Regime Switching Volatility Calibration by the Baum-Welch Method

Regime switching (also known as hidden Markov models (HMM)) volatility models provide a tractable method of modelling stochastic volatility. Currently the most popular method of regime switching calibration is the Hamilton filter. However, regime switching calibration has been tackled in engineering (particularly for speech processing) for some time using the Baum-Welch algorithm (BW), where it is the most popular and standard method of HMM calibration. A review of the Baum-Welch algorithm can be found in [Lev05], [JR91]. The BW algorithm is increasingly being applied beyond engineering applications (for instance in bioinformatics [BEDE04]) but has been hardly applied to financial modelling, especially to regime switching stochastic volatility models.

Unlike the Hamilton filter, the BW algorithm is capable of determining the entire set of HMM parameters from a sequence of observation data. Furthermore, BW is a complete estimation method since it also provides the required optimisation method to determine the parameters by MLE.

The outline of the paper is as follows. Firstly, we introduce regime switching volatility models and the Hamilton filter. In the next section we introduce the Baum-Welch method, describing the algorithm even for multivariate Gaussian mixture observations. We then conduct a numerical experiment to verify the Baum-Welch method’s to detect regimes for the S&P 500 index. We finally end with a conclusion.

Wiener process driven stochastic volatility models capture price and volatility dynamics more successfully compared to previous volatility models. Specifically, such models successfully capture the short term volatility dynamics. However, for longer term dynamics and fundamental economic changes (e.g. “credit crunch”), no mechanism existed to address the change in volatility dynamics and it has been empirically shown that volatility is related to long term and fundamental conditions. Bekaert in [BHL06] claims that volatility changes are caused by economic reforms, for example on Black Wednesday the pound sterling was withdrawn from the ERM (European Exchange Rate Mechanism), causing a sudden change in value of the pound sterling [BR02]. Schwert [Sch89] empirically shows that volatility increases during financial crises.

A class of models that address fundamental and long term volatility modelling is the regime switching model (or hidden Markov model) e.g. as discusssed in [Tim00], [EvdH97].

In fact, Schwert suggests in [Sch89] that volatility changes during the Great Depression can be accounted for by a regime change such as in Hamilton’s regime switching model [Ham89]. Regime switching is considered a tractable method of modelling price dynamics and does not violate Fama’s “Efficient Market Hypothesis” [Fam65], which claims that price processes must follow a Markov process. Hamilton [Ham89] was the first to introduce regime switching models, which was applied to specifically model fundamental economic changes.

For regime switching models, generally the return distribution rather than the continuous time process is specified. A typical example of a regime switching model is Hardy’s model [Har01]:

where • ϕ i and u i are constant for the duration of the regime;

• i denotes the current regime (also called the Markov state or hidden Markov state);

• R denotes the total number of regimes;

• a transition matrix A is specified.

For Hardy’s model the regime changes discretely in monthly time steps but stochastically, according to a Markov process.

Due to the ability of regime switching models to capture long term and fundamental changes, regime switching models are primarily focussed on modelling the long term behaviour, rather than the continuous time dynamics. Therefore regime switching models switch regimes over time periods of months, rather than switching in continuous time. Examples of regime switching models that model dynamics over shorter time periods are Valls-Pereira et al. [VPHS04], who propose a regime switching GARCH process, while Hamilton and Susmel [HS94] give a regime switching ARCH process.

Note that economic variables other than stock returns, such as inflation, can also be modelled using regime switching models.

Regime switching has been developed by various researchers. For example, Kim

and Yoo [KY95] develop a multivariate regime switching model for coincident economic indicators. Honda [Hon03] determines the optimal portfolio choice in terms of utility for assets following GBM but with continuous time regime switching mean returns.

where • q t is the Markov state (or regime) at time t of X(t);

• i and j are specific Markov states.

As time passes the process may remain or change to another state (known as state transition). The state transition probability matrix (also known as the transition kernel or stochastic matrix ) A, with elements a ij , tells us the probability of the process changing to state j given that we are now in state i, that is a ij = p(

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