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.     

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.     

Conditional higher order moments in metal asset returns

This study examines the role of higher order moments in the returns of four important metals, aluminium, copper, gold and silver, using the asymmetric GARCH (AGARCH) model with a conditional skewed generalized-t (SGT) distribution. Implications of higher order moments in metal returns are evaluated by comparing the performances of conditional value-at-risk measures obtained from the AGARCH models with SGT distributions to those obtained from the AGARCH models with normal and student-t distributions. With the exception of gold, the AGARCH model with the SGT distribution appears to have the best fit for all metals examined.     

基于贝叶斯MSSV-ST金融波动模型的股市特征及机制转移性研究

针对有偏厚尾金融随机波动模型难以刻画参数的动态时变性及结构突变的问题,设置偏态参数服从 Markov 转换过程,采用贝叶斯方法,构建带机制转移的有偏厚尾金融随机波动模型,考量股市不同波动状态间的机制转移性,捕捉股市间多重波动特性。通过设置先验分布,实现模型的贝叶斯推断,设计相应的马尔科夫链蒙特卡洛算法进行估计,并利用上证指数进行实证。结果表明：模型不仅刻画了股市的尖峰厚尾、杠杆效应等特性,发现收益率条件分布的偏度参数具有动态时变性,股市波动呈现出显著的机制转移特性,而且证实了若模型考虑波动的不同阶段性状态后,将降低持续性参数向上偏倚幅度的结论。     

Option pricing under regime switching

Abstract

This paper develops a family of option pricing models when the underlying stock price dynamic is modelled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime. Our family includes the regime switching models of Hamilton (Hamilton J 1989 Econometrica 57 357–84), in which volatility influences returns. In addition, our models allow for feedback effects from returns to volatilities. Our family also includes GARCH option models as a special limiting case. Our models are more general than GARCH models in that our variance updating schemes do not only depend on levels of volatility and asset innovations, but also allow for a second factor that is orthogonal to asset innovations. The underlying processes in our family capture the asymmetric response of volatility to good and bad news and thus permit negative (or positive) correlation between returns and volatility. We provide the theory for pricing options under such processes, present an analytical solution for the special case where returns provide no feedback to volatility levels, and develop an efficient algorithm for the computation of American option prices for the general case.     

Box-Cox stochastic volatility models with heavy-tails and correlated errors

This paper presents a Markov chain Monte Carlo (MCMC) algorithm to estimate parameters and latent stochastic processes in the asymmetric stochastic volatility (SV) model, in which the Box-Cox transformation of the squared volatility follows an autoregressive Gaussian distribution and the marginal density of asset returns has heavy-tails. We employed the Bayes factor and the Bayesian information criterion (BIC) to examine whether the Box-Cox transformation of squared volatility is favored against the log-transformation. When applying the heavy-tailed asymmetric Box-Cox transformed SV model, three competing SV models and the t-GARCH(1,1) model to continuously compounded daily returns of the Australian stock index, we find that the Box-Cox transformation of squared volatility is strongly favored by Bayes factors and BIC against the log-transformation. While both criteria strongly favor the t-GARCH(1,1) model against the heavy-tailed asymmetric Box-Cox transformed SV model and the other three competing SV models, we find that SV models fit the data better than the t-GARCH(1,1) model based on a measure of closeness between the distribution of the fitted residuals and the distribution of the model disturbance. When our model and its competing models are applied to daily returns of another five stock indices, we find that in terms of SV models, the Box-Cox transformation of squared volatility is strongly favored against the log-transformation for the five data sets.     

Skewness and leptokurtosis in GARCH-typed VaR estimation of petroleum and metal asset returns

This paper utilizes the most flexible skewed generalized t (SGT) distribution for describing petroleum and metal volatilities that are characterized by leptokurtosis and skewness in order to provide better approximations of the reality. The empirical results indicate that the forecasted Value-at-Risk (VaR) obtained using the SGT distribution provides the most accurate out-of-sample forecasts for both the petroleum and metal markets. With regard to the unconditional and conditional coverage tests, the SGT distribution produces the most appropriate VaR estimates in terms of the total number of rejections; this is followed by the nonparametric distribution, generalized error distribution (GED), and finally the normal distribution. Similarly, in the dynamic quantile test, the VaR estimates generated by the SGT and nonparametric distributions perform better than that generated by other distributions. Finally, in the superior predictive test, the SGT distribution has significantly lower capital requirements than the nonparametric distribution for most commodities.     

