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Quantification of the Model Risk in Finance and Related Problems Ismail Laachir Quantification of the model risk in finance and related problems Ismail Laachir To cite this version: Ismail Laachir. Quantification of the model risk in finance and related problems. Risk Management [q-fin.RM]. Université de Bretagne Sud, 2015. English. NNT : 2015LORIS375. tel-01305545 HAL Id: tel-01305545 https://tel.archives-ouvertes.fr/tel-01305545 Submitted on 21 Apr 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. ` ´ THESE / UNIVERSITE DE BRETAGNE SUD present´ ee´ par UFR Sciences et Sciences de l’Ing´enieur sous le sceau de l’Universit´eEurop´eennede Bretagne Ismail LAACHIR Pour obtenir le grade de : DOCTEUR DE L’UNIVERSITE´ DE BRETAGNE SUD Unite´ de Mathematiques´ Appliques´ (ENSTA ParisTech) / Mention : STIC Ecole´ Doctorale SICMA Lab-STICC (UBS) Th`esesoutenue le 02 Juillet 2015, Quantification of the model risk in devant la commission d’examen composee´ de : Mme. Monique JEANBLANC Professeur, Universite´ d’Evry´ Val d’Essonne, France / Presidente´ finance and related problems. M. Stefan ANKIRCHNER Professeur, University of Jena, Germany / Rapporteur Mme. Delphine LAUTIER Professeur, Universite´ de Paris Dauphine, France / Rapporteur M. Patrick HENAFF´ Maˆıtre de conferences,´ IAE Paris, France / Examinateur M. Claude MARTINI CEO Zeliade Systems, France / Examinateur M. Jean-Marc LE CAILLEC Professeur, TELECOM Bretagne, France / Directeur de these` M. Francesco RUSSO Professeur, ENSTA ParisTech, France / Directeur de these` Quantification of the model risk in finance and related problems Ismail Laachir 2015 Quantification of the model risk in finance and related problems Ismail Laachir 2015 ”Essentially, all models are wrong, but some are useful.” George E. P. Box Quantification of the model risk in finance and related problems Ismail Laachir 2015 0 Quantification of the model risk in finance and related problems Ismail Laachir 2015 Contents Table of Contents i Introduction 1 1 Change of numeraire in the two-marginals martingale transport problem. 17 1.1 Introduction ................................... 17 1.2 Basic left-monotone transference plan: existence and uniqueness . 20 1.2.1 Necessary conditions .......................... 20 1.2.2 Sufficient conditions .......................... 21 1.3 Change of numeraire .............................. 23 1.3.1 The symmetry operator S ....................... 23 1.3.2 The symmetric two-marginals martingale problem . 24 1.3.3 Relation to the generalized Spence-Mirrlees condition . 25 1.3.4 Symmetry and model risk ....................... 26 1.4 Construction of the basic right-monotone transference map via change of numeraire ................................... 27 1.5 Symmetry: Hobson and Klimmek [2015] revisited . 31 1.5.1 Sufficient conditions .......................... 34 1.6 Two new transference plans .......................... 38 1.6.1 What are the payoffs for which this transference plan is optimal ? 43 1.7 Applications ................................... 45 1.7.1 Symmetrized payoffs have a lower model risk . 45 1.7.2 Example: the symmetric log normal case . 46 1.8 Conclusion .................................... 48 Appendix 1.A Proof of Lemma 1.2.3 ....................... 50 Appendix 1.B Proof of Proposition 1.2.4 ..................... 50 Appendix 1.C Proof of Lemma 1.2.6 ........................ 51 Appendix 1.D Proof of Lemma 1.5.4 ........................ 52 2 Gas storage valuation and hedging. A quantification of model risk. 55 2.1 Introduction ................................... 55 2.2 Natural gas stylized facts ............................ 57 2.3 Valuation and hedging of a gas storage utility . 60 2.3.1 Gas storage specification ........................ 61 2.3.2 Dynamic programming equation ................... 63 2.3.3 Financial hedging strategy ...................... 64 2.4 Literature on price processes .......................... 66 2.4.1 Spot price processes .......................... 66 2.5 Our modeling framework ........................... 68 2.5.1 Modeling the futures curve ...................... 68 2.5.2 Modeling spot price .......................... 71 2.6 Numerical results ................................ 75 2.7 Model risk .................................... 82 2.7.1 Spot modeling .............................. 82 i Quantification of the model risk in finance and related problems Ismail Laachir 2015 ii CONTENTS 2.7.2 Model risk measure .......................... 84 2.8 Conclusion .................................... 88 Appendix 2.A Different types of gas storage facilities . 