Option Valuation ─ A First Course in Financial Mathematics

Offers a straightforward account of the principles and models of option pricing
Focuses on the (discrete time) binomial model and the (continuous time) Black-Scholes-Merton model
Develops probability theory and finance theory from first principles
Covers various types of financial derivatives, including currency forwards, put and call options, and path-dependent options (Asian, lookback, and barrier options)
Uses the notion of variation of a function to illustrate the similarities and differences between classical calculus and stochastic calculus
Presents a martingale approach to option pricing
Contains many examples and end-of-chapter exercises
Solutions manual available for qualifying instructors
Option Valuation: A First Course in Financial Mathematics provides a straightforward introduction to the mathematics and models used in the valuation of financial derivatives. It examines the principles of option pricing in detail via standard binomial and stochastic calculus models. Developing the requisite mathematical background as needed, the text presents an introduction to probability theory and stochastic calculus suitable for undergraduate students in mathematics, economics, and finance.

The first nine chapters of the book describe option valuation techniques in discrete time, focusing on the binomial model. The author shows how the binomial model offers a practical method for pricing options using relatively elementary mathematical tools. The binomial model also enables a clear, concrete exposition of fundamental principles of finance, such as arbitrage and hedging, without the distraction of complex mathematical constructs. The remaining chapters illustrate the theory in continuous time, with an emphasis on the more mathematically sophisticated Black-Scholes-Merton model.

Largely self-contained, this classroom-tested text offers a sound introduction to applied probability through a mathematical finance perspective. Numerous examples and exercises help students gain expertise with financial calculus methods and increase their general mathematical sophistication. The exercises range from routine applications to spreadsheet projects to the pricing of a variety of complex financial instruments. Hints and solutions to odd-numbered problems are given in an appendix and a full solutions manual is available for qualifying instructors.

Hugo D. Junghenn is a professor of mathematics at the George Washington University. His research interests include functional analysis and semigroups.

Interest and Present Value
Compound Interest
Annuities
Bonds
Rate of Return

Probability Spaces
Sample Spaces and Events
Discrete Probability Spaces
General Probability Spaces
Conditional Probability
Independence

Random Variables
Definition and General Properties
Discrete Random Variables
Continuous Random Variables
Joint Distributions
Independent Random Variables
Sums of Independent Random Variables

Options and Arbitrage
Arbitrage
Classification of Derivatives
Forwards
Currency Forwards
Futures
Options
Properties of Options
Dividend-Paying Stocks

Discrete-Time Portfolio Processes
Discrete-Time Stochastic Processes
Self-Financing Portfolios
Option Valuation by Portfolios

Expectation of a Random Variable
Discrete Case: Definition and Examples
Continuous Case: Definition and Examples
Properties of Expectation
Variance of a Random Variable
The Central Limit Theorem

The Binomial Model
Construction of the Binomial Model
Pricing a Claim in the Binomial Model
The Cox-Ross-Rubinstein Formula

Conditional Expectation and Discrete-Time Martingales
Definition of Conditional Expectation
Examples of Conditional Expectation
Properties of Conditional Expectation
Discrete-Time Martingales

The Binomial Model Revisited
Martingales in the Binomial Model
Change of Probability
American Claims in the Binomial Model
Stopping Times
Optimal Exercise of an American Claim
Dividends in the Binomial Model
The General Finite Market Model

Stochastic Calculus
Differential Equations
Continuous-Time Stochastic Processes
Brownian Motion
Variation of Brownian Paths
Riemann-Stieltjes Integrals
Stochastic Integrals
The Ito-Doeblin Formula
Stochastic Differential Equations

The Black-Scholes-Merton Model
The Stock Price SDE
Continuous-Time Portfolios
The Black-Scholes-Merton PDE
Properties of the BSM Call Function

Continuous-Time Martingales
Conditional Expectation
Martingales: Definition and Examples
Martingale Representation Theorem
Moment Generating Functions
Change of Probability and Girsanov? Theorem

The BSM Model Revisited
Risk-Neutral Valuation of a Derivative
Proofs of the Valuation Formulas
Valuation under P
The Feynman-Kac Representation Theorem

Other Options
Currency Options
Forward Start Options
Chooser Options
Compound Options
Path-Dependent Derivatives
Quantos
Options on Dividend-Paying Stocks
American Claims in the BSM Model

Appendix A: Sets and Counting
Appendix B: Solution of the BSM PDE
Appendix C: Analytical Properties of the BSM Call Function
Appendix D: Hints and Solutions to Odd-Numbered Problems

Bibliography
Index

Exercises appear at the end of each chapter.