# incomplete markets

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

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A closed-form formula as an infinite sum can be developed for the price of a call or put option in a jump-diffusion model. The market consisting of a stock evolving to a jump-diffusion model and a riskless bond is incomplete. In an incomplete market an option does not have a unique price. When changing measure in a jump-diffusion world, we can change the drift, the intensity of the jumps and the jump distribution but we cannot change the volatility of the underlying. 386 Incomplete markets and jump-diffusion processes Increasing jump-intensity always increases the price of a European option, which has a convex final payoff. For a digital option increasing jump-intensity can either increase or decrease the price of an option. In an incomplete market it is the market which chooses the measure. It is possible to hedge in a jump-diffusion model using options provided we assume that the market does not change its choice of measure.

Our hedging argument has shown that only prices between zero and five can be non-arbitrageable, whilst the risk-neutral argument shows that prices between zero and five are not arbitrageable. We therefore conclude that the set of arbitrage-free prices for the option is the set of prices between zero and five. The three-world universe is an example of an incomplete market, that is, a market where portfolios cannot be arranged to give precisely the desired pay-off, and it is characteristic of incomplete markets that the price of an option can only be shown to lie in an interval rather than being forced to take a precise value. The market price of such an option would then be determined within the range of possible prices by the risk-preferences of traders in the market rather than mathematics. , 3.3 Multiple time steps 3.3.1 More realism At this point, option pricing is not looking very successful - clearly we will want to price options on assets that can have more than two values in the future.

The bank can still make money by selling put options but it is doing so by taking on risk, rather than by charging for the cost of hedging, as in the Black-Scholes framework. The purchase of the put option is therefore a transference of risk from the fund manager to the bank. The market price will settle on a point where the 361 362 Incomplete markets and jump-diffusion processes banks feel that they are being adequately compensated for taking on the extra risk. A major determinant of the price is therefore risk preferences rather than arbitrage. Once we have moved to an incomplete market, there are two different issues to be addressed. The first is how to use arbitrage to bound the prices of vanilla options. The second is to determine prices for exotic options which are compatible with both the model and the prices of the vanilla options traded in the market.

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

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In other words, all these measures coincide on In addition, note that theoretical problems also arise for the case of random r. 5.11.2 Pricing for an incomplete market Mean-variance hedging Similarly to the case of the discrete time market, Definition 5.43 leads to superreplication for incomplete markets. Clearly, it is not always meaningful. Therefore, there is another popular approach for an incomplete market. Definition 5.65 (mean-variance hedging). The fair price of the option is the initial wealth X(0) such E|X(T)−ψ|2 is minimal over all admissible self-financing strategies. In many cases, this definition leads to the option price e−rTE*ψ, where E* is the expectation for a risk-neutral equivalent measure that needs to be chosen by some optimal way, since this measure is not unique for an incomplete market. This measure needs to be found via solution of an optimization problem.

Find the fair price of the option with payoff F(S1,…, ST)=max(ST−1, 0). Solution. We have For incomplete markets, Definition 3.49 leads to super-replication. That is not always meaningful. Therefore, there is another popular approach for incomplete markets. Definition 3.55 (mean-variance hedging). The fair price of the option is the initial wealth X0 such that E|XT−ψ|2 is minimal over all admissible self-financing strategies. © 2007 Nikolai Dokuchaev Discrete Time Market Models 41 In many cases, this definition leads to the option price calculated as the expectation under a risk-neutral equivalent measure which needs to be chosen by some optimal way, since a risk-neutral equivalent measure is not unique for an incomplete market. 3.11 Increasing frequency and continuous time limit In reality, prices may change and be measured very frequently.

If an equivalent risk-neutral measure is not unique then, by Theorem 5.35, the market cannot be complete, i.e., there are claims ψ that cannot be replicable. In this section, we assume that r is non-random and constant. Let be the filtration generated by the process S(t). (For this case of non-random general case, the filtration For the generated by the process (w(t), η(t) is larger than 5.11.1 Examples of incomplete markets An example with a≡r Let a(t)≡r(t), and let σ=σ(t, η), where η is a random process (or a random vector, or a random variable), independent from the driving Wiener process w(t) (for instance, η may represent another Wiener process). Clearly, any original probability measure P=Pη is a risk-neutral measure (note that for any η). Any probability measure is defined by the pair (w, η), therefore it depends on the choice of η.

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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Oxford Science Publications Lewis, A Series of articles in Wilmott magazine September 2002 to August 2004 Merton, RC 1976 Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3 125-44 What is Meant by “Complete” and “Incomplete” Markets? Short Answer A complete market is one in which a derivative product can be artificially made from more basic instruments, such as cash and the underlying asset. This usually involves dynamically rebalancing a portfolio of the simpler instruments, according to some formula or algorithm, to replicate the more complicated product, the derivative. Obviously, an incomplete market is one in which you can’t replicate the option with simpler instruments. Example The classic example is replicating an equity option, a call, say, by continuously buying or selling the equity so that you always hold the amountΔ = e−D(T −t)N (d1), in the stock, where and Long Answer A slightly more mathematical, yet still quite easily understood, description is to say that a complete market is one for which there exist the same number of linearly independent securities as there are states of the world in the future.

People can therefore disagree on the probability of a stock rising or falling but still agree on the value of an option, as long as they share the same view on the stock’s volatility. In probabilistic terms we say that in a complete market there exists a unique martingale measure, but for an incomplete market there is no unique martingale measure. The interpretation of this is that even though options are risky instruments we don’t have to specify our own degree of risk aversion in order to price them. Enough of complete markets, where can we find incomplete markets? The answer is ‘everywhere.’ In practice, all markets are incomplete because of real-world effects that violate the assumptions of the simple models. Take volatility as an example. As long as we have a lognormal equity random walk, no transaction costs, continuous hedging, perfectly divisible assets,..., and constant volatility then we have a complete market.

