Does the leverage effect affect the return distribution?
aa r X i v : . [ q -f i n . M F ] S e p Does the leverage effect affect the return distribution?
Dangxing Chen
Consortium for Data Analytics in Risk, Department of Economics, 530 Evans Hall,University of California, Berkeley, CA, 94720-3880, USA
Abstract
The leverage effect refers to the generally negative correlation between the re-turn of an asset and the changes in its volatility. There is broad agreementin the literature that the effect should be present for theoretical reasons, andit has been consistently found in empirical work. However, a few papers havepointed out a puzzle: the return distributions of many assets do not appearto be affected by the leverage effect. We analyze the determinants of the re-turn distribution and find that the impact of the leverage effect comes primarilyfrom an interaction between the leverage effect and the mean-reversion effect.When the leverage effect is large and the mean-reversion effect is small, thenthe interaction exerts a strong effect on the return distribution. However, if themean-reversion effect is large, even a large leverage effect has little effect on thereturn distribution. To better understand the impact of the interaction effect,we propose an indirect method to measure it. We apply our methodology toempirical data and find that the S&P 500 data exhibits a weak interaction ef-fect, and consequently its returns distribution is little impacted by the leverageeffect. Furthermore, the interaction effect is closely related to the size factor:small firms tend to have a strong interaction effect and large firms tend to havea weak interaction effect.
Keywords : Leverage effect; Stochastic volatility; Size; Risk.
Email address: [email protected] (Dangxing Chen) This work was supported by Southwest University of Finance and Economics through theConsortium for Data Analytics in Risk.
Preprint submitted to Elsevier
EL classification : G11, G12, G17, C58.
1. Introduction
The leverage effect refers to the observed tendency of changes in an asset’svolatility to be negatively correlated with the asset’s returns. The originalinterpretation goes back to Black (1976). The decline of the stock increases thefinancial leverage and thereby increases the volatility. Conversely, increases involatility must be compensated by increases in expected future returns, whichcan only be achieved by lowering the current stock price. There are manysubsequent papers examining the cause of the leverage effect (see, e.g., Christie,1982; Figlewski and Wang, 2000; French et al., 1987; Campbell and Hentschel,1992).Whatever the cause(s) of the leverage effect, there is broad agreement in theliterature that the effect is present. Many papers have attempted to accuratelyestimate the leverage effect (see, e.g., Wang and Mykland, 2014; Bandi and Ren`o,2012; Yu, 2005). For example, Ait-Sahalia et al. (2013) calculated that the cor-relation parameter at the high-frequency limit is ρ = − ,
77 by high-frequencydata of S&P 500 data with a robust estimation, indicating that there is a strongleverage effect. However, when stochastic volatility models (SVMs) are fittedto the S&P 500, a puzzle arises: the fitting is insensitive to the correlation pa-rameter (see, e.g., Chorro et al., 2018; Drgulescu and Yakovenko, 2002; Sepp,2008). How can it be that the return distribution is insensitive to the correla-tion parameter? This puzzle is the focus of this paper.Our studies rely on the framework of the continuous-time (CT) SVM. TheCT-SVM has been widely successful under both the risk-neutral measure (see,e.g., Sepp, 2008; Heston, 1993; Forde and Jacquier, 2009; Ahn and Gao, 1999;A¨ıt-Sahalia, 2002) and the physical measure (see, e.g., Drgulescu and Yakovenko,2002; Silva and Yakovenko, 2003; Bakshi et al., 2006). The main advantagesare that the CT-SVM is mathematically well-defined and fits the empiricaldata very well. There are also many useful discrete-time (DT) SVM (see, e.g.,2auwens et al., 2006; Duan, 1995; Lamoureux and Lastrapes, 1990). In partic-ular, the DT-SVM EGARCH model derived by Nelson (1991) is widely used inpractice. However, the CT-SVM