Society for Industrial and Applied Mathematics, 2005, -316 pp.
Mathematical finance is an old science but has become a major topic for numerical analysts since Merton [97], Black-Scholes [16] modeled financial derivatives. An excellent book for the mathematical foundation of option pricing is Lamberton and Lapeyre's [85]. Since the Black-Scholes model relies on stochastic differential equations, option pricing rapidly became an attractive topic for specialists in the theory of probability, and stochastic methods were developed first for practical applications, along with analytical closed formulas. But soon, with the rapidly growing complexity of the financial products, other numerical solutions became attractive. Applying the Monte-Carlo method to option pricing is very natural and not difficult, at least for European options, but speeding up the method by variance reduction may become tricky. Similarly, tree methods are very intuitive and fast but also rapidly become difficult as the complexity of the financial product grows.
Focusing on the Black-Scholes model, a partial differential equation is obtained by It6's calculus. It can be approximated and integrated numerically by various methods, to which a very clear and concise introduction may be found in the book by Wilmott, Howison, and Dewynne [117]: the basic idea is to approximate the partial differential equation by a system of equations with a finite number of unknowns, which may be solved numerically to obtain a discrete solution. The discrete problems can be computationally intensive. The aim of this book is neither to present financial models nor to discuss their validity; we must be very modest in this perspective, since our expertise is not here. This book is not a recipe book either, and although we have tried to be broad, many financial products such as bonds are not covered.
The purpose is rather to discuss some mode numerical techniques which we believe to be useful for simulations in finance. We are not going to dwell on Monte-Carlo and tree methods, because these have been studied very well elsewhere (see [60,116]). Weessentially focus on the finite difference (Chapter 3) and the finite element methods (Chapter 4) for the partial differential equation, trying to answer the following three questions:
Are these methods reliable?
How can their accuracy be controlled and improved?
How can these methods be implemented efficiendy?
Several applications to financial products with programs in C++ are proposed. In this book, we stress the notions of error control and adapti vity: the aim is to control a posteriori the accuracy of the numerical method, and if the desired accuracy is not reached, to refine the discretization precisely where it is necessary, i.e., most often where the solution exhibits singularities. We believe that mesh adaption based on a posteriori estimates is an important and practical tool because it is the only existing way to certify that a numerical scheme will give the solution within a given error bound. It is therefore a road for software certification. Mesh adaption greatly speeds up computer programs because grid nodes are present only where they are needed; it is particularly important for American options, because the option price as a function of time and the spot price exhibits a singularity on a curve which is itself unknown. A posteriori error estimates are the subject of Chapter 5, and adaptive methods are also used for pricing American options in Chapter 6.
Controlling the accuracy of a numerical method requires a rather complete mathematical analysis of the underlying partial differential equation: this motivates partially the theoretical results contained in Chapters 2 and 6.
Option Pricing
The Black-Scholes Equation: Mathematical Analysis
Finite Differences
The Finite Element Method
Adaptive Mesh Refinement
American Options
Sensitivities and Calibration
Calibration of Local Volatility with European Options
Calibration of Local Volatility with American Options
Mathematical finance is an old science but has become a major topic for numerical analysts since Merton [97], Black-Scholes [16] modeled financial derivatives. An excellent book for the mathematical foundation of option pricing is Lamberton and Lapeyre's [85]. Since the Black-Scholes model relies on stochastic differential equations, option pricing rapidly became an attractive topic for specialists in the theory of probability, and stochastic methods were developed first for practical applications, along with analytical closed formulas. But soon, with the rapidly growing complexity of the financial products, other numerical solutions became attractive. Applying the Monte-Carlo method to option pricing is very natural and not difficult, at least for European options, but speeding up the method by variance reduction may become tricky. Similarly, tree methods are very intuitive and fast but also rapidly become difficult as the complexity of the financial product grows.
Focusing on the Black-Scholes model, a partial differential equation is obtained by It6's calculus. It can be approximated and integrated numerically by various methods, to which a very clear and concise introduction may be found in the book by Wilmott, Howison, and Dewynne [117]: the basic idea is to approximate the partial differential equation by a system of equations with a finite number of unknowns, which may be solved numerically to obtain a discrete solution. The discrete problems can be computationally intensive. The aim of this book is neither to present financial models nor to discuss their validity; we must be very modest in this perspective, since our expertise is not here. This book is not a recipe book either, and although we have tried to be broad, many financial products such as bonds are not covered.
The purpose is rather to discuss some mode numerical techniques which we believe to be useful for simulations in finance. We are not going to dwell on Monte-Carlo and tree methods, because these have been studied very well elsewhere (see [60,116]). Weessentially focus on the finite difference (Chapter 3) and the finite element methods (Chapter 4) for the partial differential equation, trying to answer the following three questions:
Are these methods reliable?
How can their accuracy be controlled and improved?
How can these methods be implemented efficiendy?
Several applications to financial products with programs in C++ are proposed. In this book, we stress the notions of error control and adapti vity: the aim is to control a posteriori the accuracy of the numerical method, and if the desired accuracy is not reached, to refine the discretization precisely where it is necessary, i.e., most often where the solution exhibits singularities. We believe that mesh adaption based on a posteriori estimates is an important and practical tool because it is the only existing way to certify that a numerical scheme will give the solution within a given error bound. It is therefore a road for software certification. Mesh adaption greatly speeds up computer programs because grid nodes are present only where they are needed; it is particularly important for American options, because the option price as a function of time and the spot price exhibits a singularity on a curve which is itself unknown. A posteriori error estimates are the subject of Chapter 5, and adaptive methods are also used for pricing American options in Chapter 6.
Controlling the accuracy of a numerical method requires a rather complete mathematical analysis of the underlying partial differential equation: this motivates partially the theoretical results contained in Chapters 2 and 6.
Option Pricing
The Black-Scholes Equation: Mathematical Analysis
Finite Differences
The Finite Element Method
Adaptive Mesh Refinement
American Options
Sensitivities and Calibration
Calibration of Local Volatility with European Options
Calibration of Local Volatility with American Options