An accessible and mathematically rigorous resource for masters and PhD students
In Foundations of the Pricing of Financial Derivatives: Theory and Analysis two expert finance academics with professional experience deliver a practical new text for doctoral and masters’ students and also new practitioners. The book draws on the authors extensive combined experience teaching, researching, and consulting on this topic and strikes an effective balance between fine-grained quantitative detail and high-level theoretical explanations.
The authors fill the gap left by books directed at masters’-level students that often lack mathematical rigor. Further, books aimed at mathematically trained graduate students often lack quantitative explanations and critical foundational materials. Thus, this book provides the technical background required to understand the more advanced mathematics used in this discipline, in class, in research, and in practice.
Readers will also find:
- Tables, figures, line drawings, practice problems (with a solutions manual), references, and a glossary of commonly used specialist terms
- Review of material in calculus, probability theory, and asset pricing
- Coverage of both arithmetic and geometric Brownian motion
- Extensive treatment of the mathematical and economic foundations of the binomial and Black-Scholes-Merton models that explains their use and derivation, deepening readers’ understanding of these essential models
- Deep discussion of essential concepts, like arbitrage, that broaden students’ understanding of the basis for derivative pricing
- Coverage of pricing of forwards, futures, and swaps, including arbitrage-free term structures and interest rate derivatives
An effective and hands-on text for masters’-level and PhD students and beginning practitioners with an interest in financial derivatives pricing, Foundations of the Pricing of Financial Derivatives is an intuitive and accessible resource that properly balances math, theory, and practical applications to help students develop a healthy command of a difficult subject.
Table of Contents:
Preface
Chapter 1: Introduction and Overview
1.1 Motivation for this Book
1.2 What is a Derivative?
1.3 Options Versus Forwards, Futures, and Swaps
1.4 Size and Scope of the Financial Derivatives Markets
1.5 Outline and Features of the Book
1.6 Final Thoughts and Preview
1.7 Questions and Problems
Chapter 2: Boundaries, Limits, and Conditions on Option Prices
2.1 Setup, Definitions, and Arbitrage
2.2 Absolute Minimum and Maximum Values
2.3 The Value of an American Option Relative to the Value of a European Option
2.4 The Value of an Option at Expiration
2.5 The Lower Bounds of European and American Options and the Optimality of Early Exercise
2.6 Differences in Option Values by Exercise Price
2.7 The Effect of Differences in Time to Expiration
2.8 The Convexity Rule
2.9 Put-Call Parity
2.10 The Effect of Interest Rates on Option Prices
2.11 The Effect of Volatility on Option Prices
2.12 The Building Blocks of European Options
2.13 Recap and Preview
2.14 Questions and Problems
Chapter 3: Elementary Review of Mathematics for Finance
3.1 Summation Notation
3.2 Product Notation
3.3 Logarithms and Exponentials
3.4 Series Formulas
3.5 Calculus Derivatives
3.6 Integration
3.7 Differential Equations
3.8 Recap and Preview
