# volatility smile

19 results back to index

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

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

The function Of has the correct boundary value by construction and since multiplication by f (1;) commutes with differentiation in the other variables, (16.26) is satisfied by Of. 398 Stochastic volatility We can now price any option for which we know the fundamental transform. We simply numerically invert the Fourier transform at the appropriate value of T and obtain a price. 16.6 Stochastic volatility smiles Since the possibility of stochastic volatility getting large increases the probability of large movements in the underlying stock, stochastic-volatility models lead to fatter tails for the distribution of the final stock price. This leads to implied-volatility smiles which pick up out-of-the-money; that is, smile-shaped smiles! If we allow correlation between the underlying and the volatility then a skew is introduced. Roughly, if the volatility and the underlying are negatively correlated then as the stock price falls, it becomes more volatile and so out-of-the-money put options require more hedging and thus are more valuable which leads to increased implied volatilities.

Jump-diffusion smiles for time horizons of one through five years. The sharpest smile is one year, and the shallowest is five years. Spot is 100 and jumps are asymmetric with mean ratio equal to 0.8. 18.3.2 Stochastic-volatility smiles If we use a stochastic-volatility model with constant parameters the model is of log-type and again everything is defined relative to the current value of spot and time, and we obtain a functional dependence for the implied volatility of the form &(S, K, t, T) _ &(K/S, T - t). (18.4) 18.3 Dynamics implied by models 417 0.25 0.2 0.15 0.1 0.05 0 Fig. 18.3. Stochastic-volatility smiles (Heston model) for time horizons of one through five years. The sharpest smile is one year, and the shallowest is five years. Spot is 100 and volatility is uncorrelated with spot. The reversion speed is 1 and the volatility of variance is 1.

Once again we obtain an implied volatility function of the form &(S, K, t, T) = &(K/S, T - t). (18.5) Smile dynamics and the pricing of exotic options 418 0.13 0.125 0.120.1150.11 0.105 0.1 0.095 0.090.0850.08 Fig. 18.4. Stochastic-volatility smiles (Heston model) for time horizons of one through five years. The sharpest smile is one year, and the shallowest is five years. Spot is 100 and volatility is uncorrelated with spot. The reversion speed is 1 and the volatility of variance is 0.1. Initial volatility is 10%. 0.14 0.13 0.12 0.11 0.1 0.090.080.07 -I 0.06 'O 6y '10 Ab 00 0h 00 0h X00 X20 'p' bO Fig. 18.5. Stochastic-volatility smiles (Heston model) for time horizons of two through ten years. The highest smile is two years, and the bottom is ten years. Spot is 100 and volatility is uncorrelated with spot. The reversion speed is 0.1 and the volatility of variance is 1.

pages: 345 words: 86,394

Frequently Asked Questions in Quantitative Finance by Paul Wilmott

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Applied Mathematical Finance 2 117-133 Rubinstein, M 1994 Implied binomial trees. Journal of Finance 69 771-818 Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second edition. John Wiley & Sons What is the Volatility Smile? Short Answer Volatility smile is the phrase used to describe how the implied volatilities of options vary with their strikes. A smile means that out-of-the-money puts and out-of-the-money calls both have higher implied volatilities than at-the-money options. Other shapes are possible as well. A slope in the curve is called a skew. So a negative skew would be a download sloping graph of implied volatility versus strike. Example Figure 2-9: The volatility ‘smile’ for one-month SP500 options, February 2004. Long Answer Let us begin with how to calculate the implied volatilities. Start with the prices of traded vanilla options, usually the mid price between bid and offer, and all other parameters needed in the Black-Scholes formulæ, such as strikes, expirations, interest rates, dividends, except for volatilities.

This deviation from the flat-volatility Black-Scholes world tends to get more pronounced closer to expiration. A more general explanation for the volatility smile is that it incorporates the kurtosis seen in stock returns. Stock returns are not normal, stock prices are not lognormal. Both have fatter tails than you would expect from normally distributed returns. We know that, theoretically, the value of an option is the present value of the expected payoff under a risk-neutral random walk. If that risk-neutral probability density function has fat tails then you would expect option prices to be higher than Black-Scholes for very low and high strikes. Hence higher implied volatilities, and the smile. Another school of thought is that the volatility smile and skew exist because of supply and demand. Option prices come less from an analysis of probability of tail events than from simple agreement between a buyer and a seller.

pages: 408 words: 85,118

Python for Finance by Yuxing Yan

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

For example, instead of showing how to run CAPM to estimate the beta (market risk), I show you how to estimate IBM, Apple, or Walmart's betas. Rather than just presenting formulae that shows you how to estimate a portfolio's return and risk, the Python programs are given to download real-world data, form various portfolios, and then estimate their returns and risk including Value at Risk (VaR). When I was a doctoral student, I learned the basic concept of volatility smiles. However, until writing this book, I had a chance to download real-world data to draw IBM's volatility smile. [2] Preface What this book covers Chapter 1, Introduction and Installation of Python, offers a short introduction, and explains how to install Python and covers other related issues such as how to launch and quit Python. Chapter 2, Using Python as an Ordinary Calculator, presents some basic concepts and several frequently used Python built-in functions, such as basic assignment, precision, addition, subtraction, division, power function, and square root function.

