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I see the Generalized Method of Moments suggested in numerous academic papers as a way to calibrate stochastic volatility models. However, any decent trading shop is going to calibrate to observable option prices instead.

Are there any places that have used GMM in an actual trading context, say in some market where historical time series are obtainable and derivatives prices unavailable?

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GMM method is a powerful method to calibrate historically, only. Also, the historical Calibration is used in the banking industry for forecast an asset’s performances and not for replicating them.

Mathematically, it's known that historical vs options calibration is equivalent to observing an asset through two different probabilities (historical vs the neutral one). This is why you will observe Apple's trend around $\mu \approx 45\% $ under the historical proba instead of the lower interest rate in the neutral world.

Appl

For an insight on what could be done with the two probabilities see the article "P" versus "Q" of Attilio Meucci.

There are some articles who are interested in linking the two probabilities, i.e. the historical and the neutral one. The big advantage of this approach is indubitably the great amount of historical data which would provide an appreciable robustness to the neutral calibration.

But at my knowledge this is so far not used in the banking industry for reasons linked to the change of probability (weird assumptions on the risk premia) and the resulting change of model.

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Quite right. One does occasionally see historical data being used for model calibration in, e.g. loan and CLO markets where there are no derivatives to calibrate to. However I don't think any of the models employed there are sufficiently intractable to require GMM. –  Brian B Dec 5 '11 at 19:19
    
One of the most popular GMM method is the Maximum Likelihood methodology. For many models (ARCH/ GARCH/ ..) it leads to a closed-form function maximization while for the vast majority models (stochastic volatility models, ..) the maximized function is not computable analytically and need to be approached with filters. This approach - originated from robotic - was introduced recently by practitioners in the banking area. A good reference (but difficult) on this matter is the book 'Inference in Hidden Markov Models' of Cappé, Moulines and Rydén. –  Beer4All Dec 5 '11 at 21:30
    
I tend to think of GMM as a generalization of Maximum Likelihood, rather than thinking of Maximum Likelihood as an application of GMM. Mainly because ML estimation precedes GMM in so many ways. –  Brian B Dec 6 '11 at 21:13
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