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.
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.
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