Financial Mathematics: A Comprehensive Treatment in Continuous Time Volume II

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
List of Figures
Preface
Authors
I. Stochastic Calculus with Brownian Motion
1. One-Dimensional Brownian Motion and Related Processes
1.1. Multivariate Normal Distributions
1.1.1. Multivariate Normal Distribution
1.1.2. Conditional Normal Distributions
1.2. Standard Brownian Motion
1.2.1. One-Dimensional Symmetric Random Walk
1.2.2. Formal Definition and Basic Properties of Brownian Motion
1.2.3. Multivariate Distribution of Brownian Motion
1.2.4. The Markov Property and the Transition PDF
1.2.5. Quadratic Variation and Nondifferentiability of Paths
1.3. Some Processes Derived from Brownian Motion
1.3.1. Drifted Brownian Motion
1.3.2. Geometric Brownian Motion
1.3.3. Processes related by a monotonic mapping
1.3.4. Brownian Bridge
1.3.5. Gaussian Processes
1.4. First Hitting Times and Maximum and Minimum of Brownian Motion
1.4.1. The Reflection Principle: Standard Brownian Motion
1.4.2. Translated and Scaled Driftless Brownian Motion
1.4.3. Brownian Motion with Drift
1.5. Exercises
2. Introduction to Continuous-Time Stochastic Calculus
2.1. The Riemann Integral of Brownian Motion
2.1.1. The Riemann Integral
2.1.2. The Integral of a Brownian Path
2.2. The Riemann–Stieltjes Integral of Brownian Motion
2.2.1. The Riemann–Stieltjes Integral
2.2.2. Integrals w.r.t. Brownian Motion: Preliminary Discussion
2.3. The Itô Integral and Its Basic Properties
2.3.1. The Itô Integral for Simple Processes
2.3.2. The Itô Integral for General Processes
2.4. Itô Processes and Their Properties
2.4.1. Gaussian Processes Generated by Itô Integrals
2.4.2. Itô Processes
2.4.3. Quadratic and Co-Variation of Itô Processes
2.5. Itô's Formula for Functions of BM and Itô Processes
2.5.1. Itô's Formula for Functions of BM
2.5.2. An "Antiderivative" Formula for Evaluating Itô Integrals
2.5.3. Itô's Formula for Itô Processes
2.6. Stochastic Differential Equations
2.6.1. Solutions to Linear SDEs
2.6.2. Existence and Uniqueness of a Strong Solution to an SDE
2.7. The Markov Property, Martingales, Feynman–Kac Formulae, and Transition CDFs and PDFs
2.7.1. Forward Kolmogorov PDE
2.7.2. Transition CDF/PDF for Time-Homogeneous Diffusions
2.8. Radon–Nikodym Derivative Process and Girsanov's Theorem
2.8.1. Some Applications of Girsanov's Theorem
2.9. Brownian Martingale Representation Theorem
2.10. Stochastic Calculus for Multidimensional BM
2.10.1. The Itô Integral and Itô's Formula for Multiple Processes on Multidimensional BM
2.10.2. Multidimensional SDEs, Feynman–Kac Formulae, and Transition CDFs and PDFs
2.10.3. Girsanov's Theorem for Multidimensional BM
2.10.4. Martingale Representation Theorem for Multidimensional BM
2.11. Exercises
II. Continuous-Time Modelling
3. Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock
3.1. From the CRR Model to the BSM Model
3.1.1. Portfolio Strategies in the Binomial Tree Model
3.1.2. The Cox–Ross–Rubinstein Model and its Continuous-Time Limit
3.2. Replication (Hedging) and Derivative Pricing in the Simplest Black–Scholes Economy
3.2.1. Pricing Standard European Calls and Puts
3.2.2. Hedging Standard European Calls and Puts
3.2.3. Europeans with Piecewise Linear Payoffs
3.2.4. Power Options
3.2.5. Dividend Paying Stock
3.2.6. Option Pricing with the Stock Numéraire
3.3. Forward Starting, Chooser, and Compound Options
3.4. Some European-Style Path-Dependent Derivatives
3.4.1. Risk-Neutral Pricing under GBM
