Asymptotics of a Two-Scale Sto chastic Volatility Mo del y z J.P. Fouque G. Papanicolaou K.R. Sircar To Jacques-Louis Lions on the occasion of his seventieth birthday Abstract We present an asymptotic analysis of derivative prices arising from a sto chastic volatility mo del of the underlying asset price that incorp orates a separation b etween the short tick-by- tick time-scale of uctuation of the price and the longer less rapid time-scale of volatility uctuations. The mo del includes leverage or skew e ects a negative correlation b etween price and volatility sho cks, and a nonzero market price of volatility risk. The results can be used to estimate the latter parameter, which is not observable, from at-the-money Europ ean option prices. Detailed simulations and estimation of parameters are presented in [6]. 1 Intro duction 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 phenomenon, which is well- do cumented in, for example, [9 , 12 ], stands in empirical contradiction to the consistent use of a classical Black-Scholes constant volatility approach to pricing options and similar securities. 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 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. One approach, termed the implied deterministic volatility IDV approach [5, Chapter 8], is to supp ose volatility is a deterministic function of the asset price X : volatility= t; X , so that the t t sto chastic di erential equation mo deling the asset price b ecomes dX = X dt + t; X X dW : t t t t t The function C t; x giving the no-arbitrage price of a Europ ean derivative security at time t when the asset price X = x then satis es the generalized Black-Scholes PDE t 1 2 2 t; xx C + r xC C =0; C + xx x t 2 with r the constant risk free interest rate and with terminal condition appropriate for the contract. This has the nice feature that the market is complete which, in this context, means that the CNRS-CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex France, [email protected]. This work was done while visiting the Department of Mathematics, Stanford University. y Department of Mathematics, Stanford University, Stanford CA 94305, [email protected] z Department of Mathematics, University of Michigan, Ann Arb or MI 48109, [email protected] 1 derivative's risk can theoretically b e p erfectly hedged by the underlying, and there is no volatility risk premium to b e estimated. However, while attempts to infer a lo cal volatility surface from market data by tree metho ds [13 ] or relative-entropy minimization [2 ] or interp olation [15 ] have yielded interesting qualitative prop erties of the risk-neutral probability distribution used by the market to price derivatives such as excess skew and leptokurtosis in comparison to the lognormal distribution, this approach has not pro duced a stable surface that can b e used consistently and with con dence over time. For this reason, we concentrate on the \pure" sto chastic volatility approach of [8 ] reviewed in [16 , Section 1.2] or [3 ], in which volatility is mo deled as an It^o pro cess driven by a Brownian t motion that has a comp onent indep endent of the Brownian motion W driving the asset price. t There has b een a lot of analysis of sp eci c It^o mo dels in the literature [14 , 17 , 7] bynumerical and analytical metho ds, many of which have ignored skew e ects and/or the volatility risk premium for tractability. Our goal [6 ] is to estimate these parameters from market data and to test their stability over time and thus the p otential usefulness of sto chastic volatility mo dels for hedging derivatives. What is to our knowledge new here in comparison with previous empirical work on sto chastic volatility mo dels is our keeping of these two factors, use of high-frequency intraday data, and an asymptotic simpli cation of option prices predicted by the mo del that allows for easy estimation of the volatility risk premium from at-the-money market option prices. The latter exploits the separation of time-scales rst intro duced in this context in [16 ]. It is often observed that while volatility might uctuate considerably over the many months comprising the lifetime of an options contract, it do es not do so as rapidly as the sto ck price itself. That is, there are p erio ds when the volatility is high, followed by p erio ds when it is low. Within these p erio ds, there might be much uctuation of the sto ck price as usual, but the volatility can be considered relatively constantuntil its next \ma jor" uctuation. The \minor" volatility uctuations within these p erio ds are relatively insigni cant, esp ecially as far as option prices, which come from an average of a functional of p ossible paths of the volatility, are concerned. Many authors, for example [1], have prop osed nonparametric estimation of the pricing measure for derivatives. The analysis in [16 ] is indep endent of sp eci c mo deling of the volatility pro cess, but results in bands for option prices that describ e p otential volatility risk in relation to its historical auto correlation decay structure, while obviating the need to estimate the risk premium. However, the market in at-the-money Europ ean options is liquid and its historical data can b e used to estimate 1 this premium . For this reason, we shall attempt this with a mo del that is highly parametric, but complex enough to re ect an imp ortantnumb er of observed volatility features: 1. volatility is p ositive; 2. volatility is rapidly mean-reverting see for example [10 ]; 3. volatility sho cks are negatively correlated with asset price sho cks. That is, when volatility go es up, sto ck prices tend to go down and vice-versa. This is often referred to as leverage [4 ], and it at least partially accounts for a skewed distribution for the asset price that lognormal or zero-correlation sto chastic volatility mo dels do not exhibit. The skew is do cumented in empirical studies of historical sto ck data. 2 Mo del The mo del wecho ose is that volatility is the exp onential of a mean-reverting Ornstein-Uhlenbeck OU pro cess or, equivalently, log is mean-reverting OU. With a suitable initial distribution, t 1 This was suggested to us by Darrell Due. 2 the volatility pro cess is stationary and ergo dic which allows us to use averaging principles to ap- proximate the option price, separating the minor and ma jor uctuations. This mo del has b een considered in [14 ] and it is related to EGARCH mo dels which, as shown in [11 ], are weak approx- imations to the continuous-time di usion. Another mo del that is stationary and can be similarly implemented and analyzed is when is a mean-reverting Feller or Cox-Ingersoll-Ross pro cess [3 ]. t The nal ingredient is to mo del the two time scales describ ed previously. To this end, we intro duce a small parameter " > 0 describing the discrepancy between the scales, and mo del the " volatilityas = , where is exp onential OU. Thus the volatilityisthe pro cess \sp eeded- t t t=" t up" to re ect that there are many ma jor uctuations over the life of the options contract this time scale is O 1 in the usual time-unit of years, but not as many as there are minor It^o uctuations. We de ne Y := log and supp ose it satis es t t ^ dY = m Y dt + dZ t t t " " ^ for constants >0; > 0;m and Z a Brownian motion. Then Y := log is describ ed by t t t " " ^ p = dY dt + m Y dZ t t t " " p where now and have b een replaced by =" and = " to mo del rapid mean reversion and overall ^ variance of order one. Finally, to incorp orate the correlation skew e ect dhW; Z i = dt,we write t p 2 ^ Z = W + 1 Z , where W and Z are indep endent Brownian motions, to arrive at the nal t t t " sto chastic volatility mo del for the sto ck price X t " " " Y " t dX = X dt + e X dW 1 t t t t q " " 2 p 2 m Y dW + 1 dZ dt + dY = t t t t " " Then, by the usual no-arbitrage argument, as detailed for example in [16 , Section 1.3], the Europ ean " call option price C t; x; y satis es y 2 1 xe " 2y 2 " " " " " p C e x C C C + + + + r xC C t xx xy yy x 2 " 2" " p m y C = 0 3 + y " " " + C T ; x; y = x K " where is the market price of volatility risk whichwe assume constant. If C t; x; y satis es this " " " " equation then from It^o's formula C = C t; X ;Y satis es the sto chastic di erential equation t t " " " " " " Y " " " ^ t p p dC =[rC + r X C + C ]dt + e X C dW + C dZ 4 t t x y t x y " " p From this expression we see that an in nitesimal change in the volatility risk = " changes the in nitesimal rate of return of the option by times the change in volatility risk.