An Introduction to the Mathematics of Financial Derivatives, 3rd Edition

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