Knut Sølna

Multiscale stochastic volatility for equity, interest rate, and credit derivatives

Introduction 1. The Black-Scholes theory of derivative pricing 2. Introduction to stochastic volatility models 3. Volatility time scales 4. First order perturbation theory 5. Implied volatility formulas and calibration 6. Application to exotic derivatives 7. Application to American derivatives 8. Hedging strategies 9. Extensions 10. Around the Heston model 11. Other applications 12. Interest rate models 13. Credit risk I: structural models with stochastic volatility 14. Credit risk II: multiscale intensity-based models 15. Epilogue Bibliography Index.

Singular Perturbations in Option Pricing

After the celebrated Black--Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics motivated numerous works during the 1980s and 1990s. In particular, a lot of attention has been paid to stochastic volatility models in which the volatility is randomly fluctuating driven by an additional Brownian motion. We have shown in [Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000; Internat. J. Theoret. Appl. Finance, 13 (2000), pp. 101--142] 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 PDEs, this method corresponds to a singular perturbation analysis. The aim of this paper is to deal with the nonsmoothness of the payoff...

Multiscale Stochastic Volatility Asymptotics

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 make 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, [J. P. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000]), we considered the situation when the volatility is fast mean reverting. Using a...

Statistical Stability in Time Reversal

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.

Imaging Schemes for Perfectly Conducting Cracks

We consider the problem of locating perfectly conducting cracks and estimating their geometric features from multistatic 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 multistatic response matrix measurements. We present numerical experiments to illustrate some of our main findings.

A direct imaging algorithm for extended targets

We present a direct imaging algorithm for extended targets. The algorithm is based on a physical factorization of the response matrix of a transducer array. The multi-signal classification imaging function is used to visualize the results. A resolution and noise level-based thresholding strategy is developed for regularization. The algorithm is simple and efficient since no forward solver or iteration is needed. Multiple-frequency information improves both resolution and stability of the algorithm. Efficiency and robustness of the algorithm with respect to measurement noise and random medium fluctuations are demonstrated.

Stochastic Volatility Effects on Defaultable Bonds

This paper studies the effect of introducing stochastic volatility in the first‐passage structural approach to default risk. The impact of volatility time scales on the yield spread curve is analyzed. In particular it is shown that the presence of a short time scale in the volatility raises the yield spreads at short maturities. It is argued that combining first passage default modelling with multiscale stochastic volatility produces more realistic yield spreads. Moreover, this framework enables the use of perturbation techniques to derive explicit approximations which facilitate the complicated issue of calibration of parameters.

Mathematical and statistical methods for multistatic imaging

Mathematical and Probabilistic Tools.- Small Volume Expansions and Concept of Generalized Polarization Tensors.- Multistatic Configuration.- Localization and Detection Algorithms.- Dictionary Matching and Tracking Algorithms.- Imaging of Extended Targets.- Invisibility.- Numerical Implementations and Results.- References.- Index.

Multistatic Imaging of Extended Targets

In this paper we develop iterative approaches for imaging extended inclusions from multistatic response measurements at single or multiple frequencies. Assuming measurement noise, we perform a detailed stability and resolution analysis of the proposed algorithms in two different asymptotic regimes. We consider both the Born approximation in the nonmagnetic case and a high-frequency regime in the general case. Based on a high-frequency asymptotic analysis of the measurements, an algorithm for finding a good initial guess for the illuminated part of the inclusion is provided and its optimality is shown. The initial guess, obtained through standard statistical arguments, turns out to be Kirchhoff migration. We illustrate the efficiency and the limitations of the proposed algorithms with a variety of numerical examples.

A phase and space coherent direct imaging method

A direct imaging algorithm for point and extended targets is presented. The algorithm is based on a physical factorization of the response matrix of a transducer array. The factorization is used to transform a passive target problem to an active source problem and to extract principal components (tones) in a phase consistent way. The multitone imaging function can superpose multiple tones (spatial diversity/aperture of the array) and frequencies (bandwidth of the probing signal) based on phase coherence. The method is a direct imaging algorithm that is simple and efficient since no forward solver or iteration is needed. Robustness of the algorithm with respect to noise is demonstrated via numerical examples.