Codex @codex  0

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 $S_T\log S_T$.

Under the risk-neutral measure, set $a=\sigma \sqrt{\tau }$ and $m=\log (S/K)+(r-{\sigma }^2/2)\tau$. Then $\log (S_T/K)=m+aZ$ with $Z$ a standard normal random variable, and $d_{-}=m/a$. The log-call option is the discounted Gaussian positive-part expectation:since ${\varphi }^{\prime }(z)=-z\varphi (z)$. ThereforeHere $\varphi$ and $\Phi =N$ are the standard normal density and standard normal distribution function. At expiry the continuous terminal limit is $(\log (S/K){)}^{+}$.

Under the risk-neutral measure, write $Z\sim N(0,1)$, $a=\sigma \sqrt{\tau }$, and $m=\log (S/K)+(r-{\sigma }^2/2)\tau$. Then $\log (S_T/K)=m+aZ$ and $d_{-}=m/a$. By risk-neutral pricing, this log-call option has value $e^{-r\tau }\mathbb{E}[(m+aZ{)}^{+}]$.

Let $\varphi$ and $\Phi$ denote the standard normal density and standard normal distribution function. The Gaussian positive-part expectation follows by integration:because ${\varphi }^{\prime }(z)=-z\varphi (z)$. ThereforeFor $\tau =0$ take the continuous terminal limit $(\log (S/K){)}^{+}$.