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Mastering Python for Finance
Mastering Python for Finance

Mastering Python for Finance: Understand, design, and implement state-of-the-art mathematical and statistical applications used in finance with Python

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Mastering Python for Finance

Chapter 2. The Importance of Linearity in Finance

Nonlinear dynamics plays a vital role in our world. Linear models are often employed in economics due to their simplicity for studies and easier modeling capabilities. In finance, linear models are widely used to help price securities and perform optimal portfolio allocation, among other useful things. One of the significance of linearity in financial modeling is its assurance that a problem terminates at a global optimal solution.

In order to perform prediction and forecasting, regression analysis is widely used in the field of statistics to estimate relationships among variables. With an extensive mathematics library being one of Python's greatest strength, Python is frequently used as a scientific scripting language to aid in these problems. Modules such as the SciPy and NumPy packages contain a variety of linear regression functions for data scientists to work with.

In traditional portfolio management, the allocation of...

The capital asset pricing model and the security market line

Many financial literatures devote exclusive discussions to the capital asset pricing model (CAPM). In this section, we will explore the key concepts that highlight the importance of linearity in finance.

In the famous CAPM, the relationship between risk and rates of returns in a security is described as follows:

The capital asset pricing model and the security market line

For a security The capital asset pricing model and the security market line, its returns is defined as The capital asset pricing model and the security market line and its beta as The capital asset pricing model and the security market line. The CAPM defines the return of the security as the sum of the risk-free rate The capital asset pricing model and the security market line and the multiplication of its beta with the risk premium. The risk premium can be thought of as the market portfolio's excess returns exclusive of the risk-free rate. The following figure is a visual representation of the CAPM:

The capital asset pricing model and the security market line

Beta is a measure of the systematic risk of a stock; a risk that cannot be diversified away. In essence, it describes the sensitivity of stock returns with respect to movements in the market. For example, a stock with a beta of zero produces no excess returns...

The Arbitrage Pricing Theory model

The CAPM suffers from several limitations, such as the use of a mean-variance framework and the fact that returns are captured by one risk factor—the market risk factor. In a well-diversified portfolio, the unsystematic risk of various stocks cancels out and is essentially eliminated.

The Arbitrage Pricing Theory (APT) model was put forward to address these shortcomings and offers a general approach of determining the asset prices other than the mean and variances.

The APT model assumes that the security returns are generated according to multiple factor models, which consist of a linear combination of several systematic risk factors. Such factors could be the inflation rate, GDP growth rate, real interest rates, or dividends.

The equilibrium asset pricing equation according to the APT model is as follows:

The Arbitrage Pricing Theory model

Here, The Arbitrage Pricing Theory model is the expected rate of return on security The Arbitrage Pricing Theory model, The Arbitrage Pricing Theory model is the expected return on stock The Arbitrage Pricing Theory model if all factors are negligible, The Arbitrage Pricing Theory model is the sensitivity of the The Arbitrage Pricing Theory modelth...

Multivariate linear regression of factor models

Many Python packages such as SciPy come with several variants of regression functions. In particular, the statsmodels package is a complement to SciPy with descriptive statistics and estimation of statistical models. The official page for statsmodels is http://statsmodels.sourceforge.net/.

In this example, we will use the ols function of the statsmodels module to perform an ordinary least squares regression and view its summary.

Let's assume that you have implemented an APT model with seven factors that return the values of Y. Consider the following set of data collected over 9 time periods, Multivariate linear regression of factor models to Multivariate linear regression of factor models. X1 to X7 are independent variables observed at each period. The regression problem is therefore structured as:Multivariate linear regression of factor models.

A simple ordinary least squares regression on values of X and Y can be performed with the following code:

""" Least squares regression with statsmodels """
import numpy as np
import statsmodels.api as sm
...

Linear optimization

In the CAPM and APT pricing theories, we assumed linearity in the models and solved for expected security prices using regressions in Python.

As the number of securities in our portfolio increases, certain limitations are introduced as well. A portfolio manager would find himself constrained by these rules in pursing certain objectives mandated by investors.

