Introduction to Financial Mathematics: With Computer Applications

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface to the Instructor
Preface to the Student
Acknowledgments
About the Authors
1. Introduction to Financial Derivatives and Valuation
1.1 Foundations in Economics and Finance
1.1.1 The Role of Exchanging Real Assets
1.1.2 The Role of Financial Assets
1.1.3 The Roles of Financial Mathematics and Financial Derivatives
1.1.4 The Roles of Arbitragers, Hedgers, and Speculators
1.2 Introduction to the Valuation of Financial Contracts
1.2.1 Market Prices, Risk, and Randomness
1.2.2 Expected Value as a Foundation of Asset Valuation
1.2.3 Discounted Expected Value and the Time Value of Money
1.2.4 Adjusting for Risk through Discounting
1.2.5 Valuing Financial Derivatives
1.3 Details Regarding Forward Contracts
1.3.1 Spot Prices and the Mechanics of Forward Contracts
1.3.2 Cash Settlement as an Alternative to Actual Physical Settlement
1.3.3 Forward Contracts and Futures Contracts
1.3.4 Applications of Forward Contracts
1.4 The Arbitrage-Free Pricing of Forward Contracts
1.4.1 Efficient and Frictionless Markets as a Foundation for Forward Price Models
1.4.2 Simplified Arbitrage-Free Model
1.4.3 An Arbitrage-Free Model of Forward Prices
1.4.4 Analytics of Forward Contracts
1.5 Introduction to Option Contracts
1.5.1 The Mechanics and Terms of Options
1.5.2 Applications of Financial Options
1.5.3 The Payoffs of Calls and Puts
1.5.4 Profit and Loss Diagrams for Analyzing Option Exposures
1.5.5 Applications of Options
1.5.6 Option Hedging Strategies
1.6 Put-Call Parity and Arbitrage
1.7 Arbitrage-Free Binomial Option Valuation
1.8 Compound Options and Option Strategies
1.8.1 Popular Option Strategies with Multiple Positions
1.8.2 Option Spread Strategies
1.8.3 Option Combination Strategies
1.8.4 Other Options
1.9 The Limits to Arbitrage and Complete Markets
1.10 Computing Systems: Mathematica and R
Chapter Summary
Chapter Demonstrating Exercises
End-of-Chapter Problems (Fundamental Problems)
Notes
2. Introduction to Interest Rates, Bonds, and Equities
2.1 The Time Value of Money and Interest Rates
2.1.1 Future Values and Interest Rates
2.1.2 Interest Rates, Future Values, and Compounding Methods
2.1.3 Discounting and Present Values
2.1.4 Valuation of Pure Discount Bonds
2.1.5 Coupon Bond Yields and Prices
2.1.6 Major Types of Fixed-Income Rates
2.2 Term Structures, Yield Curves, and Forward Rates
2.2.1 The Bootstrap Method for Treasury Zero Coupon Interest Rates and the Term Structure of Interest Rates
2.2.2 Coupon Bond Yield Curves
2.2.3 Forward Interest Rates
2.2.4 Forward Interest Rate Structures
2.3 Fixed-Income Risk Measurement and Management
2.3.1 Duration
2.3.2 Convexity
2.3.3 Interest Rate Risk Management
2.3.4 Introduction to Credit Risk and Default Risk
2.4 Equities and Equity Valuation
2.4.1 Equity as a Residual Claimant in a Firm's Capital Structure
2.4.2 The Dividend Discount Model
2.4.3 Technical and Fundamental and Analysis
2.4.4 Informational Market Efficiency
2.4.5 Dividend Policy and Theory
2.5 Asset Pricing and Equity Risk Management
2.5.1 Two-Asset Portfolios and Markowitz Diversification
2.5.2 Investor Preferences and Market Equilibrium
2.5.3 Systematic and Diversifiable Risk
