Estimating long term volatility regimes with Markov chains
Sep 22, 2005
No matter how sophisticated a math model you use, we believe that long term volatility cannot be predicted at all, let alone making a precise forecast. But we do believe that given the volatility tracking indices available in the market, the probability of asset or equity volatility moving from one state (say, a high volatility state) to another state (say, a low volatility) can be estimated with some accuracy.
It is a matter of great debate, not only amongst the option traders, but also amongst academics, regulators and the investment community at large, as to how the asset volatilities in general and equity volatilities in particular will move in the coming months and years. Option and derivatives traders are beholden to the future forecast of volatility as their entire pricing structure depends on it. Yet, we all know that there is no well defined deterministic equation which models volatility or even relates it to the movement in the underlying asset. We know that volatility moves in clusters and is somehow related to the range of an asset price rather than the asset price. The well known VIX, and our own proprietary volatility indices, BRiXX™ (for Asian markets), show that implied volatility in most major equity markets is inversely proportional to the level of the asset.
We have recently tried to apply a simple Markov chain analysis to look at the probability of moving from a low volatility state to high volatility state and vice versa.
Let us say that there are only two states in this world. A state of low volatility which we call state 1 and a high volatility state which we call state 2. This can be a very real life problem. One can study the VIX volatility index for the last 20 years or so for the U.S. equity markets and determine that volatility alternates between two regimes of high and low volatility. Let us say that after careful analysis of the VIX index (or maybe from some other detailed analysis) we have determined that the holding rates and the jump rates for these two states of volatility are as follows:
This means that the mean exit time for volatility from state 1 is 0.5 and the mean exit time for volatility from state 2 is 0.1667. The jump rates and are related to the probability that the volatility will move from one state to the other for the first time.
Our problem is simply to find a transition matrix (as a function of time) with the limit of time tends to infinity, i.e. . We know that the transition matrix is related to the fundamental matrix by the following equation:
The fundamental matrix can be found in terms of the eigenvalues and eigenvectors of the matrix Q, where and the matrix elements are given by our estimates;
Finding out the eigenvalues and the eigenvectors gives us the value of the transition matrix as:
And taking the limit of the above equation we get:
The form of this matrix shows that the long term probability of the states of volatility. Both terms of the first column in the above transition matrix are ¾ or 0.75, signifying that the long term probability that the volatility is in low volatility state (state 1) is 0.75, no matter whether or not it was in a low volatility state initially. Similarly, from the second column of the transition matrix tell us that the long term probability that the volatility will be in high volatility state (state 2) regardless so whether it was in that state initially is ¼ or 0.25.
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