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Mathematics of Computational Finance.pdf
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The goal of this book is to provide a relatively short (but sufficiently rigorous and hopefully readable) account of basic numerical methods used in pricing derivative financial products — one of the essential problems of financial mathematics. Rigorous pricing methods started in the 1970s with the celebrated Black–Scholes model (also called the Black–Scholes–Merton model) and became the basis for the pricing of thousands of complex financial products. Many excellent books deal with the numerical implementations of these pricing methods. However, the majority is only focused on the numerical algorithms used in pricing. This book originated from a course given to mathematics students at the University of Warsaw. This audience obliged the lecturer to speak not only about algorithms but also about the mathematical foundations behind these algorithms: convergence, stability, and error estimates. The main body of the text concentrates on theoretical material. Numerous examples illustrate the introduced theoretical concepts using simple financial models. In contrast to this theoretical analysis, the problem sections at the end of each chapter are very practical. The exercises at the end of each chapter deal with many more advanced financial models and numerical algorithms applied to them. Some problems analyze the theoretical aspects of the described algorithms, but many require computer implementations and analysis of the obtained results. Several exercises extend the material of the corresponding chapter by presenting additional (less frequently used) algorithms or supplementary material. It can be said that the main body of the book teaches the reader how to analyze numerical algorithms, and the problem sections serve to improve their computational skills.
In writing this book, I assumed that the reader possesses a broad knowledge of continuous finance and its mathematical models. Thus, there are no introductory chapters on quantitative finance. The introductory material in some chapters establishes the terminology used in the book and cannot be treated by the reader as a source of knowledge on quantitative finance. All the described numerical methods begin with a presentation of the analytical problem described as a system of stochastic or partial differential equations without discussing their financial motivation. The described numerical methods are quite general, and the experienced reader can recognize that they cover not only the Black–Scholes model but also some advanced local and stochastic volatility models. On the other hand, the reader whose acquaintance with quantitative finance is limited to the Black–Scholes model can ignore the multidimensional approach because the results are also valid for simple models.
The fundamental principle of numerical analysis is that one can compute only solutions that do exist. Following this principle, I have included in the book theorems that guarantee the existence (and possibly uniqueness) of solutions to equations that are the subject of subsequent numerical analysis. This goal is easy to achieve for financial models formulated in probabilistic terms (random variables, stochastic processes, stochastic differential equations) because they are standard tools used in continuous finance. In addition, the most advanced result I use is the existence and properties of solutions for the stochastic differential equation of the Itˆo type, the results belonging to the usual prerequisites for any course of quantitative finance. The situation is completely different for financial models described by partial differential equations (PDEs). First, a course in PDE theory is rarely required as a prerequisite for financial courses. Second, to formulate a theorem on the existence of solutions for a PDE model, we need advanced tools: weak derivatives, Sobolev’s spaces, weak solutions, etc. Therefore, I have decided to give a thorough introduction to PDEs together with the Feynman–Kac theorem, which gives a rigorous passage from the stochastic to the PDE formulation of financial models. Finally, presenting proofs of existence for PDEs has some didactic aspects, as these proofs are, in many cases, prototypes for the design of numerical methods and their proofs of convergence. Hence, before passing to operations with multiple indices of a numerical algorithm, the reader can see the proof idea in a clean Banach’s space formulation.
