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An Introduction to Exotic Option Pricing

Book Description

In an easy-to-understand, nontechnical yet mathematically elegant manner, An Introduction to Exotic Option Pricing shows how to price exotic options, including complex ones, without performing complicated integrations or formally solving partial differential equations (PDEs). The author incorporates much of his own unpublished work, including ideas and techniques new to the general quantitative finance community.

The first part of the text presents the necessary financial, mathematical, and statistical background, covering both standard and specialized topics. Using no-arbitrage concepts, the Black–Scholes model, and the fundamental theorem of asset pricing, the author develops such specialized methods as the principle of static replication, the Gaussian shift theorem, and the method of images. A key feature is the application of the Gaussian shift theorem and its multivariate extension to price exotic options without needing a single integration.

The second part focuses on applications to exotic option pricing, including dual-expiry, multi-asset rainbow, barrier, lookback, and Asian options. Pushing Black–Scholes option pricing to its limits, the author introduces a powerful formula for pricing a class of multi-asset, multiperiod derivatives. He gives full details of the calculations involved in pricing all of the exotic options.

Taking an applied mathematics approach, this book illustrates how to use straightforward techniques to price a wide range of exotic options within the Black–Scholes framework. These methods can even be used as control variates in a Monte Carlo simulation of a stochastic volatility model.

