On January 15th, the one-day return to holding Swiss Francs from a Euro perspective was 16.9%. This is a high one-day return for any currency pair, but appears cataclysmic given the extremely low return volatility of the Swiss Franc from a Euro perspective in recent months. This one-day jump was a “239-sigma event” meaning that the magnitude of this return was 239 times the recent return volatility (using a 90-day historical estimate of volatility). In fact, in the period just before the sudden jump, the sample volatility of this exchange rate was even lower. Using a shorter 20-day volatility estimate, the sudden jump was a 400-sigma event.

It is interesting how closely the time-series behaviour of this exchange rate matches the predictions of Krugman’s 1991 model of a government-implemented exchange rate limit, in which traders credibly believe that the authorities will prevent the exchange rate from piercing the exchange rate limit. As the fundamentals for the exchange rate made the Swiss Franc greatly undervalued, the traded exchange rate settled down just near the government-imposed limit, with very low volatility. And then suddenly the credible promise became a non-promise.

Chalk one up for Krugman, in terms of the elegant fit between his theoretical model and this recent market experience. Several forex trading firms went bust, but they should have had better risk management systems.

*Related*

Daniel Gros pointed out, a few years back, that the Swiss exchange rate cap was a prop to the Euro.

http://www.voxeu.org/article/overlooked-currency-war-europe

In some of the FX shops the business model more or less requires them to retain exposure to potential, but unlikely within tenure in post, large losses.

The correct strategic call on situations like the Swiss CB peg is usually to be patient enough to stay away, but the business imperitive motivates people to pick up the pennies.

@unfeasiblycharming – Your analysis sounds correct to me. I would be careful to call it a “cap” rather than a “peg” but that is me being academic.

The FX shops had a business model that offered leverage of up to 500 to dumb money punters of whom 85% became defunct financially and had to be replaced by new punters. They were too busy in the moment to see the flaws in their risk modeling.

I wonder how good S&P’s models are now with a likely pickup in volatility on the horizon.

So it was a 200+ sigma event ….

But you then say that FX firms should have had better risk controls?

It was massively irresponsible by the Swiss; considering they gave official reassurances regarding the maintenance of the Floor just one week previously, the move would be akin to Janet Yellen hiking US rates by 300 basis points next week – just because she felt like it. Such a move would blow up the World – and no “risk controls” would catch that one either.

@Be a Debaser — It was a 200-sigma event using a rolling window sample estimate of volatility. A risk management operation relying on that for risk control is virtually bound to fail eventually. There are infrequent large jumps in forex markets, and the sample volatility was also pushed down by the Krugman-type boundary effect. When a exchange rate hits a credible cap its sample volatility falls to near zero.

Perhaps I am being pedantic here but doesn’t a claim like 200 sigma require tge underlying to be normally distributed. And don’t we all know already that asset prices are fat tailed. So can we stop with the x-sigma nonsense?

@Garo — An x-sigma event is just the ratio of the realized return to the estimated standard deviation. Having a standard deviation and computing a ratio does not require that the return is normally distributed. This is not even the true standard deviation it is just the recent sample standard deviation. The true standard deviation (of the non-normal return distribution) is much higher than the recent historical volatility (the sample standard deviation over a period with no jumps). A claim of 200 sigma absolutely does NOT require a normal distribution. Quite the opposite — it would show non-normality if such a test were needed.

@ Gregory

how often would a 200 sigma event be expected to happen? Once in every what ?

and is it as bad as one Goldman on the scale of financial ineptitude ie

“We were seeing things that were 25-standard deviation events, several days in a row”

From Chebychev’s Inequality if one uses the true standard deviation then a 200-sigma event can only happen on average once every 40,000 days. But under the Krugman process + nonzero prob of collapse of the cap the sample standard deviation in a sample with no jumps is badly biased downward, particularly when the rate is smoothly pasted near the cap. Using that theory the 200 sigma event using sample standard deviation does not seem so low-prob.

@Gregory…but wouldn’t that 40,000 days only be true if you have a real daily sigma. And if you measure over a short term, with low variance, do you?

@Hugh Sheehly — Yes, exactly; that was what I meant in my last comment just above yours, but perhaps it was stated too obscurely. If you sample measure over a short term, with this type of process and no jumps, it is not the true (square root of) variance.

@gregory Yes, thought so. But wanted to be sure I was getting good point.

I guess, putting it even more bluntly, Chebychev works on the data after you have it. Not before. It doesn’t limit what the next data point will be. Or something.

And what I was trying to say was “getting it right”…..

Darn spell check

@ GC

I think Garo has a point. When that quip about 7 sigma events was used rhetorically it was in the context of the Normal distribution – where it is a 1 in a million, million chance. On Chubby’s formula 7 sigmas merely sets the upper bound at 1 in 49 which is hardly so rhetorical. Seafood, my Excel is incapable of evaluating a 200 sigma Normal event but my guess it would cover the whole universe of space time events so far and still give negligible chance.

Anybody who modelled CHF markets on anything like a normal distribution shouldn’t be let out. A naïve model would say that on the one hand we have a Normal type model of the behaviour of market participants but overlaid by a model of the behaviour a dominant market setter who makes one off decisions from time to time, the last one back in Sept 2011. Now in the run up to last week’s move any model which used short term estimates of the parameters would have no sightings of this second factor and would thus rate its chances as Nil. The reverse of course is true today.

But the error would be in the model for the distribution not in the estimate of the single parameter. As it happens Chubby would allow a model distribution faithful to the sample sigma which put the chances of the jump at 1 in 13 years, which does not seem too outlandish.

@ seafood

Actually one can put bounds on the probability of a 200 sigma Normal Distribution event.

The probability of between 200 sigmas and 1,000,200 sigmas is less than 1000000 x (1/2pi)* exp(-20,000). Now exp (-1) is less than 1/3 x 10 so that gives our probability as being less than 10^-6,000 and when we think that the size of the hydrogen atom is 10^-10 metres I think my earlier guess actually overstates this probability.

@the second

I don’t think the normal distribution is safe when Central Banks are trying to keep a lid on volatility.

Trying to objectify risk is stupid at times like these given the emotion that drives markets.

let us just conclude by saying that if you were using 90 day HV to write puts on EURCHF Curncy you got what you deserved.