Risk-Neutral Pricing of a Log-Return Payoff

Summary

The document considers a derivative that pays the continuously compounded return from the initial asset price to its terminal price, scaled by the maturity. It asks how to price that payoff when the asset follows a lognormal model with stated expected return and volatility. The answer switches to the risk-neutral measure, where the log return has mean equal to the risk-free rate minus half the variance rate, multiplied by time, and variance equal to volatility squared times time. Discounting the expected payoff then gives the displayed value at inception.

The key pricing idea is to take the expectation under the risk-neutral measure rather than use the asset's physical expected return, then discount at the risk-free rate. The result depends on the model assumptions and the stated payoff; the reply does not derive the change of measure or discuss dividends, alternative market dynamics, or whether the payoff can be replicated in a particular market. It is a compact worked result rather than a broader option-pricing treatment.

Key ideas

• The payoff is the terminal log return divided by the time to maturity.
• Under the risk-neutral measure, the log return has a drift adjusted by half the variance rate.
• The derivative value is the discounted risk-neutral expectation of its payoff.
• The displayed result assumes the lognormal model and risk-free rate specified in the answer.

Tags

# Pricing of a derivative using Risk Neutral Valuation.


# 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.

## Answer by LocalVolatility (score 2, accepted)

https://quant.stackexchange.com/a/37498

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}

Thus,

\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}

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.