Particle MCMC for a time changed Ĺevy process


Thesis Type: Doctorate

Institution Of The Thesis: Middle East Technical University, Turkey

Approval Date: 2015

Thesis Language: English

Student: Ayhan Yüksel

Consultant: AZİZE HAYFAVİ

Abstract:

For almost any type of financial modelling exercise, the most fundamental problem is findingsuitablestochasticprocessesthatcapturetheobservedbehaviourofassetprices well. Stochastic volatility models, and their extensions with jumps, are class of flexible models that can capture such empirical dynamics quite well. However this richer modelling environment comes at the expense of estimation challenges. Estimation of these flexible models involves some additional challenges that do not exist for simpler ones. In this thesis, motivated by models with stochastic volatility and jumps, simulation based Bayesian inference methods are analyzed. First we discuss different Markov Chain Monte Carlo (MCMC) approaches in detail, develop various algorithms and implement them for a basic stochastic volatility model. Next we turn our attention to on-line inference and analyze particle filtering methods. We begin with a simple particle filter and then discuss methods to improve the basic filter. We also develop MonteCarloalgorithmstoimplementparticlefiltersforourstochasticvolatilitymodel. More advanced financial models typically include many latent random variables and complicated likelihood functions where standard MCMC methods may fail to efficientlyestimatethem. Asamoreeffectivealternative,wediscussedtheParticleMCMC methods recently proposed by C. Andrieu, A. Doucet, and R. Holenstein (Particle Markov Chain Monte Carlo. Journal of the Royal Statistical Society: Series B 72 (3), 2010, pp 269-342). Particle MCMC methods combine two strands of simulation based Bayesian inference, namely, particle filtering and MCMC, and offer a powerful tool for estimating complex financial models. The theoretical foundations for particle MCMC as well as various samplers proposed in the literature are analyzed in the thesis. In the final part of the thesis, we develop MCMC and particle MCMC methods for a stock price model with a time changed Ĺevy process. We assume that the stock pricefollowsaHeston-typestochasticvolatilityplusvariance-gammajumpsinreturns. Variance-Gamma process is an infinite activity finite variation Ĺevy process obtained bysubordinatinganarithmeticBrownianmotionwithaGammaprocess. Themodelis quiteflexibleinitsnatureandcancapturemostoftheobservedcharacteristicsofstock prices. Our main contribution to existing academic literature is the efficient particle MCMC algorithms that are developed for the Ĺevy based model. We compare MCMC and particle MCMC algorithms in an empirical implementation using S&P500 Index with 15 years of data. The results indicate that the particle MCMC algorithm is a more efficient alternative to standardMCMCandtypicallygivessmallerstandarderrorsand lower autocorrelations.