1st Edition

Computational Methods in Finance

By Ali Hirsa Copyright 2013
    444 Pages 73 B/W Illustrations
    by CRC Press

    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
    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.


    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