This book prepares students to execute the quantitative and computational needs of the finance industry. The quantitative methods are explained in detail with examples from real financial problems like option pricing, risk management, portfolio selection, etc. Codes are provided in R programming language to execute the methods. Tables and figures, often with real data, illustrate the codes. References to related work are intended to aid the reader to pursue areas of specific interest in further detail. The comprehensive background with economic, statistical, mathematical, and computational theory strengthens the understanding. The coverage is broad, and linkages between different sections are explained. The primary audience is graduate students, while it should also be accessible to advanced undergraduates. Practitioners working in the finance industry will also benefit.
This book covers all aspects of computation, namely numerical, simulation, and statistical, in a single volume. Numerical procedures, their advantages, applications in finance and execution in R, are presented in Part I. Despite the advantages, it is not always possible to solve financial problems numerically. In such cases, simulation methods are very useful. These are presented in Part II. The final part concentrates on statistical methods. These enable the reader to train the economic models to real data, test the suitability of the model, and forecast important quantities like risk. The basic statistical topics of descriptive, inferential, multivariate, and time series analysis are presented with their applications in finance. A whole chapter is devoted to the quantification of risk and another to high-frequency data. Two chapters expose the reader to cutting-edge Machine Learning techniques. Two topics related to simulation, namely Bayesian Monte Carlo and resampling methods, are included in the third section as they require some basic statistical knowledge. Necessary theory of mathematical finance and extreme value, as well as an introduction to R, is presented in the Appendix.
