Yip, I stole the graph from an econometrics website, and then edited on the sigmas and other text. I'm not sure what periodicity they used when they calculated the distros. This type of graph can be found in many texts looking at this subject (in front of me I have Poon's book of forecasting vol and Taleb's Dynamic Hedging - I think that you would find Taleb useful. Both have diagrams very similar to the one I posted.) If stock price motion is truly Brownian (or Brownian with drift) would it really matter what periodicity you used?
I back tested the SPX butterflies for Jan 2001 to Jan 2002 - 12 trades with a next profit of $1564 on ~$4000 max risk. I included slippage and commissions. Not too bad. There was a lot of dVol during that period, and I can imagine that putting the trades on only when volty is high might improve things further. Ugly
What if the distribution in question is actually a Cauchy instead of merely leptokurtotic and log-normal? What would that imply about where you should buy or sell options? What would it imply about the volatility and the theoretical pricing model? --segv
This is the very graph that I think is going to be having the most influence on SPX over the next 6 months. Some predict that FED has no choice now but to lower interest rates since we are under the 50 level and they have a high statistical likelihood of stepping in with stimulus again. With the dollar getting pummeled overseas and this graph it looks like the FED is between a rock and a hard spot. Life is going to be very "interesting" over the next few quarters and options writers may get some fun volatility swings soon. TS
This "resonates" well with this journal's principal theme. Personally I don't buy into the well behaved statistical theory implied with all these statistical characterizations. I think the statistical distribution "chooses" to be what it is based on its underlying population character. And that I think is more an irrational and non-linear human behavior that is fundamentally fear & greed based. So stochastic theory probably is closer to reality. But I am only schooled modestly in all these statistical areas to be "dangerous" to my own portfolio - this can be very heady stuff. But the possibility of a Cauchy is interesting if not fanciful. If this were true or predominantly true then it implies to me that we have no concept of a mean nor "fat tails". That should make it fairly easy to consistently profit using the same IC methodologies explored by the thread starter at the 1.5 - 2 sigma level would it not? But I think the peakiness and unexpectedly high frequency of the outlier events in the markets are pretty strong evidence that we are closer to a leptokurtotic character. I am not well enough versed in the pricing model to comment on its adequacy in either case since I generally do not subscribe to it at all except as a "ball park estimator". I mean after all the volatility smile and the non-symmetrical vol skew between equities and currencies tells me its got serious problems (as if there is no correlation between equities and underlying currency?). My opinion is that emotion drives IV and price structure more than anything and I don't think it can ever be mathematically modeled as a realistic continuous function without some kind of estimate or predictor of its own error function. I think Markov processes could be used to predict an error band. We use to do this sort of thing a lot in engineering applications driven by real-time data. But they usually required a "human in the loop" acting as an inexpensive artificial intelligence agent to make the final decisions. I am contemplating designing a real-time "decision support" software tool to give myself a trading edge. Just my two cents. TS
Cool point. Everybody grab your calculators! There are a lot of neat things about the Cauchy-Lorentz distro. From the point of view of options trading, there are (at least) 2 big take home points: 1) Fat tails are real. That is, large movements occur far more commonly than predicted by the lognormal distro in the BSM equation. 2) It isn't possible to fully hedge these large moves To try to answer your question Segv, if the distro is Cauchian rather than Gaussian, it implies that FOTM options are undervalued with respect to their probability of moneyness, and should be bought. Perhaps you could play the spread between the prices predicted by the 2 distro models ...
Perhaps the takeaway strategy for most of us is not so much that these options should be bought, but that selling them is dangerous - and should be avoided. Mark
I'm definitely exhausting my memory of undergrad math, but I seem to recall that Cauchy distrobutions have unlimited variance. To me this implies fat tails are more probable? I could be out to lunch on this one. I haven't looked at this stuff for years and as Donna pointed out, I've only got half a brain!