Alternative statistical distributions for estimating value-at-risk: theory and evidence

A number of applications presume that asset returns are normally distributed, even though they are widely known to be skewed leptokurtic and fat-tailed and excess kurtosis. This leads to the underestimation or overestimation of the true value-at-risk (VaR). This study utilizes a composite trapezoid rule, a numerical integral method, for estimating quantiles on the skewed generalized t distribution (SGT) which permits returns innovation to flexibly treat skewness, leptokurtosis and fat tails. Daily spot prices of the thirteen stock indices in North America, Europe and Asia provide data for examining the one-day-ahead VaR forecasting performance of the GARCH model with normal, student??s t and SGT distributions. Empirical results indicate that the SGT provides a good fit to the empirical distribution of the log-returns followed by student??s t and normal distributions. Moreover, for all confidence levels, all models tend to underestimate real market risk. Furthermore, the GARCH-based model, with SGT distributional setting, generates the most conservative VaR forecasts followed by student??s t and normal distributions for a long position. Consequently, it appears reasonable to conclude that, from the viewpoint of accuracy, the influence of both skewness and fat-tails effects (SGT) is more important than only the effect of fat-tails (student??s t) on VaR estimates in stock markets for a long position.     

Robust estimation with flexible parametric distributions: estimation of utility stock betas

The distributions of stock returns and capital asset pricing model (CAPM) regression residuals are typically characterized by skewness and kurtosis. We apply four flexible probability density functions (pdfs) to model possible skewness and kurtosis in estimating the parameters of the CAPM and compare the corresponding estimates with ordinary least squares (OLS) and other symmetric distribution estimates. Estimation using the flexible pdfs provides more efficient results than OLS when the errors are non-normal and similar results when the errors are normal. Large estimation differences correspond to clear departures from normality. Our results show that OLS is not the best estimator of betas using this type of data. Our results suggest that the use of OLS CAPM betas may lead to erroneous estimates of the cost of capital for public utility stocks.     

SKEWNESS AND KURTOSIS IN S&P 500 INDEX RETURNS IMPLIED BY OPTION PRICES

The Black-Scholes (1973) model frequently misprices deep-in-the-money and deep-out-of-the-money options. Practitioners popularly refer to these strike price biases as volatility smiles. In this paper we examine a method to extend the Black-Scholes model to account for biases induced by nonnormal skewness and kurtosis in stock return distributions. The method adapts a Gram-Charlier series expansion of the normal density function to provide skewness and kurtosis adjustment terms for the Black-Scholes formula. Using this method, we estimate option-implied coefficients of skewness and kurtosis in S&P 500 stock index returns. We find significant nonnormal skewness and kurtosis implied by option prices.     

Asymmetric and leptokurtic distribution for heteroscedastic asset returns: The SU-normal distribution

This paper proposes the S_U-normal distribution to describe non-normality features embedded in financial time series, such as: asymmetry and fat tails. Applying the S_U-normal distribution to the estimation of univariate and multivariate GARCH models, we test its validity in capturing asymmetry and excess kurtosis of heteroscedastic asset returns. We find that the S_U-normal distribution outperforms the normal and Student-t distributions for describing both the entire shape of the conditional distribution and the extreme tail shape of daily exchange rates and stock returns. The goodness-of-fit (GoF) results indicate that the skewness and excess kurtosis are better captured by the S_U-normal distribution. The exceeding ratio (ER) test results indicate that the S_U-normal is superior to the normal and Student-t distributions, which consistently underestimate both the lower and upper extreme tails, and tend to overestimate the lower tail in general.     

Implied volatility skews and stock return skewness and kurtosis implied by stock option prices

The Black-Scholes* option pricing model is commonly applied to value a wide range of option contracts. However, the model often inconsistently prices deep in-the-money and deep out-of-the-money options. Options professionals refer to this well-known phenomenon as a volatility ‘skew’ or ‘smile’. In this paper, we examine an extension of the Black-Scholes model developed by Corrado and Su that suggests skewness and kurtosis in the option-implied distributions of stock returns as the source of volatility skews. Adapting their methodology, we estimate option-implied coefficients of skewness and kurtosis for four actively traded stock options. We find significantly nonnormal skewness and kurtosis in the option-implied distributions of stock returns.     

