Pricing Derivative Securities 2nd Edition by Thomas Wake Epps 9812700331 9789812700339
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ISBN 10: 9812700331
ISBN 13: 9789812700339
Author: Thomas Wake Epps
This book presents techniques for valuing derivative securities at a level suitable for practitioners, students in doctoral programs in economics and finance, and those in masters-level programs in financial mathematics and computational finance. It provides the necessary mathematical tools from analysis, probability theory, the theory of stochastic processes, and stochastic calculus, making extensive use of examples. It also covers pricing theory, with emphasis on martingale methods. The chapters are organized around the assumptions made about the dynamics of underlying price processes. Readers begin with simple, discrete-time models that require little mathematical sophistication, proceed to the basic Black-Scholes theory, and then advance to continuous-time models with multiple risk sources. The second edition takes account of the major developments in the field since 2000. New topics include the use of simulation to price American-style derivatives, a new one-step approach to pricing options by inverting characteristic functions, and models that allow jumps in volatility and Markov-driven changes in regime. The new chapter on interest-rate derivatives includes extensive coverage of the LIBOR market model and an introduction to the modeling of credit risk. As a supplement to the text, the book contains an accompanying CD-ROM with user-friendly FORTRAN, C++, and VBA program components.
Table of contents:
Part I – Preliminaries
1. Introduction and Overview
1.1 A Tour of Derivatives and Markets
1.1.1 Forward Contracts
1.1.2 Futures
1.1.3 “Vanilla” Options
1.1.4 Other Derivative Products
1.2 An Overview of Derivatives Pricing
1.2.1 Replication: Static and Dynamic
1.2.2 Approaches to Valuation when Replication is Possible
1.2.3 Markets: Complete and Otherwise
1.2.4 Derivatives Pricing in Incomplete Markets
2. Mathematical Preparation 21
2.1 Analytical Tools
2.1.1 Order Notation
2.1.2 Series Expansions and Finite Sums
2.1.3 Measures
2.1.4 Measurable Functions
2.1.5 Variation and Absolute Continuity of Functions
2.1.6 Integration
2.1.7 Change of Measure: Radon-Nikodym Theorem
2.1.8 Special Functions and Integral Transforms
2.2 Probability
2.2.1 Probability Spaces
2.2.2 Random Variables and Their Distributions
2.2.3 Mathematical Expectation
2.2.4 Radon-Nikodym for Probability Measures
2.2.5 Conditional Probability and Expectation
2.2.6 Stochastic Convergence
2.2.7 Models for Distributions
2.2.8 Introduction to Stochastic Processes
3. Tools for Continuous-Time Models
3.1 Wiener Processes
3.1.1 Definition and Background
3.1.2 Essential Properties
3.2 Ito Integrals and Processes
3.2.1 A Motivating Example
3.2.2 Integrals with Respect to Brownian Motions
3.2.3 Ito Processes
3.3 Ito’s Formula
3.3.1 The Result, and Some Intuition
3.3.2 Outline of Proof
3.3.3 Functions of Time and an Ito Process
3.3.4 Illustrations
3.3.5 Functions of Higher-Dimensional Processes
3.3.6 Self-Financing Portfolios in Continuous Time
3.4 Tools for Martingale Pricing
3.4.1 Girsanov’s Theorem and Changes of Measure
3.4.2 Representation of Martingales
3.4.3 Numeraires, Changes of Numeraire, and Changes of Measure
3.5 Tools for Discontinuous Processes
3.5.1 J Processes
3.5.2 More General Processes
Part II – Pricing Theory
4. Dynamics-Free Pricing
4.1 Bond Prices and Interest Rates
4.1.1 Spot Bond Prices and Rates
4.1.2 Forward Bond Prices and Rates
4.1.3 Uncertainty in Future Bond Prices and Rates
4.2 Forwards and Futures
4.2.1 Forward Prices and Values of Forward Contracts
4.2.2 Determining Futures Prices
4.2.3 Illustrations and Caveats
4.2.4 A Preview of Martingale Pricing
4.3 Options
4.3.1 Payoff Distributions for European Options
4.3.2 Put-Call Parity
4.3.3 Bounds on Option Prices
4.3.4 How Prices Vary with T, X, and St
5. Pricing under Bernoulli Dynamics
5.1 The Structure of Bernoulli Dynamics
5.2 Replication and Binomial Pricing
5.3 Interpreting the Binomial Solution
5.3.1 The P.D.E. Interpretation
5.3.2 The Risk-Neutral or Martingale Interpretation
5.4 Specific Applications
5.4.1 European Stock Options
5.4.2 Futures and Futures Options
5.4.3 American-Style Derivatives
5.4.4 Derivatives on Assets That Pay Dividends
5.5 Implementing the Binomial Method
5.5.1 Modeling the Dynamics
5.5.2 Efficient Calculation
5.6 Inferring Trees from Prices of Traded Options
5.6.1 Assessing the Implicit Risk-Neutral Distribution of ST
5.6.2 Building the Tree
5.6.3 Appraisal
6. Black-Scholes Dynamics
6.1 The Structure of Black-Scholes Dynamics
6.2 Approaches to Arbitrage-Free Pricing
6.2.1 The Differential-Equation Approach
6.2.2 The Equivalent-Martingale Approach
6.3 Applications
