Extreme Asymmetric Volatility, Leverage, Feedback and Asset Prices

Asymmetric volatility in equity markets has been widely documented in finance, where two competing explanations, as considered in Bekaert and Wu (2000), are the financial leverage and the volatility feedback hypothesis. We explicitly test for the role of both hypotheses in explaining extreme daily U.S. equity market movements during the period January 1990 to September 2008. To this aim, we examine asymmetric volatility based on a novel model of market returns, implied market volatility and volatility of volatility. We then test for extreme asymmetry and the distinct predictions of both hypotheses. Our results document significant extreme asymmetric volatility. This effect is contemporaneous, consistent with both hypotheses, and it is important for large market declines. We further derive aggregate asset pricing implications under extreme volatility feedback. Given our results, asymmetric volatility, which includes the effect of volatility feedback at extreme levels, is shown to play an important role in explaining substantial equity market declines.

Asymmetric volatility in equity markets has been widely documented in finance, stating that returns and volatility are negatively related and that this relation is more pronounced for negative returns. In other words, large volatility increases and large market declines tend to coincide. This striking phenomenon is relevant to financial stability and to the occurrence of periods of market stress (see for example IMF (2003)). It may also affect the overall economy, as the following foresighted statement illustrates: "Members of the Federal Reserve's policy-setting committee worried at their most recent meeting that housing and financial market stress could trigger a nasty slide in the economy. (Reuters, April 8, 2008)". Asymmetric equity market volatility is important for at least three reasons. First, it is an important characteristic of the market volatility dynamics, has asset pricing implications and is a characteristic of priced risk factors.
Second, it plays an important role in risk prediction, hedging and option pricing.
Finally, asymmetric volatility implies negatively skewed return distributions, i.e. it may help to explain some of the probability of large losses.
In the present paper, we address asymmetric volatility when markets face extreme downside price movements. Two prominent competing economic hypotheses, which aim at an explanation of the phenomenon, are the so-called "leverage effect" hypothesis and the "volatility feedback effect" hypothesis; see Black (1976), Campbell and Hentschel (1992) and Bekaert and Wu (2000). While it is known that volatility changes behave asymmetrically with security price changes, it is not yet documented whether such behavior may also be consid-ered as a driver of periods of market stress. Campbell and Hentschel (1992) point out that volatility feedback is presumably more important during periods of market stress. Therefore, the phenomenon may help to explain the severity of large market declines. Their results are based on a model of autoregressive conditional heteroskedasticity (ARCH), which is not designed to model extreme behavior. We test whether the asymmetric price-volatility relation prevails when markets face extreme downside price shocks and whether the risk of large market volatility shocks is priced.
Our paper covers three novel areas. First, we propose a model of the distribution of market returns and observable implied volatility. We do this for the U.S. equity market and the market volatility index (VIX). Our model includes market returns, volatility as well as volatility of volatility. We allow for asymmetric volatility based on a threshold model. Volatility of volatility is modeled by an asymmetric ARCH-process. The random innovations of our model are given by a pair of unexpected return and volatility shocks. These shocks may exhibit cross-sectional dependence.
Second, we contribute to the literature by examining volatility asymmetry under normal periods within the dynamic conditional correlation (DCC) approach of Engle (2002). For periods of market stress, we consider extreme dependence in the context of extreme value theory (EVT). We apply EVT in order to perform distinct tests of the implications of the volatility feedback hypothesis for large unexpected return and volatility shocks. The feedback hypothesis predicts that periods of pronounced volatility increases together with extreme negative returns are not simply outlying observations. In contrast, they should be systematically dependent. While market volatility is heavily subject to jump risk (see e.g. Todorov and Tauchen (2008)), volatility feedback is known to potentially amplify large negative return shocks (see Campbell and Hentschel (1992)). In sum, when volatility feedback is at work, volatility jumps and large negative return shocks amplify each other. This could lead to a situation, where the market would be predicted to melt down given a large initial positive shock to volatility. We empirically examine the possibility of such an extreme event, which is a component to systemic market risk.
Finally, we contribute to the empirical asset pricing literature by outlining pricing implications of extreme volatility feedback. As pointed out for example in Li (2004) and Engle and Mistry (2007), intertemporal asset pricing models predict volatility asymmetry once market volatility is a priced risk factor. We consider the empirical asset pricing implications of extreme volatility feedback in more detail.
Our daily sample covers daily U.S. market returns and VIX volatility during the period 1990 to 2008. It includes several periods of market stress, including the Russian debt crisis and the recent financial crisis. For the univariate series, our empirical results indicate a statistically insignificant leverage effect, i.e. lagged market returns do not significantly affect VIX volatility. However, we find a significant hyper-level leverage effect. Given this latter effect, volatility of volatility is asymmetric in that past positive volatility shocks drive positive shocks to volatility of volatility. Our empirical findings on the dependence of unexpected return and volatility shocks further document that time-variation is an important characteristic of asymmetric market volatility. As such, return and volatility shocks exhibit negative correlation throughout our sample, but this relation is heavily time-varying, showing remarkable swings in its magnitude. Negative dependence increases under stress, i.e. for market declines, but not for market upswings. Turning to an examination of the tails of the joint return-volatility distribution, we confirm this asymmetric behavior by a strict test of tail independence. Our results are shown to be robust with respect to a number of parameter choices.
We document systemic market risk for large market declines and test for the role of both hypotheses in explaining extreme market movements. We thereby address the distinct return-volatility causality of each hypothesis. While we do not find return-volatility causality in either direction, our results suggest that both hypotheses play an important role in the contemporaneous explanation of extreme asymmetric volatility. We conclude that periods of market stress have substantial impact on asymmetric market volatility. The relation between market risk and return appears strong during periods of market stress. We point out the asset pricing implications under extreme volatility feedback, which helps to explain large market declines.
The remainder of the paper is organized as follows. Section 1 provides a review on asymmetric volatility. Our new model of market returns and implied volatility is outlined in Section 2, which also includes our methodology for the study of the dependence between volatility and returns. In Section 3, we present the data set and the empirical results. The paper concludes in Section 4.