2.5.4 The Capital Asset Pricing Model (CAPM)
2.5.5 Volatility and Beta
Chapter Summary
End-of-Chapter Problems (Fundamental Problems)
3. Fundamentals of Financial Derivative Pricing
3.1 Overview of the Three Primary Derivative Pricing Approaches
3.1.1 Valuing the Dice Game with a Tree Model
3.1.2 Valuing the Dice Game with a Monte Carlo Simulation Model
3.1.3 Valuing the Dice Game with an Analytical Model
3.1.4 Summary of Three Approaches to Valuing the Dice Game
3.2 The Binomial Tree Model Approach to Derivative Pricing
3.2.1 The Assumptions and Framework of Derivative Valuation with a Binomial Tree
3.2.2 No Arbitrage Tree Method to Value a Derivative with a One-Period Binomial Tree
3.2.3 A Numerical Example of Derivative Pricing with a One-Period Binomial Model
3.2.4 The Risk-Neutral Principle Derived in a One-Period Binomial Tree
3.2.5 P-Measures, Q-Measures, and Risk Aversion
3.2.6 The Risk-Neutral Tree Method to Value a Derivative with a Multi-Period Binomial Tree
3.2.7 Determination of Parameters u and d Using Volatility Matching
3.2.8 Using Spreadsheets to Value European and American Calls and Puts
Section Summary
Demonstration Exercises
3.3 The Geometric Brownian Motion and Monte Carlo Simulation
3.3.1 Introduction to Generalized Wiener Processes and Ito Processes
3.3.2 Wiener Processes and the Simulation of Paths
3.3.3 The Generalized Wiener Process and the Simulation of Its Paths
3.3.4 Ito Processes, Geometric Brownian Motion, and the Simulation of Its Paths
3.3.5 Finding the Distribution of a Generalized Wiener Process without Simulation
3.3.6 Review of Mathematics Regarding the Normal Distribution, Moment Generating Functions, and Taylor Expansions
3.3.7 Finding the Distribution of Geometric Brownian Motion without Simulation Using Ito's Lemma
3.3.8 Finding the Mean and Variance of ST for Geometric Brownian Motion
3.3.9 Risk Aversion vs. Risk Neutrality
Section Summary
Demonstration Exercises
3.4 Analytical Model Approaches to Derivative Pricing
3.4.1 The Black-Scholes PDE
3.4.2 The Intuition Behind the Black-Scholes PDE: Comparison of the PDE Approach and a Binomial Tree Approach
3.4.3 Boundary Conditions for the Black-Scholes PDE and the Black-Scholes Formula
3.4.4 Conclusions Concerning the Black-Scholes Formula
3.4.5 Risk-Neutral Evaluation and the Black-Scholes Formula
3.4.6 The Risk-Neutral Principle and Derivation of the Black-Scholes Formula
3.4.7 Monte Carlo Simulation and Derivative Valuation
Summary of Formulas in Derivative Valuation
Chapter Summary
End-of-Chapter Demonstration Exercises
Chapter 3 Extensions and Further Reading
End-of-Chapter Project
End-of-Chapter Problems (Fundamental Problems)
End-of-Chapter Problems (Challenging Problems)
Note
4. More about Derivative Valuation
4.1 Various Methods to Value Derivatives
4.1.1 Overview of the Three Primary Derivative Valuation Approaches
4.1.1.1 Binomial Tree
4.1.1.2 Black-Scholes Partial Differential Equation (Black-Scholes PDE)
4.1.1.3 Monte-Carlo Simulation
4.1.2 Alternative Binomial Trees
4.1.3 Using the Finite Difference Method to Solve the Black-Scholes PDE
4.1.4 Advantages vs. Disadvantages of Derivative Valuation Methods
4.2 Various Exotic Derivatives
4.2.1 Gap Options
4.2.2 Asian Options
4.2.3 Barrier Options
4.2.4 The Volatility Parameter
