Arbitrage Pricing of Contingent Claims

This book is intended as a contribution to the theory of contingent claim valuation based on arbitrage considerations. It is concerned with preference-free valuations of contingent claims (such as options written on a stock) in frictionless multiperiod securities markets that do not permit arbitrage profits. Besides the question of pricing it considers the possibility of hedging in securities markets. The research reported in this book was carried out at the Institut fUr Gesellschafts- und Wirtschaftswissenschaften, University of Bonn. While working in this field and preparing this monograph I received helpful comments and encouragement from many people, and I would like to thank all of them. Special thanks are due to Prof. Dr. Dieter Sonder­ mann. He first stimulated my interest in the theory of contingent claim valuation and commented on my work at various stages. Furthermore, I would like to thank Profs. Martin Hellwig, Peter Schonfeld and Klaus SchUrger, all University of Bonn, for helpful comments on earlier ver­ sions of this monograph. Parts of this monograph were presented at va­ rious meetings including the European Meeting of the Econometric Socie­ ty, Pisa 1983, the European Meeting of the Econometric Society, Madrid 1984, and the third Conference "Geld, Banken und Versicherungen", Karls­ ruhe 1984. I greatly appreciate comments of Profs. Stephen Ross, Yale University, and Michael Brennan, UCLA. I take this opportunity to ex­ press my indebtness to my colleague Dr. Shinichiro Nakamura for his con­ stant encouragement during the laborious process of writing this mono­ graph.

Inhaltsangabe1 Introduction.- 2 The Valuation of Contingent Claims: A Survey.- 3 Existence of Consistent Price Systems.- 3.1 The basic model.- 3.2 Arbitrage and equivalent martingale measures.- 3.3 Examples.- 3.3.1 Geometric Brownian motion.- 3.3.2 A process with a random variance.- 3.3.3 Geometric Poisson process.- 4 The Continuous-time Trading Model.- 4.1 Continuous-time self-financing trading strategies.- 4.2 A characterization of P*-attainable contingent claims.- 4.3 Classes of P*-attainable contingent claims for specific security price processes.- 4.3.1 Geometric Brownian motion.- 4.3.2 A process with a random variance.- 4.3.3 Geometric Poisson process.- 4.4 The relationship between P*-attainable contingent claims and solutions of associated differencedifferential equations.- 4.5 Complete securities market models.- 4.6 Counterexamples.- 5 Extensions of the BLACK/SCHOLES Model.- 5.1 Determination of an equivalent martingale measure.- 5.2 The relationship between the original and the discounted model.- 5.3 Completeness and the determination of self-financing trading strategies in the case of a European call option.- 5.4 Completeness and the relaxation of assumptions.- 5.5 Incompleteness caused by variations of assumptions.- 6 From Preference-free to Preference-dependent Valuations of Contingent Claims: the Hedge Approach in Incomplete Models.- 7 Conclusion.- References.- A Appendix.- A 1 Notation.- A 2 Mathematical Tools.- A 2.1 Miscellany.- A 2.2 Measure theory.- A 2.3 Stochastic calculus.