An Introduction to Exotic Option Pricing

An Introduction to Exotic Option Pricing

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Features

  • Fully derives every price formula for the exotic options
  • Develops special pricing techniques based on the no-arbitrage principle
  • Contains a significant amount of original, previously unpublished material, such as the use of log-volutions and Mellin transforms to solve the Black–Scholes PDE
  • Demystifies many esoteric issues underpinning the mathematical treatment of the subject
  • Includes challenging problems at the end of each chapter to illustrate the special pricing techniques

Solutions manual available with qualifying course adoption

Summary

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 The author incorporates much of his own unpublished work, including ideas and techniques new to the general quantitative finance community.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

TECHNICAL BACKGROUND
Financial Preliminaries
European Derivative Securities
Exotic Options
Binary Options
No-Arbitrage
Pricing Methods
The Black–Scholes PDE Method
Derivation of Black–Scholes PDE
Meaning of the Black–Scholes PDE
The Fundamental Theorem of Asset Pricing
The EMM Pricing Method
Black–Scholes and the FTAP
Effect of Dividends

Mathematical Preliminaries
Probability Spaces
Brownian Motion
Stochastic Des
Stochastic Integrals
Itô’s Lemma
Martingales
Feynman-Kac Formula
Girsanov’s Theorem
Time Varying Parameters
The Black–Scholes PDE
The BS Green’s Function
Log-Volutions

Gaussian Random Variables
Univariate Gaussian Random Variables
Gaussian Shift Theorem
Rescaled Gaussians
Gaussian Moments
Central Limit Theorem
Log-Normal Distribution
Bivariate Normal
Multivariate Gaussian Statistics
Multivariate Gaussian Shift Theorem
Multivariate Itô’s Lemma and BS-PDE
Linear Transformations of Gaussian RVs

APPLICATIONS TO EXOTIC OPTION PRICING
Simple Exotic Options
First-Order Binaries
BS-Prices for First-Order Asset and Bond Binaries
Parity Relation
European Calls and Puts
Gap and Q-Options
Capped Calls and Puts
Range Forward Contracts
Turbo Binary
The Log-Contract
Pay-at-Expiry and Money-Back Options
Corporate Bonds
Binomial Trees
Options on a Traded Account

Dual Expiry Options
Forward Start Calls and Puts
Second-Order Binaries
Second-Order Asset and Bond Binaries
Second-Order Q-Options
Compound Options
Chooser Options
Reset Options
Simple Cliquet Option

Two-Asset Rainbow Options
Two-Asset Binaries
The Exchange Option
Options on the Minimum/Maximum of Two Assets
Product and Quotient Options
ICIAM Option Competition
Executive Stock Option

Barrier Options
Introduction
Method of Images
Barrier Parity Relations
Equivalent Payoffs for Barrier Options
Call and Put Barrier Options
Barrier Option Rebates
Barrier Option Extensions
Binomial Model for Barrier Options
Partial Time Barrier Options
Double Barriers
Sequential Barrier Options
Compound Barrier Options
Outside-Barrier Options
Reflecting Barriers

Lookback Options
Introduction
Equivalent Payoffs for Lookback Options
The Generic Lookback Options m(x, y, t) and M(x, z, t)
The Standard Lookback Calls and Puts
Partial Price Lookback Options
Partial Time Lookback Options
Extreme Spread Options
Look-Barrier Options

Asian Options
Introduction
Pricing Framework
Geometric Mean Asian Options
FTAP Method for GM Asian Options
PDE Method for GM Asian Options
Discrete GM Asian Options

Exotic Multi-Options
Introduction
Matrix and Vector Notation
The M-Binary Payoff
Valuation of the M-Binary
Previous Results Revisited
Multi-Asset, One-Period Asset and Bond Binaries
Quality Options
Compound Exchange Option
Multi-Asset Barrier Options

References

Index

A Summary and Exercises appear at the end of each chapter.

Author Bio(s)

Peter Buchen is an Associate Professor of Finance at the University of Sydney Business School. Dr. Buchen is co-founder of the Sydney Financial Mathematics Workshop, has authored many publications in financial mathematics, and has taught courses in quantitative finance and derivative securities. His research focuses on mathematical methods for valuing exotic options.

Editorial Reviews

"… an excellent guide to modern financial modelling. … The author promises that you can ‘price exotic options without needing a single integration’ and keeps the promise. … The author’s background in teaching makes this book easy to read."
—Osmo Jauri, International Statistical Review, 2014

"The book presents an entertaining and captivating course in option pricing, aiming to derive closed form analytical formulas for the prices of exotic options in an elegant way, provided such a formula exists. Thanks to the machinery developed by the author and his work group, pricing formulas for even the most complex exotic options are obtained from elementary pricing formulas using elegant arguments and simple algebraic manipulations, i.e. without lengthy integrations. … a very valuable treatise on exotic option pricing in a Black-Scholes economy. In addition, every chapter concludes with a set of highly relevant and inspiring exercises."
—Tamás Mátrai, Zentralblatt MATH 1242

 
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