Author(s): Rituparna Sen, Sourish Das
Publisher: Springer
Year: 2023
Language: English
Pages: 352
Preface
Contents
About the Authors
Part I Numerical Methods
1 Preliminaries
1.1 Algorithms
1.1.1 Iterative Algorithms
1.1.2 Error
1.1.3 Stability
1.2 Interpolation of Functions
1.2.1 Polynomial Interpolation
1.2.2 Spline Approximation
1.3 Exercises
2 Vectors and Matrices
2.1 Dimension of a Vector Space
2.2 Solving Linear Equations
2.2.1 Condition Number
2.2.2 Iterative Procedures
2.3 Eigenvalues and Eigenvectors
2.4 Cholesky Factorization
2.5 Exercises
3 Solving Nonlinear Equations
3.1 Find Implied Volatility
3.2 Bisection Algorithm
3.3 Newton's Method
3.4 Secant Method
3.5 Fixed Point Method
3.6 R Functions for Root Finding
3.7 Exercises
4 Numerical Integration
4.1 Introduction
4.2 Newton–Cotes Formulas
4.2.1 Error
4.2.2 Trapezoidal Rule
4.2.3 Simpson's Rule
4.3 Gaussian Quadrature
4.4 Exercises
5 Numerical Differentiation
5.1 Introduction
5.2 Differentiation Via Interpolation
5.3 Method of Undetermined Coefficients
5.4 Exercises
6 Numerical Methods for PDE
6.1 Introduction
6.2 Finite Difference Methods
6.2.1 Explicit Scheme
6.2.2 Implicit Scheme
6.2.3 Crank–Nicolson Scheme
6.3 Solution to Black–Scholes PDE by Transformation to Heat Equation
6.4 Error
6.5 Exercises
7 Optimization
7.1 Linear Programming
7.2 Quadratic Programming
7.3 Other Methods
7.4 Exercises
Part II Simulation Methods
8 Monte Carlo Methods
8.1 Generating Sequence of Uniform[0, 1] Random Numbers
8.1.1 Linear Congruential Generator
8.1.2 Combining Generators
8.2 General Sampling Methods
8.2.1 Inverse Transform
8.2.2 Accept–Reject Method
8.3 Exercises
9 Lattice Models
9.1 Option Pricing with Binomial Lattices
9.2 Binomial Option Price as Approximate Solution to Black–Scholes Equation
9.3 Specific Models
9.4 n-Period Binomial Option Pricing Formula
9.5 Simulating from Binomial Asset Pricing Model
9.6 Log-Normal Distribution as Limit of BAPM
9.7 Pricing American Options with R
9.8 Beyond Binomial
9.9 Exercises
10 Simulating Brownian Motion
10.1 Generating Normal Random Variables and Vectors
10.1.1 Inverse Transform Method
10.1.2 Box–Muller Method
10.1.3 Generating Multivariate Normal
10.2 Simulation of Brownian Motion
10.2.1 Random Walk Construction
10.2.2 Brownian Bridge Construction
10.2.3 Principal Components Construction
10.3 Geometric Brownian Motion
10.4 Example: Path-Dependent Options
10.5 Multiple Dimensions
10.6 Exercises
11 Variance Reduction
11.1 Control Variates
11.1.1 Multiple Controls
11.1.2 Nonlinear Controls and Delta Method
11.2 Antithetic Variates
11.3 Importance Sampling
11.4 Stratified Sampling
11.4.1 Optimal Allocation
11.4.2 Stratifying a Poisson Process
11.4.3 Stratifying the Binomial Lattice
11.5 Exercises
Part III Statistical Methods
12 Descriptive Statistics
12.1 Financial Data
12.2 Graphical Representation of Qualitative Data
12.3 Numerical Representation of Quantitative Data
12.4 Graphical Representation of Quantitative Data
12.5 Assessing Normality
12.6 Measures of Relative Position
12.7 Stylized Facts
12.7.1 Gain-Loss Asymmetry
12.7.2 Aggregational Gaussianity
12.7.3 Heavy Tails
12.8 Exercises
13 Inferential Statistics
13.1 Estimation
13.1.1 Methods of Estimation
13.1.2 Properties of Estimators
13.2 Interval Estimation
13.3 Hypothesis Testing
13.4 Exercises
14 Bayesian Computation
14.1 Introduction to Bayesian Inference
14.2 Poisson Model for Count Data
14.3 Bayesian Inference with Monte Carlo for Any Model
14.4 Markov Chain Monte Carlo
14.4.1 Markov Chain
14.4.2 Markov Chain Monte Carlo Integration
14.4.3 How MCMC Works?
14.5 Exercises
15 Resampling
15.1 Jackknife
15.2 Bootstrap
15.2.1 Parametric Bootstrap
15.2.2 Bootstrap of Portfolio Returns
15.3 Cross-Validation
15.4 Exercises
16 Statistical Risk Analysis
16.1 Value at Risk
16.1.1 Gaussian Estimation
16.1.2 Modified and Robust Estimation
16.1.3 Historical Simulation
16.1.4 EVT-Based Estimation
16.2 Expected Shortfall
16.3 Other Measures of Risk
16.4 Ratios to Compare Portfolios
16.5 Back-Testing VaR Models
16.6 Exercises
17 Supervised Learning
17.1 Framework for Statistical Learning
17.1.1 Bayesian Decision Theoretic Framework
17.1.2 Learning with Empirical Risk Minimization
17.2 Regression and Classification
17.3 Supervised Learning and Generalized Linear Models
17.4 Regression
17.5 Logistic Regression
17.5.1 Sigmoid Curve Behavior of Logistic Regression
17.5.2 Non-monotonic Relation with Logistic Regression
17.6 Discriminant Analysis
17.7 Tree Structured Model
17.7.1 Decision Tree
17.7.2 Random Forest
17.8 Exercises
18 Linear Systems
18.1 Capital Market Line
18.2 Systematic Risk: Beta
18.2.1 CAPM and Beta Using R
18.2.2 Security Market Line
18.2.3 Can Beta Be Negative?
18.2.4 Beta Hedging
18.2.5 Achieving Target Beta
18.2.6 Effect of Outlier on Beta
18.3 Measuring Active Return: Alpha
18.4 Pricing Theory and System of Linear Equations
18.4.1 Factor Models and Risk Matrix
18.4.2 Benefit of Diversification
18.4.3 Solving Factor Model as System of Equation
18.4.4 Stress Testing
18.5 Summary
18.6 Exercises
19 Time Series
19.1 Components
19.2 Stationary Time Series
19.2.1 White Noise
19.2.2 Moving Average Models
19.2.3 Autoregressive Models
19.2.4 ARMA Models
19.2.5 ARIMA Models
19.3 Fitting ARIMA Model to Data
19.4 Forecasting
19.5 Exercises
20 High-Frequency Data
20.1 Infill Asymptotics in Geometric Brownian Motion
20.2 Microstructure Noise
20.3 Asynchronicity
20.4 Exercises
21 Unsupervised Learning
21.1 Introduction
21.2 Dimension Reduction
21.2.1 Multidimensional Scaling
21.2.2 Principal Component Analysis
21.3 Clustering
21.3.1 K-means Clustering
21.3.2 Hierarchical Clustering
21.4 Discussion
21.5 Exercises
Appendix A Basics of Mathematical Finance
A.1 Fundamental Theorems of Asset Pricing
A.2 Brownian Motion
A.2.1 Quadratic Variation of Brownian Paths
A.3 Ito Integral
A.4 Ito's Formula
A.5 The Black-Scholes Model
A.6 The Black–Scholes Formula for the Price of a European Call Option
Appendix B Introduction to R
B.1 R Language Essentials
B.2 Vector
B.3 Matrices and Arrays
B.4 Lists
B.5 Data Frame
B.6 Indexing
B.7 Looping
B.8 Functions
B.9 Download Stock Market Data Using R
Appendix C Extreme Value Theory (EVT) in Finance
Appendix Bibliography
Index
