Implicit probability distribution for WTI options: The Black Scholes vs. the semi-nonparametric approach

Este documento contribuye a la literatura sobre la estimación de la función de Densidad de Riesgo Neutral (RND) modelando los precios de las opciones del crudo West Texas Intermediate (WTI) que se comercializaron en el período comprendido entre enero de 2016 y enero de 2017. Para estas series, se extrae la RND implícita en los precios de las opciones aplicando el modelo tradicional Black & Scholes (1973) y el modelo semi-no paramétrico (SNP) propuesto por Backus, Foresi, Li y Wu (1997). Los resultados obtenidos muestran que cuando el precio promedio del mercado se compara con el precio teórico promedio, la especificación lognormal tiende a subestimar sistemáticamente la estimación. Por el contrario, el modelo de fijación de precios SNP, que se ajusta explícitamente a la asimetría negativa y al exceso de curtosis, da como resultado una precisión marcadamente mejorada.This paper contributes to the literature on the estimation of the Risk Neutral Density (RND) function by modeling the prices of options for West Texas Intermediate (WTI) crude oil that were traded in the period between January 2016 and January 2017. For these series we extract the implicit RND in the option prices by applying the traditional Black & Scholes (1973) model and the semi-nonparametric (SNP) model proposed by Backus, Foresi, Li, & Wu (1997). The results obtained show that when the average market price is compared to the average theoretical price, the lognormal specification tends to systematically undervalue the estimation. On the contrary, the SNP option pricing model, which explicitly adjust for negative skewness and excess kurtosis, results in markedly improved accuracy

Multivariate Approximations to Portfolio Return Distribution

This article proposes a three-step procedure to estimate portfolio return distributions under the multivariate Gram-Charlier (MGC) distribution. The method combines quasi maximum likelihood (QML) estimation for conditional means and variances and the method of moments (MM) estimation for the rest of the density parameters, including the correlation coefficients. The procedure involves consistent estimates even under density misspecification and solves the so-called ‘curse of dimensionality’ of multivariate modelling. Furthermore, the use of a MGC distribution represents a flexible and general approximation to the true distribution of portfolio returns and accounts for all its empirical regularities. An application of such procedure is performed for a portfolio composed of three European indices as an illustration. The MM estimation of the MGC (MGC-MM) is compared with the traditional maximum likelihood of both the MGC and multivariate Student’s t (benchmark) densities. A simulation on Value-at-Risk (VaR) performance for an equally weighted portfolio at 1% and 5% confidence indicates that the MGC-MM method provides reasonable approximations to the true empirical VaR. Therefore, the procedure seems to be a useful tool for risk managers and practitioners

Statistical analysis of some financial time series models

The aim of this dissertation is to study the dynamics of asset returns under both the physical measure and the risk neutral measure. It consists of two different research topics. The first topic is primarily concerned with a specific class of volatility component models. This family of models have received much attention recently, not only because of their ability to capture complex dynamics via a parsimonious parameter structure, but also because it is believed that they can handle well structural breaks or non-stationarity in asset price volatility. The first part of the dissertation focuses on their probabilistic properties and statistical inference on these models is discussed as well. The second topic pertains to the distributional approximations of risk neutral distribution of asset returns for the purpose of option pricing. Risk neutral measures are a key ingredient of financial derivative pricing. Much effort has been devoted to characterizing the risk neutral distribution pertaining to the underlying asset. The rest of the dissertation studies the Generalized Hyperbolic family of distributions and examines their applications in option pricing

Gram-Charlier Processes and Applications to Option Pricing

A Gram-Charlier distribution has a density that is a polynomial times a normal density. For option pricing this retains the tractability of the normal distribution while allowing nonzero skewness and excess kurtosis. Properties of the Gram-Charlier distributions are derived, leading to the definition of a process with independent Gram-Charlier increments, as well as formulas for option prices and their sensitivities. A procedure for simulating Gram-Charlier distributions and processes is given. Numerical illustrations show the effect of skewness and kurtosis on option prices

Enfoque semi-noparamétrico para la medición de variables positivas de colas pesadas en los campos de la economía y las finanzas

