Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2015 / 2 / d / Solution

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