Stochastic Analysis for Finance With Simulations

✍ Scribed by Geon Ho Choe

• Publisher: Springer Nature
• Year: 2016
• Tongue: English
• Leaves: 660
• Edition: 1st ed. 2016
• Category: Library

✦ Synopsis

This book is an introduction to stochastic analysis and quantitative finance; it includes both theoretical and computational methods. Topics covered are stochastic calculus, option pricing, optimal portfolio investment, and interest rate models. Also included are simulations of stochastic phenomena, numerical solutions of the Black–Scholes–Merton equation, Monte Carlo methods, and time series. Basic measure theory is used as a tool to describe probabilistic phenomena. 

The level of familiarity with computer programming is kept to a minimum. To make the book accessible to a wider audience, some background mathematical facts are included in the first part of the book and also in the appendices. This work attempts to bridge the gap between mathematics and finance by using diagrams, graphs and simulations in addition to rigorous theoretical exposition. Simulations are not only used as the computational method in quantitative finance, but they can also facilitate an intuitive and deeper understanding of theoretical concepts.  

Stochastic Analysis for Finance with Simulations is designed for readers who want to have a deeper understanding of the delicate theory of quantitative finance by doing computer simulations in addition to theoretical study. It will particularly appeal to advanced undergraduate and graduate students in mathematics and business, but not excluding practitioners in finance industry.  