3.9 Questions and Problems
Chapter 4: Elementary Review of Probability for Finance
4.1 Marginal, Conditional, and Joint Probabilities
4.2 Expectations, Variances, and Covariances of Discrete Random Variables
4.3 Continuous Random Variables
4.4 Some General Results in Probability Theory
4.5 Technical Introduction to Common Probability Distributions Used in Finance
4.6 Recap and Preview
4.7 Questions and Problems
Chapter 5: Financial Applications of Probability Distributions
5.1 The Univariate Normal Probability Distribution
5.2 Contrasting the Normal with the Lognormal Probability Distribution
5.3 Bivariate Normal Probability Distribution
5.4 The Bivariate Lognormal Probability Distribution
5.5 Recap and Preview
Appendix 5A An Excel Routine for the Bivariate Normal Probability
5.6 Questions and Problems
Chapter 6: Basic Concepts in Valuing Risky Assets and Derivatives
6.1 Valuing Risky Assets
6.2 Risk Neutral Pricing in Discrete Time
6.3 Identical Assets and the Law of One Price
6.4 Derivative Contracts
6.5 A First Look at Valuing Options
6.6 A World of Risk Averse and Risk Neutral Investors
6.7 Pricing Options Under Risk Aversion
6.8 Recap and Preview
6.9 Questions and Problems
Chapter 7: The Binomial Model
7.1 The One-Period Binomial Model for Calls
7.2 The One-Period Binomial Model for Puts
7.3 Arbitraging Price Discrepancies
7.4 The Multiperiod Model
7.5 American Options and Early Exercise in the Binomial Framework
7.6 Dividends and Recombination
7.7 Path Independence and Path Dependence
7.8 Recap and Preview
Appendix 7A Derivation of Equation
Appendix 7B Pascal’s Triangle and the Binomial Model
7.9 Questions and Problems
Chapter 8: Calculating the Greeks in the Binomial Model
8.1 Standard Approach
8.2 An Enhanced Method for Estimating Delta and Gamma
8.3 Numerical Examples
8.4 Dividends
8.5 Recap and Preview
8.5 Questions and Problems
Chapter 9: Convergence of the Binomial Model to the Black–Scholes–Merton Model
9.1 Setting up the Problem
9.2 The Hsia Proof
9.3 Put Options
9.4 Dividends
9.5 Recap and Preview
9.6 Questions and Problems
Chapter 10: The Basics of Brownian Motion and Wiener Processes
10.1 Brownian Motion
10.2 The Wiener Process
10.3 Properties of a Model of Asset Price Fluctuations
10.4 Building a Model of Asset Price Fluctuations
10.5 Simulating Brownian Motion and Wiener Processes
10.6 Formal Statement of Wiener Process Properties
10.7 Recap and Preview
Appendix 10A Simulation of the Wiener Process and the Square of the Wiener Process for Successively Smaller Time Intervals
10.8 Questions and Problems
Chapter 11: Stochastic Calculus and Itô’s Lemma
11.1 A Result from Basic Calculus
11.2 Introducing Stochastic Calculus and Itô’s Lemma
11.3 Itô’s Integral
11.4 The Integral Form of Itô’s Lemma
11.5 Some Additional Cases of Itô’s Lemma
11.6 Recap and Preview
Appendix 11A Technical Stochastic Integral Results
11A.1 Selected Stochastic Integral Results
11A.2 A General Linear Theorem
11.7 Questions and Problems
Chapter 12: Properties of the Lognormal and Normal Diffusion Processes for Modeling Assets
12.1 A Stochastic Process for the Asset Relative Return
12.2 A Stochastic Process for the Asset Price Change
12.3 Solving the Stochastic Differential Equation
12.4 Solutions to Stochastic Differential Equations are Not Always the Same as Solutions to Corresponding Ordinary Differential Equations
