Fractal Financial Markets
Are financial markets fractals? Financial markets are often modeled with a series of small normally distributed (Gaussian) changes. Such a model has great theoretical appeal, and is the core of other models, e.g. for measuring risk and pricing options. There is only one problem – capital markets clearly are not Gaussian random walks. There are a large number of arguments both against the Gaussian assumption and against the random walk assumption. Some of those argument can be explained away, but others can’t. One of the most compelling arguments against a Gaussian random walk is that markets appear to have fractal structure.
A fractal is a geometrical structure that is self-similar when scaled. A branch of a tree is often used as an example. The branch is similar to the whole tree, and if you break a twig off the branch, the twig is similar to the branch. In a true, mathematical, fractal, this scaling goes on forever, but in all real systems, there is a largest and smallest scale which exhibits fractal behavior.
A fractal always has a fractal dimension. The fractal dimension tells us what happens to the length, area or volume of the fractal when you enlarge it. Think of the length of a jagged shore line. If you measure it on a map, you may not see the small bays. At the other extreme, if you try to measure it with a ruler, you will see every stone at the shore. The smaller the structures that you measure are, the longer the shore line will seem. The fractal dimension will tell you how much longer it will become when you measure smaller structures.
A straight line, or a smooth curve have a fractal dimension of one. A filled circle has a fractal dimension of two. A fractal that you can draw on a piece of paper has a fractal dimension between one and two. (To be completely truthful, it could also have a fractal dimension below one, but that is beyond our purposes for now.)
A Gaussian random walk has a fractal dimension of exactly 1.5. And here comes the interesting part ¢ capital market data tend to have a fractal dimension of 1.4. This result does not tell us much about what capital markets look like, but it does tell us that they are not Gaussian random walks. They could still be Gaussian or they could still be random walks, but they can¢t be both.
The fractal dimension of a financial time series can be measured by constructing a bar chart and measuring the area of all the bars in the bar chart. This area in itself does not tell us anything. We are interested in how the area scales when we construct bar charts with different time frames, such as daily, weekly, monthly and yearly bar charts. If we had a Gaussian random walk, we would expect the average height of the bars to scale according to the square root of the time frame. In practice, the bars grow faster than that when we increase the scale. The scaling factor is about the same for different time frames ¢ this is typical of a fractal.
In order to be able to explain the fractal scaling we will have to give up either the Gaussian hypothesis or the random walk hypothesis. Either risks in financial markets are much larger than would be expected by the Gaussian hypothesis or the market is prone to trending. In practice, it is hard to tell which alternative is true.
If we give up the Gaussian hypothesis, we almost certainly end up with something called stable probability distributions. The normal distribution is a special case of a stable distribution, and it is the only one that has finite variance. All the other stable distributions have infinite variance. Often, risk is equated with variance, and to say we have infinite variance would mean the risk would be unlimited. This is of course not true. With stable distributions, the variance may be infinite, but the average size of, say, a daily move is not. But what happens when we don¢t have a normal distribution is that large moves become more likely, and very large moves become very much more likely. What would also happen is that theories such as CAPM would become complete non-sense, as the variance is unlimited.
We know that large moves in the capital markets are more common than would be suggested by a Gaussian random walk. This could be explained in other ways than unlimited variance, such as increased volatility or even strong trends. However, the fact that the large moves do exist as would be expected if they were to explain the fractal dimension is a strong indication that variance in the capital markets is indeed unlimited. (As always in practice, the fractal characteristic is only evident within certain limits. If we were to look at a sufficiently long financial time series, hundreds or even thousands of years perhaps, we might find that variance on that scale would be limited.)
The other possibility to explain the fractal dimension is to give up the random walk assumptions. Under a random walk, all changes are independent. At every point in time, the market is equally likely to move up as to move down, regardless of how it has moved before. A fractal dimension below 1.5 could be explained if the market where prone to trending (a straight line, the ultimate trend, would have a fractal dimension of one). A trend is not a random walk, as prices are more likely to move with the trend than against. But the trending would have to be fractal ¢ there would have to be trends at every time scale, and those trends could potentially go in different directions.
