Review the derivation of the Black-Scholes formula (5.6.18). For this exercise, assume that our stock price at time u in the future is S0eμu+Wu, where Wu has the gamma distribution with parameters αu and β with β > 1. Let r be the risk-free interest rate.
a. Prove that e−ruE(Su) = S0 if and only if μ = r − α log(β/[β − 1]).
b. Assume that μ = r − α log(β/[β − 1]). Let R be 1 minus the c.d.f. of the gamma distribution with parameters αu and 1. Prove that the risk-neutral price for the option to buy one share of the stock for the price q at time u is S0R(c[β − 1]) − qe−ruR(cβ), where c = log (q/S0) + αu log (β/β – 1) − ru.
c. Find the price for the option being considered when u = 1, q = S0, r = 0.06, α = 1, and β = 10.