103; see also credit spreads stack hedges 96 Stamenson, Michael 124–5 standard deviation 161, 193, 195, 199 Steinberg, Sol 91 stock market booms 258, 260 stock market crashes 42–3, 168, 203, 257, 259, 319 straddles or strangles 131 strategy in banking 70 stress testing 164–6 stripping of convertible bonds 253–4 structured investment products 44, 112, 115, 118, 128, 211–39, 298 structured note asset packages (SNAPs) 233 Stuart SC 18, 307, 316–18 Styblo Bleder, Tanya 153 Suharto, Thojib 81–2 Sumitomo Corporation 100, 142 Sun Tzu 61 Svensk Exportkredit (SEK) 38–9 swaps 5–10, 26, 35–40, 107, 188, 211; see also equity swaps ‘swaptions’ 205–6 Swiss Bank Corporation (SBC) 248–9 Swiss banks 108, 305 ‘Swiss cheese theory’ 176 synthetic securitization 284–5, 288–90 systemic risk 151 Takeover Panel 248–9 Taleb, Nassim 130, 136, 167 target redemption notes 225–6 tax and tax credits 171, 242–7, 260–3 Taylor, Frederick 98, 101 team-building exercises 76 team moves 149 technical analysis 60–1, 135 television programmes about money 53, 62–3 Thailand 9, 80, 302–5 13_INDEX.QXD 17/2/06 4:44 pm Page 334 334 Index Thatcher, Margaret 205 Thorp, Edward 253 tobashi trades 105–7 Tokyo Disneyland 92, 212 top managers 72–3 total return swaps 246–8, 269 tracking error 138 traders in financial products 59–65, 129–31, 135–6, 140, 148, 151, 168, 185–6, 198; see also dealers trading limits 42, 157, 201 trading rooms 53–4, 64, 68, 75–7, 184–7, 208 Trafalgar House 248 tranching 286–9, 292, 296 transparency 26, 117, 126, 129–30, 310 Treynor, Jack 111 trust investment enhanced return securities (TIERS) 216, 233 trust obligation participating securities (TOPS) 232 TXU Europe 279 UBS Global Asset Management 110, 150, 263–4, 274 uncertainty principle 122–3 unique selling propositions 118 unit trusts 109 university education 187 unspecified fund obligations (UFOs) 292 ‘upfronting’ of income 139, 151 Valéry, Paul 163 valuation 64, 142–6 value at risk (VAR) concept 160–7, 173 value investing 111 Vanguard 116 vanity bonds 230 variance 161 Vietnam War 182, 195 Virgin Islands 233–4 Vivendi 247–8 volatility of bond prices 197 of interest rates 144–5 of share prices 161–8, 172–5, 192–3, 199 Volcker, Paul 20, 33 ‘warehouses’ 40–2, 139 warrants arbitrage 99–101 weather, bonds linked to 212, 320 Weatherstone, Dennis 72, 268 Weil, Gotscal & Manges 298 Weill, Sandy 174 Westdeutsche Genosenschafts Zentralbank 143 Westminster Group 34–5 Westpac 261–2 Wheat, Allen 70, 72, 106, 167 Wojniflower, Albert 62 World Bank 4, 36, 38 World Food Programme 320 Worldcom 250, 298 Wriston, Walter 71 WTI (West Texas Intermediate) contracts 28–30 yield curves 103, 188–9, 213, 215 yield enhancement 112, 213, 269 ‘yield hogs’ 43 zaiteku 98–101, 104–5 zero coupon bonds 221–2, 257–8


pages: 543 words: 157,991

All the Devils Are Here by Bethany McLean

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Asian financial crisis, asset-backed security, bank run, Black-Scholes formula, break the buck, call centre, collateralized debt obligation, corporate governance, corporate raider, Credit Default Swap, credit default swaps / collateralized debt obligations, diversification, Exxon Valdez, fear of failure, financial innovation, fixed income, high net worth, Home mortgage interest deduction, interest rate swap, laissez-faire capitalism, Long Term Capital Management, margin call, market bubble, market fundamentalism, Maui Hawaii, money market fund, moral hazard, mortgage debt, Northern Rock, Own Your Own Home, Ponzi scheme, quantitative trading / quantitative finance, race to the bottom, risk/return, Ronald Reagan, Rosa Parks, shareholder value, short selling, South Sea Bubble, statistical model, telemarketer, too big to fail, value at risk, zero-sum game

