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

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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 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 meetingdates...... 77 5.8 Overnight rates from a discounting curve built using 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 (t,u,v)...... 127 CPN,j − D 7.4 The additive spread FX (t,u,v) FX (t,u,v) for different levels of D risk-free rate FX (t,u,v) in the multiplicative stochastic spread framework ...... 131 viii List of Figures ix

8.1 Visual representation of the transition matrix of the EUR discounting for collateral in USD with Fed Fund rates ...... 173 8.2 Differences in swap rates for different collateral discounting curves. Swapwithtenorsbetween1and30years...... 176 8.3 Differences in swap rates for different collateral discounting curves . . . . 177 8.4 Differences in spreads for different collateral discounting curves...... 178 C.1 Instabilityoffinitedifferencecomputationofderivative...... 224 List of Tables

3.1 The maximum changes of three months rates over a three month period–maximumincreaseandmaximumdecrease...... 35 3.2 Sensitivity to the change of each market quote input (rescale to a onebasispointchange)...... 37 4.1 Sensitivity to market quotes for a 100 million notional using differentinterpolationschemes...... 48 5.1 Inverse Jacobian matrix for simplified two USD curves built with twounits...... 62 5.2 Inverse Jacobian matrices for simplified three AUD curves built withtwounits...... 64 5.3 Sensitivities of a EUR/JPY cross- in a framework where curves have been built from cross-currency instruments ...... 66 5.4 Sensitivities of a EUR/JPY cross-currency swap in a framework where curves have been built from cross-currency instruments ...... 66 5.5 Sensitivities of a EUR/JPY cross-currency swap in a framework where curves have been built from cross-currency instruments ...... 67 5.6 Sensitivities of a EUR/JPY cross-currency swap in a framework where curves have been built from cross-currency instruments ...... 68 5.7 First part of the table: two USD curves – discounting and three months – built with OIS for the first and FRA and IRS for the second. Second part of the table: two USD curves – discounting and Libor three months – built with Fed Funds futures, OIS and Fed Fund swaps for the first and STIR futures and IRS for the second . . . . . 80 5.8 Three curves in EUR: discounting, forward three months andforwardsixmonthsEuribor...... 81 5.9 Three curves in AUD: discounting, forward BBSW three months andforwardBBSWsixmonths...... 82 5.10 Three curves in JPY: discounting, forward three months and forwardsixmonths...... 83 5.11 Three curves in EUR: discounting, forward three months and forwardsixmonths...... 83 6.1 Paymentdelayconvexityadjustmentimpact...... 85 6.2 Error between exact arithmetic average and proposed approximation.Resultsforthreemonthcoupons...... 92

x List of Tables xi

6.3 Performance for exact and approximated formulas of the arithmeticaveragecoupons...... 92 6.4 Difference in future price and swap present value for different expiries,tenorandmoneyness...... 98 6.5 Swap/futuresnominaltotalsensitivityhedging...... 99 6.6 Swap/futures/OIS sensitivities and hedging efficiency for 2Y futures . . . 100 6.7 Swap/futures/OIS sensitivities and hedging efficiency for 30Y futures . . 100 7.1 Call/Put and Floor/Cap comparison ...... 137 8.1 Summarised representation of the dependency of the collateral curvetotheothercurvesoftheexample...... 173 B.1 Ibor-likeindexes...... 199 B.2 Overnightindexesforthemaincurrencies...... 201 B.3 FRA dates with differences between end of the accrual period and endoftheunderlyingfixingdepositperiod...... 203 B.4 InterestratefuturesonIbordetailsandcodes...... 204 B.5 Ratefuturesmonthcodes...... 205 B.6 Vanillaswapconventions...... 210 B.7 Overnightindexedswapconventions...... 211 B.8 Vanillaswapconventions...... 216 B.9 Conventionalcurrencystrength...... 216 C.1 The generic code for a function computing a single value ...... 226 C.2 The generic code for a function computing a single value and its algorithmic differentiation code ...... 227 C.3 Time to build curves with and without algorithmic differentiation . . . . 228 Preface

The first lines of this book were written in 2006. At the time the term multi-curve framework, which is used for the book’s title, had not been coined and my idea was only to write a couple of pages for a note. In the meantime, August 2007 changed the course of writing on interest rate curve modelling for ever.