Since the book is written mainly for students of mathematics or mathematically oriented students of economics or natural sciences who study quantitative finance for academic or professional purposes, the presentation is quite advanced, taking for granted a broad knowledge of analysis, probability, and statistics with some orientation in stochastic processes. The entire presentation is restricted to pricing derivative instruments in a financial model described by the Itˆo SDE and the corresponding parabolic PDE. This simplification makes it easier to achieve the main goal of presenting complete proofs. These complete proofs are compilations from many sources, sometimes with my original additions. References to these sources in the body of the text are rather limited. They usually indicate the paper or book that inspired my writing and/or give the reader a broader account of the discussed topic. Of course, I could not succeed in writing the proofs of all the theorems. There are many theorems without proof. For such theorems, I usually quote the reference in which the proof can be found. Essentially, I omit proofs that do not belong to the area of numerical analysis, and the reason for skipping them is twofold: first, some proofs belong to the field of mathematics, which is very distant from the topic of the book, and presenting them will require a significant amount of auxiliary material (this is particularly the case of the chapter on American options); second, some proofs need advanced tools far beyond the knowledge that I can expect from the reader. In some places, I have made a compromise by just quoting a result necessary in the proof without mentioning where such a result comes from. The theorems are formulated in full generality with possibly weak assumptions. However, the presented proofs are written for the most elementary cases: one-dimensional, with constant coefficients, smooth data, etc. The reason for such an approach is purely didactic: I intend to present to the reader the main idea that remains behind the proof, which can be lost in multiple indices, splittings into subdomains, smoothing of data and coefficients, etc., required for the proof of the sharp version.
The interest of this book lies in the computation of derivative prices. Because financial models rely on stochastic differential equations, applying Monte Carlo methods to derivative pricing is very natural. Similarly, tree methods are very intuitive and fast. Applying the Feynman–Kac theorem, we can replace probabilistic methods of derivative pricing with solutions of partial differential equations, which gives rise to a large class of numerical methods for PDEs. All of these computational approaches are described in the book. However, the presentation is far from complete. After all, this is just a textbook and not a monograph, and the author has selected the most popular methods and algorithms for presentation. Of course, the selection is biased by the author’s experience in computational finance. The algorithms and methods of this book have been practically tested by myself and my students during many years of teaching computational finance. The experienced reader can ask why the book is limited to models in which randomness is described by Wiener processes and there are no models with jumps and L´evy processes. The reasons are numerous. One of these is that including L´evy processes will require writing a thorough monograph instead of a medium-sized textbook. Such a monograph is beyond the aspirations of the author.
The book is organized as follows. After the introductory Chapter 1, in which binomial and trinomial trees are presented, Chapter 2 addresses the generation of pseudo-random numbers. This chapter explains how samples from a given distribution can be generated using pseudo-random numbers. Algorithms for generating normal deviates receive particular attention in this chapter. Chapter 3 begins with crude Monte Carlo. Then the variance reduction technique is presented. The chapter concludes with the computation of Greeks, i.e., the sensitivities of prices to model parameters. In Chapter 4, numerical solutions of stochastic differential equations are discussed with proofs of convergence for the Euler and Milstein schemes and error estimates. A large part of the book is devoted to numerical solutions of partial differential equations that arise in finance. As an introduction to the topic, Chapter 5 collects fundamental facts from the theory of weak solutions of PDEs in Sobolev’s spaces. Chapter 6 focuses on finite difference methods for partial differential equations of the parabolic type. The most popular finite difference schemes are described, and their accuracy (order of approximation) and stability are proved. Finite element methods are discussed in Chapter 7 for both elliptic and parabolic problems. This chapter proves that finite element approximations converge to the corresponding solutions of differential problems. Chapter 8 is devoted to American options. Both Monte Carlo and PDE methods of pricing are presented. The Monte Carlo approach is limited to the description of the well-known algorithm of Longstaff and Schwartz. Then a careful analysis of its convergence is provided. In the final part of this chapter, the variational approach to pricing American options is discussed. After a short introduction to variational inequalities related to American options, the presentation focuses on two popular numerical methods: the projected SOR and the penalty method. For both of these methods, the proofs of stability and convergence are given.
As I have mentioned before, this book originated from a course in computational finance. However, the scope of the book is too broad to fit into a single course. A reasonable selection of material from Chapters 2–6 can be lectured in one course. On the other hand, the chapter on American options, supplemented by more information on optimal stopping problems and variational inequalities in continuous time, can be sufficient for a special course.
I would like to express my gratitude to several people who have influenced my writing of this book. I owe a particular debt to Piotr Kowalczyk, with whom I have been lecturing computational finance to several cohorts of students at the University of Warsaw. I gained much from my colleagues in the Department of Mathematics at the University of Warsaw, whose broad knowledge and competence helped me navigate the corners of this project. I would also like to acknowledge the students who, during the courses given in the past years, have helped to improve the quality of the book through their questions, comments, and feedback. Finally, many thanks to my wife Iwona and son Jan for their encouragement and support, and to Iwona for her tolerance and patience in allowing me to complete the project.
