The Future Development of Exchange Rates

As described in my post We valuate THE swap, the main driver for the coupons of the structured leg that the city would have to pay to the bank is the exchange rate between the Swiss Franc and the EUR. As soon as 1 EUR is traded below 1.54 CHF (on the fixing dates; let us name this rate x), the rate to be paid would increase from 0.065% to 0.065% + (1.54-x)/x. Thus an FX rate of 1.4 would give 0.065%+(1.54-1.4)/1.4= 10.065%.

To analyze the future distribution of exchange rates, a model is needed. The most simple one is the so-called Garman-Kohlhagen model (a generalization of Black-Scholes for FX rates). It states that the random walk for the rate F satisfies
The denomination currency of the swap is CHF, therefore the domestic rate has to be chosen as the Swiss Franc rate(s), the foreigen rate as the EUR rate(s), independent of the accounting currency of the city. As soon as a present value is obtained (in CHF), it can be converted to the EUR present value by applying the spot FX rate. As usual dW is the increment of the Wiener process and sigma the volatility of the FX rate (see also Getting Numbers for THE Swap.

As we have seen in A Short History of Floating Rates, historically the CHF rates were always lower (at least since the EUR exists) than the EUR rates, leading therefore to a negative drift rate (domestic minus foreign rate) and therefore to the expectation that the EUR will decrease compared to the CHF. (For the specialists in measure theory: in the risk neutral measure).

When we start a scenario generation at an FX rate of 1.65, we obtain with a volatility of 2% these possible outcomes after 10 years.

Random paths as described above. 100 time steps of 0.1 years each

Next week: Distirbution properties for the Garman Kohlhagen model