Logarithmic stock payoff
= Logarithmic stock payoff
{title2=$f(S)=S\log S$}
Under the <Black-Scholes model> with interest rate $\rho$, volatility $\sigma$ and remaining time $\Delta$, <risk-neutral pricing> gives $V(S)=S[\log S+(\rho+\sigma^2/2)\Delta]$. This follows by differentiating the lognormal exponential moment. Its <option delta> is $\log S+1+(\rho+\sigma^2/2)\Delta$ and its second stock derivative is $1/S$.