Book description
Versatile for Several Interrelated Courses at the Undergraduate and Graduate Levels
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.
Table of contents
- Preliminaries
- Preface
-
Part I Introduction to Pricing and Management of Financial Securities
- Chapter 1 Mathematics of Compounding
- Chapter 2 Primer on Pricing Risky Securities
- Chapter 3 Portfolio Management
-
Chapter 4 Primer on Derivative Securities
- 4.1 Forward Contracts
- 4.2 Basic Options Theory
- 4.3 Basics of Option Pricing
- 4.4 Exercises
-
Part II Discrete-Time Modelling
-
Chapter 5 Single-Period Arrow–Debreu Models
- 5.1 Specification of the Model
- 5.2 Analysis of the Arrow–Debreu Model
- 5.3 No-Arbitrage Asset Pricing
- 5.4 Pricing in an Incomplete Market
- 5.5 Change of Numéraire
- 5.6 Exercises
-
Chapter 6 Introduction to Discrete-Time Stochastic Calculus
- 6.1 A Multi-Period Binomial Probability Model
- 6.2 Information Flow
- 6.3 Conditional Expectation and Martingales
- 6.4 Exercises
-
Chapter 7 Replication and Pricing in the Binomial Tree Model
- 7.1 The Standard Binomial Tree Model
- 7.2 Self-Financing Strategies and Their Value Processes
- 7.3 Dynamic Replication in the Binomial Tree Model
- 7.4 Pricing and Hedging Non-Path-Dependent Derivatives
- 7.5 Pricing Formulae for Standard European Options
- 7.6 Pricing and Hedging Path-Dependent Derivatives
- 7.7 American Options
- 7.8 Exercises
- Chapter 8 General Multi-Asset Multi-Period Model
-
Chapter 5 Single-Period Arrow–Debreu Models
-
Part III Continuous-Time Modelling
- Chapter 9 Essentials of General Probability Theory
- Chapter 10 One-Dimensional Brownian Motion and Related Processes
-
Chapter 11 Introduction to Continuous-Time Stochastic Calculus
- 11.1 The Riemann Integral of Brownian Motion
- 11.2 The Riemann–Stieltjes Integral of Brownian Motion
- 11.3 The Itô Integral and Its Basic Properties
- 11.4 Itô Processes and Their Properties
- 11.5 Itô's Formula for Functions of BM and Itô Processes
- 11.6 Stochastic Differential Equations
- 11.7 The Markov Property, Feynman–Kac Formulae, and Transition CDFs and PDFs
- 11.8 Radon–Nikodym Derivative Process and Girsanov's Theorem
- 11.9 Brownian Martingale Representation Theorem
- 11.10 Stochastic Calculus for Multidimensional BM
- 11.11 Exercises
- Chapter 12 Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock
- Chapter 13 Risk-Neutral Pricing in a Multi-Asset Economy
- Chapter 14 American Options
- Chapter 15 Interest-Rate Modelling and Derivative Pricing
- Chapter 16 Alternative Models of Asset Price Dynamics
-
Part IV Computational Techniques
-
Chapter 17 Introduction to Monte Carlo and Simulation Methods
- 17.1 Introduction
- 17.2 Generation of Uniformly Distributed Random Numbers
- 17.3 Generation of Nonuniformly Distributed Random Numbers
- 17.4 Simulation of Random Processes
- 17.5 Variance Reduction Methods
- 17.6 Exercises
- References
-
Chapter 18 Numerical Applications to Derivative Pricing
- 18.1 Overview of Deterministic Numerical Methods
- 18.2 Pricing European Options
- 18.3 Pricing Early-Exercise and Path-Dependent Options
- Appendix: Some Useful Integral Identities and Symmetry Properties of Normal Random Variables
- Glossary of Symbols and Abbreviations
- References
-
Chapter 17 Introduction to Monte Carlo and Simulation Methods
Product information
- Title: Financial Mathematics
- Author(s):
- Release date: October 2018
- Publisher(s): Chapman and Hall/CRC
- ISBN: 9781315360485
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