Financial Mathematics: From Discrete to Continuous Time is a study of the mathematical ideas and techniques that are important to the two main arms of the area of financial mathematics: portfolio optimization and derivative valuation. The text is authored for courses taken by advanced undergraduates, MBA, or other students in quantitative finance programs.
The approach will be mathematically correct but informal, sometimes omitting proofs of the more difficult results and stressing practical results and interpretation. The text will not be dependent on any particular technology, but it will be laced with examples requiring the numerical and graphical power of the machine.
The text illustrates simulation techniques to stand in for analytical techniques when the latter are impractical. There will be an electronic version of the text that integrates Mathematica functionality into the development, making full use of the computational and simulation tools that this program provides. Prerequisites are good courses in mathematical probability, acquaintance with statistical estimation, and a grounding in matrix algebra.
The highlights of the text are:
- A thorough presentation of the problem of portfolio optimization, leading in a natural way to the Capital Market Theory
- Dynamic programming and the optimal portfolio selection-consumption problem through time
- An intuitive approach to Brownian motion and stochastic integral models for continuous time problems
- The Black-Scholes equation for simple European option values, derived in several different ways
- A chapter on several types of exotic options
- Material on the management of risk in several contexts
Author(s): Kevin J. Hastings
Series: Chapman and Hall/CRC Financial Mathematics Series
Publisher: CRC Press/Chapman & Hall
Year: 2022
Language: English
Pages: 429
City: Boca Raton
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Author
1. Review of Preliminaries
1.1. Risky Assets
1.1.1. Single and Multiple Discrete Time Periods
1.1.2. Continuous-Time Processes
1.1.3. Martingales
1.2. Risk Aversion and Portfolios of Assets
1.2.1. Risk Aversion Constant
1.2.2. The Portfolio Problem
1.3. Expectation, Variance, and Covariance
1.3.1. One Variable Expectation
1.3.2. Expectation for Multiple Random Variables
1.3.3. Variance of a Linear Combination
1.4. Simple Portfolio Optimization
1.5. Derivative Assets and Arbitrage
1.5.1. Futures
1.5.2. Arbitrage and Futures
1.5.3. Options
1.6. Valuation of Derivatives in a Single Time Period
1.6.1. Replicating Portfolios
1.6.2. Risk-Neutral Valuation
2. Portfolio Selection and CAPM Theory
2.1. Portfolio Optimization with Multiple Assets
2.1.1. Lagrange Multipliers
2.1.2. Qualitative Behavior
2.1.3. Correlated Assets
2.1.4. Portfolio Separation and the Market Portfolio
2.2. Capital Market Theory, Part I
2.2.1. Linear Algebraic Approach
2.2.2. Efficient Mean-Standard Deviation Frontier
2.3. Capital Market Theory, Part II
2.3.1. Capital Market Line
2.3.2. CAPM Formula; Asset β
2.3.3. Systematic and Non-Systematic Risk; Pricing Using CAPM
2.4. Utility Theory
2.4.1. Securities and Axioms for Investor Behavior
2.4.2. Indifference Curves, Certainty Equivalent, Risk Aversion
2.4.3. Examples of Utility Functions
2.4.4. Absolute and Relative Risk Aversion
2.4.5. Utility Maximization
2.5. Multiple Period Portfolio Problems
2.5.1. Problem Description and Dynamic Programming Approach
2.5.2. Examples
2.5.3. Optimal Portfolios and Martingales
3. Discrete-Time Derivatives Valuation
3.1. Options Pricing for Multiple Time Periods
3.1.1. Introduction
3.1.2. Valuation by Chaining
3.1.3. Valuation by Martingales
3.2.Key Ideas of Discrete Probability, Part I
3.2.1. Algebras and Measurability
3.2.2. Independence
3.3. Key Ideas of Discrete Probability, Part II
3.3.1. Conditional Expectation
3.3.2. Application to Pricing Models
3.4. Fundamental Theorems of Options Pricing
3.4.1. The Market Model
3.4.2. Gain, Arbitrage, and Attainability
3.4.3. Martingale Measures and the Fundamental Theorems
3.5. Valuation of Non-Vanilla Options
3.5.1. American and Bermudan Options
3.5.2. Barrier Options
3.5.3. Asian Options
3.5.4. Two-Asset Derivatives
3.6. Derivatives Pricing by Simulation
3.6.1. Setup and Algorithm
3.6.2. Examples
3.7. From Discrete to Continuous Time (A Preview)
4. Continuous Probability Models
4.1. Continuous Distributions and Expectation
4.1.1. Densities and Cumulative Distribution Functions
4.1.2. Expectation
4.1.3. Normal and Lognormal Distributions
4.2. Joint Distributions
4.2.1. Basic Ideas
4.2.2. Marginal and Conditional Distributions
4.2.3. Independence
4.2.4. Covariance and Correlation
4.2.5. Bivariate Normal Distribution
4.3. Measurability and Conditional Expectation
4.3.1. Sigma Algebras
4.3.2. Random Variables and Measurability
4.3.3. Continuous Conditional Expectation
4.4. Brownian Motion and Geometric Brownian Motion
4.4.1. Random Walk Processes
4.4.2. Standard Brownian Motion
4.4.3. Non-Standard Brownian Motion
4.4.4. Geometric Brownian Motion
4.4.5. Brownian Motion and Binomial Branch Processes
4.5. Introduction to Stochastic Differential Equations
4.5.1. Meaning of the General SDE
4.5.2. Ito's Formula
4.5.3. Geometric Brownian Motion as Solution
5. Derivative Valuation in Continuous Time
5.1. Black-Scholes Via Limits
5.1.1. Black-Scholes Formula
5.1.2. Limiting Approach
5.1.3. Put Options and Put-Call Parity
5.2. Black-Scholes Via Martingales
5.2.1. Trading Strategies and Martingale Valuation
5.2.2. Martingale Measures
5.2.3. Asset and Bond Binaries
5.2.4. Other Binary Derivatives
5.3. Black-Scholes Via Differential Equations
5.3.1. Deriving the PDE
5.3.2. Boundary Conditions; Solving the PDE
5.4. Checking Black-Scholes Assumptions
5.4.1. Normality of Rates of Return
5.4.2. Stability of Parameters
5.4.3. Independence of Rates of Return
A Multivariate Normal Distribution
A.1. Review of Matrix Concepts
A.2. Multivariate Normal Distribution
B Answers to Selected Exercises
Bibliography
Index
