Introduction

Publisher Summary

This chapter presents an introduction to this book. The book is structured into three parts. The first part, “Fundamentals,” begins with an introduction to numerical analysis, so one discusses computer arithmetic, approximation errors, how to solve linear equations, how to approximate derivatives, and other topics. The second part, “Simulation,” starts with chapters on how to generate random numbers and how to model dependence. There is also a short example of how to implement an agent-based model. By means of several case studies, applications will illustrate how these models can be used for generating more realistic price processes and how simulation can help to develop an intuitive understanding of econometric models and solve practical problems. The third part, “Optimization,” deals with optimization problems in finance. Many relevant optimization models cannot be solved with standard methods that are readily available in software packages.

Numerical methods in finance; Financial Modeling; Model accuracy; Model risk; Approximation; Optimization

This chapter gives an overview of the book and lays out several principles by which financial problems should be approached.We discuss financial models and software; we also provide a brief discussion of the various sources of approximation error in financial modeling. We stress that numerical methods in finance are tools, that is, they are means, not ends. The appropriateness and quality of such tools needs to be judged with respect to the application at hand. In a nutshell, we argue that quantitative models are useful in finance and that effective tools to solve them are needed. But we also emphasize the need to remember the limitations imposed by any given model and, even more so, by the difficulty to estimate and forecast financial quantities. Several examples are given to illustrate these points.

1.1 About this book

The origin of this book lies in the city of Geneva in Switzerland, so it seems only appropriate to start there. In case you do not know Switzerland, here is a map.

Imagine you stand at the wonderful lakefront of Geneva, with a view on Mont Blanc, and you decide to take a run to Lake Zurich. We admit that this is unlikely, given that the distance is 250 km. But it is just an example. Also, imagine you could run the whole distance with a speed of 20 km per hour which is about the speed that a world-class marathon runner could manage (we ignore the fact that Switzerland is quite a mountainous region). It would still take you more than 12 hours to arrive in Zurich. Now, imagine you took a plane with an average speed of 800 km per hour. This is about 40 times faster, and you would arrive in less than half an hour. What is our point? First, few people would run from Geneva to Zurich. But what we wanted to give you is an idea of speedup. It is just forty in this case, not a large number, but it makes a difference.

This book is not about traveling but about quantitative methods in a specific field, computational finance. Quantitative analysis has become ever more important in scientific research and the industry over the last decades, and in our view, it has much changed. On the one hand, ever more data is collected and stored today, waiting to be analyzed. At the same time, computing power has increased dramatically. If we measure the speed of a computer by the number of operations it can perform in a second, then computers have improved by a factor of perhaps 1,000,000 since the early 1980s. 1 If traveling had matched this speedup, we could go from Geneva to Zurich in /100 of a second. Better yet, we do not talk about supercomputers here but about the kind of computers we have on our desktops. (And at least in the past, the power of what was called a supercomputer at one point in time was available for private users only a few years later.)

Computational finance is not a well-defined discipline. It is the intersection of financial economics, scientific computing, econometrics, software engineering, and many other fields. Its goal is to better understand prices and markets (if you are an academic), and to make money (if you are more practically inclined). This book is about tools for computational finance, with an emphasis on optimization techniques. In fact, we will focus on a particular type of optimization methods: heuristics. The theme of the book is the practical, computational application of financial theory; we do not deal with the theory itself; in some cases, there may not even be much established theory. Hence, we will not develop ideas theoretically but consider quantitative analysis as a primarily computational discipline, in which applications are put into software form and tested empirically. For every topic in this book, we discuss the implementation of methods; algorithms will be given in pseudocode, Matlab or R code will be provided for all important examples.2

The readers we have in mind are students (Master or PhD level) in programs on quantitative and computational finance, and researchers in finance but also practitioners in banks and other financial companies. The book could be the main reference for courses in computational finance programs or an additional reference for quantitative or mathematical finance courses. Many of the chapters have the flavor of case studies. From a pedagogical view, this allows students to learn the required steps for tackling specific problems; these steps can then be applied to other problems. From the practical side the selected problems will, we hope, be relevant enough to make the book a reference on how to implement a given technique. In fact, the ideal, prototypic kind of reader that we have in mind—let us call him “the analyst” —does not only work on theory, but his job requires a close interaction between theoretical ideas and computation with data.

The book is structured into three parts. The first part, “Fundamentals,” begins with an introduction to numerical analysis, so we discuss computer arithmetic, approximation errors, how to solve linear equations, how to approximate derivatives, and other topics. These discussions will serve as a reference to which later chapters often come back. For instance, the Greeks for option pricing models can often be computed by finite differences. Numerical methods are rarely discussed in standard finance books (Brandimarte, 2006, is an exception), even though they are the basis for empirically applying and testing models. Beyond the initial chapters, our book will not discuss numerical methods in an abstract way. For a general treatment see, for instance, Heath (2005) or the concise and very recommended summary given by Trefethen (2008b); we will only discuss numerical aspects that are relevant to our applications. Further chapters will explain deterministic numerical methods to price instruments, namely trees and finite difference methods, and their application to standard products.

The second part, “Simulation,” starts with chapters on how to generate random numbers and how to model dependence. Chapter 8 discusses how time series processes, simple assets, and portfolios can be simulated. There is also a short example on how to implement an agent-based model. By means of several case studies, applications will illustrate how these models can be used for generating more realistic price processes and how simulation can help to develop an intuitive understanding of econometric models and solve practical problems.

The third part, “Optimization” deals with optimization problems in finance. Many relevant optimization models cannot be solved with standard methods that are readily available in software packages. Hence, standard methods will only be briefly touched. We will rather show how to apply another class of techniques, so-called heuristics. Chapter 11 will detail building blocks of numerical optimization algorithms. Though this includes gradient-based techniques like Newton's method, emphasis will be put on more robust techniques like direct search. Chapter 12, “Heuristic Methods in a Nutshell,” will then move from these building blocks to specific techniques, and give an overview of heuristic methods. The remaining chapters will deal with specific problems from portfolio optimization, the estimation of econometric models, and the calibration of option pricing models.

1.2 Principles

We start with some principles, or guidelines, for the applications in the book.