3.4.2. Pricing Single Barrier Options
3.4.3. Pricing Lookback Options
3.5. Structural Credit Risk Models
3.5.1. The Merton Model
3.5.2. The Black–Cox Model
3.6. Exercises
4. Risk-Neutral Pricing in a Multi-Asset Economy
4.1. General Multi-Asset Market Model: Replication and Risk-Neutral Pricing
4.2. Equivalent Martingale Measures: Derivative Pricing with General Numéraire Assets
4.3. Black–Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives
4.3.1. Standard European Option Pricing for Multi-Stock GBM
4.3.2. Explicit Pricing Formulae for the GBM Model
4.3.3. Cross-Currency Option Valuation
4.3.4. Option Valuation with General Numéraire Assets
4.4. Exercises
5. American Options
5.1. Basic Properties of Early-Exercise Options
5.2. Arbitrage-Free Pricing of American Options
5.2.1. Optimal Stopping Formulation and Early-Exercise Boundary
5.2.2. The Smooth Pasting Condition
5.2.3. Put-Call Symmetry Relation
5.2.4. Dynamic Programming Approach for Bermudan Options
5.3. Perpetual American Options
5.3.1. Pricing a Perpetual Put Option
5.3.2. Pricing a Perpetual Call Option
5.4. Finite-Expiration American Options
5.4.1. The PDE Formulation
5.4.2. The Integral Equation Formulation
5.5. Exercises
6. Interest-Rate Modelling and Derivative Pricing
6.1. Basic Fixed Income Instruments
6.1.1. Bonds
6.1.2. Forward Rates
6.1.3. Arbitrage-Free Pricing
6.1.4. Fixed Income Derivatives
6.2. Single-Factor Models
6.2.1. Diffusion Models for the Short Rate Process
6.2.2. PDE for the Zero-Coupon Bond Value
6.2.3. Affine Term Structure Models
6.2.4. The Ho–Lee Model
6.2.5. The Vasiček Model
6.2.6. The Cox–Ingersoll–Ross Model
6.3. Heath–Jarrow–Morton Formulation
6.3.1. HJM under Risk-Neutral Measure
6.3.2. Relationship between HJM and Affine Yield Models
6.4. Multifactor Affine Term Structure Models
6.4.1. Gaussian Multifactor Models
6.4.2. Equivalent Classes of Affine Models
6.5. Pricing Derivatives under Forward Measures
6.5.1. Forward Measures
6.5.2. Pricing Stock Options under Stochastic Interest Rates
6.5.3. Pricing Options on Zero-Coupon Bonds
6.6. LIBOR Model
6.6.1. LIBOR Rates
6.6.2. The Brace–Gatarek–Musiela Model of LIBOR Rates
6.6.3. Pricing Caplets, Caps, and Swaps
6.7. Exercises
7. Alternative Models of Asset Price Dynamics
7.1. Characteristic Functions
7.1.1. Definition and Properties
7.1.2. Recovering the Distribution Function
7.1.3. Pricing Standard European Options
7.1.4. The Carr–Madan Method for Pricing Vanilla Options
7.2. Stochastic Volatility Diffusion Models
7.2.1. Local Volatility Models
7.2.2. Constant Elasticity of Variance Model
7.3. The Heston model
7.3.1. Solution to the Ricatti Equation
7.3.2. Implied Volatility for the Heston Model
7.4. Models with Jumps
7.4.1. The Poisson Process
7.4.2. Jump Diffusion Models with a Compound Poisson Component
7.4.3. The Merton Jump Diffusion Model
7.4.4. Characteristic Function for a Jump Diffusion Process
7.4.5. Change of Measure for Jump Diffusion Processes
7.4.6. The Variance Gamma Model
7.5. Exercises
A. Essentials of General Probability Theory
A.1. Random Variables and Lebesgue Integration
A.2. Multidimensional Lebesgue Integration
A.3. Multiple Random Variables and Joint Distributions
A.4. Conditioning
A.5. Changing Probability Measures
B. Some Useful Integral (Expectation) Identities and Symmetry Properties of Normal Random Variables
C. Answers and Hints to Exercises
C.1. Chapter 1
C.2. Chapter 2
C.3. Chapter 3
C.4. Chapter 4
C.5. Chapter 5
C.6. Chapter 6
C.7. Chapter 7
D. Glossary of Symbols and Abbreviations
References
Index