Thesis Type: Doctorate
Institution Of The Thesis: Middle East Technical University, Turkey
Approval Date: 2015
Thesis Language: English
Student: Ayhan Yüksel
Consultant: AZİZE HAYFAVİAbstract:
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.