If financial markets behaved in the way assumed by the Black-Scholes option pricing model, crashes would never happen. This is because at the heart of the Black-Scholes model is the assumption that price changes are normally distributed, i.e. that they fit the bell curve. However, as anyone who experienced Black Monday or the 1998 events in U.S. dollar/Japanese yen knows, the risks that are disregarded by the model cannot be ignored in real life.
The volatility smile provides an ad-hoc correction to the model for option traders, but the true problem is simply that the normal distribution is the wrong shape for calculating the probability of large moves. These are precisely the events a prudent risk manager must consider to answer the important questions:
* What is the probability of a given loss, even when no such loss has ever occurred before?
* What is the level of loss at a given probability?
TAIL RISK
The tails are the parts of a distribution that deal with these extreme moves. True financial distributions of returns tend to be more sharply peaked and have longer tails than the normal distribution predicts. This means that very small moves and very large moves are more likely than the normal distribution says they should be, while medium-sized moves are less likely. For example, by comparing daily returns in the dollar/yen foreign exchange rate to the best fit normal distribution, we can see that:
(1) Very small moves happen more often than the normal distribution would indicate.
(2) Extreme moves like the largest ever move (-5.5% on Oct. 7, 1998) have a tiny chance of happening, according to the normal
distribution--about one day every 10 million years!
(3) However, extreme moves happen far more often than the normal distribution indicates.
WHAT DISTRIBUTION SHOULD WE USE?
While the normal distribution differs from true financial distributions in several respects, there is only one place where the difference becomes a critical problem. This is in the tail where the largest moves lie.
Luckily there is a branch of mathematics, called Extreme Value Theory (EVT), which is especially designed to deal with extreme situations like the large market moves found in tail distributions. EVT shows that all far tail distributions found in the markets can be well fitted by the Generalized Pareto Distribution (GPD). So, it does not matter if the precise nature of financial distributions is unknown, we can always use the GPD to fit the tail.
To illustrate this improved fit, we home in on the dollar/yen loss tail in Figure 1. We can see that the GPD fits the yen data very well, even for the largest ever daily down move of 5.5%. The normal distribution would have told us to expect this kind of move only once in 10 million years, while the GPD give a likelihood of once every 19 years--far more consistent with history.
EXAMPLE OF RISK ANALYSIS FOR DOLLAR/YEN DAILY MOVES
The GPD will not be of much use for everyday risk analysis if all we can do with it is calculate the probability of very large moves. We also need to be able to use it in less extreme situations. A good example of such a situation is a move of 2% or more in dollar/yen. A move of this size in a day is an important event, but it happens frequently enough that market makers cannot afford to ignore the possibility that it might happen. How often would we expect this kind of move to occur?
In Table 1, we show how likely different moves are in dollar/yen. The first column tells us how likely the moves are as calculated by the normal distribution. The second uses the GPD to get a better estimate. The third shows us how often the moves actually do occur. For clarity the figures are approximated to familiar timescales.
The results for a 2% move and a 3% move are particularly interesting. The normal distribution gives a very unrealistic picture for risks that are frequent enough to need precautionary measures.
OTHER APPLICATIONS OF EXTREME VALUE THEORY AND THE GPD
One of the most useful features of the GPD is that it can be used to say how likely losses are, even when no such losses have ever occurred. It can be used in all situations where extreme moves occur, to give an estimate of how likely such moves are. Examples of other situations where it could be used to calculate loss likelihood are:
* Any forex rates (including emerging markets)
* Equity indices
* Interest rates
* Portfolio P/L
* Insurance data
* Balance sheet simulation.
SUMMARY
* The normal distribution can severely underestimate the probability of large moves
* For risk purposes, it is critical to know the tail of the distribution
* The GPD fits the tail distributions of financial data to a high accuracy, much better than the normal distribution
* Risk analysis using the GPD can give accurate loss estimates even when such losses have not yet occurred.
This week's Learning Curve was written by Jessica James& Chris Attfield in the strategic risk management advisory group at Bank One in London.
