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On the Dependence Structure Between S&P500, Vix and Implicit Interexpectile Differences
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... As part of these measures of skewness, interexpectile distances (also called interexpectile ranges) appear quite naturally; they have also been used in a finance context by Bellini et al. (2018bBellini et al. ( , 2020Bellini et al. ( , 2021. We introduce orders of variability based on slightly more general quantities and show that the so-called weak expectile dispersive order is equivalent to the dilation order. ...
... These scale measures obviously preserve the ≤ we−disp ordering, and, hence, also the dilation order; they have already appeared as a scaling factor in the definition of s 2 . Moreover, (implicit) interexpectile differences have been used to extract information about the risk-neutral distribution of a financial index by Bellini et al. (2018bBellini et al. ( , 2020. ...
... By definition, the interquantile range preserves the dispersive order. However, it is not consistent with the dilation order, which may be seen as a disadvantage in specific applications (Bellini et al. 2020). ...
Recently, expectile-based measures of skewness akin to well-known quantile-based skewness measures have been introduced, and it has been shown that these measures possess quite promising properties (Eberl and Klar in Comput Stat Data Anal 146:106939, 2020; Scand J Stat, 2021, https://doi.org/10.1111/sjos.12518). However, it remained unanswered whether they preserve the convex transformation order of van Zwet, which is sometimes seen as a basic requirement for a measure of skewness. It is one of the aims of the present work to answer this question in the affirmative. These measures of skewness are scaled using interexpectile distances. We introduce orders of variability based on these quantities and show that the so-called weak expectile dispersive order is equivalent to the dilation order. Further, we analyze the statistical properties of empirical interexpectile ranges in some detail.
... This paper presents an extensive comparison between option-implied quantile and optionimplied expectile curves, derived from the quotation of weekly out-of-the-money options on the S&P 500. Implicit quantile and expectile curves, introduced in Barone Adesi (2016), Barone Adesi et al. (2019b), and Bellini et al. (2019), are here computed through a fully nonparametric methodology, and thus are virtually insensitive to discretization and truncation error; the empirical results found on this dataset indeed demonstrate the viability of the approach. The option-implied implicit quantile and expectile curves fully determine the risk-neutral distribution, and can be used as a starting point for the for the computation of more complicated functionals. ...
We compute nonparametric and forward-looking option-implied quantile and expectile curves, and we study their properties on a 5-year dataset of weekly options written on the S&P 500 Index. After studying the dynamics of the single curves and their joint behaviour, we investigate the potentiality of these quantities for risk management and forecasting purposes. As an alternative form of variability mesaures, we compute option-implied interquantile and interexpectile differences, that are compared with a weekly VIX-like index. In terms of forecasting power we investigate how different quantities related to the implicit quantile and expectile curves predict future logreturns and future realized variances.
We show how to compute the expectiles of the risk-neutral distribution from the prices of European call and put options. Empirical properties of these implicit expectiles are studied on a data-set of closing daily prices of FTSE MIB index options. We introduce the interexpectile difference , for , and suggest that it is a natural measure of the variability of the risk-neutral distribution. We investigate its theoretical and empirical properties and compare it with the VIX index computed by CBOE. We also discuss a theoretical comparison with implicit VaR and CVaR introduced in Barone Adesi [J. Risk Financ. Manage., 2016, 9].
In this paper we introduce the expectile order, defined by X ≤eY if eα(X) ≤eα(Y) for each α ∈ (0, 1), where eα denotes the α-expectile. We show that the expectile order is equivalent to the pointwise ordering of the Omega ratios, and we derive several necessary and sufficient conditions. In the case of equal means, the expectile order can be easily characterized by means of the stop-loss transform; in the more general case of different means we provide some sufficient conditions. In contrast with the more common stochastic orders such as ≤st and ≤cx, the expectile order is not generated by a class of utility functions and is not closed with respect to convolutions. As an illustration, we compare the ≤st, ≤icx and ≤e orders in the family of Lomax distributions and compare Lomax distributions fitted to real world data of natural disasters in the U.S. caused by different sources of weather risk like storms or floods.
For nearly every major stock market there exist equity and implied volatility indices. These play important roles within finance: be it as a benchmark, a measure of general uncertainty or a way of investing or hedging. It is well known in the academic literature, that correlations and higher moments between different indices tend to vary in time. However, to the best of our knowledge, no one has yet considered a global setup including both, equity and implied volatility indices of various continents, and allowing for a changing dependence structure. We aim to close this gap by applying Markov-switching R-vine models to investigate the existence of different, global dependence regimes. In particular, we identify times of "normal" and "abnormal" states within a data set consisting of North-American, European and Asian indices. Our results confirm the existence of joint points in time at which global regime switching takes place.
1. What are Copulas? 2. Which Rules for Handling Copulas Do I Need? 3. How to Measure Dependence? 4. What are Popular Families or Copulas? 5. How to Stimulate Multivariate Distributions? 6. How to Estimate Parameters of a Multivariate Model? 7. How to Deal with Uncertainty Concerning Dependence? 8. How to Construct a Portfolio-Default Model?
This study investigates the forecasting power of implied volatility indices on forward looking returns. Prior studies document that negative innovations to returns are associated with increasing implied volatility of the underlying indices; thus, suggesting a possible relationship between extremely high levels of implied volatility and positive short term returns. We investigate this issue by examining the predictive power of three implied volatility indices, VIX, VXN and VDAX, on the underlying index returns. We extend previous research by also focusing on characterised selected stocks and examine the relationship between implied volatility indices and future returns across different sectors and classified portfolios. Our findings suggest that implied volatility indices are good predictors of 20-days and 60-days forward looking returns and illustrate insignificant predictive power for very short term (1-day and 5-days) returns.
Prior studies find that the CBOE volatility index (VIX) predicts returns on stock market indices, suggesting implied volatilities measured by VIX are a risk factor affecting security returns or an indicator of market inefficiency. We extend prior work in three important ways. First, we investigate the relationship between future returns and current implied volatility levels and innovations. Second, we examine portfolios sorted on book-to-market equity, size, and beta. Third, we control for the four Fama and French [Fama, E., French, K., 1993. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, 3–56.] and Carhart [Carhart, M., 1997. On persistence in mutual fund performance. Journal of Finance, 52, 57–82.] factors. We find that VIX-related variables have strong predictive ability.
In the statistical literature many generalizations of quantiles have been considered so far. Some of these generalized quantiles are natural candidates for risk measures that could be of interest in actuarial science as well. In this paper we investigate in particular the case of M-quantiles. It turns out that they share many properties with risk measures and that the special case of a so called expectile leads to an interesting coherent risk measure that is investigated here in detail. Expectiles can also been seen as an unusual special case of a zero utility premium principle with a non differentiable utility function. Robustness properties of expectiles and their relation to classical quantiles are also considered.