This is because there are now more states of the world than there are linearly independent securities. In reality, we don’t know what volatility will be in the future so markets are incomplete. We also get incomplete markets if the underlying follows a jump-diffusion process. Again more possible states than there are underlying securities. Another common reason for getting incompleteness is if the underlying or one of the variables governing the behaviour of the underlying is random. Options on terrorist acts cannot be hedged since terrorist acts aren’t traded (to my knowledge at least). We still have to price contracts even in incomplete markets, so what can we do? There are two main ideas here. One is to price the actuarial way, the other is to try to make all option prices consistent with each other. The actuarial way is to look at pricing in some average sense.

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Derivatives Markets by David Goldenberg

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Unfortunately, contrary to intuition, the no-arbitrage condition is not enough machinery to uniquely price derivative securities. That is, no-arbitrage is not a sufficient condition for uniquely pricing derivative securities. There are generally many linear, positive pricing mechanisms, all consistent with no-arbitrage, for non-replicable ﬁnancial claims! OPTION PRICING IN DISCRETE TIME, PART 1 451 Another way to say this is that in an incomplete market, there can be many equally valid (no-arbitrage) pricing mechanisms. We already have an inkling of this through our study of ROP. Rational option pricing, which is exclusively based on the foundation of the no-arbitrage principle, produces pricing relationships (such as European Put-Call Parity) and pricing bounds or ranges into which rational option prices must fall. ROP does not produce speciﬁc, unique prices.

Therefore, no risk premium for pricing the option is needed; a risk premium is built into the pricing of the underlying stock, St (see Chapter 17). There is no immediate and completely adequate empirical ﬁx for the constant assumption, except to throw out Black–Scholes’ assumption of a stationary log-normal diffusion, and search for a viable (smile-consistent) underlying stochastic process among the vast set of alternatives, many of which will lead to incomplete markets. Black–Scholes and its modiﬁcations, however, still have tremendous appeal, especially among traders, who use Black–Scholes calibrated to an implied volatility surface. Traders use ATM options to imply volatility, since these are the most liquid, and therefore most informative about future volatility. Furthermore, there are exotic and American options for which the lognormal GBM remains the workhorse.

Because the non-hedgeable risks are precisely those that cannot be diversiﬁed away (hedged out) by attempting to replicate the contingent claim, they would RISK-NEUTRAL VALUATION 601 command risk premia in the contingent claim price. These would, of necessity, show up in arbitrage-free pricing formulas for the contingent claim. This also renders such contingent claim prices not preference-free. We can summarize this as follows. In incomplete markets, non-replicable claims could be priced in a manner that is arbitrage-free (EMMs exist), and yet not preference-free. Furthermore, there would be multiple arbitrage-free, non-preference-free valuations of non-replicable claims. We know this is true, because FTAP2 tells us that a claim is replicable if and only if there is a unique EMM for the discounted price process. We will also see this in another, more practical manner below. 17.1.5 Black–Scholes’ Contribution A few more observations about Black–Scholes are in order.

Monte Carlo Simulation and Finance by Don L. McLeish

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SOME BASIC THEORY OF FINANCE While it is not too diﬃcult to solve this system in this case one can see that with more branches and more derivatives, this non-linear system of equations becomes diﬃcult very quickly. What do we do if we observe market prices for only two derivatives defined on this stock, and only two parameters can be obtained from the market information? This is an example of what is called an incomplete market, a market in which the risk neutral distribution is not uniquely specified by market information. In general when we have fewer equations than parameters in a model, there are really only two choices (a) Simplify the model so that the number of unknown parameters and the number of equations match. (b) Determine additional natural criteria or constraints that the parameters must satisfy. In this case, for example, one might prefer a model in which the probability of a step up or down depends on the time, but not on the current price of the stock.

SOME BASIC THEORY OF FINANCE Theorem 9 shows that this exponentially tilted distribution has the property of being the closest to the original measure P while satisfying the condition that the normalized sequence of stock prices forms a martingale. There is a considerable literature exploring the links between entropy and risk-neutral valuation of derivatives. See for example Gerber and Shiu (1994), Avellaneda et. al (1997), Gulko(1998), Samperi (1998). In a complete or incomplete market, risk-neutral valuation may be carried out using a martingale measure which maximizes entropy or minimizes cross-entropy subject to some natural constraints including the martingale constraint. For example it is easy to show that when interest rates r are constant, Q is the risk-neutral measure for pricing derivatives on a stock with stock price process St , t = 0, 1, ... if and only if it is the probability measure minimizing H(Q, P ) subject to the martingale constraint 1 St+1 ]. 1̄ + r St = EQ [ (2.23) There is a continuous time analogue of (2.22) as well which we can anticipate by inspecting the form of the solution.

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Mathematics for Economics and Finance by Michael Harrison, Patrick Waldron

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PRICING STATE-CONTINGENT CLAIMS Theorem 5.4.1 If there are M complex securities (M = N ) and the payoff matrix Y is non-singular, then markets are complete. Proof Suppose the optimal trade for consumer i state j is xij − eij . Then can invert Y to work out optimal trades in terms of complex securities. Q.E.D. An (N + 1)st security would be redundant. Either a singular square matrix or < N complex securities would lead to incomplete markets. So far, we have made no assumptions about the form of the utility function, written purely as u (x0 , x1 , x2 , . . . , xN ) , where x0 represents the quantity consumed at date 0 and xi (i > 0) represents the quantity consumed at date 1 if state i materialises. 5.4.1 Completion of markets using options Assume that there exists a state index portfolio, Y , yielding different non-zero payoffs in each state (i.e. a portfolio with a different payout in each state of nature, possibly one mimicking aggregate consumption).

