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...

Abstract. We consider the problem of locating perfectly conducting cracks and estimating their geometric features from multi-static response matrix measurements at a single or multiple frequencies. A main objective is to design specific crack detection rules and to analyze their receiver operating characteristics and the associated signal-to-noise ratios. In this paper we introduce an analytic framework that uses asymptotic expansions which are uniform with respect to the wavelength-to-crack size ratio in combination with a hypothesis test based formulation to construct specific procedures for detection of perfectly conducting cracks. A central ingredient in our approach is the use of random matrix theory to characterize the signal space associated with the multi-static response matrix measurements. We present numerical experiments to illustrate some of our main findings.

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Abstract. When a signal is emitted from a source, recorded by an array of transducers, time reversed and re-emitted into the medium, it will refocus approximately on the source location. We analyze the refocusing resolution in a high frequency, remote sensing regime, and show that, because of multiple scattering, in an inhomogeneous or random medium it can improve beyond the diffraction limit. We also show that the back-propagated signal from a spatially localized narrow-band source is self-averaging, or statistically stable, and relate this to the self-averaging properties of functionals of the Wigner distribution in phase space. Time reversal from spatially distributed sources is self-averaging only for broad-band signals. The array of transducers operates in a remote-sensing regime so we analyze time reversal with the parabolic or paraxial wave equation.