Mathematical Finance: Theory, Modeling, ImplementationISBN: 9780470047224
544 pages
September 2007

Description
The evergrowing use of derivative products makes it essential for financial industry practitioners to have a solid understanding of derivative pricing. To cope with the growing complexity, narrowing margins, and shortening lifecycle of the individual derivative product, an efficient, yet modular, implementation of the pricing algorithms is necessary. Mathematical Finance is the first book to harmonize the theory, modeling, and implementation of today's most prevalent pricing models under one convenient cover. Building a bridge from academia to practice, this selfcontained text applies theoretical concepts to realworld examples and introduces stateoftheart, objectoriented programming techniques that equip the reader with the conceptual and illustrative tools needed to understand and develop successful derivative pricing models.
Utilizing almost twenty years of academic and industry experience, the author discusses the mathematical concepts that are the foundation of commonly used derivative pricing models, and insightful Motivation and Interpretation sections for each concept are presented to further illustrate the relationship between theory and practice. Indepth coverage of the common characteristics found amongst successful pricing models are provided in addition to key techniques and tips for the construction of these models. The opportunity to interactively explore the book's principal ideas and methodologies is made possible via a related Web site that features interactive Java experiments and exercises.
While a high standard of mathematical precision is retained, Mathematical Finance emphasizes practical motivations, interpretations, and results and is an excellent textbook for students in mathematical finance, computational finance, and derivative pricing courses at the upper undergraduate or beginning graduate level. It also serves as a valuable reference for professionals in the banking, insurance, and asset management industries.
Table of Contents
1.1 Theory, Modeling and Implementation.
1.2 Interest Rate Models and Interest Rate Derivatives.
1.3 How to Read this Book.
1.3.1 Abridged Versions.
1.3.2 Special Sections.
1.3.3 Notation.
I: FOUNDATIONS.
2. Foundations.
2.1 Probability Theory.
2.2 Stochastic Processes.
2.3 Filtration.
2.4 Brownian Motion.
2.5 Wiener Measure, Canonical Setup.
2.6 Itô Calculus.
2.6.1 Itô Integral.
2.6.2 Itô Process.
2.6.3 Itô Lemma and Product Rule.
2.7 Brownian Motion with Instantaneous Correlation.
2.8 Martingales.
2.8.1 Martingale Representation Theorem.
2.9 Change of Measure (Girsanov, Cameron, Martin).
2.10 Stochastic Integration.
2.11 Partial Differential Equations (PDE).
2.11.1 FeynmanKac Theorem .
2.12 List of Symbols.
3. Replication.
3.1 Replication Strategies.
3.1.1 Introduction.
3.1.2 Replication in a discrete Model.
3.2 Foundations: Equivalent Martingale Measure.
3.2.1 Challenge and Solution Outline.
3.2.2 Steps towards the Universal Pricing Theorem.
3.3 Excursus: Relative Prices and Risk Neutral Measures.
3.3.1 Why relative prices?
3.3.2 Risk Neutral Measure.
II: FIRST APPLICATIONS.
4. Pricing of a European Stock Option under the BlackScholes Model.
5. Excursus: The Density of the Underlying of a European Call Option.
6. Excursus: Interpolation of European Option Prices.
6.1 NoArbitrage Conditions for Interpolated Prices.
6.2 Arbitrage Violations through Interpolation.
6.2.1 Example (1): Interpolation of four Prices.
6.2.2 Example (2): Interpolation of two Prices.
6.3 ArbitrageFree Interpolation of European Option Prices.
7. Hedging in Continuous and Discrete Time and the Greeks.
7.1 Introduction.
7.2 Deriving the Replications Strategy from Pricing Theory.
7.2.1 Deriving the Replication Strategy under the Assumption of a Locally Riskless Product.
7.2.2 The BlackScholes Differential Equation.
7.2.3 The Derivative V(t) as a Function of its Underlyings S _{i}(t).
7.2.4 Example: Replication Portfolio and PDE under a BlackScholes Model.
7.3 Greeks.
7.3.1 Greeks of a European CallOption under the BlackScholes model.
7.4 Hedging in Discrete Time: Delta and DeltaGamma Hedging.
7.4.1 Delta Hedging.
7.4.2 Error Propagation.
7.4.3 DeltaGamma Hedging.
7.4.4 Vega Hedging.
7.5 Hedging in Discrete Time: Minimizing the Residual Error (BouchaudSornette Method).
7.5.1 Minimizing the Residual Error at Maturity T.
7.5.2 Minimizing the Residual Error in each Time Step.
III: INTEREST RATE STRUCTURES, INTEREST RATE PRODUCTS AND ANALYTIC PRICING FORMULAS.