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The Euro: How a Common Currency Threatens the Future of Europe by Joseph E. Stiglitz, Alex Hyde-White

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,” IMF Staff Discussion Note 11/08, April 8, 2011, available at https://www.imf.org/external/pubs/ft/sdn/2011/sdn1108.pdf; and Jonathan D. Ostry, Andrew Berg, and Charalambos G. Tsangarides, “Redistribution, Inequality, and Growth,” IMF Staff Discussion Note 14/02, February 2014, available at https://www.imf.org/external/pubs/ft/sdn/2014/sdn1402.pdf. 14 Bruce C. Greenwald and Joseph E. Stiglitz, “Externalities in Economies with Imperfect Information and Incomplete Markets,” Quarterly Journal of Economics 101, no. 2 (1986): 229–64. 15 I became particularly engaged in “the economics of crises” during my time at the World Bank and wrote extensively on the subject, both alone and with my colleagues at the World Bank. A popular account is provided in Globalization and Its Discontents. See also my articles “Lessons from the Global Financial Crisis,” in Global Financial Crises: Lessons from Recent Events, ed.

Neyman [Berkeley: University of California Press, 1951], pp. 507–32; and Gerard Debreu, “Valuation Equilibrium and Pareto Optimum,” Proceedings of the National Academy of Sciences 40, no. 7 [1954]: 588–92; and Debreu, The Theory of Value [New Haven, CT: Yale University Press, 1959.]) The circumstances that they identified where markets did not lead to efficiency were called market failures. Subsequently, Greenwald and Stiglitz showed that whenever information was imperfect and markets incomplete—essentially always—markets were not efficient (“Externalities in Economies with Imperfect Information and Incomplete Markets”). Of course, even earlier, Keynes had emphasized that markets do not by themselves maintain full employment. 34 See James Edward Meade, The Theory of International Economic Policy, vol. 2, Trade and Welfare (London: Oxford University Press, 1955); and Richard G. Lipsey and Kelvin Lancaster, “The General Theory of Second Best,” Review of Economic Studies 24, no. 1 (1956): 11–32. 35 See David Newbery and J.

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The Irrational Economist: Making Decisions in a Dangerous World by Erwann Michel-Kerjan, Paul Slovic

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K. Hsee (2000). “The Affection Effect in Insurance Decisions.” Journal of Risk and Uncertainty 20, no. 2: 149-159. Kunreuther, H., and M. V. Pauly (2000). NBER Reporter, March 22. Pagán, J. A., and M. V. Pauly (2006). “Community-Level Uninsurance and Unmet Medical Needs of Insured and Uninsured Adults.” Health Services Research 41, no. 3: 788-803. Schlesinger, H., and N. Doherty (1985). “Incomplete Markets for Insurance: An Overview.” Journal of Risk and Insurance 52: 402-423. 17 The Hold-Up Problem Why It Is Urgent to Rethink the Economics of Disaster Insurance Protection W. KIP VISCUSI As other contributors to this book have suggested, how people make decisions involving risk and uncertainty and how economists think people should make these decisions are often quite different matters.

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Tools for Computational Finance by Rüdiger Seydel