usually has a more flexible form than the DT-SVM, and the discretization from continuous time to discrete time doesn’t seemalways appropriate. For instance, in empirical data, the annualized marginalvariance may grow over time; this is not allowed in the EGARCH model. There-fore, in this paper, we would only focus on the CT-SVM.Under our framework, using the stochastic volatility model of Heston (1993)as an example, we see that the leverage effect is important, but it need not havea strong impact on the return distribution. In particular, the impact of a strongleverage effect on the return distribution can be negated by a strong mean-reversion effect. Hence, when studying the distribution properties, one mustconsider the interaction effect between the leverage effect and the mean-reversioneffect. The interaction effect is not directly measurable, since volatility is latentand the Brownian motion is not observable. We propose an indirect method,relying on calculating the dynamics of the marginal variance. A direct impact ofthis measurement is that the annualized marginal variance will grow over timefor a strong interaction effect, but barely change for a weak interaction effect.This phenomenon also provides a better understanding of the performance ofthe square-root-of-time rule (SRTR) (see, e.g., Danielsson and Zigrand, 2006;Wang et al., 2011; Chen and Anderson, 2018). It is well known that, in thepresence of a strong mean-reversion effect, SRTR underpredicts annual volatilitywhen current volatility is low, and overpredicts annual volatility when currentvolatility is high. We show that, even with a weak mean-reversion effect, ifthe leverage effect is strong, then the interaction effect has a strong impact onthe return distribution. As a result, the SRTR tends to underpredict annualvolatility on average.We apply our methodology to study the relationship between the interac-tion effect with firm size. The empirical evidence indicates that the interactioneffect is strong for small firms but weak for large firms. As a consequence, theannualized marginal variance grows over time for small firms but barely change3or large firms.Finally, we aim to explain the leverage effect puzzle in S&P 500. By usingthe generalized hyperbolic distribution (see Eberlein et al., 1995), a distributionthat arises from the CT-SVM when ρ = 0, fits the S&P 500 return distributionvery well, despite the fact that the S&P 500 exhibits a strong leverage effect.By our method, we show that the interaction effect of the S&P 500 is weak,answering the puzzle.The paper is organized as follows. In Section 2, we introduce our basicframework of the stochastic volatility model, followed with a detailed example,the Heston model. Section 3 documents the presence of the leverage effectpuzzle, and provides the explanation and the solution. The relationship of theinteraction effect with firm size is explored in Section 4. Section 5 studies theleverage effect puzzle in S&P 500. Section 6 concludes.
2. Stochastic volatility model
We assume the price of a security follows the stochastic differential equation(SDE) dS t = rS t dt + p V t S t d e B t , (2.1) dV t = µ ( V t ) dt + σ ( V t ) dW t . (2.2)where e B t and W t are two standard Brownian motions with E [ d e B t dW t ] = ρ dt ,and r is the rate of return. Note that ρ = lim s → Corr( V t + s − V t , X t + s − X t ) (2.3)so that the leverage effect is summarized by the correlation parameter ρ underthe model (2.1) and (2.2). It is convenient to apply the Gram-Schmidt processto rewrite the price dynamics in terms of two independent Brownian motions dS t = rS t dt + ρ p V t S t dW t + p − ρ p V t S t dB t . (2.4) Estimates of the mean rate of return are notoriously noisy even over periods of years ordecades. Since we cannot make meaningful estimates of r t , we might as well assume it asconstant. X t = ln( S t ) can be derived by Itˆo’s Lemma, dX t = (cid:18) r − V t (cid:19) dt + ρ p V t dW t + p − ρ p V t dB t . (2.5)Notice that, for greater generality, we don’t specify the detailed form of thevariance process. The choice of µ ( V t ) and σ ( V t ) can be quite flexible. Weassume that the initial variance V > V t is a stationary process.We will focus on the marginal return distribution, since it can be directlyestimated from the empirical data. We derive some asymptotic properties ofthe marginal return distribution. We consider the centralized and scaled returndistribution e X t = X t − E [ X t ] √ t . (2.6)At a short horizon t →
