Is regime-switching a cure for equity fat tails?
Posted on 17-08-2009 | 0 comments
The standard textbook model for equity returns is one which assumes that the (log) equity return in excess of the risk-free rate is normally distributed with a given mean and volatility. It is well known that this model fails to capture some of the empirical properties of equity returns, in particular the observed likelihood of extremely low (and high) returns will be much higher than predicted by the model (the so-called equity ‘fat-tails’). It also does not generate the bunching through time of extreme returns observed in real-world markets.
The inability of the Normal distribution to realistically describe observed behavior has a number of results:
- The model is generally unable to accurately price equity options with different strike prices at the same time; the ‘correct’ volatility to use in order to reproduce a market price will tend to vary by moneyness of the option, to compensate for the mismatch between ‘true’ and modelled equity return distributions. The model may hence not be suitable for the market-consistent valuation of certain liability profiles.
- If the mean and volatility of the normal distribution is calibrated to produce realistic average returns and return volatility, the tail percentiles of the equity return distribution are unlikely to accurately reflect historical data. Understating the likelihood of extreme events may be particularly problematic for real-world modelling work where the tails of the distribution have a material impact on the result (such as 1-in-200 Value-at-Risk calculations). The model only has two parameters so we can only fit exactly two characteristics of the real-world distribution.
A regime-switching equity model is a relatively simple technique that attempts to capture fat-tailed distributions, and is one that Barrie & Hibbert have employed in the past.
The basic idea is that, rather than describing equity returns with one normal distribution, we use a combination of two (or more) normal distributions, each with a different volatility and mean. This fits with the intuitive idea that equity markets ‘flip’ between states of low volatility (which prevails most of the time) and high volatility which occurs less frequently. Most importantly, calibrations can be developed that provide a relatively accurate fit to historical equity return data.
However, there are some important drawbacks; the assumption that volatility switches between two (or more) distinct regimes rather than changes continuously is unrealistic and means that projected option prices simply switch between two different states. Introducing further states can provide more realism but introduces a significant degree of complexity (and subjectivity) into the calibration process.
It is also not possible to capture the so-called ‘gearing effect’ where poor equity returns tend to be associated with increases in equity volatility. For these reasons, a regime-switching model would not be an ideal choice for anyone performing a real-world projection of a market-consistent balance sheet, particularly if using a replicating portfolio as a proxy: this is because the model would produce unrealistic distributions for the prices of any replicating assets that are valued using equity volatilities, such as equity options.
Note also that where the model is used for projecting at different time increments, a different calibration must be provided for each set-up. These calibrations will not normally be consistent with each other.
Finally, it may be necessary to model more than one risk driver using a regime-switching process. For example, a domestic equity index and an overseas equity index may both follow a regime-switching equity model, each with a different calibration. Should these regimes switch independently of each other, at the same time or with some specified correlation?
At Barrie & Hibbert we tend to favour models where equity volatility is itself stochastic, which provide a richer description of volatility behaviour that can also be correlated to other economic variables, such as equity returns themselves. The addition of random jumps in the equity price, where the timing and size of the jump is also stochastic, allows us to generate fat-tailed equity return distributions as well as realistic distributions of option prices.
This is the approach adopted in our recently-developed Stochastic Volatility Jump Diffusion (SVJD) equity model. Whilst any model is ultimately a simplification of reality and model choice is sometimes subjective, experience has taught us that there are compelling arguments why a model such as SVJD, rather than regime-switching, should be the model of choice for anyone wishing to capture the key risk drivers associated with equity exposure.
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