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Index Absolute error 93 Accuracy 20, 49, 107, 116, 161–165, 177 Adapted 50, 259 Admissible trading 262 Algorithm 10 – American options 154, 158–159 – Assembling 192 – Binomial method 55 – Box–Muller method 73 – Brennan-Schwartz 181 – Correlated random variables 76 – Crank–Nicolson method 138 – Distribution function 53 – Euler discretization of an SDE 33, 91 – Fibonacci generator 67 – Finite elements 197–198 – Implied volatility 54 – Interpolation 169 – Inversion of the standard normal distribution 281 – Lax-Wendroﬀ 232 – Linear congruential generator 62 – Marsaglias polar method 74 – Milstein integrator 99 – Monte Carlo simulation 105 – Projection SOR 156 – Quadratic approximation 171 – Radical–inverse function 83, 90 – Variance 53–54 – Wiener process 27 Analytic methods 124 Antithetic variates 86, 108–109, 111 Arbitrage 4–5, 9, 15, 23, 37, 52, 59, 124, 141, 143, 180, 212, 220, 241–243, 245, 260–262, 277 ARCH 52 Artiﬁcial viscosity, see Numerical dissipation Asian option, see Option Assembling 191–192, 196, 205 Autonomous 95–96 Average option, see Option, Asian Bachelier 25, 51 Backward diﬀerence 127, 133, 135, 152 Backward diﬀerence formula (BDF) 127, 154, 175, 181 Backward time centered space (BTCS) 134 Barrier option, see Option Basis function 185–193, 195, 201, 205–206, 221 Basis representation 185 Bernoulli experiment 45, 256 Bias 107, 256 Bifurcation 52 Bilinear form 199–204 Binary option, see Option Binomial coeﬃcient 257 Binomial distribution 56, 257 Binomial method 12–21, 49, 55–58, 103, 115, 123, 177, 211, 213, 236 Bisection 69, 268 Bivariate integral 173 Black–Merton–Scholes approach 8, 34, 49, 50–51, 103, 116, 138 Black–Scholes equation 9, 48, 49, 53, 59, 60, 116, 118, 123–126, 135, 138, 146–148, 169–170, 172, 174, 177, 179–181, 209, 211–213, 215, 220–224, 226, 231, 235, 237, 244–246, 248–249 Black–Scholes formula 10, 19, 41, 54, 55, 103, 105, 164, 166–169, 171, 177, 211, 213, 246–247 Bond 5, 37, 49, 59, 239–240, 242, 244–245, 267 Boundary conditions 9, 53, 123, 125, 129–130, 136, 138–145, 151, 153, 159, 161, 172, 174–175, 179–181, 186, 294 Index 188, 193–194, 198–201, 204, 211–213, 217–218, 235, 274 Bounds on options 4–5, 7–8, 59, 113–115, 121, 139–141, 167–168, 277–279 Box–Muller method 72–75, 85, 117 Bridge 102, 116, 118, 121, 236 Brownian motion 25–26, see Wiener process Bubnov 186 Business time 52 Calculus of variations 200 Calibration 38, 52, 54 Call, see Option Cancellation 54 Cauchy convergence 31, 275 Cauchy distribution 88 Céa 202 Centered time centered space (CTCS) 232 Central Limit Theorem 69, 78, 256 Chain rule 40, 71, 95 Chaos 52 Cholesky decomposition 75–76, 270 Chooser option, see Option Classical solution, see Strong solution Collateralized mortgage obligation (CMO) 78 Collocation 187 Commodities 239 Complementarity 124, 149–155, 175, 178, 194 Compound option, see Option Compound Poisson process 47 Conditional expectation 14, 115, 259 Conforming element 205 Congruential generator 62–68, 85, 87 Conservation law 231, 236 Contact point 141–142, 145, 147 Continuation function 115 Continuation region 142, 146–147, 169, 194, 249 Continuum 11–12, 128 Control variate 86, 109–110, 119 Convection 223–224, 226 Convection–diﬀusion problem 209, 222–226, 231 Convergence 20, 81, 155, 157, 165, 268, 270–272 Convergence in the mean 29–31, 257 Convergence order, see Order of error Correlated random variable 75–77, 89 Correlation 39–40, 63, 68, 108, 110, 212–213 Courant–Friedrichs–Lewy (CFL) condition 227, 232 Courant number 225–227 Covariance 75–76, 100, 108, 110, 212, 254–255 Cox 12, 38, 49 Crank–Nicolson method 135–140, 152, 154, 158, 160–161, 164, 174–175, 178, 196, 221, 235 Cryer 154–157, 272 Cubic spline 206 Curse of dimension 86, 212–213 DAX 55 Decomposition of a matrix 134, 138, 159, 268–270 Delta 25, 49–50, 116, 246–247 Density function 41–43, 57, 70–77, 88, 90, 102–103, 116, 248, 254–255 Derivatives 240–242 Diﬀerentiable (smooth) 27–28, 40–41, 50, 86, 96, 103, 126, 135, 143, 147, 154, 160–161, 164–165, 175–177, 184, 188, 194–195, 200, 265, 272 Diﬀusion 32–33, 50, 223–224, 226, 228, 233, 236 Dirac’s delta function 187 Discounting 36, 50, 56, 103–105, 138, 263 Discrepancy 61, 79–84, 86, 89, 117, 119 Discrete monitoring 220–221 Discretization 11–13, 27, 33, 35, 126–127, 150, 152, 184 Discretization error 12, 91, 106, 109, 117, 135, 161–162, 172 Dispersion 226, 230–231 Dissipation 231, see Numerical dissipation Distribution 61, 63, 69–71, 88, 90, 92, 103, 167, 213, 248 Distribution function 254, see also Distribution Dividend 5, 9, 14, 21, 118, 123–124, 139, 142–146, 151, 163, 167, 169, 174, 176, 178–179, 213, 224, 243, 246, 251, 277, 279 Dow Jones Industrial Average 1, 26 Drift 27, 32–33, 36–38, 106, 259–262 Index Dynamic programming Dynamical system 52 19, 115 Early exercise 5, 7, 19, 23, 49, 105, 111, 115, 123, 214 Early–exercise curve (Free boundary) Sf 7, 112–113, 124, 140–149, 151–152, 154, 159–160, 168–173, 175–177, 180–181, 195, 249–251 Eﬃcient market 25, 242 Eigenmode 225 Eigenvalue 131–132, 137, 156, 225, 236, 268, 272 Element matrix 191–192, 206 Elliptic 202 Equivalent diﬀerential equation 230 Error control 11–12, 55, 161–165 Error damping 131 Error function 53 Error projection 202 Error propagation 131 Estimate 78, 256 Euler diﬀerential equation 125 Euler discretization method 33, 91, 93–94, 98, 101, 106, 109, 118, 119, 174–175 Excess return 36 Exercise of an option 1–3, 5–6, 105, 115, 143, 167, 173, 180, 241 Exotic option, see Option Expectation 14, 39–41, 51, 56, 77–78, 89, 94–95, 100, 103, 105, 120, 212, 248, 254–257 Expiration 2, see also Maturity Explicit method 91, 128–134, 152, 174–175 Exponential distribution 46–47, 71 Extrapolation 20, 49, 165–166, 173, 182, 267 Faure sequence 83, 86 Feynman 118 Fibonacci generator 