Motivation and Overview.
8. Interest Rate Structures.
8.1 Introduction.
8.1.1 Fixing Times and Tenor Times.
8.2 Definitions.
8.3 Interest Rate Curve Bootstrapping.
8.4 Interpolation of Interest Rate Curves.
8.5 Implementation.
9. Simple Interest Rate Products.
9.1 Interest Rate Products Part 1: Products without Optionality.
9.1.1 Fix, Floating and Swap.
9.1.2 MoneyMarket Account.
9.2 Interest Rate Products Part 2: Simple Options.
9.2.1 Cap, Floor, Swaption.
9.2.2 Foreign Caplet, Quanto.
10. The Black Model for a Caplet.
11. Pricing of a Quanto Caplet (Modeling the FFX).
11.1 Choice of Numéraire.
12. Exotic Derivatives.
12.1 Prototypical Product Properties.
12.2 Interest Rate Products Part 3: Exotic Interest Rate Derivatives.
12.2.1 Structured Bond, Structured Swap, Zero Structure.
12.2.2 Bermudan Option.
12.2.3 Bermudan Callable and Bermudan Cancelable.
12.2.4 Compound Options.
12.2.5 Trigger Products.
12.2.6 Structured Coupons.
12.2.7 Shout Options.
12.3 Product Toolbox.
IV: DISCRETIZATION AND NUMERICAL VALUATION METHODS.
Motivation and Overview.
13. Discretization of time and state space.
13.1 Discretization of Time: The Euler and the Milstein Scheme.
13.1.1 Definitions.
13.1.2 TimeDiscretization of a Lognormal Process.
13.2 Discretization of Paths (MonteCarlo Simulation) .
13.2.1 MonteCarlo Simulation.
13.2.2 Weighted MonteCarlo Simulation.
13.2.3 Implementation.
13.2.4 Review.
13.3 Discretization of State Space.
13.3.1 Definitions.
13.3.2 BackwardAlgorithm.
13.3.3 Review.
13.4 Path Simulation through a Lattice: Two Layers.
14. Numerical Methods for Partial Differential Equations.
15. Pricing Bermudan Options in a Monte Carlo Simulation.
15.1 Introduction.
15.2 Bermudan Options: Notation.
15.2.1 Bermudan Callable.
15.2.2 Relative Prices.
15.3 Bermudan Option as Optimal Exercise Problem.
15.3.1 Bermudan Option Value as single (unconditioned) Expectation: The Optimal Exercise Value.
15.4 Bermudan Option Pricing  The Backward Algorithm.
15.5 Resimulation.
15.6 Perfect Foresight.
15.7 Conditional Expectation as Functional Dependence.
15.8 Binning.
15.8.1 Binning as a LeastSquare Regression.
15.9 Foresight Bias.
15.10 Regression Methods  Least Square MonteCarlo.
15.10.1 Least Square Approximation of the Conditional Expectation.
15.10.2 Example: Evaluation of a Bermudan Option on a Stock (Backward Algorithm with Conditional Expectation Estimator).
15.10.3 Example: Evaluation of a Bermudan Callable.
15.10.4 Implementation.
15.10.5 Binning as linear LeastSquare Regression.
15.11 Optimization Methods.
15.11.1 Andersen Algorithm for Bermudan Swaptions.
15.11.2 Review of the Threshold Optimization Method.
15.11.3 Optimization of Exercise Strategy: A more general Formulation.
15.11.4 Comparison of Optimization Method and Regression.
Method.
