The Pricing Of Exchange Option With Delays

In the Black and Scholes formula, the fitness of the model, which bases on the assumption of constant volatility, has been questioned since empirical evidence shows that the stock price actually depends on past and present time in a way no predictable. The pricing of European options have been studied in many ways when the underly-ing stock price follows a stochastic differential delay equation. However, they mostly research one risk asset with delay. In the paper, using Girsanov theorem, martingale representation theorem, equivalent martingale measure, as well the option pricing for-mula, we study exchange option pricing when the two underlying stocks price exist delay. In the paper, our works are following:1. Arriojas, Hu, Mohammed et al. derived a Black-Scholes-type formula for a call option value in the market with one stock price having time-delay. We further expand to study exchange option pricing with two underlying assets. First of all, we obtain an equivalent martingale measure using Girsanov theorem by choosing an numeraire unit. Then we obtain self-financial trading strategy and a closed form solution of the exchange option price by the conditional expectation of the discounted terminal payoff under the equivalent martingale measure.2. If two risk assets pay continuous dividends, we obtain an equivalent martingale measure by change of numeraire unit and equivalent martingale, and we prove the completeness of the market. Then we develop an explicit formula for pricing exchange option in a sub-interval of exchange option life.