# Pricing of a derivative using Risk Neutral Valuation.

I am new to option pricing and following problem came up that I don't understand how to handle.

A derivative will pay out dollar amount equal to $$\frac1T\ln \frac{S_T}{S_0}$$ at maturity, where $S_T$ is distributed log-normally, and the expected return is $\mu$ and volatility is $\sigma$ and $T$ is the time. So what is the price of the derivative using risk neutral valuation..

I know I have to use a stock and a derivative to make a risk neutral portfolio, but not really sure how to proceed.

Under the risk-neutral probability measure $\mathbb{Q}$, the logarithmic return is normally distributed with
• In Hull the expression uses $\sigma$ and not $\sigma^2$ and $\mu$ instead of $r$. Also what you did here seems like you calculated the present value of the expected return we will get in the future. Can you clarify these two things? Dec 26 '17 at 13:19
• 1) Under the risk-neutral probability measure, the expected return on the stock is $r$ not $\mu$. 2) If Hull has a $\sigma$ in the final expression, then this is wrong. 3) Discounting because $\left( r - \frac{1}{2} \sigma^2 \right)$ is the expected payoff at time $T$ while you want the present value. Dec 26 '17 at 13:21