The two output datasets with the Pandas' pickle format can be downloaded from http://canisius.edu/~yany/callsFeb2014.pickle and http://canisius.edu/~yany/putsFeb2014.pickle: from pandas.io.data import Options import datetime import pandas as pd def call_data(tickrr,exp_date): x = Options(ticker,'yahoo') data= x.get_call_data(expiry=exp_date) return data ticker='IBM' exp_date=datetime.date(2014,2,28) c=call_data(ticker,exp_date) print c.head() callsFeb2014=pd.DataFrame(c,columns=['Strike','Symbol','Chg','Bid','Ask', 'Vol','Open Int']) callsFeb2014.to_pickle('c:/temp/callsFeb2014.pickle') def put_data(tickrr,exp_date): x = Options(ticker,'yahoo') data= x.get_put_data(expiry=exp_date) return data [ 359 ] Volatility Measures and GARCH p=put_data(ticker,exp_date) putsFeb2014=pd.DataFrame(p,columns=['Strike','Symbol','Chg','Bid','Ask',' Vol','Open Int']) putsFeb2014.to_pickle('c:/temp/putsFeb2014.pickle') Volatility smile and skewness Obviously, each stock should possess just one volatility. However, when estimating implied volatility, different strike prices might offer us different implied volatilities. More specifically, the implied volatility based on out-of-the-money options, atthe-money options, and in-the-money options might be quite different. Volatility smile is the shape going down then up with the exercise prices, while the volatility skewness is downward or upward sloping. The key is that investors' sentiments and the supply and demand relationship have a fundamental impact on the volatility skewness.

Finance 300 The put-call ratio 300 The put-call ratio for a short period with a trend 302 Summary 303 Exercises 304 [ vii ] Table of Contents Chapter 11: Monte Carlo Simulation and Options Generating random numbers from a standard normal distribution Drawing random samples from a normal (Gaussian) distribution Generating random numbers with a seed Generating n random numbers from a normal distribution Histogram for a normal distribution Graphical presentation of a lognormal distribution Generating random numbers from a uniform distribution Using simulation to estimate the pi value Generating random numbers from a Poisson distribution Selecting m stocks randomly from n given stocks Bootstrapping with/without replacements Distribution of annual returns Simulation of stock price movements Graphical presentation of stock prices at options' maturity dates Finding an efficient portfolio and frontier Finding an efficient frontier based on two stocks Impact of different correlations 307 308 309 309 310 310 311 312 313 315 315 317 319 320 322 324 324 326 Constructing an efficient frontier with n stocks 329 Geometric versus arithmetic mean 332 Long-term return forecasting 333 Pricing a call using simulation 334 Exotic options 335 Using the Monte Carlo simulation to price average options 335 Pricing barrier options using the Monte Carlo simulation 337 Barrier in-and-out parity 339 Graphical presentation of an up-and-out and up-and-in parity 340 Pricing lookback options with floating strikes 342 Using the Sobol sequence to improve the efficiency 344 Summary 344 Exercises 345 Chapter 12: Volatility Measures and GARCH Conventional volatility measure – standard deviation Tests of normality Estimating fat tails Lower partial standard deviation Test of equivalency of volatility over two periods Test of heteroskedasticity, Breusch, and Pagan (1979) Retrieving option data from Yahoo! Finance Volatility smile and skewness Graphical presentation of volatility clustering [ viii ] 347 348 349 350 352 354 355 358 360 362 Table of Contents The ARCH model 363 Simulating an ARCH (1) process 364 The GARCH (Generalized ARCH) model 365 Simulating a GARCH process 366 Simulating a GARCH (p,q) process using modified garchSim() 367 GJR_GARCH by Glosten, Jagannanthan, and Runkle (1993) 369 Summary 373 Exercises 373 Index 375 [ ix ] Preface It is our firm belief that an ambitious student major in finance should learn at least one computer language.

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

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

For instance, there is some empirical evidences that the Black-Scholes formula gives a better estimate of at-the-money options (i.e., when K~S(t)), and a larger error for in-the-money and outof-the-money options (i.e., the historical volatility is closer to the implied volatility when K~S(t)). In addition, different models for evolution of random volatility are often tested by comparing the shape of volatility smiles resulting from simulation with the volatility smile obtained from real market data. 7.2 Calculation of implied volatility By Lemma 7.3, any Black-Scholes price V(σ) for call or put is a monotonic increasing function in σ. An example of its shape is given in Figure 7.1. The code that creates this graph is given below. MATLAB code for representation of the Black-Scholes price as a function of volatility function[v]=volprice(N,eps,s,K,T,r) v=zeros(1,N); op=v; for k=1:N, v(k)=eps*k; op(k)=call (x, K, v (k), T,r); end; plot(v(1:N), op(1:N),'b-'); Since V is monotonic, finding the root of the scalar equation numerical problem.