[ES]En diferentes campos del conocimiento, los seres humanos se han interesado por analizar el comportamiento de diferentes fenómenos con el fin de comprenderlos y anticiparse al futuro. Al respecto, la modelización adecuada de las variables que determinan un fenómeno es fundamental. Como metodologías de análisis, la caracterización de una variable aleatoria mediante su función de densidad (pdf) y su ajuste a la distribución empírica de una serie puede realizarse mediante distintos enfoques que van desde una perspectiva paramétrica basada en una distribución de frecuencias con forma funcional conocida a un enfoque no paramétrico. En esta tesis se considera un enfoque semi-noparamétrico, en donde la función de distribución de probabilidad desconocida se modeliza a partir de una expansión de series de polinomios ortogonales. Se estudian las expansiones de Gram-Charlier y en particular, se propone el uso de una distribución log semi-noparamétrica (log-SNP) que anida a la lognormal. Se muestra que la distribución log-SNP permite mejoras de ajuste significativas al modelizar variables económicas y financieras puesto que permite la incorporación de parámetros adicionales a los de una distribución paramétrica tradicional como la lognormal

Implied Volatility: A theoretical study on explaining the stylized facts of implied volatility using the utility indifference model.

The breakthrough Black-Scholes (BS) model predicts a horizontal line when plotting the implied volatility (IV) against the strike price. However, empirical studies uncovered that the implied volatility derived from option market prices in the BS model varies with strike prices and time to maturity, leading to the identification of three stylized facts that the BS model fails to explain. First, the IV curves exhibit a smile/smirk pattern, with an upward-sloping term structure for at-the-money options. Second, option prices tend to reflect higher implied volatility compared to the realized volatility of asset returns. Third, the negative skewness implied by options prices is greater in absolute terms compared with the realized skewness. Consequently, numerous sophisticated models have been developed to address these stylized facts. Nevertheless, traditional models often fall short of fully explaining all aspects of these phenomena. This study introduces a novel approach to the utility indifference model by incorporating behavioral utility functions to provide a more accurate representation of these anomalies. To evaluate the model’s performance, the standard function used in expected utility theory and behavioral utility functions are tested under both normal and Normal Inverse Gaussian (NIG) distributions. The findings indicate that the conventional utility function fails to capture the observed smirk patterns. In contrast, the behavioral utility function generates the IV smirks that closely align with empirical shapes, even under the normal distribution. These results highlight the effectiveness of the utility indifference model with behavioral utility functions in explaining these stylized facts that standard models struggle to reproduce

iCOS: Option-Implied COS Method

This paper proposes the option-implied Fourier-cosine method, iCOS, for non-parametric estimation of risk-neutral densities, option prices, and option sensitivities. The iCOS method leverages the Fourier-based COS technique, proposed by Fang and Oosterlee (2008), by utilizing the option-implied cosine series coefficients. Notably, this procedure does not rely on any model assumptions about the underlying asset price dynamics, it is fully non-parametric, and it does not involve any numerical optimization. These features make it rather general and computationally appealing. Furthermore, we derive the asymptotic properties of the proposed non-parametric estimators and study their finite-sample behavior in Monte Carlo simulations. Our empirical analysis using S&P 500 index options and Amazon equity options illustrates the effectiveness of the iCOS method in extracting valuable information from option prices under different market conditions

Essays on Asset Pricing Using Option-Implied Information

The forward-looking nature of the options market makes it an ideal environment for investigating the determinants and the information content of investors' expectations about the future. Therefore, this thesis explores the interrelations arising between the macroeconomic and stock market environment, and the S&P 500 index options market. First, we examine how investors' sentiment driven by macroeconomic fundamentals and investors' erroneous beliefs impact the risk-neutral skewness. Our findings reveal that the macroeconomic fundamentals component of investor sentiment is the main driving force of risk neutral skewness throughout the whole sample period, while the error in investors' beliefs has limited explanatory power and only during the earlier years examined. Moreover, we show that the fundamentals component of investor sentiment affects differently the prices of call and put options. Second, we extend the concept of risk-neutral skewness by creating measures of forward skewness and gauge their predictive ability for a wide range of macroeconomic variables, asset prices, as well as systemic risk, crash risk, and uncertainty variables. Overall, we document that forward skewness encapsulates important information about future macroeconomic and financial market conditions for horizons up to one year ahead over and above forward variance. Third, we propose a novel measure of dispersion in expectations that is derived from the dispersion of options' trading volume across strike prices. We show that dispersion consistently forecasts negative excess market returns, for horizons up to two years ahead, exhibiting a predictive ability comparable to that of the variance risk premium and outperforming all other variables considered. This thesis contributes to the asset pricing and macro-finance literature by unravelling the determinants of the pricing kernel, showing that the call and put options markets are segmented and revealing that option prices and trading volume have significant forecasting ability for many aspects of the macroeconomic and financial environment. In that respect our findings are of particular interest not only to academics but also to investors and policy makers