But as a risk manager, Guldimann was often confronted with the problem of what to do when a trader wanted to increase his limit. “How should I know if he should get his increase?” Guldimann says. “All I could do is ask around. Is he a good guy? Does he know what he’s doing? It was ridiculous.” There was never any question about how Guldimann and his team would approach this task. They would use statistics and probability theories that had long been popular on Wall Street. (The Black-Scholes formula, for example, developed in the early 1970s for pricing options, had become one of the linchpins of modern Wall Street.) The quants swarming Wall Street were all steeped in those theories—this was the essential building block of virtually everything they did. They knew no other way to approach the subject. Sure enough, Value at Risk, or VaR, the model the J.P. Morgan quants came up with after years of trial and error, was built on a key tenet of the mathematics of probability, called Gaussian distribution.

See Federal bailouts Bair, Sheila Baker, Richard Bankers Trust, swap deal lawsuit Bank of America Countrywide acquired by Merrill Lynch acquired by subprime branches, closing Barnes, Roy Bartiromo, Maria Basel Committee on Banking Supervision, capital reserves rule Basis Yield Alpha Fund Bear Stearns ABS index Bank of America lawsuit CDOs foreclosures, plan to prevent hedge funds, collapse of High-Grade Structured Credit Fund High-Grade Structured Credit Strategies Enhanced Leverage Limited Partnership J.P. Morgan acquisition of Beattie, Richard Behavioral economics Beneficial Bensinger, Steve Bernanke, Ben Berson, David Birnbaum, Josh BlackRock Black-Scholes formula Black swans Blankfein, Lloyd during collapse compensation from Goldman Goldman Sachs under Blue sky laws, MBSs exemption from Blum, Michael Blumenthal Stephen BNC Mortgage BNP Paribas Bolten, Joshua Bomchill, Mark Bond, Kit Bond ratings CDOs and credit enhancements failures, examples of public trust in ratings shopping system of for tranches Value at Risk (VaR) applied to See also Moody’s; Standard & Poor’s Bonuses, post-TARP Born, Brooksley biographical information derivatives, regulatory efforts style/personality of Bothwell, James Bowsher, Charles Bradbury, Darcy Brandt, Amy Breit, John Brennan, Mary Elizabeth Brickell, Mark Brightpoint fraud Broad Index Secured Trust Offering (BISTRO) AIG FP credit protection features of Broderick, Craig Bronfman, Edward and Peter Bruce, Kenneth Buffett, Warren Burry, Michael Bush, George W.


pages: 545 words: 137,789

How Markets Fail: The Logic of Economic Calamities by John Cassidy

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Albert Einstein, Andrei Shleifer, anti-communist, asset allocation, asset-backed security, availability heuristic, bank run, banking crisis, Benoit Mandelbrot, Berlin Wall, Bernie Madoff, Black-Scholes formula, Bretton Woods, British Empire, capital asset pricing model, centralized clearinghouse, collateralized debt obligation, Columbine, conceptual framework, Corn Laws, corporate raider, correlation coefficient, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, crony capitalism, Daniel Kahneman / Amos Tversky, debt deflation, diversification, Elliott wave, Eugene Fama: efficient market hypothesis, financial deregulation, financial innovation, Financial Instability Hypothesis, financial intermediation, full employment, George Akerlof, global supply chain, Gunnar Myrdal, Haight Ashbury, hiring and firing, Hyman Minsky, income per capita, incomplete markets, index fund, information asymmetry, Intergovernmental Panel on Climate Change (IPCC), invisible hand, John Nash: game theory, John von Neumann, Joseph Schumpeter, Kenneth Arrow, laissez-faire capitalism, Landlord’s Game, liquidity trap, London Interbank Offered Rate, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, market bubble, market clearing, mental accounting, Mikhail Gorbachev, money market fund, Mont Pelerin Society, moral hazard, mortgage debt, Myron Scholes, Naomi Klein, negative equity, Network effects, Nick Leeson, Northern Rock, paradox of thrift, Pareto efficiency, Paul Samuelson, Ponzi scheme, price discrimination, price stability, principal–agent problem, profit maximization, quantitative trading / quantitative finance, race to the bottom, Ralph Nader, RAND corporation, random walk, Renaissance Technologies, rent control, Richard Thaler, risk tolerance, risk-adjusted returns, road to serfdom, Robert Shiller, Robert Shiller, Ronald Coase, Ronald Reagan, shareholder value, short selling, Silicon Valley, South Sea Bubble, sovereign wealth fund, statistical model, technology bubble, The Chicago School, The Great Moderation, The Market for Lemons, The Wealth of Nations by Adam Smith, too big to fail, transaction costs, unorthodox policies, value at risk, Vanguard fund, Vilfredo Pareto, wealth creators, zero-sum game

