Empirical Assessment of an Intertemporal Option Pricing Model with Latent variables

This paper assesses the empirical performance of an intertemporal option pricing model with latent variables which generalizes the Hull-White stochastic volatility formula. Using this generalized formula in an ad-hoc fashion to extract two implicit parameters and forecast next day S&P 500 option prices, we obtain similar pricing errors than with implied volatility alone as in the Hull-White case. When we specialize this model to an equilibrium recursive utility model, we show through simulations that option prices are more informative than stock prices about the structural parameters of the model. We also show that a simple method of moments with a panel of option prices provides good estimates of the parameters of the model. This lays the ground for an empirical assessment of this equilibrium model with S&P 500 option prices in terms of pricing errors.

Risk Aversion, Intertemporal Substitution, and Option Pricing

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

Asymmetric Smiles, Leverage Effects and Structural Parameters.

In this paper, we characterize the asymmetries of the smile through multiple leverage effects in a stochastic dynamic asset pricing framework. The dependence between price movements and future volatility is introduced through a set of latent state variables. These latent variables can capture not only the volatility risk and the interest rate risk which potentially affect option prices, but also any kind of correlation risk and jump risk. The standard financial leverage effect is produced by a cross-correlation effect between the state variables which enter into the stochastic volatility process of the stock price and the stock price process itself. However, we provide a more general framework where asymmetric implied volatility curves result from any source of instantaneous correlation between the state variables and either the return on the stock or the stochastic discount factor. In order to draw the shapes of the implied volatility curves generated by a model with latent variables, we specify an equilibrium-based stochastic discount factor with time non-separable preferences. When we calibrate this model to empirically reasonable values of the parameters, we are able to reproduce the various types of implied volatility curves inferred from option market data.

Stochastic Volatility.

This paper prepared for the Handbook of Statistics (Vol.14: Statistical Methods in Finance), surveys the subject of stochastic volatility. the following subjects are covered: volatility in financial markets (instantaneous volatility of asset returns, implied volatilities in option prices and related stylized facts), statistical modelling in discrete and continuous time and, finally, statistical inference (methods of moments, quasi-maximum likelihood, likelihood-based and bayesian methods and indirect inference).

Trois essais sur la liquidité : ses effets sur les primes de risque, les anticipations et l'asymétrie des risques financiers

Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal

Modélisation du carnet d'ordres limites et prévision de séries temporelles

Le contenu de cette thèse est divisé de la façon suivante. Après un premier chapitre d’introduction, le Chapitre 2 est consacré à introduire aussi simplement que possible certaines des théories qui seront utilisées dans les deux premiers articles. Dans un premier temps, nous discuterons des points importants pour la construction de l’intégrale stochastique par rapport aux semimartingales avec paramètre spatial. Ensuite, nous décrirons les principaux résultats de la théorie de l’évaluation en monde neutre au risque et, finalement, nous donnerons une brève description d’une méthode d’optimisation connue sous le nom de dualité. Les Chapitres 3 et 4 traitent de la modélisation de l’illiquidité et font l’objet de deux articles. Le premier propose un modèle en temps continu pour la structure et le comportement du carnet d’ordres limites. Le comportement du portefeuille d’un investisseur utilisant des ordres de marché est déduit et des conditions permettant d’éliminer les possibilités d’arbitrages sont données. Grâce à la formule d’Itô généralisée il est aussi possible d’écrire la valeur du portefeuille comme une équation différentielle stochastique. Un exemple complet de modèle de marché est présenté de même qu’une méthode de calibrage. Dans le deuxième article, écrit en collaboration avec Bruno Rémillard, nous proposons un modèle similaire mais cette fois-ci en temps discret. La question de tarification des produits dérivés est étudiée et des solutions pour le prix des options européennes de vente et d’achat sont données sous forme explicite. Des conditions spécifiques à ce modèle qui permettent d’éliminer l’arbitrage sont aussi données. Grâce à la méthode duale, nous montrons qu’il est aussi possible d’écrire le prix des options européennes comme un problème d’optimisation d’une espérance sur en ensemble de mesures de probabilité. Le Chapitre 5 contient le troisième article de la thèse et porte sur un sujet différent. Dans cet article, aussi écrit en collaboration avec Bruno Rémillard, nous proposons une méthode de prévision des séries temporelles basée sur les copules multivariées. Afin de mieux comprendre le gain en performance que donne cette méthode, nous étudions à l’aide d’expériences numériques l’effet de la force et la structure de dépendance sur les prévisions. Puisque les copules permettent d’isoler la structure de dépendance et les distributions marginales, nous étudions l’impact de différentes distributions marginales sur la performance des prévisions. Finalement, nous étudions aussi l’effet des erreurs d’estimation sur la performance des prévisions. Dans tous les cas, nous comparons la performance des prévisions en utilisant des prévisions provenant d’une série bivariée et d’une série univariée, ce qui permet d’illustrer l’avantage de cette méthode. Dans un intérêt plus pratique, nous présentons une application complète sur des données financières.

