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Stochastic Volatility Correction to Black-Scholes

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Stochastic Volatility Correction to Black-Scholes Sto chastic Volatility Correction to Black-Scholes y z Jean-Pierre Fouque George Papanicolaou K. Ronnie Sircar July 1999; revised August 1999. This version January 2000. Sto chastic volatility mo dels have b ecome p opular for derivative pricing and hedging in the last ten years as the existence of a non at implied volatility surface or term-structure has b een noticed and b ecome more pronounced, esp ecially since the 1987 crash. This phe- 1 nomenon, whichiswell-do cumented , stands in empirical contradiction to the consistent use of a classical Black-Scholes constant volatility approach to pricing options and similar se- curities. However, it is clearly desirable to maintain as many of the features as p ossible that have contributed to this mo del's p opularity and longevity, and the natural extension pursued b oth in the literature and in practice has b een to mo dify the sp eci cation of volatility in the sto chastic dynamics of the underlying asset price mo del. There are many stories b ehind why we should mo del volatilityto be a random pro cess. For example, it could simply represent estimation uncertainty, or it can arise as a friction from transaction costs, or it could simulate non-Gaussian heavy-tailed returns distributions. In other words, sto chastic volatilityis a far-reaching extension of the Black-Scholes lognormal mo del, describing a much more complex market. Any extended mo del must also sp ecify what data it is to be calibrated from. The pure Black-Scholes pro cedure of estimating from historical sto ck data only is not p ossible in an incomplete market if one takes the view as we shall that the market selects a unique risk-neutral derivative pricing measure, from a family of p ossible measures, which re ects its degree of "crash-o-phobia". This pricing measure is re ected in traded at-the-money Europ ean options prices, so, as is common practice, this \smile data" is used for calibration. Parameter estimation and stability of the estimates in time presents the ma jor mathe- matical and practical challenge here. Without a formula for option prices under a particular sto chastic volatility mo del, estimating the risk-neutral parameters is computationally inten- sive we have to run a tree or simulations at each step in an iterative search pro cedure. Often mo dels are chosen so that there is a closed-form solution, and this usually means tak- ing the volatility to b e indep endent of the Brownian motion driving the sto ck price, whereas Department of Mathematics, NC State University, Raleigh NC 27695-8205, [email protected]. y Department of Mathematics, Stanford University, Stanford CA 94305, [email protected] z Department of Mathematics, University of Michigan, Ann Arb or MI 48109-1109, [email protected]. Work supp orted by NSF grant DMS-9803169. 1 See for example, Jackwerth, J. and Rubinstein, M, 1996, Recovering Probability Distributions from Contemporaneous Security Prices, Journal of Finance 515, pages 1611-1631, and Rubinstein, M., 1985, Nonparametric Tests of Alternative Option Pricing Models, Journal of Finance 402, pages 455-480. 1 common exp erience and empirical evidence suggests a negative correlation: when volatility go es up, sto ck prices tend to go down. Now supp ose that we have estimated the parameters. How go o d will these estimates b e tomorrow? It is p ossible to haveavery tight tover a short time, but often these break down 2 signi cantly thereafter. This is certainly a problem with estimates of volatility surfaces , where unlike in sto chastic volatility mo deling, volatility is mo deled to be a function of time and sto ck price with no indep endent randomness. We present here a new approach to sto chastic volatility that has the following features: It requires that volatility b e mean-reverting, but, other than that, do es not dep end in an essential way on how the volatility is mo deled. It translates the slop e and intercept of the implied volatility skew into information ab out the correlation b etween volatility and sto ck price sho cks and the market's volatil- ity risk premium. It simpli es enormously the parameter estimation problem. It gives a recip e for pricing and hedging other derivatives in a sto chastic volatility en- vironmentby identifying their e ective approximating derivative security in a constant