An Introduction to the Mathematics of Financial Derivatives is a popular, intuitive text that eases the transition between basic summaries of financial engineering to more advanced treatments using stochastic calculus. Requiring only a basic knowledge of calculus and probability, it takes readers on a tour of advanced financial engineering. This classic title has been revised by Ali Hirsa, who accentuates its well-known strengths while introducing new subjects, updating others, and bringing new continuity to the whole. Popular with readers because it emphasizes intuition and common sense, An Introduction to the Mathematics of Financial Derivatives remains the only "introductory" text that can appeal to people outside the mathematics and physics communities as it explains the hows and whys of practical finance problems.
Key Features
- Facilitates readers' understanding of underlying mathematical and theoretical models by presenting a mixture of theory and applications with hands-on learning
- Presented intuitively, breaking up complex mathematics concepts into easily understood notions
- Encourages use of discrete chapters as complementary readings on different topics, offering flexibility in learning and teaching
List of Symbols and Acronyms
Chapter 1. Financial Derivatives—A Brief Introduction
Abstract
1.1 Introduction
1.2 Definitions
1.3 Types of Derivatives
1.4 Forwards and Futures
1.5 Options
1.6 Swaps
1.7 Conclusion
1.8 References
1.9 Exercises
Chapter 2. A Primer on the Arbitrage Theorem
Abstract
2.1 Introduction
2.2 Notation
2.3 A Numerical Example
2.4 An Application: Lattice Models
2.5 Payouts and Foreign Currencies
2.6 Some Generalizations
2.7 Conclusions: A Methodology for Pricing Assets
2.8 References
2.9 Appendix: Generalization of the Arbitrage Theorem
2.10 Exercises
Chapter 3. Review of Deterministic Calculus
Abstract
3.1 Introduction
3.2 Some Tools of Standard Calculus
3.3 Functions
3.4 Convergence and Limit
3.5 Partial Derivatives
3.6 Conclusions
3.7 References
3.8 Exercises
Chapter 4. Pricing Derivatives: Models and Notation
Abstract
4.1 Introduction
4.2 Pricing Functions
4.3 Application: Another Pricing Model
4.4 The Problem
4.5 Conclusions
4.6 References
4.7 Exercises
Chapter 5. Tools in Probability Theory
Abstract
5.1 Introduction
5.2 Probability
5.3 Moments
5.4 Conditional Expectations
5.5 Some Important Models
5.6 Exponential Distribution
5.7 Gamma distribution
5.8 Markov Processes and Their Relevance
5.9 Convergence of Random Variables
5.10 Conclusions
5.11 References
5.12 Exercises
Chapter 6. Martingales and Martingale Representations
Abstract
6.1 Introduction
6.2 Definitions
6.3 The Use of Martingales in Asset Pricing
6.4 Relevance of Martingales in Stochastic Modeling
6.5 Properties of Martingale Trajectories
6.6 Examples of Martingales
6.7 The Simplest Martingale
6.8 Martingale Representations
6.9 The First Stochastic Integral
6.10 Martingale Methods and Pricing
6.11 A Pricing Methodology
6.12 Conclusions
6.13 References
6.14 Exercises
Chapter 7. Differentiation in Stochastic Environments
Abstract
7.1 Introduction
7.2 Motivation
7.3 A Framework for Discussing Differentiation
7.4 The “Size” of Incremental Errors
7.5 One Implication
7.6 Putting the Results Together
7.7 Conclusion
7.8 References
7.9 Exercises
Chapter 8. The Wiener Process, Lévy Processes, and Rare Events in Financial Markets