Asymmetric Volatility
In this section we give a brief review on asymmetric volatility. This review is intended to clarify some of the controversial points in the literature. While the term "asymmetric volatility" is sometimes not used consistently, the following definition appears consistent with most of the literature: Asymmetric volatility states that returns and volatility are negatively related and that the relation is more pronounced for negative returns. In detail, this definition covers three distinct characteristics, namely • (i) a negative relation between (past) realized returns and volatility, • (ii) a positive relation between expected market returns and volatility and • (iii) asymmetry, i.e. relation (i) is more pronounced for negative returns.
The first empirical finding, (i), is linked to the discussion of the "leverage effect" hypothesis. The hypothesis can be traced back to the work of Black (1976) and Christie (1982). It states that, once market prices of equity drop, financial leverage of the firm increases and, as a result, the volatility of equity price changes increases. Christie (1982) examines the relation for riskless debt as well as for risky debt within the framework of a structural credit risk model. For increasing leverage, he predicts a monotonically (non-monotonically) decreasing strength in the negative relation between stock returns and volatility for the riskless (risky) debt model. Hence, while the leverage hypothesis explains a negative relation (i), even with risky debt, the observation of asymmetry, (iii), would require additional assumptions. To our knowledge, there is no such attempt in the literature so far, and this may in part explain why most studies conclude that the leverage effect cannot fully explain the phenomenon of asymmetric volatility; see also Aydemir, Gallmeyer, and Hollifield (2006).
The second and third characteristic, (ii) and (iii), both follow within an intertemporal asset pricing context as given for example by Merton (1973Merton ( , 1980. Assuming further that volatility is persistent (as documented in the literature, see e.g. Bollerslev, Chou, and Kroner (1992)), an increase in conditional stock market volatility predicts increased future volatility, which may increase required future market returns, and thus lower present prices. Note that this argumentation can also explain characteristics (i) and (ii) above. It is the cornerstone of the volatility feedback effect hypothesis, which was proposed (also) by Black (1976) as well as by Pindyck (1984), French, Schwert, and Stambaugh (1987) and Campbell and Hentschel (1992).
As Campbell and Hentschel (1992) argue, volatility feedback can explain observation (iii), i.e. it may explain why volatility is asymmetric. Given a large piece of unexpected good news, i.e. a large positive return shock, volatility increases and feedback yields a decrease in prices, which in turn dampens the initial positive return shock. However, given a large piece of unexpected bad news, i.e. a large negative return shock, volatility increases and feedback yields an additional decrease in prices, which in turn amplifies the initial negative return shock. As a result, volatility is asymmetric, because volatility feedback amplifies negative stock returns while it dampens positive returns. Consistent with findings in the empirical literature, volatility feedback can explain why market returns tend to be negatively skewed and why changes in market volatility typically show positive skewness. We argue that the feedback effect has to converge and eventually reach a steady state, as otherwise-given an initial large piece of bad news-the market could be predicted to melt down. Systemic market risk is, among others, described by the probability of such an extreme event.
Given both hypotheses above, it can be supposed that leverage and feedback jointly drive asymmetric volatility in equity markets. Bekaert and Wu (2000) provide a first empirical study on the explanation of asymmetric equity market volatility based on leverage as well as feedback effects. They conclude that leverage models cannot explain the observed patterns (on the firm as well as on the aggregate market level), while volatility feedback appears more plausible. Bae, Kim, and Nelson (2007) support these findings. Controlling for leverage, they show that in contrast to temporary volatility changes, only persistent volatility changes feed back to stock returns. Economic models of asymmetric volatility may explain equilibrium return/volatility dynamics driven by feedback (as e.g. in Wu (2001)) or driven by leverage (as e.g. in Aydemir, Gallmeyer, and Hollifield (2006)).
Econometric models of asymmetric volatility improve the modeling and forecasting of conditional volatility by incorporating observation (iii) above. Frequently, these models also make predictions of the conditional risk-return relation (ii). The most widespread class of models is ARCH models (see e.g. Bollerslev, Chou, and Kroner (1992)). Asymmetric model versions include the Nelson (1991) EGARCH model and threshold ARCH models including for example the GJR-GARCH model of Glosten, Jagannathan, and Runkle (1993), the model of Zakoian (1994) and the model of Engle and Ng (1993). Regime switching models with asymmetry are used by Kim, Morley, and Nelson (2004) and by Bae, Recent research indicates that asymmetric volatility is more closely linked to asset pricing issues than previously considered given the two competing traditional hypotheses above. Along these lines, Avramov, Chordia and Goyal (2006) document that order flow can explain asymmetric volatility, where, conditional on a negative return, volatility is higher after large volume days. Their em-pirical findings also support the authors' hypothesis that large contrarian trades decrease volatility while large herding trades increase volatility. Another tradingbased explanation of asymmetric volatility is given by Wagner and Marsh (2005) who extend the GJR-GARCH model via asymmetric volume effects. Their findings show that asymmetric market volatility and a positive conditional riskreturn relation-which is weak under the original GJR-specification-both become significant when the GJR-model accounts for volume shocks. Smith (2007) shows that incorporating volatility feedback in a stochastic volatility model transforms a weak negative risk-return relation into a statistically significant positive relation.
As Beakert and Wu (2000) point out, if relation (ii) between conditional market volatility and expected return is not positive, the validity of a time-varying market risk premium hypothesis is in doubt. In fact, once changes in conditional expected market returns are related to persistent changes in today's conditional market volatility, volatility risk is a priced risk factor; see also Li (2004) and Engle and Mistry (2007). As (ii) implies (iii) above via the volatility feedback effect, Engle and Mistry suggest that testing for volatility asymmetry is equivalent to testing for priced risk factors. Overall, the above findings suggest that leverage and volatility feedback as well as liquidity and private information flow may help to explain the characteristics of asymmetric volatility. Volatility feedback appears central and relies on a positive relation between expected market returns and market volatility. As a consequence, asymmetric volatility is not only an important characteristic of the market volatility dynamics but a fundamental characteristic of priced risk factors.

