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Library of Congress Cataloging-in-Publication Data:
Names: Sadr, Amir, author.
Title: Mathematical techniques in finance : an introduction / Amir Sadr.
Description: Hoboken, New Jersey : John Wiley & Sons, Inc., [2022] | Series: Wiley finance series | Includes index.
Identifiers: LCCN 2022000565 (print) | LCCN 2022000566 (ebook) | ISBN 9781119838401 (cloth) | ISBN 9781119838425 (adobe pdf) | ISBN 9781119838418 (epub)
Subjects: LCSH: Finance–Mathematical models.
Classification: LCC HG106 .S23 2022 (print) | LCC HG106 (ebook) | DDC 332.01/5195–dc23/eng/20220112
LC record available at https://lccn.loc.gov/2022000565
LC ebook record available at https://lccn.loc.gov/2022000566
Cover Design: Wiley
Cover Image: © StationaryTraveller/Getty
To my students
Preface
Finance as a distinct field from economics is generally defined as the science or study of the management of funds. The creation of credit, savings, investments, banking institutions, financial markets and products, and risk management all fall under the purview of finance. The unifying themes in finance are time, risk, and money.
Mathematical or quantitative finance is the application of mathematics to these core areas. While simple arithmetic was enough for accounting and keeping ledgers and double‐entry bookkeeping, Louis Bachelier's doctoral thesis, Théorie de la spéculation and published in 1900, used Brownian motion to study stock prices, and is widely recognized as the beginning of quantitative finance. Since then, the use of increasingly sophisticated and specialized mathematics has created the modern field of quantitative finance encompassing investment theory, asset pricing, derivatives, financial data science, and the emerging area of crypto assets and Decentralized Finance (DeFi).
BACKGROUND
This book is the collection of my lecture notes for an elective senior level undergraduate course on mathematics of finance at NYU Courant. The mostly senior and some first year graduate students come from different majors with an even distribution of mathematics, engineering, economics, and business majors. The prerequisites for the book are the same as the ones for the course: basic calculus, probability, and linear algebra. The goal of the book is to introduce the mathematical techniques used in different areas of finance and highlight their usage by drawing from actual markets and products.
BOOK STRUCTURE
A simple definition of finance would be the study of money; quantitative finance could be thought of as the mathematics of money. While reductive and simplistic, this book uses this metaphor and follows the money across different markets to motivate and introduce concepts and mathematical techniques.
Bonds
In Chapter 2, we start with the basic building blocks of interest rates and time value of money to price and discount future cash flows for fixed income and bond markets. The concept of compound interest and its limit as continuous compounding is the first foray into mathematics of finance. Coupon bonds make regular interest payments, and we introduce the Geometric series to derive the classic bond price‐yield formula.
As there is generally no closed form formula for implied calculations such as implied yield or volatility given a bond or option price, these calculations require numerical root‐solving methods and we present the Newton‐Raphson method and the more robust and popular bisection method.
The concept of risk is introduced by considering the bond price sensitivity to interest rates. The Taylor series expansion of a function provides the first and second order sensitivities leading to duration and convexity for bonds in Chapter 2, and delta and gamma for options in Chapter 6. Similar first and second order measures are the basis of the mean‐variance theory of portfolio selection in Chapter 3.
In the United States, households hold the largest amount of net worth, followed by firms, while the U.S. government runs a negative balance and is in debt. Most of
