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:
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.
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?