Using Peak-Over-Threshold EVT for Portfolio VaR and Expected Shortfall
Summary
The document outlines how extreme value theory can estimate tail risk, especially for a portfolio of long and short stock options. It recommends the peak-over-threshold approach: select sufficiently extreme losses, fit a Generalized Pareto Distribution, and use the fitted tail to estimate Value at Risk and Expected Shortfall. It also advises estimating standard errors so the uncertainty of these estimates is visible.
Option nonlinearities can limit the method. If extreme portfolio outcomes coincide with an option entering the money and its gamma changing materially, a tail model fitted only to portfolio returns may not capture the behavior reliably. Options that behave more linearly over the relevant large moves may be more suitable. The discussion points to R packages and references for learning and implementation, but provides no worked calculation, parameter choices, or empirical comparison, so it is guidance rather than a validated recipe.
Key ideas
- Peak-over-threshold EVT fits a Generalized Pareto Distribution to losses beyond a selected threshold.
- The fitted tail can be used to estimate VaR and Expected Shortfall.
- Standard errors help convey how uncertain the tail-risk estimates are.
- Option gamma and payoff nonlinearity can undermine estimates based only on portfolio returns.
- R has packages that support extreme value analysis.
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Full text
# Extreme Value Theory in Risk Management # Extreme Value Theory in Risk Management 1 - I am trying to understand the concept of EVT and how we are able to calculate VaR and ES from that. I would like to understand the maths in more detail. 2 - If I have a portfolio of Long and Short options on various stocks, I can create the historical PnL distribution of this portfolio with the help of historical stock price data. Now for this portfolio if I want to use EVT to calculate VaR and ES, how do I go about that? How can I do it with R or by hand? It will be great if you can point me to some videos or materials which handles this topic from the very basics. ## Answer by salisboss (score 2, accepted) https://quant.stackexchange.com/a/29998 To answer part 1: For my MFE capstone project on EVT on interest rate swaps I used Practical Methods of Financial Engineering and Risk Management: Tools for Modern Financial Professionals by Rupak Chatterjee (my advisor). Chapter 8 gives a good step by step intro to Power Laws and EVT. It also includes some quick Excel demos. ## Answer by RiskyScientist (score 4) https://quant.stackexchange.com/a/29972 EVT has pluses and minuses, but (under certain conditions) provides the best estimate of extreme quantile returns in a portfolio given the data available. Probably the simplest and easiest way to do this is to use the peak over threshold method and fit the Generalized Pareto Distribution (GPD). The GPD is very convenient for calculating VaR and ES. A good reference by an expert is this paper by McNeil. You should make sure to estimate the standard error around your estimate so that you know how reliable the result is. In terms of a long/short portfolio of stock options, you should check whether the option strikes are such that the options are likely to have a material amount of gamma when your portfolio experiences very large moves. For example, if we only had a single stock option in the portfolio and the 99.9th percentile portfolio return happens just when the stock option starts to get in the money, it is likely that EVT will struggle to give a good estimate due to the option nonlinearity (the EVT doesn't get to know which instruments are in your portfolio, only the portfolio returns). But if the options are struck reasonably close to the money, then they will likely act in a linear way when the large returns from the time series are applied to them. You can probably code something in Excel, but R has a lot of useful packages for EVT - I recommend RStudio (free) as an interface to R. ## Answer by Con Fluentsy (score 1) https://quant.stackexchange.com/a/53275 Multiple packages are available in R for extreme value analysis.If you are looking at extreme value theory in regards to stock prices there is full implementation of libraries in the rMetrics team's fExtremes library in the R statistical script language. This and others are covered in the book on R for finance by Gilleland, Ribatet, and Stephenson (2013). In chapter 8 of Introduction to Quantitative Finance by PACKT publishing, they present the evir package, and a complete working example of cash shortfall. Bernard Pfaff's Book Financial Risk Modeling and Portfolio Optimization in R covers the topic in depth also.
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