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.

1. First things first, lets get a benchmark. Price the call options with strikes that range from 100 to 2000 USD (at 100 USD increases), for a total of 20 options using the Black and Scholes formula. Use a volatility parameter of 20% (i.e., the long run volatility that was used for the stochastic volatility model). We are going to call those the BS prices.