Extreme Value Theory for Tail-Related Risk Measures

Many fields of modern science and engineering have to deal with events which are rare but have significant consequences. Extreme value theory is considered to provide the basis for the statistical modeling of such extremes. The potential of extreme value theory applied to financial problems has only been recognized recently. This paper aims at introducing the fundamentals of extreme value theory as well as practical aspects for estimating and assessing statistical models for tail-related risk measures.Extreme Value Theory; Generalized Pareto Distribution, Generalized Extreme Value Distribution; Quantile Estimation, Risk Measures; Maximum Likelihood Estimation; Profile Likelihood Confidence Intervals.

A Heuristic Approach to Portfolio Optimization

Constraints on downside risk, measured by shortfall probability, expected shortfall, semi-variance etc., lead to optimal asset allocations which differ from the meanvariance optimum. The resulting optimization problem can become quite complex as it exhibits multiple local extrema and discontinuities, in particular if we also introduce constraints restricting the trading variables to integers, constraints on the holding size of assets or on the maximum number of different assets in the portfolio. In such situations classical optimization methods fail to work efficiently and heuristic optimization techniques can be the only way out. The paper shows how a particular optimization heuristic, called threshold accepting, can be successfully used to solve complex portfolio choice problems.Portfolio Optimization; Downside Risk Measures;Heuristic Optimization Threshold Accepting.

Serial and Parallel Krylov Methods for Implicit Finite Difference Schemes Arising in Multivariate Option Pricing

This paper investigates computational and implementation issues for the valuation of options on three underlying assets, focusing on the use of the finite difference methods. We demonstrate that implicit methods, which have good convergence and stability prooperties, can now be implemented efficiently due to the recent development of techniques that allow the efficient solution of large and sparse linear systems. In the trivariate option valuation problem, we use nonstationary iterative methods (also called Krylov methods) for the solution of the large and sparse linear systems arising while using implicit methods. Krylov methods are investigated both in serial and in parallel implementations. Computational results show that the parallel implementation is particularly efficient if a fine grid space is needed.Multivariate option pricing, finite difference methods; Krylov methods; parallel Krylov methods

An Application of Extreme Value Theory for Measuring Financial Risk

Assessing the probability of rare and extreme events is an important issue in the risk management of financial portfolios. Extreme value theory provides the solid fundamentals needed for the statistical modelling of such events and the computation of extreme risk measures. The focus of the paper is on the use of extreme value theory to compute tail risk measures and the related confidence intervals, applying it to several major stock market indice

A Data-Driven Optimization Heuristic for Downside Risk Minimization

In practical portfolio choice models risk is often defined as VaR, expected short-fall, maximum loss, Omega function, etc. and is computed from simulated future scenarios of the portfolio value. It is well known that the minimization of these functions can not, in general, be performed with standard methods. We present a multi-purpose data-driven optimization heuristic capable to deal efficiently with a variety of risk functions and practical constraints on the positions in the portfolio. The efficiency and robustness of the heuristic is illustrated by solving a collection of real world portfolio optimization problems using different risk functions such as VaR, expected shortfall, maximum loss and Omega function with the same algorithm.Portfolio optimization, Heuristic optimization, Threshold accepting, Downside risk