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Multi Factor Stochastic Volatility for Interest Rates Modeling Ernesto Palidda

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Multi Factor Stochastic Volatility for Interest Rates Modeling Ernesto Palidda Multi factor stochastic volatility for interest rates modeling Ernesto Palidda To cite this version: Ernesto Palidda. Multi factor stochastic volatility for interest rates modeling. General Mathematics [math.GM]. Université Paris-Est, 2015. English. NNT : 2015PESC1027. tel-01272433 HAL Id: tel-01272433 https://pastel.archives-ouvertes.fr/tel-01272433 Submitted on 10 Feb 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thse pr´esent´eepour obtenir le grade de Docteur de l’Universit´eParis-Est Sp´ecialit´e:Math´ematiques par Ernesto Palidda Ecole Doctorale : Mathematiques´ et Sciences et Technologies de l’Information et de la Communication Mod´elisation du smile de volatilit´epour les produits d´eriv´esde taux d’int´erˆet. Th`esesoutenue le 29 mai 2015 devant le jury compos´ede: Damiano Brigo Rapporteur Martino Grasselli Rapporteur Nicole El Karoui Examinateur Christophe Michel Examinateur Bernard Lapeyre Directeur de th`ese Aur´elien Alfonsi Co-Directeur de th`ese 2 Contents I Models for managing interest rates derivatives 29 1 The interest rates market 31 1.1 Reference rates and products . 31 1.1.1 Reference rates . 31 1.1.2 Linearproducts.............................. 32 1.1.3 Nonlinearityepisode1: optionality . 33 1.2 Recent developments . 35 1.2.1 The IBOR-OIS and other basis . 35 1.2.2 The impact of collateralization in the derivative pricing . ..... 36 1.2.3 Whichdiscountingcurve? . 37 1.2.4 Priceadjustments............................. 39 1.3 Valuation principles and quotation conventions: pre and post 2008 crisis . 40 1.3.1 Non-linearity episode 2: hidden optionality . .. 42 1.3.2 Viewing swaption as basket options . 43 1.3.3 Different settlements . 45 1.4 Peculiarities.................................... 46 1.4.1 Market representation of option prices . 46 1.4.2 Slidingunderlyings ............................ 47 1.5 Some empirical facts about the interest rates market . 47 1.5.1 PCAoftheswaptionvolatilitycube. 48 2 Term structure models 53 2.1 Arbitrage free framework . 55 2.1.1 Arbitrage-free constraints . 55 2.1.2 Change of probabilities and forward neutral measures . .. 56 2.2 Modeling with state variables . 57 2.3 Affinetermstructuremodels........................... 59 2.4 Examples of affine term structure models . 60 2.4.1 The multi-dimensional linear Gaussian model . 60 2.4.2 ThequadraticGaussianmodel. 65 2.5 Markovfunctionalmodels ............................ 70 2.5.1 Calibrationprocedure .......................... 71 2.5.2 Yield curve dynamics and implied volatility . 72 3 2.6 Volatilityandhedginginfactorialmodels. ..... 72 2.6.1 Thenatureofvolatility. 73 2.6.2 The implications of local volatility: an introduction to hedging and modelrisk................................. 73 2.7 Hedging Interest rates derivatives: theory . 76 2.7.1 Hedging in a general AOA framework . 77 2.7.2 Hedginginfactorialmodels . 78 2.7.3 Hedging with market securities . 78 2.7.4 Being consistent with market practice . 80 2.7.5 Continuousframework .......................... 82 II An affine stochastic variance-covariance model 85 3 Modeling dependence through affine process: theory and Monte Carlo framework 87 3.1 A stochastic variance-covariance Ornstein-Uhlenbeck process . 88 3.1.1 The infinitesimal generator on Rp +(R)............... 89 ×Sd 3.1.2 The Laplace transform of affine processes . 91 3.1.3 A detailed study of the matrix Riccati differential equation . .. 94 3.1.4 Numerical resolution of the MRDE with time dependent coefficients. 97 3.1.5 Some identities in law for affine processes . 97 3.2 Second order discretization schemes for Monte Carlo simulation . 98 3.2.1 Some preliminary results on discretization schemes for SDEs . 99 3.2.2 High order discretization schemes for L ................. 102 3.2.3 A second order scheme . 103 3.2.4 A faster second order scheme when Ω In +(R).......... 105 − d ∈Sd 3.2.5 Numericalresults ............................. 106 4 A stochastic variance-covariance affine term structure model 111 4.1 Modeldefinition.................................. 112 4.1.1 Modelreduction.............................. 115 4.1.2 Change of measure and Laplace transform . 116 4.2 Numericalframework............................... 118 4.2.1 Series expansion of distribution . 118 4.2.2 Transformpricing............................. 123 4.2.3 Numericalresults ............................. 127 4.3 Volatility expansions for caplets and swaptions . ..... 