In particular, the price of the call option with the strike price K and expiration time T is HBS,c(S(0), K, T, σ1, r)+(1−p)HBS,c(S(0), K, T, σ2, r). (7.6) Figure 7.2 represents the implied volatility calculated for this market using the code given below. It can be seen that even this very simple volatility model generates a volatility smile. Figure 7.2 Implied volatility, the market with random volatility from Example 5.63. It is assumed that (N, eps, s, T, r)=(24, 0.125, 2, 1, 0.1). © 2007 Nikolai Dokuchaev Mathematical Finance 138 MATLAB code for modelling of volatility smile (Example 5.63) function[v]=volsmile(N,eps,s,T,r) p=0.5; v1=0.3; v2=0.7; N1=100; eps1=1/100; v=zeros(1,N); op=v; K=v; K0=max(0.1, sN*eps/2); for k=1:N K(k)=K0+eps*k; op(k)=p*call(x,K(k),v1,T,r)+(1p)*call(x,K(k),v2,T,r); v(k)=impliedvol(N1,eps1,op(k),s,K(k),T,r); end; plot(K(1:N),v(1:N),'ro'); 7.4 Problems for two different stocks at time t=0, i=1, Problem 7.7 Assume that we observe prices 2.

In this case, we can conclude that the Black-Scholes model does not describe the real market perfectly, and its imperfections can be characterized by the gap between the historical and implied volatilities. Varying K and T gives different patterns for implied volatility. Similarly, the evolving price S(t) gives different patterns for implied volatility for different t for a given K. The most famous pattern is the so-called volatility smile (or volatility skew) that describes dependence of σimp on K. Very often these patterns have the shape of a smile (or sometimes skew). These shapes are carefully studied in finance, and very often they are used by decision-makers. They are considered to be important market indicators, and there are some empirical rules about how to use them in option pricing. For instance, there is some empirical evidences that the Black-Scholes formula gives a better estimate of at-the-money options (i.e., when K~S(t)), and a larger error for in-the-money and outof-the-money options (i.e., the historical volatility is closer to the implied volatility when K~S(t)).

pages: 338 words: 106,936

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

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Whereas in the years leading up to the crash the Black-Scholes model seemed to get options prices exactly right, in virtually all contexts and all markets, after the crash certain discrepancies began to appear. These discrepancies are often called the volatility smile because of their distinctive shape in certain graphs. The smile appeared suddenly and presented a major mystery for financial engineers in the early 1990s, when its prevalence was first recognized. Notably, Emanuel Derman came up with a way of modifying the Black-Scholes model to account for the volatility smile, though he never came up with a principled reason why the Black-Scholes model had stopped working. Mandelbrot’s work, however, offers a compelling explanation for the volatility smile. One way of interpreting the smile is as an indication that the market believes large shifts in prices are more likely than the Black-Scholes model assumes.

During 1977 and 1978, Greenbaum, Struve, and a small team of proto-quants worked out a modified Black-Scholes model that took into account things like sudden jumps in prices, which can lead to fat tails. O’Connor was famously successful, first in options and then in other derivatives — in part because the modified Black-Scholes model tended to outperform the standard one. Remarkably, according to Struve, O’Connor was aware of the volatility smile from very early on. That is, even before the crash of 1987, there were small, potentially exploitable discrepancies between the Black-Scholes model and market prices. Later, when the 1987 crash did occur, O’Connor survived. There’s another, deeper concern about the market revolution initiated by Black and his followers that many people worried about in 1987 and that has become quite stark in the wake of the most recent crisis.

As Osborne in particular emphasized, the models that resulted were only as good as the assumptions that went in. Sometimes assumptions that are usually excellent quickly become lousy as market conditions change. For this reason, the O’Connor story has an important moral. Many histories suggest that the 1987 crash rocked the financial world because it was so entirely unexpected — impossible to anticipate, in fact, given the prevailing market models. The sudden appearance of the volatility smile is taken as evidence that models can work for a while and then suddenly stop working, which in turn is supposed to undermine the reliability of the whole market-modeling enterprise. If models that work today can break tomorrow, with no warning and no explanation, why should anyone ever trust physicists on Wall Street? But this just isn’t right. By carefully thinking through the simplest model and complicating it as appropriate — in essence, by accounting for fat tails — O’Connor was able to anticipate the conditions under which Black-Scholes would break down, and to adopt a strategy that allowed the firm to weather an event like the 1987 crash.

pages: 447 words: 104,258

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

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Moreover, statistics explain that variances are additive.3 So to compute the forward volatility σ1, 2 from period t1 to period t2, we need the (spot) volatilities σ1 on t1 and σ2 on t2: Example. From the data used for the previous volatility curve (S&P 500, ATM options), by using the above formula, we can compute the 3-month forward implied volatility after 6 months: the data are: spot 6-month volatility @ 18.56% spot 9-month volatility @ 18.86% giving (with 6 months and 9 months = 0.5 and 0.75 year respectively): 12.1.3 The volatility smile Stochastic models for underlyings are essentially based on the hypothesis of normal distribution of the log returns. This has proven a robust approach, and makes easier the calculations. A contrario, developing models based on a more general distribution presents: the difficulty of selecting what would be the adequate alternative distribution: this point is developed in Chapter 15, Section 15.1; the disadvantage that the kth moments are less and less stationary over time with k increasing (cf.

