# Reading Track. Mathematical Finance Arlie O. Petters and Xiaoying Dong, *An Introduction to Mathematical Finance with Applications. Understanding and Building Financial Intuition* (Springer Undergraduate Texts in Mathematics and Technology, 2016). The full text is available through the Mines library at [link.springer.com (Mines access)](https://link-springer-com.mines.idm.oclc.org/book/10.1007/978-1-4939-3783-7). Off campus you will be asked for your Mines login. > [!info]- Why this book > Two reasons. First, its opening chapters are a translation layer. Chapter 1 defines the vocabulary of banks, rates, and securities markets, and Chapter 2 builds the time value of money from scratch, so that the finance words map onto mathematics you already own before any new mathematics appears. That translation step is the hard part of the finance projects students have tried in the past, and this book does it well. Second, Chapter 5 models security prices with binomial trees, which are the random walks we consider in this course wearing a different vocabulary, and Chapter 6 takes the continuum limit to geometric Brownian motion, which is where the [[MATH310F26 - Reading Track - Stochastic Differential Equations|Stochastic Differential Equations track]] lands. So the book meets the course exactly where the course is going. > [!quote]- From the preface. Prerequisites and audience > **Prerequisites.** The required mathematics consists of introductory courses on multivariable calculus, probability, and linear algebra. Along the way, we introduce additional mathematical tools as needed, e.g., some measure theory is presented from scratch. > > *No background in finance is assumed.* As noted above, the necessary financial concepts and tools are introduced in the text, with the first chapter giving an overview of several common finance terminologies associated with securities and securities markets. > > Our book does *not* require computer programming. In our experience, finance courses based on computer programming are best taken after students have developed a fundamental understanding of the theoretical architecture of financial models. > > **Audience.** The text is aimed at *advanced undergraduates* and *master's degree students* who are either new to finance or want a more rigorous treatment of the mathematical models used in finance. The students typically are from economics, mathematics, engineering, physics, and computer science. > > We also believe that a faculty member who is teaching finance for the first time will find this introduction readily manageable. ## Map of the book | Chapter | What it is for us | |---|---| | 1. Preliminaries on Financial Markets | The vocabulary. Banks and rates, securities markets, economic indicators. Read it once, then keep it open as a glossary. | | 2. The Time Value of Money | Compound interest is a geometric recurrence. Net present value and internal rate of return are the mathematics behind "is this degree worth it" and "which investment strategy." Annuities, stock and bond valuation. | | 3. Markowitz Portfolio Theory | Risk and return as mean and variance. The efficient frontier is a constrained optimization. A good model-review chapter. | | 4. Capital Market Theory and Portfolio Risk Measures | Value at risk and the linear factor models. The risk-management questions live here. | | 5. Binomial Trees and Security Pricing Modeling | The random walk in log price. See the once-over below. | | 6. Stochastic Calculus and Geometric Brownian Motion Model | Brownian motion, quadratic variation, the Itô integral and formula, geometric Brownian motion, and Section 6.10, "BM as a limit of simple symmetric RW," which is our Random-Walks note in the book's notation. | | 7. Derivatives | Forwards, futures, swaps, options. Needed only if your question is about a derivative contract. | | 8. The BSM Model and European Option Pricing | Black-Scholes-Merton, risk-neutral pricing, hedging, and the model against market data (Section 8.8), which is the honest "where does it fail" chapter. | The library link above gets you every chapter. The chapters that align with the course are 1, 2, 5, and 6, in that order. ## A quick once-over. Binomial trees and security pricing The binomial tree is the simplest model of a price that moves randomly in time, and it is a random walk with a change of variables. **The tree.** Start at price $S_0$. Over each time step of length $\Delta t$ the price either goes up by a factor $u$ or down by a factor $d$, with probability $p$ of going up. After $n$ steps, if $k$ of them were up moves, the price is $S_n = S_0\, u^{k} d^{\,n-k}, \qquad k \sim \text{Binomial}(n, p).$ Take the logarithm and the multiplication becomes addition, $\ln S_n = \ln S_0 + k \ln u + (n-k)\ln d,$ which is a walk on the line with steps $\ln u$ and $\ln d$. That is the Plinko board. The ball falls through $n$ rows of pegs, bounces left or right at each, and the bin it lands in is the price. **The Cox-Ross-Rubinstein choice.** Section 5.2 picks $u = e^{\sigma\sqrt{\Delta t}}$ and $d = 1/u$, where $\sigma$ is the volatility. Notice the $\sqrt{\Delta t}$. It is the same scaling $\Delta y \propto \sqrt{\Delta t}$ that our Random-Walks note identifies as the one that makes the continuum limit exist. Section 5.3 takes that limit and gets geometric Brownian motion, $dS = \mu S\, dt + \sigma S\, dW,$ the standard stock-price model, and Chapter 6 develops the calculus needed to work with it. **Pricing on the tree.** The book's second idea is that you can price a contract on the tree by working backward from its payoff at the last step. The trick is to replace $p$ by the risk-neutral probability $q = \frac{e^{r\Delta t} - d}{u - d},$ where $r$ is the risk-free rate, and then the price today is the expected payoff under $q$, discounted. Backward induction on the tree is a recurrence relation, one step per level, which is why a spreadsheet can price an option. The F25 [[Binomial Tree Guided Discovery Activity]] walks through this with your own numbers. **What to take from it.** A binomial tree is a discrete-time random walk, a geometric Brownian motion is its continuous-time limit, and pricing is expectation under a cleverly chosen probability. If you can say those three sentences with the symbols attached, you own Chapters 5 and 6 at the level this course needs. <!-- Related. [[MATH310F26 - Reading Track - Stochastic Differential Equations]] for the continuous-time mathematics in our own notation.