# MCMC Estimation: In Stochastic Volatility Model

Søren Svinth Aagaard & Frederik Pedersen

Student thesis: Master thesis

## Abstract

In this project, we will introduce the simulation technique known as Markov Chain Monte Carlo (MCMC), including Metropolis-Hastings and Gibbs Sampler. These techniques depends upon theory about stochastic processes, epsically Markov chains, which will be introduced aswell. MCMC techniques are methods for sampling from probability distributions using Markov chains and are used in data modelling for Bayesian inference and numerical integration. This thesis emphasizes the practical applications regarding financial data, including two exchange rates in DKK/SEK and USD/EUR along with the Nordic40 stockindex. By using this data, we will estimate the parameters in a discrete time stochastic volatility model by applying the shown theory in accordance with MCMC. Stochastic volatility (SV) models are the corner stones of modelling and prediction of time variant volatility in financial markets. Since data is typically only observable at discrete moments, discrete time formulations for SV models become as relevant as continuous time based formulations. The primary focus of this thesis will be to explain the two main MCMC algorithms, Gibbs Sampler and Metropolis-Hastings. These will be related to the estimation procedure in a discrete time SV model. Throughout this paper, practical appplications for the theoretical examples will be presented with in-depth theory regarding the two MCMC algorithms. Furthermore, a connection will be drawn between the two algorithms via simulated data. Subsequenty, we will perform a comparison with the built-in MCMC algorithms in the statistical computing software, R. From the analyses of the financial data, we could conclude that the SV model can be estimated by use of MCMC techniques, but that the credbility of these estimates is determined by the start prior distributions. The biggest disadvantage for the MCMC techniques is that they are built on Bayesian statistics, which fundamentally concerns the phenomenon ”prior,” that is to say previous assumptions and knowledge. As these are often subjective, misleading results and conclusions might occur, which we account for throughout this paper.

Educations MSc in Business Administration and Mathematical Business Economics, (Graduate Programme) Final Thesis Danish 2019 160 Anders Rønn-Nielsen