12.5 Finding the Expected Future Asset Price
12.5 Geometric Brownian Motion or Arithmetic Brownian Motion?
12.6 Recap and Preview
12.7 Questions and Problems
Chapter 13: Deriving the Black–Scholes–Merton Model
13.1 Derivation of the European Call Option Pricing Formula
13.2 The European Put Option Pricing Formula
13.3 Deriving the Black–Scholes–Merton Model as an Expected Value
13.4 Deriving the Black–Scholes–Merton Model as the Solution of a Partial Differential Equation
13.5 Decomposing the Black–Scholes–Merton Model into Binary Options
13.6 Black–Scholes–Merton Option Pricing when there are Dividends
13.7 Selected Black–Scholes–Merton Model Limiting Results
13.8 Computing the Black–Scholes–Merton Option Pricing Model Values
13.9 Recap and Preview
Appendix 13.A Deriving the Arithmetic Brownian Motion Option Pricing Model
13.10 Questions and Problems
Chapter 14: The Greeks in the Black–Scholes–Merton Model
14.1 Delta: The First Derivative with Respect to the Underlying Price
14.2 Gamma: The Second Derivative with Respect to the Underlying Price
14.3 Theta: The First Derivative with Respect to Time
14.4 Verifying the Solution of the Partial Differential Equation
14.5 Selected Other Partial Derivatives of the Black–Scholes–Merton Model
14.6 Partial Derivatives of the Black–Scholes–Merton European Put Option Pricing Model
14.7 Incorporating Dividends
14.8 Greek Sensitivities
14.8 Elasticities
14.8 Extended Greeks of the Black–Scholes–Merton Option Pricing Model
14.9 Recap and Preview
14.10 Questions and Problems
Chapter 15: Girsanov’s Theorem in Option Pricing
15.1 The Martingale Representation Theorem
15.2 Introducing the Radon-Nikodym Derivative by Changing the Drift for a Single Random Variable
15.3 A Complete Probability Space
15.4 Formal Statement of Girsanov’s Theorem
15.5 Changing the Drift in a Continuous Time Stochastic Process
15.6 Changing the Drift of an Asset Price Process
15.7 Recap and Preview
15.8 Questions and Problems
Chapter 16: Connecting Discrete and Continuous Brownian Motions
16.1 Brownian Motion in a Discrete World
16.2 Moving from a Discrete to a Continuous World
16.3 Changing the Probability Measure with the Radon-Nikodym Derivative in Discrete Time
16.4 The Kolmogorov Equations
16.5 Recap and Preview
16.6 Questions and Problems
Chapter 17: Applying Linear Homogeneity to Option Pricing
17.1 Introduction to Exchange Options
17.2 Homogeneous Functions
17.3 Euler’s Rule
17.4 Using Linear Homogeneity and Euler’s Rule to Derive the Black–Scholes–Merton Model
17.5 Exchange Option Pricing
17.6 Spread Options
17.7 Forward-Start Options
17.8 Recap and Preview
Appendix 17A Linear Homogeneity and the Arithmetic Brownian Motion Model
Appendix 17B Multivariate Itô’s Lemma
Appendix 17C Greeks of the Exchange Option Model
17.7 Questions and Problems
Chapter 18: Compound Option Pricing
18.1 Equity as an Option
18.2 Valuing an Option on the Equity as a Compound Option
18.3 Compound Option Boundary Conditions and Parities
18.4 Geske’s Approach to Valuing a Call on a Call
18.5 Characteristics of Geske’s Call on Call Option
18.6 Geske’s Call on Call Option Model and Linear Homogeneity
18.7 Generalized Compound Option Pricing Model
18.8 Installment Options
18.9 Recap and Preview
Appendix 18A Selected Greeks of the Compound Option
18.10 Questions and Problems
Chapter 19: American Call Option Pricing
19.1 Closed-Form American Call Pricing: Roll–Geske–Whaley
19.2 The Two-Payment Case
19.3 Recap and Preview
Appendix 19A Numerical Example of the One-Dividend Model
19.4 Questions and Problems
Chapter 20: American Put Option Pricing
20.1 The Nature of the Problem of Pricing an American Put
20.2 The American Put as a Series of Compound Options
20.3 Recap and Preview
20.4 Questions and Problems
Chapter 21: Min-Max Option Pricing
21.1 Characteristics of Stulz’ Min-Max Option
21.2 Pricing the Call on the Min
21.3 Other Related Options
21.4 Recap and Preview
Appendix 21A Multivariate Feynman-Kac Theorem
