Risk-Neutral Valuation Book

Contents Preface to the Second Edition Preface to the First Edition
1. Derivative Background
1.1 Financial Markets and Instruments
1.1.1 Derivative Instruments
1.1.2 Underlying Securities
1.1.3 Markets
1.1.4 Types of Traders
1.1.5 Modeling Assumptions
1.2 Arbitrage
1.3 Arbitrage Relationships
1.3.1 Fundamental Determinants of Option Values
1.3.2 Arbitrage Bounds
1.4 Single-period Market Models
1.4.1 A Fundamental Example
1.4.2 A Single-period Model
1.4.3 A Few Financial-economic Considerations Exercises

2. Probability Background
2.1 Measure
2.2 Integral
2.3 Probability
2.4 Equivalent Measures and Radon-Nikodym Derivatives
2.5 Conditional Expectation
2.6 Modes of Convergence
2.7 Convolution and Characteristic Functions
2.8 The Central Limit Theorem
2.9 Asset Return Distributions
2.10 In.nite Divisibility and the Ĺevy-Khintchine Formula
2.11 Elliptically Contoured Distributions
2.12 Hyberbolic Distributions Exercises

3. Shastic Processes in Discrete Time
3.1 Information and Filtrations
3.2 Discrete-parameter Shastic Processes
3.3 De.nition and Basic Properties of Martingales
3.4 Martingale Transforms
3.5 Stopping Times and Optional Stopping
3.6 The Snell Envelope and Optimal Stopping
3.7 Spaces of Martingales
3.8 Markov Chains Exercises

4. Mathematical Finance in Discrete Time
4.1 The Model
4.2 Existence of Equivalent Martingale Measures
4.2.1 The No-arbitrage Condition
4.2.2 Risk-Neutral Pricing
4.3 Complete Markets: Uniqueness of EMMs
4.4 The Fundamental Theorem of Asset Pricing: Risk-Neutral Valuation
4.5 The Cox-Ross-Rubinstein Model
4.5.1 Model Structure
4.5.2 Risk-neutral Pricing
4.5.3 Hedging
4.6 Binomial Approximations
4.6.1 Model Structure
4.6.2 The Black-Scholes Option Pricing Formula
4.6.3 Further Limiting Models
4.7 American Options
4.7.1 Theory
4.7.2 American Options in the CRR Model
4.8 Further Contingent Claim Valuation in Discrete Time
4.8.1 Barrier Options
4.8.2 Lookback Options
4.8.3 A Three-period Example
4.9 Multifactor Models
4.9.1 Extended Binomial Model
4.9.2 Multinomial Models Exercises

5. Shastic Processes in Continuous Time
5.1 Filtrations; Finite-dimensional Distributions
5.2 Classes of Processes
5.2.1 Martingales
5.2.2 Gaussian Processes
5.2.3 Markov Processes
5.2.4 Diffusions
5.3 Brownian Motion
5.3.1 Definition and Existence
5.3.2 Quadratic Variation of Brownian Motion
5.3.3 Properties of Brownian Motion
5.3.4 Brownian Motion in Shastic Modeling
5.4 Point Processes
5.4.1 Exponential Distribution
5.4.2 The Poisson Process
5.4.3 Compound Poisson Processes
5.4.4 Renewal Processes
5.5 Levy Processes
5.5.1 Distributions
5.5.2 Levy Processes
5.5.3 Levy Processes and the Levy-Khintchine Formula
5.6 Shastic Integrals; Ito Calculus
5.6.1 Shastic Integration
5.6.2 Ito’s Lemma
5.6.3 Geometric Brownian Motion
5.7 Shastic Calculus for Black-Scholes Models
5.8 Shastic Differential Equations
5.9 Likelihood Estimation for Diffusions
5.10 Martingales, Local Martingales and Semi-martingales
5.10.1 Definitions
5.10.2 Semi-martingale Calculus
5.10.3 Shastic Exponentials
5.10.4 Semi-martingale Characteristics
5.11 Weak Convergence of Shastic Processes
5.11.1 The Spaces Cd and Dd
5.11.2 Definition and Motivation
5.11.3 Basic Theorems of Weak Convergence
5.11.4 Weak Convergence Results for Shastic Integrals Exercises