0, we have e X t | V → N (0 , V ) . (2.7)With this expression, the related moments can be calculatedVar[ e X t ] = E [ V ] , (2.8)Skewness[ e X t ] = 0 , (2.9)Kurtosis[ e X t ] = 3 + 3 Var[ V ] E [ V ] . (2.10)Empirically, as one increases the time scale over which returns are calculated,their distributions looks more and more like a Gaussian distribution, as discussedby Cont (2001) . At a long horizon t → ∞ , under some mild conditions (seePeligrad, 1986) , which we believe is satisfied empirically, we have e X t → N (cid:18) , Var[ X t ] t (cid:19) . (2.11)The central moments of the Gaussian distribution are well known: E [ e X pt ] = , if p is odd , (cid:16) Var[ X t ] t (cid:17) p/ ( p − , if p is even . ( p − , . . . , ( p − .1. Heston model Throughout the paper, we will frequently use the stochastic volatility modelof Heston (1993), which has the advantage of providing explicit expressions formany distributional properties. The equation is written as dX t = (cid:18) r − V t (cid:19) dt + ρ p V t dW t + p − ρ p V t dB t , (2.12) dV t = κ ( θ − V t ) dt + σ p V t dW t . (2.13)The dynamic of the variance process in the Heston model is also known as theCox-Ingersoll-Ross (CIR) process (see Cox et al., 2005). We assume in whatfollows that the Feller condition 2 κθ > σ holds, which guarantees that thevariance process is always strictly positive.The marginal density function can be written in terms of the Fourier integral(see Drgulescu and Yakovenko, 2002) P t ( x ) = 12 π Z + ∞−∞ e ip x x + F t ( p x ) dp x , (2.14)with F t ( p x ) = κθσ Γ t − κθσ ln (cid:20) cosh (cid:18) Ω t (cid:19) + Ω − Γ + 2 κ Γ2 κ Γ sinh (cid:18) Ω t (cid:19)(cid:21) , (2.15)Γ = κ + iρσp x , (2.16)Ω = p Γ + σ ( p x − ip x ) . (2.17)With this, the marginal density function of return can be recovered via the fastFourier transform (see, e.g., Valsa and Branˇcik, 1998; Abate and Whitt, 1995)Many formulas regarding the marginal moments can then be derived fromFormula (2.14). For an intuitive understanding, we consider the first four mo-ments. The marginal expectation is given by E [ X t ] = rt − θt . (2.18)The marginal variance is given byVar[ X t ] = E (cid:20)Z t V s ds (cid:21) + 14 Var (cid:20)Z t V s ds (cid:21) − ρ E (cid:20)(cid:18)Z t V s ds (cid:19) (cid:18)Z t p V s dW s (cid:19)(cid:21) . (2.19)6nd the expression for these terms are E (cid:20)Z t V s ds (cid:21) = θt, (2.20) E (cid:20)(cid:18)Z t V s ds (cid:19) (cid:18)Z t p V s dW s (cid:19)(cid:21) = θ σκ (cid:20) t + e − κt − κ (cid:21) , (2.21)Var (cid:20)Z t V s ds (cid:21) = θ σ κ (cid:20) t + e − κt − κ (cid:21) . (2.22)The marginal skewness isSkewness[ X t ] = E [( X t − E [ X t ]) ]Var[ X t ] / , (2.23)where E [( X t − E [ X t ]) ] = 38 κ e − κt θσ ( − κρ + σ )( − κ + 8 κρσ + 4 tκ ρσ − σ − tκσ (2.24)+ e κt ( − κ ( − tκ ) + 4 κ ( − tκ ) ρσ + (2 − tκ ) σ )) . (2.25)The marginal kurtosis isKurtosis[ X t ] = E [( X t − E [ X t ]) ]Var[ X t ] , (2.26)where E [( X t − E [ X t ]) ] = 332 κ e − κt θ ( σ ( − κρ + σ ) (2 θκ + σ )+ 4 e κt σ ( − tθκ ρ + 4 κ ((2 + tθ ) κ + 4( θ ( − tκ ) + 2 κ (2 + tκ )) ρ ) σ − κ ρ ( θ ( − tκ ) + 6 κ (2 + tκ ) + 2 κ (6 + tκ (4 + tκ )) ρ ) σ + κ ( θ ( − tκ ) + 4 κ (6 + 3 tκ + (34 + tκ (24 + 5 tκ )) ρ )) σ − κ (7 + tκ (5 + tκ )) ρσ + (7 + tκ (5 + tκ )) σ )+ e κt (2 t θκ (4 κ − κρσ + σ ) + 2 tκσ (4 κ − κρσ + σ )(8 θκ ρ + 2 κ ( − θ + 2 κ + 8 κρ ) σ − κρσ + 5 σ )+ σ ( − κ (1 + 8 ρ ) − σ + 2 κσ ( θ + 116 ρσ ) + 32 κ ρ ( θρ + 12(1 + ρ ) σ ) − κ σ (6 σ + ρ ( θ + 35 ρσ ))))) . . The interaction effect To motivate the analysis that follows, we start with a straightforward ar-tificial example to illustrate the leverage effect puzzle. Figure 1 compares themarginal densities of the return in the Heston model (2.14) with and withoutthe leverage effect for two sets of Heston parameters κ, σ . In the first examplewith κ = 16 and σ = 0 .
