Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

1. 9. Pricing a Security Derivative. Consider an economy consisting of a risk-free asset and a stock price (risky asset). At time , the risk-free asset and the stock price have the following diffusion processes
   

   where is the risk-free rate, is the continuous dividend yield, is the stock price growth rate, is the stock price volatility (which are all time dependent) and is a -standard Wiener process on the probability space .

   At time , we consider a trader who has a portfolio valued at holding ...