Martingale Methods Application In Exchange Option Princing

Exchange option is one of common exotic options, refers to the option holders at maturity T moment has the right to (but not required) exchange one asset for another. While the term structure of interest rates and behaviors play key roles in financial asset pricing process. Frist, the paper discusses the pricing formula of exchange options under the not random risk-free rate. Using Girsanov theorems, two relevant Brownian motions under the original measure transform to equivalent martingale measure, which is risk neutral measures. Then use Bayes law to eliminate some random items and turn to be integral problem. Second, assume interest rates model is Vasicek, and the risk-free rate and two underlying assets of the exchange option are related or not related. Then using the similar methods to get the pricing formula of exchange option. And when the risk-free rate and two underlying assets are related, need to orthogonally decompose the ralvant Brownian motion. Finally, we can get three pricing formulas from the three conditions ordinally inclusive. Thus, we get a general pricing formula of the exchange option.