Chapter 10, Section 10.2.4), that is, practically speaking, in function of the option delta.4 The largest returns correspond indeed to a |Δ| getting closer to 0 or to 100%, such as that in Figure 12.8. As a consequence, quoting an option may involve the determination of an implied volatility, which not only depends on the nature of the underlying and of the maturity of the option, but also of the spread between the strike price and the underlying spot price at that moment. The shape of the relationship such as on the graph explains why this feature has been called the “volatility smile”. If the actual distribution presents no fat tails, but some asymmetry, first, the market may well quote different implied volatility levels for calls and for puts, implying thus a kind of market consensus for a directional trend in the underlying evolution; second, the market can quote different implied volatilities for DOTM calls and DITM puts on the one hand, that is corresponding to lower underlying spot prices, than for DITM calls and DOTM puts on the other hand, corresponding to higher underlying spot prices.

Instead of starting from the process of the underlying spot as in Eq. 12.2, we start from a series of Ft forward underlying prices or rates, and the SABR model consists of the following system For a given instrument, the parameters α, β and ρ need to be calibrated on the corresponding volatility curve, that is, they must fit the market data, including observed options implied volatilities. In particular the SABR model allows for taking into account observed volatility smiles.7 As an alternative to the Heston and SABR models, let us also mention the one8 consisting – instead of starting from Eq. 12.3 – in considering the following relationship: that creates a third stochastic process Z3 that is independent (uncorrelated) with Z1. Provided some hypothesis can be reasonably made about ρ1, 2, presumably as a function of σt, the model allows for a Monte Carlo simulation.

pages: 313 words: 34,042

Tools for Computational Finance by Rüdiger Seydel

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

With Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1968). [Ad75] R.A. Adams: Sobolev Spaces. Academic Press, New York (1975). [AiC97] F. AitSahlia, P. Carr: American options: A comparison of numerical methods. In [RT97] (1997) p. 67-87. [AnA00] L. Andersen, J. Andreasen: Jump diﬀusion process: Volatility smile ﬁtting and numerical methods for option pricing. Review Derivatives Research 4 (2000) 231-262. [AB97] L.B.G. Andersen, R. Brotherton-Ratcliﬀe: The equity option volatility smile: an implicit ﬁnite-diﬀerence approach. J. Computational Finance 1,2 (1997/1998) 5–38. [AnéG00] T. Ané, H. Geman: Order ﬂow, transaction clock, and normality of asset returns. J. of Finance 55 (2000) 2259-2284. [Ar74] L. Arnold: Stochastic Diﬀerential Equations (Theory and Applications). Wiley, New York (1974).

The derivative f (xk ) can be approximated by the diﬀerence quotient f (xk ) − f (xk−1 ) . xk − xk−1 c) For the resulting secant iteration invent a stopping criterion that requires smallness of both |f (xk )| and |xk − xk−1 |. Calculate the implied volatilities for the data T − t = 0.211 , S0 = 5290.36 , r = 0.0328 Exercises 55 and the pairs K, V from Table 1.3 (for more data see www.compfin.de). Enter for each calculated value of σ the point (K, σ) into a ﬁgure, joining the points with straight lines. (You will notice a convex shape of the curve. This shape has lead to call this phenomenon volatility smile.) Table 1.3. Calls on the DAX on 4. Jan 1999 K V 6000 80.2 6200 47.1 6300 35.9 6350 31.3 6400 27.7 6600 16.6 6800 11.4 Exercise 1.6 Price Evolution for the Binomial Method For β from (1.11) and u = β + β 2 − 1 show " ! √ " ! u = exp σ ∆t + O (∆t)3 . Exercise 1.7 Implementing the Binomial Method Design and implement an algorithm for calculating the value V (M ) of a European or American option.

Summarizing the Black-Scholes equation to ∂V + LBS = 0 ∂t (4.60) where LBS represents the other terms of the equation, see Section 4.5.3, motivates an interpretation of the ﬁnite-diﬀerence schemes in the light of numerical ODEs. There the forward approach is known as explicit Euler method and the backward approach as implicit Euler method. on Section 4.3: Crank and Nicolson suggested their approach in 1947. Theorem 4.4 discusses three main principles of numerical analysis, namely order (of convergence), stability, and eﬃciency. A Crank-Nicolson variant has been developed that is consistent with the volatility smile, which reﬂects the dependence of the volatility on the strike [AB97]. Notes and Comments 175 In view of the representation (4.12) the Crank-Nicolson approach corresponds to the ODE trapezoidal rule. Following these lines suggests to apply other ODE approaches, some of which lead to methods that relate more than two time levels. In particular, backward diﬀerence formula (BDF) are of interest, which evaluate L at only one time level.

