BANKERS, MARKETS INVESTORS an academic and professional review

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1 n 119 July-August 2012 ISSN euros revue-banque.fr BANKERS, MARKETS INVESTORS an academic and professional review ARTICLES 4 Option Pricing and Hedging in the Presence of Cross-Hedge Risk Lionel Martellini, EDHEC Business School, EDHEC-Risk Institute Vincent Milhau, EDHEC-Risk Institute 19 Optimal Economic Capital and Investment Decisions for a Non-Life Insurance Company Selim Mankaï, EconomiX, Université Paris Ouest Nanterre la Défense Catherine Bruneau, Université Paris 1 Panthéon Sorbonne 31 Implied Distribution as a Function of the Volatility Smile Bertrand TAVIN, Université Paris 1 Panthéon Sorbonne (PRISM) FOCUS ON Regulatory Governance Mechanisms in Banks Jerome Maati, Université de Lille Nord de France LEM Christine Maati-Sauvez, Université de Lille Nord de France IDP In partnership with Association française de finance

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3 Article submission : authors guideline Bankers, Markets & Investors aim is to make up-to-date scientific research in financial matters available to members of the profession. Articles can focus on all matters related to financial markets. Financial models can be applied to a wide range of subjects: financial assets evaluation, risk analysis, investment policies, microstructure and markets organisation. Applications to corporate finance, assets and liabilities management and strategy of financial intermediaries are welcome. Analysis of new products or contracts, which concern retail as well as corporate markets, new technical applications and new developments of financial markets can also be dealt with as far as their evaluation and usefulness are analysed. Articles submitted for publication will be read by a scientific committee of two competent experts. The committee will be particularly attentive to the explanations given on the concepts from an economic, financial and mathematical point of view. Any kind of illustrations and graphs are welcome. A significant part of the article will concentrate on the possible practical applications of the methods and concepts which are presented, and will insist on implications in related financial areas. Empirical studies should mention their sources of information, or else provide the data drawn upon in appended notes. Articles can be written exclusively in English, in straight forward correct language. Mathematical developments should be limited to essential notions, and originals aspects, if any, are to be detailed in a note. Reference to scientific literature shall also be limited to essential matters. The length of the articles should not exceed 20 pages (ideally 14 to 15 pages, including graphs and notes). Shorter articles are naturally welcome. Each article submitted has to include an abstract in English of less than 500 characters. The articles (.doc,.rtf,.txt, but no.pdf or.tex) have to be sent by to hauvette@revue-banque.fr. Strategic Committee Francis Candylaftis/Eurizon Capital, Bernard Dumas/Université de Lausanne, Thierry Foucault/HEC, René Karsenti/ICMA, Denis Kessler/Scor, André Lévy-Lang, Bertrand de Mazières/BEI, Théo Nijman/Université de Tilburg, Tom Steenkamp/ABP Investments, Mike Wright/Université de Nottingham Editorial Committee Editor : Jean-François Boulier/ Aviva Investors France Sanvi Avouyi-Dovi/Banque de France, Bruno Biais/Université Toulouse 1, Alain Chevalier/ESCP-EAP, Philippe Desbrières/IAE Dijon, Nicole EL KAROUI/École polytechnique, Antoine Frachot/Groupe des écoles nationales d économie et statistique (GENES), Edith Ginglinger/Université Paris-Dauphine, Ulrich Hege/HEC, Monique Jeanblanc/Université d Evry, Lionel Martellini/Edhec, Patrice Poncet/ Essec, Prof. Flavio PRESSACCO/Facolta di Economia di Udine, Nizar Touzi/École polytechnique Reading Committee Hervé ALEXANDRE/Université Paris-Dauphine, Franck BANCEL/ESCP Europe, Lorenzo BERGOMI/SG CIB, Bruno-Rolland BERNARD/LVMH, Éric de BODT/ESA Lille, Hubert de la BRUSLERIE/Université Paris I, Gérard CHARREAUX/IAE Dijon, Stéphane CRÉPEY/Université d Évry, Michel DIETSCH/IEP Strasbourg, Patrice Fontaine/Eurofidai, Jacques Hamon/CEREG-Université Paris-Dauphine, Hélène HARASTY/Lombard Odier Darier Hentsch & Cie, Maria-Laura HARTPENCE/HSBC AM, Hervé LE BIHAN/Banque de France, Frédéric LOBEZ/ESA Lille, Christophe MOUSSU/ESCP-EAP, Fabrice PANSARD/AMF, François QUITTARD-PINON/ISFA Université Lyon 1, Catherine Refait-Alexandre/CRESE-Université Franche-Comté, Patrick ROGER/Université Louis-Pasteur Strasbourg, Patrick ROUSSEAU/IAE Aix-en-Provence, Alain SCHATT/IAE Dijon, Éric SEVERIN/OSTL Lille 1, Jacques SIKORAV/BNP Paribas, Grégory TAILLARD/HSBC AM. 18 rue La Fayette Paris According to French Law (loi du 11 mars 1957 sur la propriété artistique et littéraire) no part of Bankers, Markets & Investors articles may be reproduced in any form or by any means without prior written permission of Revue Banque SARL. Managing Director : Valérie Ohannessian General Secretary : Élisabeth Coulomb Subediting : Alain de Seze (54 17) ; Christine Hauvette (54 10); Emmanuel Gonzalez (54 12) ; Alexandra Démétriadis (54 18) and DESK Subscription : NPAI - REVUE BANQUE 39 rue Marcelin Berthelot Drancy Cedex Tel. : Fax : revue-banque.abo@npai.fr CPPAP n 0613 T printer : SPEI (Pulnoy, France) Copyright deposit 3 rd quarter Bankers, Markets & Investors 2 Bankers, Markets & Investors n 119 july-august 2012