Methodology
Our methodology captures the joint distribution of returns and implied volatility. We thereby treat returns and implied volatility as observable variables, while the volatility of volatility is unobservable. We model market returns parsimoniously with constant mean return and with the VIX as observable implied market volatility. The VIX conditional variance is modeled as a GJR-GARCH process. Hence, the model includes a standard "leverage"-effect between lagged market returns and changes in implied volatility and with a volatility of volatility, which is in turn asymmetric. The model is driven by unexpected shocks to market returns and implied market volatility.
Our aim is to examine the nature of asymmetric market volatility in our above model setting. We do this by a detailed study of the negative dependence between the unexpected return and volatility shocks. In order to examine correlation, we consider DCC in the multivariate GARCH framework of Engle (2002) and Engle and Sheppard (2001). This allows us to characterize the time-varying nature of the negative relation between realized returns and implied volatility.
In order to examine the tail behavior of large unexpected shocks to returns and volatility, we apply models of bivariate EVT. 2 A general discussion of EVT with a background in finance is by Embrechts, Klüppelberg and Mikosch (1997).
Summaries on univariate and bivariate results can be found in Coles (2001) and Beirlant, Goegebeur, Segers, and Teugels (2004), for example. EVT analysis allows us to study asymmetric volatility under market stress and to test the implications of the feedback hypothesis.

