FX Option Values Under Garman Kohlhagen

In my recent post, I presented the probability density (in the risk neutral measure) for the future distribution of FX rates under the Garman Kohlhagen model.

The value of a European option is then given by
(1) calculate the expected value of the option payoff under this risk neutral measure, and then
(2) discount it by the domestic rate.

As a formula, this reads as

 with V(F,T) denoting the payoff and the factor before the integral used for discounting.

Assume that F0=1.62, the rate difference (domestic CHF - foreign EUR rate) is -1.5% and the time horizon T-t is 5 years. Then the probability density as a function of the volatility sigma behaves like this.

Starting (in the animation) with a volatility of 1.9%, the cumulative density of being below 1.54 first decreases (until 6.9%) and then increases again.

Next: VaR and Expected Shortfall