Finally, in the early 1970s, Black, a man of few words, and Scholes, a voluble Canadian, derived a simple formula that related the price of an option to the volatility of the underlying stock. By coincidence, the paper that contained the Black-Scholes option pricing formula was published in May 1973, a month after the opening of the Chicago Board Options Exchange. To compute the value of an option using the Black-Scholes formula all you needed, in addition to the strike price, the current price, and the duration of the option, was the interest rate on government bonds, the standard deviation of the stock, and a table of the normal distribution. By the end of 1973, you didn’t even need a pen and paper to do the calculation: Texas Instruments had introduced a calculator that did it for you. That was only the beginning.

Mutual funds were able to insure themselves against the risk of corporations defaulting on their bonds, banks could insure themselves against some of their lenders defaulting, and insurance companies could insure against the chances of a freak hurricane leaving them with enormous claims from their policyholders. In each of these areas, the key was the development of mathematical methods to price risk. Almost all of these methods relied, to some extent, on the Black-Scholes formula and the bell curve. Simply by invoking the ghost of Louis Bachelier, it was possible to take much of the danger out of finance. Or was it? As far back as the 1960s and ’70s, some academics and Wall Street practitioners didn’t buy into the coin-tossing view of finance. Many old-school bankers and traders were put off by the mathematical demands it came with, but numbered among the skeptics were also some technically adept economists, including Sanford Grossman, of Wharton, and Joseph Stiglitz, who is now at Columbia.


pages: 504 words: 139,137

Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined by Lasse Heje Pedersen

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activist fund / activist shareholder / activist investor, algorithmic trading, Andrei Shleifer, asset allocation, backtesting, bank run, banking crisis, barriers to entry, Black-Scholes formula, Brownian motion, buy low sell high, capital asset pricing model, commodity trading advisor, conceptual framework, corporate governance, credit crunch, Credit Default Swap, currency peg, David Ricardo: comparative advantage, declining real wages, discounted cash flows, diversification, diversified portfolio, Emanuel Derman, equity premium, Eugene Fama: efficient market hypothesis, fixed income, Flash crash, floating exchange rates, frictionless, frictionless market, Gordon Gekko, implied volatility, index arbitrage, index fund, interest rate swap, late capitalism, law of one price, Long Term Capital Management, margin call, market clearing, market design, market friction, merger arbitrage, money market fund, mortgage debt, Myron Scholes, New Journalism, paper trading, passive investing, price discovery process, price stability, purchasing power parity, quantitative easing, quantitative trading / quantitative finance, random walk, Renaissance Technologies, Richard Thaler, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, selection bias, shareholder value, Sharpe ratio, short selling, sovereign wealth fund, statistical arbitrage, statistical model, survivorship bias, systematic trading, technology bubble, time value of money, total factor productivity, transaction costs, value at risk, Vanguard fund, yield curve, zero-coupon bond

It’s one thing to look at the water from afar; it’s another thing to look at it right up close. From afar, it looks pretty calm; up close, it looks chaotic. I felt that the experience of the chaotic world, married to my theoretical abilities, would allow me to gain unique perspectives. For that reason, I gravitated to work for a while at Salomon Brothers. LHP: When most people think about the Black–Scholes formula, they think first about equity options, but you focused on fixed income arbitrage—why? MS: Right. The fascination with fixed-income arbitrage came about after many years of thinking about the idea that there are natural segmented clienteles. Insurance companies and pension funds tend to be at the longer end of the interest rate curve. Macro hedge funds try to express their macro views at the 10-year part of the curve, and other issuers—mortgage issuers and so forth—are also at the 10–15 year part of the curve.

