Let $S$ be a log normal distribution with $\ln(S)$ distributed as $ \mathcal{N}(0,\sigma^{2})$. I need to calculate $E(\max(S-k,0))$ where $k$ is some fixed positive constant.
My attempt:
$E[\max(S-k,0)]$ = $E(S\mid S>k)\Pr(S>k)$ but how to calculate this conditional expectation.
Your derivation is correct. Further, this is the simplest case of conditioning, namely related to a constant, and we call that a "truncation", since no other random variable is involved.
Truncated densities are formulated in a simple way, involving the distribution function: if $S$ is truncated from below at $k$ with density $\tilde f$, and if $f$ is the untruncated density and $F$ the untruncated distribution function, then
$$\tilde f(s) = \frac{f(s)}{1-F(k)}, \;\;s \in (k,\infty)$$
It follows that
$$E(S \mid S>k) = (1-F(k))^{-1}\int_k^{\infty} sf(s) ds$$