Extreme downside risk and market turbulence

We investigate the dynamics of the relationship between returns and extreme downside risk in different states of the market by combining the framework of Bali et al. [Is there an intertemporal relation between downside risk and expected returns? Journal of Financial and Quantitative Analysis, 2009, 44, 883–909] with a Markov switching mechanism. We show that the risk-return relationship identified by Bali et al. (2009) is highly significant in the low volatility state but disappears during periods of market turbulence. This is puzzling since it is during such periods that downside risk should be most prominent. We show that the absence of the risk-return relationship in the high volatility state is due to leverage and volatility feedback effects arising from increased persistence in volatility. To better filter out these effects, we propose a simple modification that yields a positive tail risk-return relationship in all states of market volatility.     

Risk Measurement Performance of Alternative Distribution Functions

This paper evaluates the performance of three extreme value distributions, i.e., generalized Pareto distribution (GPD), generalized extreme value distribution (GEV), and Box‐Cox‐GEV, and four skewed fat‐tailed distributions, i.e., skewed generalized error distribution (SGED), skewed generalized t (SGT), exponential generalized beta of the second kind (EGB2), and inverse hyperbolic sign (IHS) in estimating conditional and unconditional value at risk (VaR) thresholds. The results provide strong evidence that the SGT, EGB2, and IHS distributions perform as well as the more specialized extreme value distributions in modeling the tail behavior of portfolio returns. All three distributions produce similar VaR thresholds and perform better than the SGED and the normal distribution in approximating the extreme tails of the return distribution. The conditional coverage and the out‐of‐sample performance tests show that the actual VaR thresholds are time varying to a degree not captured by unconditional VaR measures. In light of the fact that VaR type measures are employed in many different types of financial and insurance applications including the determination of capital requirements, capital reserves, the setting of insurance deductibles, the setting of reinsurance cedance levels, as well as the estimation of expected claims and expected losses, these results are important to financial managers, actuaries, and insurance practitioners.     

On the distribution of government bond returns: evidence from the EMU

This paper assesses the statistical distribution of daily EMU bond returns for the period 1999–2012. The normality assumption is tested and clearly rejected for all European countries and maturities. Although skewness plays a minor role in this departure from normality, it is mainly due to the excess kurtosis of bond returns. Therefore, we test the Student’s t, skewed Student’s t, and stable distribution that exhibit this feature. The financial crisis leads to a structural break in the time series. We account for this and retest the alternative distributions. A value-at-risk application underlines the importance of our findings for investors. In sum, excess kurtosis in bond returns is essential for risk management, and the stable distribution captures this feature best.     

Earnings quality and the heterogeneous relation between earnings and stock returns

We adopt a heterogeneous regime switching method to examine the informativeness of accounting earnings for stock returns. We identify two distinct time-series regimes in terms of the relation between earnings and returns. In the low volatility regime (typical of bull markets), earnings are moderately informative for stock returns. But in high volatility market conditions (typical of financial crisis), earnings are strongly related to returns. Our evidence suggests that earnings are more informative to investors when uncertainty and risk is high which is consistent with the idea that during market downturns investors rely more on fundamental information about the firm. Next, we identify groups of firms that follow similar regime dynamics. We find that the importance of accounting earnings for returns in each of the market regimes varies across firms: certain firms spend more time in a regime where their earnings are highly relevant to returns, and other firms spend more time in a regime where earnings are moderately relevant to returns. We also show that firms with poorer accrual quality have a greater probability of belonging to the high volatility regime.     

Implicit expectiles and measures of implied volatility

It is well known that the normal distribution is inadequate in capturing the skewed and heavy-tailed behaviour of exchange rate returns. To this end, various flexible distributions that are capable of modelling the asymmetric and tailed behaviour of returns have been proposed. In this paper, we investigate the performance of the generalized lambda distribution (GLD) to capture the skewed and leptokurtic behaviour of exchange rate returns. We do this by conducting a comprehensive numerical study to compare the performance of the GLD against the performances of the skewed t distribution, the unbounded Johnson family of distributions and the normal inverse Gaussian (NIG) distribution. Our results suggest that in terms of the value-at-risk and expected shortfall, the GLD shows at least similar performance to the skewed t distribution and the NIG distribution. Considering the ease in GLD’s use for random variate generation in Monte Carlo simulations, we conclude that the GLD can be a good alternative in various financial applications where modelling of the heavy tail behaviour is critical.     

Accurate minimum capital risk requirements: A comparison of several approaches

In this paper we estimate, for several investment horizons, minimum capital risk requirements for short and long positions, using the unconditional distribution of three daily indexes futures returns and a set of short and long memory stochastic volatility and GARCH-type models. We consider the possibility that errors follow a t-Student distribution in order to capture the kurtosis of the returns’ series. The results suggest that accurate modelling of extreme observations obtained for long and short trading investment positions is possible with an autoregressive stochastic volatility model. Moreover, modelling futures returns with a long memory stochastic volatility model produces, in general, excessive volatility persistence, and consequently, leads to large minimum capital risk requirement estimates. Finally, the models’ predictive ability is assessed with the help of out-of-sample conditional tests.     

