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.

3. Now we are going to compare the stochastic volatility model to the log-normal pricing 2 model (i.e., Black and Scholes): plot the implied volatilities obtained from the SVM prices and those obtained from the BS prices against their respective strike levels.