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Option Valuation with Conditional Heteroskedasticity and Non-Normality

  • Peter Christoffersen
  • Redouane Elkamhi
  • Bruno Feunou
  • Kris Jacobs

We provide results for the valuation of European style contingent claims for a large class of specifications of the underlying asset returns. Our valuation results obtain in a discrete time, infinite state-space setup using the no-arbitrage principle and an equivalent martingale measure. Our approach allows for general forms of heteroskedasticity in returns, and valuation results for homoskedastic processes can be obtained as a special case. It also allows for conditional non-normal return innovations, which is critically important because heteroskedasticity alone does not suffice to capture the option smirk. We analyze a class of equivalent martingale measures for which the resulting risk-neutral return dynamics are from the same family of distributions as the physical return dynamics. In this case, our framework nests the valuation results obtained by Duan (1995) and Heston and Nandi (2000) by allowing for a time-varying price of risk and non-normal innovations. We provide extensions of these results to more general equivalent martingale measures and to discrete time stochastic volatility models, and we analyze the relation between our results and those obtained for continuous time models. Nous présentons les résultats d'une étude portant sur l'évaluation de créances éventuelles de style européen pour une grande variété de caractéristiques liées au rendement des actifs sous-jacents. Les résultats de notre évaluation proposent en temps discret une formule état-espace infinie, à partir du principe de non-arbitrage et d'une mesure de martingale équivalente. Notre approche permet de tenir compte de formes générales d'hétéroscédasticité dans les rendements et d'obtenir, dans des cas spéciaux, des résultats d'évaluation liés aux processus homoscédastiques. Elle permet aussi de considérer les innovations conditionnellement non normales en matière de rendement, ce qui représente un facteur critique, compte tenu du fait que l'hétéroscédasticité ne permet pas, à elle seule, de saisir pleinement le caractère ironique de l'option. Nous analysons une catégorie de mesures de martingale équivalentes dont la dynamique du rendement risque-neutre obtenu est de la même famille de distribution que la dynamique du rendement physique. Dans ce cas, notre cadre d'étude soutient les résultats d'évaluation obtenus par Duan (1995) et par Heston et Nandi (2000) et tient compte du coût du risque variant dans le temps et des innovations non normales. Nous étendons ces résultats aux mesures de martingale équivalentes plus générales et aux modèles de volatilité stochastique en temps discret et analysons aussi la relation entre nos résultats et ceux obtenus dans le cas des modèles en temps continu.

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Paper provided by CIRANO in its series CIRANO Working Papers with number 2009s-32.

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Length: 53 pages
Date of creation: 01 Aug 2009
Date of revision:
Handle: RePEc:cir:cirwor:2009s-32
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