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## Book Description

Financial Mathematics: A Comprehensive Treatment provides a unified, self-contained account of the main theory and application of methods behind modern-day financial mathematics. Tested and refined through years of the authors’ teaching experiences, the book encompasses a breadth of topics, from introductory to more advanced ones.

Accessible to undergraduate students in mathematics, finance, actuarial science, economics, and related quantitative areas, much of the text covers essential material for core curriculum courses on financial mathematics. Some of the more advanced topics, such as formal derivative pricing theory, stochastic calculus, Monte Carlo simulation, and numerical methods, can be used in courses at the graduate level. Researchers and practitioners in quantitative finance will also benefit from the combination of analytical and numerical methods for solving various derivative pricing problems.

With an abundance of examples, problems, and fully worked out solutions, the text introduces the financial theory and relevant mathematical methods in a mathematically rigorous yet engaging way. Unlike similar texts in the field, this one presents multiple problem-solving approaches, linking related comprehensive techniques for pricing different types of financial derivatives. The book provides complete coverage of both discrete- and continuous-time financial models that form the cornerstones of financial derivative pricing theory. It also presents a self-contained introduction to stochastic calculus and martingale theory, which are key fundamental elements in quantitative finance.

1. Preliminaries
2. Preface
3. Part I Introduction to Pricing and Management of Financial Securities
1. Chapter 1 Mathematics of Compounding
1. 1.1 Interest and Return
2. 1.2 Time Value of Money and Cash Flows
3. 1.3 Annuities
1. 1.3.1 Simple Annuities
2. 1.3.2 Determining the Term of an Annuity
3. 1.3.3 General Annuities
4. 1.3.4 Perpetuities
5. 1.3.5 Continuous Annuities
4. 1.4 Bonds
5. 1.5 Yield Rates
1. 1.5.1 Internal Rate of Return and Evaluation Criteria
2. 1.5.2 Determining Yield Rates for Bonds
3. 1.5.3 Approximation Methods
4. 1.5.4 The Yield Curve
6. 1.6 Exercises
2. Chapter 2 Primer on Pricing Risky Securities
1. 2.1 Stocks and Stock Price Models
2. 2.2 Basic Price Models
3. 2.3 Arbitrage and Risk-Neutral Pricing
4. 2.4 Value at Risk
5. 2.5 Dividend Paying Stock
6. 2.6 Exercises
3. Chapter 3 Portfolio Management
1. 3.1 Expected Utility Functions
1. 3.1.1 Utility Functions
2. 3.1.2 Mean-Variance Criterion
2. 3.2 Portfolio Optimization for Two Assets
1. 3.2.1 Portfolio of Two Assets
2. 3.2.2 Portfolio Lines
3. 3.2.3 The Minimum Variance Portfolio
4. 3.2.4 Selection of Optimal Portfolios
3. 3.3 Portfolio Optimization for N Assets
4. 3.4 The Capital Asset Pricing Model
5. 3.5 Exercises
4. Chapter 4 Primer on Derivative Securities
1. 4.1 Forward Contracts
1. 4.1.1 No-Arbitrage Evaluation of Forward Contracts
2. 4.1.2 Value of a Forward Contract
2. 4.2 Basic Options Theory
1. 4.2.1 Concept of an Option Contract
2. 4.2.2 Put-Call Parities
3. 4.2.3 Properties of European Options
4. 4.2.4 Early Exercise and American Options
5. 4.2.5 Nonstandard European Options
3. 4.3 Basics of Option Pricing
1. 4.3.1 Pricing of European-Style Derivatives in the Binomial Tree Model
2. 4.3.2 Pricing of American Options in the Binomial Tree Model
