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.