# Large deviations theory in finance

In probability theory, the theory of large deviations concerns the asymptotic behavior of remote tails of sequences of probability distributions.

A related post says:

Large deviations theory is concerned with the behavior of the tails of the distribution of a r.v. as a parameter N becomes large. The Central Limit theorem tells you what happens to the mean, the LD Theory tells you about the tails.

Not sure what $$N$$ refers to that becomes large, but what are some common and practical applications of large deviations theory in finance? (Practical as in usable within a well-known financial model, as opposed to theoretical, pointless meanderings about stochastic process behaviors.)

Or better, what is the most popular measure or metric put out by large deviations theory (in the same way that the measure called expected shortfall (CVaR) was borne from extreme value theory)?

Please no links to papers from google search unless you plan to explain what those papers actually contain.

• I don't know for sure but it's relation to crashes come to mind. I would look at material of Nassim Taleb or Didier Sornette.. – mark leeds Nov 8 '20 at 17:54
• I think large deviations theory feature most prominently in risk management, and perhaps not as much directly to finance as to say, actuarial maths, dealing with insurance claims for very rare events. One approach to model tail risk for heavy tailed distribution you can fit a Generalized Pareto Distribution to just the tail part of the EDF and in that way get more realistic probability measures for outcomes more extreme than those previously observed (e.g. 99.9% VaR of 300 sample occurrences). You could look up Peaks Over Threshold method – Oscar Nov 9 '20 at 8:25
• I don't think large deviations theory has seen enough development yet to be of use in finance. There are some advances in large deviations for martingales that might be of interest (e.g. over brownian filtration), but that's about it. – Always Right Never Left Nov 10 '20 at 15:01
• Just a comment: let us not mix Large Deviation Theory (LDT) with Extreme Value Theory (EVT), see eg. quant.stackexchange.com/questions/22706/…. – Daneel Olivaw Nov 10 '20 at 15:10
• Extreme changepoint detection theory is far more prevalent in finance than large deviations theory, in fact, I have never seen a quant, not an actuary use it. Regarding extreme changepoint detection, see the following topics; Runs tests, control theory and the Shiryaev Roberts statistic. Books on Crashes by Didier Sornette, William Ziemba, forget Taleb populist post ergo hoc clairvoyant, large on commentary, low on real practical real life advice re: risk management. – Con Fluentsy Nov 12 '20 at 0:11