Computational Methods in Finance (Hardback) book cover

Computational Methods in Finance

By Ali Hirsa

© 2012 – CRC Press

444 pages | 73 B/W Illus.

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Hardback: 9781439829578
pub: 2012-09-05
eBook (VitalSource) : 9781466576049
pub: 2016-04-19
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About the Book

As today’s financial products have become more complex, quantitative analysts, financial engineers, and others in the financial industry now require robust techniques for numerical analysis. Covering advanced quantitative techniques, Computational Methods in Finance explains how to solve complex functional equations through numerical methods.

The first part of the book describes pricing methods for numerous derivatives under a variety of models. The book reviews common processes for modeling assets in different markets. It then examines many computational approaches for pricing derivatives. These include transform techniques, such as the fast Fourier transform, the fractional fast Fourier transform, the Fourier-cosine method, and saddlepoint method; the finite difference method for solving PDEs in the diffusion framework and PIDEs in the pure jump framework; and Monte Carlo simulation.

The next part focuses on essential steps in real-world derivative pricing. The author discusses how to calibrate model parameters so that model prices are compatible with market prices. He also covers various filtering techniques and their implementations and gives examples of filtering and parameter estimation.

Developed from the author’s courses at Columbia University and the Courant Institute of New York University, this self-contained text is designed for graduate students in financial engineering and mathematical finance as well as practitioners in the financial industry. It will help readers accurately price a vast array of derivatives.


"The depth and breadth of this stand-alone textbook on computational methods in finance is astonishing. It brings together a full-spectrum of methods with many practical examples. … the purpose of the book is to aid the understanding and solving of current problems in computational finance. … an excellent synthesis of numerical methods needed for solving practical problems in finance. This book provides plenty of exercises and realistic case studies. Those who work through them will gain a deep understanding of the modern computational methods in finance. This uniquely comprehensive and well-written book will undoubtedly prove invaluable to many researchers and practitioners. In addition, it seems to be an excellent teaching book."

—Lasse Koskinen, International Statistical Review (2013), 81

"… there are several sections on topics that are rarely treated in textbooks: saddle point approximations, numerical solution of PIDEs, and others. There is also extensive material on model calibration, including interest rate models and filtering approaches. The book is a very comprehensive and useful reference for anyone, even with limited mathematical background, who wishes to quickly understand techniques from computational finance."

—Stefan Gerhold, Zentralblatt MATH 1260

"A natural polymath, the author is at once a teacher, a trader, a quant, and now an author of a book for the ages. The content reflects the author’s vast experience teaching master’s level courses at Columbia and NYU, while simultaneously researching and trading on quantitative finance in leading banks and hedge funds."

—Dr. Peter Carr, Global Head of Market Modeling, Morgan Stanley, and Executive Director of Masters in Math Finance, NYU Courant Institute of Mathematical Sciences

"A long-time expert in computational finance, Ali Hirsa brings his excellent expository skills to bear on not just one technique but the whole panoply, from finite difference solutions to PDEs/PIDEs through simulation to calibration and parameter estimation."

—Emanuel Derman, professor at Columbia University and author of Models Behaving Badly

Table of Contents

I Pricing and Valuation

Stochastic Processes and Risk-Neutral Pricing

Characteristic Function

Stochastic Models of Asset Prices

Valuing Derivatives under Various Measures

Types of Derivatives

Derivatives Pricing via Transform Techniques

Derivatives Pricing via the Fast Fourier Transform

Fractional Fast Fourier Transform

Derivatives Pricing via the Fourier-Cosine (COS) Method

Cosine Method for Path-Dependent Options

Saddlepoint Method

Introduction to Finite Differences

Taylor Expansion

Finite Difference Method

Stability Analysis

Derivative Approximation by Finite Differences: A Generic Approach

Matrix Equations Solver

Derivative Pricing via Numerical Solutions of PDEs

Option Pricing under the Generalized Black-Scholes PDE

Boundary Conditions and Critical Points

Nonuniform Grid Points

Dimension Reduction

Pricing Path-Dependent Options in a Diffusion Framework

Forward PDEs

Finite Differences in Higher Dimensions

Derivative Pricing via Numerical Solutions of PIDEs

Numerical Solution of PIDEs (a Generic Example)

American Options

PIDE Solutions for Lévy Processes

Forward PIDEs

Calculation of g1 and g2

Simulation Methods for Derivatives Pricing

Random Number Generation

Samples from Various Distributions

Models of Dependence

Brownian Bridge

Monte Carlo Integration

Numerical Integration of Stochastic Differential Equations

Simulating SDEs under Different Models

Output/Simulation Analysis

Variance Reduction Techniques

II Calibration and Estimation

Model Calibration

Calibration Formulation

Calibration of a Single Underlier Model

Interest Rate Models

Model Risk

Optimization and Optimization Methodology

Construction of the Discount Curve

Arbitrage Restrictions on Option Premiums

Interest Rate Definitions

Filtering and Parameter Estimation


The Likelihood Function

Kalman Filter

Non-Linear Filters

Extended Kalman Filter

Unscented Kalman Filter

Square Root Unscented Kalman Filter (SR UKF)

Particle Filter

Markov Chain Monte Carlo (MCMC)



Problems appear at the end of each chapter.

About the Author

Ali Hirsa is head of Analytical Trading Strategy at Caspian Capital Management. Dr. Hirsa is also an adjunct professor at Columbia University and NYU’s Courant Institute of Mathematical Sciences.

About the Series

Chapman and Hall/CRC Financial Mathematics Series

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
BUSINESS & ECONOMICS / Operations Research
MATHEMATICS / Probability & Statistics / General