3. 4.3.3 Option Pricing in the Log-Normal Model: The Black–Scholes– Merton Formula
4. 4.3.4 Greeks and Hedging of Options
5. 4.3.5 Black–Scholes Equation
4. 4.4 Exercises
4. Part II Discrete-Time Modelling
1. Chapter 5 Single-Period Arrow–Debreu Models
1. 5.1 Specification of the Model
2. 5.2 Analysis of the Arrow–Debreu Model
3. 5.3 No-Arbitrage Asset Pricing
1. 5.3.1 The Law of One Price
2. 5.3.2 Arbitrage
3. 5.3.3 The First Fundamental Theorem of Asset Pricing
4. 5.3.4 Risk-Neutral Probabilities
5. 5.3.5 The Second Fundamental Theorem of Asset Pricing
6. 5.3.6 Investment Portfolio Optimization
4. 5.4 Pricing in an Incomplete Market
5. 5.5 Change of Numéraire
6. 5.6 Exercises
2. Chapter 6 Introduction to Discrete-Time Stochastic Calculus
1. 6.1 A Multi-Period Binomial Probability Model
1. 6.1.1 The Binomial Probability Space
2. 6.1.2 Random Processes
2. 6.2 Information Flow
1. 6.2.1 Partitions and Their Refinements
2. 6.2.2 Sigma-Algebras
3. 6.2.3 Filtration
4. 6.2.4 Filtered Probability Space
3. 6.3 Conditional Expectation and Martingales
1. 6.3.1 Measurability of Random Variables and Processes
2. 6.3.2 Conditional Expectations
3. 6.3.3 Properties of Conditional Expectations
4. 6.3.4 Conditioning in the Binomial Model
5. 6.3.5 Sub-, Super-, and True Martingales
6. 6.3.6 Classification of Stochastic Processes
7. 6.3.7 Stopping Times
4. 6.4 Exercises
3. Chapter 7 Replication and Pricing in the Binomial Tree Model
1. 7.1 The Standard Binomial Tree Model
2. 7.2 Self-Financing Strategies and Their Value Processes
3. 7.3 Dynamic Replication in the Binomial Tree Model
4. 7.4 Pricing and Hedging Non-Path-Dependent Derivatives
5. 7.5 Pricing Formulae for Standard European Options
6. 7.6 Pricing and Hedging Path-Dependent Derivatives
7. 7.7 American Options
8. 7.8 Exercises
4. Chapter 8 General Multi-Asset Multi-Period Model
1. 8.1 Main Elements of the Model
2. 8.2 Assets, Portfolios, and Strategies
3. 8.3 Fundamental Theorems of Asset Pricing
4. 8.4 Examples of Discrete-Time Models
5. 8.5 Exercises
5. Part III Continuous-Time Modelling
1. Chapter 9 Essentials of General Probability Theory
2. Chapter 10 One-Dimensional Brownian Motion and Related Processes
1. 10.1 Multivariate Normal Distributions
2. 10.2 Standard Brownian Motion
3. 10.3 Some Processes Derived from Brownian Motion
4. 10.4 First Hitting Times and Maximum and Minimum of Brownian Motion
5. 10.5 Exercises
3. Chapter 11 Introduction to Continuous-Time Stochastic Calculus
1. 11.1 The Riemann Integral of Brownian Motion
2. 11.2 The Riemann–Stieltjes Integral of Brownian Motion
3. 11.3 The Itô Integral and Its Basic Properties
4. 11.4 Itô Processes and Their Properties
5. 11.5 Itô's Formula for Functions of BM and Itô Processes
6. 11.6 Stochastic Differential Equations
7. 11.7 The Markov Property, Feynman–Kac Formulae, and Transition CDFs and PDFs
8. 11.8 Radon–Nikodym Derivative Process and Girsanov's Theorem
9. 11.9 Brownian Martingale Representation Theorem
10. 11.10 Stochastic Calculus for Multidimensional BM
11. 11.11 Exercises
4. Chapter 12 Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock
1. 12.1 Replication (Hedging) and Derivative Pricing in the Simplest Black–Scholes Economy
1. 12.1.1 Pricing Standard European Calls and Puts
2. 12.1.2 Hedging Standard European Calls and Puts
3. 12.1.3 Europeans with Piecewise Linear Payoffs
4. 12.1.4 Power Options
5. 12.1.5 Dividend Paying Stock
