Mathematical Finance – Theory, Modeling and Implementation

Christian Fries, PhD, is Lecturer of Mathematical Finance at the University of Frankfurt and head of financial model development at DZ Bank AG Frankfurt, both located in Germany. With extensive knowledge in various programming languages, Dr. Fries has conducted quantitative analysis and overseen the implementation of mathematical modeling platforms at numerous financial institutions. His research interests within the field of mathematical finance include the LIBOR Market Model, Efficient Calculation of Risk Measures with Monte-Carlo Methods, Pricing of Bermudan Options with Monte-Carlo Methods, and Markov Functional Models.

1. Introduction. 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 Ito Calculus. 2.6.1 Ito Integral. 2.6.2 Ito Process. 2.6.3 Ito 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 Feynman-Kac 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 Black-Scholes Model. 5. Excursus: The Density of the Underlying of a European Call Option. 6. Excursus: Interpolation of European Option Prices. 6.1 No-Arbitrage 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 Arbitrage-Free 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 Black-Scholes 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 Black-Scholes Model. 7.3 Greeks. 7.3.1 Greeks of a European Call-Option under the Black-Scholes model. 7.4 Hedging in Discrete Time: Delta and Delta-Gamma Hedging. 7.4.1 Delta Hedging. 7.4.2 Error Propagation. 7.4.3 Delta-Gamma Hedging. 7.4.4 Vega Hedging. 7.5 Hedging in Discrete Time: Minimizing the Residual Error (Bouchaud-Sornette 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 Money-Market 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 Numeraire. 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 Time-Discretization of a Lognormal Process. 13.2 Discretization of Paths (Monte-Carlo Simulation) . 13.2.1 Monte-Carlo Simulation. 13.2.2 Weighted Monte-Carlo Simulation. 13.2.3 Implementation. 13.2.4 Review. 13.3 Discretization of State Space. 13.3.1 Definitions. 13.3.2 Backward-Algorithm. 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 Re-simulation. 15.6 Perfect Foresight. 15.7 Conditional Expectation as Functional Dependence. 15.8 Binning. 15.8.1 Binning as a Least-Square Regression. 15.9 Foresight Bias. 15.10 Regression Methods - Least Square Monte-Carlo. 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 Least-Square 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 Primal-Dual Method: Upper and Lower Bound. 16. Pricing Path-Dependent 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 Monte-Carlo Simulation. 17.2.2 Sensitivities from Monte-Carlo 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 Black-Scholes 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 Black-Scholes 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 Monte-Carlo weights. 18.2 Sensitivities by Finite Differences on a Proxy Simulation Scheme. 18.2.1 Localization. 18.2.2 Object-Oriented 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 Non-Linear 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(Tm(t)+1;t) . 19.1.3 Discretization and (Monte-Carlo) 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 De-correlation in a One-Factor 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.Heath-Jarrow-Morton Framework: Foundations. 22.1 Introduction. 22.2 The Market Price of Risk. 22.3 Overview: Some Common Models. 22.4 Implementations. 22.4.1 Monte-Carlo Implementation of Short-Rate Models. 22.4.2 Lattice Implementation of Short-Rate Models. 23. Short-Rate Models. 23.1 Short Rate Models in the HJM Framework. 23.1.1 Example: The Ho-Lee Model in the HJM Framework. 23.1.2 Example: The Hull-White 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 Heath-Jarrow-Morton Framwork: Immersion of Short-Rate 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. Ritchken-Sakarasubramanian 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 Black-Scholes Model. 26.2.3 Numerical Calibration to a Full Two-Dimensional 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 numeraire in a Markov-Functional 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 Spot-Measure. 28.1.2 Implementation. 28.2 Equity Hybrid LIBOR Market Model. 28.2.1 Derivation of the Drift Term under Spot-Measure. 28.2.2 Implementation. 28.3 Equity-Hybrid Cross-Currency 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(t) Class. 29.5 Libraries. 29.5.1 Java(t)2 Platform, Standard Edition (j2se). 29.5.2 Java(t)2 Platform, Enterprise Edition (j2ee). 29.5.3 Colt. 29.5.4 Commons-Math: The Jakarta Mathematics Library. 29.6 Some Final Remarks. 29.6.1 Object Oriented Design (OOD) / Unified Modeling Language. PART VII: Implementation 30. Object-Oriented Implementatin in JavaTM. 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 n-dimensional Brownian Motion. B.3 Factor Decomposition - Generation of Correlated Brownian Motion. B.4 Factor Reduction. B.5 Optimization (one-dimensional): Golden Section Search. B.6 Convolution with Normal Density. C: Exercises. D: List of Symbols. E: Java(t) Source Code (Selection). E.1 Java(t) Classes for Chapter 29. List of Figures. List of Tables. List of Listings. Bibliography. Index.