An Introduction to the Mathematics of Financial Derivatives

3rd Edition

Authors: Ali Hirsa
Hardcover ISBN: 9780123846822
eBook ISBN: 9780123846839
Imprint: Academic Press
Published Date: 12th December 2013
Page Count: 480
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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


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


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


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


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


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


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


6.1 Introduction

6.2 Definitions

6.3 The Use of Martingales in Asset Pricing

6.4 Relevance of Martingales in Stoch


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© Academic Press 2014
Academic Press
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About the Author

Ali Hirsa

Ali Hirsa is managing partner at Sauma Capital, LLC. Previously he was partner and head of analytical trading strategy at Caspian Capital Management, LLC. Prior to joining Caspian, Ali worked as a quant at Morgan Stanley, Banc of America Securities, and Prudential Securities. He is also an adjunct associate professor of financial engineering at Columbia University since 2000 and Courant Institute of New York University in the mathematics of finance program since 2004. Ali is the author of Computational Methods in Finance, Chapman & Hall/CRC 2012 and the co-author of An Introduction to Mathematics of Financial Derivatives, Academic Press 2013 and is the editor of Journal of Investment Strategies. He has several publications and is a frequent speaker at academic and practitioner conferences. Ali received his Ph.D. in applied mathematics from University of Maryland at College Park under the supervision of Professors Howard C. Elman and Dilip B. Madan. He currently serves as a trustee at University of Maryland College Park Foundation.

Affiliations and Expertise

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


"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…", 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."