The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

CHAPTER 1

Introduction to the Techniques of Derivative Modeling

1.1. INTRODUCTION

“How do I model derivatives?”

For the desk quantitative analyst (“desk-quant”), the quantitative-programmer, the quantitative-trader or risk analyst—at a firm actively trading, risk managing, or even auditing books of derivatives—who needs to know the basic answer to this question, it is contained in a toolkit of well-established mathematical models and techniques. These professionals might be prepared to forgo mathematical rigor and the security against subtle errors that a thorough foundation could perhaps provide. They might be prepared to take a few shortcuts to get at these techniques relying on another team member or risk-group reviewer who has the thorough mathematical background to provide a safety net against subtle errors and misunderstandings. But the desk quant, the programmer, and the trader do not have to forgo everything. This book is aimed at such readers who want a good quick grasp of these techniques.

This first chapter reviews in coarse outline the two most typical techniques for theoretically modeling and pricing derivatives and takes the first motivational step of mentioning the model of the process for stock that underpins the first technique and the simplest implementations of the second. The remainder of the book will take the reader through the mathematical tools that underpin these two and many other techniques.

1.2. MODELS

1.2.1. What Is a Derivative?

The archetypal example ...