Festina lente. Latin saying Personal translation: Haste slowly. Why did it take me so long?

Chacun sa methode...´ Moi, je travaille en dormant et la solution de tous les problemes,` je la trouve en revant.(Eachhisownmethod...Myself,Iworksleepingandthesolutionˆ to all problems, I find it dreaming). Drˆole de drame (1937) – Marcel Carn´e This means a lot of nights spent working to dream up all these pages. The starting point of the reflection was my quest to answer the question ‘What is the present value of an FRA()?’ in a convincing way. I could not find a satisfactory answer in the literature. The answers I could find were either ‘it is trivial’, or a description of a replication argument for which it was not acceptable to discuss the numerous hidden hypotheses. Discussing the hypothesis was not politically correct as a scientist nor as a business executive. For the former, it questioned a foundation of quantitative finance, and thus the developments built on those foundations. For the latter, it was not seen positively by board members and supervisors as it casted doubts on accounting figures – maybe rightly so – and had legal implications. The reason for my interest in discounting is that

Gentlemen prefer bonds. Andrew Mellon – 1855–1937 Ironically, the first article to come out of those reflections, titled Irony in deriva- tives discounting, was published just one month before the now famous August 2007. The publication was not a prediction of what would happen in the deriva- tive market just after and should not be seen as a premonition. Neither should it be seen as a cause of the crisis. It was nevertheless an indication of inconsistencies in xii Preface xiii the practice of pricing that were not answered by a coherent theoretical framework. The book, which started, unbeknown to its author, seven years ago, is intended to be a description of the current status of the subject, which is now called the multi- curve framework. It borrows from the developments of numerous practitioners and academics working on the subject. The length of the bibliography, with most of the references dating from 2009 and after, is a witness of the activity on the subject over the last years. The book profited from fruitful conversations with and comments on drafts by Marco Bianchetti, Damiano Brigo, Antonio Castagna, Stethane´ Crepey,´ Christophe Laforge, Andrea Macrina, Fabio Mercurio, Massimo Morini, and Chyng Wen Tee. This book would not have been possible without the support of OpenGamma. The company encouraged me to complete the book and gave me the time to do it. The quality of the book was improved by discussion on its content with OpenGamma colleagues: Casey Clement, Yukinori Iwashita, Elaine McLeod, Soila Patajoki, Joan Puig, Richard White, and Arroub Zine-Eddine. It is also my pleasure in this book to follow in the footsteps of giants in the art of relevant and irrelevant quotes. I have done my best in that regard, but I can not claim to match them. Beyond the written literature, another source of inspiration for the development of this book was the participation in numerous practitioner and academic con- ferences and the perspicacious questions and remarks by their attendants. While preparing the final version of this book, I also had the pleasure of lecturing on the subject in the mathematical finance masters program at University College Lon- don. The book profited from the course preparation and from the questions, naive in appearance but fundamentally deep, asked by the participants. In the book I use the term ‘we’ with the general and vague meaning of the author, the quantitative finance research community and the readers. The terms ‘I’, ‘me’ or ‘my’ are used with the precise meaning of the author personally with his opinions and bias. The term ‘I’ should be used as a warning sign that the sentence contains opinions and maybe not only facts. You have been warned! Hopefully the reader will find as much interest in the subject as I have over the last seven years. Hopefully he will also be intrigued, surprised, amused and maybe amazed at some of the subject facets. Oh, by the way, my quest is still on! I’m still asking myself ‘What is the present value of an FRA?’, even if the question has changed to ‘What is the collateral quote ofanFRA?’.A questionthatisthesourceofinsomnia ...andmaybethestarting point of another book in a couple of years.

Enjoy!