Market Models with Stochastic Volatility

Sammanfattning: Financial Markets is an interesting wide range area of research in Financial Engineering. In this thesis, which consists of an introduction, six papers and appendices, we deal with market models with stochastic volatility in order to understand some financial derivatives, mainly European options. Stochastic volatility models appear as a response to the weakness of the constant volatility models. Paper A is presented as a survey of different models where the volatility is itself a stochastic process and we present the techniques of pricing European options. Comparing single factor stochastic volatility models to constant factor volatility models, it seems evident that the stochastic volatility models represent nicely the movement of the asset price and its relations with changes in the risk. However, these models fail to explain the large fluctuations in the volatility levels and slope. We propose also a new model which is a variation of the Chiarella and Ziveyi model and we use the first order asymptotic expansion methods to determine the price of European options. Multiscale stochastic volatility models can capture the smile and skew of volatilities and therefore describe more accurately the movements of the trading prices. In paper B , we present an asymptotic expansion for the option price. We provide experimental and numerical studies on investigating the accuracy of the approximation formulae given by this asymptotic expansion. We present also a procedure for calibrating the parameters produced by our first-order asymptotic approximation formulae. Our approximated option prices are compared to the approximation obtained by Chiarella and Ziveyi. In paper C , we implement and analyze the Regime-Switching GARCH model using real NordPool Electricity spot data. We allow the model parameters to switch between a regular regime and a non-regular regime, which is justified by the so-called structural break behaviour of electricity price series. In splitting the two regimes we consider three criteria, namely the intercountry price difference criterion, the capacity/flow difference criterion and the spikes-in-Finland criterion. We study the correlation relationships among these criteria using the mean-square contingency coefficient and the co-occurrence measure. We also estimate our model parameters and present empirical validity of the model. In paper D , we consider a market model with four correlated factors and two stochastic volatilities which is the same model as the one introduced in paper A and used in paper B . An advanced Monte Carlo method is used to find the no-arbitrage price of the European call option in the considered model. In paper E , we forecast the stochastic volatility for exchange rates using Exponential Weighted Moving Average (EWMA) model and study the effect of the out-of-sample periods and also the effect of the decay factor on the forecasts. In Paper F , considering a two-dimensional Black-Scholes equation, we compare the performances between the Crank-Nicolson scheme and the lognormality condition when pricing the European options. We do this by studying the effects of different parameters.