 Log-call option

A European option paying the positive part of the logarithm of terminal stock price divided by a positive strike $K$. In the dividend-free Black-Scholes model, that logarithm has a normal distribution under the risk-neutral measure. The Gaussian positive-part expectation gives $V=e^{-r\tau }[m\Phi (m/a)+a\varphi (m/a)]$, where $m=\log (S/K)+(r-{\sigma }^2/2)\tau$ and $a=\sigma \sqrt{\tau }$. This differs from the untruncated logarithmic payoff and from the payoff $S_T\log S_T$.