Hidden Semi-Markov Models, from Bruno Ombreux

March 4, 2008 |

The recent mention of GARCH reminded me of another complicated acronym which could be interesting to specs: HSMM, which stands for Hidden Semi-Markov Models.

The acronym makes it sound complicated, but it is a very simple concept:

1. The market alternates between different regimes, for instance greed and fear

2. Each regime has its own mean and variance, for instance (positive drift, low variance) and (negative drift, high variance)

3. The switch between regimes is probabilistic (with constant probabilities, which is a simplification — it is only a model, after all)

4. Regime duration, aka sojourn time, is some random variable

The fourth point is what differentiates HMM from HSMM, Markov from Semi-Markov. The later is more general in terms of sojourn time distribution.

I feel this type of model is a welcome addition, not alternative, to GARCH in the trader's toolbox:

1. GARCH fits the facts very well but does not really provide any explanation. HSMM fit the facts and provides a physical explanation, for instance alternating moods of greed and fear. Actually, empirical price behavior is incredibly well explained by this simple alternation of different regimes.

2. GARCH focuses on predicting variance. Therefore it is very useful to option traders. HSMM focus on predicting regimes. That is, they focus on the counter's nemesis: ever-changing-cycles. They are useful to all people concerned with the shelf-life of their edges.

What is the probability to be in one cycle? What is the probability to remain in this cycle? Those are questions that HSMM try to answer. Because they are in their infancy, they often provide more questions than answers. It is an open research area. But they can already shed some light on the matter, and it is not always optimistic.

The best fit to empirical decay of squared returns autocorrelations — a long and wordy statement — is provided by negative binomial sojourn times. One interpretation of the negative binomial distribution, is as a not well behaved generalization of the relatively well behaved Poisson distribution. With Poisson, mean = variance. With the negative binomial, mean < variance. If so, then it is not easy at all to forecast regime duration.

However, it is as interesting to know what one can't trade as to know what one can trade. And it is always possible that some more encouraging results could be achieved by looking at other markets, other timeframes, and other phenomena, e.g. backwardation/contango switches in energy markets.

By the way, I am reading Dr McDonnell's excellent book Optimal Portfolio Modeling. As usual, Phil is clear, clever and very right. Everybody should read the book.

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