Abstract
8.1 Introduction
8.2 Two Generic Models
8.3 SDE in Discrete Intervals, Again
8.4 Characterizing Rare and Normal Events
8.5 A Model for Rare Events
8.6 Moments that Matter
8.7 Conclusions
8.8 Rare and Normal Events in Practice
8.9 References
8.10 Exercises
Chapter 9. Integration in Stochastic Environments
Abstract
9.1 Introduction
9.2 The Ito Integral
9.3 Properties of the Ito Integral
9.4 Other Properties of the Itô Integral
9.5 Integrals with Respect to Jump Processes
9.6 Conclusion
9.7 References
9.8 Exercises
Chapter 10. Itô’s Lemma
Abstract
10.1 Introduction
10.2 Types of Derivatives
10.3 Ito’s Lemma
10.4 The Ito Formula
10.5 Uses of Ito’s Lemma
10.6 Integral Form of Ito’s Lemma
10.7 Ito’s Formula in More Complex Settings
10.8 Conclusion
10.9 References
10.10 Exercises
Chapter 11. The Dynamics of Derivative Prices
Abstract
11.1 Introduction
11.2 A Geometric Description of Paths Implied by SDEs
11.3 Solution of SDEs
11.4 Major Models of SDEs
11.5 Stochastic Volatility
Pure Jump Framework
11.6 Conclusions
11.7 References
11.8 Exercises
Chapter 12. Pricing Derivative Products: Partial Differential Equations
Abstract
12.1 Introduction
12.2 Forming Risk-Free Portfolios
12.3 Accuracy of the Method
12.4 Partial Differential Equations
12.5 Classification of PDEs
12.6 A Reminder: Bivariate, Second-Degree Equations
12.7 Types of PDEs
12.8 Pricing Under Variance Gamma Model
12.9 Conclusions
12.10 References
12.11 Exercises
Chapter 13. PDEs and PIDEs—An Application
Abstract
13.1 Introduction
13.2 The Black–Scholes PDE
13.3 Local Volatility Model
13.4 Partial Integro-Differential Equations (PIDEs)
13.5 PDEs/PIDEs in Asset Pricing
13.6 Exotic Options
13.7 Solving PDEs/PIDEs in Practice
13.8 Conclusions
13.9 References
13.10 Exercises
Chapter 14. Pricing Derivative Products: Equivalent Martingale Measures
Abstract
14.1 Translations of Probabilities
14.2 Changing Means
14.3 The Girsanov Theorem
14.4 Statement of the Girsanov Theorem
14.5 A Discussion of the Girsanov Theorem
14.6 Which Probabilities?
14.7 A Method for Generating Equivalent Probabilities
14.8 Conclusion
14.9 References
14.10 Exercises
Chapter 15. Equivalent Martingale Measures
Abstract
15.1 Introduction
15.2 A Martingale Measure
15.3 Converting Asset Prices into Martingales
15.4 Application: The Black–Scholes Formula
15.5 Comparing Martingale and PDE Approaches
15.6 Conclusions
15.7 References
15.8 Exercises
Chapter 16. New Results and Tools for Interest-Sensitive Securities
Abstract
16.1 Introduction
16.2 A Summary
16.3 Interest Rate Derivatives
16.4 Complications
16.5 Conclusions
16.6 References
16.7 Exercises
Chapter 17. Arbitrage Theorem in a New Setting
Abstract
17.1 Introduction
17.2 A Model for New Instruments
17.3 Other Equivalent Martingale Measures
17.4 Conclusion
17.5 References
17.6 Exercises
Chapter 18. Modeling Term Structure and Related Concepts
Abstract
18.1 Introduction
18.2 Main Concepts
18.3 A Bond Pricing Equation
18.4 Forward Rates and Bond Prices
18.5 Conclusions: Relevance of the Relationships
18.6 References
18.7 Exercises
Chapter 19. Classical and HJM Approach to Fixed Income
Abstract
19.1 Introduction
19.2 The Classical Approach
19.3 The HJM Approach to Term Structure
19.4 How to Fit to Initial Term Structure
19.5 Conclusion
19.6 References
19.7 Exercises
Chapter 20. Classical PDE Analysis for Interest Rate Derivatives
20.1 Introduction
20.2 The Framework
20.3 Market Price of Interest Rate Risk
20.4 Derivation of the PDE
20.5 Closed-Form Solutions of the PDE
20.6 Conclusion
20.7 References
20.8 Exercises
Chapter 21. Relating Conditional Expectations to PDEs
21.1 Introduction
21.2 From Conditional Expectations to PDEs
21.3 From PDEs to Conditional Expectations
21.4 Generators, Feynman–Kac Formula, and Other Tools
21.5 Feynman–Kac Formula
21.6 Conclusions
21.7 References
21.8 Exercises
Chapter 22. Pricing Derivatives via Fourier Transform Technique
Abstract
22.1 Derivatives Pricing via the Fourier Transform
22.2 Findings and Observations
22.3 Conclusions
22.4 Problems
Chapter 23. Credit Spread and Credit Derivatives
Abstract
23.1 Standard Contracts
23.2 Pricing of Credit Default Swaps
23.3 Pricing Multi-name Credit Products
23.4 Credit Spread Obtained from Options Market
23.5 Problems
Chapter 24. Stopping Times and American-Type Securities
24.1 Introduction
24.2 Why Study Stopping Times?
24.3 Stopping Times
24.4 Uses of Stopping Times
24.5 A Simplified Setting
24.6 A Simple Example
24.7 Stopping Times and Martingales
24.8 Conclusions
24.9 References
24.10 Exercises
Chapter 25. Overview of Calibration and Estimation Techniques
Abstract
25.1 Calibration Formulation
25.2 Underlying Models
25.3 Overview of Filtering and Estimation
25.4 Exercises
References
Index
Choudhry. An Introduction to Credit Derivatives, 2nd edition. 9780080982953. 2013. 158 pp., $89.95.
Kosowski/Neftci. Principles of Financial Engineering, 3rd edition. 9780123869685. March 2014. 700 pp., $99.95.
Sundaresan. Fixed Income Markets and Their Derivatives, 4th edition. 9780123850935. March 2014. 600 pp., $99.95.
Upper-division undergraduates and graduate students seeking an introduction to the mathematics and concepts underlying financial derivatives in specific and investment vehicles (options, futures, and other financial engineering products) in general.
