# 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
$$\ln \left( \frac{S_T}{S_0} \right) \sim \mathcal{N} \left( \left( r - \frac{1}{2} \sigma^2 \right) T, \sigma^2 T \right).$$
• 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? – Piyush Divyanakar 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. – LocalVolatility Dec 26 '17 at 13:21