Recently, I presented the Garman Kohlhagen stochastic differential equation (in The Future Development of Exchange Rates). When all parameters are constant, this SDE can be solved by applying methods from Ito calculus.
When the initial exchange rate is F0 at time t, then the probability density (in the risk-neutral measure) for the stochastic variable F (at time T) us given by
We can cisualize this (e.g.by the following Mathematica code
Here, we have set a (low) annual volatiliy of 2.5%, an interest rate difference of 1.5%, and an initial value for the exchange rate of 1.65. The plot shows the probability density for F after 5 years under these settings.
The NIntegrate command finally calculates the probability (again in the risk neutral measure) that F lies below 1.54.
Next: FX option values.