Risk Aversion, Intertemporal Substitution, and Option Pricing

9801

GARCIA, René

RENAULT, Éric

Risk Aversion, Intertemporal Substitution, and

Option Pricing

Département de sciences économiques Université de Montréal

Faculté des arts et des sciences C.P. 6128, succursale Centre-Ville Montréal (Québec) H3C 3J7 Canada http://www.sceco.umontreal.ca SCECO-information@UMontreal.CA Téléphone : (514) 343-6539 Télécopieur : (514) 343-7221

Ce cahier a également été publié par le Centre interuniversitaire de recherche en économie quantitative (CIREQ) sous le numéro 0198.

This working paper was also published by the Center for Interuniversity Research in Quantitative Economics (CIREQ), under number 0198.

CAHIER 9801

RISK AVERSION, INTERTEMPORAL SUBSTITUTION, AND OPTION PRICING

René GARCIA1 and Éric RENAULT2

1

Centre de recherche et développement en économique (C.R.D.E.) and Département de sciences économiques, Université de Montréal, and CIRANO

2

Université Paris IX-Dauphine and CREST-INSEE

February 1998

_______________

The authors would like to thank Larry Epstein, Jean-Pierre Florens, Ali Lazrak, Thomas McCurdy, Angelo Melino and Nizar Touzi for their useful comments. The first author gratefully acknowledges financial support from the Fonds pour la formation de chercheurs et l’aide à la recherche of Québec (FCAR) and the PARADI research program funded by the Canadian International Development Agency (CIDA). The second author thanks CIRANO and the C.R.D.E. for financial support.

RÉSUMÉ

Dans le présent article, on propose un cadre stochastique général et un modèle d’évaluation d’actifs financiers à l’équilibre qui mettent en évidence les rôles respectifs de l’élasticité de substitution intertemporelle et de l’aversion pour le risque dans le prix de marché des options. Nous précisons en particulier les conditions statistiques sous lesquelles les formules d’évaluation d’options dépendent ou non explicitement des paramètres de préférence, en particulier quand ces paramètres ne sont pas cachés dans les prix de l’actif sous-jacent et d’une obligation, comme c’est le cas dans les modèles standards de Black et Scholes (BS) ou de Hull et White (HW). Plusieurs effets de causalité instantanée, du type effet de levier, expliquent l’occurrence non redondante des paramètres de préférence dans les prix d’options. On prouve aussi que les modèles d’évaluation d’actifs financiers les plus classiques (CAPM pour les actions, BS ou HW où les prix d’options ne font pas apparaître les paramètres de préférence) sont fondés sur les mêmes hypothèses stochastiques (typiquement l’absence d’effet de levier), indépendamment des valeurs des paramètres de préférence. Même si notre formule générale d’évaluation d’options dépend dans certains cas explicitement des paramètres de préférence, on n’oublie pas que la formule BS est dominante à la fois comme modèle théorique de référence et comme instrument de gestion. Une autre contribution de l’article est la validation théorique de ce rôle de référence. Ainsi, dans la mesure où on accepte une propriété essentielle des prix d’options, à savoir leur homogénéité de degré un par rapport au couple formé par le prix de l’actif sous-jacent et le prix d’exercice, on peut montrer que les hypothèses statistiques nécessaires et suffisantes pour l’homogénéité donnent à l’équilibre des prix d’options qui conservent l’essentiel de la forme fonctionnelle de BS. Cette forme fonctionnelle nous permet de mettre en évidence certaines propriétés importantes du « sourire » de volatilité, c’est-à-dire de la représentation graphique des volatilités implicites de BS en fonction de la position de l’option par rapport à la monnaie. On montre d’abord que l’asymétrie de ce sourire est équivalente à une forme particulière d’asymétrie de la mesure de martingale équivalente. Enfin, cette asymétrie correspond précisément au cas où il existerait soit une prime sur un risque instantané de taux d’intérêt, soit un effet de levier généralisé, soit les deux, en d’autres termes lorsque la formule d’évaluation d’options dépend explicitement des paramètres de préférence. En conclusion, le message principal pour la gestion d’options résultant de notre analyse est que l’évidence d’une asymétrie dans le sourire de volatilité signale l’importance de la prise en compte des paramètres de préférence dans les formules d’évaluation d’options.

Mots clés : causalité, chaînes de Markov cachées, utilité non séparable, évaluation d’options par modèle d’équilibre, utilité récursive, volatilité implicite de Black-Scholes, sourire de volatilité

ABSTRACT

This paper develops a general stochastic framework and an equilibrium asset pricing model that make clear how attitudes towards intertemporal substitution and risk matter for option pricing. In particular, we show under which statistical conditions option pricing formulas are not preference-free, in other words, when preferences are not hidden in the stock and bond prices as they are in the standard Black and Scholes (BS) or Hull and White (HW) pricing formulas. The dependence of option prices on preference parameters comes from several instantaneous causality effects such as the so-called leverage effect. We also emphasize that the most standard asset pricing models (CAPM for the stock and BS or HW preference-free option pricing) are valid under the same stochastic setting (typically the absence of leverage effect), regardless of preference parameter values. Even though we propose a general non-preference-free option pricing formula, we always keep in mind that the BS formula is dominant both as a theoretical reference model and as a tool for practitioners. Another contribution of the paper is to characterize why the BS formula is such a benchmark. We show that, as soon as we are ready to accept a basic property of option prices, namely their homogeneity of degree one with respect to the pair formed by the underlying stock price and the strike price, the necessary statistical hypotheses for homogeneity provide BS-shaped option prices in equilibrium. This BS-shaped option-pricing formula allows us to derive interesting characterizations of the volatility smile, that is, the pattern of BS implicit volatilities as a function of the option moneyness. First, the asymmetry of the smile is shown to be equivalent to a particular form of asymmetry of the equivalent martingale measure. Second, this asymmetry appears precisely when there is either a premium on an instantaneous interest rate risk or on a generalized leverage effect or both, in other words, whenever the option pricing formula is not preference-free. Therefore, the main conclusion of our analysis for practitioners should be that an asymmetric smile is indicative of the relevance of preference parameters to price options.

Key words : causality, hidden Markov chains, non-separable utility, equilibrium option pricing, recursive utility, Black-Scholes implicit volatility, smile effect

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