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The Mathematics of Financial Derivatives

Book Description

Finance is one of the fastest growing areas in the modern banking and corporate world. This, together with the sophistication of modern financial products, provides a rapidly growing impetus for new mathematical models and modern mathematical methods; the area is an expanding source for novel and relevant 'real-world' mathematics. In this book the authors describe the modelling of financial derivative products from an applied mathematician's viewpoint, from modelling through analysis to elementary computation. A unified approach to modelling derivative products as partial differential equations is presented, using numerical solutions where appropriate. Some mathematics is assumed, but clear explanations are provided for material beyond elementary calculus, probability, and algebra. Over 140 exercises are included. This volume will become the standard introduction to this exciting new field for advanced undergraduate students.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. Part One: Basic Option Theory
    1. 1 An Introduction to Options and Markets
      1. 1.1 Introduction
      2. 1.2 What is an Option?
      3. 1.3 Reading the Financial Press
      4. 1.4 What are Options For?
      5. 1.5 Other Types of Option
      6. 1.6 Forward and Futures Contracts
      7. 1.7 Interest Rates and Present Value
    2. 2 Asset Price Random Walks
      1. 2.1 Introduction
      2. 2.2 A Simple Model for Asset Prices
      3. 2.3 Itô’s Lemma
      4. 2.4 The Elimination of Randomness
    3. 3 The Black-Scholes Model
      1. 3.1 Introduction
      2. 3.2 Arbitrage
      3. 3.3 Option Values, Payoffs and Strategies
      4. 3.4 Put-call Parity
      5. 3.5 The Black-Scholes Analysis
      6. 3.6 The Black-Scholes Equation
      7. 3.7 Boundary and Final Conditions
      8. 3.8 The Black-Scholes Formulæ
      9. 3.9 Hedging in Practice
      10. 3.10 Implied Volatility
    4. 4 Partial Differential Equations
      1. 4.1 Introduction
      2. 4.2 The Diffusion Equation
      3. 4.3 Initial and Boundary Conditions
      4. 4.4 Forward versus Backward
    5. 5 The Black-Scholes Formulæ
      1. 5.1 Introduction
      2. 5.2 Similarity Solutions
      3. 5.3 An Initial Value Problem
      4. 5.4 The Formulæ Derived
      5. 5.5 Binary Options
      6. 5.6 Risk Neutrality
    6. 6 Variations on the Black-Scholes Model
      1. 6.1 Introduction
      2. 6.2 Options on Dividend-paying Assets
      3. 6.3 Forward and Futures Contracts
      4. 6.4 Options on Futures
      5. 6.5 Time-dependent Parameters
    7. 7 American Options
      1. 7.1 Introduction
      2. 7.2 The Obstacle Problem
      3. 7.3 American Options as Free Boundary Problems
      4. 7.4 The American Put
      5. 7.5 Other American Options
      6. 7.6 Linear Complementarity Problems
      7. 7.7 The American Call with Dividends
  7. Part Two: Numerical Methods
    1. 8 Finite-difference Methods
      1. 8.1 Introduction
      2. 8.2 Finite-difference Approximations
      3. 8.3 The Finite-difference Mesh
      4. 8.4 The Explicit Finite-difference Method
      5. 8.5 Implicit Finite-difference Methods
      6. 8.6 The Fully-implicit Method
      7. 8.7 The Crank–Nicolson Method
    2. 9 Methods for American Options
      1. 9.1 Introduction
      2. 9.2 Finite-difference Formulation
      3. 9.3 The Constrained Matrix Problem
      4. 9.4 Projected SOR
      5. 9.5 The Time-stepping Algorithm
      6. 9.6 Numerical Examples
      7. 9.7 Convergence of the Method
    3. 10 Binomial Methods
      1. 10.1 Introduction
      2. 10.2 The Discrete Random Walk
      3. 10.3 Valuing the Option
      4. 10.4 European Options
      5. 10.5 American Options
      6. 10.6 Dividend Yields
  8. Part Three: Further Option Theory
    1. 11 Exotic and Path-dependent Options
      1. 11.1 Introduction
      2. 11.2 Compound Options: Options on Options
      3. 11.3 Chooser Options
      4. 11.4 Barrier Options
      5. 11.5 Asian Options
      6. 11.6 Lookback Options
    2. 12 Barrier Options
      1. 12.1 Introduction
      2. 12.2 Knock-outs
      3. 12.3 Knock-ins
    3. 13 A Unifying Framework for Path-dependent Options
      1. 13.1 Introduction
      2. 13.2 Time Integrals of the Random Walk
      3. 13.3 Discrete Sampling
    4. 14 Asian Options
      1. 14.1 Introduction
      2. 14.2 Continuously Sampled Averages
      3. 14.3 Similarity Reductions
      4. 14.4 The Average Strike Option
      5. 14.5 Average Rate Options
      6. 14.6 Discretely Sampled Averages
    5. 15 Lookback Options
      1. 15.1 Introduction
      2. 15.2 Continuous Sampling of the Maximum
      3. 15.3 Discrete Sampling of the Maximum
      4. 15.4 Similarity Reductions
      5. 15.5 Some Numerical Examples
      6. 15.6 Two ‘Perpetual Options’
    6. 16 Options with Transaction Costs
      1. 16.1 Introduction
      2. 16.2 Discrete Hedging
      3. 16.3 Portfolios of Options
  9. Part Four: Interest Rate Derivative Products
    1. 17 Interest Rate Derivatives
      1. 17.1 Introduction
      2. 17.2 Basics of Bond Pricing
      3. 17.3 The Yield Curve
      4. 17.4 Stochastic Interest Rates
      5. 17.5 The Bond Pricing Equation
      6. 17.6 Solutions of the Bond Pricing Equation
      7. 17.7 The Extended Vasicek Model of Hull & White
      8. 17.8 Bond Options
      9. 17.9 Other Interest Rate Products
    2. 18 Convertible Bonds
      1. 18.1 Introduction
      2. 18.2 Convertible Bonds
      3. 18.3 Convertible Bonds with Random Interest Rate
  10. Hints to Selected Exercises
  11. Bibliography
  12. Index