6.3.1 Forward Contracts
6.3.2 European Options on Primary Assets
6.3.3 Extensions of the Black-Scholes Theory
6.4 Properties of Black-Scholes Formulas
6.4.1 Symmetry and Put-Call Parity
6.4.2 Extreme Values and Comparative Statics
6.4.3 Implicit Volatility
6.4.4 Delta Hedging and Synthetic Options
6.4.5 Instantaneous Risks and Expected Returns of European Options
6.4.6 Holding-Period Returns for European Options
7. American Options and “Exotics”
7.1 American Options
7.1.1 Calls on Stocks Paying Lump-Sum Dividends
7.1.2 Options on Assets Paying Continuous Dividends
7.1.3 Indefinitely Lived American Options
7.2 Compound and Extendable Options
7.2.1 Options on Options
7.2.2 Options with Extra Lives
7.3 Other Path-Independent Claims
7.3.1 Digital Options
7.3.2 Threshold Options
7.3.3 “As-You-Like-It” or “Chooser” Options
7.3.4 Forward-Start Options
7.3.5 Options on the Max or Min
7.3.6 Quantos
7.4 Path-Dependent Options
7.4.1 Extrema of Brownian Paths
7.4.2 Lookback Options
7.4.3 Barrier Options
7.4.4 Ladder Options
7.4.5 Asian Options
8. Models with Uncertain Volatility
8.1 Empirical Motivation
8.1.1 Brownian Motion Does Not Fit Underlying Prices
8.1.2 Black-Scholes No Longer Fits Option Prices
8.2 Price-Dependent Volatility
8.2.1 Qualitative Features of Derivatives Prices
8.2.2 Two Specific Models
8.2.3 Numerical Methods
8.2.4 Limitations of Price-Dependent Volatility
8.2.5 Incorporating Dependence on Past Prices
8.3 Stochastic-Volatility Models
8.3.1 Nonuniqueness of Arbitrage-Free Prices
8.3.2 Specific S.V. Models
8.4 Computational Issues
8.4.1 Inverting C.f.s
8.4.2 Two One-Step Approaches
9. Discontinuous Processes
9.1 Derivatives with Random Payoff Times
9.2 Derivatives on Mixed Jump/Diffusions
9.2.1 Jumps Plus Constant-Volatility Diffusions
9.2.2 Nonuniqueness of the Martingale Measure
9.2.3 European Options under Jump Dynamics
9.2.4 Properties of Jump-Dynamics Option Prices
9.2.5 Options Subject to Early Exercise
9.3 Jumps Plus Stochastic Volatility
9.3.1 The S.V.-Jump Model
9.3.2 Further Variations
9.4 Pure-Jump Models
9.4.1 The Variance-Gamma Model
9.4.2 The Hyperbolic Model
9.4.3 A Levy Process with Finite Levy Measure
9.4.4 Modeling Prices as Branching Processes
9.4.5 Assessing the Pure-Jump Models
9.5 A Markov-Switching Model
10. Interest-Rate Dynamics
10.1 Preliminaries
10.1.1 A Summary of Basic Concepts
10.1.2 Spot and Forward Measures
10.1.3 A Preview of Things to Come
10.2 Spot-Rate Models
10.2.1 Bond Prices under Vasicek
10.2.2 Bond Prices under Cox, Ingersoll, Ross
10.3 A Forward-Rate Model
10.3.1 The One-Factor HJM Model
10.3.2 Allowing Additional Risk Sources
10.3.3 Implementation and Applications
10.4 The LIBOR Market Model
10.4.1 Deriving Black’s Formulas
10.4.2 Applying the Model
10.5 Modeling Default Risk
10.5.1 Endogenous Risk: The Black-Scholes-Merton Model
10.5.2 Exogenous Default Risk
Part III – Computational Methods
11. Simulation
11.1 Generating Pseudorandom Deviates
11.1.1 Uniform Deviates
11.1.2 Deviates from Other Distributions
11.2 Variance-Reduction Techniques
11.2.1 Stratified Sampling
11.2.2 Importance Sampling
11.2.3 Antithetic Variates
11.2.4 Control Variates
11.2.5 Richardson Extrapolation
11.3 Applications
11.3.1 “Basket” Options
11.3.2 European Options under Stochastic Volatility
11.3.3 Lookback Options under Stochastic Volatility
11.3.4 American-Style Derivatives
12. Solving P.D.E.s Numerically
12.1 Setting Up for Solution
12.1.1 Approximating the Derivatives
12.1.2 Constructing a Discrete Time/Price Grid
12.1.3 Specifying Boundary Conditions
12.2 Obtaining a Solution
12.2.1 The Explicit Method
12.2.2 A First-Order Implicit Method
12.2.3 Crank-Nicolson’s Second-Order Implicit Method
12.2.4 Comparison of Methods
12.3 Extensions
12.3.1 More General P.D.E.s
12.3.2 Allowing for Lump-Sum Dividends
13. Programs
13.1 Generate and Test Random Deviates
13.1.1 Generating Uniform Deviates
13.1.2 Generating Poisson Deviates
13.1.3 Generating Normal Deviates
13.1.4 Testing for Randomness
13.1.5 Testing for Uniformity
13.1.6 Anderson-Darling Test for Normality
13.1.7 ICF Test for Normality
13.2 General Computation
13.2.1 Standard Normal CDF
13.2.2 Expectation of Function of Normal Variate
13.2.3 Standard Inversion of Characteristic Function
13.2.4 Inversion of Characteristic Function by FFT
13.3 Discrete-Time Pricing
13.3.1 Binomial Pricing
13.3.2 Solving PDEs under Geometric Brownian Motion
13.3.3 Crank-Nicolson Solution of General PDE
13.4 Continuous-Time Pricing
13.4.1 Shell for Black-Scholes with Input/Output
13.4.2 Basic Black-Scholes Routine
13.4.3 Pricing under the C.E.V. Model
13.4.4 Pricing a Compound Option
13.4.5 Pricing an Extendable Option
13.4.6 Pricing under Jump Dynamics
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Tags: Thomas Wake Epps, Pricing, Derivative, Securities