Pricing Exchange Options With Stochastic Interest Rate By Martingale Method

Option is a kind of important financial derivatives. Since Black and Scholes established the famous option pricing model B-S model in1973, option pricing theory has got fast development. In addition to the popular European and American pricing option pricing research, there are also a lot of research on exotic options pricing, such as exchange option pricing. Based on Black and Scholes’s European option pricing theory, Margrabe got the exchange option pricing formula for the first time by using partial differential equation method in1978. But in Margrabe’s formula, risk-free interest rate was considered to be constant, which was not in accordance with the reality.This paper reviewes the relevant basic knowledge of mathematics and B-S pricing formula, and on that basis discusses the situation of constant interest rate. Then the interest rate for stochastic volatility model is analyzed, considering the Brownian motion of stochastic interest rate model not associated with the Brownian motions of two underlying asset as well as the Brownian motion of stochastic interest rate model associated with the Brownian motions of two underlying asset. Finally a pricing formula is deduced by the method of numeraire transform and the pricing of exchange option in a complete market will not be affected by stochastic interest rate is verified.