Computational finance. An introductory course with R.

*(English)*Zbl 1309.91001
Atlantis Studies in Computational Finance and Financial Engineering 1. Amsterdam: Atlantis Press (ISBN 978-94-6239-069-0/hbk; 978-94-6239-070-6/ebook). x, 301 p. (2014).

Finance is about managing money. There are several instruments for dealing with money. There are mathematical models for the optimal selection of financial portfolios and for predicting their assets’ future values. Also computers can help us with all the necessary real-world data crunching to exploit the formal models to unprecedented possibilities.

The aim of this book is to give knowledge required for all numerical methods, theorems, algorithms and optimization heuristics geared to the solution of problems in economics and finance. Also the subject area includes the computational resolution of financial problems.

Chapter 1 – an arbidget introduction to finance – is intended for giving the reader the minimum background on the fundamentals of finance. It is an outlook on the most common financial instruments and the places where these are traded. There is an introduction to investment strategies, portfolio management and basic asset pricing. Section 1.1 is devoted to the problems of financial securities. In Section 1.2, some notions, as a price and value of stocks, an arbitrage and risk-neutral valuation principle, European options, risk-neutral valuation, are presented and discussed. In Section 1.3 some remarks about R lab are given.

In Chapter 2, the author briefly reviews some of the fundamental concepts of statistics, such as moments of a distribution, distribution and density functions, likelihood methods and other tools that are necessary for the analysis of returns and financial time series. In Section 2.1, the notions of time series of returns as simple returns, \(\tau\)-period simple gross returns, log returns, returns with dividends, excess returns are given and discussed. In Section 2.2, a series of distributions, as the normal distribution, distributions of financial returns, the log-normal distribution, the uniform distribution are presented and discussed. The central limit theorem is given. The concept of moments of a random variable is given. In Section 2.3, the notions of stationarity, covariance of two random variables and autocovariance are presented. In Section 2.4, the idea for designing models for predicting future values is developed. In Section 2.5, the idea for maximum likelihood methods, as a technique in statistics, is presented. In Section 2.6, the notions of volatility, scaling the volatility estimates and time dependent weighted volatility are developed. In Section 2.7, some R lab demonstrations for going some descriptive statistics of financial returns are presented.

In Chapter 3, some correlations, causalities and similarities are considered. In Section 3.1, a series of correlations, as a measure of association, are considered. For example, the linear correlation and the rank correlation are developed. In Section 3.3, the notions of the causality as a Granger causality, nonparametric Granger causality are developed. In Section 3.3, the basics of data clustering, clustering methods, hierarchical clustering, partitional clustering and graph based clustering are presented. In Section 3.4, several empirical properties of asset returns using a variety of tools for statistical analysis are learned.

Chapter 4 presents some basic discrete-time models for financial time series. In Section 4.1, some notes on trend and seasonality are presented. Section 4.2 is devoted to present linear processes and autoregressive moving average models. In Section 4.3, some nonlinear models as the autoregressive conditional heteroscedastic model of order \(p\) and the generalized autoregressive conditional heteroscedasticity model of order \(p\) are considered. In Section 4.4, two new nonlinear models as neural networks and support vector machines are developed and illustrated. In Section 4.5, tests for nonlinearity and tests of model performance are given.

Chapter 5 presents Brownian motion, also known as Wiener process. On top of the Brownian motion other more elaborated continuous processes are built, that explain better the behavior of stock prices. Also, the geometric Brownian motion which is at the core of the Black-Scholes option pricing formula, is considered. In Section 5.1, a general review of continuous-time processes is presented. The concepts of the Wiener process, the generalized Wiener process or arithmetic Brownian motion are given. The Itō lemma for the geometric Brownian motion is proved. In Section 5.2, the Black-Scholes model for pricing European options is given and details of the binomial free option pricing model are presented. In Section 5.3, Monte Carlo variation of derivatives, as a numerical approach to valuing securities, is developed. The Euler-Monte Carlo method to calculate the European options with its convergence is given. Also, a Monte Carlo method to calculate the price of Asian options is given. In Section 5.4, some computer lab and problems are presented.

