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Computational Methods in Finance

This text addresses a variety of numerical methods for pricing derivative contracts, including Fourier techniques, finite differences, numerical simulation, and Monte Carlo simulation methods--one of the first books to cover all of these techniques. After presenting the basics of pricing techniques, it covers key concepts of calibration and parameter estimation. Written by a popular professor at Columbia University and NYU's Courant Institute, the book is suitable for any graduate course on computational finance in financial engineering and financial mathematics programs as well as for practitioners interested in computational methods in finance.
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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 Levy 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 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.

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.


"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

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