8, we see a strong impact of the leverage effect. In fact,with ρ = −
1, the marginal density is quite negatively skewed. In the secondexample, with κ = 1 and σ = 0 .
02, we see that the two marginal densities with ρ = 0 and ρ = − -0.2 0 0.20510152025303540 [ , , ]=[16,0.02,0.8] =0=-1 -0.2 0 0.20510152025303540 [ , , ]=[1,0.02,0.02] =0=-1 Figure 1: The marginal density of the Heston model at 5-day horizon with and without theleverage effect .2. Interaction effect When the leverage effect is not present, i.e., ρ = 0, the solution to the SDE(2.5) can be written as X t (cid:12)(cid:12)(cid:12)(cid:12) Z t V s ds = N (cid:18) r − Z t V s ds, Z t V s ds (cid:19) . (3.1)Now consider another case when there is a strong mean-reversion effect suchthat V t is independent of W t , then the equation (3.1) also holds and the impactof the leverage effect is not observed. This situation could occurs, for example,when the effect from the diffusion term is relatively weak. Returning to theexample of the Heston model in Figure 1, let’s see the impact of the leverageeffect on the marginal moments of return. At a short horizon t ∼
0, the followingsimple approximations follow from equations (2.20), (2.21), and (2.22) E [ X t ] ∼ rt − θt , Var[ X t ] ∼ θt, Skewness[ X t ] ∼ σ (cid:18) ρ − σκ (cid:19) r tθ , Kurtosis[ X t ] ∼ σ κθ . Note the expectation, variance, and kurtosis are not affected by the leverageeffect, but the skewness is. Even with a strong leverage effect, if σ is small (i.e.the variance process V t is not very volatile), the skewness will be close to 0 andthe impact of the leverage effect will be hard to observe.Hence, a strong leverage effect can be negated by a strong mean-reversioneffect. When studying return distributions, it is essential to incorporate the in-teraction effect, namely the interaction between the mean-reversion and leverageeffects. Since the variance is latent and the Brownian motion is not observable,we propose an indirect method to measure the interaction effect. We need an accurate, robust, and interpretable measure of the interactioneffect. Commonly used measurements in summary statistics include the location9e.g., mean, median), dispersion (e.g., standard deviation), and shape (skewnessand kurtosis). For these measurements, calculations typically rely on momentsor quantiles. It would be natural to focus on the moment-based calculations,since these can be readily interpreted in terms of the SDE. Unfortunately, theestimation of moments in financial data may be inaccurate. For example, con-sider the expectation of return E [ X t ] = rt − θt . At a short horizon, the standarddeviation of the return p Var[ X t ] ∼ √ θt has a much larger magnitude than itsexpectation. As a consequence, we cannot accurately estimate the mean returneven with about 100 years of daily return, due to the slow convergence underthe central limit theorem. The lack of robustness in the calculation of skewnessand kurtosis in financial data using moments is documented in Kim and White(2004) and Bonato (2011).Therefore, among these common measurements, we focus on the calculationof the marginal variance since it can be estimated accurately and is easily in-terpretable. Recall that by algebra, the marginal variance can be decomposedinto three terms:Var[ X t ] = E (cid:20)Z t V s ds (cid:21) + 14 Var (cid:20)Z t V s ds (cid:21) − ρ E (cid:20)(cid:18)Z t V s ds (cid:19) (cid:18)Z t p V s dW s (cid:19)(cid:21) . (3.2)For simplicity, we use the following notation for these terms:EIV t = E (cid:20)Z t V s ds (cid:21) , (3.3)VIV t = 14 Var (cid:20)Z t V s ds (cid:21) , (3.4)EMIV t = − ρ E (cid:20)(cid:18)Z t V s ds (cid:19) (cid:18)Z t p V s dW s (cid:19)(cid:21) . (3.5)Note that the EIV t is scalable with respect to time, i.e., EIV t = ts EIV s . Dueto the mean-reversion effect, empirically, we believe the VIV t is negligible com-paring to the EIV t , i.e., VIV t ≪ EIV t . Then the EMIV t comes from thecontribution from the interaction effect and is the term we want to measure.The EMIV t should be close to 0 for a weak interaction effect, far away from 0for a strong interaction effect. 10or a more intuitive understanding of these formulas, we use the Hestonmodel as an example, taking formulas from Equations (2.20), (2.21), and (2.22).At a short horizon t ∼ E (cid:20)Z t V s ds (cid:21) ∼ θt, (3.6) E (cid:20)(cid:18)Z t V s ds (cid:19) (cid:18)Z t p V s dW s (cid:19)(cid:21) ∼ θ σ t , (3.7)Var (cid:20)Z t V s ds (cid:21) ∼ θ σ κ t . (3.8)Thus, at a short horizon t ∼