Table of Contents

  1. Preliminaries
  2. Symbols and Abbreviations
  3. Preface
  4. Part I Technical Background
    1. Chapter 1 Financial Prelinaries
      1. 1.1 European Derivative Securities
      2. 1.2 Exotic Options
      3. 1.3 Binary Options
      4. 1.4 No-Arbitrage
        1. 1.4.1 The Law of One Price
        2. 1.4.2 The Principle of Static Replication
        3. 1.4.3 Parity Relations
      5. 1.5 Pricing Methods
      6. 1.6 The Black-Scholes PDE Method
      7. 1.7 Derivation of Black-Scholes PDE
      8. 1.8 Meaning of the Black-Scholes PDE
        1. 1.8.1 Other Derivatives and the BS-PDE
      9. 1.9 The Fundamental Theorem of Asset Pricing
      10. 1.10 The EMM Pricing Method
        1. 1.10.1 The Martingale Restriction
      11. 1.11 Black-Scholes and the FTAP
      12. 1.12 Effect of Dividends
          1. Cost-of-Carry Model
      13. 1.13 Summary
      14. Exercise Problems
    2. Chapter 2 Mathematical Preliminaries
      1. 2.1 Probability Spaces
          1. The Tower Law
      2. 2.2 Brownian Motion
      3. 2.3 Stochastic DE’s
        1. 2.3.1 Arithmetic Brownian Motion
      4. 2.4 Stochastic Integrals
          1. Itô’s Isometry
      5. 2.5 Ito’s Lemma
        1. 2.5.1 Geometrical Brownian Motion
        2. 2.5.2 Ito’s Product and Quotient Rules
      6. 2.6 Martingales
        1. 2.6.1 Martingale Representation Theorem
      7. 2.7 Feynman-Kac Formula
      8. 2.8 Girsanov’s Theorem
      9. 2.9 Time Varying Parameters
      10. 2.10 The Black-Scholes PDE
      11. 2.11 The BS Green’s Function
      12. 2.12 Log-Volutions
        1. 2.12.1 The Mellin Transform
      13. 2.13 Summary
      14. Exercise Problems
    3. Chapter 3 Gaussian Random Variables
      1. 3.1 Univariate Gaussian Random Variables
      2. 3.2 Gaussian Shift Theorem
      3. 3.3 Rescaled Gaussians
      4. 3.4 Gaussian Moments
        1. 3.4.1 Sums of Independent Gaussians
      5. 3.5 Central Limit Theorem
      6. 3.6 Log-Normal Distribution
      7. 3.7 Bivariate Normal
        1. 3.7.1 Gaussian Shift Theorem (Bivariate Case)
      8. 3.8 Multi-Variate Gaussian Statistics
      9. 3.9 Multi-Variate Gaussian Shift Theorem
      10. 3.10 Multi-Variate Ito’s Lemma and BS-PDE
      11. 3.11 Linear Transformations of Gaussian RVs
      12. 3.12 Summary
      13. Exercise Problems
        1. Figure 3.1
        2. Figure 3.2
  5. Part II Applications to Exotic Option Pricing
    1. Chapter 4 Simple Exotic Options
      1. 4.1 First-Order Binaries
      2. 4.2 BS-Prices for First-Order Asset and Bond Binaries
      3. 4.3 Parity Relation
      4. 4.4 European Calls and Puts
      5. 4.5 Gap and Q-Options
      6. 4.6 Capped Calls and Puts
      7. 4.7 Range Forward Contracts
      8. 4.8 Turbo Binary
      9. 4.9 The Log-Contract
      10. 4.10 Pay-at-Expiry and Money-Back Options
      11. 4.11 Corporate Bonds
      12. 4.12 Binomial Trees
      13. 4.13 Options on a Traded Account
      14. 4.14 Summary
      15. Exercise Problems
        1. Figure 4.1
        2. Figure 4.2
        3. Figure 4.3
    2. Chapter 5 Dual Expiry Options
      1. 5.1 Forward Start Calls and Puts
      2. 5.2 Second-Order Binaries
      3. 5.3 Second-Order Asset and Bond Binaries
      4. 5.4 Second-Order Q-Options
      5. 5.5 Compound Options
      6. 5.6 Chooser Options
      7. 5.7 Reset Options
      8. 5.8 Simple Cliquet Option
      9. 5.9 Summary
      10. Exercise Problems
        1. Figure 5.1
        2. Figure 5.2
        3. Figure 5.3
        4. Figure 5.4
    3. Chapter 6 Two-Asset Rainbow Options
      1. 6.1 Two-Asset Binaries
      2. 6.2 The Exchange Option
      3. 6.3 Options on the Minimum/Maximum of Two Assets
      4. 6.4 Product and Quotient Options
      5. 6.5 ICIAM Option Competition
      6. 6.6 Executive Stock Option
      7. 6.7 Summary
      8. Exercise Problems
        1. Figure 6.1
        2. Figure 6.2
        3. Figure 6.3
    4. Chapter 7 Barrier Options
      1. 7.1 Introduction
      2. 7.2 Method of Images
      3. 7.3 Barrier Parity Relations
      4. 7.4 Equivalent Payoffs for Barrier Options
      5. 7.5 Call and Put Barrier Options
      6. 7.6 Barrier Option Rebates
      7. 7.7 Barrier Option Extensions
          1. 1. Dividend Yield
          2. 2. Exponential Barriers
      8. 7.8 Binomial Model for Barrier Options
      9. 7.9 Partial Time Barrier Options
        1. 7.9.1 Start-out, Partial Time Barrier Options
        2. 7.9.2 End-out, Partial Time Barrier Options
      10. 7.10 Double Barriers
        1. 7.10.1 Proof of MoI for Double Barrier Options
        2. 7.10.2 Double Barrier Calls and Puts
      11. 7.11 Sequential Barrier Options
      12. 7.12 Compound Barrier Options
      13. 7.13 Outside Barrier Options
      14. 7.14 Reflecting Barriers
      15. 7.15 Summary
      16. Exercise Problems
        1. Figure 7.1
    5. Chapter 8 Lookback Options
      1. 8.1 Introduction
      2. 8.2 Equivalent Payoffs for Lookback Options
      3. 8.3 The Generic Lookback Options m(x,y,t) and M(x,z,t)
      4. 8.4 The Standard Lookback Calls and Puts
      5. 8.5 Partial Price Lookback Options
      6. 8.6 Partial Time Lookback Options
      7. 8.7 Extreme Spread Options
      8. 8.8 Look-Barrier Options
      9. 8.9 Summary
      10. Exercise Problems
        1. Figure 8.1
    6. Chapter 9 Asian Options
      1. 9.1 Introduction
      2. 9.2 Pricing Framework
      3. 9.3 Geometric Mean Asian Options
      4. 9.4 FTAP Method for GM Asian Options
        1. 9.4.1 GM Asian Calls and Puts
      5. 9.5 PDE Method for GM Asian Options
        1. 9.5.1 Fixed Strike GM Asian Calls
        2. 9.5.2 Floating Strike GM Asian Calls
      6. 9.6 Discrete GM Asian Options
      7. 9.7 Summary
      8. Exercise Problems
        1. Figure 9.1
    7. Chapter 10 Exotic Multi-Options
      1. 10.1 Introduction
      2. 10.2 Matrix and Vector Notation
      3. 10.3 The M-Binary Payoff
      4. 10.4 Valuation of the M-Binary
      5. 10.5 Previous Results Revisited
      6. 10.6 Multi-Asset, 1-Period Asset and Bond Binaries
      7. 10.7 Quality Options
      8. 10.8 Compound Exchange Option
      9. 10.9 Multi-Asset Barrier Options
      10. 10.10 Summary
      11. Exercise Problems
    8. References