Market Returns, Volatility and Volatility of Volatility
Our model of market returns, implied market volatility and volatility of volatility is based on a discrete time approach for the dynamics of returns and volatility, which allows for a double-asymmetric process.
For the periods, t = 1, ..., T , the market returns R t are given as where µ ∈ R and the return shocks are independent and identically dis- 3 In order to allow for time series dependence in returns and for time-variation in expected returns, we may e.g. set the conditional return expectation in (2.1) as: µ = µ 1 +µ 2 R t−1 +µ 3 σ t−1 . However, our empirical investigations of this equation, which are unreported here, show that the coefficients for lagged returns and lagged volatility turn out to be insignificant based on White heteroskedasticity consistent estimates of the standard errors. This confirms the statement of Black (1976) as reproduced in footnote 1 and the findings of Hibbert, Daigler and Dupoyet (2008) who state that the return-implied volatility relation is foremost a contempo-In equation (2.1), the time-varying market volatility, σ t , is assumed to be directly observable via VIX implied volatility. 4 Turning next to VIX implied market volatility, we model logarithmic changes in volatility as an autoregressive process of order p ≥ 1, AR(p). We use lagged return innovations of equation (2.1) as explanatory variables, where a threshold specification allows volatility to react asymmetrically to past return shocks. In detail, we assume where the Z t represent white noise innovations, Z t ∼ iidF Z (0, 1). We can rearrange the above equation and obtain a model for the logarithmic volatility. We have: For δ 1 = 0, it follows that the above equation has a unit root. With δ j = 0, j = 1, ..., p, the process is a non-stationary random walk, i.e. an integrated volatility raneous relation. Also, we consider large innovations in returns in this paper, which deviate heavily from a constant as well as from time-varying expected market returns. We therefore conclude that the simple constant mean model proves to be sufficient for our purposes. 4 The new VIX represents the risk neutral expectation of future market volatility and thereby contains a variance risk premium, see e.g. Bollerslev, Sizova and Tauchen (2008). A straightforward extension of our approach would be to represent σ t by the VIX and an estimate of a constant variance risk premium, such as the Carr  process. Otherwise, for −2 < δ 1 < 0, a stationary solution may result. For the remaining parameters we assume −1 < δ j < 1, j = 2, ..., p.
We may specify various parametric functions f in equation (2.2). As motivated by threshold ARCH models, we choose the following three alternative piecewise linear specifications where I {A} denotes the indicator variable for the event A and θ, ω 0 , ω 1 , ω 2 ∈ R.
The above specifications allow implied market volatility to react asymmetrically to return shocks. In particular, the models may explain skewness in the distribution of volatility changes. The typical specification, f 1 , distinguishes between positive and negative lagged returns. Specification f 2 follows f 1 but depends only on lagged return shocks not volatilities. Specification f 3 allows for a threshold θ = 0 in specification f 1 , i.e. we may examine asymmetry for two arbitrary return intervals (−∞; θ) and [θ; ∞).
On the subsequent level, we allow for heteroskedasticity in logarithmic volatility changes, which are given in equation (2.2). The conditional variance of logarithmic volatility, φ 2 t , is assumed to follow an asymmetric GJR-GARCH equation of the form Note that a positive variance φ 2 t is normally assured under the sufficient param- The iid innovations (Y t , Z t ) drive the above bivariate return-volatility model, which is defined by equations (2.1), (2.2) and (2.3). The innovations may exhibit cross-sectional dependence as characterized by their joint distribution function F. We address the joint distribution of (Y t , Z t ) in the following.

Time-Varying Asymmetric Volatility
Given our above model of the market dynamics, we now turn to a model of the joint distribution of unexpected market returns and variance changes. Assuming a time-varying contemporaneous correlation between returns and variance, we can write The unexpected returns and unexpected volatility changes have standardized marginal distributions and their contemporaneous correlation (or covariance in this setting) is given by ρ t . The time-varying correlations describe volatility asymmetry under normal market conditions. We now assume DCC-dynamics for the conditional correlations, i.e.
In equation (2.6), the conditional covariance, q t , and the conditional variances, q Y,t and q Z,t , each follow GARCH(1,1)-type processes which equals E(Y t Z t ) for standardized variables.