4.3 Dividends
4.3.1 How Dividends Affect Various Valuation Formulas
4.3.2 Dividends and Options on Foreign Currency or Futures Contracts
4.4 More on Forward Values, Futures Contracts, and Futures Prices
4.4.1 Forward Value to the Buyer
4.4.2 Forward Contracts vs. Futures Contracts
Chapter Summary
Chapter 4 Extensions and Further Reading:
End-of-Chapter Project
End of Chapter Problems (Fundamental Problems)
End of Chapter Problems (Challenging Problems)
Note
5. Risk Management and Hedging Strategies
5.1 Simple Hedging Using Forward and Futures Contracts
5.1.1 One-to-One Hedging of Commodity Risks Using Forward Contracts
5.1.2 Natural Hedgers, Speculators, and Keynes' Theory of Commodity Returns
5.1.3 Hedging Multiple Commodity Risks
5.1.4 Hedging of Financial Risks Using Forward Contracts
5.1.5 Hedging of Currency Risks Using Forward Contracts
5.1.6 Hedging Using Futures Instead of Forwards
5.2 One-to-h Hedging Using Forward and Futures Contracts
5.2.1 Return-Based Hedging Using Forward Contracts
5.2.2 Unit Value-Based Hedging Using Forward Contracts
5.2.3 Dynamic Hedging
5.2.4 Hedging with Perfectly Correlated Assets and Differing Leverage
5.2.5 Hedging with Imperfectly Correlated Assets
5.3 Delta-Hedging of Options Portfolio Using the Underlying Asset
5.3.1 Introduction to Continuous-Time Delta-Hedging Involving Options
5.3.2 An Example without Delta Hedging
5.3.3 Delta-Hedging an Option Position Using Its Underlying Asset
5.3.4 Delta-Hedging a Portfolio Containing Options by Matching the Deltas
5.4 Delta-Hedging Using Futures
5.4.1 The Delta of a Forward Contract
5.4.2 The Delta of a Futures Contract
5.4.3 Delta of an Option on a Futures Contract
5.4.4 Examples of Delta-Hedging of Options on Futures
5.5 Greek Letters Hedging of Other Option Risk Factors
5.5.1 Vega and Vega Neutral Hedging
5.5.2 Rho and Theta
5.5.3 Advantages and Limitations of Delta-Hedging
5.5.4 Gamma as a Second-Order Greek and Gamma-Neutral Hedging
5.5.5 An Example of Delta-Hedging, Vega-Hedging, and Gamma-Hedging
5.5.6 Relationships between Delta, Gamma, and Theta
5.6 Duration-Hedging of Fixed-Income Instruments and Interest Rate Risk Using Bonds and Bond Futures
5.6.1 Duration and Duration-Hedging
5.6.2 Duration-Hedging Using Bond Futures
5.6.2.1 Duration-Hedging a Bond Portfolio Using Bond Futures
5.6.2.2 Duration-Hedging Bank Loan/deposit Interest Rate Risk Using Bond Futures
5.6.2.3 Various Examples
5.6.3 Dollar Duration
5.6.4 Duration Hedging with Second-Order Partial Derivatives
5.7 Hedging Various Risks Using Swap Contracts
Hedging Formulas Summary
Chapter Summary
Chapter 5 Extensions and Further Reading
End-of-Chapter Project
How to Perform Arbitrage in the Real World—A Naïve Example
End-of-Chapter Problems (Fundamental Problems)
End-of-Chapter Problems (Challenging Problems)
6. Portfolio Management
6.1 Markowitz Frontier
6.1.1 Analytical Solution for the Markowitz Frontier without a Risk-Free Asset
6.1.2 Analytical Solution for the Markowitz Frontier When There Is a Risk-Free Asset
6.1.3 The Capital Market Line, CAPM, and Security Market Line
6.2 Portfolio Optimization: Linear and Quadratic Programming
6.2.1 Finding the Markowitz Frontier Using Quadratic Programming