✦ Table of Contents

Preface
Acknowledgements
Contents
List of Figures
List of Tables
List of Simulations
Acronyms
List of Symbols
Part I Introduction to Financial Mathematics
1 Fundamental Concepts
1.1 Risk
1.2 Time Value of Money
1.3 No Arbitrage Principle
1.4 Arbitrage Free Market
1.5 Risk-Neutral Pricing and Martingale Measures
1.6 The One Period Binomial Tree Model
1.7 Models in Finance
Exercises
2 Financial Derivatives
2.1 Forward Contracts and Futures
2.2 Options
2.3 Put-Call Parity
2.4 Relations Among Option Pricing Methods
Exercises
Part II Probability Theory
3 The Lebesgue Integral
3.1 Measures
3.2 Simple Functions
3.3 The Lebesgue Integral
3.4 Inequalities
3.5 The Radon–Nikodym Theorem
3.6 Computer Experiments
Exercises
4 Basic Probability Theory
4.1 Measure and Probability
4.2 Characteristic Functions
4.3 Independent Random Variables
4.4 Change of Variables
4.5 The Law of Large Numbers
4.6 The Central Limit Theorem
4.7 Statistical Ideas
4.8 Computer Experiments
Exercises
5 Conditional Expectation
5.1 Conditional Expectation Given an Event
5.2 Conditional Expectation with Respect to a σ-Algebra
5.3 Conditional Expectation with Respect to a Random Variable
5.4 Computer Experiments
Exercises
6 Stochastic Processes
6.1 Stochastic Processes
6.2 Predictable Processes
6.3 Martingales
6.4 Stopping Time
6.5 Computer Experiments
Exercises
Part III Brownian Motion
7 Brownian Motion
7.1 Brownian Motion as a Stochastic Process
7.2 Sample Paths of Brownian Motion
7.3 Brownian Motion and Martingales
7.4 Computer Experiments
Exercises
8 Girsanov's Theorem
8.1 Motivation
8.2 Equivalent Probability Measure
8.3 Brownian Motion with Drift
8.4 Computer Experiments
Exercises
9 The Reflection Principle of Brownian Motion
9.1 The Reflection Property of Brownian Motion
9.2 The Maximum of Brownian Motion
9.3 The Maximum of Brownian Motion with Drift
9.4 Computer Experiments
Part IV Itô Calculus
10 The Itô Integral
10.1 Definition of the Itô Integral
10.2 The Martingale Property of the Itô Integral
10.3 Stochastic Integrals with Respect to a Martingale
10.4 The Martingale Representation Theorem
10.5 Computer Experiments
Exercises
11 The Itô Formula
11.1 Motivation for the Itô Formula
11.2 The Itô Formula: Basic Form
11.3 The Itô Formula: General Form
11.4 Multidimensional Brownian Motion and the Itô Formula
11.5 Computer Experiments
Exercises
12 Stochastic Differential Equations
12.1 Strong Solutions
12.2 Weak Solutions
12.3 Brownian Bridges
12.4 Computer Experiments
Exercises
13 The Feynman–Kac Theorem
13.1 The Feynman–Kac Theorem
13.2 Application to the Black–Scholes–Merton Equation
13.3 The Kolmogorov Equations
13.4 Computer Experiments
Part V Option Pricing Methods
14 The Binomial Tree Method for Option Pricing
14.1 Motivation for the Binomial Tree Method
14.2 The One Period Binomial Tree Method
14.2.1 Pricing by Hedging
14.2.2 Pricing by Replication
14.3 The Multiperiod Binomial Tree Method
14.4 Convergence to the Black–Scholes–Merton Formula
14.5 Computer Experiments
Exercises
15 The Black–Scholes–Merton Differential Equation
15.1 Derivation of the Black–Scholes–Merton Differential Equation
Assumptions on the Underlying Asset
Assumptions on the Financial Market
15.2 Price of a European Call Option
15.3 Greeks
15.4 Solution by the Laplace Transform
15.5 Computer Experiments
Exercises
16 The Martingale Method
16.1 Option Pricing by the Martingale Method
16.2 The Probability Distribution of Asset Price
16.3 The Black–Scholes–Merton Formula
16.4 Derivation of the Black–Scholes–Merton Equation
16.5 Delta Hedging
16.6 Computer Experiments
Exercises
Part VI Examples of Option Pricing
17 Pricing of Vanilla Options
17.1 Stocks with a Dividend
17.2 Bonds with Coupons
17.3 Binary Options
Exercises
17.4 Computer Experiments
18 Pricing of Exotic Options
18.1 Asian Options
18.2 Barrier Options
18.3 Computer Experiments
19 American Options
19.1 American Call Options
19.2 American Put Options
19.3 The Least Squares Method of Longstaff and Schwartz
19.4 Computer Experiments
Part VII Portfolio Management
20 The Capital Asset Pricing Model
20.1 Return Rate and the Covariance Matrix
20.2 Portfolios of Two Assets and More
20.3 An Application of the Lagrange Multiplier Method
20.4 Minimum Variance Line
20.5 The Efficient Frontier
20.6 The Market Portfolio
20.7 The Beta Coefficient
20.8 Computer Experiments
Exercises
21 Dynamic Programming
21.1 The Hamilton–Jacobi–Bellman Equation
21.2 Portfolio Management for Optimal Consumption
21.3 Computer Experiments
Part VIII Interest Rate Models
22 Bond Pricing
22.1 Periodic and Continuous Compounding
22.2 Zero Coupon Interest Rates
22.3 Term Structure of Interest Rates
22.4 Forward Rates
22.5 Yield to Maturity
22.6 Duration
22.7 Definitions of Various Interest Rates
22.8 The Fundamental Equation for Bond Pricing
22.9 Computer Experiments
Exercises
23 Interest Rate Models
23.1 Short Rate Models
23.2 The Vasicek Model
23.3 The Cox–Ingersoll–Ross Model
23.4 The Ho–Lee Model
23.5 The Hull–White Model
23.6 Computer Experiments
Exercises
24 Numeraires
24.1 Change of Numeraire for a Binomial Tree Model
24.2 Change of Numeraire for Continuous Time
24.3 Numeraires for Pricing of Interest Rate Derivatives
Part IX Computational Methods
25 Numerical Estimation of Volatility
25.1 Historical Volatility
25.2 Implied Volatility
25.2.1 The Bisection Method
25.2.2 The Newton–Raphson Method
25.3 Computer Experiments
Exercises
26 Time Series
26.1 The Cobweb Model
26.2 The Spectral Theory of Time Series
26.3 Autoregressive and Moving Average Models
26.4 Time Series Models for Volatility
26.5 Computer Experiments
Exercises
27 Random Numbers
27.1 What Is a Monte Carlo Method?
27.2 Uniformly Distributed Random Numbers
27.3 Testing Random Number Generators
27.4 Normally Distributed Random Numbers
27.5 Computer Experiments
Exercises
28 The Monte Carlo Method for Option Pricing
28.1 The Antithetic Variate Method
28.2 The Control Variate Method
28.3 The Importance Sampling Method
28.4 Computer Experiments
Exercises
29 Numerical Solution of the Black–Scholes–Merton Equation
29.1 Difference Operators
29.2 Grid and Finite Difference Methods
29.2.1 Explicit Method
29.2.2 Implicit Method
29.2.3 Crank–Nicolson Method
29.3 Numerical Methods for the Black–Scholes–Merton Equation
29.4 Stability
29.5 Computer Experiments
Exercises
30 Numerical Solution of Stochastic Differential Equations
30.1 Discretization of Stochastic Differential Equations
30.2 Stochastic Taylor Series
30.2.1 Taylor Series for an Ordinary Differential Equation
30.2.2 Taylor Series for a Stochastic Differential Equation
30.3 The Euler Scheme
30.4 The Milstein Scheme
30.5 Computer Experiments
Exercises
A Basic Analysis
A.1 Sets and Functions
A.2 Metric Spaces
A.3 Continuous Functions
A.4 Bounded Linear Transformations
A.5 Extension of a Function
A.6 Differentiation of a Function
B Linear Algebra
B.1 Vectors
B.2 Matrices
B.3 The Method of Least Squares
B.4 Symmetric Matrices
B.5 Principal Component Analysis (PCA)
B.6 Tridiagonal Matrices
B.7 Convergence of Iterative Algorithms
C Ordinary Differential Equations
C.1 Linear Differential Equations with Constant Coefficients
C.2 Linear Differential Equations with Nonconstant Coefficients
C.3 Nonlinear Differential Equations
C.4 Ordinary Differential Equations Defined by Vector Fields
C.5 Infinitesimal Generators for Vector Fields
D Diffusion Equations
D.1 Examples of Partial Differential Equations
D.2 The Fourier Transform
D.3 The Laplace Transform
D.4 The Boundary Value Problem for Diffusion Equations
E Entropy
E.1 What Is Entropy?
E.2 The Maximum Entropy Principle
F Matlab Programming
F.1 How to Start
F.2 Random Numbers
F.3 Vectors and Matrices
F.4 Tridiagonal Matrices
F.5 Loops for Iterative Algorithms
F.6 How to Plot Graphs
F.7 Curve Fitting
F.8 How to Define a Function
F.9 Statistics
Solutions for Selected Problems
Problems of Chap. 1
Problems of Chap. 2
Problems of Chap. 3
Problems of Chap. 4
Problems of Chap. 5
Problems of Chap. 6
Problems of Chap. 7
Problems of Chap. 8
Problems of Chap. 10
Problems of Chap. 11
Problems of Chap. 12
Problems of Chap. 15
Problems of Chap. 16
Problems of Chap. 17
Problems of Chap. 20
Problems of Chap. 22
Problems of Chap. 23
Problems of Chap. 26
Problems of Chap. 27
Problems of Chap. 28
Problems of Chap. 29
Glossary
References
Index