VaR and Expected Shortfall for Managing the FX Swap Risk

In my recent post on Garman Kohlhagen option values, I presented the probability density (in the risk neutral measure) for the Garman Kohlhagen model.

The above plot shows the probability density of the FX value in 5 years after the start date, when we start at F0=1.65, when the interest rate differenct between CHF and EUR is 1.5% per year and the annual volatility is 2.5%. The y-axes is drawn at the the magic 1.54, meaning that left of the axes the City has to pay additional coupons.

The vertical line (at 1.39194) is chosen in such a way that the filled area measures exactly 0.05. Being a probability density, the complete area (from zero to infinity) has to be 1.
Therefore, the value F=1.39194 (at T=5, with vol=2.5%) is that value for which 5% of the FX random walk results are worse (from the point of view of the city) and 95% are better. This is the definition of the 95% Value at Risk.

In that case, the interest rate (at T=5) the City has to pay is
VaR=(1.54-1.39194)/1.39194 = 10.637%
(plus the basis fee of 0.065%).

The filled area describes those cases that are worse than the 95%VaR, and the 95% expected shortfall is defined as the expectation of losses given that the outcomes are worse than the 95% VaR. Expected shortfall is used to study the tail influence in risk management.

Here we obtain for the expected shortfall (at T=5, vol=2.5%):
ES=13.28% (plus basis fee).

In both cases (VaR and ES) we did not discount the future payments.

Obviously, these extreme outcomes (95% are better) depend heavily on the volatility and on the initial value of F. Remember that in 2011 extreme (historical) volatilities of 16% were observed.

Next: Value at Risk and Expected Shortfall for the combination of 20 coupon payments instead of one isolated coupon.

Should Quants Learn More About Deep Learning?

A few months ago I asked should quants learn more about machine learning?

Last month Google acquired DeepMind Technologies with the interest in Deep Learning. It seems that in the view of Google deep learning deals with the use of neural networks to build powerful general purpose learning algorithms.

Some writers distinguish deep learning from machine learning, especial because its was less explicit and not supervised, but unsupervised.

Cross Country Skiing and Quant Finance

I returned from the Antholz Valley yesterday. It was a fantastic week. The map above show the tracks of the Antholz Biathlon Centre. At the world cup event tracks 5 or 6 are used by the top female and male biathletes.

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

Garman Kohlhagen Analyzed

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.

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

Cool Business - My Relation to the Next Future

I could not resist and take a week off enjoying cross country skiing in the Antholz Valley with its Biathlon Center offering professional trails - the same as used at the Biathlon Wold Cup (again).

In seeking optimal risk it is important to select the right trails, length and duration, steepness and speed to have fun, but avoid hazardous actions.

Getting Numbers for the Swap

Today in the late afternoon, I had the privilege to talk (as announced in earlier posts) at the Johannes Kepler symposium, and I really enjoyed it. A very heterogeneous audience of maybe 60 persons - academic staff, bank professionals, colleagues, consultanta, interested high school students, lawyers, personal friends, students (in alphabetical order) - gave me a good time and a few challenging discusiions.

When valuating the swap, I tried to use only publicly available data. I estimated volatilities from historical data, obtained Libor rates from the Federal Bank of St. Louis and exchange rates from the Austian National Bank.

The historical volatility (one year moving window, no fading memory) of the EUR CHF exchange rate had lowest values around 2 percent and highest values of 16%.

FX volatility estimated from historical data.

Observe that the collapse of Lehman Brothers led to not so high volatilities as the EUR crisis did. From the beginning of the Swiss National Bank intervention to support a EUR=1.2 CHF rate, volatilites became very low again.  

Symbols or Numbers - Pure vs Applied Maths?

When in London, we enjoyed a meeting with Paul Wilmott. Among other things we talked about mathematical creativity and whether computer based learning can create the creativity. And about the creativity in applied mathematics, that is often questioned by pure mathematicians.

Mathematical Quality Still Matters - Multiple Feedback in London

Is there a difference between quant finance 11 years ago and now? Some say, it became more boring, because the simplification of instruments moved emphasis away from modeling to data management. I always disagreed and the week in London confirmed: modeling, and correct model solving matters more than ever.