The role of autoregressive conditional skewness and kurtosis in the estimation of conditional VaR

This paper investigates the role of high-order moments in the estimation of conditional value at risk (VaR). We use the skewed generalized t distribution (SGT) with time-varying parameters to provide an accurate characterization of the tails of the standardized return distribution. We allow the high-order moments of the SGT density to depend on the past information set, and hence relax the conventional assumption in conditional VaR calculation that the distribution of standardized returns is iid. The maximum likelihood estimates show that the time-varying conditional volatility, skewness, tail-thickness, and peakedness parameters of the SGT density are statistically significant. The in-sample and out-of-sample performance results indicate that the conditional SGT-GARCH approach with autoregressive conditional skewness and kurtosis provides very accurate and robust estimates of the actual VaR thresholds.     

On the distribution and conditional heteroscedasticity in Taiwan stock prices

This paper studies the distribution and conditional heteroscedasticity in stock returns on the Taiwan stock market. Apart from the normal distribution, in order to explain the leptokurtosis and skewness observed in the stock return distribution, we also examine the Student-t, the Poisson–normal, and the mixed-normal distributions, which are essentially a mixture of normal distributions, as conditional distributions in the stock return process. We also use the ARMA (1,1) model to adjust the serial correlation, and adopt the GJR–generalized autoregressive conditional heteroscedasticity (GARCH (1,1)) model to account for the conditional heterscedasticity in the return process. The empirical results show that the mixed–normal–GARCH model is the most probable specification for Taiwan stock returns. The results also show that skewness seems to be diversifiable through portfolio. Thus the normal–GARCH or the Student-t–GARCH model which involves symmetric conditional distribution may be a reasonable model to describe the stock portfolio return process¹.     

Some extensions of the CAPM for individual assets

There is ample evidence that stock returns exhibit non-normal distributions with high skewness and excess kurtosis. Experimental evidence has shown that investors like positive skewness, dislike extreme losses and show high levels of prudence. This has motivated the introduction of the four-moment capital asset pricing model (CAPM). This extension, however, has not been able to successfully explain average returns. Our paper argues that a number of pitfalls may have contributed to the weak and conflicting empirical results found in the literature. We investigate whether conditional models, whether models that use individual stocks rather than portfolios and whether models that extend both the moment and factor dimension can improve on more traditional static, portfolio-based, mean–variance models. More importantly, we find that the use of a scaled coskewness measure in cross-section regression is likely to be spurious because of the possibility for the market skewness to be close to zero, at least for some periods. We provide a simple solution to this problem.     

CALL OPTION VALUATION FOR DISCRETE NORMAL MIXTURES

In this study a mixture call option pricing model is derived to examine the impact of non-normal underlying returns densities. Observed fat-tailed and skewed distributions are assumed to be the result of independent Gaussian processes with nonstationary parameters, modeled by discrete k-component independent normal mixtures. The mixture model provides an exact solution with intuitive appeal using weighted sums of Black-Scholes (B-S) solutions. Simulating returns densities representative of equity securities, significant mispricing by B-S is found in low-priced at- and out-of-the-money near-term options. The lower the variance and the higher the leptokurtosis and positive skewness of the underlying returns, the more pronounced is this mispricing. Values of in-the-money options and options with several weeks or more to expiration are closely approximated by B-S.     

Pricing options on scenario trees

We examine valuation procedures that can be applied to incorporate options in scenario-based portfolio optimization models. Stochastic programming models use discrete scenarios to represent the stochastic evolution of asset prices. At issue is the adoption of suitable procedures to price options on the basis of the postulated discrete distributions of asset prices so as to ensure internally consistent portfolio optimization models. We adapt and implement two methods to price European options in accordance with discrete distributions represented by scenario trees and assess their performance with numerical tests. We consider features of option prices that are observed in practice. We find that asymmetries and/or leptokurtic features in the distribution of the underlying materially affect option prices; we quantify the impact of higher moments (skewness and excess kurtosis) on option prices. We demonstrate through empirical tests using market prices of the S&P500 stock index and options on the index that the proposed procedures consistently approximate the observed prices of options under different market regimes, especially for deep out-of-the-money options.     

