Numerical Methods in Finance

Abstract

The book series features volumes of selected contributions from workshops and conferences in all areas of current research activity in mathematics. Besides an overall evaluation, at the hands of the publisher, of the interest, scientific quality, and timeliness of each proposal, every individual contribution is refereed to standards comparable to those of leading mathematics journals. This series thus proposes to the research community well-edited and authoritative reports on newest developments in the most interesting and promising areas of mathematical research today. PrefaceThe history of mathematical and numerical finance starts in 1900, with the seminal thesis of Louis Bachelier, Théorie de la Spéculation, which introduced Brownian motion in order to model stock price movements and evaluate options. Not only did this remarkable work modeled the randomness of stock prices in a mathematical framework germane to the popular Nobel Prize in Economics winning solution proposed by Fischer Black, Myron Scholes and Robert Merton in 1973, but it also laid the foundation for some key concepts of stochastic analysis.The celebrated Black-Scholes-Merton pricing paradigm which took the financial industry by storm, is not limited to the Samuelson's geometric Brownian motion model. However, it is based on a series of unrealistic assumptions, including Gaussian return fluctuations, constant volatility, risk-free interest rates, full liquidity, absence of frictions, no price impact from large or frequent trades, . . . , and the list could go on. Furthermore, the original pricing arguments do not directly apply to derivatives with non-European exercises such as American options, without another level of sophistication and approximation.The last two decades have seen a rapid development of increasingly realistic and sophisticated stochastic models and methods for pricing, hedging and risk management in rapidly growing markets, with more unfathomable financial products. Modern finance is becoming increasingly technical, requiring the use of complicated mathematical models, and involving numerical techniques based on theoretical results from subfields of mathematics ranging from stochastic analysis, dynamical system theory, nonlinear integro-differential equations, game theory, optimal control and dynamic programming, to statistical learning and information theory. Situated at the confluence of applied mathematics, computer sciences and economics, quantitative finance distinguishes itself through its wide range of themes, and its interaction with a broad spectrum of scientific domains.Any attempt at capturing all the fundamental developments which occurred in quantitative finance would simply be an impossible undertaking. With this volume, we aim at something less ambitious, more focused, and hopefully more useful, offering a collection of representative articles in the area of computational finance. Our objective is to bring financial professionals, economists and mathematicians v vi Preface i