How do you model the dynamics of asset prices? A self-reflection (Epilogue of 3)

As the title suggests, this was supposed to be a three part series. But I feel like I have left out so much that I wanted to discuss, so although the first 3 parts have already provided an overview on what I would call the "classical" quant modeling approach (loosely speaking, one that focuses on describing the asset price time series with a random process), I would take this opportunity to talk about what lies beyond such approach.

We are interested particularly in capturing the tails of the distribution. What about putting the tail on the center stage? That is the spirit of extreme value theory. It has had wide application in natural disaster forecast. The basic idea is to posit a probability distribution that specifically fits the tail of the distribution. Contrast this with, say, the stochastic volatility approach in which we tried to fatten the tail, while still keeping our focus on the 'body' of the return distribution. Now we are directly and un-apologetically modeling the tail itself. The advantage is obvious: we have more freedom in fitting the tails closely by dedicating the calibration to it. The short coming is more subtle. In order to perform extreme value analysis, we have to specify the threshold at where the tail starts, so that we can attach a distribution to it. If you define daily return of  -20% as the threshold, then you can use some distribution to catch the sub-sample of daily return less than -20%. But how do you choose the threshold? Choose it too deep into the body and you miss the point of using extreme value models. After all, we are trying to focus on the tail by not worrying about modeling the boring part of the distribution. On the other hand, choose it too far off into the tail and you suffer by not having enough historical data points for an accurate calibration. The other issue is that extreme value analysis would not solve the "you don't know what you don't know" problem. You pick a threshold, you do the fitting, and chances are sooner or later a much worse draw down would take you by surprise, proving to you that the tail is even fatter that you have assumed.

Now you might say, "Hey, isn't that a necessary evil of any perceivable model in finance? That one has to make assumptions and wait to be surprised?" And I tend to agree to the sad fact. Perhaps except for if we follow a completely different paradigm. Recently some researchers having been looking at crash forecasting using a wide range of unconventional tools. They range from fractal analysis that studies the (change in) scaling of time series; agent-based models that take a bottom-up approach trying to produce asset price dynamics by considering (grossly simplified) behaviors of traders; and critical point/phase change models that are inspired by Ising model in solid state physics. The common theme is the acknowledgement of market non-linearity and the lack of a single, forever stable market state. Unfortunately there is no straight forward way to connect these models to option prices, but with the advancements of numerical methods we may some day see wider applications of them.