For each bond they provide the coupon, the conversion ratio, the conversion value—all the salient terms of the instrument. Based on the market prices in that book, there appeared to be some bonds that were mispriced. I made it my mission to educate myself on these instruments and to understand the pricing and trading of convertibles. LHP: Was the insight just based on some back of the envelope calculations, or did you need to already appreciate something like the Black–Scholes Formula or the binomial option pricing model at that time? KG: Back of the envelope, some common sense, and a bit of naïveté as to the dynamics around why these mispricings might exist. Many mispricings were driven by the inability to borrow the underlying common stock and therefore the convertible bond traded close to conversion value because the arbitrage was difficult. Nonetheless, I didn’t understand these dynamics at the moment.

Stock Market Wizards: Interviews With America's Top Stock Traders by Jack D. Schwager

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Asian financial crisis, banking crisis, barriers to entry, beat the dealer, Black-Scholes formula, commodity trading advisor, computer vision, East Village, Edward Thorp, financial independence, fixed income, implied volatility, index fund, Jeff Bezos, John Meriwether, John von Neumann, locking in a profit, Long Term Capital Management, margin call, money market fund, Myron Scholes, paper trading, passive investing, pattern recognition, random walk, risk tolerance, risk-adjusted returns, short selling, Silicon Valley, statistical arbitrage, the scientific method, transaction costs, Y2K

The problem is that the Almighty is not giving me or anyone else the probability distribution for the price of IBM a month from now. The standard approach, which is based on the Black-Scholes formula, assumes that the probability distribution will conform to a normal curve [the familiar bell-shaped curve frequently used to depict probabilities, such as the probability distribution of IQ scores among the population]. The critical statement is that it "assumes a normal probability distribution." Who ran out and told these guys that was the correct probability distribution? Where did they get this idea? [The Black-Scholes formula (or one of its variations) is the widely used equation for deriving an option's theoretical value. An implicit assump*A probability distribution is simply a curve that shows the probabilities of some event occurring—in this case, the probabilities of a given stock being at any price on the option expiration date.


pages: 261 words: 10,785

The Lights in the Tunnel by Martin Ford

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Albert Einstein, Bill Joy: nanobots, Black-Scholes formula, call centre, cloud computing, collateralized debt obligation, commoditize, creative destruction, credit crunch, double helix, en.wikipedia.org, factory automation, full employment, income inequality, index card, industrial robot, inventory management, invisible hand, Isaac Newton, job automation, John Markoff, John Maynard Keynes: Economic Possibilities for our Grandchildren, John Maynard Keynes: technological unemployment, knowledge worker, low skilled workers, mass immigration, moral hazard, pattern recognition, prediction markets, Productivity paradox, Ray Kurzweil, Search for Extraterrestrial Intelligence, Silicon Valley, Stephen Hawking, strong AI, superintelligent machines, technological singularity, Thomas L Friedman, Turing test, Vernor Vinge, War on Poverty

Stock options, which represent the right to buy or sell a stock at a given price at some point in the future, had been traded on markets for some time, but no one knew how to calculate a precise value for them. In the years that followed, and especially during the 1980s, a large number of people originally trained as physicists or mathematicians began to take much higher paying jobs on Wall Street. These guys (they were virtually all men) were referred to as “quants.” The quants started working with the Black-Scholes formula and expanded it in new ways. They turned their formulas into computer programs and gradually began to create new types of derivatives based on stocks, bonds, indexes and many other securities or combinations of securities.14 Copyrighted Material – Paperback/Kindle available @ Amazon Acceleration / 45 As their computers got faster and faster, the quants were able to do more and more. They created new exotic derivatives based on strange combinations of things.


pages: 204 words: 58,565

Keeping Up With the Quants: Your Guide to Understanding and Using Analytics by Thomas H. Davenport, Jinho Kim