volatilityenvironment. This includes barriers, Asians and Americans. It pro duces parameter estimates from implied volatilityskews that are stable. This is achieved by exploiting the mean-reverting b ehaviour of volatility and the much- noted observation that volatility is p ersistent. Framework The sto ck price S satis es t t0 dS = S dt + S dW ; t t t t t where is the volatility pro cess. To incorp orate the correlation with the Brownian t t0 motion W which leads to the implied volatility skew in these mo dels, it is convenient t to take to be a di usion pro cess to o, although it can have jumps as well. To x t ideas, we shall write volatility as a p ositive function of a mean-reverting Gaussian Ornstein- Uhlenbeck pro cess: = f Y , where t t q 2 dY = m Y dt + dW + ; 1 dZ t t t t with Z an indep endent Brownian motion the source of the additional randomness and 2 See Dumas, B., Fleming, J. and Whaley, R., 1998, ImpliedVolatility Functions: Empirical Tests, Journal of Finance 536, pages 2059-2106 for an empirical study of this and, for a mathematical analysis, Lee, R., 1999, Local Volatilities under Stochastic Volatility, to app ear in International Journal of Theoretical and Applied Finance. 2 = the rate of mean-reversion; m = the long-run mean of Y; = the \v-vol"; = the correlation co ecient : Our ob jective is to analyze the e ect of sto chastic volatility in the basic Black-Scholes mo del. Therefore, we assume that these parameters, as well as the rate of return are, for simplicity, constant. For the same reason we assume that f Y do es not dep end on t t explicitly. In fact we do not need to sp ecify f in detail since the mean reversion asymptotics give results that are insensitive to all but a few general features of f . The precise mo del for the pro cess Y driving the volatilitydoes not matter either, so long as it is an ergo dic t pro cess like the Ornstein-Uhlenbeck pro cess ab ove. Three observable quantities emerge from the asymptotics and they are the only ones that must b e calibrated from historical data and the term structure of volatility. These quantities are complicated functions of the primitive mo del parameters, the function f and the market price of volatility risk intro duced b elow, which need not be calibrated separately. This is a new approach to the study of sto chastic volatility mo dels. Volatility Persistence It is often noted in empirical studies of sto ck prices that volatility is p ersistent or bursty - for days at a time it is high and then, for a similar length of time, it is low. However, over the lifetime of a derivative contract a few months, there are many such p erio ds, and lo oked at on this timescale, volatility is uctuating fast, but not as fast as the rapidly changing sto ck price. In terms of our mo del, wesay that the volatility pro cess is fast mean-reverting relativeto the yearly timescale, but slow mean-reverting by the tick-tick timescale. Since the derivative pricing and hedging problems we study are p osed over the former p erio d, we shall say that volatility exhibits fast mean-reversion without explicitly mentioning the longer timescale of reference. 1 The rate of mean-reversion is governed by the parameter , in annualized units of years . 2 Fast mean-reversion means that is in fact large and that =2 , the variance of the invariant distribution of the OU pro cess, is a stable O 1 constant. As an illustration, Figure 1 shows simulated volatility paths for the mo del ab ove in which = 1 on the top and = 200 b elow. In practice the volatility pro cess is not directly observable. In fact the true observable is the de-meaned returns pro cess dS t dt = dW ; t t S t at discrete times. In Figure 2 we show simulated tra jectories of this returns pro cess corre- sp onding to these volatility tra jectories in the rst two graphs. We observe that the size of the uctuation in the returns pro cess of the top picture is relatively constant over time, while for the large picture, in the middle, it is changing a lot. This is exactly what we call fast mean-reverting or p ersistent sto chastic volatility. 3 =1 0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 = 200 0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Figure 1: Simulated volatility for smal l and large rates of mean-reversion for the OU model, with y the choice f y =e . Note how volatility \clusters" in the latter case. In the b ottom graph of Figure 2, we show the returns pro cess for the S&P 500 over the rst ve months of 1996.
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