4 Abstracts Option Pricing and Hedging in the Presence of Cross-Hedge Risk 4 Lionel Martellini, EDHEC Business School, EDHEC-Risk Institute Vincent Milhau, EDHEC-Risk Institute This paper addresses the question of option pricing and hedging when the underlying asset is not available for dynamic trading, and some other asset is used as a substitute. We first provide an overview of the various hedging methodologies that can be used in this incomplete market setting, distinguishing between self-financing and non-self-financing strategies. Focussing on a local risk-minimization criterion, we present an analytical expression for the optimal hedging strategy and the corresponding option price. We also provide a quantitative measure of the residual risk over the life of the option. We find that the use of the optimal strategy induces a much smaller replication error compared to the replication error induced by a naive Black-Scholes strategy, especially for low levels of the correlation between the underlying asset and the substitute. In the absence of transaction costs, we also find that cross hedge risk is more substantial than the risk induced by discrete trading for reasonable parameter values. While this result implies that trading in the substitute can only be rationalized for exceedingly high correlations, the presence of (higher levels of) transaction costs is likely, however, to make trading in the actual underlying asset a prohibitively costly alternative. Keywords: Option pricing; Cross-hedge risk; Incomplete markets. JEL codes: G12; G13 Optimal Economic Capital and Investment Decisions for a Non-Life Insurance Company 19 Selim Mankaï, EconomiX, Université Paris Ouest Nanterre la Défense Catherine Bruneau, Université Paris 1 Panthéon Sorbonne This paper develops a theoretical portfolio-based approach to find the economic capital and the optimal assets allocation. The model maximizes a generalized version of the RORAC ratio, under a zero-cvar constraint. An efficient optimization procedure is proposed. To illustrate the theoretical developments, we give a numerical application and conduct sensitivity analysis. Capital adjustment appears to be more profitable than cash holding. This result holds irrespectively of the choice of the additional capital allocation. The risk magnitude and the initial capital level influence the capital adjustment decisions. Keywords: Portfolio selection; Risk management; Stochastic fractional programming; Monte Carlo simulation ; Non-life insurance. JEL: C61; D81; G11; G28; G3 Implied Distribution as a Function of the Volatility Smile 31 Bertrand Tavin, Université Paris 1 Panthéon Sorbonne (PRISM) In this paper we obtain the risk-neutral density of an underlying asset price as a function of its option implied volatility. We derive a known expression for the density and decompose it into a sum of lognormal and adjustment terms. We also derive no-arbitrage conditions on the volatility smile. We then explain how to use the results. Our methodology is applied to the pricing of a portfolio of digital options. It is then applied to the fitting of a distribution for log-return modelling. Keywords: Option pricing; No arbitrage; Risk-neutral distribution; Implied volatility smile. JEL: C14; C52; G13 FOCUS ON Regulatory Governance Mechanisms in Banks 43 Jerome Maati, Université de Lille Nord de France LEM Christine Maati-Sauvez, Université de Lille Nord de France IDP This paper reviews the regulatory governance mechanisms that increase banking stability. Deposit insurance is one such mechanism; however, its design may generate a moral hazard. There is consensus on the impact of deposit insurance on increasing risk-taking, but not on the effect of the capital adequacy ratio or on regulation restraining competition. Moreover, supervisory practices that require the disclosure of accurate and transparent information by banks may be more effective than official supervisory powers that monitor credit allocation. Keywords: Corporate governance; Bank; Literature review; Risk-taking, Regulatory mechanisms; Board of directors; Internal controls. JEL: G28; G34 bankers, markets & investors n 119 july-august

