Given the following model for the evolution of a stock price in the risk-neutral world In S(T) = In S + (r - sigma^2/2)T + sigma Squareroot TN(0, 1), and that the value of an European put is given by P(S, K, r, sigma, T) = e^-rT E((K - S(T))_+). Derive the Black-Scholes formula for the value of a put option: P(S, K, r, sigma, T) = -SN(-d_1) + Ke^-rT N(-d_2), where S = initial value of the stock, K = strike price, sigma = volatility, r = risk free interest, T = expiry time, d_1 = log (S/K) + (r + 1/2 sigma^2)T/sigma Squareroot T, d_2 = log (S/K) + (r + 1/2 sigma^2)T/sigma Squareroot T, and N(x) = 1/Squareroot 2 pi integral^x_-infinity e^-t^2/2 dt. Given the following model for the evolution of a stock price in the risk-neutral world In S(T) = In S + (r - sigma^2/2)T + sigma Squareroot TN(0, 1), and that the value of an European put is given by P(S, K, r, sigma, T) = e^-rT E((K - S(T))_+). Derive the Black-Scholes formula for the value of a put option: P(S, K, r, sigma, T) = -SN(-d_1) + Ke^-rT N(-d_2), where S = initial value of the stock, K = strike price, sigma = volatility, r = risk free interest, T = expiry time, d_1 = log (S/K) + (r + 1/2 sigma^2)T/sigma Squareroot T, d_2 = log (S/K) + (r + 1/2 sigma^2)T/sigma Squareroot T, and N(x) = 1/Squareroot 2 pi integral^x_-infinity e^-t^2/2 dt