Financial markets do not behave the same way every day. Periods of calm can be followed by periods of extreme volatility. GARCH and regime-switching models give us ways to describe this changing behaviour.
We have already seen that financial returns can have fat tails and skewness. But there is another important feature: volatility itself changes over time.
Markets can spend weeks or months in relatively calm conditions and then suddenly experience a period of large daily movements.
This creates a phenomenon known as volatility clustering.
The chart below simulates returns where volatility moves between periods of calm and periods of turbulence.
GARCH stands for Generalized Autoregressive Conditional Heteroskedasticity.
The name is complicated, but the underlying idea is relatively simple: today's volatility depends partly on recent volatility and recent returns.
A basic GARCH(1,1) model can be written as:
The three components have intuitive interpretations.
The long-run baseline level of variance in the model.
How strongly volatility reacts to a recent shock.
How persistent volatility is once it has changed.
The unexpected return or shock observed in the previous period.
The following simulation allows us to change the parameters controlling how quickly volatility reacts to shocks and how persistent it is.
One reason GARCH models are useful is that they allow us to estimate conditional volatility.
Instead of assuming that tomorrow's volatility is always the same, the model can respond to what has happened recently.
GARCH is very good at modelling changing volatility, but it does not necessarily tell us why volatility has changed.
Imagine that markets move between fundamentally different environments:
Lower volatility, smaller daily movements and relatively stable market conditions.
Higher volatility, larger movements and much greater uncertainty.
This leads naturally to the idea of regime switching.
A regime-switching model assumes that the market can exist in one of several underlying states, or regimes.
We do not necessarily observe the regime directly. Instead, we observe returns and infer which regime is most likely.
The model also estimates the probability of moving from one regime to another.
Here we simulate a market that switches between a low-volatility and high-volatility regime.
Typical daily volatility in the simulated calm regime.
Typical daily volatility in the simulated stress regime.
Probability of remaining in the current regime.
The two approaches solve related but different problems.
Volatility responds continuously to recent shocks and gradually returns toward its long-run level.
The market can move between distinct underlying environments.
Volatility can evolve dynamically while also having different target levels in different regimes.
We can extend the idea to three states.
Low target variance and relatively small movements.
Intermediate volatility and more typical market conditions.
High target variance and much larger potential movements.
This is where the concepts become directly relevant to our financial modelling work.
A simple Monte Carlo model might assume that investment returns have constant volatility and come from one distribution.
A more realistic model can allow:
We have now travelled quite a long way from the original idea of probability.
Each step adds another layer of realism.
The important lesson is that the mathematics is not just about complicated formulas. Each model is trying to capture something we can actually observe in the real world.
These ideas provide the mathematical foundation for the more advanced financial simulations elsewhere on the site.
The next step is to take these concepts and apply them to an actual fitted financial return model.