Volatility smile

Résumé

Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices (and thus implied volatilities) than what is suggested by standard option pricing models. These options are said to be either deep in-the-money or out-of-the-money. Graphing implied volatilities against strike prices for a given expiry produces a skewed "smile" instead of the expected flat surface. The pattern differs across various markets. Equity options traded in American markets did not show a volatility smile before the Crash of 1987 but began showing one afterwards. It is believed that investor reassessments of the probabilities of fat-tail have led to higher prices for out-of-the-money options. This anomaly implies deficiencie

Source officielle

https://en.wikipedia.org/wiki/Volatility_smile

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En finance, une option est un produit dérivé qui établit un contrat entre un acheteur et un vendeur. L'acheteur de l'option obtient le droit, et non pas l'obligation, d'acheter (call) ou de vendre (pu

Le modèle de Black-Scholes est utilisé pour désigner deux concepts très proches :

le modèle Black-Scholes ou modèle Black-Scholes-Merton qui est un modèle mathématique du marché pour une action, dan

La volatilité (en finance) est l'ampleur des variations du cours d'un actif financier. Elle sert de paramètre de quantification du risque de rendement et de prix d'un actif financier. Lorsque la vol

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An Option Pricing Formula for the GARCH Diffusion Model

Giovanni Barone-Adesi, Claudia Ravanelli

We derive analytically the first four conditional moments of the integrated variance implied by the GARCH diffusion process. From these moments we obtain an analytical closed-form approximation formula to price European options under the GARCH diffusion model. Using Monte Carlo simulations, we show that this approximation formula is accurate for a large set of reasonable parameters. Finally, we use the closed-form option pricing solution to shed light on the qualitative properties of implied volatility surfaces induced by GARCH diffusion models.

Elsevier2005

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A GARCH Option Pricing Model with Filtered Historical Simulation

Giovanni Barone-Adesi, Loriano Mancini

We propose a new method for pricing options based on GARCH models with filtered historical innovations. In an incomplete market framework, we allow for different distributions of historical and pricing return dynamics enhancing the model flexibility to fit market option prices. An extensive empirical analysis based on S&P 500 index options shows that our model outperforms other competing GARCH pricing models and ad hoc Black-Scholes models. We show that the flexible change of measure, the asymmetric GARCH volatility and the nonparametric innovation distribution induce the accurate pricing performance of our model. Using a nonparametric approach, we obtain decreasing state price densities per unit probability as suggested by economic theory and corroborating our GARCH pricing model. Implied volatility smiles appear to be explained by asymmetric volatility and negative skewness of filtered historical innovations.

2008

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This thesis focuses on non-parametric covariance estimation for random surfaces, i.e.~functional data on a two-dimensional domain. Non-parametric covariance estimation lies at the heart of functional data analysis, andconsiderations of statistical and computational efficiency often compel the use of separability of the covariance, when working with random surfaces. We seek to provide efficient alternatives to this ambivalent assumption.In Chapter 2, we study a setting where the covariance structure may fail to be separable locally -- either due to noise contamination or due to the presence of a non-separable short-range dependent signal component. That is, the covariance is an additive perturbation of a separable component by a non-separable but banded component. We introduce non-parametric estimators hinging on shifted partial tracing -- a novel concept enjoying strong denoising properties. We illustrate the usefulness of the proposed methodology on a data set of mortality surfaces.In Chapter 3, we propose a distinctive decomposition of the covariance, which allows us to understand separability as an unconventional form of low-rankness. From this perspective, a separable covariance has rank one. Allowing for a higher rank suggests a structured class in which any covariance can be approximated up to an arbitrary precision. The key notion of the partial inner product allows us to generalize the power iteration method to general Hilbert spaces and estimate the aforementioned decomposition from data. Truncation and retention of the leading terms automatically induces a non-parametric estimator of the covariance, whose parsimony is dictated by the truncation level. Advantages of this approach, allowing for estimation beyond separability, are demonstrated on the task of classification of EEG signals.While Chapters 2 and 3 propose several generalizations of separability in the densely sampled regime, Chapter 4 deals with the sparse regime, where the latent surfaces are observed only at few irregular locations. Here, a separable covariance estimator based on local linear smoothers is proposed, which is the first non-parametric utilization of separability in the sparse regime. The assumption of separability reduces the intrinsically four-dimensional smoothing problem into several two-dimensional smoothers and allows the proposed estimator to retain the classical minimax-optimal convergence rate for two-dimensional smoothers. The proposed methodology is used for a qualitative analysis of implied volatility surfaces corresponding to call options, and for prediction of the latent surfaces based on information from the entire data set, allowing for uncertainty quantification. Our quantitative results show that the proposed methodology outperforms the common approach of pre-smoothing every implied volatility surface separately.Throughout the thesis, we put emphasis on computational aspects, since those are the main reason behind the immense popularity of separability. We show that the covariance structures of Chapters 2 and 3 come with no (asymptotic) computational overhead relative to assuming separability. In fact, the proposed covariance structures can be estimated and manipulated with the same asymptotic costs as the separable model. In particular, we develop numerical algorithms that can be used for efficient inversion, as required e.g.~for prediction. All the methods are implemented in R and available on~GitHub.

EPFL2022

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EPFL2022