2. 12.2 Forward Starting and Compound Options
3. 12.3 Some European-Style Path-Dependent Derivatives
4. 12.4 Exercises
5. Chapter 13 Risk-Neutral Pricing in a Multi-Asset Economy
1. 13.1 General Multi-Asset Market Model: Replication and Risk-Neutral Pricing
2. 13.2 Black–Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives within a General Diffusion Model
1. 13.2.1 Standard European Option Pricing for Multi-Stock GBM
2. 13.2.2 Explicit Pricing Formulae for the GBM Model
3. 13.2.3 Cross-Currency Option Valuation
3. 13.3 Equivalent Martingale Measures: Derivative Pricing with General Numéraire Assets
4. 13.4 Exercises
6. Chapter 14 American Options
1. 14.1 Basic Properties of Early-Exercise Options
2. 14.2 Arbitrage-Free Pricing of American Options
3. 14.3 Perpetual American Options
4. 14.4 Finite-Expiration American Options
5. 14.5 Exercises
7. Chapter 15 Interest-Rate Modelling and Derivative Pricing
1. 15.1 Basic Fixed Income Instruments
1. 15.1.1 Bonds
2. 15.1.2 Forward Rates
3. 15.1.3 Arbitrage-Free Pricing
4. 15.1.4 Fixed Income Derivatives
2. 15.2 Single-Factor Models
3. 15.3 Heath–Jarrow–Morton Formulation
1. 15.3.1 HJM under Risk-Neutral Measure
2. 15.3.2 Relationship between HJM and Affine Yield Models
4. 15.4 Multifactor Affine Term Structure Models
5. 15.5 Pricing Derivatives under Forward Measures
6. 15.6 LIBOR Model
7. 15.7 Exercises
8. Chapter 16 Alternative Models of Asset Price Dynamics
1. 16.1 Stochastic Volatility Diffusion Models
1. 16.1.1 Local Volatility Models
2. 16.1.2 Constant Elasticity of Variance Model
3. 16.1.3 The Heston Model
2. 16.2 Models with Jumps
3. 16.3 Exercises
6. Part IV Computational Techniques
1. Chapter 17 Introduction to Monte Carlo and Simulation Methods
1. 17.1 Introduction
2. 17.2 Generation of Uniformly Distributed Random Numbers
3. 17.3 Generation of Nonuniformly Distributed Random Numbers
1. 17.3.1 Transformations of Random Variables
2. 17.3.2 Inversion Method
3. 17.3.3 Composition Methods
4. 17.3.4 Acceptance-Rejection Methods
5. 17.3.5 Multivariate Sampling
4. 17.4 Simulation of Random Processes
1. 17.4.1 Simulation of Brownian Processes
2. 17.4.2 Simulation of Gaussian Processes
3. 17.4.3 Diffusion Processes: Exact Simulation Methods
4. 17.4.4 Diffusion Processes: Approximation Schemes
5. 17.4.5 Simulation of Processes with Jumps
5. 17.5 Variance Reduction Methods
6. 17.6 Exercises
7. References
2. Chapter 18 Numerical Applications to Derivative Pricing
1. 18.1 Overview of Deterministic Numerical Methods
2. 18.1.2 Finite-Difference Methods
1. 18.1.2.1 Finite-Difference Approximations for ODEs
2. 18.1.2.2 Second-Order Linear PDEs
3. 18.1.2.3 Finite-Difference Approximations for the Heat Equation
4. 18.1.2.4 Stability Analysis
2. 18.2 Pricing European Options
1. 18.2.1 Pricing European Options by Quadrature Rules
2. 18.2.2 Pricing European Options by the Monte Carlo Method
3. 18.2.3 Pricing European Options by Tree Methods
4. 18.2.4 Pricing European Options by PDEs
1. 18.2.4.1 Pricing by the Heat Equation
2. 18.2.4.2 Pricing by the Black–Scholes PDE
5. 18.2.5 Calibration of Asset Price Models to Empirical Data
3. 18.3 Pricing Early-Exercise and Path-Dependent Options
1. 18.3.1 Pricing American and Bermudan Options
2. 18.3.2 Pricing Asian Options
3. Appendix: Some Useful Integral Identities and Symmetry Properties of Normal Random Variables
4. Glossary of Symbols and Abbreviations
5. References