Quantitative Finance

A Simulation-Based Introduction Using Excel

Matt Davison

• Publisher: Chapman and Hall/CRC
• Year: 2014
• ISBN:
  9781498782661
• Edition: 1
• Topic: Business
• Subtopic: Finance
• Index: Business

Teach Your Students How to Become Successful Working QuantsQuantitative Finance: A Simulation-Based Introduction Using Excel provides an introduction to financial mathematics for students in applied mathematics, financial engineering, actuarial science, and business administration. The text not only enables students to practice with the basic techn

Chapter 1

Introduction

This is an introductory book on financial mathematics. It is neither mathematically rigorous nor incredibly realistic about actual markets. It is designed to be a book that helps its reader to gain a certain amount of practice with the basic techniques of financial mathematics, but more importantly, to gain significant intuition about what they are, what they mean, how they work, and, perhaps, most importantly, what happens when they stop working.

To me, the whole beautiful idea of financial mathematics is that it recognizes that risk is something that can be shaped and altered. This idea is the source of many of the most famous ideas of financial mathematics, from Markowitz’s mean variance portfolio theory, to the Black Scholes equation, and to the invention of collateralized debt obligations.

All three ideas are about risk, but all three ideas are also about portfolios. Mean variance portfolio theory is famously about, not so much picking the right stocks, but about assembling the right teams of stocks. Collateralized debt obligations are about creating a cash waterfall that unequally distributes risk so that those who want high risk and high return can have it, and those who want low risk and are OK with accepting the corresponding low return can have that as well. (This was the idea, in any case.)

The Black Scholes idea, that we can price derivative securities by creating portfolios that entirely remove all risk, has been very successful. It has explained why market participants with widely varying attitudes toward risk can nonetheless all agree on the same price for a stock option.

But, in terms of teaching our subject, perhaps, the Black Scholes model has been a bit too successful. For a newcomer to the subject, there is perhaps not sufficient ground between the idea that, under certain idealized circumstances, all risk can be removed from an option-containing portfolio, to the idea that risk is something we do not need to consider at all.

To get away from that, I have chosen to begin my book with some chapters discussing risk and return, and classical decision theory, and to use a lot of examples from simple bond portfolios to build the required intuition. The tool used to look at these portfolios of bonds is not only pencil and paper calculations, but spreadsheets simulating all the messy glory of the things that can happen, not nice well-behaved mean and variances.

Why have I chosen to use spreadsheets? In part, because I find them beautifully visual in their display of all the guts and details of a portfolio, and in part because all working quants must have at least some familiarity with this tool.

Students working through math books are, or at least should be, aware of the fact that they need to do so with a pencil in hand, checking the derivations and working the exercises. Students working on this book should try to build the spreadsheets as well. We have provided the spreadsheets in a web link, but it is much better if you try to build your own first and go to these spreadsheets only if you get stuck.

Financial math is successful because it has the risk idea described earlier. It is also successful because it can draw on some powerful mathematical tools. These tools come from probability theory, from stochastic calculus, and from the calculus/differential equations sequence of courses. I have deliberately begun the book with a problem of calculating mortgage payments, not in the standard way, but with a difference equation setup, to get people thinking along the right lines from the very start. (It is also easier to reach students who already know all the formulas from a standard “annuities and bonds” kind of course if you at least give them some new tools.) I have designed the book to be, as much as possible, accessible to readers with a relatively modest level of mathematical background, perhaps, that of a typical engineering graduate. I have taught parts of this book at the 3rd year level to applied mathematicians and engineers, other parts to 4th year actuarial science students, and yet other parts to MBA (master of business administration) students. Some relatively advanced ways of using mathematics are present in the pages of this book, but I have always tried to motivate and explain as I go.

The overall structure of the book is as follows. After an introduction on risk, return, and decision making under uncertainty, and traditional discounted cash flow project analysis in Chapters 2 through 4, we begin to study mortgages, bonds, and annuities using a blend of Excel simulation (Chapters 5, 6, and 10) and difference equation or algebraic formalism (Chapters 5, 6, and 9).

We then devote four chapters to some of the nitty-gritty details of how interest rate markets work in Chapters 11 through 13, and an empirical look at how we might model bond prices in Chapter 14. After a short return to financial matters in Chapter 15, on mean variance portfolio optimization and a bit about the capital asset pricing model, we provide a short qualitative introduction to options in Chapter 16, and use them in our discussion of value at risk (VaR) in Chapter 17. An illustration of how VaR can be gamed by unscrupulous risk-hiding traders is included in this chapter.