Extreme Asymmetric Volatility
In order to characterize asymmetric volatility under periods of stress, we consider an EVT approach. We proceed in two steps, considering the marginal distributions first and then the dependence via an EVT-copula approach.
In the first step, in order to model the extreme behavior of the iid univariate changes, Y t = Y and Z t = Z, we assume that the marginal distributions of F , denoted e.g. by F X , can by characterized by results from EVT, which makes only weak assumptions about the distribution and requires that the univariate changes are iid (see e.g. Embrechts, Klüppelberg and Mikosch (1997)). Without loss of generality, we have and we restrict our discussion to upper tails in the following. We study return maxima Y , return minima −Y , volatility maxima Z and volatility minima −Z, We use a convenient Pareto tail approximation for excesses of a sufficiently high threshold, u > 0. EVT shows that a suitable asymptotic approximation is the Generalized Pareto Distribution (GPD), which is parameterized by a tail index ξ ∈ R and a scaling parameter β > 0. Conditional on X > u, a random number of excesses, X − u > 0, follows a GPD tail approximation. Considering the case ξ ≥ 0, the GPD is given by: In the second step, we model the joint extreme behavior of the unexpected with 0 < α ≤ 1. Independence is reached when α = 1 and perfect dependence when α → 0. A generalization of the Gumbel copula allows for asymmetric dependence with where 0 < α ≤ 1 and θ 1 ≥ 0, θ 2 ≤ 1; see Tawn (1988). Under the asymmetric model, symmetry is obtained under θ 1 = θ 2 = 1 and independence is reached when either α = 1, θ 1 = 0 or θ 2 = 0.
While the unexpected changes (Y, Z) may exhibit significant dependence, dependence could vanish gradually, leaving them asymptotically independent in the joint tail. Such behavior would not be suitable to the above threshold model, which is based on the assumption that extreme value dependence holds above the thresholds u Y and u Z ; see Ledford and Tawn (1996). Formally, consider the limiting conditional probability where the case χ = 0 denotes asymptotic independence and χ > 0 denotes asymptotic dependence. It can be shown that χ = 2 − 2 A (1/2) for any limiting distribution H in equation (2.8); see e.g. Coles (2001, p.164). Hence, it follows that χ = 2 − 2 α for model (2.9) and that In order to examine the above two cases of extreme dependence, we use two approaches from the literature. First, Coles, Heffernan and Tawn (1999) define the following pair (2.13) They show that, in the case of asymptotic independence, i.e. when χ = 0, it follows, χ(u) → 0 and χ(u) → const. < 1. Under asymptotic dependence, i.e.

Hypotheses
In this section we address asymmetric volatility by formulating testable hypotheses for unexpected return and volatility shocks. We first consider hypotheses, which are implied by both, feedback and leverage. Then we consider extreme volatility feedback as opposed to leverage. In total, we put up seven hypotheses, which we subsequently consider in our empirical investigation.

Feedback and Leverage
Following Section 1, a first testable hypothesis, which is implied by both feedback and leverage, is that of a contemporaneous negative relation (i) of Section 1. This implies a negative correlation ρ between unexpected return and volatility shocks, i.e. we would at least attempt to reject By examining the tails of the joint distribution, we test the hypothesis of dependence between the extremes of (Y t , Z t ). We formulate the 'boom' scenario, H 0 : "(Y t , −Z t ) are asymptotically independent " and the 'crash' scenario, H 0 : "(−Y t , Z t ) are asymptotically independent".
In the case of extreme asymmetric market volatility, we expect to reject H 0 .

Feedback versus Leverage
Testing the leverage effect hypothesis against the feedback effect hypothesis is clearly non-trivial as both hypotheses predict a contemporaneous negative volatility-return relationship. Given our discussion in Section 1, we argue that asymmetry, i.e. point (iii) in Section 1, is particularly supported by the feedback hypothesis of Campbell and Hentschel (1992), but not by the leverage models of Christie (1982). That is, under extreme volatility feedback we would particularly predict that the dependence between Y and −Z in H (2) 0 is weaker than that between −Y and Z in H 0 .
Furthermore, as discussed in Section 1, there is agreement in the literature that the leverage and the feedback effect hypotheses differ in their causality. 5 In order to test for potential causality, we consider whether lagged unexpected price shocks relate to unexpected volatility shocks of opposite sign and vice versa.
This forms the following two hypotheses: as a null hypothesis in order to test for leverage and H as a null hypothesis in order to test for feedback. Additionally, we may test for asymmetry within the contemporaneous dependence relation as put up by 5 See e.g. Black (1976), Beakert and Wu (2000) and Bollerslev, Litvinova, and Tauchen (2006). Note that causality is a difficult concept in economics where the fact that some event A regularly occurs before another event B does not imply that "A causes B" in a strict sense. See e.g. the discussion in Black (1976).

Empirical Investigation
This section first introduces the data set used in our empirical study. It then outlines how the model of Section 2 is implemented. Finally, we describe our estimation results, consider the hypotheses of Section 2.4 and then give an interpretation of our findings. The VIX has rapidly become the benchmark for stock market volatility since its introduction in 1993 and may be seen as an "investor fear gauge" following Whaley (2000). The original VIX was built as a proxy of at-the-money implied volatility with a one-month time to maturity. The CBOE decided to amend the index construction in 2003; see CBOE (2003). The new VIX is no longer computed from at-the-money strikes only, but also includes a set of out-of-the money calls and puts. Its computation is directly based on index option prices rather than on single implied volatilities and the underlying is the broader S&P 500 index instead of the S&P 100 index. Given the changes in index construction, the new VIX can be regarded as an approximation of the variance swap rate.
Daily VIX implied market volatility is given by VIX allows us to study asymmetric volatility under the risk-neutral measure, as formally, the VIX represents the risk-neutral expectation of future annualized 30day quadratic variation in S&P 500 index returns; see e.g. Carr and Wu (2006) and Todorov and Tauchen (2008). As a consequence, we expect differences in the behavior of volatility under the physical and under the risk-neutral probability measure, which are given by the variance risk premium; see e.g. Bollerslev, Sizova and Tauchen (2008).

