Pricing and Hedging of Derivative Securities

Pricing and Hedging of Derivative Securities

Pricing and Hedging of Derivative Securities

Pricing and Hedging of Derivative Securities

Synopsis

The theory of pricing and hedging of derivative securities is mathematically sophisticated. This book is an introduction to the use of advanced probability theory in financial economics, presenting the necessary mathematics in a precise and rigorous manner. Professor Nielsen concentrates on three main areas: the theory of continuous-time stochastic processes, a notorious barrier to the understanding of probability theory in finance; the general theory of trading, pricing, and hedging in continuous time, using the martingale approach; and a detailed look at the BlackScholes and the Gaussian one-factor models of the term structure of interest rates. His book enables the reader to read the journal literature with confidence, apply the methods to new problems, or to do original research in the field.

Excerpt

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, Itô processes and Itô'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 . . .

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