Sujet de la thèse : Particle Methods in Finance
Sous la direction de : Raphaël DOUADY@: firstname.lastname@example.org 2013-2014 Master student of ENSTA ParisTech, Quantitative Finance2012-2013 Master student of University of Paris 1, Operational research and optimisation.2012-2012 Exchange student at the Bielefeld University2011-2012 First year of Master’s program in applied mathematics in Finance at the University of Paris 1Research interests: Counterparty credit risk modelling(CVA), Monte-Carlo Algorithms and Stochastic Differential Equations. Abstract
The thesis introduces simulation techniques that are based on particle methods and it consists of two parts, namely rare event simulation and a homotopy transport for stochastic volatility model estimation.
Particle methods, that generalize hidden Markov models, are widely used in different fields such as signal processing, biology, rare events estimation, finance, etc. There are a number of approaches that are based on Monte Carlo methods that allow to approximate a target density such as Markov Chain Monte Carlo (MCMC), sequential Monte Carlo (SMC). We apply SMC algorithms to estimate default probabilities in a stable process based intensity process to compute a credit value adjustment (CVA) with a wrong way risk (WWR). We propose a novel approach to estimate rare events, which is based on the generation of Markov Chains by simulating the Hamiltonian system. We demonstrate the properties, that allows us to have ergodic Markov Chain and show the performance of our approach on the example that we encounter in option pricing.
In the second part, we aim at numerically estimating a stochastic volatility model, and consider it in the context of a transportation problem, when we would like to find « an optimal transport map » that pushes forward the measure. In a filtering context, we understand it as the transportation of particles from a prior to a posterior distribution in pseudotime. We also proposed to reweight transported particles, so as we can direct to the area, where particles with high weights are concentrated. We showed the application of our method on the example of option pricing with Stein-Stein stochastic volatility model and illustrated the bias and variance.