0, we have Var[ X t ] ∼ EIV t . At a long horizon t ∼ ∞ , we have E (cid:20)Z t V s ds (cid:21) ∼ θt, (3.9) E (cid:20)(cid:18)Z t V s ds (cid:19) (cid:18)Z t p V s dW s (cid:19)(cid:21) ∼ θt σκ , (3.10)Var (cid:20)Z t V s ds (cid:21) ∼ θt σ κ . (3.11)Thus, the contribution from the EMIV t can be observed. In practice, the calcu-lation is done by EMIV t ≈ Var[ X t ] − ts Var[ X s ] for t > s . Note that each term isscaled by a factor σκ . This term serves as a mean-reversion factor in the Hestonmodel.From the calculation of the interaction effect by EMIV t , there is a directimpact: the annualized marginal variance will grow over time until it convergeswhen the interaction effect is strong, but it will barely change when the interac-tion effect is weak. This phenomenon suggests that in practice, if one observestwo securities with the same average variance at a short horizon, it doesn’t im-ply that these two securities have the same volatility. If the interaction effect isstrong for one and weak for another one, the security with the strong interactioneffect will become more volatile over time.Another advantage of measuring the marginal variance is we know the cen-tralized and scaled return e X t would eventually converge to Gaussian (2.11).Suppose the information of marginal variance is known, we then know the11symptotic distribution of the return. In practice, the asymptotic distribu-tion of the return can be easily obtained for a security with a weak interactioneffect. It is common to use the SRTR (see Danielsson and Zigrand, 2006) to extrap-olate the conditional variance of the log return Var[ X t | V ]. Under the SVM, theSRTR serves as the constant approximation to the conditional varianceVar[ X t | V ] ≈ V t. We discuss the impact of the mean-reversion effect and the leverage effect onthe performance of the SRTR, along with the Heston model as an example inFigure 2. It is well known that, because it ignores the mean-reversion effect, theSRTR tends to under-predict for small initial variance but over-predict for largeinitial variance. As shown in the first example in Figure 2, due to the mean-reversion effect, the conditional variance is flatter than the SRTR. But this isnot the whole story; one must also take the interaction effect into account. Ifthe mean-reversion effect is weak and the leverage effect is strong, the resultingstrong interaction effect leads SRTR tends to under-predict on average, as shownin the second example of Figure 2. This also explains the downward-biasedprediction using the SRTR observed by Wang et al. (2011). In the case of aweak mean-reversion effect and a weak leverage effect, the SRTR serves as agood approximation, as plotted in the third example in Figure 2.12 V strong mean-reversion[ , , , ]=[5,0.02,0.2,-1] V tVar[X t |V ] 0 0.02 0.04 0.06 0.08 0.1 V weak mean-reversion and strong leverage[ , , , ]=[0.1,0.02,1,-1] V tVar[X t |V ] 0 0.02 0.04 0.06 0.08 0.1 V weak mean-reversion and weak leverage[ , , , ]=[0.1,0.02,1,0] V tVar[X t |V ] Figure 2: Performance of the square-root-of-time rule by Heston model for different cases at2-month horizon
4. Relationship of the interaction effect with firm size
Here, we use five value-weighted size portfolios constructed by Fama andFrench , sorted by market capitalization, price times shares outstanding. AllNYSE, AMEX, and NASDAQ stocks are included. We use daily data from July1926 to January 2019.To study the interaction effect, we compare the dynamics of the marginalvariance to the EIV t over time in Figure 3. From the figure, we observe a cleartrend. The gap between the marginal variance at time t and EIV t grows overtime for small firms but is nearly zero for large firms. This implies that theinteraction effect is stronger for small firms than larger firms. The data is downloaded from the website http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html days -3 Var[X t ]EIV t days -3 Var[X t ]EIV t days -3 Var[X t ]EIV t days -3 Var[X t ]EIV t days -3 Var[X t ]EIV t Figure 3: The interaction effect for portfolios constructed by size
