Wipplinger, Evert

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Financ Mark Portf Manag (2011) 25: 107–108DOI 10.1007/s11408-010-0148-4B O O K R E V I E WYuri Kabanov and Mher Safarin: Marketswith transaction costsSpringer, 2009Evert WipplingerPublished online: 21 January 2011© Swiss Society for Financial Market Research 2011Classic financial models, such as the Black–Scholes one, rely on markets withoutfrictions. It is well known that in complete markets without frictions, options can bepriced by replication. However, trading in real markets can involve substantial costs.Markets with Transaction Costs provides the mathematical background for dealingwith proportional transaction costs. It is the first book to compile the results from asubstantial body of literature coherently in a single volume, and thus may be valuableto researchers looking for a theoretical foundation for their models. The book’s sub-title, “Mathematical Theory,” is important for it is a mathematically rigorous book.Therefore, the financial or economic generalist may gain little from the book apartfrom an in-depth discussion of Leland’s (1985) seminal model. The book covers threemajor topics: the approximate replication of contingent claims, no-arbitrage pricingtheory with transaction costs, and an optimal control problem.In the first chapter, “Approximate Hedging,” Kabanov and Safarin review the Le-land (1985) model and provide additional proofs. They discuss the problem of ap-proximating the terminal payoff of options with a hedging strategy when transactioncosts depend on the number of hedging portfolio revisions. Because of the practi-cal relevance of the Leland (1985) model, this chapter may be of interest to a moregeneral audience.The second chapter is an exposition of discrete-time arbitrage theory in marketswithout frictions, which is extended to markets with transaction costs in the thirdchapter. Inspired by Schachermayer (2004), the authors investigate the fact that astraightforward extension of the fundamental theorem of asset pricing does not holdin general for infinite-state markets with frictions. They proceed to discuss alternativeE. Wipplinger ()Swiss Institute of Banking and Finance, University of St. Gallen, Rosenbergstrasse 52, 9000St. Gallen, Switzerlande-mail: evert.wipplinger@unisg.ch108 E. Wipplingerformulations and super-replicating portfolios for hedging American and Europeanoptions.The fourth chapter deals with Merton’s (1971) seminal model of choice betweenconsumption and investment when transaction costs are considered (Davis and Nor-man 1990). In this setting, the model becomes much more complex and the authorscover the theory of viscosity solutions first proposed by Shreve and Soner (1994).Some of the original articles cited in this volume are 20 years old, but the authorshave incorporated up-to-date results. Except for the last part of the book, most of thechapters can be read and understood as stand-alone pieces. The notation, exposition,and style of the book are typical of mathematical texts but, fortunately, intermediatesteps are usually provided. The authors also provide additional comments and re-marks, making the results much more accessible. The “Bibliographical Comments,”where the authors relate the sections of the book to the original literature, are espe-cially helpful. Personally, I would have preferred these to be incorporated into themain text and not as a separate section at the end of the book because, ultimately, itis this section that allows the reader to access the subject beyond the results selectedby the authors. It is also evident that Kabanov and Safarin do some editorializing inthe less mathematical sections when they present their own view of the field and howit has evolved. These parts are interesting, but the bread and butter of the text are themathematical proofs. Hence, as is true of any text dealing with complex problems inmathematical finance, the reader needs to be acquainted with its foundations, such asmartingale theory. Of some help in this regard is an appendix containing auxiliary re-sults. By compiling the results and presenting them within a greater scope, Kabanovand Safarin have succeeded in creating the first book to summarize some of the majorcontributions in the mathematical finance literature on proportional transaction costs.They tackle a variety of problems but never lose focus. Because of the challenging na-ture of its topic, the book is obviously not aimed at an audience lacking a backgroundin mathematical finance, but to those who have such knowledge, I recommend it as astepping stone to the research field of transaction costs.ReferencesDavis, M.H.A., Norman, A.R.: Portfolio selection with transaction costs. Math. Oper. Res. 15, 676–713(1990)Leland, H.E.: Option pricing and replication with transactions costs. J. Finance 40, 1283–1301 (1985)Merton, R.C.: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3,373–413 (1971)Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finitediscrete time. Math. Finance 14, 19–48 (2004)Shreve, S., Soner, H.: Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4,609–693 (1994)

Yuri Kabanov and Mher Safarin: Markets withtransaction costs: Springer, 2009

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Yuri Kabanov and Mher Safarin: Markets withtransaction costs: Springer, 2009

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