Description
“In order to make any progress, it is necessary to think of approximate techniques, and above all, numerical algorithms … Once again, what became a major endeavor of mine, the computational solution of complex functional equations, was entered into quite diffidently. I had never been interested in numerical analysis up to that point. Like most mathematicians of my generation, I had been brought up to scorn this utilitarian activity. Numerical solution was considered the last resort of an incompetent mathematician. The opposite, of course, is true. Once working in this area, it is very quickly realized that far more ability and sophistication is required to obtain a numerical solution than to establish the usual existence and uniqueness theorems. It is far more difficult to obtain an effective algorithm than one that stops with a demonstration of validity. A final goal of any scientific theory must be the derivation of numbers.” This is an excerpt1 from Eye of the Hurricane [30] on page 185 by Richard Bellman. It seems appropriate to start the preface with this quote considering advances in quantitative finance would have been impossible without utilizing computational/numerical techniques and their impact on the evolution of the field in recent years.
In most applications and physical phenomena, we are in search of a solution that happens to be an approximation of the true solution. As a result, some sort of a computational method/technique or a numerical procedure is a must. In quantitative finance, aside from a few cases with an analytical or a semi-analytical solution, we typically wind up with an approximation as well. As today’s financial products have become more complex, quantitative analysts, financial engineers, and others in the financial industry now require robust techniques for numerical solutions. Computational finance has been a field that has been growing tremendously and intricacy of products and markets suggests there will be an even higher demand in the field.
This book is based on lecture notes I have used in my courses at Columbia University and my course at the Courant Institute of New York University. The selection of topics has been influenced by students and market requirements throughout my teaching over the years. Rama Cont, my colleague and friend, suggested to incorporate these notes into a textbook and referred me to the publisher.
My goal has been to write a textbook on computational methods in finance bringing together a full-spectrum of methods and schemes for pricing of derivatives contracts and related products, simulation, model calibration and parameter estimation with many practical examples. This book is intended for first/second year graduate students in the financial engineering or mathematics of finance field as well as practitioners, quants, researchers, technologists implementing models, and those who are interested in the field. My intention has been to keep the book self-contained and stand-alone.
Overall I have been pretty informal about theory.2 The aim has not been to get into detail on stochastic calculus or martingales pricing as they are not prerequisites for understanding the procedures in the book. Yet in some cases it has been unavoidable, and I try to give sufficient explanation so that the reader can proceed without any need to delve into the derivation or the theory behind it.