Estimation of the Stochastic Volatility Model: A Markov Chain Monte Carlo Approach

Abstract

Stochastic volatility (SV) is an important part on how to modelling and predict time-variant volatility, which can not be observed in the financial market. Financial data is typically only observable in discrete-time meaning models describing stochastic volatility in discrete-time is equally relevant to models in continuous-time.
In this thesis, a method for estimating the stochastic volatility model will be introduced. It is a little bit more difficult for the SV model compared to the well-known GARCH model because the likelihood of the SV model is not directly available in a closed form. This requires a differ-ent approach than the usual one in maximum likelihood estimation. The method used in this thesis is a simulation technique known as Markov Chain Monte Carlo (MCMC), which exploits the advantages of the properties for Markov Chain and Monte Carlo. The MCMC technique is based on Bayesian inference, which gives us a probability distribution that allows us to generate samples from. These samples shape a Markov Chain, and for this part, Metropolis-Hastings and Gibbs sampler will be introduced as well. They give us two different methods on how to generate samples.
The main purpose of this thesis is to present the simplest form of the discrete-time SV and how statistical inference can be reached through full implementation of MCMC in R. Moreover, to the theoretical part of the two algorithms, it will be illustrated how they work in practice and a test case on the final implemented model will be performed. Finally, an analysis of a financial asset will be made, where a comparison will be made with the well-known GARCH model.
Our results show that the SV model was not fully capable to capture the over kurtosis we are seeing in returns of financial assets. But overall it did capture the structure of the absolute returns.
MCMC has some downsides, as they are based on Bayesian statistics, the result of the estima-tors will depend on the choice of iterations, burn-in period, start values of the parameters and the start prior distribution.