89 Appendix 2.B Futures-based valuation methods . 90 3 BSDEs, c`adl`agmartingale problems and mean-variance hedging under basis risk. 93 3.1 Introduction ................................... 93 3.2 Strong inhomogeneous martingale problem . 97 3.2.1 General considerations ......................... 97 3.2.2 The case of Markov semigroup .................... 99 3.2.3 Diffusion processes . 106 3.2.4 Variant of diffusion processes . 107 3.2.5 Exponential of additive processes . 108 3.3 The basic BSDE and the deterministic problem . 117 3.3.1 General framework . 117 3.3.2 The forward-backward case and the deterministic problem . 117 3.3.3 Illustration 1: the Markov semigroup case . 120 3.3.4 Illustration 2: the diffusion case . 121 3.4 Explicit solution for Follmer-Schweizer¨ decomposition in the basis risk context ......................................122 3.4.1 General considerations . 122 3.4.2 Application: exponential of additive processes . 125 3.4.3 Diffusion processes . 134 Appendix 3.A Proof of Proposition 3.2.8 . 141 Appendix 3.B Proof of Theorem 3.2.18 . 143 Acknowledgements 151 Bibliography 161 Quantification of the model risk in finance and related problems Ismail Laachir 2015 Introduction The actors in any financial market are exposed to a myriad of risks. Derivative con- tracts allow the transfer of a specific risk from a party that wishes to cover that risk to a party willing to be exposed to it. For example, a call option covers its buyer from the rise of the underlying price. The rapid growth of the derivatives market during the last century generated an extensive interest in several associated financial risks, e.g. the market risk, the counterparty credit risk, the liquidity risk etc. Historically, the market risk was the first which was taken into account. It denotes the risk that the price variations of the underlying financial assets impact the party’s portfolio composed of multiple positions in derivatives contracts. The seminal work of Merton [1973] and Black and Scholes [1973] postulated the geometric Brownian motion as a stochastic model for the underlying price dynamics and constructed pric- ing and hedging strategies for basic derivative contracts. Since then, the (derivative- related) market risk has been managed by proposing a long stream of models (stochas- tic volatility, local volatility, jump-diffusion etc.), which were supposed to correctly describe the dynamics of the option underlying, such as interest rates, equities, infla- tion, currencies etc. In addition to market risk, derivatives contracts carry other financial risks, as coun- terparty credit risk and liquidity risk. The recent subprime crisis stressed the importance of the counterparty credit risk when managing derivatives. This constitutes the risk that a counterparty may not be able to fulfill its engagements in a derivative contract, for example in the case of bankruptcy. This risk is generally mitigated through the clearing houses for the exchange traded derivatives or through tailored collateral ar- rangements for over-the-counter products. Typical instruments for risk reduction are credit derivatives as CDS. Additionally to the uncertainty on the counterparty credit worthiness, the liquidity risk is also a critical risk for derivatives dealers. It represents the risk that the strategies that a company decides to implement may become inappli- cable because of a lack of liquidity of the underlying. For instance, a hedging strategy of an exotic option may become too expensive and inefficient, because the trading in the underlying asset and/or the vanilla instruments may suffer from an unexpected reduced market liquidity, producing a bid/ask spread widening. The quantification of all the three financial risks described above requires the pos- tulation of a complex dynamic model, which for instance, involves the volatility mod- eling for the market risk or the default probability modeling for the counterparty credit risk. Assuming the validity of that model, the specification of a pricing and hedging strategy ensures, theoretically, that the market risk is managed and quantified. How- ever, as pointed out by Hobson [2011], “although market risk (the known unknown) is eliminated, model risk (the unknown unknown) remains”. Indeed, the initial and subjective choice of the model and its practical use may become a source of ambi- guity and give rise to a new risk concept, i.e. the model uncertainty. Quoting the European Directive (2013/36/EU), “model risk means the potential loss an institution 1 Quantification of the model
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