Monte Carlo valuation with log returns: European and Asian options

In short

• Price model (geometric Brownian motion): $r = (r_f - \tfrac{1}{2}\sigma^2)\Delta t + \sigma\sqrt{\Delta t}\,Z$, then $P_t = P_{t-1}e^{r}$. Prices stay positive and are lognormal.
• Under risk-neutral valuation the drift is the risk-free rate (adjusted by $-\tfrac{1}{2}\sigma^2$), each path is equally probable, and the average payoff is discounted at the risk-free rate: $V_0 = E[\text{Payoff}]e^{-r_fT}$.
• A European call depends only on the final price, so one step to maturity is enough; with many trials the estimate converges to the Black–Scholes–Merton value.
• An Asian option pays $\max(P_T - \bar P, 0)$, where $\bar P$ is the average price over its life: it is path dependent, so each path needs every daily price.
• Annual volatility scales to a step of $\Delta t$ years by $\sigma\sqrt{\Delta t}$ (daily: $\sqrt{1/252}$).
• More trials mean more precision but more run time; a single path tells you almost nothing.