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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 consumer finance assets and liabilities are in the form of level pay home mortgage, student, and auto loans. These products can still be tackled by the application of the Geometric series, and we can calculate various measures such as average life and time to pay a given fraction of the loan via these formulas. A large part of consumer home mortgage loans are securitized as mortgage‐backed securities by companies originally set up by the U.S. government to promote home ownership and student loans. The footprint of these giants in the financial markets is large and is the main driver of structured finance. We introduce tools and techniques to quantify the negative convexity risk due to prepayments for these markets.

While the analytical price‐yield formula for bonds, loans, and mortgage‐backed securities can provide pricing and risk measures for single products in isolation, a variety of bonds and fixed income products trade simultaneously in markets giving rise to different yield and spread curves. We introduce the bootstrap and interpolation methods to handle yields curves and overlapping cash flows of multiple instruments in a consistent manner.

Stocks, Investments

In Chapter 3, we focus on investments and the interplay between risk‐free and risky assets. We present the St. Petersburg paradox to motivate the concept of utility and to highlight the problem of investment choice, ranking, and decision‐making under uncertainty. We introduce the concept of risk‐preference and show the personalist nature of ranking of random payoffs. We present utility theory and its axioms, certainty‐equivalent lotteries, and different measures of risk‐preference (risk‐taking, risk‐aversion, risk‐neutrality) as characterized by the utility function. Utility functions representing different classes of Arrow‐Pratt measures (CARA, CRRA, HARA) are introduced and discussed.

The mean‐variance theory of portfolio selection draws from the techniques of constrained and convex optimization, and we discuss and show the method of Lagrange multipliers in various calculations such as the minimum‐variance portfolio, minimum‐variance frontier, and tangency (market) portfolio. The seminal CAPM formula relating the excess return of an asset to that of the market portfolio is derived by using the chain rule and properties of the hyperbola of feasible portfolios.

Moving from equilibrium results, we next introduce statistical techniques such as regression, factor models, and PCA to find common drivers of asset returns and statistical measures such as the alpha and beta of portfolio performance. Trading strategies such as pairs trading and mean‐reversion trades are based on these methods. We conclude by showing the use of recurrence equations and optimization techniques for risk and money management leading to the gambler's ruin formula and Kelly's ratio.

Forwards, Futures

In Chapter 4, we introduce the forward contract as the gateway product to more complicated contingent claims and options and derivatives. The basic cash‐and‐carry argument shows the method of static replication and arbitrage pricing. This method is used to compute forward prices in equities with discrete dividends or dividend yields, forward exchange rate via covered interest parity, and forward rates in interest rate markets.

Risk‐Neutral Option Pricing

Chapter 5 presents the building blocks of the modern risk‐neutral pricing framework. Starting with a simple one‐step binomial model, we flesh out the full details of the replication of a contingent claim via the underlying asset and a loan and show that a contingent claim's replication price can be computed by taking expectations in a risk‐neutral setting. This basic building block is extended to multiple steps through dynamic hedging of a self‐financing replicating portfolio, leading to martingale relative prices and the fundamental theorems of asset pricing for complete and arbitrage‐free economies.

Option Pricing

In Chapter 6, we use the risk‐neutral framework to derive the Black‐Scholes‐Merton (BSM) option pricing formula by modeling asset returns as the continuous‐time limit of a random walk, that is a Brownian motion with risk‐adjusted drift. We recover and investigate the underlying replicating portfolio by considering the option Greeks: delta, gamma, theta. The interplay between these is shown by applying the Ito's lemma to the diffusion process driving an underlying asset and its derivative, leading to the BSM partial differential equation and its solution via methods from the classical boundary value heat equations.

We discuss the Cox‐Ross‐Rubinstein (CRR) model as a popular and practical computational method for pricing options that can also be used to compute the price of options with early exercise features via the backward induction algorithm from dynamic programming. For path‐dependent options such as barrier or averaging options, we present numerical models such as the Monte Carlo simulation models and variance reduction techniques.

Interest Rate Derivatives

Chapter 7 introduces interest rate swaps and their derivatives used in structured finance. A plain vanilla swap can be priced via a static replication argument from a bootstrapped discount factor curve. In practice, simple European options on swaps and interest rate products are priced and risk‐managed via the normal version of Black's formula for futures. We introduce this model under the risk‐neutral pricing framework and show the pricing of the mainstream cap/floors, European swaptions, and CMS products. For complex derivatives, one needs a model for the evolution of multiple maturity zero‐coupon bonds in a risk‐neutral framework. We present the popular Hull‐White mean‐reverting model for the short rate and show the typical implementation methods and techniques, such as the forward induction method for yield curve inversion. We show the pricing of Bermudan swaptions via these lattice models. We conclude our discussion by presenting methods for calculating interest rate curve risk and VaR.

Exercises and Python Projects

The end‐of‐chapter exercises are based on real‐world markets and products and delve deeper into some financial products and highlight the details of applying the techniques to them. All exercises can be solved by using a spreadsheet package like Excel. The Python projects are longer problems and can be done by small groups of students as a term project.

It is my hope that by the end of this book, readers have obtained a good toolkit of mathematical techniques, methods, and models used in financial markets and products, and their interest is piqued for a deeper journey into quantitative finance.

—Amir Sadr

New York, New York

December 2021