5 Option Pricing and Hedging in the Presence of Cross-Hedge Risk LIONEL MARTELLINI Professor of Finance at EDHEC Business School and Scientific Director of EDHEC-Risk Institute VINCENT MILHAU* Deputy Scientific Director of EDHEC-Risk Institute This paper addresses the question of dynamic hedging of an option when the underlying asset is not available for trading, and some other asset, or portfolio, is used as a substitute. The unavailability of the underlying asset may be due for example to liquidity constraints, legal constraints, high market frictions, etc. If the substitute asset were perfectly correlated with the actual underlying asset, no further risk would be introduced, since one could offset any gain or loss in the option position by dynamically trading the substitute asset. In general, however, correlation is not perfect, and the unavailability of the underlying asset induces some form of dynamic incompleteness in that perfect replication is no longer possible with a self-financing strategy. A typical example of what is known as cross-hedge risk can be found in index option markets, where S&P100 options are systematically hedged using dynamic trading in S&P500 futures contracts, because investors can not trade in the actual underlying asset. 1 In a similar spirit, investors are typically more willing to trade in Treasury bonds when dynamically hedging contingent positions on corporate bonds, mainly for liquidity reasons. Other examples include hedging a book of equity options with index futures, or hedging bond options with T-Bond futures. Another example is the case of basket options; even when one can trade the basket components, for efficiency reasons, one may still prefer to use a correlated index for pricing and hedging purposes. More generally, one can extend the analysis to encompass the question of vega hedging an option using some other option on the same underlying asset, but with a different strike price and/or time to maturity. In this case, cross-hedge risk is induced by stochastic deformations of the term and strike structure of implied volatilities. One last example would relate to the hedging of options written on illiquid underlying hedge fund porttolios, where some kind of an imperfect passive replication vehicle would be used in the dynamic hedging process. * The author for correspondence is Lionel Martellini. He can be reached at EDHEC Risk Institute, 400 Promenade des Anglais, BP 3116, Nice Cedex 3 France. Ph: +33 (0) Fax: +33 (0) lionel.martellini@edhec.edu. Given that the presence of cross-hedge risk induces a specific form of market incompleteness, it is impossible to price a contingent claim by arbitrage considerations only. Because there are an infinite number of equivalent martingale measures (EMMs), one may simply obtain bounds on the price of a given security (see for example [3], who provide a binomial valuation framework for options written on non-tradable assets). In most cases, however, such bounds are too wide to be of any practical use. 2 In this context, [1] and [2] propose to derive tighter bounds on options on non traded assets by imposing a restriction more stringent than the mere absence of arbitrage. In particular, [2] rule out «good deals», that is essentially portfolios with too high Sharpe ratios, while [1] rule out «quasi-arbitrage» opportunities, that is portfolios with a gain-loss ratio above a certain value. In order to further specify a unique price for the option in the absence of complete markets, one needs to introduce an optimality criterion that the strategy should satisfy. In a continuous-time setting, various criteria have been proposed in the literature. A key distinction can be made between self-financing and non-self-financing strategies. Because of market incompleteness, self-financing strategies cannot lead to perfect replication of the option payoff, and a reasonable criterion consists in minimizing the expected quadratic error. This objective has been introduced by [4] and has been the focus of several subsequent papers (see e.g. [9]), who show that the corresponding price can be expressed as an expectation of the option payoff under a so-called variance-optimal measure. While relatively intuitive, this approach suffers from a number of shortcomings, including the absence of time-consistency and the fact that it equally penalizes positive and negative net terminal wealth level. To address these questions, [11] have proposed to cast the problem into an expected utility framework, where the writer of the option has preferences over her terminal net wealth. This framework naturally leads to the notion of indifference price, which is defined as a certainty equivalent, namely the additional initial wealth that the writer of the option should be given so as to generate the same expected utility that would be achieved without the 4 Bankers, Markets & Investors nº 119 july-august 2012

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