Risk Latte - A Double Mean Reverting Model for analyzing Stochastic Volatility

A Double Mean Reverting Model for analyzing Stochastic Volatility

Team latte
September 2, 2010

Recently, we came across a math model of stochastic volatility that is quite intuitive and at the same time very practical. This is the Double Mean Reverting Stochastic Volatility (DMR-SV) model. To the best of our knowledge, the first DMR-SV model was introduced by Jim Gatheral. In a more generalized case, a DMR-SV model can be based on Buehler’s “Variance Curve Models”. In a Double Mean Reverting Stochastic Volatility model (DMR-SV) the short variance is a mean reverting stochastic process whose “mean reversion” itself is stochastic. Such behaviour is quite often seen in the financial markets.

In this article we talk about a generalized Double Mean Reverting Stochastic Volatility model which is calibrated using Buehler’s Variance Curve model.

First let’s talk very briefly about, Buehler’s Variance Curve Model.

Hans Buehler, who is currently a senior quant at Deutsche Bank in London, introduced the Variance Curve models in 2006. The key idea was to get a spot stochastic volatility model based on observable Variance Swap curves in the market.

Simply put: variance curve – the floating leg of a variance swap – are observable in the market and therefore, forward variance is observable from the variance swap curves. This is further be numerically approximated. From forward variance we can get the implied short variance. By following an approach very similar to the Heath Jarrow Morton (HJM) approach for modelling the forward interest rates Buehler derives implied spot volatility from the forward variance.

The beauty of Buehler’s approach is to show that a reasonable specification of variance swaps in the market is the key to modelling stochastic volatility. We’ll talk in detail about Variance Curve model of Buehler elsewhere on this site.

Now let’s talk for the Double Mean Reverting Stochastic Volatility (DMR-SV) model for Stochastic Volatility. Rather than discussing Gatheral’s model we would discuss a generalized Double Mean Reverting Stochastic Volatility model that relies on Buehler’s Variance Curve model to calibrate the parameters.

A generalized DMR-SV model has the form:

In the above model, the respective Weiner processes are given by:

In the above, the respective correlations are given by and .

The calibration of the initial states , and along with the mean reversion speeds and are done by fitting a double linearly mean reverting variance curve functional to the observed variance swap market data. This is where Buehler’s Variance Curve model and the techniques to estimate Variance Curve functionals come in.


Reference:
(1) Hans Buehler, Consistent Variance Curves, Finance and Stochastics (2006);
(2) A Double Mean Reverting Stochastic Volatility Model with Concurrent Jumps by Chenxu Li, Columbia University, October 2009

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