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A Course in Financial Calculus

Book Description

Finance provides a dramatic example of the successful application of advanced mathematical techniques to the practical problem of pricing financial derivatives. This self-contained 2002 text is designed for first courses in financial calculus aimed at students with a good background in mathematics. Key concepts such as martingales and change of measure are introduced in the discrete time framework, allowing an accessible account of Brownian motion and stochastic calculus: proofs in the continuous-time world follow naturally. The Black-Scholes pricing formula is first derived in the simplest financial context. The second half of the book is then devoted to increasing the financial sophistication of the models and instruments. The final chapter introduces more advanced topics including stock price models with jumps, and stochastic volatility. A valuable feature is the large number of exercises and examples, designed to test technique and illustrate how the methods and concepts can be applied to realistic financial questions.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. 1 Single Period Models
    1. Summary
    2. 1.1 Some Definitions from Finance
    3. 1.2 Pricing a Forward
    4. 1.3 The One-step Binary Model
    5. 1.4 A Ternary Model
    6. 1.5 A Characterisation of no Arbitrage
    7. 1.6 The Risk-neutral Probability Measure
    8. Exercises
  7. 2 Binomial Trees and Discrete Parameter Martingales
    1. Summary
    2. 2.1 The Multiperiod Binary Model
    3. 2.2 American Options
    4. 2.3 Discrete Parameter Martingales and Markov Processes
    5. 2.4 Some Important Martingale Theorems
    6. 2.5 The Binomial Representation Theorem
    7. 2.6 Overture to Continuous Models
    8. Exercises
  8. 3 Brownian Motion
    1. Summary
    2. 3.1 Definition of the Process
    3. 3.2 Lévy’s Construction of Brownian Motion
    4. 3.3 The Reflection Principle and Scaling
    5. 3.4 Martingales in Continuous Time
    6. Exercises
  9. 4 Stochastic Calculus
    1. Summary
    2. 4.1 Stock Prices are not Differentiable
    3. 4.2 Stochastic Integration
    4. 4.3 Itô’s Formula
    5. 4.4 Integration by Parts and a Stochastic Fubini Theorem
    6. 4.5 The Girsanov Theorem
    7. 4.6 The Brownian Martingale Representation Theorem
    8. 4.7 Why Geometric Brownian Motion?
    9. 4.8 The Feynman–Kac Representation
    10. Exercises
  10. 5 The Black–Scholes Model
    1. Summary
    2. 5.1 The Basic Black–Scholes Model
    3. 5.2 Black–Scholes Price and Hedge for European Options
    4. 5.3 Foreign Exchange
    5. 5.4 Dividends
    6. 5.5 Bonds
    7. 5.6 Market Price of Risk
    8. Exercises
  11. 6 Different Payoffs
    1. Summary
    2. 6.1 European Options with Discontinuous Payoffs
    3. 6.2 Multistage Options
    4. 6.3 Lookbacks and Barriers
    5. 6.4 Asian Options
    6. 6.5 American Options
    7. Exercises
  12. 7 Bigger Models
    1. Summary
    2. 7.1 General Stock Model
    3. 7.2 Multiple Stock Models
    4. 7.3 Asset Prices with Jumps
    5. 7.4 Model Error
    6. Exercises
  13. Bibliography
  14. Notation
  15. Index