67–68, 85, 88, 90 Filtration 111, 253, 258–259 Financial engineering 240 Finite diﬀerences 11, Chapter 4, 183, 188, 197–198, 222–227, 230–234, 236 Finite elements 11, Chapter 5, 212, 236 Finite–volume method 236 Fixed–point equation 270 Foreign exchange 244 Forward 240–244 295 Forward diﬀerence 129, 133, 135 Forward time backward space (FTBS) 227, 237 Forward time centered space (FTCS) 225–227, 229, 231 Fourier mode 225, 230–231 Fractal interpolation 102 Free boundary problem 140–149, 152 Frequency 230 Front ﬁxing 146, 175, 177, 181 Function spaces 188, 199–200, 272–275 Future 240–242 Galerkin 183, 186, 188, 192, 206 GARCH 52 Gaussian elimination 268–270 Gaussian process 25, 51 Gauß–Seidel method 179, 271–272 Geometric Brownian motion (GBM) 9, 34, 36, 41–43, 45, 47, 49, 51, 102, 121, 124, 138, 212, 215, 244, 262 Gerschgorin 137, 268 Girsanov 260 Godunov 236 Greek 116, 246–247 Grid 11–17, 49, 58, 79, 126–128, 150, 152, 161, 165, 178, 183–184, 193, 195, 222, 227, 231, 235 Halton sequence 83–84, 116–117 Hat function 187–191, 196, 198, 201, 203–206 Harrison 260 Hedging 5, 22, 25, 49, 240–241, 244–248 Hermite 206, 270 High–contact condition 145, 148, 170, 173, 176, 180 High resolution 209, 231–235 Hilbert space 274–275 Histogram 35, 37, 42–44, 58 Hitting time 112, 121 Hlawka 81, 84, 86 Holder 1–4, 23, 142, 180 Horner scheme 265 Implicit method 133–134, 136, 160, 174–175, 178, 235 Implied volatility 36, 54 Importance sampling 119 Incomplete market 262 296 Index Independent random variable 26–27, 46–47, 73, 78, 89, 101, 118, 248, 255–256 Inequalities 123, 146–152, 175, 194, 196, 203, 243 Ingersoll 38 Initial conditions 125, 129, 131, 151, 153, 174 Inner product 186, 199, 274–275 Integrability 188, 200–201, 273 Integral representation 42, 103, 138, 177 Integration by parts 119, 149, 188, 193–194, 199, 273 Interest rate r 4–6, 9, 14–15, 23–24, 37–38, 49, 52, 116, 224, 240–241, 243 Interpolation 12, 88, 102, 167–169, 184, 190, 203–204, 221, 265–267 Intrinsic value 2, 3, 115 Inversion method 69–70, 72, 85, 88, 117, 280 Isometry 31, 120 Iteration 267, 270–271 Itô integral 31–32, 50, 91, 119 Itô Lemma, see Lemma of Itô Itô process 32, 40–41, 59, 91, 257, 260–261 Itô–Taylor expansion 96–97 Jacobi matrix 72–73, 101, 267 Jacobi method 271–272 Jump 45, 220–221, 247 Jump diﬀusion 47–48, 52, 247–248 Jump process 45–48, 51–52, 247–248 Kac 118 Koksma 81, 84, 86 Kuhn–Tucker theorem Kurtosis 51 156–157 Lack of smoothness 147, 154, 160, 164, 175, 177 Landau symbol XVII, 267 Lattice method, see Binomial method Law of large numbers 78, 256 Lax–Friedrichs scheme 227, 233–234, 237 Lax–Wendroﬀ scheme 231–234, 236–237 Leap frog 236 Least squares 115, 187 Lebesgue integral 32, 78, 273 Lehmer generator 85 Lemma of Céa 202–203 Lemma of Itô 40–42, 43–44, 51, 57, 97, 119, 212, 215, 244–245, 247, 257, 260, 262 Lemma of Lax–Milgram 202 Lévy process 52, 85 LIBOR 240 Limiter 234–235 Linear element, see Hat function Local discretization error 136 Lognormal 42, 48, 51, 57, 102–103, 213, 248 Long position 4, 22, 243–244 Lookback option, see Option Low discrepancy 82, see Discrepancy Malliavin 116 Market model 8–9, see Model of the Market Market price of risk 36, 262 Markov Chain Monte Carlo (MCMC) 86 Markov process 25, 46 Marsaglia method 73–76, 85, 88, 117 Martingale 24, 27, 31, 37, 50, 118, 259–262 Maruyama 33 Mass matrix 194, 205 Maturity (expiration) T 1, 3, 5–6, 16, 21, 52, 54, 111, 121, 125, 138, 159, 166, 169, 175, 239–245, 251, 267, 279 Mean reversion 37–39 Mean square error 107 Measurable 253, 259 Merton 48, 49 Mersenne twister 85 Method of lines 172 Milstein 98–99, 109, 121 Minimization 150, 156–157, 178, 187, 195, 200 Mode, see Fourier mode Model error 161–162 Model of the Market 8–10, 25, 161, 242, 262 Model problem – −u = f 192, 200–201, 203 – ut + aux = buxx 224–226, 228 – ut + aux = 0 226–227, 230–235 Modulo congruence 62 Molecule 129, 134–135 Moment 51, 57, 94, 100–101, 120, 254 Index Monotonicity of a numerical scheme 232–236 Monte Carlo method 11, 40, 61, 77–82, 85–86, 89, 102–118, 121, 211–213, 236, 237, 263 Multifactor model 39, 116, 210–213 Multigrid 178 Newton’s method 54, 69, 166, 171, 235, 267–268 Nicolson, see Crank Niederreiter sequence 83, 86 Nitsche 204 No–arbitrage principle 4, 242–243, see Arbitrage Nobel Prize 49, 51 Node 16, 128–129, 213, 227 Nonconstant coeﬃcients 44, 224, 246 Norm 202–204, 268, 270, 272–275 Normal distribution 10, 25–28, 35, 40–41, 46–47, 51–53, 56, 61, 69–76, 87–91, 101, 106, 108, 117, 120, 167, 170, 173, 246, 255–256, 279–280 Numerical dissipation 228, 233–236 Obstacle problem 148–151, 175, 194, 200 One–factor model 39 One–period model 21–25, 115 Option 1, 41, 103, 240–241 – American 2–8, 10, 19–21, 23, 55, 105, 111–115, 123, 139, 140–148, 151–165, 167, 173, 176–177, 179–182, 194–195, 210, 213, 214, 249–252, 277–279 – Asian 126, 209–211, 214–221, 224, 235–237 – average, see Asian option – barrier 210–211, 235–237 – Basket 211–212, 237 – Bermudan 115 – binary 210 – call 1, 3–5, 9–10, 17, 19, 52, 55, 59, 103, 123, 125, 138–139, 142–143, 145–147, 151–152, 159, 181, 210, 212, 216–217, 221–222, 228, 241, 246–247, 249, 251–252, 277–279 – chooser 210 – compound 173, 210 – European 2, 5, 8–9, 19–20, 42, 52, 54, 55, 103–106, 111, 113, 121, 123, 139–143, 146, 159, 163–164, 167–168, 297 173, 175, 210–211, 214, 216–217, 221–223, 228, 245–247, 263, 277–279 – exotic 11, 177, 209–221, 235–237 – in the money 104 – lookback 107, 210–211, 235–236 – out of the money 104 – path–dependent 16, 21, 107, 111, 210–211, 214, 235–236 – perpetual 145, 179, 249 – put 1, 3–8, 18–21, 42, 52, 57, 103, 105–106, 112, 121, 138–139, 141–144, 147, 151–152, 159, 163, 167, 173, 176–177, 179–180, 182, 210, 218, 223, 241, 246–247, 249–252, 277–279 – rainbow 211–212 – vanilla (standard) 1, 59, 209–210 Order of error 20, 79, 91, 