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A Benchmark Approach to Quantitative Finance, by Eckhard Platen and David Heath

This book presents a comprehensive and unifying treatment of Quantitative Finance based on the so-called benchmark approach that has been developed over recent years by E. Platen and co-workers. This approach is characterized by the central role assigned to the growth optimal portfolio (GOP): for portfolio optimization it serves as a benchmark, while for pricing and hedging it can be taken as numeraire, whereby the real world (physical) measure P becomes the pricing measure, namely GOP denominated prices are P martingales. This latter aspect leads to various advantages, for example when it comes to using estimation procedures (filtering and/or parameter estimation). The major advantage is that it allows pricing and hedging to be performed also in situations when a classical local martingale measure does not exist. In this sense the benchmark approach is also in line with current investigations that take into account the fact that in various financial applications a local martingale may not be a true martingale and, in particular, the density process for a candidate (local) martingale measure may be a strict supermartingale. An important result in this context concerns furthermore the fact that a market may be complete without necessarily requiring the existence of a martingale measure. Typical examples to illustrate this situation in a diffusion-typemodel involve Bessel processes that are described in a systematic way in the book. Finally, the benchmark approach allows one also to establish a link between current mathematical finance methodologies and the more classical CAPM-based ones that are still in wide use, especially among practitioners. We describe now in more detail the contents of the book that consists of fifteen chapters. The first seven chapters, as well as part of chapters eight, nine and fifteen cover background material that for many aspects goes beyond what can be found in standard textbooks, in particular in view of the specific topic of the book. Chapters eight and nine deal with derivative pricing. While chapter eight presents an overview of standard derivative pricing, in chapter nine various alternative results for pricing according to the benchmark approach are discussed, in particular ‘fair pricing/real world pricing’ where GOP-denominated prices are martingales under the real world probability measure. Examples are given where the candidate density for a martingale measure is a strict local martingale, but the GOP still exists, demonstrating fair pricing to be more general than classical risk neutral pricing. Chapter ten is mainly devoted to the GOP and it is shown that it can be obtained without reference to a specific model, in particular, it can be approximated by well-diversified portfolios. Chapter eleven is dedicated to portfolio optimization thereby extending a range of classical results in this area. A central concept here is that of a ‘locally optimal portfolio’ and a two-fund type result is shown, namely that a locally optimal portfolio can 2011, Springer