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.

Heuristic Optimisation in Financial Modelling

There is a large number of optimisation problems in theoretical and applied finance that are difficult to solve as they exhibit multiple local optima or are not ‘well- behaved’ in other ways (eg, discontinuities in the objective function). One way to deal with such problems is to adjust and to simplify them, for instance by dropping constraints, until they can be solved with standard numerical methods. This paper argues that an alternative approach is the application of optimisation heuristics like Simulated Annealing or Genetic Algorithms. These methods have been shown to be capable to handle non-convex optimisation problems with all kinds of constraints. To motivate the use of such techniques in finance, the paper presents several actual problems where classical methods fail. Next, several well-known heuristic techniques that may be deployed in such cases are described. Since such presentations are quite general, the paper describes in some detail how a particular problem, portfolio selection, can be tackled by a particular heuristic method, Threshold Accepting. Finally, the stochastics of the solutions obtained from heuristics are discussed. It is shown, again for the example from portfolio selection, how this random character of the solutions can be exploited to inform the distribution of computations.Optimisation heuristics, Financial Optimisation, Portfolio Optimisation

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.

Implementing Binomial Trees

This paper details the implementation of binomial tree methods for the pricing of European and American options. Pseudocode and sample programmes for Matlab and R are given.Option pricing, Binomial trees, Numerical methods, Matlab, R

Robust regression with optimisation heuristics

Linear regression is widely-used in finance. While the standard method to obtain parameter estimates, Least Squares, has very appealing theoretical and numerical properties, obtained estimates are often unstable in the presence of extreme observations which are rather common in financial time series. One approach to deal with such extreme observations is the application of robust or resistant estimators, like Least Quantile of Squares estimators. Unfortunately, for many such alternative approaches, the estimation is much more difficult than in the Least Squares case, as the objective function is not convex and often has many local optima. We apply different heuristic methods like Differential Evolution, Particle Swarm and Threshold Accepting to obtain parameter estimates. Particular emphasis is put on the convergence properties of these techniques for fixed computational resources, and the techniques’ sensitivity for different parameter settings.Optimisation heuristics, Robust Regression, Least Median of Squares

A note on ‘good starting values’ in numerical optimisation

Many optimisation problems in finance and economics have multiple local optima or discontinuities in their objective functions. In such cases it is stressed that ‘good starting points are important’. We look into a particular example: calibrating a yield curve model. We find that while ‘good starting values’ suggested in the literature produce parameters that are indeed ‘good’, a simple best-of-n–restarts strategy with random starting points gives results that are never worse, but better in many cases.

Calibrating Option Pricing Models with Heuristics

Calibrating option pricing models to market prices often leads to optimisation problems to which standard methods (like such based on gradients) cannot be applied. We investigate two models: Heston’s stochastic volatility model, and Bates’s model which also includes jumps. We discuss how to price options under these models, and how to calibrate the parameters of the models with heuristic techniques.

Indirect Estimation of the Parameters of Agent Based Models of Financial Markets

Agent based models take into account limited rational behaviour of individuals acting on financial markets. Explicit simulation of this behaviour and the resulting interac-tion of individuals provide a description of aggregate financial market time series. Al-though the outcomes of such simulations often exhibit similarities with real financial market time series, methods for explicit validation are required. This paper proposes validation using simulation based indirect estimation. It uses typical characteristic moments of financial market data to assess the similarity of simulation outcomes. Fur-thermore, the parameters of the agent based models can be estimated by maximizing this similarity. The paper presents details of this estimation approach and first results for the US–$/DM exchange rate.Agent Based Models; Indirect Estimation; Validation

Calibrating the Nelson–Siegel–Svensson model

The Nelson–Siegel–Svensson model is widely-used for modelling the yield curve, yet many authors have reported ‘numerical difficulties’ when calibrating the model. We argue that the problem is twofold: firstly, the optimisation problem is not convex and has multiple local optima. Hence standard methods that are readily available in statistical packages are not appropriate. We implement and test an optimisation heuristic, Differential Evolution, and show that it is capable of reliably solving the model. Secondly, we also stress that in certain ranges of the parameters, the model is badly conditioned, thus estimated parameters are unstable given small perturbations of the data. We discuss to what extent these difficulties affect applications of the model.