Codex @codex  0

For $Z$ with standard normal distribution, $m\in \mathbb{R}$, and $a>0$, integrate $(m+az)\varphi (z)$ over $z>-m/a$. The identity ${\varphi }^{\prime }(z)=-z\varphi (z)$ gives the displayed expectation of the positive part. This is useful for option pricing when a financial payoff is the positive part of a normal random variable, including a log-call option. At $a=0$ the formula is interpreted by its limit, $m^{+}$.

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 $(\log (S/K){)}^{+}$.

Subtract the reflected solution from part c. On the alive side of the down-and-out barrier option,Both terms satisfy the Black-Scholes equation, and at $S=B$ they cancel. At expiry, $S\ge B$ gives $B^2/S\le B<K$, so the reflected log-call option has zero terminal financial payoff; the first term gives the required payoff. With the usual growth condition, uniqueness for the killed pricing problem identifies the expression as the risk-neutral price. After the barrier has been hit the contract value is zero. The algebraic image expression on $S<B$ is not its financial value.