2
1 Introduction
the students’ “toolbox” with additional ways to solve problems. Basically, it should
open their eyes to all the exciting methods out there that just wait to be applied to
problems that come their way.
Before delving into the subject matter, however, we have to familiarize ourselves
with the language and the concepts used in finance, to which the second chapter
is devoted. In the third chapter, we discuss how to pick a collection of stocks—a
portfolio—in such a way that the risk of losses is minimized. This is a wonderful application of variational methods with constraints, the same methods used to
analyze mechanical systems. This analysis is quasi-static, a restriction we relax in
the fourth chapter, where we consider stochastic processes and their description by
Langevin and Fokker-Planck equations. This chapter sets the stage to derive and
solve the Black-Scholes equation and to some of its applications in the fifth and
sixth chapter. Since stock values are recorded over time, they constitute time series
of data, from which we try to extract information by fitting regression models. This
fitting of models to data is very common in physics and is the topic of the seventh
and eights chapter, where we explore and illustrate reliability and robustness of the
fitting as well as forecasting into the future. In Chap. 9 we have a look at financial
bubbles and crashes by first discussing historical crashes and their possible reasons.
This leads to questioning the basis of the previous analysis and we encounter fattailed distributions, fractals, and the central-limit theorem in attempts to understand
these often dramatic events. In Chap. 10 we return to less disruptive topics and
apply Feynman’s path integrals to quantum and to financial systems and solve them
using Monte-Carlo methods. All systems covered up to this point followed some
intrinsic dynamics. In the eleventh chapter we develop methods to influence and
control these system ourselves by adjusting external parameters in order to optimize
some performance measure. This gives us a chance to review the relation between
Lagrangian and Hamiltonian mechanics and leads to the subject of optimal control
theory. Here we pick examples from mechanics and simple macroeconomic models.
In the final chapter we focus on cryptocurrencies and explore the underlying concepts
from information theory and cryptography before discussing two currencies: Bitcoin
and Ethereum. We close the chapter with a discussion of the threat that quantum
computers might pose to the cryptographic methods. MATLAB
® is used throughout
the book to illustrate various numerical methods. Useful functions are collected in
an appendix and are available as electronic supplementary material (ESM) from the
book’s web page at https://www.springer.com/9783030636425. At the end of each
chapter, the reader will find a number of exercises whose solutions are discussed
in the final chapter. The associated MATLAB examples are also available from the
book’s web page.
With the plan laid out, let’s jump right into the main part of the book and review
basic concepts and the language used in finance.
For product information on MATLAB
® , please contact:
The Mathworks, Inc.
3 Apple Hill Drive
Natick, MA 01760-2098 USA
Tel: 508-647-7000
1 Introduction
the students’ “toolbox” with additional ways to solve problems. Basically, it should
open their eyes to all the exciting methods out there that just wait to be applied to
problems that come their way.
Before delving into the subject matter, however, we have to familiarize ourselves
with the language and the concepts used in finance, to which the second chapter
is devoted. In the third chapter, we discuss how to pick a collection of stocks—a
portfolio—in such a way that the risk of losses is minimized. This is a wonderful application of variational methods with constraints, the same methods used to
analyze mechanical systems. This analysis is quasi-static, a restriction we relax in
the fourth chapter, where we consider stochastic processes and their description by
Langevin and Fokker-Planck equations. This chapter sets the stage to derive and
solve the Black-Scholes equation and to some of its applications in the fifth and
sixth chapter. Since stock values are recorded over time, they constitute time series
of data, from which we try to extract information by fitting regression models. This
fitting of models to data is very common in physics and is the topic of the seventh
and eights chapter, where we explore and illustrate reliability and robustness of the
fitting as well as forecasting into the future. In Chap. 9 we have a look at financial
bubbles and crashes by first discussing historical crashes and their possible reasons.
This leads to questioning the basis of the previous analysis and we encounter fattailed distributions, fractals, and the central-limit theorem in attempts to understand
these often dramatic events. In Chap. 10 we return to less disruptive topics and
apply Feynman’s path integrals to quantum and to financial systems and solve them
using Monte-Carlo methods. All systems covered up to this point followed some
intrinsic dynamics. In the eleventh chapter we develop methods to influence and
control these system ourselves by adjusting external parameters in order to optimize
some performance measure. This gives us a chance to review the relation between
Lagrangian and Hamiltonian mechanics and leads to the subject of optimal control
theory. Here we pick examples from mechanics and simple macroeconomic models.
In the final chapter we focus on cryptocurrencies and explore the underlying concepts
from information theory and cryptography before discussing two currencies: Bitcoin
and Ethereum. We close the chapter with a discussion of the threat that quantum
computers might pose to the cryptographic methods. MATLAB
® is used throughout
the book to illustrate various numerical methods. Useful functions are collected in
an appendix and are available as electronic supplementary material (ESM) from the
book’s web page at https://www.springer.com/9783030636425. At the end of each
chapter, the reader will find a number of exercises whose solutions are discussed
in the final chapter. The associated MATLAB examples are also available from the
book’s web page.
With the plan laid out, let’s jump right into the main part of the book and review
basic concepts and the language used in finance.
For product information on MATLAB
® , please contact:
The Mathworks, Inc.
3 Apple Hill Drive
Natick, MA 01760-2098 USA
Tel: 508-647-7000