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2019 | Buch

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Mathematics of Finance

An Intuitive Introduction

verfasst von: Prof. Donald G. Saari

Verlag: Springer International Publishing

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This textbook invites the reader to develop a holistic grounding in mathematical finance, where concepts and intuition play as important a role as powerful mathematical tools. Financial interactions are characterized by a vast amount of data and uncertainty; navigating the inherent dangers and hidden opportunities requires a keen understanding of what techniques to apply and when. By exploring the conceptual foundations of options pricing, the author equips readers to choose their tools with a critical eye and adapt to emerging challenges.

Introducing the basics of gambles through realistic scenarios, the text goes on to build the core financial techniques of Puts, Calls, hedging, and arbitrage. Chapters on modeling and probability lead into the centerpiece: the Black–Scholes equation. Omitting the mechanics of solving Black–Scholes itself, the presentation instead focuses on an in-depth analysis of its derivation and solutions. Advanced topics that follow include the Greeks, American options, and embellishments. Throughout, the author presents topics in an engaging conversational style. “Intuition breaks” frequently prompt students to set aside mathematical details and think critically about the relevance of tools in context.

Mathematics of Finance is ideal for undergraduates from a variety of backgrounds, including mathematics, economics, statistics, data science, and computer science. Students should have experience with the standard calculus sequence, as well as a familiarity with differential equations and probability. No financial expertise is assumed of student or instructor; in fact, the text’s deep connection to mathematical ideas makes it suitable for a math capstone course. A complete set of the author’s lecture videos is available on YouTube, providing a comprehensive supplementary resource for a course or independent study.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries via Gambles

Abstract

Before tackling the complexities of the financial market and encountering unfamiliar words such as “options,” “hedging,” “arbitrage,” “Puts,” and “Calls,” consider a simpler issue that, in fact, captures much of what will be discussed. Suppose next Sunday there will be a football game between the Vikings from Minneapolis and the Packers from Green Bay.

Donald G. Saari

Chapter 2. Options

Abstract

A valued lesson from Chapter 1 is that a way to reduce risk is to bet on both sides of an issue. To apply this theme to finance, appropriate market devices, with features similar to what has been discussed, are identified.

Donald G. Saari

Chapter 3. Modeling

Abstract

The football example demonstrated the wisdom of treating tacit assumptions with care: They may be false. Yes, the sum of the probabilities over all events equals one. But when dealing with different individuals, a selective sum of their “implicit probabilities” could differ from unity—which can either hurt or help you.

Donald G. Saari

Chapter 4. Some Probability

Abstract

Imagine what personal advantage could accrue if we knew, weeks in advance, tomorrow’s price of IBM or Uber stock! Such information would identify what Calls, Puts, or other financial activities to put into place to support an early retirement.

Donald G. Saari

Chapter 5. The Black–Scholes Equation

Abstract

We now are ready to derive the important Black–Scholes Equation [1], which is widely used to determine pricing of Calls and Puts! An outline is given next; details are developed in the next chapter.

Donald G. Saari

Chapter 6. Solutions of Black–Scholes

Abstract

Because many introductory courses in partial differential equations solve the heat equation, it is not necessary to do so here. Instead, the emphasis will be to explain the various terms embedded in the solution: All reflect those change of variables that converted the Black–Scholes Equation into the heat equation. These changes must be reinserted into a heat equation solution to transform it into a Black–Scholes solution.

Donald G. Saari

Chapter 7. Partial Information: The Greeks

Abstract

Applying the Equation 6.1 argument to P _E(S, t), along with the boundary condition $P_E(S, T) = \max (E-S, 0)$, makes it reasonable to expect that

$$\displaystyle P_E(S, t) = Ee^{-r(T-t)}\times [\mathrm{modifying terms}] - S\times [\mathrm{modifying terms}]. $$

This the case. The actual P _E(S, t) solution follows immediately from our powerful friend the Put–Call Parity Equation.

Donald G. Saari

Chapter 8. Sketching and the American Options

Abstract

Although C _E(S, t) and P _E(S, t) have been analyzed and described in various ways, something is missing. Similar to where a description of the moon shining over a lake nested in the snow-capped mountains is a poor substitute for an actual picture, what is needed is a portrait of these options. And so, we now give an outline how to sketch the graphs. Of added value, the method can be adopted to sketch our new acquaintances, the Greeks, to better understand where they exercise power.

Donald G. Saari

Chapter 9. Embellishments

Abstract

A delight of this topic is that it is possible to go on and on and on. But closure is needed somewhere, and it is with this chapter. The farewell message is to stress that the powerful tools derived in this book can be used elsewhere: This concluding chapter suggests how and where. At this stage, for instance, the reader probably can develop at least a partial explanation for other topics encountered on the market. It may take some imagination to find a surrogate for inflation, or pollution, or …, but being able to do so is what offers a personal advantage.

Donald G. Saari

Backmatter

Titel: Mathematics of Finance
verfasst von: Prof. Donald G. Saari
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-25443-8
Print ISBN: 978-3-030-25442-1
DOI: https://doi.org/10.1007/978-3-030-25443-8