pages: 313 words: 101,403

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

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Modeling mortgages Salomon's skill at quantitative marketing Mercifully laid off ■ CHAPTER 13 CIVILIZATION AND ITS DISCONTENTS 203 Goldman as home Heading the Quantitative Strategies Group Equity derivatives The Nikkei puts and exotic options Nothing beats working closely with traders Financial engineering becomes a real field CHAPTER 14 LAUGHTER IN THE DARK 225 The puzzle of the volatility smile Beyond Black-Scholes: the race to develop local-volatility models of options The right model is hard to find CHAPTER 15 THE SNOWS OF YESTERYEAR 251 Will Street consolidates Clothing goes casual Moving from equity derivatives to firmwide risk The bursting of the Internet bubble Taking my leave CHAPTER 16 THE GREAT PRETENDER 265 Full circle, back to Columbia Physics and finance redux ■ Different endeavors require different degrees of precision ■ Financial models as gedanken experiments Acknowledgments 271 About the Author 272 Index 273 MODELING THE WORLD If mathematics is the Queen of Sciences, as the great mathematician Karl Friedrich Gauss christened it in the nineteenth century, then physics is king.

Like stableboys who shut the barn door after the horse has bolted, investors who lived through the 1987 crash were now willing to pay up for future insurance against the risks they had previously suffered. By 1990 there were similar smiles or skews in all equity markets. Before 1987, in contrast, more light-heartedly naive options markets were happy to charge about the same implied volatility for all strikes, as illustrated by the dashed line in Figure 14.1. Figure 14.1 A typical implied volatility smile for three-month options on the Nikkei index in late 1994. The dashed line shows the lack of skew that was common prior to the 1987 crash. It was not only three-month implied volatilities that were skewed. A similar effect was visible for options of any expiration, so that implied volatility varied not only with strike but also with expiration. We began to plot this double variation of implied volatility in both the time and strike dimension as a two-dimensional implied volatility surface.

Although we did live to see local volatility become a household word and a textbook topic, I discovered that it was much more difficult than I had imagined to create a truly successful financial model. Local volatility was an improvement on Black-Scholes in that it could account for the smile, but it had three genuine failings. First, our new model excluded the possibility that an index or stock could jump, and most market participants nowadays regard that possibility as the main factor determining the shape of the very short-term volatility smile. Our very first attempt to model the smile had indeed involved such jumps. We were never fond of jump models-since jumps are too violent and discontinuous to be hedged, when you include them you lose much of the coherence of the Black-Scholes model. But jumps are real, and omitting then made our model less realistic. Second, implied trees were difficult to calibrate. Often, as you tried to build progressively finer-meshed trees for better computational accuracy, the local volatility surface grew wild, displaying unrealistic peaks and troughs as it varied from point to point.

pages: 374 words: 114,600

The Quants by Scott Patterson

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Traders soon came up with a name for this phenomenon: the “volatility smile.” It was the grim memory of Black Monday grinning sinisterly from within the very prices that underpinned the market. The volatility smile disobeyed the orderly world of “no arbitrage” laid out by Black-Scholes and modern portfolio theory, since it implied that traders could make a lot of money by selling these out-of-the-money puts. If the puts were too expensive for the risk they carried (according to the formula), the smart move would be to sell them hand over fist. Eventually that would drive the price down to where it should be. But, oddly, traders weren’t doing that. They were presumably frightened that another crash like Black Monday could wipe them out. They never got over the fear. The volatility smile persists to this day. The volatility smile perplexed Wall Street’s quants.

For Mom and Pop Contents The Players 1 ALL IN 2 THE GODFATHER: ED THORP 3 BEAT THE MARKET 4 THE VOLATILITY SMILE 5 FOUR OF A KIND 6 THE WOLF 7 THE MONEY GRID 8 LIVING THE DREAM 9 “I KEEP MY FINGERS CROSSED FOR THE FUTURE” 10 THE AUGUST FACTOR 11 THE DOOMSDAY CLOCK 12 A FLAW 13 THE DEVIL’S WORK 14 DARK POOLS Notes Glossary Acknowledgments The Players Peter Muller, outspokenly eccentric manager of Morgan Stanley’s secretive hedge fund PDT. A whip-smart mathematician who occasionally took to New York’s subways to play his keyboard for commuters, in 2007 Muller had just returned to his hedge fund after a long sabbatical, with grand plans of expanding operations and juicing returns even further. Ken Griffin, tough-as-nails manager of Chicago hedge fund Citadel Investment Group, one of the largest and most successful funds in the world.

Gerry Bamberger discovered stat arb: The section on the discovery of statistical arbitrage is based almost entirely on interviews with Gerry Bamberger, Nunzio Tartaglia, and several other members of the original Morgan Stanley group that discovered and spread stat arb across Wall Street. Previous mention of this group can be found in Demon of Our Own Design, by Richard Bookstaber (John Wiley & Sons, 2007). Morgan had hired Shaw: The account of Shaw’s departure from Morgan are based on interviews with Nunzio Tartaglia and others who were at APT. 4 THE VOLATILITY SMILE Sometime around midnight: Many details of Black Monday were found in numerous Wall Street Journal articles written during the time, including “The Crash of ’87—Before the Fall: Speculative Fever Ran High in the 10 Months Prior to Black Monday,” by James B. Stewart and Daniel Hertzberg, December 11, 1987. Other details, including the description at the opening of the chapter, were found in An Engine, Not a Camera: How Financial Models Shape Markets, by Donald MacKenzie (MIT Press, 2006), and The Age of Turbulence: Adventures in a New World, by Alan Greenspan (Penguin 2007), 105.

