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Mathematical Finance: Theory, Modeling, Implementation

Christian Fries
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Ita Cirovic Donev
, on

Mathematical finance, as a field, has grown substantially over the past several years. New financial products are derived quite frequently. This in turn encourages research and hence the publishing of new books (not necessarily with new content). This book fits this description.

The first part of the book covers concepts from measure theory, stochastic processes and integration and replication strategies. It would be beneficial to have some graduate background in these areas, as the author does not provide any motivation or explanation of the results; the main results are just listed.

The second part of the book gives first applications, i.e. the pricing methods of a European stock option under the Black-Scholes model. The examples provided are quite detailed; the mathematical exposition is not hand-waved. Appropriate figures further illustrate the examples and concepts presented.

The remaining part of the book focuses on fixed income products. The author provides theory, modeling and implementation of the concepts. To quote from the back cover, “Mathemtical Finance is the first book to harmonize the theory, modeling, and implementation of today’s most prevalent pricing models under one convenient cover.”

I don’t quite agree with this statement, as the author does not spend too much time explaining the theory. The motivation is quite weak and the derivation of the concepts is almost non existent. The explanations are limited only to the presentation of the results. The section that precedes each chapter gives some motivation. For this type of the book, intended primarily for students, I feel this is not adequate. Modeling is the aspect presented in the most detailed form. I expected much more from the implementation part. For practitioners this should be one of the most interesting parts, but apart from some explanations of the ways to define classes and objects followed by smaller examples there is no further guidance.

Throughout the book the author provides notes for additional reading material and experiments. These should be quite appealing to students and practitioners. The author provides additional experiments and Java applets on the book’s website.

Overall, this is more of a cookbook for the fixed income products from the mathematical finance perspective. The book should be very useful to practitioners and students with appropriate background as it would serve as a reminder of the results. For beginners, this could potentially be a very confusing text due to the lack of detail in the derivation of the results. However, along with some additional references, it could serve as a good reference book on the topic.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.

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.


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 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.


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.


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 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.


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.


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.


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 numéraire 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™ 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 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™ Source Code (Selection).

E.1 Java™ Classes for Chapter 29.

List of Figures.

List of Tables.

List of Listings.