Linear optimization helps you overcome the problem of portfolio allocation. Optimization focuses on minimizing or maximizing the value of the objective functions. The examples are maximizing returns and minimizing volatility. These objectives are usually governed by certain regulations, such as no short-selling rule, limits on the number of securities to be invested, and so on.

Unfortunately, in Python there is no single official package that supports this solution. However, there are third-party packages available with the implementation of the simplex algorithm for linear programming. For the purpose of this demonstration...

Solving linear equations using matrices

In the previous section, we looked at solving a system of linear equations with inequality constraints. If a set of systematic linear equations has constraints that are deterministic, we can represent the problem as matrices and apply matrix algebra. Matrix methods represent multiple linear equations in a compact manner while using the existing matrix library functions.

Suppose we would like to build a portfolio consisting of 3 securities, Solving linear equations using matrices, Solving linear equations using matrices and Solving linear equations using matrices. The allocation of the portfolio must meet certain constraints: it must consist of 6 units of a long position in security a. With every 2 units of security a, 1 unit of security b, and 1 unit of security c invested, the net position must be 4 units long. With every 1 unit of security a, 3 units of security b, and 2 units of security c invested, the net position must be long 5 units.

To find out the number of securities to invest in, we can frame the problem mathematically as follows:

Solving linear equations using matrices
Solving linear equations using matrices
Solving linear equations using matrices

With all of the coefficients...

The capital asset pricing model and the security market line


Many financial literatures devote exclusive discussions to the capital asset pricing model (CAPM). In this section, we will explore the key concepts that highlight the importance of linearity in finance.

In the famous CAPM, the relationship between risk and rates of returns in a security is described as follows:

For a security , its returns is defined as and its beta as . The CAPM defines the return of the security as the sum of the risk-free rate and the multiplication of its beta with the risk premium. The risk premium can be thought of as the market portfolio's excess returns exclusive of the risk-free rate. The following figure is a visual representation of the CAPM:

Beta is a measure of the systematic risk of a stock; a risk that cannot be diversified away. In essence, it describes the sensitivity of stock returns with respect to movements in the market. For example, a stock with a beta of zero produces no excess returns regardless...

The Arbitrage Pricing Theory model


The CAPM suffers from several limitations, such as the use of a mean-variance framework and the fact that returns are captured by one risk factor—the market risk factor. In a well-diversified portfolio, the unsystematic risk of various stocks cancels out and is essentially eliminated.

The Arbitrage Pricing Theory (APT) model was put forward to address these shortcomings and offers a general approach of determining the asset prices other than the mean and variances.

The APT model assumes that the security returns are generated according to multiple factor models, which consist of a linear combination of several systematic risk factors. Such factors could be the inflation rate, GDP growth rate, real interest rates, or dividends.

The equilibrium asset pricing equation according to the APT model is as follows:

Here, is the expected rate of return on security , is the expected return on stock if all factors are negligible, is the sensitivity of the th asset...

Multivariate linear regression of factor models


Many Python packages such as SciPy come with several variants of regression functions. In particular, the statsmodels package is a complement to SciPy with descriptive statistics and estimation of statistical models. The official page for statsmodels is http://statsmodels.sourceforge.net/.

In this example, we will use the ols function of the statsmodels module to perform an ordinary least squares regression and view its summary.

Let's assume that you have implemented an APT model with seven factors that return the values of Y. Consider the following set of data collected over 9 time periods, to . X1 to X7 are independent variables observed at each period. The regression problem is therefore structured as:.

A simple ordinary least squares regression on values of X and Y can be performed with the following code:

""" Least squares regression with statsmodels """
import numpy as np
import statsmodels.api as sm

# Generate some sample data
num_periods...

Linear optimization


In the CAPM and APT pricing theories, we assumed linearity in the models and solved for expected security prices using regressions in Python.

As the number of securities in our portfolio increases, certain limitations are introduced as well. A portfolio manager would find himself constrained by these rules in pursing certain objectives mandated by investors.

