Bayesian Option Pricing Using Mixed Normal Heteroskedasticity Models

While stochastic volatility models improve on the option pricing error when compared to the Black-Scholes-Merton model, mispricings remain. This paper uses mixed normal heteroskedasticity models to price options. Our model allows for significant negative skewness and time varying higher order moments of the risk neutral distribution. Parameter inference using Gibbs sampling is explained and we detail how to compute risk neutral predictive densities taking into account parameter uncertainty. When forecasting out-of-sample options on the S&P 500 index, substantial improvements are found compared to a benchmark model in terms of dollar losses and the ability to explain the smirk in implied volatilities.Bayesian inference, option pricing, finite mixture models, out-of-sample prediction, GARCH models

Multivariate Option Pricing with Time Varying Volatility and Correlations

In recent years multivariate models for asset returns have received much attention, in particular this is the case for models with time varying volatility. In this paper we consider models of this class and examine their potential when it comes to option pricing. Specifically, we derive the risk neutral dynamics for a general class of multivariate heteroskedastic models, and we provide a feasible way to price options in this framework. Our framework can be used irrespective of the assumed underlying distribution and dynamics, and it nests several important special cases. We provide an application to options on the minimum of two indices. Our results show that not only is correlation important for these options but so is allowing this correlation to be dynamic. Moreover, we show that for the general model exposure to correlation risk carries an important premium, and when this is neglected option prices are estimated with errors. Finally, we show that when neglecting the non-Gaussian features of the data, option prices are also estimated with large errors.Multivariate risk premia, option pricing, GARCH models

Bayesian Option Pricing Using Mixed Normal Heteroskedasticity Models

While stochastic volatility models improve on the option pricing error when compared to the Black-Scholes-Merton model, mispricings remain. This paper uses mixed normal heteroskedasticity models to price options. Our model allows for significant negative skewness and time varying higher order moments of the risk neutral distribution. Parameter inference using Gibbs sampling is explained and we detail how to compute risk neutral predictive densities taking into account parameter uncertainty. When forecasting out-of-sample options on the S&P 500 index, substantial improvements are found compared to a benchmark model in terms of dollar losses and the ability to explain the smirk in implied volatilities. Les modèles à volatilité stochastique apportent des améliorations en ce qui a trait à l’erreur d’établissement des prix des options comparativement au modèle de Black-Scholes-Merton. Toutefois, la fixation incorrecte des prix persiste. Le présent document a recours à des modèles mixtes avec hétéroscédasticité normale pour fixer les prix des options. Notre modèle permet de tenir compte de l’asymétrie négative importante et des moments d’ordre élevé variant dans le temps liés à la distribution du risque nul. Nous expliquons l’inférence des paramètres selon l’échantillonnage de Gibbs et détaillons la façon de traiter les densités prédictives de risque neutre en prenant en considération l’incertitude des paramètres. Dans le cas des prévisions concernant les options hors-échantillonnage sur l’indice S&P 500, nous constatons des améliorations importantes, par rapport à un modèle de référence, en termes de pertes exprimées en dollars et de capacité d’expliquer l’ironie des volatilités implicites.Bayesian inference, option pricing, finite mixture models, out-of-sample prediction, GARCH models, Inférence bayésienne, fixation du prix des options, modèles à mélanges finis, prédiction hors-échantillon, modèles GARCH.

Bayesian option pricing using mixed normal heteroskedasticity models

Bayesian inference, option pricing, finite mixture models, out-of-sample prediction, GARCH models

Option pricing with asymmetric heteroskedastic normal mixture models

This paper uses asymmetric heteroskedastic normal mixture models to fit return data and to price options. The models can be estimated straightforwardly by maximum likelihood, have high statistical fit when used on S&P 500 index return data, and allow for substantial negative skewness and time varying higher order moments of the risk neutral distribution. When forecasting out-of-sample a large set of index options between 1996 and 2009, substantial improvements are found compared to several benchmark models in terms of dollar losses and the ability to explain the smirk in implied volatilities. Overall, the dollar root mean squared error of the best performing benchmark component model is 39% larger than for the mixture model. When considering the recent financial crisis this difference increases to 69%.asymmetric heteroskadastic models, finite mixture models, option pricing, out-of- sample prediction, statistical fit