Appendix 21B An Alternative Derivation of the Min Max Option Model
21.5 Questions and Problems
Chapter 22: Pricing Forwards, Futures, and Options on Forwards and Futures
22.1 Forward Contracts
22.2 Pricing Futures Contracts
22.3 Options on Forwards and Futures
22.4 Recap and Preview
22.5 Questions and Problems
Chapter 23: Monte Carlo Simulation
23.1 Standard Monte Carlo Simulation of the Lognormal Diffusion
23.2 Reducing the Standard Error
23.3 Simulation with More than One Random Variable
23.4 Recap and Preview
23.5 Questions and Problems
Chapter 24: Finite Difference Methods
24.1 Setting up the Finite Difference Problem
24.2 The Explicit Finite Difference Method
24.3 The Implicit Finite Difference Method
24.4 Finite Difference Put Option Pricing
24.5 Dividends and Early Exercise
24.6 Recap and Preview
24.7 Questions and Problems
Chapter 25: The Term Structure of Interest Rates
25.1 The Unbiased Expectations Hypothesis
25.2 The Local Expectations Hypothesis
25.3 The Difference Between the Local and Unbiased Expectations Hypotheses
25.4 Other Term Structure of Interest Rate Hypotheses
25.5 Recap and Preview
25.6 Questions and Problems
Chapter 26: Interest Rate Contracts: Forward Rate Agreements, Swaps, and Options
26.1 Interest Rate Forwards
26.2 Interest Rate Swaps
26.3 Interest Rate Options
26.4 Recap and Preview
26.5 Questions and Problems
Chapter 27: Fitting an Arbitrage-Free Term Structure Model
27.1 Basic Structure of the HJM Model
27.2 Discretizing the HJM Model
27.3 Fitting a Binomial Tree to the Heath-Jarrow-Morton Model
27.4 Filling in the Remainder of the HJM Binomial Tree
27.5 Recap and Preview
27.6 Questions and Problems
Chapter 28. Pricing Fixed-Income Securities and Derivatives Using an Arbitrage-Free Binomial Tree
28.1 Zero Coupon Bonds
28.2 Coupon Bonds
28.3 Options on Zero-Coupon Bonds
28.4 Options on Coupon Bonds
28.5 Callable Bonds
28.6 Forward Rate Agreements (FRAs)
28.7 Interest Rate Swaps
28.8 Interest Rate Options
28.9 Interest Rate Swaptions
28.10 Interest Rate Futures
28.11 Recap and Preview
28.12 Questions and Problems
Chapter 29: Option Prices and the Prices of State-Contingent Claims
29.1 Pure Assets in the Market
29.2 Pricing Pure and Complex Assets
29.3 Numerical Example
29.4 State Pricing and Options in a Binomial Framework
29.5 State Pricing and Options in Continuous Time
29.6 Recap and Preview
29.7 Questions and Problems
Chapter 30: Option Prices and Expected Returns
30.1 The Basic Framework
30.2 Expected Returns on Options
30.3 Volatilities of Options
30.4 Options and the Capital Asset Pricing Model
30.5 Options and the Sharpe Ratio
30.6 The Stochastic Process Followed by the Option
30.7 Recap and Preview
30.8 Questions and Problems
Chapter 31: Implied Volatility and the Volatility Smile
31.1 Historical Volatility and the VIX
31.2 An Example of Implied Volatility
31.3 The Volatility Surface
31.4 The Perfect Substitutability of Options
31.5 Other Attempts to Explain the Implied Volatility Smile
31.6 How Practitioners Use the Implied Volatility Surface
31.7 Recap and Preview
31.8 Questions and Problems
Chapter 32: Pricing Foreign Currency Options
32.1 Definition of Terms
32.2 Option Payoffs
32.3 Valuation of the Options
32.4 Probability of Exercise
32.5 Some Terminology Confusion
32.6 Recap
32.7 Questions and Problems
References
Symbols Used
About the Author :
ROBERT E. BROOKS, PHD, CFA, is Professor Emeritus of Finance at the University of Alabama. He is the President of Financial Risk Management, LLC, a quantitative finance consulting firm. He is the author of several books and maintains a YouTube channel, @FRMHelpForYou.
DON M. CHANCE, PHD, CFA, holds the James C. Flores Endowed Chair of MBA Studies and is Professor of Finance at the E.J. Ourso College of Business at Louisiana State University. He is the author of four books on derivatives and risk management. His consulting firm is Omega Risk Advisors, LLC, and his website is donchance.com.
