Pricing a Log Contract on a Forward Under Risk-Neutral Dynamics

Summary

The discussion explains how to price a derivative whose payoff is the logarithm of a forward price. Under the risk-neutral measure, the forward price follows a diffusion with zero drift and volatility proportional to its level. Applying Itô’s lemma gives this dynamic, after which the log-contract pricing expression for a spot asset can be adapted by setting the interest-rate drift to zero.

This yields a forward log-contract value equal to the logarithm of the current forward price less one half the variance rate times the remaining term. The response clarifies why this is not obtained by simply multiplying the spot log-contract price by a discount factor: the forward’s risk-neutral drift differs from the spot’s. The explanation assumes the stated diffusion model and does not discuss complications such as stochastic volatility, jumps, or other market frictions.

Key ideas

• Under the risk-neutral measure, the forward price has zero drift in the stated model.
• Itô’s lemma provides the forward price dynamics used to value the log payoff.
• The spot log-contract formula adapts to a forward by using a zero interest-rate drift.
• The resulting value depends on the current forward price and the variance over the remaining term.

Tags

# Derive the price of log contract


# Derive the price of log contract












I am reading the Neuberger [1999] Log Contract paper and really confused on the log contract. So if the payoff is $\ln(S_T)$, then we can easily solve the price of such derivative: $$f_t^s = e^{-r(T-t)}[\ln(S_t)+(r-\frac{1}{2}\sigma^2)(T-t)]$$ So my question is that when we have log contract whose underlying is $F_t = S_te^{r(T-t)}$, how do we derive the price of derivative with payoff $\ln(F_T)$, as indicated in the paper: $$f_t^F = \ln(F_t) -\frac{1}{2}\sigma^2(T-t)$$ It looks like $f_t^F = f_t^se^{r(T-t)}$, but why since natural log is a nonlinear tranformation.

## Answer by NN2 (score 1, accepted)

https://quant.stackexchange.com/a/68973

Applying the Ito lemma, you prove easily that the dynamics of $F_t$ in risk-neutral measure $\Bbb Q$ is $$ \frac{dF_t}{F_t} = \sigma dW_t $$ (the drift is $0\cdot dt$, in stead of $r\cdot dt$ as in the dynamics of $S_t$)

Thus, it suffices to apply the formula of (Neuberger, 1999) to derive the price of the derivative with payoff $\ln (F_T)$ by replacing $r = 0$.

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