6.2.2 Finding the Markowitz Frontier Using Quadratic Programming When Short Sales Are Disallowed
6.2.3 Testing for Arbitrage Using Linear Programming
6.3 The Optimal Growth Portfolio
6.3.1 The Portfolio of Maximal Growth Rate without the Risk-Free Asset
6.3.2 The Portfolio of Maximal Growth Rate Including the Risk-Free Asset
Chapter Summary
Chapter 6 Extensions and Further Reading
End-of-Chapter Project
End-of-Chapter Problems (Fundamental Problems)
End-of-Chapter Problems (Challenging Problems)
References
7. Interest Rate Derivatives Modeling and Risk Management in the HJM Framework
7.1 Zero Coupon Bonds, Forward Rates, Short Rates, and Money Markets in a Deterministic World
7.2 Risk-Neutral Valuation in a One-Factor HJM Model
7.2.1 Evolvement of Zero Coupon Bonds, Forward Rates, Short Rates, and Money Market Funds
7.2.2 HJM One-Factor Model
7.3 Valuation and Risk Management of Options on Bonds in One-Factor HJM Model
7.4 Valuation and Risk Management of Caps, Floors, and Interest Rate Swaps in a One-Factor HJM Model
7.4.1 Pricing and Hedging a Cap
7.4.2 Pricing and Hedging a Swap
7.5 Valuation and Risk Management of Forward and Futures Contracts on a Bond in a One-Factor HJM Model
7.6 Calibrating the HJM Model Parameters from the Market Prices
7.6.1 Stripping the Initial Forward Rate from Coupon Bonds
7.6.2 Building the Forward Rate and Volatility Trees
7.6.3 Building a Forward Tree and Valuing Interest Rate Derivatives
Chapter Summary
Extensions and Further Reading
End-of-Chapter Problems (Fundamental Problems)
Note
Reference
8. Credit Risk and Credit Derivatives
8.1 Default Probabilities and Extract Default Probabilities from Market
8.1.1 Types of Default Probabilities
8.1.2 Deriving Default Probabilities from N-year Risk-Free Zero Coupon Bonds and Defaultable Zero Coupon Bonds
8.1.3 Deriving Default Probabilities from Coupon Bond Prices
8.1.4 Deriving Default Probabilities from Asset Swaps
8.1.5 Statistical, Historical, and Risk-Neutral Default Probabilities
Section Summary
Demonstration Exercises
8.2 Single-Name Credit Derivatives
8.2.1 Credit Default Swaps (CDSs)
8.2.2 CDS Valuation Using the Analytical Method
8.2.3 CDS Valuation Using Simulation
8.2.4 Calibrating Piecewise Constant Default Probabilities from Multiple CDS Spreads
Summary of Single-Name Credit Derivatives
Demonstration Exercises
8.3 Collateralized Debt Obligations (CDOs) and a CDO Model with Independent Bond Defaults
8.3.1 Overview of CDO Tranches and Structuring of Cash Flows
8.3.2 Economic Roles and Risks of CDOs
8.3.3 An Example of a Cash CDO Structure
8.3.4 Synthetic Unfunded CDO Structures
8.3.5 A Simplified Simulation based CDO Model with Independent Bond Defaults
8.4 Collateralized Debt Obligations (CDOs) Model with Positively Correlated Bond Defaults
8.4.1 The Copula Method and the Mathematical Background for CDO Pricing
8.4.2 CDO Pricing Based on Simulation and the Copula Model
8.4.3 CDO Spreads as a Function of the Correlation Coefficient
8.4.4 Recovery Does Matter in CDO Spreads and in Global Financial Crisis
8.5 CDO Squareds
8.6 The Financial Crisis and Credit Derivatives
Chapter Summary
Demonstration Exercises
Extensions and Further Reading
End-of-Chapter Project
End-of-Chapter Problems (Fundamental Problems)
End-of-Chapter Problems (Challenging Problems)
Notes
Index