93–94, 98, 127, 135–136, 154, 161–162, 174, 178, 183, 198, 202–204, 268 Orthogonality 186, 205 Oscillations 223–224, 226, 229–236 Parabolic PDE 126 Parallelization 118 Pareto 51 Partial diﬀerential equation (PDE) 9–11, 123–126, 213 Partial integro-diﬀerential equation (PIDE) 48, 248 Partition of a domain 185 Path (Trajectory) 25, 33, 35, 40, 45, 92, 104, 106, 113–115, 118–119, 121 Path–(in)dependent, see Option Payoﬀ 2–4, 7–9, 17, 19, 21, 23-24, 56–57, 59, 103–105, 110, 113, 115, 138, 140–143, 145–148, 160, 168–170, 173, 177, 210–212, 214–217, 221, 235, 241, 245, 249–251, 263 Péclet number 174, 223–227, 236 Penalty method 178 Period of random numbers 62–63, 67 Phase shift 230–231 Pliska 260 Poincaré 203 Poisson distribution 45, 257 Poisson process 45–48, 52, 259 Pole behavior 70, 266 Polygon 190, 201, 203–204, 265 Polynomial 184, 188, 201, 204–206, 226, 265–266, 280 Portfolio 22–25, 28, 49, 52, 59–60, 180, 212, 237, 239, 242, 244–247 Power method 272 298 Index Preconditioner 271–272 Premium 1–2, 4, 169, 177, 241, 243 Present value 267 Probability 14–15, 22–24, 36, 45–46, 57, 63, 69, 75, 79, 90, 103, 173, 253–257, 259–260 Proﬁt 4, 143, 242–243 Projection SOR 154–158, 160, 198, 272 Pseudo–random number 61 Put, see Option Put–call parity 5, 52, 139, 171, 179, 246, 279 Quadratic approximation 169–171 Quadrature 53, 82, 104, 266, 280 Quasi Monte Carlo 84 Quasi–random number 61, 82, 116–117 Radical–inverse function 83 Radon–Nikodym 260 Rainbow option, see Option Random number 27, 40, 47, 61–90, 106, 108, 114, 116–119 Random variable 25, 56, 100–101, 105, 108, 111, 120, 253, 259 RANDU 66 Rational approximation 70, 88, 266, 280 Rayleigh–Ritz principle 202, 205 Regression 168 Relaxation parameter 155, 176, 271–272 Replication portfolio 49–50, 245, 247, 263 Residual 185–186 Return 33, 35–36, 42–43, 51–52, 58, 212, 242 Riemann–(Stieltjes–) integral 29, 50 Risk 4, 5, 36, 43, 48, 52, 239–242, 246 Risk free, Risk neutral 5, 14, 18, 21–25, 36–37, 49–50, 57, 102–103, 115, 143, 167, 240, 242–244, 246 Ross 12, 38, 49 Rounding error 12, 54, 85, 117, 131, 133, 161–162, 236 Rubinstein 12, 49 Runge–Kutta method 99–100 Sample 61, 63, 69, 90, 253 Sample variance 63, 256 Sampling error 78–79, 107, 117 Samuelson 51 Scholes 49, see Black Schwarzian inequality 201, 203, 275 SDE, see Stochastic Diﬀerential Equation Secant method 54, 69, 268 Seed 62, 68, 93, 105–106 Self–ﬁnancing 50, 60, 242, 247, 260–261, 263 Separation 195, 216–217, 222 Semi–discretization 13, 115, 127, 180 Short position 4, 22, 52, 243–244 Short sale 243 Shuﬄing 67 Similarity reduction 235 Simple process 31 Simulation 33, 40, 47, 61, 93, 102–106, 111, 115, 121, 213 Singular matrix 193 Smooth, see Diﬀerentiable Sobol sequence 83, 86 Sobolev space 200–201, 274 Software 10, 280–282 SOR 155, 158–159, 179, 181, 198, 271–272 Sparse matrix 187, 201 Spectral method 178, 205 Spectral radius 131, 270 Spline 188, 201, 206, 265 Spot market 239 Spot price 6, 13, 239, 243–244 Spurious 209, 223–236 Square integrable 201, 273 Square mean 257 Stability 11–12, 118, 126, 130–135, 137, 140, 174, 174, 178, 224–232, 235–237 Staggered grid 231–232 Standard deviation 5, 58, 78, 106, 254 State–price process 260–262 Star discrepancy 81 Step length 33, 91, 93, 102, 106, 118, 128, 133, 160, 226 Stiﬀness 118, 191 Stiﬀness matrix 194, 205 Stochastic diﬀerential equation (SDE) 11, 32–44, 47–48, 51, 91–96, 104–107, 110–111, 115–116, 118–119, 121, 124, 215–216, 235, 244, 260–261 Stochastic integral 28–31, 39, 50 Stochastic process 6, 10, 25–32, 45, 50–52, 57, 91, 102, 213 Index Stochastic Taylor expansion 95–99 Stock 1, 33, 37, 41–42, 51, 58–59, 239, 241, 244–245 Stopping time 111–113, 142 Stopping region 142–143, 146–147, 159 Stratiﬁed sampling 85 Stratonovich integral 50 Strike price K 1–2, 5–6, 49, 52, 142–143, 159, 161, 167, 174, 207, 215, 241, 279 Strong convergence 94–95, 99–100, 118 Strong (classical) solution 92, 118, 195, 199–200 Subdomain 184–185, 187, 190–191 Subordinator 52 Support 70–72, 90, 188, 201, 206 Swap 240–242 Symmetry of put and call 252 Tail of a distribution 43, 51–52 Taylor expansion 41, 126–127, 136, 183, 230, 258 Terminal condition 9, 53, 169, 221, 224, 246 Test function, see Weighting function Total variation diminishing (TVD) 232–237 Trading strategy 28, 262–263 Trajectory, see Path Transaction costs 4, 9–10, 52 Transformations 42, 53, 69–74, 76, 88, 124–125, 128, 139, 146, 151, 154, 174, 181, 209, 216, 224, 237 Trapezoidal rule 175, 266 Trapezoidal sum (composite rule) 79, 248, 266 Traveling wave, see Wave Tree 12–13, 16–18, 21, 213 Tree method 49, 213, see Binomial method Trial function, see Basis function Tridiagonal matrix 130, 132, 134–135, 137, 181, 192, 198, 269–270 Trinomial model 21, 175, 213 Truncation error 299 161, 172 Underlying 1–2, 5, 58 Uniform distribution 61–74, 77–79, 88, 90, 255 Upwind scheme 209, 226–234, 237 Value at Risk 52, 116 Value function 9 Van der Corput sequence 82–83, 86 Van Leer 235 Variance 14–15, 40–43, 51, 53, 78–79, 87, 94–95, 100, 107–108, 110, 121, 254–257 Variance reduction 85, 108–111, 116, 119 Variation 29, 85, 215 Variational problem 149–151, 194, 200, 202 Vasicek 38 Vieta 17 Volatility 5–6, 9, 15, 17, 34, 36–40, 43, 52, 54, 58, 92, 102, 107, 116, 164, 212, 224, 226, 235, 246 Volatility smile 55, 174 Von Neumann stability 225, 227, 235–237 Wave 226, 230–231 Wave number 225–226, 231 Wavelet 206 Weak convergence 94–95, 100–101, 109, 118 Weak derivative 273–274 Weak solution 92, 120, 195, 199–202 Weighted residuals 183–187 Weighting function 186 White noise 32, 51 Wiener process (Brownian motion) Wt 25–34, 36, 38–44, 47, 50–52, 56, 57, 91–93, 96–102, 105, 119–121, 212, 215, 257–261 Writer 1–2, 4 Yield to maturity 267 Universitext Aguilar, M.; Gitler, S.; Prieto, C.: Algebraic Topology from a Homotopical Viewpoint Bonnans, J.

