Stochastic Volatility Surface Estimation Suhas Nayak and George Papanicolaou May 9, 2006 Abstract We propose a method for calibrating a volatility surface that matches option prices using an entropy-inspired framework. Starting with a stochastic volatility model for asset prices, we cast the estimation problem as a variational one and we derive a Hamilton-Jacobi-Bellman (HJB) equation for the volatility surface. We study the asymptotics of the HJB equation assuming that the stochastic volatility model exhibits fast mean-reversion. From the asymptotic solution of the HJB equation we get an estimate of the stochastic volatility surface. We also incorporate uncertainty in quoted derivative prices through a penalty term, i.e. by softening the constraints in the HJB equation. We present numerical solutions of our estimation scheme. We find that, depending on the softness of the constraints, certain parameters of the volatility surface related to the implied volatility smile can be calibrated so that they are stable over time. These parameters are essentially the ones found in previous fast mean-reversion asymptotics papers by Fouque, Papanicolaou and Sircar. We find that our procedure provides a natural way of interpolating between the prior parameters and the parameters of Fouque, Papanicolaou and Sircar. 1 Introduction and Review Parameter identification for systems governed by partial differential equations is a well-studied inverse problem. In mathematical finance, finding the volatility of a risky asset from options with multiple strikes and maturities is an example. The information contained in the market is not enough to identify a pricing model and so many sets of model parameters and many types of models could potentially be compatible with the observed option prices. We study the problem of estimating a volatility surface from option prices in an incomplete market setting. 1 1.1 Complete Markets In the complete markets case, there have been several approaches to estimating volatilities from observed option prices. One may try to use the Black-Scholes partial differential equation 1 2 Ct + σ(S; t) CSS + rCS rC = 0; t < T 2 − C(S; T ) = h(S) directly to estimate σ(S; t). This was done in Andersen and Brotherton-Ratcliffe ([2], [3]). Here, h(S) is the payoff of the option. Penalization criteria like smoothness norms were used by Lagnado and Osher [18] and Jackson et. al. [16] to regularize the volatility extraction procedure. Another possible approach in the complete market setting is to use a dual of the option-pricing PDE. Dupire's equation (see [10], [11]) determines the dependence of option prices on strikes and expiration dates. It has the form 2 1 2 2 @ C @C @T C σ (T; K)K + rK = 0; T > t − 2 @K2 @K C(t; K) = h∗(K) Here, h∗(K) has the form as h(S). Since this equation has derivatives in K and not in S, it is easier to handle because derivative prices are quoted for certain expiration times and strikes. Achdou and Pironneau [1] used this equation in combination with least-squares and a regularization (penalty) term to solve the inverse problem of estimating σ(T; K). Their objective was to minimize 2 J(σ) = C(ti; Ki) ci + Jr(σ) j − j i X over σ subject to C solving Dupire's equation. Here the ci are observed derivative prices and f g 2 Jr(σ) is an appropriate Tychonoff regularization functional that involves the L -norm of σ and the L2-norm of its derivatives with respect to K and T . 1.2 Entropy-based methods Regularization may also be achieved through the use of entropy. Entropy minimization for calibrat- ing one-period asset pricing models was used by Buchen and Kelly [7], by Gulko [15], by Jackwerth and Rubinstein [17] and by Platen and Rebolledo [19]. Relative entropy-based methods can be motivated as follows. There is a range of strategies available to hedge an option. If the hedger is fully confident that the option will follow dynamics given by a known model, then it would be worthwhile to ignore observed market volatilities and hedge using the model. If, on the other hand, the hedger believes that current prices of the option fully determine the future evolution of the stock, then it would be a good idea to hedge according to 2 these current option prices. Relative entropy-based methods help bridge the gap between these two possible strategies. They provide estimates for model parameters that are close to prior information while still matching current option prices. Within the class of relative entropy methods there are two general approaches. In Avellaneda [4], a probability law is found for the risky asset that satisfies certain moment constraints, namely that it matches observed market prices of options, and is close to a a prior probability law, which can arise from historical or other econometric information. The closeness to the prior is measured by the relative entropy, which is given by dP H(P P ) = EP ln j 0 dP 0 dP where P0 is the prior distribution, and dP0 is the Radon-Nikodym derivative of P with respect to P . For H(P P ) to be finite, the measure P has to be absolutely continuous with respect to P . 0 j 0 0 So, in particular, if under P0, we assume our asset prices follow ν (0) (0) dXi(t) = σij dBj(t) + µi dt i X=1 where Bj are standard Brownian motions, then, by Girsanov's theorem, we must have that under f g P , the asset follows the price process ν (0) dXi(t) = σij dBj(t) + µidt: i X=1 (0) (0) Here µi = µi + j σij mj and mj is a market price of risk. In other words, once the volatility of the asset under the prior probability law is specified, the volatility of the asset under the new P measure must be the same. Only the market price of risk can be adjusted to give a consistent pricing measure. In this framework we assume that many different, perhaps correlated, shocks drive the evolution of each asset. Moreover, we assume that there is a good prior that describes the effect of fluctuations (0) through the volatilities σij . It is unlikely that we will have an accurate assessment of these volatilities. Yet once they are fixed in the model we are unable to deviate from them in the new probability law because of the absolute continuity that is assumed. In order for the market price of risk to correct for possible misspecification of volatilities, wild swings in its value may be necessary. The work of Carmona and Xu [8], who introduce a stochastic volatility model in an entropy framework, suffers from the same problems. There, the market price of risk associated with the stochastic volatility process is the only degree of flexibility and all other parameters are fixed prior to the estimation procedure. It is desirable, therefore, to use a different relative entropy approach, one that was introduced in [5]. They considered processes for equity of the form dSt P0 P0 = σ0;tdBt + µt dt, under P0 St 3 and dSt P P = σtdBt + µt dt, under P: St Unless σ0 = σ under P , the relative entropy of the two measures is infinite, since P and P0 are then mutually singular. Avellaneda et. al. in [5] extended the concept of relative entropy to probability measures under which the processes do not necessarily have the same volatility. They considered the most singular part of the relative entropy using a time discretization that is based on trinomial trees. With this discretization and a small σ σ expansion, they found that to highest order the − 0 relative entropy looks like (σ2 σ2)2 over each time step. − 0 Both approaches to relative entropy calibration have the same general objective. Once the form of the relative entropy is determined the objective is to minimize it over all possible P subject to the constraint that the prices of options under P match observed prices. 1.3 Stochastic volatility surface estimation Finding volatilities across strikes and expiration dates for incomplete markets is a very difficult task. We focus our attention on stochastic volatility models. These models have a large number of parameters that need to be known for pricing purposes and options can be quite sensitive to them. There are, however, several papers that deal with this issue. Broadie et al. [6], for example, fit the parameters of various stochastic volatility models. They first obtain parameters using a long-run time series of underlying asset returns. They then use the information found in option prices to estimate volatility and risk premia. This second stage involves the minimization of an objective function that is just the sum of the squares of the difference in the model-derived Black-Scholes implied volatilities and the implied volatilities that correspond to the data. Their method forces consistency in parameters between the data for the underlying asset and the data for the option prices. The problem of option pricing in a stochastic volatility setting was tackled in a series of papers by Fouque, Papanicolaou and Sircar (see [12] and the references therein). They developed a method that reduced the number of parameters that were needed for pricing and hedging to just three. Two were derived directly from the smile associated with options prices, and the other was some estimate of the underlying's volatility. They introduced a type of model where the stochastic volatility factor followed a fast mean-reverting process. The data set of options they then considered was chosen so that the asymptotics would be valid.