Assume S0 = $50, r = 0.05, σ = 0.50, and δ = 0. The Black-Scholes price for a 2-year at-the-money put is $10.906. Suppose that the stock price is lognormal but can also jump, with the number of jumps Poisson-distributed. Assume α = 0.05 (the expected return to the stock is equal to the risk-free rate), σ = 0.50, λ = 2, αJ =−0.04, σJ = 0.08.
a. Using 2000 simulations incorporating jumps, simulate the 2-year price and draw a histogram of continuously compounded returns.
b. Using Monte Carlo incorporating jumps, value a 2-year at-the-money put. Is this value significantly different from the Black-Scholes value?