Financial returns often contain heavier tails and more asymmetry than the Normal distribution can describe. The Student-t and Skew-t distributions provide progressively more realistic ways to model them.
The Normal distribution is an excellent mathematical starting point. It is simple, intuitive and completely described by its mean and standard deviation.
But financial returns often contain extreme observations that occur more frequently than a Normal model predicts.
They may also be asymmetric, with unusually large losses occurring more often than unusually large gains.
The Normal distribution is symmetrical around its mean.
The further we move from the centre, the less probable an observation becomes. Extremely large movements therefore become very unlikely.
Positive and negative movements are treated symmetrically.
Extreme observations are relatively rare.
The distribution is symmetrical.
The Student-t distribution introduces heavier tails while remaining symmetrical.
The key parameter is the number of degrees of freedom, usually written as ν.
A high number of degrees of freedom makes the Student-t distribution look increasingly like the Normal distribution.
Lower degrees of freedom create increasingly heavy tails.
The difference becomes particularly obvious when we compare the probability of extreme observations.
The centre of the distributions can look very similar while the tails behave very differently.
The degrees of freedom parameter controls how quickly the tails of the Student-t distribution decay.
With only a few degrees of freedom, very large observations remain considerably more plausible than under the Normal distribution.
As ν becomes larger, the Student-t distribution approaches the Normal distribution.
Heavy tails and a much greater probability of extreme observations.
The distribution becomes increasingly similar to the Normal distribution.
Student-t solves one major problem: fat tails.
But it does not solve another: asymmetry.
The standard Student-t distribution remains symmetrical. Large positive and negative movements are treated in the same way.
For some financial assets, this is still too restrictive.
The Skew-t distribution extends the Student-t distribution by introducing asymmetry.
It can therefore capture two important characteristics of financial returns:
This makes the Skew-t distribution particularly interesting for financial modelling.
The following simulation allows us to alter both the tail thickness and the direction of skewness.
We can now see the progression clearly.
Symmetrical with relatively low probability of extreme events.
Symmetrical but better able to represent extreme observations.
Can represent both heavy tails and asymmetry.
Suppose we use a Monte Carlo model to simulate thousands of possible investment paths.
If we draw returns from a Normal distribution, extreme losses may be underrepresented.
A Student-t distribution can increase the frequency of extreme events. A Skew-t distribution can additionally make the downside behaviour different from the upside behaviour.
This is where the mathematics becomes particularly useful.
Our pension Monte Carlo model can generate thousands of possible future investment paths. The assumptions we make about the daily return distribution directly influence those paths.
Instead of assuming that every return comes from a simple Normal distribution, we can fit a more realistic distribution to observed financial returns.
We started with probability and distributions. We have now reached the point where those ideas can be used directly in a financial simulation.
The next challenge is to understand how volatility itself changes through time — leading naturally to GARCH and regime-switching models.