Erim - 05 omslag Pietersz 9/23/05 1:41 PM Pagina 1 Design: B&T Ontwerp en advies www.b-en-t.nl Print: Haveka www.haveka.nl Pricing models for Bermudan-style interest rate 71 RAOUL PIETERSZ derivatives Bermudan-style interest rate derivatives are an important class of RAOUL PIETERSZ options. Many banking and insurance products, such as mortgages, cancellable bonds, and life insurance products, contain Bermudan interest rate options associated with early redemption or cancella- tion of the contract. The abundance of these options makes evident Pricing models for that their proper valuation and risk measurement are important to banks and insurance companies. Risk measurement allows for off- Bermudan-style setting market risk by hedging with underlying liquidly traded assets rate derivatives Pricing models for Bermudan-style interest and options. Pricing models must be arbitrage-free, and consistent with (calibrated to) prices of actively traded underlying options. interest rate derivatives Model dynamics need be consistent with the observed dynamics of the term structure of interest rates, e.g., correlation between inte- rest rates. Moreover, valuation algorithms need be efficient: Finan- cial decisions based on derivatives pricing calculations often need to be made in seconds, rather than hours or days. In recent years, a successful class of models has appeared in the literature known as market models. This thesis extends the theory of market models, in the following ways: (i) it introduces a new, efficient, and more accurate approximate pricing technique, (ii) it presents two new and fast algorithms for correlation-calibration, (iii) it develops new models that enable efficient calibration for a whole new range of deriva- tives, such as fixed-maturity Bermudan swaptions, and (iv) it presents novel empirical comparisons of the performance of existing calibra- tion techniques and models, in terms of reduction of risk. ERIM The Erasmus Research Institute of Management (ERIM) is the Research School (Onderzoekschool) in the field of management of the Erasmus University Rotterdam. The founding participants of ERIM are RSM Erasmus University and the Erasmus School of Economics. ERIM was founded in 1999 and is officially accredited by the Royal Netherlands Academy of Arts and Sciences (KNAW). The research undertaken by ERIM is focussed on the management of the firm in its environment, its intra- and inter-firm relations, and its business processes in their interdependent connections. The objective of ERIM is to carry out first rate research in manage- ment, and to offer an advanced graduate program in Research in Management. Within ERIM, over two hundred senior researchers and Ph.D. candidates are active in the different research programs. From a variety of academic backgrounds and expertises, the ERIM commu- nity is united in striving for excellence and working at the forefront of creating new business knowledge. www.erim.eur.nl ISBN 90-5892-099-2 Pricing Models for Bermudan-Style Interest Rate Derivatives 1 2 Pricing Models for Bermudan-Style Interest Rate Derivatives Waarderingsmodellen voor Bermuda-stijl rente derivaten Proefschrift ter verkrijging van de graad van doctor aan de Erasmus Universiteit Rotterdam op gezag van de rector magnificus Prof.dr. S.W.J. Lamberts en volgens besluit van het College voor Promoties. De openbare verdediging zal plaatsvinden op donderdag 8 december 2005 om 16.00 uur door Raoul Pietersz geboren te Rotterdam 3 Promotiecommissie Promotoren: Prof.dr. A.A.J. Pelsser Prof.dr. A.C.F. Vorst Overige leden: Prof.dr. P.J.F. Groenen Prof.dr. F.C.J.M. de Jong Dr.ir. M.P.E. Martens Erasmus Research Institute of Management (ERIM) Erasmus University Rotterdam Internet: http://www.erim.eur.nl ERIM Ph.D. Series Research in Management 71 ISBN 90-5892-099-2 °c 2005, Raoul Pietersz All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the author. 4 Voor Beata, Karsten en Dani¨el 5 6 Acknowledgements First and foremost, I would like to thank my promotor Antoon Pelsser. His guidance throughout the Ph.D. period has been excellent. Chapters 2, 5 and 6 were written in cooperation with Antoon. The research benefitted greatly from his invaluable suggestions, and he truly is an inspirator. Thank you Antoon. Second, I thank my promotor and former employer Ton Vorst. He is the one who suggested me to start the Ph.D. track with Antoon Pelsser, and who part-time employed me at ABN AMRO Bank, Quantitative Risk Analytics (at the time “Market Risk – Modelling and Product Analysis”), for the period September 2001 till June 2004. For all this, I am very grateful. Third, many thanks go to the other members of the small