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Black-Scholes formula, business intelligence, business process, call centre, computer age, correlation coefficient, correlation does not imply causation, Credit Default Swap, en.wikipedia.org, feminist movement, Florence Nightingale: pie chart, forensic accounting, global supply chain, Hans Rosling, hypertext link, invention of the telescope, inventory management, Jeff Bezos, margin call, Moneyball by Michael Lewis explains big data, Myron Scholes, Netflix Prize, p-value, performance metric, publish or perish, quantitative hedge fund, random walk, Renaissance Technologies, Robert Shiller, Robert Shiller, self-driving car, sentiment analysis, six sigma, Skype, statistical model, supply-chain management, text mining, the scientific method

However, Black and Scholes performed empirical tests of their theoretically derived model on a large body of call-option data in their paper “The Pricing of Options and Corporate Liabilities.”16 DATA ANALYSIS. Black and Scholes could derive a partial differential equation based on some arguments and technical assumptions (a model from calculus, not statistics). The solution to this equation was the Black-Scholes formula, which suggested how the price of a call option might be calculated as a function of a risk-free interest rate, the price variance of the asset on which the option was written, and the parameters of the option (strike price, term, and the market price of the underlying asset). The formula introduces the concept that, the higher the share price today, the higher the volatility of the share price, the higher the risk-free interest rate, the longer the time to maturity, and the lower the exercise price, then the higher the option value.


pages: 295 words: 66,824

A Mathematician Plays the Stock Market by John Allen Paulos

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Benoit Mandelbrot, Black-Scholes formula, Brownian motion, business climate, butterfly effect, capital asset pricing model, correlation coefficient, correlation does not imply causation, Daniel Kahneman / Amos Tversky, diversified portfolio, Donald Trump, double entry bookkeeping, Elliott wave, endowment effect, Erdős number, Eugene Fama: efficient market hypothesis, four colour theorem, George Gilder, global village, greed is good, index fund, intangible asset, invisible hand, Isaac Newton, John Nash: game theory, Long Term Capital Management, loss aversion, Louis Bachelier, mandelbrot fractal, margin call, mental accounting, Myron Scholes, Nash equilibrium, Network effects, passive investing, Paul Erdős, Paul Samuelson, Ponzi scheme, price anchoring, Ralph Nelson Elliott, random walk, Richard Thaler, Robert Shiller, Robert Shiller, short selling, six sigma, Stephen Hawking, survivorship bias, transaction costs, ultimatum game, Vanguard fund, Yogi Berra

And options on a stock whose volatility is high will cost more than options on stocks that barely move from quarter to quarter (just as a short man on a pogo stick is more likely to be able to peek over a nine-foot fence than a tall man who can’t jump). Less intuitive is the fact that the cost of a call option also rises with the interest rate, assuming all other parameters remain unchanged. Although there are any number of books and websites on the Black-Scholes formula, it and its variants are more likely to be used by professional traders than by gamblers, who rely on commonsense considerations and gut feel. Viewing options as pure bets, gamblers are generally as interested in carefully pricing them as casino-goers are in the payoff ratios of slot machines. The Lure of Illegal Leverage Because of the leverage possible with the purchase, sale, or mere possession of options, they sometimes attract people who aren’t content to merely play the slots but wish to stick their thumbs onto the spinning disks and directly affect the outcomes.


pages: 311 words: 99,699

Fool's Gold: How the Bold Dream of a Small Tribe at J.P. Morgan Was Corrupted by Wall Street Greed and Unleashed a Catastrophe by Gillian Tett

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accounting loophole / creative accounting, asset-backed security, bank run, banking crisis, Black-Scholes formula, break the buck, Bretton Woods, business climate, collateralized debt obligation, commoditize, creative destruction, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, diversification, easy for humans, difficult for computers, financial innovation, fixed income, housing crisis, interest rate derivative, interest rate swap, locking in a profit, Long Term Capital Management, McMansion, money market fund, mortgage debt, North Sea oil, Northern Rock, Renaissance Technologies, risk tolerance, Robert Shiller, Robert Shiller, Satyajit Das, short selling, sovereign wealth fund, statistical model, The Great Moderation, too big to fail, value at risk, yield curve

