Some Issues About Option Pricing Under Stochastic Interest Rate

The theory of option pricing is one of the greatest discover in the finance mathematics even economics domain, people can't estimate its value before it produced, but, after produced, people can grasp these invisible opportunities and better use it through quantitative estimate. The modern economic theories, the financial theories and the fiscal theories doubtless is interlinked, the basic principle and method obtained from the financial option research, particularly the evaluation about the risk and the return, is been applied extensively in the research of the modern finance theories, and, the application in the finance most concentrated embody the application of option theories. So far, the application of option theories brought various breakthroughs in the finance theories.Option pricing theory, the important part of modern finance, has promoted the prosperity of finance market. Together with the portfolio selection theory, the capital asset pricing theory, the effectiveness theory of market and acting issue, it is regarded as one of the five theory modules in modern finance.Because of relevant national policies, the economy development condition as well as stock market fluctuation and so on the many kinds of factors can cause the market rate undulation, so carry on a long-term investment use stochastic interest rate model of market to handle the stock and its option pricing problem, can more true and accurate response the price change of stock and option.This dissertation is intended to study the option pricing problems, so as to research the option pricing under stochastic interest rate by means of mathematical tools such as martingale theory and stochastic analysis, to deduce the relevant option pricing equation.The main contents of this dissertation are as follows:Chapter 1 is an introduction, which summarizes the development, the research pre- sent situation, the significance of the option pricing theories, and analyzes the research present condition of domestic and international of stochastic interest rate model.In Chapter 2, we introduce various different interest rate terms structure, and get European option pricing formulas under different interest rate models. Among them, the Vasicek interest rate and it's expanded form Hull-White interest rate are the most familiar interest rate model, they are all belong to the Gauss interest rate, the CIR interest rate is also one of the most familiar interest rate models, and it considered the interest rate stochastic, the fluctuation rate is direct proportion with the square root of its interest rate.In Chapter 3, based on a new type interest rate model, we study the option pricing under stochastic interest rate when the stock price is jump-diffusion process, and we consider the process of stock is paying dividend. The formula of option pricing is obtained when stock price is Jump-diffusion under stochastic interest rate, and the model of interest rate is an approximate sense when the short rate follows the Ito process. This approach will be adopted in evaluating option value under non-Gaussian short rate.In Chapter 4, the pricing of Asian geometric average option under stochastic interest rates is studied, assuming CIR model for the interest rates. Asian option is strongly dependent with path, usually have a long duration, the movement of interest rates becomes more important in pricing such long-dated options. The CIR interest rate model not only considered the interest rate behavior has mean-reverting but also take into account that the fluctuation rate is direct proportion with the square root of its interest rate. Finally, the method of finite difference for the Asian geometric average option under CIR interest rate model is given.In Chapter 5, we study the pricing of contingent claim of foreign exchange under Hull-White interest rate, and consider the rate of exchange to mean-reverting phenomenon, assumption exchange rate driven by Ornstein-Uhlenback process. The equivalent martingale measure is found by the transformation of martingale measure, pricing formula of European contingent claim on foreign is given by using martingale method, In addition, the price evolution of foreign currency option when interest rate and exchange rate, is examined.