Unexpected Returns and Volatility Changes
A summary of the results of estimating model (2.1-3) for our data set is given in   Table 1. They reveal several interesting features of logarithmic VIX volatility. First, as stated above, the given time-series specification allows us to remove linear and quadratic time-series dependence in log volatility changes. This is our main objective in the present study. Second, for lagged returns, our model indicates a statistically insignificant leverage effect but a significant hyper-level leverage effect. Interestingly, the asymmetric volatility effect, as measured by ω 2 , exhibits insignificant negative sign. Given that the logarithm is a strictly increasing transformation, negative parameter estimates of ω 1 and ω 2 suggest that lagged return shocks yield somewhat lower subsequent volatility levels. Hence, negative lagged return shocks predict an insignificant decrease in implied volatility. It therefore appears that the contemporaneous asymmetric negative dependence quickly reverts to a lagged weakly positive dependence, i.e. implied market volatility tends to somewhat decline one day after a negative market return shock. In sum, we do not find a significant leverage effect for lagged returns. However, on the hyper-level, we find strong asymmetry as the coefficient γ 2 estimate is significantly negative. While the significant asym- In summary, this finding lets us conclude that positive volatility shocks drive positive shocks in volatility of volatility. We graphically illustrate our results in Figure 1. The figure shows a plot of our estimation of volatility of volatility as well as plots of market returns and conditional VIX volatility. As can be seen from Figure 1, large absolute market returns may relate to jumps in both series, while the jump magnitude may be high in volatility but not in volatility of volatility and vice versa.

Time-Varying Asymmetric Volatility
Based on our discussion in Section 2, we now test for of the time-varying behavior of asymmetric volatility. There is reason to believe that the negative relation between market volatility and returns is not constant but stronger in some states of the economy than in others. Both our hypotheses, leverage and feedback, do not make clear predictions on such effects. Therefore, our empirical investigation sheds new light on this issue and estimate the DCC model according to equation (2.5). The log-likelihood function is given in Engle (2002) and Engle and Sheppard (2001), who note that conditional multivariate normality is an ML-estimation assumption, which is not required for the consistency and the asymptotic normality of the DCC estimator.  Figure 2. It can be seen that estimated correlation stays below zero throughout the sample period. Higher levels of the VIX tend to coincide with lower levels of negative correlation between return and volatility shocks, and vice versa. In other words, a higher market risk relates to higher absolute correlation, i.e. volatility asymmetry appears more pronounced when conditional volatility is high. Given this evidence of a strong negative average correlation throughout our sample, we conclude that asymmetric market volatility appears to be a persistent phenomenon with time-varying magnitude.