After he left J.P. Morgan back in 2000, he had created a consultancy group that advised governments and companies on how to use innovative financial products, such as derivatives, to their benefit. The other founding members of the group were Roberto Mendoza, another former J.P. Morgan banker, and Robert Merton, the Nobel Prize–winning economist who had helped to create the pathbreaking Black-Scholes formula that had played a crucial role in the development of derivatives. For seven long years, Hancock had extolled the virtues of financial innovation, often in the face of client skepticism. Even as the banking world reeled in shock in late 2007, he remained committed to the cause. “A lot of the problems in structured finance have not been due to too much innovation, but a failure to innovate sufficiently,” Hancock observed.


pages: 338 words: 106,936

The Physics of Wall Street: A Brief History of Predicting the Unpredictable by James Owen Weatherall

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Albert Einstein, algorithmic trading, Antoine Gombaud: Chevalier de Méré, Asian financial crisis, bank run, beat the dealer, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, butterfly effect, capital asset pricing model, Carmen Reinhart, Claude Shannon: information theory, collateralized debt obligation, collective bargaining, dark matter, Edward Lorenz: Chaos theory, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, George Akerlof, Gerolamo Cardano, Henri Poincaré, invisible hand, Isaac Newton, iterative process, John Nash: game theory, Kenneth Rogoff, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, martingale, Myron Scholes, new economy, Paul Lévy, Paul Samuelson, prediction markets, probability theory / Blaise Pascal / Pierre de Fermat, quantitative trading / quantitative finance, random walk, Renaissance Technologies, risk-adjusted returns, Robert Gordon, Robert Shiller, Robert Shiller, Ronald Coase, Sharpe ratio, short selling, Silicon Valley, South Sea Bubble, statistical arbitrage, statistical model, stochastic process, The Chicago School, The Myth of the Rational Market, tulip mania, V2 rocket, Vilfredo Pareto, volatility smile

Throughout the text, however, I will mean something very specific: a strategy by which one constantly updates the proportions of stocks and options in one’s portfolio so that the portfolio as a whole is risk-free. “. . . successfully urged the Journal of Political Economy to reconsider . . .”: The article was published as Black and Scholes (1973). See also Merton (1973) and Black and Scholes (1972, 1974). For more on the Black-Scholes formula and its generalizations and extensions, see Hull (2011) and Cox and Rubinstein (1985). “The head of that committee was James Lorie . . .”: For more on the history of the CBOE, see Markham (2002) and MacKenzie (2006). “On the first day of trading . . .”: These numbers are from Markham (2002, vol. 3, p. 52). “But volume grew at an astonishing rate . . .”: These numbers are from Ansbacher (2000, p. xii).


pages: 437 words: 132,041

Alex's Adventures in Numberland by Alex Bellos

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Andrew Wiles, Antoine Gombaud: Chevalier de Méré, beat the dealer, Black Swan, Black-Scholes formula, Claude Shannon: information theory, computer age, Daniel Kahneman / Amos Tversky, Edward Thorp, family office, forensic accounting, game design, Georg Cantor, Henri Poincaré, Isaac Newton, Myron Scholes, pattern recognition, Paul Erdős, Pierre-Simon Laplace, probability theory / Blaise Pascal / Pierre de Fermat, random walk, Richard Feynman, Richard Feynman, Rubik’s Cube, SETI@home, Steve Jobs, The Bell Curve by Richard Herrnstein and Charles Murray, traveling salesman

‘The family went on a trip to the World’s Fair in Spokane and on the way back we stopped at Harrah’s [casino] and I told my kids to give me a couple of hours because I wanted to pay for the trip – which I did.’ Beat the Dealer is not just a gambling classic. It also reverberated through the worlds of economics and finance. A generation of mathematicians inspired by Thorp’s book began to create models of the financial markets and apply betting strategies to them. Two of them, Fischer Black and Myron Scholes, created the Black-Scholes formula indicating how to price financial derivatives – Wall Street’s most famous (and infamous) equation. Thorp ushered in an era when the quantitative analyst, the ‘quant’ – the name given to the mathematicians relied on by banks to find clever ways of investing – was king. ‘Beat the Dealer was kind of the first quant book out there and it led fairly directly to quite a revolution,’ said Thorp, who can claim – with some justification – to being the first-ever quant.