We can test the trend hypothesis by taking the changes in a financial time series, e.g. the changes in a stock market index. We randomly shuffle the time series of changes and then create a new time series by adding them together again. This should eliminate the trends, and therefore should increase the fractal dimension to 1.5. In practice, we find that the fractal dimension does indeed increase, but not all the way to 1.5. However, if stable distributions were the only explanation for the low fractal dimension, we wouldn¢t expect the fractal dimension of the shuffled time series to change at all. There may, however, be other explanations than trending for the change in fractal dimension, such as structure in the volatility of the time series for example. So the results are inconclusive.
There are some excellent books written by Edgar E. Peters that argue the case of fractal structure in capital markets. They are a very good introduction to chaos theory applied to capital markets, but I would recommend anyone reading them to check out the math for himself. I am sure that he didn¢t make up the results. It is just that some of them require methods, assumptions and lines of reasoning that I do not think hold up to scrutiny. I am especially skeptical about the R/S-analysis that he is employing. While it might be good for certain purposes, I do not think that it is an appropriate general purpose tool.
Anyway, Peters has this theory, that in the intermediate run, the fractal structure is explained by what he calls fractal noise, i.e. stable probability distributions and infinite variance. In the long run the fractal structure is explained by what he calls noisy chaos, i.e. fractal trends with noise. He might be right. In the long run, capital markets should be determined by the economy, and to a large extent, the economy is deterministic, although very unpredictable, just as it should be in order to drive a fractal system. And there is certainly random noise affecting the economy all the time.
Implications of Fractal Structure in Capital Markets
The two most obvious areas where fractal structure in capital markets would be of interest is in option pricing and risk management.
Option pricing are traditionally priced using Black-Scholes formula, which is a special case of arbitrage theory. Arbitrage theory makes use of the fact that the fractal dimension is exactly 1.5 to create perfect hedges for options and other derivatives. Unfortunately, this is not possible with other fractal dimensions. Mathematically, the It¢ formula will not work. Conceptually, this means that it is still possible to hedge away the risk of small changes, but the risk of large moves will be impossible to hedge away. This is also an experience of the arbitrage community. So called delta hedging of written options is perceived as much more risky than it should be according to normal arbitrage theory.
It is difficult to generalize arbitrage theory to fractal markets. One way would be to try to create close to perfect hedges for carefully selected portfolios of options rather than for individual options. That process would arrive at pricing relationships between options, rather that at a fixed price for each option. Another way would be to use an expected current value as a proxy for the option price. Under arbitrage theory, the price of an option is the expected value of the option at expiry, assuming that there is no compensation for risk for the option (or the underlying stock). It would be possible to generalize this to the fractal case, but assuming an arbitrary market price for risk. In most cases, the price of the options would not be very much affected by the size of the market price for risk (assuming reasonable values) so this would be a pretty stable pricing model. In addition, it is intuitively appealing.
One general observation about option pricing could be made ¢ far out-of-the-money options should be much more expensive than predicted by the Black-Scholes formula. This is also the case in practice.
As mentioned before, risks are often associated with variance or volatility. In a fractal market, variance and volatility are unlimited. This means that risks are much larger than would be expected by standard theory.
One way around the problem with unlimited volatility would be to use the c parameter in the stable distributions as a measure of risk instead of volatility. For the Gaussian distribution, the c parameter is actually equivalent to volatility (with a scale factor). If all assets in a portfolio have the same fractal dimension, optimal portfolio selections exists, that are similar to the Gaussian case.
The most important part of risk management is to select the amount of risk that should be assumed given expected returns. One way to do this is to use a utility function and optimize the expected utility (in practice, position sizes much less than the optimal sizes must be used, as we only have estimates of the parameters used). With fractal markets, some unbounded utility functions could easily produce unreasonably large risks. Therefore, with fractal markets, it becomes much more important to pay attention to the utility function. If unlimited utility functions are used, the implications must be carefully considered.
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