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The Entrepreneurial State: Debunking Public vs. Private Sector Myths by Mariana Mazzucato

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Imperfections can arise for various reasons: the unwillingness of private firms to invest in areas, like basic research, from which they cannot appropriate private profits because the results are a ‘public good’ accessible to all firms (results of basic R&D as a positive externality); the fact that private firms do not factor in the cost of their pollution in setting prices (pollution as a negative externality); or the fact that the risk of certain investments is too high for any one firm to bear them all alone (leading to incomplete markets). Given these different forms of market failure, examples of the expected role of the State would include publicly funded basic research, taxes levied on polluting firms and public funding for infrastructure projects. While this framework is useful, it cannot explain the ‘visionary’ strategic role that government has played in making these investments. Indeed, the discovery of the Internet or the emergence of the nanotechnology industry did not occur because the private sector wanted something but could not find the resources to invest in it.

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Grave New World: The End of Globalization, the Return of History by Stephen D. King

A BORDERLESS WORLD A third option would be to dispense with borders altogether. The nearest we have got to this is perhaps the European Union – or, more specifically, the 19 members that make up the Eurozone. Yet the single currency project is only half-finished – and arguably only half-baked. The Eurozone has some aspects of nationhood: a single currency, a single monetary policy, a single (although incomplete) market and, for those who also happen to be members of Schengen, a common external border. Yet it lacks other aspects: there is no common fiscal policy and no common border force; the European Parliament is a weak and distant institution; and a common European defence policy has so far proved be more a matter of words than deeds. Moreover, the strategic direction of the EU is determined by a European Council that is not much more than a talking shop for the various European heads of state or government – in other words, it is a bit like a White House occupied not only by a president, but also by assorted state governors, each of whom is entitled to promote his or her legitimate point of view.

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

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Remark 3.6.8 in Karatzas and Shreve (1998). 11.6 Duality Approach For any stopping time τ ∈ S[0,T ] , we denote by τ (x) the set of portfolio/ consumption-rate processes triplets (π, C) for which (π, C, τ ) ∈ A(x). For ﬁxed τ ∈ S, we consider the utility maximization problem Vτ (x) sup (π ,C )∈ τ (x) J (x; π, C, τ ). (11.25) 301 11.6 Duality Approach Not allowing our agent to invest in the stock market (π(τ ,T ] ≡ 0) after retirement, creates an incomplete market in which the problem is difﬁcult to solve explicitly. However, under the additional Assumption (11.1) that the interest rate is locked after retirement, we can solve the optimization problem after retirement explicitly by pathwise optimization given the information available at retirement time Fτ . Equation 11.12 with π(τ ,T ] ≡ 0 implies that we have the following constraint ∞ γ (t)c(t)dt ≤ γ (τ )(X x,π ,C − ξ ), a.s. (11.26) τ In conjunction with Equation 11.15 this constraint leads to τ H (τ ) T E γ (t)c(t)dt + H (τ )ξ + H (s)c(s)ds γ (τ ) τ 0 τ x,π ,C (τ ) − ξ ) + H (τ )ξ + H (s)c(s)ds ≤ x. ≤ E H (τ )(X 0 The problem can be solved as usual, with the introduction of a Lagrange multiplier λ > 0: τ T −βt −βτ −βt e U1 (c(t))dt + e U2 (ξ ) + e U3 (c(t))dt J (x; π, C, τ ) = E τ 0 τ ≤E + e−βt Ũ1 (λeβt H (t))dt + e−βτ Ũ2 (λeβτ H (τ )) 0 T −βt e τ τ +λ·E 0 τ ≤E + τ −βt e H (τ ) H (t)c(t)ds + H (τ )ξ + γ (τ ) τ T γ (t)c(t)dt e−βt Ũ1 (λeβt H (t))dt + e−βτ Ũ2 (λeβτ H (τ )) 0 T βt H (τ ) Ũ3 λe γ (t) dt γ (τ ) βt H (τ ) Ũ3 λe γ (t) dt γ (τ ) + λx with equality if and only if ⎧ βt ⎨I1 (λe 0 < t < τ, H (t)) H (τ ) (11.27) c(t) = ⎩I3 λeβt γ (t) τ < t ≤ T. γ (τ ) T γ (s) βτ x,π ,C βs H (τ ) −ξ = ξ = I2 (λe H (τ )) and X I3 λe γ (s) ds, γ (τ ) τ γ (τ ) (11.28) 302 CHAPTER 11 The ‘‘Retirement’’ Problem τ E 0 H (τ ) H (t)c(t)dt + H (τ )ξ + γ (τ ) T τ γ (t)c(t)dt = x