15.12 Duality Method: Upper Bound for Bermudan Option Prices.
15.12.1 Foundations.
15.12.2 American Option Evaluation as Optimal Stopping Problem.
15.13 PrimalDual Method: Upper and Lower Bound.
16. Pricing PathDependent Options in a Backward Algorithm.
16.1 Evaluation of a Snowball / Memory in a Backward Algorithm.
16.2 Evaluation of a Flexi Cap in a Backward Algorithm.
17. Sensitivities (Partial Derivatives) of Monte Carlo Prices.
17.1 Introduction.
17.2 Problem Description.
17.2.1 Pricing using MonteCarlo Simulation.
17.2.2 Sensitivities from MonteCarlo Pricing.
17.2.3 Example: The Linear and the Discontinuous Payout.
17.2.4 Example: Trigger Products.
17.3 Generic Sensitivities: Bumping the Model.
17.4 Sensitivities by Finite Differences.
17.4.1 Example: Finite Differences applied to Smooth and Discontinuous Payout.
17.5 Sensitivities by Pathwise Differentiation.
17.5.1 Example: Delta of a European Option under a BlackScholes Model.
17.5.2 Pathwise Differentiation for Discontinuous Payouts.
17.6 Sensitivities by Likelihood Ratio Weighting.
17.6.1 Example: Delta of a European Option under a BlackScholes Model using Pathwise Derivative.
17.6.2 Example: Variance Increase of the Sensitivity when using Likelihood Ratio Method for Smooth Payouts.
17.7 Sensitivities by Malliavin Weighting.
17.8 Proxy Simulation Scheme.
18. Proxy Simulation Schemes for Monte Carlo Sensitivities and Importance Sampling.
18.1 Full Proxy Simulation Scheme.
18.1.1 Calculation of MonteCarlo weights.
18.2 Sensitivities by Finite Differences on a Proxy Simulation Scheme.
18.2.1 Localization.
18.2.2 ObjectOriented Design.
18.3 Importance Sampling.
18.3.1 Example.
18.4 Partial Proxy Simulation Schemes.
18.4.1 Linear Proxy Constraint.
18.4.2 Comparison to Full Proxy Scheme Method.
18.4.3 NonLinear Proxy Constraint.
18.4.4 Transition Probability from a Nonlinear Proxy Constraint.
18.4.5 Sensitivity with respect to the Diffusion Coefficients  Vega.
18.4.6 Example: LIBOR Target Redemption Note.
18.4.7 Example: CMS Target Redemption Note.
V: PRICING MODELS FOR INTEREST RATE DERIVATIVES.
19. LIBOR Market Models.
19.1 LIBOR Market Model.
19.1.1 Derivation of the Drift Term.
19.1.2 The Short Period Bond P(T_{m(t)+1};t) .
19.1.3 Discretization and (MonteCarlo) Simulation.
19.1.4 Calibration  Choice of the free Parameters.
19.1.5 Interpolation of Forward Rates in the LIBOR Market Model.
19.2 Object Oriented Design.
19.2.1 Reuse of Implementation.
19.2.2 Separation of Product and Model.
19.2.3 Abstraction of Model Parameters.
19.2.4 Abstraction of Calibration.
19.3 Swap Rate Market Models (Jamshidian 1997).
19.3.1 The Swap Measure.
19.3.2 Derivation of the Drift Term.
19.3.3 Calibration  Choice of the free Parameters.
20. Swap Rate Market Models.
20.1 Definitions.
20.2 Terminal Correlation examined in a LIBOR Market Model Example.
20.2.1 Decorrelation in a OneFactor Model.
20.2.2 Impact of the Time Structure of the Instantaneous Volatility on Caplet and Swaption Prices.
20.2.3 The Swaption Value as a Function of Forward Rates.
20.3 Terminal Correlation is dependent on the Equivalent Martingale Measure.
20.3.1 Dependence of the Terminal Density on the Martingale Measure.
21. Excursus: Instantaneous Correlation and Terminal Correlation.
21.1 Short Rate Process in the HJM Framework.
21.2 The HJM Drift Condition.
22.HeathJarrowMorton Framework: Foundations.