1. Introduction 1
1.1 Financial Market . . . . . . . . . . . . . . . . . . . . 1
1.2 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Implied Trees . . . . . . . . . . . . . . . . . . . . . . 22
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 29
2. Random Number Generators 41
2.1 Generators of Uniform Deviates . . . . . . . . . . . . 41
2.2 Nonuniform Variates . . . . . . . . . . . . . . . . . . 46
2.3 Multivariate Random Variables . . . . . . . . . . . . 54
2.4 Low-Discrepancy Sequences . . . . . . . . . . . . . . 58
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 63
3. Monte Carlo Methods 69
3.1 Monte Carlo Integration . . . . . . . . . . . . . . . . 69
3.2 Variance Reduction Methods . . . . . . . . . . . . . 73
3.2.1 Importance sampling . . . . . . . . . . . . . . 74
3.2.2 Stratified sampling . . . . . . . . . . . . . . . 80
3.2.3 Antithetic variates . . . . . . . . . . . . . . . 82
3.2.4 Control variates . . . . . . . . . . . . . . . . 84
3.3 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3.1 Finite differences . . . . . . . . . . . . . . . . 89
xiii
xiv Mathematics of Computational Finance
3.3.2 Pathwise differentiation . . . . . . . . . . . . 92
3.3.3 The likelihood ratio method . . . . . . . . . . 93
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 95
4. Integration of Stochastic Differential Equations 105
4.1 Introduction to Itˆo Stochastic Calculus . . . . . . . . 105
4.1.1 Stochastic integral . . . . . . . . . . . . . . . 106
4.1.2 Stochastic differential equations . . . . . . . . 111
4.2 Numerical Schemes for Stochastic Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . 117
4.3 Proofs of Convergence . . . . . . . . . . . . . . . . . 123
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 146
5. Introduction to Elliptic and Parabolic Equations 161
5.1 From Stochastic to Partial Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . 161
5.2 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . 167
5.2.1 Embedding theorems . . . . . . . . . . . . . . 173
5.3 Elliptic Equations of Second-Order . . . . . . . . . . 175
5.4 Parabolic Equations of Second-Order . . . . . . . . . 179
5.4.1 Galerkin approximation . . . . . . . . . . . . 184
5.5 The Black–Scholes Equation . . . . . . . . . . . . . . 191
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 203
6. Finite Difference Methods for Parabolic
Equations 211
6.1 Introduction to Finite Differences . . . . . . . . . . . 212
6.2 Convergence Analysis of Two-Level Schemes . . . . . 216
6.3 θ-schemes . . . . . . . . . . . . . . . . . . . . . . . . 225
6.4 Stability of Difference Schemes . . . . . . . . . . . . 228
6.5 Finite Differences in Many Dimensions . . . . . . . . 248
6.5.1 Generalizations of one-dimensional
methods . . . . . . . . . . . . . . . . . . . . . 248
6.5.2 Alternating direction method . . . . . . . . . 253
6.5.3 Additional topics . . . . . . . . . . . . . . . . 258
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 266
7. Finite Element Methods 279
7.1 Finite Elements for Elliptic Equations . . . . . . . . 280
7.2 Finite Elements for Parabolic Equations . . . . . . . 302
7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 323
Contents xv
8. American Options 335
8.1 Pricing American Options . . . . . . . . . . . . . . . 336
8.2 Monte Carlo Pricing . . . . . . . . . . . . . . . . . . 339
8.2.1 Convergence . . . . . . . . . . . . . . . . . . 345
8.3 Variational Inequalities . . . . . . . . . . . . . . . . . 352
8.3.1 Discrete variational inequalities . . . . . . . . 365
8.3.2 Projected SOR algorithm . . . . . . . . . . . 383
8.3.3 Penalty method . . . . . . . . . . . . . . . . . 392
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 400
References 415
Index 4
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