- 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- Contains a significant amount of original, previously unpublished material, such as the use of log-volutions and Mellin transforms to solve the Black–Scholes PDE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*

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 The author incorporates much of his own unpublished work, including ideas and techniques new to the general quantitative finance community.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

**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 PRICINGSimple Exotic Options **First-Order Binaries

BS-Prices for First-Order Asset and Bond Binaries

Parity Relation

European Calls and Puts

Gap and

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

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

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.*

**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.

"… 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