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Risk Aversion, Intertemporal Substitution, and Option Pricing

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  • René Garcia
  • Éric Renault

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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, the0501n conclusion of our analysis for practitioners should be that an asymmetric smile is indicative of the relevance of preference parameters to price options. 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 existe 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.

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 98s-02.

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Date of creation: 01 Feb 1998
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Handle: RePEc:cir:cirwor:98s-02

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Keywords: Causality; hidden Markov chains; non-separable utility; equilibrium option pricing; recursive utility; Black-Scholes implicit volatility; smile effect; 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;

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  1. Naik, Vasanttilak & Lee, Moon, 1990. "General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 493-521.
  2. Aït-Sahalia, Yacine. & Bickel, Peter J. & Stoker, Thomas M., 1994. "Goodness-of-fit tests for regression using kernel methods," Working papers 3747-94., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  3. Yaacov Z. Bergman & Bruce D. Grundy & Zvi Wiener, . "General Properties of Option Prices (Revision of 11-95) (Reprint 058)," Rodney L. White Center for Financial Research Working Papers 01-96, Wharton School Rodney L. White Center for Financial Research.
  4. S.G. Cecchetti & P. Lam & N.C. Mark, 2010. "The equity premium and the risk-free rate: matching the moments," Levine's Working Paper Archive 1396, David K. Levine.
  5. Bonomo, M. & Garcia, R., 1991. "Consumption and Equilibrium Asset Pricing: an Empirical Assessment," Cahiers de recherche 9126, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  6. Engle, Robert F. & Mustafa, Chowdhury, 1992. "Implied ARCH models from options prices," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 289-311.
  7. Kreps, David M & Porteus, Evan L, 1978. "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica, Econometric Society, vol. 46(1), pages 185-200, January.
  8. Turnbull, Stuart M & Milne, Frank, 1991. "A Simple Approach to Interest-Rate Option Pricing," Review of Financial Studies, Society for Financial Studies, vol. 4(1), pages 87-120.
  9. Ghysels, E. & Harvey, A. & Renault, E., 1996. "Stochastic Volatility," Cahiers de recherche 9613, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  10. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
  11. René Garcia & �ric Renault, 1998. "A Note on Hedging in ARCH and Stochastic Volatility Option Pricing Models," Mathematical Finance, Wiley Blackwell, vol. 8(2), pages 153-161.
  12. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October.
  13. Garcia, Rene, 1998. "Asymptotic Null Distribution of the Likelihood Ratio Test in Markov Switching Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(3), pages 763-88, August.
  14. Bossaerts, P. & Hillion, P., 1995. "Local Parametric Analysis of Hedging in Discrete Time," Discussion Paper 1995-23, Tilburg University, Center for Economic Research.
  15. Yacine Aït-Sahalia & Andrew W. Lo, 1998. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," Journal of Finance, American Finance Association, vol. 53(2), pages 499-547, 04.
  16. Duffie, Darrell & Epstein, Larry G, 1992. "Asset Pricing with Stochastic Differential Utility," Review of Financial Studies, Society for Financial Studies, vol. 5(3), pages 411-36.
  17. Dumas, Bernard J & Fleming, Jeff & Whaley, Robert E, 1996. "Implied Volatility Functions: Empirical Tests," CEPR Discussion Papers 1369, C.E.P.R. Discussion Papers.
  18. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  19. Brennan, M J, 1979. "The Pricing of Contingent Claims in Discrete Time Models," Journal of Finance, American Finance Association, vol. 34(1), pages 53-68, March.
  20. Garcia, Rene & Gencay, Ramazan, 2000. "Pricing and hedging derivative securities with neural networks and a homogeneity hint," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 93-115.
  21. Amin, Kaushik I & Ng, Victor K, 1993. " Option Valuation with Systematic Stochastic Volatility," Journal of Finance, American Finance Association, vol. 48(3), pages 881-910, July.
  22. Bernard Dumas & Jeff Fleming & Robert E. Whaley, 1996. "Implied Volatility Functions: Empirical Tests," NBER Working Papers 5500, National Bureau of Economic Research, Inc.
  23. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
  24. Garcia, R. & Bonomo, M., 1993. "Disappointment Aversion as a Solution to the Equity Premium and the Risk- Free Rate Puzzles," Cahiers de recherche 9334, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  25. Bailey, Warren & Stulz, René M., 1989. "The Pricing of Stock Index Options in a General Equilibrium Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(01), pages 1-12, March.
  26. Florens, J P & Mouchart, M, 1982. "A Note on Noncausality," Econometrica, Econometric Society, vol. 50(3), pages 583-91, May.
