Implied volatility asymptotics under affine stochastic volatility models

Abstract

This thesis is concerned with the calibration of affine stochastic volatility models with jumps. This class of models encompasses most models used in practice and captures some of the common features of market data such as jumps and heavy tail distributions of returns. Two questions arise when one wants to calibrate such a model: (a) How to check its theoretical consistency with the relevant market characteristics? (b) How to calibrate it rigorously to market data, in particular to the so-called implied volatility, which is a normalised measure of option prices? These two questions form the backbone of this thesis, since they led to the following idea: instead of calibrating a model using a computer-intensive global optimisation algorithm, it should be more efficient to use a less robust—hence faster—algorithm, but with an accurate starting point. Henceforth deriving closed-form approximation formulae for the implied-volatility should provide a way to obtain such accurate initial points, thus ensuring a faster calibration. In this thesis we propose such a calibration approach based on the time-asymptotics of affine stochastic volatility models with jumps. Mathematically since this class of models is defined via its Laplace transform, the tools we naturally use are large deviations theory as well as complex saddle-point methods. Large deviations enable us to obtain the limiting behaviour (in small or large time) of the implied volatility, and saddle-point methods are needed to obtain more accurate results on the speed of convergence. We also provide numerical evidence in order to highlight the accuracy of the closed-form approximations thus obtained, and compare them to standard pricing methods based on real calibrated data.