How would you price an option with payout ln(St) where St is the stock price at time t

Question

I know it has to be done through martingales, but I am not fully sure how to do this BSM pricing.

Answer 1

We assume that, under the risk-neutral measure, the stock price process $\{S_t, \, t\ge 0\}$ satisfies an SDE of the form \begin{align*} dS_t = S_t(rdt + \sigma dW_t), \end{align*} where $r$ is the constant interest rate, $\sigma$ is the constant volatility, and $\{W_t, \, t \ge 0\}$ is a standard Brownian motion. Then \begin{align*} S_T = S_0 e^{(r-\frac{1}{2}\sigma^2) T + \sigma W_T}. \end{align*} Moreover, the option payoff $\ln S_T$ has a value given by \begin{align*} e^{-rT} E\big(\ln S_T\big) &= e^{-rT} E\Big(\ln S_0+\Big(r-\frac{1}{2}\sigma^2\Big) T + \sigma W_T \Big)\\ &=e^{-rT} \Big [\ln S_0+\Big(r-\frac{1}{2}\sigma^2\Big) T\Big]. \end{align*}