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### Garch fitting in practice to obtain finish-level probabilities on indices

Posted: **February 4th, 2011, 11:55 am**

by **CommodityQuant**

When using Garch modelling to model real-world phenomena for example movements of stock indices, no matter how good (well-calibrated) the model is, there are bound to be different behaviours in the real-world object from an idealized Garch process (as generated by matlab's garchsim for example). My aim is to find the real-world probabilities of events -- for example the Dow index finishing above a given level. One approach is this: 1) Calibrate the Garch model by MLE, 2) Use garch theory to predict the vols. 3) Use the vols to get the probabilities. 4) Use a q-q plot to transform the theoretical probabilities to the real-world probabilities. The problem with this is that one model, being regularly recalibrated by MLE can have many different vols, which distorts the q-q method. Another method might be to linearly transform the garch-predicted vol to the real-world vol. Another view might be that all such "fudge" transformations are bad, and that the best way to proceed is just to improve the methodology for obtaining the Garch parameters -- for example, by more robust MLE methods. Any thoughts on this problem? Maybe I'm missing some terminology which could help me search better. The matter is akin to model calibration. But it isn't exactly that. Model calibration would generally mean making sure the Garch coefficients are correct. But I'm looking into, for example, obtaining conclusions such as "A Garch vol of x corresponds to a real- world vol of x + delta" and therefore transforming the probability computations after the Garch model has been run. Thanks a lot

### Garch fitting in practice to obtain finish-level probabilities on indices

Posted: **February 4th, 2011, 1:36 pm**

by **Caesaria**

QuoteOriginally posted by: CommodityQuantMy aim is to find the real-world probabilities of events -- for example the Dow index finishing above a given level. One approach is this: 1) Calibrate the Garch model by MLE, 2) Use garch theory to predict the vols. 3) Use the vols to get the probabilities. 4) Use a q-q plot to transform the theoretical probabilities to the real-world probabilities. I just don't see how you could find the real-world probability of an event here. What you are looking at implicitly is trying to predict the skew on the returns. You simply do not have enough information, the vols just tell you the \sig of the norm dist, which is symmetric. So projecting skew could be like taking an exp mov avg of your returns, which doesn't make sense (using history to predict skew?). Or the RSI indicator for technical analysis ?

### Garch fitting in practice to obtain finish-level probabilities on indices

Posted: **February 5th, 2011, 3:25 pm**

by **Richyiee**

why dont you just calculate probabilities directly ?for example the Dow index finishing above a given level = the number of days it finishes above certain level/total number of days

### Garch fitting in practice to obtain finish-level probabilities on indices

Posted: **February 5th, 2011, 6:09 pm**

by **CommodityQuant**

Thanks for the feedback. Richyee, the time intervals are tiny -- less than five minutes. Although I should have said that at the outset.Thanks,CommodityQuant

### Garch fitting in practice to obtain finish-level probabilities on indices

Posted: **February 6th, 2011, 2:56 am**

by **Richyiee**

you can still calculate probabilities directly tho ;p - number of intervals of event happening / total number of intervals