The Application of Stochastic Volatility Models: Via. Markov Chain Monte Carlo Simulation

Abstract

This thesis aims to show how to implement stochastic volatility models on financial data through Markov Chain Monte Carlo simulation, specifically to estimate and forecast the volatility of a stock and explore the advantages and disadvantages of this approach compared to more classical models like GARCH. Initially the relevant theoretical aspects will be introduced, such as Bayesian statistics and Markov chains, along with classical AR, ARCH and GARCH models and the main focus of this paper, stochastic volatility models. For the SV models, three different versions will be presented; the standard model, an SV model with 𝑡-distributed error terms and a version that can model the leverage effect often seen in financial data. This paper will make use of the simulation technique, Markov Chain Monte Carlo simulation, to estimate the stochastic volatility models. This method exploits the advantages of Markov chains and the simulation technique Monte Carlo to efficiently implement the model, for which more traditional approaches like maximum likelihood are not feasible. Specifically, this thesis will focus on the two MCMC-methods, Metropolis-Hastings and Gibbs Sampler, which will both be introduced separately and in a combined algorithm. To apply a more practical aspect to this paper, the empirical study will make use of the Rpackage stochvol, for which the algorithm will first be introduced theoretically with illustrative examples and later applied on the Danske Bank stock. To obtain the most efficient model for the following forecast, the performance of the three versions of the SV model is assessed based on convergence and autocorrelation. The best model is then used to conduct a 21 day forecast of the future volatility alongside a GARCH(1,1) model with 𝑡-distributed error terms to compare the performance. The result of this thesis shows that for the particular stock, the standard SV model performed the best of the three versions, and the following forecast likewise showed that the standard SV model performed slightly better than the implemented GARCH model. However, these results are highly depending on selected data, and the simulation choices, such as burn-in, thinning, number of iterations and prior distributions.