128 4.3.1 Price and volatility expansion for Caplets . 129 4.3.2 Price and volatility expansion for Swaptions . 134 4.3.3 Numericalresults ............................. 140 4.3.4 On the impact of the parameter ϵ .................... 142 4.3.5 Analysisoftheimpliedvolatility . 143 4 4.4 HedgingintheSCVATSM ............................ 143 4.4.1 Traitsofstochasticvolatility. 144 5 Some applications 153 5.1 A semidefinite programming approach for model calibration . ..... 154 5.1.1 Thecalibrationproblem . 154 5.1.2 TheSDPformulation. 155 5.2 Choice of the constant parameters for model calibration . 156 5.3 Calibrationresults ................................ 159 5.3.1 Calibrating x andΩ ........................... 159 5.3.2 Calibrating ∆x .............................. 162 III AquantitativeviewonALM 169 6 Notions of ALM 173 6.1 Assets and liabilities . 173 6.1.1 Some fundamental differences with a Mark to market approach . 174 6.2 Modelingschedules ................................ 174 6.3 Interestrategap ................................. 177 6.4 Modelinganddata ................................ 179 7 Hedging Interest rate risk in ALM: some examples 181 7.1 Hedging the Livret A saving accounts . 181 7.1.1 Different views of the Livret A . 182 7.1.2 RiskanalysisofLivretAproduct . 184 7.1.3 HedgingofLivretAinpractice . 187 7.2 A model for PEL saving accounts . 188 7.2.1 Productdescription............................ 188 7.2.2 Notations ................................. 189 7.2.3 Themortgage............................... 190 7.2.4 A model for the evolution of the PEL contracts . 191 7.2.5 A PDE resolution in the Vasicek model . 192 7.2.6 Numericalresults ............................. 195 7.3 Conclusionofthechapter ............................ 198 8 Hedging of options in ALM 199 8.1 Some questions about Delta hedging in ALM . 199 8.2 Delta hedging in theory: chosing the numeraire . 200 8.2.1 The classic risk-neutral delta hedging portfolio . 200 8.2.2 Alternative hedging: hedging under the forward measure . 202 8.2.3 Comparison of the risk neutral and forward neutral measure . 203 8.2.4 Hedginginpractice............................ 204 5 8.3 Numerical implementation in the Hull-White model . 206 8.3.1 The derivations of the option and hedges values in the model . 206 8.3.2 Numerical comparison of the alternative delta strategies . 208 8.3.3 Hedging performance and cash flows generation . 209 8.4 Conclusionofthechapter ............................ 211 A MF Calibration Algorithm 221 A.1 ”Anti-diagonal”Calibration . 221 A.2 ”Column”Calibration .............................. 225 B Proof of lemma 46 229 C Black-Scholes and Bachelier prices and greeks 231 C.1 Log-normalmodel................................. 231 C.2 Normalmodel................................... 231 D Gamma-Vega relationships 233 D.1 Log-normalmodel................................. 233 D.2 Normalmodel................................... 234 E Expansion of the price and volatility: details of calculations 235 E.1 Caplets price expansion . 235 E.2 Swaptionpriceexpansion............................. 238 6 Acknowledgements This PhD thesis has been a far longer journey than initially expected: looking back I value every moment of this experience, and I strongly believe it has deeply contributed to change my mindset. Although my PhD thesis officially started in October 2010, and most of the work presented here has been done after that date, I believe this journey has started in July 2008, when I first entered the Groupe de Recherche Op´erationnelle as an intern. Since then, I have been exposed to mathematical models used to manage interest rates derivatives. I believe this experience would not be so rich if it wasn’t for the people I met along the way. Some of them have deeply contributed to this work and I would like to thank them here. I am grateful to my PhD supervisor Bernard Lapeyre for being so open and receptive to the format of the PhD thesis, for its curiosity on mechanics of the banking industry, for always being available to meet and discuss the progress of my work, and for its continuous guidance during this work. I am grateful to Aur´elien Alfonsi, who co-spervised my PhD thesis, took the time to review some of the most involved computations line by line and al- lowed me to benefit of his deep understanding of the mathematical tools involved in this work. I am thankful to Abdelkoddousse Ahdida, who strongly contributed to the advances of this work, and shared with me some unique Sunday’s afternoons computing the higher order terms of prices expansions. I am grateful to Christophe Michel who has ”commissioned” the work and internally super- vised my work
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