Autoregressive stochastic volatility models with heavy-tailed distributions: A comparison with multifactor volatility models

This paper examines two asymmetric stochastic volatility models used to describe the heavy tails and volatility dependencies found in most financial returns. The first is the autoregressive stochastic volatility model with Student's t-distribution (ARSV-t), and the second is the multifactor stochastic volatility (MFSV) model. In order to estimate these models, the analysis employs the Monte Carlo likelihood (MCL) method proposed by Sandmann and Koopman [Sandmann, G., Koopman, S.J., 1998. Estimation of stochastic volatility models via Monte Carlo maximum likelihood. Journal of Econometrics 87, 271–301.]. To guarantee the positive definiteness of the sampling distribution of the MCL, the nearest covariance matrix in the Frobenius norm is used. The empirical results using returns on the S&P 500 Composite and Tokyo stock price indexes and the Japan–US exchange rate indicate that the ARSV-t model provides a better fit than the MFSV model on the basis of Akaike information criterion (AIC) and the Bayes information criterion (BIC).     

VaR and expected shortfall: a non-normal regime switching framework

We have developed a regime switching framework to compute the Value at Risk and Expected Shortfall measures. Although Value at Risk as a risk measure has been criticized by some researchers for lack of subadditivity, it is still a central tool in banking regulations and internal risk management in the finance industry. In contrast, Expected Shortfall is coherent and convex, so it is a better measure of risk than Value at Risk. Expected Shortfall is widely used in the insurance industry and has the potential to replace Value at Risk as a standard risk measure in the near future. We have proposed regime switching models to measure value at risk and expected shortfall for a single financial asset as well as financial portfolios. Our models capture the volatility clustering phenomenon and variance-independent variation in the higher moments by assuming the returns follow Student-t distributions.     

Value-at-Risk analysis for long-term interest rate futures: Fat-tail and long memory in return innovations

This article uses the FIGARCH(1,d,1) models to calculate daily Value-at-Risk (VaR) for T-bond interest rate futures returns of long and short trading positions based on the normal, Student-t, and skewed Student-t innovations distributions. The empirical results show that based on Kupiec LR failure rate tests, in-sample and out-of-sample VaR values calculated using FIGARCH(1,d,1) model with skewed Student-t innovations are more accurate than those generated using traditional GARCH(1,1) models. Moreover, we find that the in-sample values of VaR are subject to a significant positive bias, as pointed out by Inui et al. [Inui, K., Kijima, M., Kitano, A., 2003. VaR is subject to a significant positive bias, working paper].     

Evidence of feedback trading with Markov switching regimes

Previous research has concluded that the degree of return autocorrelation observed in index returns varies linearly with the volatility of the series, and that feedback traders are at least partly responsible for this phenomenon. Using daily Australian bond and equity market returns, we test this conclusion directly by using a Markov switching model for changing variance that explicitly allows the autocorrelation of returns to vary with the volatility regime. We find evidence that a significant proportion of investors in both the Australian equity and bond markets are positive feedback traders and are responsible for the observed increase in negative autocorrelation in index returns during periods of high and increasing volatility.

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.     

Dynamic correlations and volatility linkages between stocks and sukuk: Evidence from international markets

An understanding of volatility and co-movements in financial markets is important for portfolio allocation and risk management practices. The current financial crisis caused a shrinkage in values of most assets, an increased volatility and a threat to the survival of several institutional investors. Managing risks and returns within the classic portfolio theory, when correlations across securities soar, is increasingly challenging. In this paper, we investigate the volatility behavior and the co-movements between sukuk and international stock indexes. Symmetric multivariate GARCH models with dynamic conditional correlations (DCC) were estimated under Student-t distribution. We provide evidence of high correlations between sukuk and US and EU stock markets, without finding the well-known flight to quality behavior affecting Islamic bonds. We also show that volatility linkages between sukuk and regional market indexes are higher during financial crisis. We argue that investors could obtain diversification benefits including sukuk in a well-diversified equity portfolio, given their lower volatility compared to equity. But higher volatility linkages and dynamic correlations during financial crises show that they are hybrid instruments between bonds and equity. Our findings are relevant for institutional investors and asset managers that include Islamic bonds in a diversified portfolio.     

Volatility in stock returns for new EU member states: Markov regime switching model

In this paper, we investigate the volatility in stock markets for the new European Union (EU) member states of the Czech Republic, Hungary, Poland, Slovenia and Slovakia by utilising the Markov regime switching model. The model detects that there are two or three volatility states for the emerging stock markets. The result reveals that there is a tendency that the emerging stock markets move from the high volatility regime in the earlier period of transition into the low volatility regime as they move into the EU. Entry to the EU appears to be associated with a reduction of volatility in unstable emerging markets.