Pricing a European call by simulation

Outline how you would use Monte Carlo simulation to value a European call option.

Simulate risk-neutral terminal prices, average the discounted payoffs. Draw $Z_i\sim N(0,1)$, set $S_T^{(i)}=S_0e^{(r-\sigma^2/2)T+\sigma\sqrt TZ_i}$, compute $\max(S_T^{(i)}-K,0)$, and take $e^{-rT}$ times the sample mean. The standard error is the sample standard deviation over $\sqrt n$, so accuracy improves only as $1/\sqrt n$; for path-dependent payoffs simulate full paths on a time grid.