Modeling Derivatives in C++

by London, Justin

• ISBN13: 9780471654643
• ISBN10: 0471654647
• Edition: CD
• Format: Paperback
• Copyright: 2004-12-06
• Publisher: WILEY

Summary

This book is the definitive and most comprehensive guide to modeling derivatives in C++ today. Providing readers with not only the theory and math behind the models, as well as the fundamental concepts of financial engineering, but also actual robust object-oriented C++ code, this is a practical introduction to the most important derivative models used in practice today, including equity (standard and exotics including barrier, lookback, and Asian) and fixed income (bonds, caps, swaptions, swaps, credit) derivatives. The book provides complete C++ implementations for many of the most important derivatives and interest rate pricing models used on Wall Street including Hull-White, BDT, CIR, HJM, and LIBOR Market Model. London illustrates the practical and efficient implementations of these models in real-world situations and discusses the mathematical underpinnings and derivation of the models in a detailed yet accessible manner illustrated by many examples with numerical data as well as real market data. A companion CD contains quantitative libraries, tools, applications, and resources that will be of value to those doing quantitative programming and analysis in C++. Filled with practical advice and helpful tools, Modeling Derivatives in C++ will help readers succeed in understanding and implementing C++ when modeling all types of derivatives.

Author Biography

Justin London has analyzed and managed bank corporate loan portfolios using credit derivatives in the Asset Portfolio Management Group of a large bank in Chicago, Illinois. He has developed fixed-income and equity models for trading companies and his own quantitative consulting firm. London has written code and algorithms in C++ to price and hedge various equity and fixed-income derivatives with a focus on building interest rate models. In 1999, he founded Global Max Trading (GMT), a global online trading and financial technology company. A graduate of the University of Michigan, London has five degrees, including a BA in economics and mathematics, an MA in applied economics, and an MS in financial engineering, computer science, and mathematics, respectively.