Asymmetric Volatility under Stress
We now study the return and volatility shocks, Y t and Z t , which are obtained from our GJR-GARCH specification (2.2-3) under periods of market stress. We thereby follow the idea of using an approximately iid series for EVT-analysis as proposed in Diebold, Schuermann and Stroughair (1998).
In order to study the extreme behavior of each marginal series, we first consider the GPD model (2.7). We perform a detailed pre-analysis in order to insure an appropriate threshold selection. In detail, we outline the results of four different methods of threshold selection in Table 3, namely use of the mean excess function (MEF) plot, use of the threshold plot, visual inspection based on these two plots as well as using an adaptive (i.e. automated) selection method. We choose an adaptive approach which aims at minimizing asymptotic mean squared estimation error as described in Section 4.7 ii of Beirlant, Goegebeur, Segers and Teugels (2004). We use the method in order to arrive at a threshold choice which is not guided by graphical (subjective) judgment. 7 We find here that the thresh- In the next step, we follow Coles, Heffernan and Tawn (1999) as outlined in Section 2.3. The corresponding Chi-and Chi-Bar-plots are given in Figure 4. As can be seen from the resulting asymptotic behavior in these plots, 'boom' shocks (Y t , −Z t ) appear asymptotically independent as χ(u) → 0 and χ(u) → 0.4 > 0.
However, 'crash' shocks (−Y t , Z t ) appear asymptotically dependent as χ(u) → 0.6 > 0 and χ(u) → 1. Hence, we may apply the bivariate threshold model and interpret the convergence of χ(u) as a first estimate of χ. We then perform a strict test of asymptotic independence. Falk and Michel (2006) 0 since both hypotheses state that χ = 0.
Hence, empirical evidence suggests a non-zero asymptotic melt-down probability.
Finally, for the crash scenario, (−Y t , Z t ), the bivariate threshold approach of Section 2.3 serves as a valid model. Therefore, we estimate the GPD model (2.7-8) jointly via MLE following the censored maximum likelihood approach of Ledford and Tawn (1996). Figure 5 serves as an illustration of the bivariate threshold approach, which models observations in the upper right corner of the plane of daily realized return and volatility shocks. Given our adaptive threshold choice, the ML estimation results for the marginal GPD distributions are given in Table   4. The estimation results for the symmetric and the asymmetric dependence model, (2.9) and (2.10), are given in Table 5.
The results for the marginal GPD model in Table 4 Table 5 demonstrate that the boom and the crash scenario both exhibit weak evidence of asymmetry within the dependence relation. and consider only the symmetric model (2.9) in the following. The estimation results in Table 5 indicate extreme dependence for both booms as well as crashes (estimated alphas all significantly different from 1), while dependence appears slightly stronger for crashes. As outlined in Section 2.4, this finding supports extreme volatility feedback as opposed to leverage. This conclusion can still be maintained once we point out that the bivariate threshold approach serves as a strictly valid model only for the crash scenario (we reject H 0 based on the KS-test), while it supposedly yields an overestimation of extreme dependence for the boom scenario. See also Ledford and Tawn (1996) for a discussion.
In an additional step we may test for causality in the joint tail using lagged variables. Rejection of the hypothesis H However, as is turns out, lagged exceedances are rare in our sample and it is not possible to derive reliable statements or even to perform strict tests. Simply by counting events, we observe a feedback event, (−Y t > u −y , Z t−1 > u z ), five times. In contrast, a leverage event, (−Y t−1 > u −y , Z t > u z ), is observed eight times in our sample. While this could provide some evidence in favor of leverage.
Considering unreported model parameter estimates as well as Chi and Chi-bar plots, we also have to emphasize that there is no evidence of asymptotic dependence in any form. Hence, we point out that extreme asymmetric volatility is characterized by a contemporaneous (i.e. intradaily) dependence relation.
In Table 6 we present estimated high-quantile exceedance probabilities for the symmetric logistic GPD model (2.9). The conditioning exceedance values are chosen as the 98, 98.5, 99, 99.5, 99.9 and the 99.95 percent quantile of the respective marginal distributions. Reported are joint as well as conditional exceedance probabilities. Given that extreme asymmetric volatility is mostly an intradaily contemporaneous phenomenon, the conditional exceedance probabilities allow us to distinguish between intradaily leverage and feedback causality.
As the results in Table 6 show, the conditional probability of a large return shock given a large volatility shock is slightly bigger than the reverse for the 98, 98.5, 99 and 99.5 percent quantiles. However, the difference becomes smaller and the situation finally reverses for the 99.9 and 99.95 percent quantiles.
The above finding supports the statement that both leverage as well as volatility feedback may play a role in explaining extreme asymmetric volatility. This evidence is supported by the results of Table 7, where we plot the estimated high-quantile exceedance probabilities for the symmetric logistic GPD model with lagged shocks. We choose the conditioning values as the 99 and 99.9 percent quantiles and then consider the conditional probabilities of large shocks in either series given a lagged large shock in the respective other series. As can be seen from the results, e.g., a volatility shock of 2.97 or higher and a subsequent return shock of -1.75 or smaller occurs with a probability of 2.69 percent. Such event would describe a lagged extreme volatility feedback event at the 99 percent exceedance level. As can further be seen from Table 7, leverage events appear with even higher probability. In sum, we again find that both hypothesis play a role in explaining volatility asymmetry.