In Chapter 6, two popular approaches, Technical analysis and Fundamental analysis, are considered to investment, although considered as opposite paradigms of financial engineering. In Section 6.1, the concept of Technical analysis with details as chats, support and resistance levels and trends is given. In Subsection 6.1.4, the mathematical foundation for Technical analysis is developed. Section 6.2 is devoted to Fundamental analysis. This chapter finishes with computer lab and problems.

Chapter 7 considers algorithmic methods for finding approximate solutions of the basic optimization problem of minimizing or maximizing. In Section 7.2, the algorithm of the simulated annealing method is presented. In Section 7.3, the basics of genetic programming, as an extension of the genetic algorithm, is presented. Also, the algorithm for finding a technical trading rule is given. In Section 7.4, the ant colony optimization is described and presented as an algorithm. In Section 7.5, it is shown how to combine the individual heuristics to obtain hybrid heuristics. This chapter finishes again with computer lab and problems.

Chapter 8 presents the Markowitz mean-variance model for portfolio optimization and some of tools for portfolio management such as Sharpe ratio, beta of a security and the Capital Asset Pricing Model. In Section 8.1, the details of the mean-variance model of Markowitz are presented. In Section 8.2, portfolios with a risk-free asset are considered. In Section 8.3, an optimization of portfolios under different constraint sets is developed. Also the algorithm for simulated annealing optimization of portfolios is given. Section 8.4 considers the problem of portfolio selection. This chapter finishes again with computer lab and problems.

Chapter 9 presents some problems of online finance as online competitive analysis, online price search, searching for a price at random, online trading, online portfolio selection, the universal online portfolio and efficient universal online portfolio strategies. This chapter finishes again with computer lab and problems.

In Appendix A a brief introduction to R is given. A list of available packages that extend R are presented.

The aim of this book is to give knowledge required for all numerical methods, theorems, algorithms and optimization heuristics geared to the solution of problems in economics and finance. Also the subject area includes the computational resolution of financial problems.

Chapter 1 – an arbidget introduction to finance – is intended for giving the reader the minimum background on the fundamentals of finance. It is an outlook on the most common financial instruments and the places where these are traded. There is an introduction to investment strategies, portfolio management and basic asset pricing. Section 1.1 is devoted to the problems of financial securities. In Section 1.2, some notions, as a price and value of stocks, an arbitrage and risk-neutral valuation principle, European options, risk-neutral valuation, are presented and discussed. In Section 1.3 some remarks about R lab are given.

In Chapter 2, the author briefly reviews some of the fundamental concepts of statistics, such as moments of a distribution, distribution and density functions, likelihood methods and other tools that are necessary for the analysis of returns and financial time series. In Section 2.1, the notions of time series of returns as simple returns, \(\tau\)-period simple gross returns, log returns, returns with dividends, excess returns are given and discussed. In Section 2.2, a series of distributions, as the normal distribution, distributions of financial returns, the log-normal distribution, the uniform distribution are presented and discussed. The central limit theorem is given. The concept of moments of a random variable is given. In Section 2.3, the notions of stationarity, covariance of two random variables and autocovariance are presented. In Section 2.4, the idea for designing models for predicting future values is developed. In Section 2.5, the idea for maximum likelihood methods, as a technique in statistics, is presented. In Section 2.6, the notions of volatility, scaling the volatility estimates and time dependent weighted volatility are developed. In Section 2.7, some R lab demonstrations for going some descriptive statistics of financial returns are presented.

In Chapter 3, some correlations, causalities and similarities are considered. In Section 3.1, a series of correlations, as a measure of association, are considered. For example, the linear correlation and the rank correlation are developed. In Section 3.3, the notions of the causality as a Granger causality, nonparametric Granger causality are developed. In Section 3.3, the basics of data clustering, clustering methods, hierarchical clustering, partitional clustering and graph based clustering are presented. In Section 3.4, several empirical properties of asset returns using a variety of tools for statistical analysis are learned.

