Deriving the Black–Scholes Value of a Log Contract

Summary

The document explains how to value a log contract paying minus twice the logarithm of the asset price at maturity under the Black–Scholes model. It starts with the risk-neutral price dynamics, then applies Itô’s lemma to derive the drift and volatility of the log price. The terminal log price is consequently normally distributed, which makes it possible to take the risk-neutral expectation of the payoff and obtain the discounted value formula.

The explanation identifies the model assumptions behind the result: constant volatility, a risk-neutral process with specified interest and dividend yields, and continuous-time Black–Scholes dynamics. It outlines the derivation rather than providing a numerical example or comparing the result with market data. The document references a published equation but does not discuss the log contract’s applications, hedging, or behavior under stochastic volatility. The formula should therefore be understood as a model-based valuation under the stated assumptions, not as a general result for every price process.

Key ideas

• Under risk-neutral Black–Scholes dynamics, the asset has drift equal to the risk-free rate minus the dividend yield.
• Itô’s lemma gives the log price a drift reduced by half the variance rate.
• The terminal log price is normally distributed under the model.
• The log-contract value follows by discounting the risk-neutral expected payoff.

Tags

# Value of the logcontract $Q^T(t,S)$ with payoff $Q(T,S)=-2lnS_T$


# Value of the logcontract $Q^T(t,S)$ with payoff $Q(T,S)=-2lnS_T$












Why is the value of the log-contract (Neuberger ,1990) with payoff $Q(T,S) = -2\ln S$ given by $$ Q^T(t,S)=-2e^{-r(T-t)}\left(\ln S + (r-q)(T-t)-\frac{\hat\sigma^2}{2}(T-t)\right) $$ ? It is reported in the book Bergomi 2015 at equation 3.8.

## Answer by Achrbot (score 2, accepted)

https://quant.stackexchange.com/a/78424

In the Black-Scholes model, risk-neutral the dynamics of $S_t$ are given by $$ dS_t = (r-q)S_t dt + \sigma S_t dW_t. $$ Using Itô's lemma, we can find the dynamics of the log-process $$ d\log S_t = \left((r-q) - \frac{\sigma^2}{2}\right)dt + \sigma dW_t. $$

From this we can see that $\log S_T - \log S_t \sim N\left((r-q-\frac{\sigma^2}{2})(T-t), \sigma^2(T-t) \right)$, and we can derive the formula for $Q^T(t,S)$ by taking the risk-neutral expectation of $Q(T,S_T)$.

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