Pricing a Stock Payoff Equal to Its Value Times Its Logarithm

Summary

The document derives the present value of a derivative that pays the stock price multiplied by the logarithm of that price at maturity, under standard Black–Scholes assumptions. The response uses a stock-numeraire change of measure: the risk-neutral expectation of the stock-weighted payoff is expressed as the current stock price times an expectation under a measure associated with the stock.

Under that measure, Girsanov’s theorem changes the drift of the log stock process. Taking its expected value over the remaining term gives the stated pricing expression, involving the current stock, its logarithm, the risk-free rate, volatility, and time to maturity. This is a model-based derivation within the assumptions listed in the question, including no dividends and frictionless trading. The answer is concise and does not discuss extensions to dividends, alternative dynamics, or market frictions; the logarithm also depends on the price units used.

Key ideas

• The payoff is valued under the stated Black–Scholes market assumptions.
• A stock-numeraire change of measure simplifies the stock-weighted expectation.
• Girsanov’s theorem changes the drift of the log stock under the new measure.
• The resulting present value depends on the current stock price, rate, volatility, and time.
• The derivation does not address dividends, market frictions, or alternative price dynamics.

Tags

# PV of derivative that pays $S_T \ln\left(S_T\right)$ at maturity


# PV of derivative that pays $S_T \ln\left(S_T\right)$ at maturity












We have a financial derivative that pays $S_T \ln\left(S_T\right)$ at maturity $t=T$

We assume a Black-Scholes world:

- No arbitrage opportunities.

- No dividend payments from the stock $S_t$.

- Existence of a riskless asset yielding the risk free rate

- Possibility to borrow and lend infinitely at the risk-free rate.

- Possibility to buy and sell infinitely the stock $-$ even fractional amounts.

- No transaction costs.

We also assume that the stock is tradable and that the derivative is attainable $-$ we basically assume we are in the standard pricing setting.

What's the present value of this financial derivative at $t=0$ ?

My understanding is that using risk neutral measure to calculate PV of this payoff is rather difficult. We need to change the measure to simplify the calculation.

## Answer by Ivan (score 2, accepted)

https://quant.stackexchange.com/a/38583

I assume this is a homework question. You do need to change the measure and price under $Q_s$ rather than $Q$.

$E_{Q}(S_T.\ln(S_T)) = S_t.E_{Q_s}(\ln(S_T))$

Note the SDEs for $dS_t$ and hence $d\ln(S_t)$ now change by virtue of the Girsanov theorem.

In particular $d\ln(S_t) = (r+\frac{\sigma^2}{2})dt + \sigma dW^{Q_s}_t$

Taking the integral then expectation, you should find eventually that

$PV(t) = S_t(\ln(St) + (r+\frac{\sigma^2}{2})(T-t))$

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