Reference no: EM132226862

Problem 1: Evaluation of a known integral using various quadratures: In this problem we are going to compute the price of a European call option with 3 month expiry, strike 12, and implied vol 20, Assume the underlying is 10 now and the interest rate is 4%.

1. Use Black-Scholes formula to compute the price of the call analytically.

2. Calculate the price of the call numerically using the following 3 quadrature methods:
(a) Left Riemann rule
(b) Midpoint rule
(c) Gauss nodes of your choice (say explicitly why you made that choice) with the number of nodes N = 5, 10, 50, 100 and compute the calculation error as a function of N for each of the methods.

3. Estimate the experimental rate of convergence of each method and compare it with the known theoretical estimate.

4. Which method is your favorite and why?

Problem 2: Calculation of Contingent Options: Let S₁ be a random variable that takes on the value of SPY one year from now and let S₂ take on the values of SPY 6 months from now. Assume that they are jointly normally distributed with

σ₁ = 20%
σ₂ = 15%
ρ = 0.95

By ρ here we mean correlation between S₁ and S₂. Also, assume that interest rate is zero.

1. Evaluate the price of the one year call on SPY with the strike K₁ = 260. This is an example of a vanilla option.

2. Evaluate the price of the one year call on SPY with the strike K₁ = 260, contingent on SPY at 6 months being below 250. This is a contingent option.

3. Calculate the contingent option again, but with ρ = 0.8, ρ = 0.5, and ρ = 0.2.

4. Does dependence on ρ make sense?

5. Calculate the contingent option again, but with SPY at 6 months below 240, 230, and 220.

6. Does the dependence on the 6 month value make sense?

7. Under what conditions do you think the price of the contingent option will equal the price of the vanilla one?