The Theory Of Pricing Options Under Stochastic Interest Rates

The theory of pricing options is always a hot topic in the field of financial mathematics.When we talk about option pricing,the classical assumptions are that stock prices obey the distribution of geometric Brownian motion,the market is complete and frictionless,and the interest rates are constant.But interest rates in real financial markets are affected by many random factors.Therefore,we study the pricing of European options and American options respectively under the assumption of random interest rates.Firstly,we review the research status of option pricing theory and the basic theoretical knowledge needed for option pricing under stochastic interest rates,and introduce the HJM interest rate model and asset portfolio model.Secondly,in the theory of European option pricing,we find the risk neutral probability measure Q by the Girsanov theorem first,so that the discounted assets under Q is martingale process.Next,we calculate both the value process of European contingent claim by the principle of no arbitrage pricing and the hedging strategies at any given moment,and the price of European put option is calculated by the forward measure.Finally,in the theory of American option pricing,we use the Doob-Meyer martingale decomposition theorem and Martingale representation theorem to get the initial price and the value process of the American claim,which is proved to be the minimum process,and then the approximate solution of the American claim value is obtained by using penalty method.In short,under the condition of stochastic interest rates,we get the valuation of European contingent claims,European put options and American contingent claims,then get the approximate solutions of American contingent claims by using the no-arbitrage pricing principle.