Analysis of Financial Time Series by Ruey S. Tsay

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

(6.25) 6.9 JUMP DIFFUSION MODELS Empirical studies have found that the stochastic diffusion model based on Brownian motion fails to explain some characteristics of asset returns and the prices of their derivatives (e.g., the “volatility smile” of implied volatilities; see Bakshi, Cao, and Chen, 1997, and the references therein). Volatility smile is referred to as the convex function between the implied volatility and strike price of an option. Both out-ofthe-money and in-the-money options tend to have higher implied volatilities than at-the-money options especially in the foreign exchange markets. Volatility smile is less pronounced for equity options. The inadequacy of the standard stochastic diffusion model has led to the developments of alternative continuous-time models. For example, jump diffusion and stochastic volatility models have been proposed in the literature to overcome the inadequacy; see Merton (1976) and Duffie (1995).

The parameter λ governs the occurrence of the special event and is referred to as the rate or intensity of the process. A formal definition also requires that X t be a right-continuous homogeneous Markov process with left-hand limit. In this section, we discuss a simple jump diffusion model proposed by Kou (2000). This simple model enjoys several nice properties. The returns implied by the model are leptokurtic and asymmetric with respect to zero. In addition, the model can reproduce volatility smile and provide analytical formulas for the prices of many options. The model consists of two parts, with the first part being continuous and following a geometric Brownian motion and the second part being a jump process. The occurrences of jump are governed by a Poisson process, and the jump size follows a double exponential distribution. Let Pt be the price of an asset at time t. The simple jump diffusion model postulates that the price follows the stochastic differential equation nt d Pt = µdt + σ dwt + d (Ji − 1) , Pt i=1 (6.26) 245 JUMP DIFFUSION MODELS where wt is a Wiener process, n t is a Poisson process with rate λ, and {Ji } is a sequence of independent and identically distributed nonnegative random variables such that X = ln(J ) has a double exponential distribution with probability density function f X (x) = 1 −|x−κ|/η , e 2η 0 < η < 1

., and Tauchen, G. (1997), “The relative efficiency of method of moments estimators,” Working paper, Economics Department, University of North Carolina. Hull, J. C. (1997), Options, Futures, and Other Derivatives, 3rd ed. Prentice-Hall: Upper Saddle River, New Jersey. Kessler, M. (1997), “Estimation of an ergodic diffusion from discrete observations,” Scandinavian Journal of Statistics, 24, 1–19. Kou, S. (2000), “A jump diffusion model for option pricing with three properties: Leptokurtic feature, volatility smile, and analytic tractability,” working paper, Columbia University. Lo, A. W. (1988), “Maximum likelihood estimation of generalized Ito’s processes with discretely sampled data,” Econometric Theory, 4, 231–247. Merton, R. C. (1976), “Option pricing when the underlying stock returns are discontinuous,” Journal of Financial Economics, 5, 125–144. Analysis of Financial Time Series. Ruey S. Tsay Copyright  2002 John Wiley & Sons, Inc.

pages: 443 words: 51,804

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

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Chapter Eight Parameter Estimation and Calibration for Long-Memory Stochastic Volatility Models A L E X A N D R A C H RO N O P O U LO U INRIA, Nancy, France 8.1 Introduction It has been observed that there exists a discrepancy between European option prices calculated under the Black and Scholes (1973) model with constant volatility and the market-traded option prices. In general, the volatility exhibits an intermittent behavior with periods of high values and periods of low values. As a result, implied volatilities at market prices are not constant across a range of options, but vary with respect to strike prices and create the so-called volatility smile (or smirk). Stochastic volatility models were introduced in order to model this observed random behavior of the market volatility. Under such a model, the volatility is described by a stochastic process. Among the ﬁrst models in the literature were these by Taylor (1986) and Hull and White (1987), under which the volatility dynamics are described by an Ornstein–Uhlenbeck process. Classical references for speciﬁc stochastic volatility models are Ball and Roma Handbook of Modeling High-Frequency Data in Finance, First Edition.

Viens, Maria C. Mariani, and Ionuţ Florescu. © 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc. 219 220 CHAPTER 8 Estimation and calibration for LMSV (1994), Heston (1993), and Scott (1987) whereas a presentation of properties, option pricing techniques, and statistical inference methods can be found in the book by Fouque et al. (2000a). It is widely believed that volatility smiles can be explained to a great extent by stochastic volatility models. However, it has been well documented that volatility is highly persistent, which means that even for options with long maturity, there exist pronounced smile effects. Furthermore, a unit root behavior of the conditional variance process is observed, particularly when we work with high frequency data. To better describe this behavior, Comte and Renault (1998) introduced a stochastic volatility model with long memory.