Multivariate option pricing with time varying volatility and correlations

In recent years multivariate models for asset returns have received much attention, in particular this is the case for models with time varying volatility. In this paper we consider models of this class and examine their potential when it comes to option pricing. Specifically, we derive the risk neutral dynamics for a general class of multivariate heteroskedastic models, and we provide a feasible way to price options in this framework. Our framework can be used irrespective of the assumed underlying distribution and dynamics, and it nests several important special cases. We provide an application to options on the minimum of two indices. Our results show that not only is correlation important for these options but so is allowing this correlation to be dynamic. Moreover, we show that for the general model exposure to correlation risk carries an important premium, and when this is neglected option prices are estimated with errors. Finally, we show that when neglecting the non-Gaussian features of the data, option prices are also estimated with large errors.multivariate risk premia, option pricing, GARCH models

Multivariate Option Pricing with Time Varying Volatility and Correlations

In recent years multivariate models for asset returns have received much attention, in particular this is the case for models with time varying volatility. In this paper we consider models of this class and examine their potential when it comes to option pricing. Specifically, we derive the risk neutral dynamics for a general class of multivariate heteroskedastic models, and we provide a feasible way to price options in this framework. Our framework can be used irrespective of the assumed underlying distribution and dynamics, and it nests several important special cases. We provide an application to options on the minimum of two indices. Our results show that not only is correlation important for these options but so is allowing this correlation to be dynamic. Moreover, we show that for the general model exposure to correlation risk carries an important premium, and when this is neglected option prices are estimated with errors. Finally, we show that when neglecting the non-Gaussian features of the data, option prices are also estimated with large errors

Multivariate Option Pricing With Time Varying Volatility and Correlations

In recent years multivariate models for asset returns have received much attention, in particular this is the case for models with time varying volatility. In this paper we consider models of this class and examine their potential when it comes to option pricing. Specifically, we derive the risk neutral dynamics for a general class of multivariate heteroskedastic models, and we provide a feasible way to price options in this framework. Our framework can be used irrespective of the assumed underlying distribution and dynamics, and it nests several important special cases. We provide an application to options on the minimum of two indices. Our results show that not only is correlation important for these options but so is allowing this correlation to be dynamic. Moreover, we show that for the general model exposure to correlation risk carries an important premium, and when this is neglected option prices are estimated with errors. Finally, we show that when neglecting the non-Gaussian features of the data, option prices are also estimated with large errors. Au cours des récentes années, les modèles multivariés utilisés pour évaluer les rendements de l’actif ont suscité beaucoup d’intérêt, plus particulièrement les modèles qui tiennent compte de la volatilité variant dans le temps. Dans le présent document, nous explorons les modèles de cette catégorie et examinons leur potentiel en matière de fixation du prix des options. Plus précisément, nous établissons la dynamique risque neutre pour une catégorie générale de modèles hétéroscédastiques à variables multiples et proposons un moyen réaliste de fixer le prix des options à l’intérieur de cette structure. Notre cadre de référence peut être utilisé sans égard à la distribution et la dynamique sous-jacentes possibles. Il prend également en compte de nombreux cas spéciaux importants. Nous proposons une application aux options selon un minimum de deux indices. Nos résultats révèlent non seulement l’importance de la corrélation en ce qui a trait à ces options, mais aussi l’importance d’une corrélation qui soit dynamique. De plus, nous illustrons, dans le cas du modèle général, que l’exposition au risque de corrélation comporte une prime importante et que, si cet aspect est négligé, l’évaluation du prix des options est alors erronée. Enfin, nous démontrons qu’en faisant peu de cas des caractéristiques non gaussiennes des données, l’évaluation du prix des options comporte des écarts importants.Multivariate risk premia, Option pricing, GARCH models, primes de risque à variables multiples, fixation du prix des options, modèles GARCH

