Tuesday, November 30, 2004

VIX and Black-Scholes

OK, humor me here as I continue to think about volatility. Looking at the CBOE white paper on VIX, I see a plot (page 13) indicating very strong correlation between movements in the SP and the implied vol. A change in the SP of about 1% causes a 4% shift in the VIX, but with the opposite sign. Now, SP500 options are widely traded and liquid. They should provide one of the best tests of options pricing theory. But in the usual Black-Scholes model the volatility of the underlying security is a fixed input parameter - it certainly isn't supposed to be path (history) dependent. The simple log normal random walk model has its limitations - for example, there is no reason the vol shouldn't change in time (hopefully slowly) - but I'm surprised to see such clear path dependence. Of course, it's possible that the actual volatility (as opposed to implied volatility) doesn't exhibit the correlations we are discussing. But if so, there is an inefficiency in the behavior of options traders that should be arbed away!

...I've been informed that these issues are addressed using more sophisticated GARCH models. (GARCH = Generalised Autoregressive Conditional Heteroskedastic!)

This paper seems to conclude that implied vol is a good predictor of realized vol, so the correlation between market movements and vol is not a behavioral quirk of options traders (indeed, it is a quirk of the market itself).


Carson C. Chow said...

I've always felt that fixed volatility was a huge weakness in Black-Scholes. I once saw a talk where this person plotted a securities price, arbitrarily detrended it and then fit a Gaussian. I asked him how he decided between fluctuations and trends and he had no idea what I was asking. To my eye there did not seem to be an obvious time scale in the fluctuations to separate the two. You figure with the brain power the banks now have, if something is better someone must have found it by now?

Steve Hsu said...

"detrending"= fitting Black-Scholes drift term? This always bothered me as well. They are doing more sophisticated stuff now, like using stochastic vol or these GARCH models which I guess have path dependence.

The old way of extracting implied vol was to fit option prices using the Black-Scholes formula. The new way (post 2003 for VIX) is to use the prices to fit an implied probability distribution. If you think about it, each price is equivalent to a weighted integral over part of the PD. In this method there is no drift term, although the peak of the PD might be above or below the current value of the index, which is essentially drift. Of course, you assume the PD is essentially normal in the log of the underlying so you can extract just one number. I was worried that there would be a problem with the tail, which is known to be far from normal, but the algorithm defined for VIX only goes modestly out on the dist. (2 sigma at the moment), so this effect is negligible.

Carson C. Chow said...

I don't know the details of the GARCH models but to me it seems like the Black-Scholes equation which is based on a Fokker-Planck type formalism is inherently flawed for long time scales. Probably for a month to two months it's okay but the non-Gaussianity and memory effects become more important as you go longer. I don't know the industry at all of course but I bet that traders are going on pure gut instinct for pricing options longer than a year or two.

Steve Hsu said...

What is striking about the CBOE VIX data that I linked to is not just that the vol changes over long timescales, but that it changes *in response to market moves*. That is, if the SP500 has a rough week (say, it drops 3%) the implied vol shoots up 12% on average. Whereas, if there is a runup of 3%, the vol drops by 12%. This path dependence goes beyond the original skepticism I had over using a constant vol in the B-S model.

BTW, professionals I have discussed this with are well aware of the effect and hence the use of GARCH models, stochastic vol, etc.

Blog Archive