A Test for Symmetry with Leptokurtic Financial Data

Most of the tests for symmetry are developed under the (implicitor explicit) null hypothesis of normal distribution. As is wellknown, many financial data exhibit fat tails, and thereforecommonly used tests for symmetry (such as the standard test based on sample skewness) are not valid fortesting the symmetry of leptokurtic financial data. In particular,the test uses third moment, which may not be robust in presence of gross outliers. In this article wepropose a simple test for symmetry based on the Pearson typeIV family of distributions, which take account of leptokurtosisexplicitly. Our test is based on a function that is boundedover the real line, and we expect it to be more well behavedthan the test based on sample skewness (third moment). Resultsfrom our Monte Carlo study reveal that the suggested test performsvery well in finite samples both in terms of size and power.Simulation results also support our conjecture of the teststo be well behaved and robust to excess kurtosis. We apply thetest to some selected individual stock return data to illustrateits usefulness.     

Institutional Ownership and Distribution of Equity Returns

Although there is considerable evidence of the importance of skewness and kurtosis in equity returns, much less attention has been paid to their determinants. Recent theoretical and empirical advances in the literature suggest that the information structure and other market characteristics affect the nature of return distributions. One such characteristic is the degree of institutional ownership in the stock. This study hypothesizes and documents a significant inverse relationship between the degree of institutional ownership and the standard deviation, skewness, and kurtosis of equity returns.     

Fat tails in private equity fund returns: The smooth double Pareto distribution

Does the distribution of private equity returns have fat tails? A new smooth double Pareto distribution can explain the stationary distribution of private equity funds' valuation multiples. This distribution emerges from a random growth model with lognormally distributed initial fund valuations. This model endogenously generates power-law tails in the stationary cross-section. The new distribution fits the data better than competing distributions. Fat tails are particularly pronounced in venture capital funds and suggest returns with infinite variance over the lifetime of the fund. The smooth double Pareto distribution has wide applicability to growth processes with a random initial value.     

Dynamics of credit spread moments of European corporate bond indexes

Traditional quantitative credit risk models assume that changes in credit spreads are normally distributed but empirical evidence shows that they are likely to be skewed, fat-tailed, and change behaviour over time. Not taking into account such characteristics can compromise calculation of loss probabilities, pricing of credit derivatives, and profitability of trading strategies. Therefore, the aim of this study is to investigate the dynamics of higher moments of changes in credit spreads of European corporate bond indexes using extensions of GARCH type models that allow for time-varying volatility, skewness and kurtosis of changes in credit spreads as well as a regime-switching GARCH model which allows for regime shifts in the volatility of changes in credit spreads. Performance evaluation methods are used to assess which model captures the dynamics of observed distribution of the changes in credit spreads, produces superior volatility forecasts and Value-at-Risk estimates, and yields profitable trading strategies. The results presented can have significant implications for risk management, trading activities, and pricing of credit derivatives.     

Testing the bivariate mixture hypothesis using German Stock market data

According to the bivariate mixture hypothesis (BMH) as proposed by Tauchen and Pitts (1983) and Harris (1986, 1987) the daily price changes and the corresponding trading volume on speculative markets follow a joint mixture of distributions with the unobservable number of daily information events serving as the mixing variable. Using German stock market data of 15 major companies the distributional properties of the BMH is tested employing maximum-likelihood as well as generalised method of moments estimation techniques. In addition to providing a new approach for the pointwise estimation of the latent information arrival rate based on the maximum-likelihood method, we investigate the time-series properties of the BMH. the major results can be summarised as follows: (i) the distributional characteristics of the data (especially leptokurtosis and skewness in the distribution of price changes and volume respectively) cannot be explained satisfactorily by the BMH; univariate mixture models for price changes and trading volume separately reveal a possible specification error in the model; (ii) a univariate normal mixture model can account for the observed distributional characteristics of price changes; (iii) the estimated process of the latent information rate cannot fully explain the time-series characteristics of the data (especially the volatility clustering or ARCH-effects).     

