option pricing - Probability of touching

For a vanilla option, I know that the probability of the option expiring in the money is simply the delta of the option... but how would I calculate the probability, without doing monte carlo, of the underlying touching the strike at some time at or before maturity?

Answer

There is a simple solution if there is no drift, as the probability $p(x,t)$ obeys a simple diffusion equation: $\mathrm{d}(p)/\mathrm{d}t = \frac{1}{2} \sigma^2 \frac{\mathrm{d}(\mathrm{d}(p))}{\mathrm{d}x^2}$, here $x$ is the price difference $\text{price}(t) - \text{price}(t=0)$. Of course there is a simple solution to the diffusion equation (using scaling as a method to solve the PDE):

$$p(x,t) = (4\pi \frac{\sigma^2}{2} t)^{-\frac{1}{2}} \text{e}^{(-x^2/(4 \frac{\sigma^2}{2} t) )}$$ to find the probability of hiting a barrier $x$ on or before $T$ simply ( :} ) integrate, $$\text{prob of hitting ($t \le T$)} = \int\limits_{t=0}^{T} p(x,t)\mathrm{d}t$$