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

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For the past thirty or forty years, many of the brightest minds in economics have been busy examining how markets function when the unrealistic assumptions of the free market model don’t apply. For some reason, the economics of market failure has received a lot less attention than the economics of market success. Perhaps the word “failure” has such negative connotations that it offends the American psyche. For whatever reason, “market failure economics” never took off as a catchphrase. Some textbooks refer to the “economics of information,” or the “economics of incomplete markets.” Recently, the term “behavioral economics” has come into vogue. For myself, I prefer the phrase “reality-based economics,” which is the title of Part II. Reality-based economics is less unified than utopian economics: because the modern economy is labyrinthine and complicated, it encompasses many different theories, each applying to a particular market failure. These theories aren’t as general as the invisible hand, but they are more useful.

Making Globalization Work by Joseph E. Stiglitz

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I should be clear: while the intellectual groundings have been taken away from market fundamentalism, newspaper columnists and pundits—and occasionally, even a few economists—sometimes still invoke economic "science" in defense of their position. 3. This research was cited when I was awarded the Nobel Prize. 4. See Bruce Greenwald and Joseph E. Stiglitz, "Externalities in Economies with Imperfect Information and Incomplete Markets," Quarterly Journal of Economics, vol. 101, no. 2 (May 1986), pp. 229-64. 5. Joseph E. Stiglitz, The Roaring Nineties (New York: W W Norton, 2003). 6. An expression used by the philanthropist George Soros. 7. Matthew Miller, The Two Percent Solution: Fixing America's Problems in Ways Liberals and Conservatives Can Love (New York: PublicAffairs, 2003). This is how Miller phrases the issue in the prologue to his book: "We'll first step back and lay a little philosophical groundwork by examining the pervasive role of luck in 2 94 NOTES TO PAGES 1 1 –1 6 NOTES TO PAGES xvir-1 1 life, and how taking life's 'pre-birth lottery' seriously can bring the consensus we need to make progress." 8.

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Expected Returns: An Investor's Guide to Harvesting Market Rewards by Antti Ilmanen

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The model predicts a very low equity risk premium (well below 1%) due to the low observed volatility of consumption growth and low observed correlation between consumption growth and asset returns, unless an extremely high risk aversion coefficient is used. A huge academic literature has tried to reconcile this puzzle, using market frictions (borrowing constraints, limited market participation, incomplete markets, and idiosyncratic risk), non-standard utility functions (habit formation, recursive utility), modified consumption data (durable goods, luxury goods, long-term consumption risk), and biased sample explanations (survivorship bias among countries studied, absence of negative rare events in the sample, unexpected repricing of equities or bonds) as rational explanations for high observed equity outperformance—but there is little consensus to date.

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Never Let a Serious Crisis Go to Waste: How Neoliberalism Survived the Financial Meltdown by Philip Mirowski

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Neoliberalism has therefore expanded to become a comprehensive worldview, and has not been just a doctrine solely confined to economists.77 With regard to the crisis, one wing of neoliberals has appealed to natural science concepts of “complexity” to suggest that markets transcend the very possibility of management of systemic risk.78 However, the presumed relationship of the market to nature tends to be substantially different under neoliberalism than under standard neoclassical theory. In brief, neoclassical theory has a far more static conception of market ontology than do the neoliberals. In neoclassical economics, many theoretical accounts portray the market as somehow susceptible to “incompleteness” or “failure,” generally due to unexplained natural attributes of the commodities traded: these are retailed under the rubric of “externalities,” “incomplete markets,” or other “failures.” Neoliberals conventionally reject all such recourse to defects or glitches, in favor of a narrative where evolution and/or “spontaneous order” brings the market to ever more complex states of self-realization, which may escape the ken of mere humans.79 This explains why the NTC has rejected out of hand all neoclassical “market failure” explanations of the crisis. [4] A primary ambition of the neoliberal project is to redefine the shape and functions of the state, not to destroy it.

Adaptive Markets: Financial Evolution at the Speed of Thought by Andrew W. Lo

From the biological perspective, the limitations of Homo economicus are now obvious. Neuroscience and evolutionary biology confirm that rational expectations and the Efficient Markets Hypothesis capture only a portion of the full range of human behavior. That portion isn’t small or unimportant—it provides an excellent first approximation of many financial markets and circumstances, and should never be ignored—but it’s still incomplete. Market behavior, like all human behavior, is the outcome of eons of evolutionary forces. In fact, investors would be wise to adopt the Efficient Markets Hypothesis as the starting point of any business decision. Before launching a venture, asking why your particular idea should succeed, and why The Adaptive Markets Hypothesis • 177 someone else hasn’t already done it, is a valuable discipline that can save you a lot of time and money.