# Extreme Value Theory possible for portfolios with options?

Say you have a portfolio with long exposure to a few linear assets (stock indices) and short exposure to a nonlinear asset (say call options on one of the linear assets).

I am interested in modelling extreme returns (negative returns) on this total portfolio for a 1-day horizon for calculation of risk measures. For linear assets this can be done with Extreme Value Theory by fitting a Generalised Pareto Distribution (GPD) to observations over a high threshold. When it comes to portfolios including nonlinear exposure, I can only come up with two approaches which I will briefly explain:

1. Calculate the portfolio pseudo-historical returns (similar to historical simulation) by estimating tomorrows portfolio return from past returns, where the options would be priced (e.g. by Black-and-Scholes). Then lower tail of this empirical distribution would be fitted by GPD.

2. Fit GPD to both the upper and lower tail of the linear assets empirical distributions and a kernel smoother to the middle. Then do a number of Monte Carlo simulations from these, connecting the simulated returns by a copula. The option position will priced based on the simulation.

I have not found much literature on EVT and more complex portfolio.Are any of these approaches adequate for my problem?

• Are you aware of Lisa Goldberg's work on the topic? – Brian B Oct 2 '13 at 17:43
• Not particularly, but now that you mentioned it I skimmed through some of her papers and it looks interesting. Can you refer to a single paper by her where EVT is applied to options? Thank you for your tip! – Chris Oct 2 '13 at 19:36
• Last time I talked to her, she was having trouble enough applying it to equities, but it's the only decent EVT work I've ever seen in finance. – Brian B Oct 2 '13 at 19:53
• In the limit, call options on an asset are linear ( 100pct delta when deep in the money). in fact the value is (stock price - present value of exercise price). Does that simplify your problem? – dm63 Oct 8 '16 at 11:24

The history of option returns is not always relevant to what the performance will be in the future. So what you really want to do is find the underlying drivers of the portfolio returns. This leads you to a bottom-up approach. You can still apply EVT to any of the underlying drivers, but you wouldn't be fitting the portfolio returns or the individual option returns with an extreme value distribution.

There are many variations, but here is one potential approach (based on Meucci's approach):

Begin by modelling stock returns. Regress each stock's return on some benchmark returns. Then, for the benchmark and the residuals, you can fit a GARCH model and get standardized residuals. You can then fit Extreme Value Distributions to the standardized residuals. You would then be able to use Monte Carlo to simulate from the relevant distributions to get the distributions of the stocks at the relevant horizon.

The next step would be modelling the option prices. Simulating option prices in the future depends on future values of stock prices and implied volatility (ignoring changes in interest rates). What you really need to model is changes in implied volatility. Again, here you fit some kind of model to changes in implied volatility (and here you would probably need to build up a volatility surface) and use that to forecast implied volatility at the horizon. Lots of variations. You might consider that there is a correlation between stocks and that the implied volatility has a relationship with the above GARCH volatility. Anyway, then you use the stock price distributions and the implied volatility distributions to price the options at the horizon.

Finally, you would use the weights at the initial period you could the portfolio returns at the horizon. You would then be able to calculate VaR or CVaR.

I believe the distribution you are looking for is the multivariate Gumbel distribution. A paper on it can be found at http://www.sciencedirect.com/science/article/pii/S0167715215001352

An older paper from 1998, but possibly still relevant, is Beyond VaR: From Measuring Risk to Managing Risk, by Helmut Mausser and Dan Rosen. They present both a parametric and a non-parametric approach for a portfolio containing two underlying stocks with multiple puts and calls at various strikes and expiries

The paper is widely cited, so you may be able to find something more recent, e.g. using IEEE XPLORE (where I also saw it referenced).

However, if this picture speaks to you, their 1998 paper might not be a bad starting point. 