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Extreme Values, from Victor Niederhoffer
Extreme values are often useful to study, as they are unusual and often spring from, or trigger, major effects. They arise often in engineering, where faults from unusual events can cause disaster. They crop up often in insurance work, where the extreme event causes liabilities and risk to skyrocket; in traffic, networks, highways and utilities, where an overload can cause disaster; and in materials manufacturing. We all saw them in the flood control statistics that came into play with Hurricane Katrina.
Extreme values crop up everywhere in markets -- for example, in options, where the extreme value is the main thing that causes the skew, or the opportunity to make a killing, as experts buy the options that increase inordinately. They also come into play in the simple distribution of returns, where extreme values are the things that give you the full return or loss, and where avoidance of the extreme losses is key to survival or gain.
Most work on extreme values places heavy emphasis on thresholds, the large values beyond which the extreme analysis is conducted. For example, in a repeated series of measurement of the probability of defects of 10 parts, if the probability of defect is distributed uniformly from 0 to 1, the probability of finding a threshold probability above 0.90 is 0.10.
A good discussion of extreme value statistics is in "An Introduction to Statistical Modeling of Extreme Values," by Stuart Coles.
In my study of threshold work on extreme values, I have come across three good studies that have extensive and fruitful applications to markets. The papers are:
The Adya paper is about automatic methods of forecasting, and has many useful ideas about forecasting markets -- for example, the proximity to a new high.
The Cross paper shows how you can derive attributes of prices that fit with actual moves by simulation if you assume investors have two simple emotions: cowardice and fear of inaction. It's a nice attempt to show how two actual attributes of investors might lead to price movements with various parameters fitted to individual stocks and investors.
The Vesilind paper provides a nice review of yield curve regularities and time expectation theory and shows how you can build sets of inefficiencies based on these regularities into a retrospective model that yields profits in currencies and long-only fixed-income strategies.
These papers and their extensions have raised my threshold of awareness on the use of extreme value modeling and may elicit similar augmentations in yours.