pages: 512 words: 162,977

New Market Wizards: Conversations With America's Top Traders by Jack D. Schwager

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backtesting, beat the dealer, Benoit Mandelbrot, Berlin Wall, Black-Scholes formula, butterfly effect, commodity trading advisor, computerized trading, Edward Thorp, Elliott wave, fixed income, full employment, implied volatility, interest rate swap, Louis Bachelier, margin call, market clearing, market fundamentalism, money market fund, paper trading, pattern recognition, placebo effect, prediction markets, Ralph Nelson Elliott, random walk, risk tolerance, risk/return, Saturday Night Live, Sharpe ratio, the map is not the territory, transaction costs, War on Poverty

When I think about pricing an option, I may not know calculus, but in my mind I can draw a picture of how you would price an option that looks exactly like the theoretical pricing models in the textbooks. When did you first get involved in trading options? I did a little dabbling with stock options back in 1975-76 on the Chicago Board of Options Exchange, but I didn’t stay with it. I first got involved with options in a serious way with the initiation of trading in futures options. By the way, in 1975 I crammed the Black-Scholes formula into a TI52 hand-held calculator, which was capable of giving me one option price in about thirteen seconds, after I hand-inserted all the other variables. It was pretty crude, but in the land of the blind, I was the guy with one eye. When the market was in its embryonic stage, were the options seriously mispriced, and was your basic strategy aimed at taking advantage of these mispricings?


pages: 500 words: 145,005

Misbehaving: The Making of Behavioral Economics by Richard H. Thaler

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3Com Palm IPO, Albert Einstein, Alvin Roth, Amazon Mechanical Turk, Andrei Shleifer, Apple's 1984 Super Bowl advert, Atul Gawande, Berlin Wall, Bernie Madoff, Black-Scholes formula, capital asset pricing model, Cass Sunstein, Checklist Manifesto, choice architecture, clean water, cognitive dissonance, conceptual framework, constrained optimization, Daniel Kahneman / Amos Tversky, delayed gratification, diversification, diversified portfolio, Edward Glaeser, endowment effect, equity premium, Eugene Fama: efficient market hypothesis, experimental economics, Fall of the Berlin Wall, George Akerlof, hindsight bias, Home mortgage interest deduction, impulse control, index fund, information asymmetry, invisible hand, Jean Tirole, John Nash: game theory, John von Neumann, Kenneth Arrow, late fees, law of one price, libertarian paternalism, Long Term Capital Management, loss aversion, market clearing, Mason jar, mental accounting, meta analysis, meta-analysis, money market fund, More Guns, Less Crime, mortgage debt, Myron Scholes, Nash equilibrium, Nate Silver, New Journalism, nudge unit, Paul Samuelson, payday loans, Ponzi scheme, presumed consent, pre–internet, principal–agent problem, prisoner's dilemma, profit maximization, random walk, randomized controlled trial, Richard Thaler, Robert Shiller, Robert Shiller, Ronald Coase, Silicon Valley, South Sea Bubble, statistical model, Steve Jobs, technology bubble, The Chicago School, The Myth of the Rational Market, The Signal and the Noise by Nate Silver, The Wealth of Nations by Adam Smith, Thomas Kuhn: the structure of scientific revolutions, transaction costs, ultimatum game, Vilfredo Pareto, Walter Mischel, zero-sum game

Where should this leave us regarding the validity of the Becker conjecture—that the 10% of people who can do probabilities will end up in the jobs where such skills matter? At some level we might expect this conjecture to be true. All NFL players are really good at football; all copyeditors are good at spelling and grammar; all option traders can at least find the button on their calculators that can compute the Black–Scholes formula, and so forth. A competitive labor market does do a pretty good job of channeling people into jobs that suit them. But ironically, this logic may become less compelling as we move up the managerial ladder. All economists are at least pretty good at economics, but many who are chosen to be department chair fail miserably at that job. This is the famous Peter Principle: people keep getting promoted until they reach their level of incompetence.