  27. Epstein, Larry G & Zin, Stanley E, 1989. "Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework," Econometrica, Econometric Society, vol. 57(4), pages 937-69, July.
  28. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
  29. Kreps,David M. & Wallis,Kenneth F. (ed.), 1997. "Advances in Economics and Econometrics: Theory and Applications," Cambridge Books, Cambridge University Press, number 9780521589819, October.
  30. Kreps,David M. & Wallis,Kenneth F. (ed.), 1997. "Advances in Economics and Econometrics: Theory and Applications," Cambridge Books, Cambridge University Press, number 9780521589833, October.
  31. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-45, November.
  32. Bergman, Yaacov Z & Grundy, Bruce D & Wiener, Zvi, 1996. " General Properties of Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1573-1610, December.
  33. Stephen G. Cecchetti & Pok-sang Lam & Nelson C. Mark, 1988. "Mean Reversion in Equilibrium Asset Prices," NBER Working Papers 2762, National Bureau of Economic Research, Inc.
  34. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
  35. Kreps,David M. & Wallis,Kenneth F. (ed.), 1997. "Advances in Economics and Econometrics: Theory and Applications," Cambridge Books, Cambridge University Press, number 9780521589826, October.
  36. Eric Renault & Nizar Touzi, 1996. "Option Hedging And Implied Volatilities In A Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 6(3), pages 279-302.
  37. Machina, Mark J, 1989. "Dynamic Consistency and Non-expected Utility Models of Choice under Uncertainty," Journal of Economic Literature, American Economic Association, vol. 27(4), pages 1622-68, December.
  38. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1981. "A Re-examination of Traditional Hypotheses about the Term Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 36(4), pages 769-99, September.
  39. Hansen, Lars Peter & Singleton, Kenneth J, 1983. "Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns," Journal of Political Economy, University of Chicago Press, vol. 91(2), pages 249-65, April.
  40. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-84, March.
  41. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  42. Christopher A. Sims, 1980. "Martingale-Like Behavior of Prices," NBER Working Papers 0489, National Bureau of Economic Research, Inc.
  43. Jin-Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32.
  44. Bollerslev, Tim & Engle, Robert F. & Nelson, Daniel B., 1986. "Arch models," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 49, pages 2959-3038 Elsevier.
  45. repec:fth:inseep:9329 is not listed on IDEAS
  46. Florens, Jean-Pierre & Fougere, Denis, 1996. "Noncausality in Continuous Time," Econometrica, Econometric Society, vol. 64(5), pages 1195-1212, September.
  47. Gouriéroux, Christian & Monfort, Alain & Tenreiro, Carlos, 1994. "Kernel m-estimators : non parametric diagnostics for structural models," CEPREMAP Working Papers (Couverture Orange) 9405, CEPREMAP.
  48. Kaushik I. Amin & Robert A. Jarrow, 1992. "Pricing Options On Risky Assets In A Stochastic Interest Rate Economy," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 217-237.
  49. Jorion, Philippe, 1995. " Predicting Volatility in the Foreign Exchange Market," Journal of Finance, American Finance Association, vol. 50(2), pages 507-28, June.
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Cited by:
  1. GHYSELS, Eric & PATILEA, Valentin & RENAULT, Eric & TORRES, Olivier, 1997. "Nonparametric methods and option pricing," CORE Discussion Papers 1997075, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  2. Henri Bertholon & Alain Monfort & Fulvio Pegoraro, 2006. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Working Papers 2006-28, Centre de Recherche en Economie et Statistique.
  3. Garcia, Rene & Gencay, Ramazan, 2000. "Pricing and hedging derivative securities with neural networks and a homogeneity hint," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 93-115.
  4. Garcia, R. & Luger, R. & Renault, E., 2001. "Asymmetric Smiles, Leverage Effects and Structural Parameters," Cahiers de recherche 2001-09, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  5. Rene Garcia & Richard Luger & Eric Renault, 2004. "Option Prices, Preferences, and State Variables," Emory Economics 0418, Department of Economics, Emory University (Atlanta).
  6. Garcia, Rene & Luger, Richard & Renault, Eric, 2003. "Empirical assessment of an intertemporal option pricing model with latent variables," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 49-83.
  7. René Garcia & Eric Ghysels & Éric Renault, 2004. "The Econometrics of Option Pricing," CIRANO Working Papers 2004s-04, CIRANO.
  8. H. Bertholon & A. Monfort & F. Pegoraro, 2008. "Econometric Asset Pricing Modelling," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 6(4), pages 407-458, Fall.
  9. Stanislav Khrapov, 2012. "Risk Premia: Short and Long-term," Working Papers w0169, Center for Economic and Financial Research (CEFIR).
  10. Robert R. Bliss & Nikolaos Panigirtzoglou, 2001. "Recovering risk aversion from options," Working Paper Series WP-01-15, Federal Reserve Bank of Chicago.

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