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

Question

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 1

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

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