Option Pricing with Asymmetric Heteroskedastic Normal Mixture Models

This paper uses asymmetric heteroskedastic normal mixture models to fit return data and to price options. The models can be estimated straightforwardly by maximum likelihood, have high statistical fit when used on S&P 500 index return data, and allow for substantial negative skewness and time varying higher order moments of the risk neutral distribution. When forecasting out-of-sample a large set of index options between 1996 and 2009, substantial improvements are found compared to several benchmark models in terms of dollar losses and the ability to explain the smirk in implied volatilities. Overall, the dollar root mean squared error of the best performing benchmark component model is 39% larger than for the mixture model. When considering the recent financial crisis this difference increases to 69%. Dans le présent document, nous avons recours aux modèles hétéroscédastiques asymétriques avec mélange de distributions normales pour ajuster les données sur les rendements et fixer les prix des options. Les modèles peuvent être estimés directement par le maximum de vraisemblance, ils comportent un ajustement statistique élevé quand ils sont utilisés sur les données de rendement de l’indice S&P 500, et ils permettent de tenir compte d’une asymétrie négative importante et des moments d’ordre élevé variant dans le temps liés à la distribution du risque nul. Dans le cas des prévisions hors-échantillonnage concernant une vaste gamme d’options sur indice entre 1996 et 2009, nous constatons des améliorations substantielles, par rapport à plusieurs modèles de référence, en termes de pertes exprimées en dollars et de capacité d’expliquer le caractère ironique des volatilités implicites. En général, la racine de l’erreur quadratique moyenne du modèle de référence à composantes le plus efficace est 39 % plus grande que dans le cas du modèle à mélange. Dans le contexte de la récente crise financière, cette différence augmente à 69 %.Asymmetric heteroskedastic models, finite mixture models, option pricing, out-of-sample prediction, statistical fit , modèles hétéroscédastiques asymétriques, modèle à mélanges finis, fixation des prix des options, prédiction hors-échantillonnage, ajustement statistique

Artikelskrivning som eksamensform

I denne artikel vil vi diskutere, hvordan studerendes arbejde med udformning af en videnskabelig artikel som afsæt for summativ evaluering af undervisningsmoduler, kan bidrage såvel til de studerendes oplevelse af motivation som til oplevelse af, at de gives muligheder for at udvikle konkrete aka­demiske kompetencer. Vi tager udgangspunkt i en række teoretiske over­vejelser, der knytter den summative evaluering til begreber som autentiske rammer, kompetenceudvikling og motivation. Disse overvejelser supplerer vi med resultaterne af en konkret undersøgelse af et pilotforsøg gen­nem­ført ved 2. semester på kandidatuddannelsen i Idræt på Aalborg Uni­versitet. I relation til to kursusmoduler har de studerende udarbejdet en videnskabelig artikel med afsæt i indsamlet empiri. Undersøgelsen viser, at de studerende i altovervejende grad er positive over for artikelskrivning som udgangspunkt for deres evaluering. Undersøgelsen viser videre, at de studerende oplever en følelse af motivation ved bl.a. at arbejde med selv­valgte problemstillinger. Hvorvidt de studerende faktisk udviklede kom­­petencer i forløbet, som de ikke kunne udvikle ved andre evalueringsformer, er fortsat ikke afdækket men vil være et naturligt næste genstandsfelt for undersøgelse.   In this paper we discuss how setting students a scientific paper to write as part of their course module summative assessment may help motivate students and contribute to their academic competencies. We depart from theoretical considerations that connect summative evaluation to concepts of authentic frameworks, competence development and motivation, and instead argue that writing a scientific paper as part of evaluation may be both motivational to students and provide a framework for developing competencies. We supplement these observations by presenting the results from a study conducted in a 2nd semester master programme in sport science at Aalborg University, Denmark. Based on two course modules and collected empirical data the students wrote a scientific paper. The study revealed that the majority of students found writing scientific papers as part of their evaluation a positive experience. Furthermore, the study showed that students found working with a self-selected problem motivational. Whether the student developed competencies in the process, which they could not develop through other forms of evaluation, remains unclear. This would be a natural focus for further research