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.

2. Using the simulated underlying prices that you obtained from the Monte Carlo simulation, price the same 20 call options with strikes that range from 100 to 2000 USD (at 100 USD increases). Assume a correlation of = 30% between the shocks. (Note: you do not need to simulate new paths for the price and variance for each option. You can just use the same simulation to price all options.) We are going to call those the SVM prices. Next, compute the Black-Sholes implied volatility for each one of the options prices that you have computed. What that means is that you use the implied volatility formula that you set up for the last homework, but you match the BS prices to the ones you have obtained from the SVM model.