22.1 Introduction.
22.2 The Market Price of Risk.
22.3 Overview: Some Common Models.
22.4 Implementations.
22.4.1 MonteCarlo Implementation of ShortRate Models.
22.4.2 Lattice Implementation of ShortRate Models.
23. ShortRate Models.
23.1 Short Rate Models in the HJM Framework.
23.1.1 Example: The HoLee Model in the HJM Framework.
23.1.2 Example: The HullWhite Model in the HJM Framework.
23.2 LIBOR Market Model in the HJM Framework.
23.2.1 HJM Volatility Structure of the LIBOR Market Model.
23.2.2 LIBOR Market Model Drift under the QB Measure.
23.2.3 LIBOR Market Model as a Short Rate Model.
24 HeathJarrowMorton Framwork: Immersion of ShortRate Models and LIBOR Market Model.
24.1 Model.
24.2 Interpretation of the Figures.
24.3 Mean Reversion.
24.4 Factors.
24.5 Exponential Volatility Function.
24.6 Instantaneous Correlation.
25. Excursus: Shape of teh Interst Rate Curve under Mean Reversion and a Multifactor Model.
25.1 Introduction.
25.2 Cheyette Model.
26. RitchkenSakarasubramanian Framework: JHM with Low Markov Dimension.
26.1 Introduction.
26.1.1 The Markov Functional Assumption (independent of the model considered) .
26.1.2 Outline of this Chapter .
26.2 Equity Markov Functional Model.
26.2.1 Markov Functional Assumption.
26.2.2 Example: The BlackScholes Model.
26.2.3 Numerical Calibration to a Full TwoDimensional European Option Smile Surface.
26.2.4 Interest Rates.
26.2.5 Model Dynamics.
26.2.6 Implementation.
26.3 LIBOR Markov Functional Model.
26.3.1 LIBOR Markov Functional Model in Terminal Measure.
26.3.2 LIBOR Markov Functional Model in Spot Measure.
26.3.3 Remark on Implementation.
26.3.4 Change of numéraire in a MarkovFunctional Model.
26.4 Implementation: Lattice.
26.4.1 Convolution with the Normal Probability Density.
26.4.2 State space discretization.
Markov Functional Models.
PART VI: Extended Models.
27.1 Introduction  Different Types of Spreads.
27.1.1 Spread on a Coupon.
27.1.2 Credit Spread.
27.2 Defaultable Bonds.
27.3 Integrating deterministic Credit Spread into a Pricing Model.
27.3.1 Deterministic Credit Spread.
27.3.2 Implementation.
27.4 Receiver’s and Payer’s Credit Spreads.
27.4.1 Example: Defaultable Forward Starting Coupon Bond.
27.4.2 Example: Option on a Defaultable Coupon Bond.
28. Credit Spreads.
28.1 Cross Currency LIBOR Market Model.
28.1.1 Derivation of the Drift Term under SpotMeasure.
28.1.2 Implementation.
28.2 Equity Hybrid LIBOR Market Model.
28.2.1 Derivation of the Drift Term under SpotMeasure.
28.2.2 Implementation.
28.3 EquityHybrid CrossCurrency LIBOR Market Model.
28.3.1 Summary.
28.3.2 Implementation.
29. Hybrid Models.
29.1 Elements of Object Oriented Programming: Class and Objects.
29.1.1 Example: Class of a Binomial Distributed Random Variable.
29.1.2 Constructor.
29.1.3 Methods: Getter, Setter, Static Methods.
29.2 Principles of Object Oriented Programming.
29.2.1 Encapsulation and Interfaces.
29.2.2 Abstraction and Inheritance.
29.2.3 Polymorphism.
29.3 Example: A Class Structure for One Dimensional Root Finders.
29.3.1 Root Finder for General Functions.
29.3.2 Root Finder for Functions with Analytic Derivative: Newton Method.
29.3.3 Root Finder for Functions with Derivative Estimation: Secant Method.
29.4 Anatomy of a Java™ Class.
29.5 Libraries.
29.5.1 Java™2 Platform, Standard Edition (j2se).
29.5.2 Java™2 Platform, Enterprise Edition (j2ee).
29.5.3 Colt.
29.5.4 CommonsMath: The Jakarta Mathematics Library.
29.6 Some Final Remarks.
29.6.1 Object Oriented Design (OOD) / Unified Modeling Language.
PART VII: Implementation
30. ObjectOriented Implementatin in Java^{TM}.
PART VIII: Appendices.
A: A small Collection of Common Misconceptions.
B: Tools (Selection).
B.1 Linear Regression.
B.2 Generation of Random Numbers.
B.2.1 Uniform Distributed Random Variables.
B.2.2 Transformation of the Random Number Distribution via the Inverse Distribution Function.
B.2.3 Normal Distributed Random Variables.
B.2.4 Poisson Distributed Random Variables.
B.2.5 Generation of Paths of an ndimensional Brownian Motion.
B.3 Factor Decomposition  Generation of Correlated Brownian Motion.
B.4 Factor Reduction.
B.5 Optimization (onedimensional): Golden Section Search.
B.6 Convolution with Normal Density.
C: Exercises.
D: List of Symbols.
E: Java™ Source Code (Selection).
E.1 Java™ Classes for Chapter 29.
List of Figures.
List of Tables.
List of Listings.
Bibliography.
Index.
Author Information
Reviews
"An excellent textbook for students in mathematical finance, computational finance, and derivative pricing courses at the upper undergraduate or beginning graduate level." (Mathematical Reviews 2007)
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