La révalorisation des friches industrielles: une approche option réelle.

Rapport de recherche

L'évaluation par option réelle: une application au secteur du cuivre.

Rapport de recherche

Latent Variable Models for Stochastic Discount Factors.

Latent variable models in finance originate both from asset pricing theory and time series analysis. These two strands of literature appeal to two different concepts of latent structures, which are both useful to reduce the dimension of a statistical model specified for a multivariate time series of asset prices. In the CAPM or APT beta pricing models, the dimension reduction is cross-sectional in nature, while in time-series state-space models, dimension is reduced longitudinally by assuming conditional independence between consecutive returns, given a small number of state variables. In this paper, we use the concept of Stochastic Discount Factor (SDF) or pricing kernel as a unifying principle to integrate these two concepts of latent variables. Beta pricing relations amount to characterize the factors as a basis of a vectorial space for the SDF. The coefficients of the SDF with respect to the factors are specified as deterministic functions of some state variables which summarize their dynamics. In beta pricing models, it is often said that only the factorial risk is compensated since the remaining idiosyncratic risk is diversifiable. Implicitly, this argument can be interpreted as a conditional cross-sectional factor structure, that is, a conditional independence between contemporaneous returns of a large number of assets, given a small number of factors, like in standard Factor Analysis. We provide this unifying analysis in the context of conditional equilibrium beta pricing as well as asset pricing with stochastic volatility, stochastic interest rates and other state variables. We address the general issue of econometric specifications of dynamic asset pricing models, which cover the modern literature on conditionally heteroskedastic factor models as well as equilibrium-based asset pricing models with an intertemporal specification of preferences and market fundamentals. We interpret various instantaneous causality relationships between state variables and market fundamentals as leverage effects and discuss their central role relative to the validity of standard CAPM-like stock pricing and preference-free option pricing.

The Effect of Agricultural Zoning on Land Prices, Quebec, 1975-1981

Dans Ce Texte Nous Examinons les Effets de la Loi du Zonage Agricole du Quebec, Proclame En Decembre 1978 Sur le Prix du Sol Dans une Banlieu de Montreal. a L'aide de Donnees Sur les Transactions Normales Faites a Carignan et Saint-Mathias de 1975 a 1981, Nous Estimons, a L'aide des Moindres Carrees Ordinaires, une Equation de Determination du Prix Par Acre Avec Comme Variables Independantes la Dimension du Lot, la Distance de Montreal, les Services Disponibles (Egouts,...) et le Zonage Agricole (Ou Non) du Sol. Nos Resultats Nous Indiquent Que le Zonage Agricole Reduit le Prix D'un Acre de Sol.

Explaining the Persistence of Commodity Prices

This paper extends the Competitive Storage Model by incorporating prominent features of the production process and financial markets. A major limitation of this basic model is that it cannot successfully explain the degree of serial correlation observed in actual data. The proposed extensions build on the observation that in order to generate a high degree of price persistence, a model must incorporate features such that agents are willing to hold stocks more often than predicted by the basic model. We therefore allow unique characteristics of the production and trading mechanisms to provide the required incentives. Specifically, the proposed models introduce (i) gestation lags in production with heteroskedastic supply shocks, (ii) multiperiod forward contracts, and (iii) a convenience return to inventory holding. The rational expectations solutions for twelve commodities are numerically solved. Simulations are then employed to assess the effects of the above extensions on the time series properties of commodity prices. Results indicate that each of the features above partially account for the persistence and occasional spikes observed in actual data. Evidence is presented that the precautionary demand for stocks might play a substantial role in the dynamics of commodity prices.

Sampling Interval and estimated Betas : Implications for the Presence of Transitory Components in Stock Prices

We provide a theoretical framework to explain the empirical finding that the estimated betas are sensitive to the sampling interval even when using continuously compounded returns. We suppose that stock prices have both permanent and transitory components. The permanent component is a standard geometric Brownian motion while the transitory component is a stationary Ornstein-Uhlenbeck process. The discrete time representation of the beta depends on the sampling interval and two components labelled \"permanent and transitory betas\". We show that if no transitory component is present in stock prices, then no sampling interval effect occurs. However, the presence of a transitory component implies that the beta is an increasing (decreasing) function of the sampling interval for more (less) risky assets. In our framework, assets are labelled risky if their \"permanent beta\" is greater than their \"transitory beta\" and vice versa for less risky assets. Simulations show that our theoretical results provide good approximations for the means and standard deviations of estimated betas in small samples. Our results can be perceived as indirect evidence for the presence of a transitory component in stock prices, as proposed by Fama and French (1988) and Poterba and Summers (1988).