Skewness and Kurtosis Implied by Option Prices: A Correction

Corrado and Su (1996) provide skewness and kurtosis adjustment terms for the Black‐Scholes model, using a Gram‐Charlier expansion of the normal density function. In this note we provide a correction to the expression for the skewness coefficient and illustrate the effect on call option prices of the error found.     

Skewed distributions in finance and actuarial science: a review

That the returns on financial assets and insurance claims are not well described by the multivariate normal distribution is generally acknowledged in the literature. This paper presents a review of the use of the skew-normal distribution and its extensions in finance and actuarial science, highlighting known results as well as potential directions for future research. When skewness and kurtosis are present in asset returns, the skew-normal and skew-Student distributions are natural candidates in both theoretical and empirical work. Their parameterization is parsimonious and they are mathematically tractable. In finance, the distributions are interpretable in terms of the efficient markets hypothesis. Furthermore, they lead to theoretical results that are useful for portfolio selection and asset pricing. In actuarial science, the presence of skewness and kurtosis in insurance claims data is the main motivation for using the skew-normal distribution and its extensions. The skew-normal has been used in studies on risk measurement and capital allocation, which are two important research fields in actuarial science. Empirical studies consider the skew-normal distribution because of its flexibility, interpretability, and tractability. This paper comprises four main sections: an overview of skew-normal distributions; a review of skewness in finance, including asset pricing, portfolio selection, time series modeling, and a review of its applications in insurance, in which the use of alternative distribution functions is widespread. The final section summarizes some of the challenges associated with the use of skew-elliptical distributions and points out some directions for future research.     

Co-Kurtosis and Capital Asset Pricing

This paper examines the impact of co-kurtosis on asset pricing using a four-moment capital asset pricing model. It is shown that, in the presence of skewness and kurtosis in asset return distribution, the expected excess rate of return is related not only to the systematic variance but also to the systematic skewness and systematic kurtosis. Investors are compensated in higher expected return for bearing the systematic variance and the systematic kurtosis risks. Investors also forego the expected excess return for taking the benefit of increasing the systematic skewness.     

A recombining lattice option pricing model that relaxes the assumption of lognormality

Option pricing models based on an underlying lognormal distribution typically exhibit volatility smiles or smirks where the implied volatility varies by strike price. To adequately model the underlying distribution, a less restrictive model is needed. A relaxed binomial model is developed here that can account for the skewness of the underlying distribution and a relaxed trinomial model is developed that can account for the skewness and kurtosis of the underlying distribution. The new model incorporates the usual binomial and trinomial tree models as restricted special cases. Unlike previous flexible tree models, the size and probability of jumps are held constant at each node so only minor modifications in existing code for lattice models are needed to implement the new approach. Also, the new approach allows calculating implied skewness and implied kurtosis. Numerical results show that the relaxed binomial and trinomial tree models developed in this study are at least as accurate as tree models based on lognormality when the true underlying distribution is lognormal and substantially more accurate when the underlying distribution is not lognormal.     

The Empirical Distribution of UK and US Stock Returns

There is now substantial evidence that daily equity returns are not normally distributed but instead display significant leptokurtosis and, in many cases, skewness. Considerable effort has been made in order to capture these empirical characteristics using a range of ad hoc statistical distributions. In this paper, we investigate the distribution of daily, weekly and monthly equity returns in the UK and US using two very flexible families of distributions that have been recently introduced: the exponential generalised beta (EGB) and the skewed generalised- t (SGT). These distributions permit very diverse levels of skewness and kurtosis and, between them, nest many of the distributions previously considered in the literature. Both the EGB and the SGT provide a very substantial improvement over the normal distribution in both markets. Moreover, for daily returns, we strongly reject the restrictions on the EGB and SGT implied by most of the distributions that are commonly used for modelling equity returns, including the student- t , the power exponential and the logistic distributions. Instead, our preferred distributions for daily returns are the generalised- t for the US and the skewed- t for the UK, both of which are members of the SGT family. For weekly returns, our preferred distributions are the student- t for the UK and the skewed- t for the US, while for monthly returns, our preferred distributions are the EBR12 for the UK and the logistic for the US. We consider the implications of our findings for the implementation of value-at-risk, a risk management methodology that rests heavily on the distributional characteristics of returns.