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FI830 Advanced Derivatives

Homework 6

1. Monte Carlo Heston Model (10 points)

Consider a call options on the S&P 500 index with s0  = 4, 100.60, and let K = 4, 100.00. Assume a constant interest rate r = 4.5%.

(a) Assume a Heston model with parameters s = 2.75, 9  = 0.035, 7u   = 0.425, u0   = 0.19 × 0.19, and correlation coefficient o = -0.46.  Use Monte Carlo simulations (> 10, 000) to calculate option prices for call options with maturity T = 0.25, 0.5, 0.75, 1.0, 1.25, 1.5 and strikes K - 200, K - 100, K, K + 100, K + 200, one for each combination. Plot the resulting surface. 

(b) Compare your results to the analytic option prices using the code we used in class.

(c) Compare your results to prices calculated in a Black-Scholes model with Black-Scholes volatility 7  = 19.00%. Briefly describe your results. (In particular, it is interesting to calculate the implied Black-Scholes volatility of the calculated option prices under the Heston model and compare them to the (constant) Black- Scholes volatility).

(d) Compare your option prices to current prices for these options, pulled from Interactive Brokers.  Also compare implied volatility surfaces.

(e) Experiment with the parameters, and attempt to find a better fit for the observed option prices.

(f) Calculate the option Delta and Vega of the 1-year at-the-money Call option at time zero via finite differ- ences.1

(g) Derive the option prices, the Delta, and the Vega of the 1-year at-the-money Call option via quantlib.