Asset Pricing Implications of Extreme Volatility Feedback
Given the above results, extreme asymmetric volatility is found to be a contemporaneous phenomenon that is related to both, the leverage and the feedback effect. As a consequence of the latter, we find that large unexpected shocks to implied market volatility may indeed affect market returns. In other words, there is a considerable chance that large negative market price shocks happen conditional on shocks to implied market volatility. As pointed out in Li (2004) and Engle and Mistry (2007), such a possibility is consistent with results from intertemporal asset pricing models, once market volatility is a priced risk factor.
Given our results above, we now consider the empirical asset pricing implications of extreme volatility feedback in more detail.
Extreme volatility feedback may have substantial asset pricing implications.
Given an extreme volatility shock, a large negative return shock will occur with some given probability. In this case, the negative return shock has a given expectation, which is the "expected shortfall". Based on our model setting above, we can derive the asset pricing implications of extreme volatility shocks. More precisely, unexpected return and volatility shocks, (−Y t , Z t ), as defined above, allow us to derive the following: • Given a large unexpected shock to implied market volatility, Z t , which exceeds its p-percent quantile z, i.e. Z t > z, we may face a negative market shock, −Y t .
• With probability P (−Y t > −y|Z t > z) = s, extreme volatility feedback occurs: The negative market shock exceeds −y and implies an expected negative market return, or "expected shortfall", of: • With probability P (−Y t ≤ −y|Z t > z) = 1−s, no extreme feedback occurs.
In order to derive asset pricing implications of extreme volatility feedback, we make the following assumptions: • The given volatility shock, Z t > z, exceeds one of the p-percent quantiles z as given in Table 6. The table also reports the corresponding feedback probability, P (−Y t > −y|Z t > z), and the expected shortfall, E(−Y t |−Y t > −y).
• Predictions are made based on our return-volatility model (2.1-2), where the volatility shock affects returns via an update in volatility.
• We set the variables σ t and φ t in equation (  The results in Figure 6 document that the asset pricing impact increases with the volatility shock size as well as with the market volatility level. Assuming, for example, an implied market volatility at its sample mean of 19.1 percent annually, an unexpected 3-sigma market volatility jump (which occurs in one out of one hundred trading days on average) has a feedback probability of a little less than 50 percent (see Table 6). Given that extreme feedback occurs, aggregate market prices are predicted to decline by about 2.5 percent. Our conditioning assumption is relatively conservative. Assuming, other things equal, an already highly volatile (i.e. stressed) market at an annualized implied volatility of 30 percent, prices are predicted to decline by about 4 percent. Figure 6 also indicates that the asset pricing effects of feedback tend to increase in the range of increas-    Maximum likelihood parameter estimates of the Dynamic Conditional Correlation (DCC) between unexpected return and volatility shocks, Y t and Z t . The asymptotic mean test is based on the hypothesis that mean conditional correlation is equal to zero, the asymptotic variance test is based on the hypothesis that the variance of conditional correlation is equal to zero. The Jarque-Bera normality test is based on the null hypothesis that conditional correlation is normally distributed. Pearson's correlation test is based on the hypothesis that unconditional correlation is zero.   Maximum likelihood estimates of the marginal GPD model parameters according to Ledford and Tawn (1996) given the threshold choice of Table 3. Estimated standard errors are given in parenthesis. '#' denotes the number of marginal tail observations. logL is the negative logarithmic likelihood as evaluated at the parameter estimates, which are obtained via the Broyden-Fletcher-Goldfarb-Shanno method. Sample period: January 2, 1990 to September 30, 2008.  Maximum likelihood estimates of the symmetric and the asymmetric logistic GPD model dependence parameters according to Ledford and Tawn (1996) given the threshold choice of Table 3. Estimated standard errors are given in parenthesis. '#joint' denotes the number of joint tail observations. logL is the negative log likelihood as evaluated at the parameter estimates, which are obtained via the Broyden-Fletcher-Goldfarb-Shanno method. Sample period: January 2, 1990 to September 30, 2008.

Figure 2: VIX and Conditional Correlation between Return and Volatility Shocks
Daily conditional VIX market volatility (VIX) and dynamic conditional correlation (DCC) estimates of the correlation between S&P 500 index return shocks and VIX volatility shocks. Sample period: January 2, 1990 to September 30, 2008.

Figure 3: Mean Excess Function Plots for Return and Volatility Shocks
Mean excess function plots for daily S&P 500 index return shocks, Y, and VIX volatility shocks, Z. Large unexpected markets increases, Y, jointly with volatility decreases, −Z, are denoted as 'boom'. Large unexpected markets decreases, −Y, jointly with volatility increases, Z, are denoted as 'crash'. Sample period: January 2, 1990 to September 30, 2008. The figure plots daily realized negative S&P 500 index return shocks, −y, and VIX volatility shocks, z, together with the marginal threshold for each series under the visual threshold choice of Table 3, where u -y = 1.2 and u z = 1.5. Joint exceedances of the thresholds occur with an empirical probability of 3.21 percent. Sample period: January 2, 1990 to September 30, 2008. The figure plots estimated daily negative S&P 500 market price impacts under the occurrence of extreme volatility feedback. Initial volatility shocks, Z>z, exceed the given 98, 98.5, 99, 99.5, 99.9 and 99.95 percent quantiles z, respectively (see Table 6). For each impact curve, the initial implied market volatility is set to an annual percentage level of 19