6. Mathematical Finance in Continuous Time
6.1 Continuous-time Financial Market Models
6.1.1 The Financial Market Model
6.1.2 Equivalent Martingale Measures
6.1.3 Risk-neutral Pricing
6.1.4 Changes of Numeraire
6.2 The Generalized Black-Scholes Model
6.2.1 The Model
6.2.2 Pricing and Hedging Contingent Claims
6.2.3 The Greeks
6.2.4 Volatility
6.3 Further Contingent Claim Valuation
6.3.1 American Options
6.3.2 Asian Options
6.3.3 Barrier Options
6.3.4 Lookback Options
6.3.5 Binary Options
6.4 Discrete- versus Continuous-time Market Models
6.4.1 Discrete- to Continuous-time Convergence Reconsidered
6.4.2 Finite Market Approximations
6.4.3 Examples of Finite Market Approximations
6.4.4 Contiguity
6.5 Further Applications of the Risk-neutral Valuation Principle
6.5.1 Futures Markets
6.5.2 Currency Markets Exercises

7. Incomplete Markets
7.1 Pricing in Incomplete Markets
7.1.1 A General Option-Pricing Formula
7.1.2 The Esscher Measure
7.2 Hedging in Incomplete Markets
7.2.1 Quadratic Principles
7.2.2 The Financial Market Model
7.2.3 Equivalent Martingale Measures
7.2.4 Hedging Contingent Claims
7.2.5 Mean-variance Hedging and the Minimal ELMM
7.2.6 Explicit Example
7.2.7 Quadratic Principles in Insurance
7.3 Shastic Volatility Models
7.4 Models Driven by Levy Processes
7.4.1 Introduction
7.4.2 General Levy-process Based Financial Market Model
7.4.3 Existence of Equivalent Martingale Measures
7.4.4 Hyperbolic Models: The Hyperbolic Levy Process

8. Interest Rate Theory
8.1 The Bond Market
8.1.1 The Term Structure of Interest Rates
8.1.2 Mathematical Modelling
8.1.3 Bond Pricing, .
8.2 Short-rate Models
8.2.1 The Term-structure Equation
8.2.2 Martingale Modelling
8.2.3 Extensions: Multi-Factor Models
8.3 Heath-Jarrow-Morton Methodology
8.3.1 The Heath-Jarrow-Morton Model Class
8.3.2 Forward Risk-neutral Martingale Measures
8.3.3 Completeness
8.4 Pricing and Hedging Contingent Claims
8.4.1 Short-rate Models
8.4.2 Gaussian HJM Framework
8.4.3 Swaps
8.4.4 Caps
8.5 Market Models of LIBOR- and Swap-rates
8.5.1 Description of the Economy
8.5.2 LIBOR Dynamics Under the Forward LIBOR Measure
8.5.3 The Spot LIBOR Measure
8.5.4 Valuation of Caplets and Floorlets in the LMM
8.5.5 The Swap Market Model
8.5.6 The Relation Between LIBOR- and Swap-market Models
8.6 Potential Models and the Flesaker-Hughston Framework
8.6.1 Pricing Kernels and Potentials
8.6.2 The Flesaker-Hughston Framework Exercises

9. Credit Risk
9.1 Aspects of Credit Risk
9.1.1 The Market
9.1.2 What Is Credit Risk?
9.1.3 Portfolio Risk Models
9.2 Basic Credit Risk Modeling
9.3 Structural Models
9.3.1 Merton’s Model
9.3.2 A Jump-di.usion Model
9.3.3 Structural Model with Premature Default
9.3.4 Structural Model with Shastic Interest Rates
9.3.5 Optimal Capital Structure – Leland’s Approach
9.4 Reduced Form Models
9.4.1 Intensity-based Valuation of a Defaultable Claim
9.4.2 Rating-based Models
9.5 Credit Derivatives
9.6 Portfolio Credit Risk Models
9.7 Collateralized Debt Obligations (CDOs)
9.7.1 Introduction
9.7.2 Review of Modelling Methods A. Hilbert Space B. Projections and Conditional Expectations C. The Separating Hyperplane Theorem Bibliograpy Index