For a more quantitative understanding, we also calculate the summary statis-tics. Define RIV t as the proportion of the EMIV t to the marginal varianceVar[ X t ], RIV t = Var[ X t ] − EIV t Var[ X t ] . The summary statistics at the 25-day time horizon are given in Table 1. Fromthe first column, it is observed that the EIV t is larger for small firms thanlarge firms, implying that at a short horizon, small firms are more volatile thanlarge firms, which is not surprising. From the second column, we see that theinteraction effect is stronger for small firms than large firms. In particular, the14nnualized marginal variance roughly doubles for the smallest firms (Var[ X t ] =(1 − . − EIV t ), but barely changes for largest firms at the 25-day horizon. Table 1: Summary statistics of the interaction effect for the portfolios constructed by size
Portfolios EIV RIV −
20% 0.037 0.5120% −
40% 0.035 0.4140% −
60% 0.032 0.3660% −
80% 0.030 0.2780% − Var[ V ] E [ V ] .Hence, at a short horizon, the excess kurtosis measures the volatility of thevariance process V t . The excess kurtosis is closely related to the mean-reversioneffect: if the variance process V t is not volatile or the mean-reversion effect isstrong, the excess kurtosis will be small.Unfortunately, as pointed out by Kim and White (2004) and Bonato (2011),the measurement of kurtosis by moments is not robust for stock returns. As analternative, we use the quantile-based measurement developed in Crow and Siddiqui(1967). The centered coefficient isExcess Kurtosis CS = F − (0 . − F − (0 . F − (0 . − F − (0 . − . , (4.1)where F is the empirical cumulative distribution function. Since we have ap-proximately 24,000 daily observations, the 0.025 and 0.975 quantiles can bemeasured with reasonable accuracy. This formula is applied to size portfoliosin Table 2. The results indicate that for small firms, in addition to the pricedynamics, the variance process V t is also more volatile than for large firms.This also implies that the mean-reversion effect is weaker for small firms thanfor large firms and therefore helps to explain why the interaction effect for isstronger for small firms than for large firms.15 able 2: Summary statistics of the marginal variance, skewness, and kurtosis for the portfoliosconstructed by size at 1-day horizon Portfolios Annualized marginal variance Skewness H Excess Kurtosis CS −
20% 0.039 -0.12 2.2220% −
40% 0.036 -0.11 2.0440% −
60% 0.034 -0.11 1.9660% −
80% 0.032 -0.093 1.8980% − H = F − (1 − α ) + F − ( α ) − F − (0 . F − (1 − α ) − F − ( α ) (4.2)with the choice of α = 0 .
05. The result is given in Table 2. The result indicatesthat the return distribution is more negatively skewed for small firms than largefirms.
5. Disentangling the puzzle for S&P 500
In this section, we explain the observed puzzle that the fitting of the S&P500 is insensitive to the correlation parameter ρ , even though the empirical dataindicates that there is a strong leverage effect ρ ∼ − . Y follows a generalized hyperbolicdistribution we write Y ∼ H ( λ, α, β, δ, µ ) . p Y ( x ) = ( γ/δ ) λ √ πK λ ( δγ ) K λ − ( α p δ + ( x − µ ) )( p δ + ( x − µ ) /α ) β − λ e ( x − µ ) , where γ = α − β , and K λ is the modified Bessel function of the third kindwith index λ . The parameter domain for the class of GH distributions is givenby δ ≥ , α > , α > β , if λ > ,δ > , α > , α > β , if λ = 0 ,δ > , α ≥ , α ≥ β , if λ < . The generalized inverse Gaussian (GIG) distribution (see Seshadri, 2004) isclosely related to GH. If V follows a GIG distribution we write V ∼ GIG ( λ, δ, γ ) . (5.1)The PDF of a GIG distribution is given by p V ( x ) = ( γ/δ ) λ K λ ( δγ ) x λ − e − ( δ x − + γ x ) , x > . (5.2)The parameter domain for the class of GIG distributions is given by δ > , γ ≥ , if λ < ,δ > , γ > , if λ = 0 ,δ ≥ , γ > , if λ > . The GH distribution was originally derived in Barndorff-Nielsen (1977); it is aGaussian variance-mean mixture where the mixing distribution is GIG. In otherwords, if Y | V = v ∼ N ( µ + βv, v ) , and V ∼ GIG( λ, δ, λ ), then the marginal distribution of Y will be GH, Y ∼ H ( λ, α, β, δ, µ ), where α = β + γ . To connect the GIG to the SDE, Sørensen(1997) considers the SDE dV t = (cid:16) β V α − t − β V αt + β V α − t (cid:17) dt + κV αt dW t , (5.3)17here β = κ ( λ −