Table of Contents

Preface	xiii
Acknowledgments	xix
Black-Scholes and Pricing Fundamentals	1	(44)
Forward Contracts	1	(3)
Black-Scholes Partial Differential Equation	4	(6)
Risk-Neutral Pricing	10	(7)
Black-Scholes and Diffusion Process Implementation	17	(13)
American Options	30	(3)
Fundamental Pricing Formulas	33	(2)
Change of Numeraire	35	(3)
Girsanov's Theorem	38	(3)
The Forward Measure	41	(1)
The Choice of Numeraire	42	(3)
Monte Carlo Simulation	45	(78)
Monte Carlo	45	(2)
Generating Sample Paths and Normal Deviates	47	(3)
Generating Correlated Normal Random Variables	50	(6)
Quasi-Random Sequences	56	(11)
Variance Reduction and Control Variate Techniques	67	(2)
Monte Carlo Implementation	69	(7)
Hedge Control Variates	76	(8)
Path-Dependent Valuation	84	(8)
Brownian Bridge Technique	92	(6)
Jump-Diffusion Process and Constant Elasticity of Variance Diffusion Model	98	(4)
Object-Oriented Monte Carlo Approach	102	(21)
Binomial Trees	123	(42)
Use of Binomial Trees	123	(8)
Cox-Ross-Rubinstein Binomial Tree	131	(1)
Jarrow-Rudd Binomial Tree	132	(1)
General Tree	133	(2)
Dividend Payments	135	(2)
American Exercise	137	(1)
Binomial Tree Implementation	138	(2)
Computing Hedge Statistics	140	(4)
Binomial Model with Time-Varying Volatility	144	(1)
Two-Variable Binomial Process	145	(5)
Valuation of Convertible Bonds	150	(15)
Trinomial Trees	165	(18)
Use of Trinomial Trees	165	(1)
Jarrow-Rudd Trinomial Tree	166	(2)
Cox-Ross-Rubinstein Trinomial Tree	168	(1)
Optimal Choice of λ	169	(1)
Trinomial Tree Implementations	170	(4)
Approximating Diffusion Processes with Trinomial Trees	174	(4)
Implied Trees	178	(5)
Finite-Difference Methods	183	(63)
Explicit Difference Methods	183	(3)
Explicit Finite-Difference Implementation	186	(5)
Implicit Difference Method	191	(3)
LU Decomposition Method	194	(2)
Implicit Difference Method Implementation	196	(6)
Object-Oriented Finite-Difference Implementation	202	(30)
Iterative Methods	232	(3)
Crank-Nicolson Scheme	235	(6)
Alternating Direction Implicit Method	241	(5)
Exotic Options	246	(28)
Barrier Options	246	(9)
Barrier Option Implementation	255	(3)
Asian Options	258	(1)
Geometric Averaging	258	(2)
Arithmetic Averaging	260	(1)
Seasoned Asian Options	261	(1)
Lookback Options	262	(3)
Implementation of Floating Lookback Option	265	(3)
Implementation of Fixed Lookback Option	268	(6)
Stochastic Volatility	274	(50)
Implied Volatility	274	(2)
Volatility Skews and Smiles	276	(7)
Empirical Explanations	283	(1)
Implied Volatility Surfaces	284	(19)
One-Factor Models	303	(2)
Constant Elasticity of Variance Models	305	(2)
Recovering Implied Volatility Surfaces	307	(2)
Local Volatility Surfaces	309	(4)
Jump-Diffusion Models	313	(2)
Two-Factor Models	315	(6)
Hedging with Stochastic Volatility	321	(3)
Statistical Models	324	(43)
Overview	324	(5)
Moving Average Models	329	(2)
Exponential Moving Average Models	331	(3)
GARCH Models	334	(3)
Asymmetric GARCH	337	(3)
GARCH Models for High-Frequency Data	340	(13)
Estimation Problems	353	(1)
GARCH Option Pricing Model	354	(8)
GARCH Forecasting	362	(5)
Stochastic Multifactor Models	367	(28)
Change of Measure for Independent Random Variables	368	(2)
Change of Measure for Correlated Random Variables	370	(1)
N-Dimensional Random Walks and Brownian Motion	371	(2)
N-Dimensional Generalized Wiener Process	373	(1)
Multivariate Diffusion Processes	374	(1)
Monte Carlo Simulation of Multivariate Diffusion Processes	375	(1)
N-Dimensional Lognormal Process	376	(12)
Ito's Lemma for Functions of Vector-Valued Diffusion Processes	388	(1)
Principal Component Analysis	389	(6)
Single-Factor Interest Rate Models	395	(72)
Short Rate Process	398	(1)
Deriving the Bond Pricing Partial Differential Equation	399	(2)
Risk-Neutral Drift of the Short Rate	401	(1)
Single-Factor Models	402	(2)
Vasicek Model	404	(7)
Pricing Zero-Coupon Bonds in the Vasicek Model	411	(9)
Pricing European Options on Zero-Coupon Bonds with Vasicek	420	(5)
