Stochastic modelling in finance

• Main Author: Baduraliya, Chaminda Hasitha
• Published: University of Strathclyde 2012
• Subjects: 332.011
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.570587

The trading of financial derivatives and products in financial markets has influenced the development of the world economy. Over the last few decades, a rapid growth in complex financial systems, which can generate unstable conditions in financial markets, has been observed. Therefore models are being developed to study and examine the uncertainty surrounding these financial systems in different circumstances. The important milestone of this work can be traced to the Black-Scholes formula for option pricing which was published in 1973 and revolutionized the financial industry by introducing the no-arbitrage principle [8]. This model assumed that the average rates of return and volatility are constant, however, this is not realistic. Therefore, several models have been developed, based on pragmatic studies, which generalize the Black-Scholes formula to acquire more knowledge for these financial systems. In this project, we will focus on Stochastic Differential Equations (SDEs) models in finance which do not have explicit solutions so far. In particular, Lewis [47] developed the mean-reverting-theta processes which can not only model the volatility but also the asset price. Therefore, we will establish the Euler-Maruyama (EM) numerical schemes to approximate the solution to this model and show that the EM approximate solution will converge in probability to the true solution under certain conditions. The convergence property of the corresponding step process will be examined under the same conditions to determine its application in finance. In addition, the Markov-switching format of this model can be used to explain some erratic situations observed in financial data. Under the same conditions on parameters of mean-reverting-theta model, the Markov-switching model will be examined to show that the EM approximate solution to this model will converge in probability to the true solution. Although previous models fit to a certain type of financial data, they can not be used to explain behaviour of the unpredictable abrupt structural changes in financial markets. However, the mean-reverting-theta stochastic volatility model driven by a Poisson jump process explains some of this phenomenon. Therefore, we will examine the analytical properties of EM approximate solutions to this model for two conditions of the parameters theta and beta. Since it is possible to obtain a more generalized formula for this stochastic volatility jump process, by incorporating a hybrid concept into this SDE model, we will consider the mean-reverting-theta volatility model with Poisson jumps driven by two independent Markov processes. Existing financial instruments are not strong enough to examine the convergence property of the approximate solution to this model. Therefore, we will establish EM approximate solutions to this model and examine their convergence property, when we assume similar parameter conditions to the mean-reverting-theta model. Finally, we will show that these approximate solutions of the SDE models can be used to evaluate financial quantities, options and bonds for example.