Few people in the financial world would claim to be able to predict the markets. While there may be a few lone geniuses who can say with confidence where a rate will be after 24 hours, less-gifted analysts would be happy to be able to quantify a range in which the rate might lie after that time. This estimation of ranges and likely movements is intimately connected with volatility.
The definition of market volatility is the standard deviation of daily returns, over a specified time period. Its value lies chiefly in the fact that, unlike so many other market parameters, it varies relatively slowly over time. Thus by calculating the volatility for the last month, it is possible to produce a reasonable estimate of the volatility for the forthcoming month or so. Additionally, volatility is an essential parameter for option pricing, and so future volatility is "estimated" by market traders on a daily basis in order to value options. This estimated volatility is called "implied volatility," and may also be used as a likely indicator of volatility in the future.
The main use of volatility, apart from option pricing, is for risk estimation. While few, if any, analysts can say that a rate will go to a specific level at a specific time, it is common practice to estimate the chance of such an event occurring. Thus it becomes possible to say, for example, "Using last month's volatility, we have a 5% chance of the U.S. dollar falling below JPY105 in the next month." This is a quantified risk in fact it is a value at risk number - and thus can be used in ongoing risk calculations. If implied volatility is used, the risk manager will be estimating market risk by using the options market, which he or she may believe is a more sensitive indicator of future events than just using the historical volatility.
NORMAL DISTRIBUTION
The simplest way of using volatility to estimate future risk uses the normal distribution. This method is quick and simple, and could be implemented as follows:
1) Take a data set of market rates.
2) Calculate the daily returns (percentage changes) for a specific time period T.
3) Find the standard deviation of these percentage changes.
4) Use the normal distribution to find the probability of different events for example, there is about a 5% chance that rates will move outside 2 standard deviations in time T.
5) Assume that rates undergo normal Brownian motion to estimate these probabilities for different time periods. This entails a simple 'scaling' by the square root of time. For example, for a time period 2T, volatility will be (check)2T, for 3T volatility will be (check)3T, etc.
The primary disadvantage of this method is, of course, that rates are generally not normally distributed and tend not to undergo Brownian motion. Usually distributions of market returns tend to be fat-tailed, and volatility does not increase with time period as fast as the Brownian motion assumption indicates.
EMPIRICAL DISTRIBUTION
It is better, if possible, to use empirical distributions. This entails actually calculating the distribution from the data set and finding the probabilities from the data. Also, the root T scaling can no longer be used. Instead one needs to do a different calculation for each time period, using historical data sets of different sizes. However, empirical distributions give a much better estimate of market risk and should be used wherever possible. VaR calculations done using empirical distributions are much more reliable than those which use the normal distribution.
IMPLIED DISTRIBUTION
If desired, implied distributions from the options market may also be used. The volatilities for different time scales are extracted from the prices of options which have different maturity dates. If only at-the-money option prices are available, it is necessary to assume a normal distribution (though the Brownian motion assumption is not necessary). If the full volatility smile data is available then an implied distribution may be calculated, though this is a more complex process and is not often performed.
USING VOLATILITY CONES
Volatility cones are a visual combination of a number of different risk parameters. They can be produced using any of normal, implied or empirical distributions. We can consider the euro/U.S. dollar rate as an example. If we assume that we know nothing about direction, but are interested in the range in which it might move, then we can calculate the range in which it might lie with 95% probability for various different time periods. Thus we would look at all the weekly moves in our historical data set, and say what the 95% range was for the next week's move. We would then do the same calculation for moves for one month, two months, three months, and so forth, until we had a set of 95% ranges which covered all periods in the future of interest to us. An example of such a set of ranges is given in the table below, and graphed in the chart to the right.
It can be seen that the "cone" arises naturally from the way that the ranges vary with time period. The graph was produced using the normal distribution, using 10 years of daily euro/dollar data, where the DEM is used as a proxy for the euro for prior to the euro's launch. It could equally well have been produced using the empirical method or implied volatilities.
Volatility cones have a number of different advantages over simpler risk parameters like VaR numbers. They encapsulate far more data in a visually striking form, and give information about ranges rather than just loss limits. Moreover, their use is not limited to market rates other important parameters like sales volumes could equally well be analysed using this method. And they are simple to understand, which is one of the major reasons for the rise in popularity of the simple VaR number.
This week's Learning Curve was written by Jessica Jamesand Chris Attfield, in the strategic risk management advisory group at Bank Onein London.
