Pricing and Hedging of Derivative Securities. By Lars Tyge Nielsen. Textbook in continuous-time finance theory. Oxford University Press, 1999.
This book is an introduction to pricing and hedging of derivative securities for academics and practitioners. It has grown out of my doctoral course in continuous-time finance theory at INSEAD. It can be used as a text in graduate programs in finance, mathematical finance, economics, mathematical economics, financial engineering, or pure or applied mathematics. It can also be used as a reference, or for self-study. I have used various versions of the manuscript in lecture series at Tilburg University and at New York University’s Stern School of Business, as well as in public and private executive courses in the mathematics of derivative securities.
The theory of pricing and hedging of derivative securities is mathematically sophisticated and requires the use of advanced probability theory. My aim has been to make the mathematics available in a precise and rigorous manner, even though the focus of the book is on financial economics. Exposure to the mathematics is necessary in order to give the reader the background to read the journal literature with confidence, apply the methods to new problems, or to do original research in the field.
An area of probability theory which is particularly important is the
theory of continuous-time stochastic processes. It is an essential prerequisite for continuous-time finance, it is not easily accessible, and it has for a long time formed a barrier of entry into the field. One of my purposes in writing this book has been to help break down that barrier and make it possible for the reader actually to learn this material.
The book begins with three chapters on stochastic processes, stochastic integration with respect to Wiener processes, Ito processes and Ito’s Lemma, Girsanov’s Theorem, the Martingale Representation Theorem, and Gaussian processes. I have put quite a lot of effort into deciding what to include and what not to include in these chapters. The guiding principle has been that all the stochastic process theory needed later on in the book should be explained here, while on the other hand very little material should be covered that is not useful in finance.
The theory of stochastic processes uses measure and integration theory, the relevant parts of which are covered in two appendices. Depending on his or her interests and background, the reader may begin by reading those appendices or may alternatively just use them as a reference. For most people, I would recommend the latter option. Go easy on the measure theory to begin with unless you already know it well. After having read the main body of the book, you may be motivated to look deeper into measure and integration, and you should in fact do so if you are seriously interested in continuous-time finance. In that case, I hope you will find the appendices to be an efficient introduction to the subject.
Building on the foundations laid in the three chapters on stochastic
processes, the following two chapters cover the general theory of trading, pricing, and hedging in continuous time, using the martingale approach. The exposition emphasizes the concepts of prices of risk, state price processes, and risk-adjusted probabilities.
The two last chapters are devoted to applications. Rather than cover
a large number of applications in survey form, I have chosen to include
only a small number but develop them in detail. The applications that I
have selected are the Black-Scholes model and the Gaussian one-factor models of the term structure of interest rates. I believe it is more important for the reader to see a few topics developed in depth than to get a broad but less detailed overview. In any case, most instructors will want to choose their own applications from the journal literature.
In order to keep the length of the book finite, it has been necessary
not only to limit the number of applications but also to omit some topics
from the fundamental theory. Most notably, I have not covered stochastic differential equations and the associated partial differential equations beyond the heat equation and the Black-Scholes PDE. Stochastic differential equations are an extremely useful tool in financial economics, but their theory is more subtle than most finance books would lead you to suspect, and it requires a fairly lengthy mathematical presentation. There was no way these equations could be included and be given a satisfactory treatment in this book.
My vision is that the reader will appreciate learning some mathematics
without being overburdened with it, will get a certain sense of satisfaction from the general theory and the applications, and in the end will be hungry for more.
I would like to thank a number of people who have influenced this book.
I initially learned continuous-time finance from Chi-fu Huang after
having tried but failed to learn it from other sources. His lecture notes
have had a significant influence on the way this book is structured.
The students who took my course at INSEAD gave me feedback and comments which have been incorporated in successive revisions of the manuscript. They include Joao Amaro de Matos (New University of Lisbon), Benoit Leleux (Babson College), Saugata Banerjee (Koc University), Fatma Lajeri Koc University), Jesus Saa-Requejo, Finn Erling Bendixen, P. Raghavendra Rau (Purdue University), Pedro Santa-Clara Gomes (University of California, Los Angeles), Salvatore Cantale (Tulane University), Yrjoe Koskinen (Stockholm School of Economics), Aris Stouraitis, Georges Huebner (University of Liege and Limburg University), Stephen Sapp, Arturo Bris (Yale University), Aine NiShuilleabhain, Dmitry Lukin (University of California, Irvine), Jos van Bommel (Babson College), Swookeun Jeon, Dima Leshchinskii, Neil Brisley, Arzu Ozoguz, Merih Sevilir, and Andrei Simonov.
Georges Constantinides read a version of the manuscript and gave me
written comments. So did David Nachman and Steven Raymar, who used the manuscript in their courses, and Alexander Reisz, who followed my lecture series at New York University. Maria Vassalou used the manuscript in a doctoral course at Columbia University and gave me detailed feedback on the substance as well as on how to restructure some of the material to make it more teachable.
Finally, a practical remark on the organization of the material. Certain
sections, subsections, theorems, propositions, proofs etc. are marked with an asterisk “(*)”. This means that they can be skipped on a first reading. If a theorem or proposition or the like is marked by (*), then the proof can be skipped too.