Hull-White Extended Vasicek Model	425	(4)
European Options on Coupon-Bearing Bonds	429	(2)
Cox-Ingersoll-Ross Model	431	(5)
Extended (Time-Homogenous) CIR Model	436	(2)
Black-Derman-Toy Short Rate Model	438	(1)
Black's Model to Price Caps	439	(4)
Black's Model to Price Swaptions	443	(5)
Pricing Caps, Caplets, and Swaptions with Short Rate Models	448	(7)
Valuation of Swaps	455	(2)
Calibration in Practice	457	(10)
Tree-Building Procedures	467	(87)
Building Binomial Short Rate Trees for Black, Derman, and Toy	468	(3)
Building the BDT Tree Calibrated to the Yield Curve	471	(5)
Building the BDT Tree Calibrated to the Yield and Volatility Curve	476	(9)
Building a Hull-White Tree Consistent with the Yield Curve	485	(10)
Building a Lognormal Hull-White (Black-Karasinski) Tree	495	(6)
Building Trees Fitted to Yield and Volatility Curves	501	(8)
Vasicek and Black-Karasinski Models	509	(6)
Cox-Ingersoll-Ross Implementation	515	(5)
A General Deterministic-Shift Extension	520	(4)
Shift-Extended Vasicek Model	524	(17)
Shift-Extended Cox-Ingersoll-Ross Model	541	(8)
Pricing Fixed Income Derivatives with the Models	549	(5)
Two-Factor Models and the Heath-Jarrow-Morton Model	554	(76)
The Two-Factor Gaussian G2++ Model	556	(7)
Building a G2++ Tree	563	(12)
Two-Factor Hull-White Model	575	(4)
Heath-Jarrow-Morton Model	579	(5)
Pricing Discount Bond Options with Gaussian HJM	584	(1)
Pricing Discount Bond Options in General HJM	585	(1)
Single-Factor HJM Discrete-State Model	586	(5)
Arbitrage-Free Restrictions in a Single-Factor Model	591	(4)
Computation of Arbitrage-Free Term Structure Evolutions	595	(3)
Single-Factor HJM Implementation	598	(8)
Synthetic Swap Valuation	606	(6)
Two-Factor HJM Model	612	(4)
Two-Factor HJM Model Implementation	616	(4)
The Ritchken and Sankarasubramanian Model	620	(3)
RS Spot Rate Process	623	(1)
Li-Ritchken-Sankarasubramanian Model	624	(2)
Implementing an LRS Trinomial Tree	626	(4)
Libor Market Models	630	(80)
LIBOR Market Models	632	(4)
Specifications of the Instantaneous Volatility of Forward Rates	636	(4)
Implementation of Hull-White LIBOR Market Model	640	(1)
Calibration of LIBOR Market Model to Caps	641	(1)
Pricing Swaptions with Lognormal Forward-Swap Model	642	(4)
Approximate Swaption Pricing with Hull-White Approach	646	(2)
LFM Formula for Swaption Volatilities	648	(2)
Monte Carlo Pricing of Swaptions Using LFM	650	(5)
Improved Monte Carlo Pricing of Swaptions with a Predictor-Corrector	655	(8)
Incompatibility between LSM and LSF	663	(2)
Instantaneous and Terminal Correlation Structures	665	(4)
Calibration to Swaption Prices	669	(1)
Connecting Caplet and S x 1-Swaption Volatilities	670	(3)
Including Caplet Smile in LFM	673	(4)
Stochastic Extension of LIBOR Market Model	677	(11)
Computing Greeks in Forward LIBOR Models	688	(22)
Bermudan and Exotic Interest Rate Derivatives	710	(61)
Bermudan Swaptions	710	(3)
Implementation for Bermudan Swaptions	713	(5)
Andersen's Method	718	(3)
Longstaff and Schwartz Method	721	(9)
Stochastic Mesh Method	730	(3)
Valuation of Range Notes	733	(9)
Valuation of Index-Amortizing Swaps	742	(10)
Valuation of Trigger Swaps	752	(2)
Quanto Derivatives	754	(6)
Gaussian Quadrature	760	(11)
APPENDIX A Probability Review	771	(12)
Probability Spaces	771	(2)
Continuous Probability Spaces	773	(1)
Single Random Variables	773	(1)
Binomial Random Variables	774	(1)
Normal Random Variables	775	(1)
Conditional Expectations	776	(2)
Probability Limit Theorems	778	(1)
Multidimensional Case	779	(1)
Dirac's Delta Function	780	(3)
APPENDIX B Stochastic Calculus Review	783	(10)
Brownian Motion	783	(1)
Brownian Motion with Drift and Volatility	784	(1)
Stochastic Integrals	785	(3)
Ito's Formula	788	(1)
Geometric Brownian Motion	789	(1)
Stochastic Leibnitz Rule	789	(1)
Quadratic Variation and Covariation	790	(3)
References	793	(10)
About the CD-ROM	803	(4)
GNU General Public License	807	(6)
Index	813