A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 1993;6(2):327–343. 14. Avellaneda M, Zhu Y. Risk neutral stochastic volatility model. Int J Theor Appl Finance 1998;1(2):289–310. 15. Berestycki H, Busca J, Florent I. Computing the implied volatility in stochastic volatility models. Commu Pure and Appl Math 2004;57(10):1352 –1373. 16. Andersen L, Andreasen J. Jump-diffusion processes: volatility smile ﬁtting and numerical methods for option pricing. Rev Deriv Res 2000;4:231–262. 17. Cont R, Tankov P. Financial modelling with jumps processes. CRC Financial mathematics series. Chapman & Hall, Boca Raton, Florida; 2003. References 381 18. Florescu I. Stochastic volatility stock price: approximation and valuation using a recombining tree. sharp estimation of the almost sure lyapunov exponent estimation for the anderson model in continuous space.

pages: 320 words: 33,385

Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

If further data on 10-delta strangles and risk reversals are available, two more points can be added to the implied volatility smile: 10 = 50 + ST10 + 21 RR10 90 = 50 + ST10 − 21 RR10 (I.5.13) A more precise interpolation and extrapolation method is then to fit a quartic polynomial to the ATM, 25-delta and 10-delta data: this is left as an exercise to the reader. I.5.3.3 Cubic Splines: Application to Yield Curves Spline interpolation is a special type of piecewise polynomial interpolation that is usually more accurate than ordinary polynomial interpolation, even when the spline polynomials have quite low degree. In this section we consider cubic splines, since these are the lowest degree splines with attractive properties and are in use by many financial institutions, for instance for yield curve fitting and for volatility smile surface interpolation. We aim to interpolate a function fx using a cubic spline.

Example I.5.5: Fitting a 25-delta currency option smile Fit a smile to the following quotes and hence interpolate the 10-delta and 90-delta volatilities: ATM forward straddle: 25-delta straddle: 25-delta risk reversal: 18% 2% 1% Solution By (I.5.12), 25 = 205% and 75 = 195%. We therefore require a b and c such that the quadratic fx = ax2 + bx + c passes through the points f025 = 0205 f050 = 018 f075 = 0195 This leads to the equations a + 4b + 16c = 16 × 0205 a + 2b + 4c = 4 × 018 9a + 12b + 16c = 16 × 195 In the spreadsheet for this example we find the solution a = −0320 b = −0340 c = 0270 Figure I.5.8 plots the fitted implied volatility smile and from this we can read off the interpolated values of the 10-delta and the 90-delta volatilities as 23.9% and 22.3%, respectively. In general, fitting the quadratic fx = ax2 + bx + c through three points, f025 = 25 f050 = 50 f075 = 75 gives the following solution for the coefficients: ⎛ ⎞ ⎛ ⎞−1 ⎛ ⎞ a 1 4 16 1625 ⎝b⎠ = ⎝1 2 4 ⎠ ⎝ 450 ⎠ c 9 12 16 1675 Numerical Methods in Finance 197 28.0% 27.0% 26.0% 25.0% 24.0% 23.0% 22.0% 21.0% 20.0% 19.0% 18.0% 17.0% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure I.5.8 Fitting a currency smile But when only three implied volatilities are available a quadratic interpolation tends to underestimate the market quotes at more extreme delta values.

Note that we tend to use the terminology estimation when we are finding the parameters of a discrete time model and calibration when we are finding the parameters of a continuous time model. For instance, by finding the best fit to historical time series data we ‘estimate’ the parameters of a GARCH model. On the other hand, we ‘calibrate’ the parameters of the Heston (1993) option pricing model by finding the best fit to a current ‘snapshot’ of market data, such as an implied volatility smile. Let y = y1 yn denote the data that we want to fit;10 denote the parameters that we want to estimate/calibrate to the data; and f x = f1 fn denote the function that we want to fit to y, which depends on the parameters and possibly other data x.11 The calibration (or estimation) problem is to find the parameters so that each fi is as close as possible to yi , for i = 1 2 n.

pages: 349 words: 134,041

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

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

The standard option pricing model did not actually tally with market prices of options. Options where the strike was a long way away from the DAS_C07.QXP 8/7/06 198 4:45 PM Page 198 Tr a d e r s , G u n s & M o n e y current price (deep in-the-money or deep out-of-the-money options) seemed to be priced using a higher volatility. Researchers built complex models incorporating the volatility ‘surface’ with an embedded volatility ‘smile’ to price these options. You will remember how NatWest misplaced their ‘smile’, losing tens of millions of pounds in the process. The volume of option pricing research from academics and quants in investment banks reached pandemic proportions. It was the age of proceduralism. Ever-increasing computer power and the demand for fast solutions for complex products drove a shift to mechanical application of procedure.

A variation involved the investor receiving two times the initial investment and the average performance of the two or more of the worst performing stocks within the basket. Another structure was six years. The investor’s initial investment was guaranteed. The investor got a return based on the average performance of a basket of six stocks. At specific intervals, the best performing stock in the basket would be removed from the basket for future periods. Dealers chuckled about selling volatility, volatility smiles, trading correlation within the basket and taking on forward volatility and correlation risks. Platoons of quants crunched numbers to price and hedge the structures. Did the investor understand what they were buying? They voted with their money. They bought the stuff in large licks. Large fund managers steered clear in the main. They talked about modelling problems, lack of transparent pricing inputs and lack of liquidity.

