VARIANCE OPTIMAL CAP PRICING MODELS J.P. LAURENT 1 and O. SCAILLET 2 T his version : December 1999 First version : December 1997 A bst r act We propose new closed-form pricing formulas for int erest rat e opt ions which guarant ee per- fect compatibility with volatility smiles. These cap pricing formulas are computed under variance optimal measures in the framework of the market model or the Gaussian model and achieve an exact calibrat ion of observed market prices. T hey are present ed in a general set t ing allowing t o st udy model and num¶eraire choice e®ect s on t he comput ed prices. We show that price sensitivities of pricing formulas with respect to observed option prices are equal to mean-variance hedging portfolio holdings. We also show that the theoretical cap prices are equal to the cost of the hedging portfolio, thus closing the gap between pricing and hedging. A numerical example and an empirical application on market data are given t o illust rat e t he pract ical use of t he calibrat ion procedure. keywords : discount bond option, cap pricing formula, volatility smile, variance optimal mea- sure, implied pricing model. JEL : G12, G13, C40. 1 ISFA Actuarial School, University of Lyon and CREST. E-mail: laurent.jeanpaul@online.fr 2 Institut d'Administration et de Gestion and D¶epartement des Sciences Economiques, Universit¶e Catholique de Louvain. E-mail: scaillet@ires.ucl.ac.be An earlier version of this paper has circulated under the title \ Cross price forecasts with interest rate options" , and is an abridged version of DP IRES 9902 and DP CREST 9907. We have bene¯ ted of numerous comments and discussions from T. De Pauw, B. Dumas, A. Frachot, C. Gouri¶eroux, F. Jouneau, F. Magnien, F. Quittard-Pinon, V. Metz, P. Poncet, O. Renault; helpful comments from R. Anderson, M. Avellaneda, N. El Karoui, G. Laroque, D. Leisen, D. Madan, M. Musiela, V. Patilea, H. Pham, H. Polemarchakis, J.L. Prigent, R. Roll, T. de Roquemaurel, N. Touzi and participants of the AFFI meeting (Aix-en-Provence), EFA meeting (Vienna) and of seminars at Gerzensee, CORE, University of Cergy-Pontoise, Institut Henri Poincar¶e, Caisse Aut onome de Re¯ nancement and Paribas are also acknowledged. Of course, t he usual disclaimers apply. We thank Paribas for kindly providing the market data used in the empirical section. T he second aut hor grat efully acknowledges t he support of t he grant \ Act ions de recherche concert ¶ees" 93/ 98-162 of the Ministry of Scienti¯ c Research (Belgian French Speaking Community). 1