My question is related to conditional expected value of lognormal variable (more precisely conditinal expected value of geometric brownian motion). If intitial value is $S_0$, after time $\tau$ the distribution of $S_\tau$ will be the following: $$ln(S_\tau)\sim N(ln(S_0)+(\mu-\frac{1}{2}\sigma^2)\tau,\sigma^2\tau)$$ I am interested in the conditional expected value of $S_\tau$, if $S_\tau$ is above X. $$E(S_\tau|S_\tau>X)=?$$
A possible solution for this (where $\Phi$ is standard normal cdf): $$E(S_\tau|S_\tau>X)=S_0 * exp(r \cdot \tau) * \frac{\Phi(d_1)}{\Phi(d_2)}$$ where $$d_2=\frac{-ln(X/S_0)-(\mu-\frac{1}{2}\sigma^2)\tau}{\sigma * \sqrt{\tau}}$$ $$d_1=d_2+\sigma * \sqrt{\tau}$$
Is there a "nicer", more compact closed form solution of conditional expected value? For example with only one $\Phi()$ function?
