m theory - GARCH Models, Fat Tails, & the Behavior of Financial Markets

January 23rd, 2008 · No Comments

m theory -

For those who do not like mathematics, turn away now!

If you have been following the financial markets, you will see that the volatility has shot through the roof and most of the indices are deep in the red. One of the crucial ways of modelling the behaviour of financial markets is to use GARCH models (Generalized AutoRegressive Conditional Heteroscedastic) to model the time series with time varying volatility. For example, these models are particularly useful for modelling when the distribution has fat tails.

What are fat tails? Well, we are not talking about sheep here, it is just a statistical term for a bit of the probability of certain events happening. Confused? So am I. Let me try to explain, if you take your classmates and then arrange them in the following manner in a line. Plonk the average in height people in the middle and cluster them according to the number. So you have a big thick clump in the middle. On the right, you have very few tall people and on the left, you have very few short people. In general, take anything in nature, you would get this “normal” distribution of “stuff”. It looks like the bell curve. Size of lemons. Amount of hair. So if you take a bunch of lemons or a bunch of guys, you will find that the majority will be around the mean, with progressively lesser numbers as you move away from the mean. At the far end, you will have one hirsute gorilla and on the other hand, you will have Apurva Agnihotri (chap looks like a boiled potato to me!). In other words, the tails, or ends of the distribution are for very rare occurrences.

So how does it matter? Well, if you were trying to figure out what is the size of next lemon you are going to pick up, then you need to have an idea about how does the distribution lie. If you know the distribution, then you can predict, with a fair degree of accuracy, as to what is the size of the next lemon you pick up. (Replace lemon with head hair, ladies, and you see the usefulness of this theory - helps you avoid lemons). Similarly, in the financial markets, knowing the distribution helps us to know how the markets will behave. Each market, product, currency, stock etc. has its own distribution and it changes in time.

And boyo, do we have fat tails or do we? For example, we saw events which were 25 standard deviation events last summer. This means that the last time this could have happened, we were covered in hair, used bone implements and grunted at each other. In other words, these events are frankly not predictable and if you did predict it using standard mathematics, these would only occur once every zillions of years. Instead of th
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