We will apply the Monte Carlo simulation method. Data: we will price a call option on the S&P 500 index. The index level at the close of yesterday was equal to 1,065. Assume an annualized long term volatility for the index of 20% per year. The 1-year LIBOR rate is at 1.25%. For the stochastic volatility process take a = 0.95 and c = 0.85. Objective: we want to price several European call options on the S&P 500 index with maturity equal to 1 year (250 trading days) and strike prices between 100 and 2000. We will do this under different correlation scenarios. The purpose is to show how a stochastic volatility process can generate model prices that exhibit a Black-Scholes implied volatility smile.

4. Now construct the implied distributions from the SVM and BS prices (you should have two distributions, one for each). Please use the following formula:

(k2) = ((c(k3)c(k2)/(k3-k2)) (c(k2)c(k1)/(k2-k1))) / (k2-k1)

(Note: the implied distribution that you get from the BS prices is the proper benchmark to look at. It is basically the better version of trying to overlay the normal distribution as we did in the homework.) Plot both distribution against the strike price.