Extreme Value Theory (EVT) is the mathematics behind extreme events. Some important results have analogies with the Central Limit Theorem, but instead of being about averages they are about extremes. Of course, whether one should even be talking about probabilities when talking about crashes is another matter. It's probably safer to look at worst-case scenarios.

Example

(Taken from McNeil, 1998.) Fit a Frechet distribution to the 28 annual maxima of the SP500 index returns from 1960 to October /6th 1987, the business day before the '87 crash. In this dataset the largest fall was 'just' 6.7%. Now calculate the probability of various returns. For example, a 50-year return level is the level which on average should only be exceeded in one year every 50 years. The Frechet distribution gives the result as 24%. One business day later the index falls 20.4%.

Long answer

Modern techniques for estimating tail risk use Extreme Value Theory. The idea is to more accurately represent the outer limits of returns distributions since this is where the most important risk is. Throw normal distributions away, their tails are far too thin to capture the frequent market crashes (and rallies).

One important EVT result concerns the distribution of maxima and minima and is used in calculations such as in the example above.

If Xi are independent, identically distributed random variables and

then the distribution of x converges to

When § = 0 this is a Gumbel distribution, §< 0 a Weibull and § > 0 a Frechet. Frechet is the one of interest in finance because it is associated with fat tails.

The role of theorems about extremes is similar to that of the Central Limit Theorem for sums/averages.

References and Further Reading

McNeil, A 1998 On extremes and crashes. Risk magazine January 99

What is CrashMetrics?

Short answer

CrashMetrics is a stress-testing methodology for evaluating portfolio performance in the event of extreme movements in financial markets. Like CAPM it relates moves in individual stocks to the moves in one or more indices but only during large moves. It is applicable to portfolios of equities and equity derivatives.

Example

Your portfolio contains many individual stocks and many derivatives of different types. It is perfectly constructed to profit from your view on the market and its volatility. But what if there is a dramatic fall in the market, perhaps 5%? What will the effect be on your P&L? And if the fall is 10%, 20%... ?

Long answer

CrashMetrics is a very simple risk-management tool for examining the effects of a large move in the market as a whole. It is therefore of use for studying times when diversification does not work.

If your portfolio consists of a single underlying equity and its derivatives, then the change in its value during a crash, 8TI, can be written as

where F() is the 'formula' for the portfolio, meaning option-pricing formulae for all of the derivatives and equity in the portfolio, and 8S is the change in the underlying.

In CrashMetrics the risk in this portfolio is measured as the worst case over some range of equity moves:

This is the number that would be quoted as the possible downside during a dramatic move in the market. This downside can be reduced by adding derivatives to the portfolio in an optimal fashion. This is called Platinum Hedging. For example, if you want to use some out-of-the-money puts to make this worst case not so bad, then you could optimize by choosing X so that the worst case of

represents an acceptable level of downside risk. Here F*()is the 'formula' for the change in value of the hedging contract, C is the 'cost' associated with each hedging contract and X is the quantity of the contract which is to be determined. In practice there would be many such hedging contracts, not necessarily just an out-of-the-money put, so you would sum over all of them and then optimize.

CrashMetrics deals with any number of underlyings by exploiting the high degree of correlation between equities during extreme markets. We can relate the return on the ith stock to the return on a representative index, x, during a crash by

where Ki is a constant crash coefficient. For example, if the kappa for stock XYZ is 1.2 it means that when the index falls by 10% XYZ will fall by 12%. The crash coefficient therefore allows a portfolio with many underlyings to be interpreted during a crash as a portfolio on a single underlying, the index. We therefore consider the worst case of

as our measure of downside risk. Note that this is really just a function of the one variable x and so it is very easy to plot the change in the portfolio against x, the return on the index.

Again Platinum Hedging can be applied when we have many underlyings. We must consider the worst case of

where F is the original portfolio and the Fks are the M available hedging contracts.

CrashMetrics is very robust because

• it does not use unstable parameters such as volatilities or correlations

• it does not rely on probabilities, instead considers worst cases.

CrashMetrics is a good risk tool because

• it is very simple and fast to implement

• it can be used to optimize portfolio insurance against market crashes

CrashMetrics is used for

• analysing derivatives portfolios under the threat of a crash

• optimizing portfolio insurance

• reporting risk

• providing trading limits to avoid intolerable performance during a crash.

References and Further Reading

Hua, P & Wilmott, P 1997 Crash courses. Risk magazine 10 (6) 64-67

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

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