Chapter 4 presents some basic discrete-time models for financial time series. In Section 4.1, some notes on trend and seasonality are presented. Section 4.2 is devoted to present linear processes and autoregressive moving average models. In Section 4.3, some nonlinear models as the autoregressive conditional heteroscedastic model of order \(p\) and the generalized autoregressive conditional heteroscedasticity model of order \(p\) are considered. In Section 4.4, two new nonlinear models as neural networks and support vector machines are developed and illustrated. In Section 4.5, tests for nonlinearity and tests of model performance are given.

Chapter 5 presents Brownian motion, also known as Wiener process. On top of the Brownian motion other more elaborated continuous processes are built, that explain better the behavior of stock prices. Also, the geometric Brownian motion which is at the core of the Black-Scholes option pricing formula, is considered. In Section 5.1, a general review of continuous-time processes is presented. The concepts of the Wiener process, the generalized Wiener process or arithmetic Brownian motion are given. The Itō lemma for the geometric Brownian motion is proved. In Section 5.2, the Black-Scholes model for pricing European options is given and details of the binomial free option pricing model are presented. In Section 5.3, Monte Carlo variation of derivatives, as a numerical approach to valuing securities, is developed. The Euler-Monte Carlo method to calculate the European options with its convergence is given. Also, a Monte Carlo method to calculate the price of Asian options is given. In Section 5.4, some computer lab and problems are presented.

In Chapter 6, two popular approaches, Technical analysis and Fundamental analysis, are considered to investment, although considered as opposite paradigms of financial engineering. In Section 6.1, the concept of Technical analysis with details as chats, support and resistance levels and trends is given. In Subsection 6.1.4, the mathematical foundation for Technical analysis is developed. Section 6.2 is devoted to Fundamental analysis. This chapter finishes with computer lab and problems.

Chapter 7 considers algorithmic methods for finding approximate solutions of the basic optimization problem of minimizing or maximizing. In Section 7.2, the algorithm of the simulated annealing method is presented. In Section 7.3, the basics of genetic programming, as an extension of the genetic algorithm, is presented. Also, the algorithm for finding a technical trading rule is given. In Section 7.4, the ant colony optimization is described and presented as an algorithm. In Section 7.5, it is shown how to combine the individual heuristics to obtain hybrid heuristics. This chapter finishes again with computer lab and problems.

Chapter 8 presents the Markowitz mean-variance model for portfolio optimization and some of tools for portfolio management such as Sharpe ratio, beta of a security and the Capital Asset Pricing Model. In Section 8.1, the details of the mean-variance model of Markowitz are presented. In Section 8.2, portfolios with a risk-free asset are considered. In Section 8.3, an optimization of portfolios under different constraint sets is developed. Also the algorithm for simulated annealing optimization of portfolios is given. Section 8.4 considers the problem of portfolio selection. This chapter finishes again with computer lab and problems.

Chapter 9 presents some problems of online finance as online competitive analysis, online price search, searching for a price at random, online trading, online portfolio selection, the universal online portfolio and efficient universal online portfolio strategies. This chapter finishes again with computer lab and problems.

In Appendix A a brief introduction to R is given. A list of available packages that extend R are presented.

Reviewer: Vassil Grozdanov (Blagoevgrad)

##### MSC:

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |

91G60 | Numerical methods (including Monte Carlo methods) |

91-08 | Computational methods for problems pertaining to game theory, economics, and finance |

91-04 | Software, source code, etc. for problems pertaining to game theory, economics, and finance |

91G10 | Portfolio theory |

91G20 | Derivative securities (option pricing, hedging, etc.) |

91G70 | Statistical methods; risk measures |

91B84 | Economic time series analysis |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

65C05 | Monte Carlo methods |

90C59 | Approximation methods and heuristics in mathematical programming |

90C90 | Applications of mathematical programming |