The Quality of Option Prices Forecasts: A Dynamic Approach

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The Quality of Option Prices Forecasts: A Dynamic Approach? Gunther Capelle-Blancardy Emmanuel Jurczenkoz Bertrand Mailletx April 2000 Preliminary draft - Not to be quoted Abstract Many empirical studies pointed out that the Black-Scholes (1973) model leads to a wrong valuation of deep in-the-money and deep out- the-money options. These biases are usually attributed to the hypothesis of log-normality of the underlying asset return density. In order to remove these biases, Jarrow and Rudd (1982) propose to use an Edgeworth ex- pansion for the state price density. This approach takes into account the skewness and the kurtosis characterizing the return densities. Using high frequency data from the Bourse de Paris S.A. database, we examine the explicative and predictive power of the Jarrow and Rudd (1982) model. We …nd that this model improves the pricing of CAC 40 Index option (PXL). We wonder, in a last section, if the comparison between mod- els lead to the same conclusion when considering a dynamic framework. For this purpose, we determine a way of modelizing conditional moments whatever they are implicit or not. Keywords: Option Pricing Models, Edgeworth Expansion, Volatility Forecast. J.E.L. Classi…cation: G.10, G.12, G.13. ?We thank Bernard Bensaïd, Rama Cont, Jean-Paul Laurent, Thierry Michel and Christophe Villa for their comments on earlier drafts of this paper.

  • distribution function

  • comparison between

  • asset return

  • return densities

  • options when

  • density function

  • volatility forecast

  • option pricing

  • model leads

  • underlying asset


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TheQualityofOptionPricesForecasts:ADynamicApproach¤Gunther Capelle-BlancardyEmmanuelJurczenkozBertrand MailletxApril 2000Preliminarydraft-Nottobequoted
A bstractMany empirical studies pointed out that the Black-Scholes (1973)model leads to a wrong valuation of deep in-the-money and deep out-the-money options. These biases are usually attributed to the hypothesisof log-normality of theunderlying asset return density. In order to removethesebiases,JarrowandRudd(1982)proposetouseanEdgeworthex-pansion for the state price density. This approach takes into account theskewness and the kurtosis characterizing the return densities. Using highfrequency data from theBourse de Paris S.A.database, we examine theexplicative and predictive power of the J arrow and Rudd (1982) model.We …nd that this model improves the pricing of CAC 40 Index option(PXL).Wewonder,inalastsection,ifthecomparisonbetweenmod-els lead to the same conclusion when considering a dynamic framework.For this purpose, we determine a way of modelizing conditional momentswhatever they are implicit or not.K eywords: Option Pricing Models, Edgeworth Expansion, VolatilityForecast.J .E .L . C lassi…cation: G.10, G.12, G.13.
¤We thank Bernard Bensaïd, Rama Cont, J ean-Paul Laurent, T hierry Michel andChristophe V illa for their comments on earlier drafts of this paper. T he usual disclaimersapply. T his paper is a preliminary draft and the dynamic part of the paper is still in progress.yT EA M - ESA 8059 du CNRS - Paris IPanthéon-SorbonneUniversity.E-mail:gunther@univ-paris1.fr. Corresponding author: T EAM-M SE, 106-112 Bd de l’Hôpital 75647Paris Cedex 13 FRANCE. Tel: (33 1) 44 07 82 71/ 70 (facsimile).zT EA M - ESA 8059 du CNRS - Paris IPanthéon-SorbonneUniversity.E-mail: ejur-czenko@aol.com.xT EA M - ESA 8059 du CNRS - Paris IPanthéon-SorbonneUniversity,ESCP andA .A.Advisors (ABN-AMRO Group). E-mail: bmaillet@univ-paris1.fr.1
TheQualityofOptionPricesForecasts:ADynamicApproach
1 IntroductionThe Black and Scholes (1973) model is certainly the most used in economy.But its formula is somehow inconsistent with stylised empirical facts. Indeed,empirical studies often report that there are systematic errors between theo-retical prices computed with the Black and Scholes (1973) model, and pricesobserved on markets. In particular, the Black and Scholes (1973) model under-estimates deep out-of-the-money option and overestimates deep in-the-moneyoptions. Therefore, the implied volatility function, representing the volatilityimpliedbythetheoreticalformulaevaluatedatdi¤erentstrikes,isU-shaped:this is the well known phenomenon called “smile”.Thisresult isgenerally attributed totheirrealistichypothesisof log-normalityof the underlying asset return combine with a constant volatility assumption.Indeed,ifrareeventsaremorefrequentthanitissupposedintheGausssiancase, then the price of deep in-the-money options and deep out-of-the-moneyoptions will be higher than the Black and Scholes (1973) model predicts. Themisspeci…cation of the distribution function tails leads, therefore, to more im-portant implied volatilities for options whose strikes are far from the currentprice. Moreover, thedensity might not besymetrical and, oncemore, this leadsto a di¤erent implied volatility for options deeply in-the-money or deeply out-of-the-money. In this case, a “smirk” appears instead of the traditional smile.Toillustratethisphenomenon,weusetheBlackandScholes(1973)modeltocompute the implied volatility of the French index CAC 40 call options quotedon the MONEP1on the 14th, J anuary 1997. The term to expiration of theseoptionsisMarch1997.Figure1representstheimpliedvolatilityfunctiononthe y-axis whilst the moneyness is on the x-axis.2- Please, insert Figure 1 somewhere here -Inordertoavoidthisdrawback,someauthorshaveconsidereddi¤erentmodelizations of the process governing the underlying asset movements. Forinstance,ajumpprocesshadbeenchosenbyMerton(1976)andmorerecentlyby Bates (1996a, 1996b), whilst Hull and White (1987), Stein and Stein (1991)and Heston (1993) considered stochastic volatility models. Thus, under theassumption of no-correlation between volatility and underlying asset return,Renault and Touzi (1996) have shown that Black and Scholes (1973) pricingformulaoverestimatesat-the-money option pricesand underestimatestheotherswhen considering a di¤usion model. But this kind of approach is not perfectly1I.e.the French Derivatives M arket. M ONEP stands forMarché des Options Négociablesde Paris.£¡K e¡r ¿¡¢2T he moneyness is de…ned such as: [100S0=Ke¡r ¿]
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