committee; Patrick Groenen, Frank de Jong, and, Martin Martens. Also, I express my gratitude to the other members of the committee; Lane Hughston, Farshid Jamshidian, and, Thierry Post. Fourth, special thanks go to Marcel van Regenmortel, for teaching me many technical and exciting aspects of interest rate derivatives pricing, and for part-time employing me at Product Development Group, Quantitative Analytics, ABN AMRO Bank, from July 2004 onwards. Chapters 5 and 7 were written in cooperation with Marcel. Fifth, I am grateful to Patrick Groenen, for co-authoring Chapter 3. Sixth, I thank Igor Grubiˇsi´cfor co-authoring Chapter 4. I am much obliged to my colleagues at Erasmus University Rotterdam: Jaap Spronk for support through the Erasmus Center for Financial Research (ECFR); Martin Martens for his help while I was co-teaching one of his example classes and for suggesting me as a lecturer at the Rotterdam School of Management (RSM); Winfried Hallerbach for help with preparation of RSM lectures; Wilfred Mijnhardt for guiding me through the publication process; Tineke Kurtz, Tineke van der Vhee, Elli Hoek van Dijke, and Ella Boniuk, for efficiently aiding me in administrative matters; and, Mari¨elleSonnenberg, for being a pleasant room-mate. During the Ph.D. period I have been able to present my work at leading international conferences. I am very grateful for the financial support that made this possible, received from Erasmus Research Institute of Management (ERIM), from the Econometric Institute (EI), and from ECFR. 7 viii ACKNOWLEDGEMENTS A special thank you to past and present managers at ABN AMRO Bank: Ton Vorst, Dick Boswinkel, Nam Kyoo Boots, Marcel van Regenmortel, Bernt van Linder and Geert Ceuppens. Thank you to past and present colleagues at Product Development Group: Nicolas Carr´ein Amsterdam, and Thilo Roßberg and Russell Barker in London. Thanks to members of CAL: Nancy Appels, Reinier Bosman, Danny Wester, Frank Putman, Andr´e Roukema, Jelper Striet, and Willem van der Zwart. Thank you to past and present colleagues at ‘Product Analysis’: Steffen Lukas, Benjamin Schiessl´e,Martijn van der Voort, Lukas Phaf, Alice Gee, Rutger Pijls, Drona Kandhai, and Alex Zilber. A thank you to colleagues that have become friends: Dion Hautvast, Bram Warmenhoven, and Glyn Baker. I am also grateful to a number of anonymous referees and to Riccardo Rebonato and Mark Joshi who provided valuable comments and suggestions to earlier versions of the papers that form the basis for this thesis. Many thanks to Frank de Jong and Joanne Kennedy for providing much appreciated feedback to my research proposal. Finally, I would like to thank my parents for always being there for me. I thank my son Karsten for bringing so much joy to my life. I thank Beata for her love, support, and kindness. Raoul Pietersz February 7th 2005, Amsterdam 8 Contents Acknowledgements vii Notation xix Outline xxiii 1 Introduction 1 1.1 Arbitrage-free pricing .............................. 1 1.1.1 Use of models in practice ........................ 4 1.2 Interest rate markets and options ....................... 6 1.2.1 Linear products: Deposits, bonds, and swaps ............. 7 1.2.2 Interest rate options: Caps, floors, and swaptions .......... 8 1.3 Interest rate derivatives pricing models .................... 11 1.3.1 Short rate models ............................ 12 1.3.2 Market models ............................. 15 1.3.3 Markov-functional models ....................... 16 1.4 American option pricing with Monte Carlo simulation . 17 2 Risk-managing Bermudan swaptions in a LIBOR model 19 2.1 Introduction ................................... 19 2.2 Recalibration approach ............................. 21 2.3 Explanation ................................... 24 2.4 Swap vega and the swap market model .................... 27 2.5 Alternative method for calculating swap vega . 29 2.6 Numerical results ................................ 30 2.7 Comparison with the swap market model ................... 30 2.8 Conclusions ................................... 32 2.A Appendix: Negative vega two-stock Bermudan options . 34 9 x CONTENTS 3 Rank reduction of correlation matrices by majorization 39 3.1 Introduction ................................... 39 3.2 Literature review ................................ 43 3.2.1 Modified PCA .............................. 44 3.2.2 Majorization ............................... 44 3.2.3 Geometric programming ........................ 45 3.2.4 Alternating projections without normal correction . 45 3.2.5 Lagrange multipliers .......................... 46 3.2.6 Parametrization ............................. 46 3.2.7 Alternating projections with normal