Linear optimization helps you overcome the problem of portfolio allocation. Optimization focuses on minimizing or maximizing the value of the objective functions. The examples are maximizing returns and minimizing volatility. These objectives are usually governed by certain regulations, such as no short-selling rule, limits on the number of securities to be invested, and so on.

Unfortunately, in Python there is no single official package that supports this solution. However, there are third-party packages available with the implementation of the simplex algorithm for linear programming. For the purpose of this demonstration...

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Description

If you are an undergraduate or graduate student, a beginner to algorithmic development and research, or a software developer in the financial industry who is interested in using Python for quantitative methods in finance, this is the book for you. It would be helpful to have a bit of familiarity with basic Python usage, but no prior experience is required.

Who is this book for?

If you are an undergraduate or graduate student, a beginner to algorithmic development and research, or a software developer in the financial industry who is interested in using Python for quantitative methods in finance, this is the book for you. It would be helpful to have a bit of familiarity with basic Python usage, but no prior experience is required.

What you will learn

  • Perform interactive computing with IPython Notebook
  • Solve linear equations of financial models and perform ordinary least squares regression
  • Explore nonlinear modeling and solutions for optimum points using rootfinding algorithms and solvers
  • Discover different types of numerical procedures used in pricing options
  • Model fixedincome instruments with bonds and interest rates
  • Manage big data with NoSQL and perform analytics with Hadoop
  • Build a highfrequency algorithmic trading platform with Python
  • Create an eventdriven backtesting tool and measure your strategies

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Publication date : Apr 29, 2015
Length: 340 pages
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Product Details

Publication date : Apr 29, 2015
Length: 340 pages
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Language : English
ISBN-13 : 9781784397876
Category :
Languages :
Concepts :

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Table of Contents

11 Chapters
1. Python for Financial Applications Chevron down icon Chevron up icon
2. The Importance of Linearity in Finance Chevron down icon Chevron up icon
3. Nonlinearity in Finance Chevron down icon Chevron up icon
4. Numerical Procedures Chevron down icon Chevron up icon
5. Interest Rates and Derivatives Chevron down icon Chevron up icon
6. Interactive Financial Analytics with Python and VSTOXX Chevron down icon Chevron up icon
7. Big Data with Python Chevron down icon Chevron up icon
8. Algorithmic Trading Chevron down icon Chevron up icon
9. Backtesting Chevron down icon Chevron up icon
10. Excel with Python Chevron down icon Chevron up icon
Index Chevron down icon Chevron up icon

Customer reviews

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Full star icon Full star icon Full star icon Half star icon Empty star icon 3.3
(7 Ratings)
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2 star 28.6%
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Duncan W. Robinson Jun 14, 2015
Full star icon Full star icon Full star icon Full star icon Full star icon 5
Wow… good stuff. Mastering Python for Finance really delivers. It does a fine job blending academic theory with a practitioner’s penchant for useful applications, delivering Python code all the way to guide the reader along his / her journey. I have not worked all of the examples yet, but I am looking forward to doing so – especially the sections that use Hadoop & Python together.
Amazon Verified review Amazon
Federico Han Feb 28, 2018
Full star icon Full star icon Full star icon Full star icon Empty star icon 4
A pretty good survey on many object-oriented implementations common in Finance.Plenty of applications and examples -some indeed with errors , though on the positive it is a useful learning experience to do some minor debugging !-.As with many practical reference books try to buy the print version, this is not bedtime reading stuff.
Amazon Verified review Amazon
Kindleのお客様 Jul 06, 2016
Full star icon Full star icon Full star icon Full star icon Empty star icon 4
金融工学の基礎を一通り習得した人が、pythonでコードをかけるようになるための一冊。目次の内容に対して、特に数学的な証明等なしで天下り的にPythonのコードが羅列されているので、ファイナンスの入門者にはお勧めしない。費用対効果は高い。
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Martin Galli Jan 01, 2017
Full star icon Full star icon Full star icon Empty star icon Empty star icon 3
Bon livre en général mais dommage qu'il soit en Python 2.7 et non Python3.5. Sinon, les solutions sont très bien expliquées et l'organisation est claire.
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Ho Yan Chan May 02, 2018
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ok
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