A First Course in Stochastic Calculus

Book Preface

As a former trader and quant, and now a teacher of quants, I can say that the book I have been waiting for has finally arrived: an introduction to stochastic calculus with financial applications in mind, shorn of unnecessary details. The respect that Louis-Pierre Arguin (L-P to his friends) clearly holds for the reader of this book is evident throughout; while the writing is conversational, it is also concise. This book is “action-packed” — every page has something of real value.
It is self-evident that stochastic calculus is a prerequisite to understand the current mathematical finance literature, in particular the literature on option pricing. And these days, not only quants but also traders, programmers, and even some salespeople are familiar with this literature. This book provides the best possible introduction to stochastic calculus, as suitable for those encountering the subject for the first time as it is for people like me who may wish to refresh their understanding of a particular topic. Also, it is worth emphasizing that while this book is introductory, everything is treated with rigor and written in the language of the contemporary literature, preparing readers well for the challenges of further reading.
Mathematicians and physicists may argue over the value of intuition, but in finan-cial practice, intuition is absolutely key. For example, finance practitioners often need to be able to quickly judge how to alter the terms of a transaction, typically to reduce costs while still meeting the needs of the client. So how does one acquire good intu-ition? Experience obviously is one part of the answer. But having a good teacher is more efficient and less expensive: L-P is renowned as a great teacher, and this book fully lives up to his reputation. His intuitive explanations of stochastic calculus are backed up and reinforced by the many numerical projects and exercises he provides.
I clearly recall my first introduction to the central limit theorem from an under-graduate experiment with a radioactive source in a physics lab at the University of Glasgow. Still today, in my mind’s eye, I can see the histogram of counts changing from Poisson to Gaussian as I bring the source closer to the detector. There is no sub-stitute for this kind of training; I have the feeling that L-P betrays his background as a physicist with his emphasis on simulation in the numerical projects. L-P builds the kind of intuition that can be felt in your bones.
Another prerequisite in the practice of quantitative finance is programming skill, and the language of choice is Python. All of the numerical projects use Python and many Python hints are provided. I suspect that many readers of this book may be more comfortable writing Python than reading or writing mathematical proofs — for them, this book will be a godsend.
L-P is to be congratulated for writing this wonderful book. I can see myself refer-ring to it for many years to come.
Jim Gatheral
Presidential Professor of Mathematics
Baruch College, City University of New York