Interest Rate Modelling in the Multi-curve Framework Interest Rate Modelling in the Multi-curve Framework Foundations, Evolution and Implementation Marc Henrard OpenGamma, London, UK © Marc Henrard 2014 Softcover reprint of the hardcover 1st edition 2014 978-1-137-37465-3 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his rights to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2014 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010. Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries ISBN 978-1-349-47704-3 ISBN 978-1-137-37466-0 (eBook) DOI 10.1057/9781137374660 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. 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Contents List of Figures .............................................viii List of Tables .............................................. x Preface .................................................. xii 1Introduction........................................... 1 1.1 9 August 2007 . 1 1.2 Foundations,evolution,andimplementation................. 2 1.3 Standardtextbookframework........................... 4 1.4 Theprecursors...................................... 5 1.5 Earlymulti-curveframeworkliterature..................... 6 1.6 Collateral and funding . 10 1.7 Howtoreadthisbook................................ 10 1.8 Whatisnotinthisbook............................... 11 2 The Multi-curve Framework Foundations ...................... 12 2.1 One-curveworld.................................... 12 2.2 Discountingcurve................................... 15 2.3 Forwardcurves..................................... 16 2.4 Interestrateswap.................................... 19 2.5 Forwardrateagreement............................... 22 2.6 STIRFutures....................................... 24 2.7 Overnightindexedswaps.............................. 26 2.8 Forexandcross-currencyswaps.......................... 29 3 Variation on a Theme ..................................... 31 3.1 Forwardcurvesthroughpseudo-discountfactors.............. 32 3.2 Directforwardcurves................................. 34 3.3 Futuresmulti-curveframework.......................... 37 4Interpolation.......................................... 41 4.1 Whattointerpolate?.................................. 43 4.2 Impactofinterpolation................................ 44 5 Curve Calibration ....................................... 50 5.1 Introduction....................................... 50 5.2 Whattocalibrate?................................... 54 5.3 Calibration........................................ 55 5.4 Discountingcurve................................... 57 5.5 Indexfixing........................................ 58 v vi Contents 5.6 Root-findingandJacobian............................. 60 5.7 Instrumentsandcurveentanglement...................... 61 5.8 Currencydependency................................. 67 5.9 Spreadcurves...................................... 69 5.10Functionalcurves.................................... 74 5.11Interpolationonexternallyprovideddates................... 76 5.12Combiningseveraleffects.............................. 77 5.13Examplesofstandardcurves............................ 79 6MoreInstruments....................................... 84 6.1 Overnightindexedswaps.............................. 84 6.2 Iborcouponswithdatemismatches....................... 86 6.3 Compoundedcoupons................................ 88 6.4 FederalFundsswaps.................................. 90 6.5 FederalFundsfutures................................. 95 6.6 Deliverableswapsfutures.............................. 96 6.7 Portfoliohedging....................................101 7 Options and Spread Modelling ..............................105 7.1 Shortratemodels....................................106 7.2 Spreadsdescription..................................107 7.3 Constantmultiplicativespread...........................110 7.4 Ibor forward rate modelling . 113 7.5 Swap rate modelling ..................................115 7.6 ParsimoniousHJMmulti-curveframework..................118 7.7 Additivestochasticspread..............................126 7.8 Multiplicativestochasticspread..........................129 7.9 Libormarketmodelonmultiplecurves.....................142 8 Collateral and Funding ....................................145 8.1 Introduction.......................................145 8.2 Collateral: rate, asset or both . 147 8.3 Multi-curve framework with collateral . 165 8.4 Modelling with collateral: collateral HJM model . 180 Appendix A. Gaussian HJM ...................................190 A.1 Model...........................................190 A.2 Genericresults......................................191 A.3 Specialcases.......................................194 A.4 MonteCarlo(Hull-White).............................196 A.5 Miscellaneous . 197 Appendix B. Conventions ....................................198 B.1 Iborindexes.......................................198 B.2 Overnightindexes...................................200 Contents vii B.3 Forwardrateagreement...............................202 B.4 STIRfutures.......................................203 B.5 Coupons..........................................206 B.6 Legs.............................................208 B.7 Swaps............................................209 B.8 Interestrateswaps:fixedforIbor.........................209 B.9 Overnightindexedswaps..............................210 B.10Basisswap:IborforIbor...............................212 B.11Basisswap:Iborforovernight...........................213 B.12FederalFundsswaps..................................213 B.13Presentvaluequoteddeliverableswapfutures................214 B.14Forexswaps........................................215 B.15Cross-currencyswaps:IborforIbor.......................216 Appendix C. Implementation in a library .........................218 C.1 Curveuniverseobject.................................218 C.2 Curvecalibration....................................220 C.3 Algorithmic differentiation . 223 Bibliography ..............................................232 Index ...................................................238 List of Figures 1.1 Relationship between swap spreads and multi-curves related literature . 8 3.1 Forward Ibor three months rates computed using pseudo-discount factors in black and direct forward rate curve in dotted line . 34 3.2 Forward Ibor three months rates computed using pseudo-discount factors (in black) and direct forward rate curve (in grey) with naturalcubicsplineinterpolation........................... 36 4.1 Impactofinterpolation.................................. 45 4.2 Forward rates for several interpolation schemes on pseudo-discountfactors................................. 46 5.1 Graphical display of the inverse Jacobian for two USD curves built usingFederalFundsswaps................................ 63 5.2 Spread-over-existingwithsparsedata........................ 70 5.3 Spread-over-existingasextrapolationscheme................... 71 5.4 Turn-of-the-yearandturn-of-the-quarter..................... 73 5.5 Curve using two different interpolation mechanisms: log-linear on discount factor up to three months (step function on instantaneous rates)anddouble-quadraticonratesabove.................... 74 5.6 Functional curve described by a Nelson-Siegel function . 75 5.7 Overnight rates from a discounting curve built using central bank meetingdates......................................... 77 5.8 Overnight rates from a discounting curve built using central bank meetingdatesandTOYeffect.............................. 78 5.9 Overnight rates and forward six month rates from the forward Euriborsixmonthcurve................................. 78 6.1 Timing adjustment for Ibor rate payments in case of dates mismatches . 88 7.1 Representation of the rates relative dynamic and the implied spread dynamicintheparsimoniousHJMapproach...................120 7.2 Spread levels for different values of the underlying standard normal random variable on the horizontal axis and different levels of displacementontheverticalaxis............................121 CPN,j − D 7.3 The additive spread FX (t,u,v) FX (t,u,v) for different levels of D risk-free rate FX