# An Introduction to the Mathematics of Financial Derivatives

## 3rd Edition

**Authors:**Ali Hirsa

**eBook ISBN:**9780123846839

**Hardcover ISBN:**9780123846822

**Imprint:**Academic Press

**Published Date:**12th December 2013

**Page Count:**454

## Description

*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

## Readership

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.

## Table of Contents

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

## Details

- No. of pages:
- 454

- Language:
- English

- Copyright:
- © Academic Press 2014

- Published:
- 12th December 2013

- Imprint:
- Academic Press

- eBook ISBN:
- 9780123846839

- Hardcover ISBN:
- 9780123846822

## About the Author

### Ali Hirsa

Ali Hirsa is a professor and co-director of financial engineering at the Industrial Engineering & Operations Research at Columbia University. He is also Managing Partner at Sauma Capital, LLC and Senior Advisor at DV Trading, LLC where he was Managing Director and Global Head of Quantitative Strategy from June 2016 to August 2017. Ali was a Fellow at Courant Institute of New York University in the Mathematics of Finance Program from 2004 to 2014. He is co-inventor of “Methods for Post Trade Allocation” (US Patent 8,799,146). The method focuses on allocation of filled orders (post-trade) on any security to multiple managed accounts which has to be fair and unbiased. Current existing methods lead to biases and the invention provides a solution to this problem.

### Affiliations and Expertise

Columbia University, New York; and New York University, New York, USA

## Reviews

*"This text introduces quantitative tools used in pricing financial derivatives to those with basic knowledge of calculus and probability. It reviews basic derivative instruments, the arbitrage theorem, and deterministic calculus, and describes models and notation in pricing derivatives, tools in probability theory, martingales and martingale representations, differentiation in stochastic environments, the Wiener and Lévy processes and rare events in financial markets…"--**ProtoView.com,* February 2014*"Ali Hirsa has done a superb job with this third edition of the very popular Neftci's An Introduction to the Mathematics of Financial Derivatives. New chapters and sections have been added covering in particular credit derivatives (Chapter 23) and jump processes and the associated partial integro-differential equations. The new material on numerical methods, in particular on Fourier techniques (Chapter 22) and calibration (Chapter 25), and added examples and exercises are very welcome. Overall, this new edition offers substantially more that the previous one in all of its chapters. This is a unique sophisticated introduction to financial mathematics accessible to a wide audience. Truly remarkable!"--***Jean-Pierre Fouque, University of California, Santa Barbara**"The publication of this expansive and erudite text in a new edition by one of the most highly respected scholars in the field should be a welcome event for practitioners and academics alike."--

**Lars Tyge Nielsen, Columbia University**

"There are many books on mathematics, probability, and stochastic calculus, but relatively few focus entirely on the pricing and hedging of financial derivatives. I have used the second edition for finance and financial engineering classes for years, and will continue with the third edition; the book will no doubt remain a valuable reference for industry practitioners as well."--

**Robert L. Kimmel, National University of Singapore**

"An excellent introduction to a wide range of topics in pricing financial derivatives with highly accessible mathematical treatment. Its heuristic style in explaining basic mathematical concepts relevant to financial markets greatly facilitates understanding the fundamentals of derivative pricing."--

**Seppo Pynnonen, Unversity of Vaasa**

"What makes this introductory text unique for students or practitioners without a major in mathematics or physics is that it provides the most helpful heuristics while clearly stating how or why the concepts are useful for practical problems in finance. The timely additions on credit derivatives and PDEs provide considerable value-added in comparison to the second edition."--**Mishael Milaković, University of Bamberg**