Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations. These can be used in equity markets, for instance, to link the prices of path-dependent exotic instruments to market implied volatilities. The methods are also used for interest rate and credit derivatives. Other applications considered include variance-reduction techniques, portfolio optimization, forward-looking estimation of CAPM 'beta', and the Heston model and generalizations of it. 'Off-the-shelf' formulas and calibration tools are provided to ease the transition for practitioners who adopt this new method. The attention to detail and explicit presentation make this also an excellent text for a graduate course in financial and applied mathematics.

Abstract. In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical Black-Scholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index say and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena makes it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work, see for instance [5], we considered the situation when the volatility is fast mean reverting. Using a singular perturbation expansion we derived an approximation for option prices. We also provided a calibration method using observed option prices as represented by the so-called term structure of implied volatility. Our analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. The combination of regular and singular perturbations approach that we set forth in this paper deals with this case. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parametrization of the implied volatility surface. In particular, the introduction of the slow factor gives a much better fit for options with longer maturities. We use option data to illustrate our results and show how exotic option prices also can be approximated using our multiscale perturbation approach. Introduction.No-arbitrage prices of options written on a risky asset are mathematical expectations of present values of the payoffs of these contracts. These expectations are in fact computed with respect to one of the so-called risk-neutral probability measures, under which the discounted price of the underlying asset is a martingale. In a Markovian context these expectations, as functions of time, the current value of the underlying asset and the volatility level, are solutions of parabolic PDE's with final conditions at maturity times. These conditions are given by the contracts payoffs, and various boundary conditions are imposed depending on the nature of the contracts.In [5] we considered a class of models where volatility is a mean-reverting diffusion with an intrinsic fast time-scale, i.e. a process which decorrelates rapidly and fluctuates on a fine time-scale. Using a singular perturbation technique on the pricing PDE, we were able to show that the option price is in fact a perturbation of the Black-Scholes price with an effective constant volatility. Moreover we derived a simple explicit expression for the first correction in the singular perturbation expansion. We have shown that this correction is universal in this class of models and that it inv...

We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its timescales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility problem for the value function, which is well understood. When volatility is fast mean-reverting, this is a singular perturbation problem for a nonlinear Hamilton-Jacobi-Bellman partial differential equation, while when volatility is slowly varying, it is a regular perturbation. These analyses can be combined for multifactor multiscale stochastic volatility models. The asymptotics shares remarkable similarities with the linear option pricing problem, which follows from some new properties of the Merton risk tolerance function. We give examples in the family of mixture of power utilities and also use our asymptotic analysis to suggest a "practical" strategy that does not require tracking the fast-moving volatility. In this paper, we present formal derivations of asymptotic approximations, and we provide a convergence proof in the case of power utility and single-factor stochastic volatility. We assess our approximation in a particular case where there is an explicit solution.

Abstract. After the celebrated Black-Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics has been the motivation of numerous works during the Eighties and Nineties. In particular, a lot of attention has been paid to stochastic volatility models where the volatility is randomly fluctuating driven by an additional Brownian motion. We have shown in [2,3] that, in the presence of a separation of time scales, between the main observed process and the volatility driving process, asymptotic methods are very efficient in capturing the effects of random volatility in simple robust corrections to constant volatility formulas. From the point of view of partial differential equations this method corresponds to a singular perturbation analysis. The aim of this paper is to deal with the nonsmoothness of the payoff function inherent to option pricing. We present the case of call options for which the payoff function forms an angle at the strike price. This case is important since these are the typical instruments used in the calibration of pricing models. We establish the pointwise accuracy of the corrected Black-Scholes price by using an appropriate payoff regularization which is removed simultaneously as the asymptotics is performed.1. Introduction. Stochastic volatility models in financial mathematics can be thought of as a Brownian-type particle (the stock price) moving in an environment where the diffusion coefficient is randomly fluctuating in time according to some ergodic (mean-reverting) diffusion process. We then have two Brownian motions, one driving the motion of the particle and the other driving the fluctuations of the medium. In the context of Physics there is no natural correlation between these two Brownian motions since they do not "live" in the same space. In the context of Finance they jointly define the dynamics of the stock price under its physical probability measure or an equivalent risk-neutral martingale measure. Correlation between them is perfectly natural. There are economic arguments for a negative correlation or leverage effect between stock price and volatility shocks, and from common experience and empirical studies, asset prices tend to go down when volatility goes up.The diffusion equation appears as a contingent claim pricing equation, its terminal condition being the payoff of the claim. We refer to [5] or [6] for surveys on stochastic volatility. When volatility is fast mean-reverting, on a time-scale smaller than typical maturities, one can perform a singular perturbation analysis of the pricing PDE. As we have shown in [2], this expansion reveals a first correction made of two terms: one is directly associated with the market price of volatility risk and the other is proportional to the correlation coefficient between the two Brownian motions involved. We refer to [2] for a detailed account of evidence of a fast scale in volatility and the use of this asymptotics to parametrize the evolution ...

We present a valuation framework that captures the main characteristics of employee stock options (ESOs), which financial regulations now require to be expensed in firms' accounting statements. The value of these options is much less than Black-Scholes prices for corresponding market-traded options due to the suboptimal exercising strategies of the holders, which arise from risk aversion, trading and hedging constraints, and job termination risk. We analyze the combined effect of all of these factors along with the standard ESO features of multiple exercising rights, and vesting periods. This leads to the study of a chain of nonlinear free-boundary problems of reaction-diffusion type. We find that job termination risk, vesting, finite maturity and non-zero interest rates are significant contributors to the ESO cost. However, we find that in the presence of vesting, the impact of allowing multiple exercise rights on ESO cost is negligible.