1st Edition

# Computational Methods in Finance

444 Pages 73 B/W Illustrations
by CRC Press

444 Pages
by CRC Press

Also available as eBook on:

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.

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

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
Interest Rate Definitions

Filtering and Parameter Estimation
Filtering
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)

References

Index

Problems appear at the end of each chapter.

### Biography

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.

"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