Particle Methods for Local Stochastic Volatility Models

Historically volatility models are used in option pricing in order to improve the results of theclassic framework proposed by Black and Scholes. This approach is extended by local stochastic volatility models that generalize both local volatility and stochastic volatility models. The aim of this work is to study the calibrated local stochastic volatility models through nonlinear stochastic differential equations in the sense of McKean and to analyze the design of suitable particle methods for their numerical approximations. This thesis is comprised of four chapters. Chapter 1 covers the theoretical background of local stochastic volatility models, starting with the classic Black and Scholes model. The paper provides a review of the framework of local volatility and derivation of the Dupire formula, as well as the examination of the theory on local stochastic volatility models, considering both framework and the well-posedness problem of a calibrated LSVM. In Chapter 2 we state the main theoretical and practical aspects concerning McKean stochastic differential equations. Based on stochastic calculus techniques, the paper proposes alternative proofs of well-posedness of McKean SDE, quantitative propagation of chaos property and the quantification of the error of Euler-Maruyama discretization scheme for a general McKean SDE. In addition, the particle method for the local stochastic volatility models is considered. Chapter 3 comprises the propositions of convergence rates for smooth particle systems for local stochastic volatility models, in particular, for the Euler-Maruyama discretization scheme, the particle approximation and the mean-field limit. The problem of existence and uniqueness of a solution for mentioned SDEs is also considered. Chapter 4 contains the results of the numerical simulations of the McKean SDE, local stochastic volatility models using the market data and the related problem of calculation of local volatility.