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

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

I later used this idea in a paper coauthored with Michael Ong called, ironically, “Digitals Defused,” which appeared in the December 1995 issue of Risk magazine. 17. It was actually published twice. First in Goldman Sachs Quantitative Strategies Research Notes, January 1994, “The Volatility Smile and Its Implied Tree,” Derman and Kani, and then later in a paper called “Riding on a Smile,” Risk, 7, no. 2 (1994), pp. 32–39. This paper and many of Derman’s other papers are available from his website, www.ederman.com. JWPR007-Lindsey 348 May 18, 2007 11:41 note s 18. This was also published twice. First in Goldman Sachs Quantitative Strategies Research Notes, February 1996, “Implied Trinomial Trees of the Volatility Smile,” Derman, Kani, and Chriss, and then in a paper by the same name published in the Summer 1996 in the Journal of Derivatives. 19. I published this as “Transatlantic Trees,” Risk 9, no. 7 (1996). 20.

pages: 819 words: 181,185

Derivatives Markets by David Goldenberg

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

Unfortunately, whether we use the historical volatility estimator or the implied volatility estimator, we are still stuck with the constant assumption. If is constant, then it is also constant across options with different exercise prices and IV should not depend upon which exercise price K is used to estimate it. This turns out not to be empirically true, at least since the market crash of 1987, and it generates a volatility smile, and its variations. A vast literature has developed around explaining volatility smiles and its variations. We can’t cover that here, but we can look at the economic reasons for expecting not to be a constant. 16.8 NON-CONSTANT VOLATILITY MODELS 16.8.1 Empirical Features of Volatility Some of the general empirical features that imply some degree of volatility predictability are clustering (persistence),which means that periods of high (low) volatility seem to persist, and reversion to the mean, which means that too high (too low) volatility seems to correct itself by moving back to the normal, longrun average level.

A Primer for the Mathematics of Financial Engineering by Dan Stefanica

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

At any point in time, several options with different strikes and maturities may be traded. If the lognormal assumption were true, then the implied volatilities corresponding to all these options should be equal. However, this does not happen. Usually, the implied volatility of either deep out of the money or deep in the money options is higher than the implied volatility of at the money options. This phenomenon is called the volatility smile. Another possible pattern for implied volatility is the volatility skew, when, e.g., the implied volatility of deep in the money options is smaller than the implied volatility of at the money options, which in turn is smaller than the implied volatility of deep out of the money options. We restrict our attention to solving the problem (8.66) corresponding to the call option, i.e., with the function f(x) given by (8.67), using Newton's method.

pages: 364 words: 101,286

The Misbehavior of Markets by Benoit Mandelbrot

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

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. The most common approach is to try merely fixing the old formula. Software to correct the “volatility smile,” the U-shaped pattern that Black-Scholes volatility errors often trace on graph paper, is now standard. Many adopt the GARCH methods mentioned earlier; while these produce better results than Black-Scholes alone, they are still not accurate. Some approaches mix ideas similar to mine with those of others. For instance, Morgan Stanley has used what is called a “variance gamma process” to value its own options books at the end of each trading day.

pages: 461 words: 128,421

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

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

The universe has only existed an estimated 12 billion years; the New York Stock Exchange was, as of October 1987, 170 years old.13 Either stock market investors were desperately, spectacularly, unimaginably unlucky that October day, or the bell curve did not come remotely close to representing the true nature of financial market risk. This realization came quickly to some options traders. After October 19, options prices displayed what came to be called a “volatility smile.” By turning the Black-Scholes equation around, one can calculate the implied volatility of any stock from the price of its options. Put options allow one to sell a share of stock at a preset price. 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.

pages: 1,088 words: 228,743

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

Amazon: amazon.comamazon.co.ukamazon.deamazon.fr

If we recognize the time-varying nature of the volatility premium, careful estimation of the two components becomes important, preferably using more robust “model-free” approaches. By market convention, implied volatilities are often quoted based on the B-S formula and reflect its underlying assumptions. Unequal B-S implied volatilities across strike prices (smile or smirk) may be interpreted as market recognition of prevalent non-normalities and other deviations from B-S assumptions. A symmetric volatility smile could be explained by fat tails in the real-world asset return distribution; while an asymmetric skew/smirk could be explained by greater downside volatility (the market’s greater propensity to jump downwards rather than upwards). Since the 1987 equity market crash, the skew for equity index options has been so extreme that investor preferences likely have contributed to it. Either a rational risk premium (such as portfolio insurance preferences leading to greater risk aversion after down markets) or investors’ irrational crash fears could explain the excess of skew implied in index option prices over that observed in realized market returns. 15.2 HISTORICAL PERFORMANCE OF VOLATILITY-TRADING STRATEGIES Covered-call-writing vs. volatility/variance-selling strategies Perhaps the most popular option strategy is covered call writing (CCW): holding a stock or a stock index and writing a call against it.