金价数字 call：计算 ITM 概率

Let $G(\cdot)$ denote the gold price.
Now is time $t$, and time $T$ is six months from now.

The naive (and incorrect) step is to conclude that a volatility of $\sigma = 60$ per annum translates to a six-month volatility of $30$.

In fact, volatility grows with the square root of the term. Thus, $60$ per year translates to about $\sqrt{\frac{1}{2}}\times 60\approx 42$ per half-year.

How do we find the probability that the option finishes in-the-money, $P(G(T) > 430)$?

With $r=0$ there is no drift in the risk-neutral world, so the distribution of $G(T)$ is centered on $G(t)=400$, with standard deviation roughly $42$.