Enrico Schumann

Heuristic Methods in a Nutshell

Heuristics are a powerful class of algorithms for numerical optimization. We explain the principles by which these methods operate and detail several specific techniques like Simulated Annealing or Differential Evolution; the construction of hybrids is briefly described, too.We provide sample code for various algorithms. Almost all heuristics are stochastic algorithms, hence we explain how the solutions obtained from heuristics can be evaluated, and what factors (e.g., computational resources) influence a solutions quality. We also discuss several practical issues: how constraints can be included in algorithms, and why we should care about efficient implementations. All discussion is kept informal, and examples and sample code are provided.

Chapter 9 – Portfolio optimization with “Threshold Accepting”: a practical guide

Recent years have seen a proliferation of new risk and performance measures in investment management. These measures take into account stylized facts of financial time series like fat tails or asymmetric return distributions. In practice, these measures are mostly used for ex post performance evaluation, only rarely for explicit portfolio optimization. One reason is that, other than in the case of classical mean–variance portfolio selection, the optimization under these new risk measures is more difficult since the resulting problems are often not convex and can thus not be solved with standard methods. This chapter describes a simple but effective optimization technique called “Threshold Accepting (TA),” which is versatile enough to be applied to different objective functions and constraints, essentially without restrictions on their functional form. This technique is capable of optimizing portfolios under various recently proposed performance or (downside) risk measures, like value at risk, drawdown, Expected Shortfall, the Sortino ratio, or Omega, while not requiring any parametric assumptions for the data, i.e., the technique works directly on the empirical distribution function of portfolio returns. This chapter gives an introduction to TA and details how to move from a general description of the algorithm to a practical implementation for portfolio selection problems.

Accuracy and Precision in Finance

Modern finance, both theoretical and practical, makes extensive use of mathematical reasoning and modelling. But this reliance on exact methods comes with unfortunate side effects: numerical precision is emphasised, but without an equal insistence on empirical accuracy; methods that do not offer numerical precision are shunned from the beginning. This is most severe in those fields of finance in which theories build on optimisation. We argue that precision in modelling is only useful up to a certain level; we can do with approximations and low precision as long as models are sufficiently accurate for the purpose at hand. We illustrate this view through a concrete example: selecting financial portfolios with so-called heuristic methods. Heuristics provide only approximations to an optimisation models solution, but we show that their precision is sufficient for all practical purposes. In compensation for their lack of precision, heuristics allow researchers to solve more-accurate models.

Heuristics for Portfolio Selection

Portfolio selection is about combining assets such that investors’ financial goals and needs are best satisfied. When operators and academics translate this actual problem into optimisation models, they face two restrictions: the models need to be empirically meaningful, and the models need to be soluble. This chapter will focus on the second restriction. Many optimisation models are difficult to solve because they have multiple local optima or are ‘badly-behaved’ in other ways. But on modern computers such models can still be handled, through so-called heuristics. To motivate the use of heuristic techniques in finance, we present examples from portfolio selection in which standard optimisation methods fail. We then outline the principles by which heuristics work. To make that discussion more concrete, we describe a simple but effective optimisation technique called Threshold Accepting and how it can be used for constructing portfolios. We also summarise the results of an empirical study on hedge-fund replication.

Optimization Problems in Finance

Optimization models are at the core of many financial applications. The formulation of such models is driven by actual financial goals and needs, but it is also restricted by empirical difficulties (e.g., to forecast required quantities) and by the need to finally come up with the models solution. Many financial optimization models are difficult to solve because objective functions have multiple local optima, are noisy, or are not well-behaved in other ways. Researchers and operators in finance often go a long to way to shape models such that they can be solved with standard methods, thereby giving up relevant aspects of the original problem.We argue that it is preferable to use appropriate methods, heuristics, which are able to compute solutions even to problems that are infeasible for standard methods. The chapter is concluded by several examples of difficult financial models; some of these examples will be revisited in later chapters.

Numerical Analysis in a Nutshell

Numerical computation is not just about translating mathematical expressions into a programming language: on a computer real numbers cannot, in general, be represented exactly, and one cannot “go to the limit“, i.e., make quantities infinitely large or small. In this chapter we present some fundamentals of computer arithmetic that should help to gain intuition about where situations can arise that will compromise the quality of the result of a numerical computation, and eventually how to avoid these situations.We also discuss the notions of numerical instability and ill-conditioning. The concept of algorithmic complexity, that is how to measure the efficiency of the set of instructions needed to compute a numerical result, is also presented.

Finite Difference Methods

Many finance problems, in particular in option pricing, lead to the formulation of differential equations. Analytic solutions to such equations are generally difficult to find or, worse, may often not exist. However, solutions can easily be approximated with satisfactory precision by resorting to a numerical method in which derivatives are replaced by finite differences. After a short general classification of differential equations the chapter describes different schemes for solving (partial) differential equations that arise in option pricing models. The chapter emphasizes practical and computational aspects, thus the implementation of the methods is described in detail. Methods and algorithms are presented in the form of pseudocode and Matlab programs; a number of illustrations is provided including the pricing of American options.

Linear Equations and Least Squares Problems

Linear systems arise very frequently in quantitative modeling and are at the core of classical optimization and Least Squares problems. There are two broad classes of methods, direct and iterative ones, for the solution of linear systems.Within these classes there exists a variety of algorithms; the appropriate choice for a given problem depends on the structure and quantification of the coefficient matrix of the linear system to be solved. This chapter explains the main algorithms and provides pseudocode for them, it also presents a classification of the problems and a discussion about what method is most efficient for a given problem. Sparse linear systems are part of the discussion. For several methods Matlab code is provided together with numerical illustrations.

A Gentle Introduction to Financial Simulation

Abstract In finance, models serve many different purposes: Econometric models capture relationships and dependencies; risk models assess likelihoods for certain outcomes; pricing models are on the quest for the fair value; microstructure models try to analyze how decisions and behavior on an individual level aggregate to macro-level market phenomena; and so on. This chapter showcases a diverse set of models and how they can be useful in a simulation framework. The examples range from single period portfolio and asset pricing problems, over multiperiod return and risk processes to a simple agent-based model. The chapter also highlights how simulation can improve understanding the workings—and shortcomings—of popular models.

Financial Simulation at Work: Some Case Studies

Simulation has become a popular method for many areas in finance. Sometimes, it is the only feasible approach to tackle certain problems. This chapter provides three case studies which illustrates this. The first example is constant proportion portfolio insurance (CPPI), an investment strategy aimed to participate from price increases as good as possible while limiting losses. In principle, the portfolio structure ought to be adjusted continuously; in practise, long stretches of time can pass without re-calibration. Simulation can help to assess the gap risk caused by the low readjustment frequencies. Next, extreme value theory is given a closer look. The application of this theoretically appealing concept is hampered by real world limitations; simulations can help to understand these limitations and how to deal with them. The largest part of this chapter is finally devoted to option pricing under non-standard situations. Real world price series are often far from a geometric Brownian motion, the core of most traditional derivative pricing models. Based on examples for return series with jumps, the reader is guided through various simulation techniques for pricing.