Pricing Financial Instruments The Finite Difference Method

by Tavella, Domingo; Randall, Curt

• ISBN13: 9780471197607
• ISBN10: 0471197602
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-04-21
• Publisher: Wiley

Summary

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Author Biography

<b>DOMINGO TAVELLA, Ph.D.</b>, is President of Octanti Associates, a consulting firm in risk management and financial systems design. He is the founder and chief editor of the Journal of Computational Finance, and has pioneered the application of advanced numerical techniques in pricing and risk analysis in the financial and insurance industries. Before founding Octanti Associates, Dr. Tavella was director of financial engineering at Integral Development Corporation and vice president at Bankers Trust Securities. Prior to that, he was a scientist at Stanford University and NASA Ames Research Center.

Table of Contents

Introduction	1	(22)
Stochastic Processes	3	(10)
Markov Processes	5	(3)
Stochastic Differential Equations	8	(1)
Ito's Formula	9	(1)
Ito's Formula for Processes with Jumps	10	(3)
Arbitrage Pricing Theory	13	(8)
Change of Measure	16	(5)
References	21	(2)
The Pricing Equations	23	(38)
European Derivatives	24	(6)
Hedging Portfolio Approach	24	(3)
Feynman-Kac Approach	27	(3)
The Pricing Equation in the Presence of Jumps	30	(9)
An Application of Jump Processes: Credit Derivatives	34	(3)
Defaultable Bonds	37	(1)
Full Protection Credit Put	38	(1)
American Derivatives	39	(6)
Relationship between European and American Derivatives	40	(2)
American Options as Dynamic Optimization Problems	42	(1)
Conditions at Exercise Boundaries	43	(1)
Linear Complementarity Formulation of American Option Pricing	44	(1)
Path Dependency	45	(3)
Discrete Sampling of Path Dependency	47	(1)
Dimensionality Reduction	48	(8)
Reformulating the Underlying Processes in a Different Measure	49	(1)
Currency Translated Options	50	(6)
Equations for the Hedging Parameters	56	(4)
Computation of Greeks by Direct Discretization	57	(1)
Computation of Greeks through Their Governing Equations	57	(3)
References	60	(1)
Analysis of Finite Difference Methods	61	(49)
Motivation	61	(6)
Constructing Finite Difference Approximations	67	(3)
Stability Analysis: Matrix Approach	70	(20)
Space Discretization	71	(2)
Time Discretization	73	(4)
Analysis of Specific Algorithms	77	(9)
Eigenvalue Analysis of the Black-Scholes Equation	86	(4)
Stability Analysis: Fourier Approach	90	(3)
Implementation of the Time Advancement	93	(7)
Solving Sparse Systems of Linear Equations	94	(6)
Finite Difference Approach to American Options	100	(5)
The Linear Complementarity Problem	101	(4)
Distortions Induced by Discretization	105	(2)
Strategies for Complex Derivative Structures	107	(1)
References	108	(2)
Special Issues	110	(46)
Effect of Payoff Discontinuities on Convergence	110	(4)
Implementing Jump Conditions	114	(6)
Boundary Conditions	120	(12)
Boundary Conditions in One Dimension	121	(9)
Boundary Conditions in Multiple Dimensions	130	(2)
Continuous and Discrete Sampling Models for Path-Dependent Options	132	(9)
Continuous Sampling	132	(4)
Discrete Sampling	136	(5)
Performance of Solvers for Multidimensional Problems	141	(6)
Numerical Solution of PIDEs: Jump-Diffusion and Pure Jump Models	147	(8)
References	155	(1)
Coordinate Transformations	156	(27)
One-Dimensional, Time-Independent Transformations	157	(15)
Transformations Place Grid Points at Selected Positions	160	(7)
Transformations That Concentrate Grid Points	167	(5)
One-Dimensional, Time-Dependent Transformations	172	(1)
Multidimensional, Time-Independent Transformations	173	(9)
Factored Multidimensional, Time-Independent Transformations	174	(1)
General Multidimensional, Time-Independent Transformations	175	(2)
Multidimensional Linear Transformations	177	(5)
References	182	(1)
Numerical Examples	183	(48)
Barrier Options	183	(13)
Time-Dependent Barriers	183	(4)
Nonuniform Grids and Discrete Sampling	187	(9)
Discretely Sampled Parisian Options	196	(6)
A Leveraged Knockin Put	202	(4)
Discretely Sampled Asian Options	206	(6)
Stochastic Volatility	212	(2)
Convertible Bond	214	(4)
Simple Fixed Income Instruments: Forward Swap	218	(5)
Credit Derivatives	223	(5)
References	228	(3)
Index	231