= Gaussian positive-part expectation
{c}
{title2=$\mathbb E[(m+aZ)^+]=m\Phi(m/a)+a\phi(m/a)$}
For $Z$ with <standard normal distribution>, $m\in\mathbb R$, and $a>0$, integrate $(m+az)\phi(z)$ over $z>-m/a$. The identity $\phi'(z)=-z\phi(z)$ 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>, $m^+$.
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