Assume we are in the midst of the financial crisis in October 2008. Your firm is considering the purchase of a 10 year put option on the S&P 500 Index. You are analyzing the pricing of this option and you would like to incorporate dissimilar deterministic assumptions for the volatility, the interest rate and the dividend yield for each future year. Let us use the following notation:

 

A) Describe how you would use market data to estimate the future interest rates, future Index volatilities and dividend yields. Describe how to use Monte Carlo simulation to estimate the price of this option. Assume the initial index level is S0 and the strike
price is  K. Assume you are writing the instructions for a programmer  who wil implement the program.

B) Explain in detail an efficient variance reduction technique to improve the efficiency of the algorithm.  
 
C) Explain carefully how you could numerically estimate the delta of this put option.