1) + κ α, β = ( κγ ) , β = ( κδ ) . This SDE hasa stationary distribution which is GIG. Note that if α = , then the diffusionprocess is the solution to dV t = (cid:18) β − β V t + β V t (cid:19) dt + κ p V t dW t , (5.4)which is the CIR-process with an additional β V t in the drift term. Empirically,the volatility of the market is never zero. In the CIR-process, the variancecan become zero if the Feller condition is violated, and imposing the Fellercondition limits our ability to calibrate the model to data. The additional β V t term prevents the variance from becoming zero, while leaving more freedom tofit the data.The fitting of the empirical distribution of the S&P 500 by GH is shownin Figure 4. Visually, the fit is virtually perfect. Quantitatively, we apply theKolmorogov-Smirnov test (see, e.g., Weiss, 1978; Chicheportiche and Bouchaud,2011) to verify with daily returns drawn one-month apart so that the data isonly very weakly dependent. The test fails to reject the null hypothesis thatthe sample is drawn from the GH distribution at the 5% significance level,confirming that we have a good fit. This experiment, which shows that it ispossible to fit the S&P 500 return distribution with the GH distribution, whichis derived under the assumption of zero leverage effect, confirms that the leverageeffect has only a very weak impact on the S&P 500 return distribution.18 empirical distributiongeneralized hyperbolic distribution Figure 4: The fitness of the S&P 500 by the generalized hyperbolic distribution at the dailyscale empirical distributionuniform distribution Figure 5: The Kolmorogov-Smirnov test test of the S&P 500 at the daily scale
We now turn to the interaction effect for the S&P 500. The comparisonof the dynamics of marginal variances with EIV t is plotted in Figure 6. Themarginal variance is quite close to EIV t , indicating that the interaction effectis quite weak. We believe this weak interaction effect is the reason that theleverage effect has little impact on the distribution for S&P 500.To back up our story, we check the mean-reversion property of the S&P500. We found that the interaction effect is weak for the S&P 500. Fromthe literature, the leverage effect is strong. By our analysis, this implies thatthe mean-reversion effect for S&P 500 must be strong. The excess kurtosis,calculated using Equation (4.1) of S&P 500 is relatively low, at 1.72, whichconfirms that the mean-reversion effect is strong, consistent with our analysis. The result is quite similar to that for large firms, 1.73. This is not surprising since theS&P 500 is composed essentially of the 500 largest firms by market capitalization. days -3 Var[X t ]EIV t Figure 6: The interaction effect of the S&P 500 over time
6. Conclusion
In this paper, we study the impact of the leverage effect on the return dis-tribution. In particular, we focus on the marginal distribution since it can bedirectly formed from the empirical data. We find that the leverage effect is im-portant, but it need not have a big impact on the return distribution. A strongleverage effect can be negated by a strong mean-reversion effect. Hence, whenstudying the return distribution properties, one must consider the interactioneffect, which is the mixture of the leverage effect and the mean-reversion effect.The interaction effect is not directly measurable. Our measurement relieson the dynamics of the marginal variance, which can be estimated accurately.A direct impact of our measurement is that the annualized marginal variance ofthe return will grow over time until it converges for a strong interaction effect,but will barely change for a weak interaction effect. The study of the interactioneffect also shed light on the performance of the SRTR. Even with a weak mean-21eversion effect, if the interaction effect is strong, the SRTR will not give anaccurate approximation.When applying our methodology to empirical data, we observed some in-teresting phenomena. We found that the interaction effect is stronger for smallfirms than large firms. We resolved the puzzle that fitting the S&P 500 returndistribution is insensitive to the leverage effect. By employing the GH distribu-tion, a marginal distribution for a class of stochastic volatility model with noleverage effect, we confirmed that the S&P 500 return distribution can be fittedwell without the leverage effect. By our method, we found the interaction effectfor S&P 500 is weak, which we believe is the explanation to the puzzle.
Acknowledgments
We thank Robert Anderson and Lisa Goldberg for helpful discussions andcomments.