# Stokastiske Volatilitetsmodeller: Bayesiansk Statistik og MCMC-estimation i Stokastiske Volatilitetsmodeller

Michelle Porsby Barth & Magnus Meinertz Hansen

Student thesis: Master thesis

## Abstract

In this paper, we explain and test the implementation of stochastic volatility models (SV-models) using financial data. This is done by introducing discrete-time Markov Chain the-ory on a general state space, followed by an implementation and illustration of Markov Chain Monte Carlo (MCMC) methods, which include the Gibbs Sampler and the Metropolis-Hastings-algorithm. After an introduction to Bayesian statistics, the algorithms are used for statistical inference. When the needed theory and methods has been provided, the common stochastic volatility model is introduced. It is explained how to carefully select the appur-tenant prior distributions of the model’s parameters, as well as hyperparameters, which leads to the deduction of the posterior distributions of the parameters.
The introduced MCMC-algorithms are combined and implemented in R in order to estimate the parameters in the SV-model as well as the latent volatility proces. Our implementation is tested against an already implemented estimation of the model, found in the stochvol R-package. First the two implementations are tested using simulated data, where the true values of the parameters are known. Both implementations provide precise estimates of the model’s parameters. The big test for the implimentations, is to estimate the latent volatility proces and parameters using sqrt-log-returns of the Vestas-stock going back 20 years. Here the models provide different estimates, due to the different implementations’ assumptions. Furthermore, a forecasting process is implemented to test the two different sets of estimated parameters. Extentions to the stochastic volatility model is discussed and tested, where two types are used on the Vestas-stock. It is dicussed and illustrated how different extensions provide different oppurtunities to model properties of financial time series. The uses for the SV-model in financial markets is then described to give an idea of the possibilities when implementing a SV-model.
It is concluded that the SV-model is sensitive to the starting-values and the hyperparameters. If realistic values are chosen, the two implementations will still give different results, due to the different prior distributions; one can overcome this problem by adding more datapoints. Looking at the results from the two models, we conclude that the implementation in the stochvol-package is more robust and gives more realistic estimates, but that our implemen-tation provide more flexibility for uses in the financial sector and has great potential for optimization.

Educations MSc in Business Administration and Management Science, (Graduate Programme) Final Thesis Danish 2020 132 Anders Rønn-Nielsen