= Log-call option
{title2=$(\log(S_T/K))^+$}
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 $V=e^{-r\tau}[m\Phi(m/a)+a\phi(m/a)]$, 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 $S_T\log S_T$.
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