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.


1 Answer 1


Under the risk-neutral probability measure $\mathbb{Q}$, the logarithmic return is normally distributed with

\begin{equation} \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). \end{equation}


\begin{eqnarray} V_0 & = & \frac{1}{T} e^{-r T} \mathbb{E}_\mathbb{Q} \left[ \ln \left( \frac{S_T}{S_0} \right) \right]\\ & = & e^{-r T} \left( r - \frac{1}{2} \sigma^2 \right). \end{eqnarray}

  • $\begingroup$ 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? $\endgroup$ Dec 26, 2017 at 13:19
  • 1
    $\begingroup$ 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. $\endgroup$ Dec 26, 2017 at 13:21
  • $\begingroup$ checked hull, the expression is correct, also i am assuming that second term in the expression of normal distribution is the variance and not the standard deviation. $\endgroup$ Dec 27, 2017 at 3:57
  • $\begingroup$ Yes - in my case this is the variance. From what I’ve seen, this is the more common notation though you do indeed find both. $\endgroup$ Dec 27, 2017 at 6:56

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