In the long, but very important, Chapter 18, we build up binomial model tools for pricing options. These motivate the consideration of discrete random walks that we analyze and compare with their continuous analogs in Chapter 19. Chapter 20 introduces the rudiments of stochastic calculus in a very nonrigorous, taught like it was first year calculus, manner, followed by Chapter 21 that discusses how to simulate geometric Brownian motion, a very practical subject for the working quant.

Chapters 22 through 30 are devoted to a thorough discussion of options pricing, mostly in continuous time. Chapter 22 extends this binomial hedging argument to continuous time, deriving the famous Black Scholes partial differential equation. This equation is solved in Chapter 23, both to reproduce the Black Scholes formula for a call and to show that all European options pricing (at least in markets where all risk may be hedged) reduces to computing the present value of the expected value of cash flows with respect to a risk-neutral measure. A short Chapter 24 presents put call parity and uses it to price puts, after which some fairly advanced ways of approximating Black Scholes solutions are introduced in Chapter 25.

Chapter 26 develops a spreadsheet that simulates the continuous-time delta-hedging process, necessarily using discrete time steps. The impact of this spreadsheet is nothing short of fantastic—it shows at the click of a button how well the model really works, even if some of the assumptions fail. (We only hedge discretely, and we can also add transaction costs.)

Returning to more analytic tools, the impact of dividends on European options is studied in Chapter 27, while Chapters 28 and 29 present some results about the challenging early exercise or American option topic. Chapter 30 closes out our discussion of equity options, at least in complete markets, with a short discussion of options on multiple underlying assets, culminating in the pricing of a Margrabe exchange option.

The remaining two chapters close some loops. Chapter 31 presents stochastic models of the yield curve, with a focus on one factor affine term structure models in general and the Vasicek model in particular. This is important to rectify a possible misconception from readers of Chapters 11 through 13 that interest rates really only vary across a term and not over time!

The book ends with Chapter 32; a very simple chapter explaining some of the ideas of incomplete markets using simple discrete models.

I have taught a number of different courses from this book. An introductory course in financial mathematics for 3rd year undergra...

Table of contents

• Preface
• Author
• Chapter 1 - Introduction
• Chapter 2 - Intuition about Uncertainty and Risk
• Chapter 3 - The Classical Approach to Decision Making under Uncertainty
• Chapter 4 - Valuing Investment Opportunities: The Discounted Cash Flow Method
• Chapter 5 - Repaying Loans over Time
• Chapter 6 - Bond Pricing with Default: Using Simulations
• Chapter 7 - Bond Pricing with Default: Using Difference Equations
• Chapter 8 - Difference Equations for Life Annuities
• Chapter 9 - Tranching and Collateralized Debt Obligations
• Chapter 10 - Bond CDOs: More than Two Bonds, Correlation, and Simulation
• Chapter 11 - Fundamentals of Fixed Income Markets
• Chapter 12 - Yield Curves and Bond Risk Measures
• Chapter 13 - Forward Rates
• Chapter 14 - Modeling Stock Prices
• Chapter 15 - Mean Variance Portfolio Optimization
• Chapter 16 - A Qualitative Introduction to Options
• Chapter 17 - Value at Risk
• Chapter 18 - Pricing Options Using Binomial Trees
• Chapter 19 - Random Walks
• Chapter 20 - Basic Stochastic Calculus
• Chapter 21 - Simulating Geometric Brownian Motion
• Chapter 22 - Black Scholes PDE for Pricing Options in Continuous Time
• Chapter 23 - Solving the Black Scholes PDE
• Chapter 24 - Pricing Put Options Using Put Call Parity
• Chapter 25 - Some Approximate Values of the Black Scholes Call Formula
• Chapter 26 - Simulating Delta Hedging
• Chapter 27 - Black Scholes with Dividends
• Chapter 28 - American Options
• Chapter 29 - Pricing the Perpetual American Put and Call
• Chapter 30 - Options on Multiple Underlying Assets
• Chapter 31 - Interest Rate Models
• Chapter 32 - Incomplete Markets
• Appendix 1: Probability Theory Basics
• Appendix 2: Proof of DeMoivre–Laplace Theorem
• Appendix 3: Naming Variables in Excel
• Appendix 4: Building VBA Macros from Excel