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Model Error in Contingent Claim Models Dynamic Evaluation

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  • Éric Jacquier

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  • Robert Jarrow

Abstract

We formally incorporate parameter uncertainty and model error in the estimation of contingent claim models and the formulation of forecasts. This allows an inference on any function of interest (option values, bias functions, hedge ratios) consistent with the uncertainty in both parameters and models. We show how to recover the exact posterior distributions of the parameters or any function of interest. It is crucial to obtain exact posterior or predictive densities because the most likely implementation, a frequent updating setup, results in small samples and requires the incorporation of specific prior information. We develop Markov Chain Monte Carlo estimators to solve the estimation problem posed. We provide both within sample and predictive model specification tests which can be used in dynamic testing or trading systems, making use of both the cross-sectional and time series information in the options data. Finally, we generalize the error distribution by allowing for the (small) probability that an observation has a larger error. For each observation, this produces the probability of its being an outlier, and may help distinguish market from model error. We apply these new techniques to equity options. When model error is taken into account, the black-Scholes appears very robust, in contrast with previous studies which at best only incorporated parameter uncertainty. We then extend the base model, e.g., Black-Scholes, by polynomial functions of parameters. This allows for intuitive specification tests. The Black-Scholes in-sample error properties can be improved by the use of these simple extended models but this does not result in major improvements in out of sample predictions. The differences between these models may be important however because, as we document it, they produce different functions of economic interest such as hedge ratios, probability of mispricing. Nous incorporons formellement l'incertitude des paramètres et l'erreur de modèle dans l'estimation des modèles d'option et la formulation de prévisions. Ceci permet l'inférence de fonctions d'intérêt (prix de l'option, biais, ratios) cohérentes avec l'incertitude des paramètres et du modèle. Nous montrons comment extraire la distribution postérieure exacte (de fonctions) des paramètres. Ceci est crucial parce que l'utilisation la plus probable, réestimation périodique des paramètres, est analogues à des échantillons de petite taille et demande l'incorporation d'informations a priori spécifiques. Nous développons des modèles Monte Carlo de chaînes markoviennes afin de résoudre les problèmes d'estimation posés. Nous fournissons des tests de spécification, à la fois pour l'échantillon et le modèle prédictif, qui peuvent être utilisés pour les tests dynamiques et les systèmes de trading en utilisant l'information en coupe transversale et temporelle des données d'option. Finalement, nous généralisons la distribution d'erreurs en tenant compte de la (faible) probabilité qu'une observation ait une plus grande probabilité d'erreur. Cela fournit pour chaque observation la probabilité d'une donnée aberrante et peut aider à différencier erreur de modèle et erreur de marché. Nous appliquons ces nouvelles techniques aux options d'équité. Quand l'erreur de modèle est prise en considération, le Black-Scholes apparaît très robuste, en contraste avec les études précédentes qui, au mieux, incluait l'erreur de paramètre. Après, nous étendons le modèle de base, i.e. Black-Schles, par des fonctions polynomiales des paramètres. Cela permet des tests intuitifs de spécification. Les erreurs en échantillon du B-S sont améliorées par l'utilisation de ces simples modèles étendus,0501s cela n'apporte pas d'amélioration majeure dans les prédictions hors-échantillon. Quoi qu'il en soit, les différences entre ces modèles peuvent être importantes parcequ'elles produisent différentes fonctions d'intérêt telles que les ratios et la probabilité d'erreur d'évaluation.

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 96s-12.

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Date of creation: 01 Mar 1996
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Handle: RePEc:cir:cirwor:96s-12

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  1. Jarrow, Robert & Rudd, Andrew, 1982. "Approximate option valuation for arbitrary stochastic processes," Journal of Financial Economics, Elsevier, vol. 10(3), pages 347-369, November.
  2. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
  3. Geweke, John, 1989. "Bayesian Inference in Econometric Models Using Monte Carlo Integration," Econometrica, Econometric Society, vol. 57(6), pages 1317-39, November.
  4. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 2002. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 69-87, January.
  5. Bossaerts, Peter & Hillion, Pierre, 1997. "Local parametric analysis of hedging in discrete time," Journal of Econometrics, Elsevier, vol. 81(1), pages 243-272, November.
  6. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
  7. 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.
  8. Schotman, Peter, 1996. "A Bayesian approach to the empirical valuation of bond options," Journal of Econometrics, Elsevier, vol. 75(1), pages 183-215, November.
  9. John F. Geweke, 1994. "Bayesian comparison of econometric models," Working Papers 532, Federal Reserve Bank of Minneapolis.
  10. Robert F. Engle & Alex Kane & Jaesun Noh, 1993. "Index-Option Pricing with Stochastic Volatility and the Value of Accurate Variance Forecasts," NBER Working Papers 4519, National Bureau of Economic Research, Inc.
  11. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
  12. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
  13. Whaley, Robert E., 1982. "Valuation of American call options on dividend-paying stocks : Empirical tests," Journal of Financial Economics, Elsevier, vol. 10(1), pages 29-58, March.
  14. MacBeth, James D & Merville, Larry J, 1979. "An Empirical Examination of the Black-Scholes Call Option Pricing Model," Journal of Finance, American Finance Association, vol. 34(5), pages 1173-86, December.
  15. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
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Cited by:
  1. Mikhail Chernov & Eric Ghysels, 1998. "What Data Should Be Used to Price Options?," CIRANO Working Papers 98s-22, CIRANO.

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