NBER WORKING PAPER SERIES THE LEVERAGE EFFECT PUZZLE: DISENTANGLING SOURCES OF BIAS AT HIGH FREQUENCY. Yacine Ait-Sahalia Jianqing Fan Yingying Li

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1 NBER WORKING PAPER SERIES THE LEVERAGE EFFECT PUZZLE: DISENTANGLING SOURCES OF BIAS AT HIGH FREQUENCY Yacine Ait-Sahalia Jianqing Fan Yingying Li Working Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cabridge, MA Noveber 2011 Aït-Sahalia's research was supported by NSF grant SES Fan's research was supported by NSF grants DMS and DMS Li's research was supported by the Bendhei Center for Finance at Princeton University and the RGC grant DAG09/10.BM12 at Hong Kong University of Science and Technology. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Econoic Research. NBER working papers are circulated for discussion and coent purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accopanies official NBER publications by Yacine Ait-Sahalia, Jianqing Fan, and Yingying Li. All rights reserved. Short sections of text, not to exceed two paragraphs, ay be quoted without explicit perission provided that full credit, including notice, is given to the source.

2 The Leverage Effect Puzzle: Disentangling Sources of Bias at High Frequency Yacine Ait-Sahalia, Jianqing Fan, and Yingying Li NBER Working Paper No Noveber 2011 JEL No. C22,G12 ABSTRACT The leverage effect refers to the generally negative correlation between an asset return and its changes of volatility. A natural estiate consists in using the epirical correlation between the daily returns and the changes of daily volatility estiated fro high-frequency data. The puzzle lies in the fact that such an intuitively natural estiate yields nearly zero correlation for ost assets tested, despite the any econoic reasons for expecting the estiated correlation to be negative. To better understand the sources of the puzzle, we analyze the different asyptotic biases that are involved in high frequency estiation of the leverage effect, including biases due to discretization errors, to soothing errors in estiating spot volatilities, to estiation error, and to arket icrostructure noise. This decoposition enables us to propose novel bias correction ethods for estiating the leverage effect. Yacine Ait-Sahalia Departent of Econoics Fisher Hall Princeton University Princeton, NJ and NBER yacine@princeton.edu Yingying Li Departent of Inforation Systes, Business Statis Hong Kong University of Science and Technology yyli@ust.hk Jianqing Fan Bendhei Center for Finance 26 Prospect Ave Princeton NJ jqfan@princeton.edu

3 1 Introduction The leverage effect refers to the observed tendency of an asset s volatility to be negatively correlated with the asset s returns. Typically, rising asset prices are accopanied by declining volatility, and vice versa. The ter leverage refers to one possible econoic interpretation of this phenoenon, developed in Black (1976) and Christie (1982): as asset prices decline, copanies becoe echanically ore leveraged since the relative value of their debt rises relative to that of their equity. As a result, it is natural to expect that their stock becoes riskier, hence ore volatile. While this is only a hypothesis, this explanation is sufficiently prevalent in the literature that the ter leverage effect has been adopted to describe the statistical regularity in question. It has also been docuented that the effect is generally asyetric: other things equal, declines in stock prices are accopanied by larger increases in volatility than the decline in volatility that accopanies rising stock arkets (see, e.g., Nelson (1991) and Engle and Ng (1993)). Various discrete-tie odels with a leverage effect have been estiated by Yu (2005). The agnitude of the effect however sees too large to be attributable solely to an increase in financial leverage: Figlewski and Wang (2000) noted aong other findings that there is no apparent effect on volatility when leverage changes because of a change in debt or nuber of shares, only when stock prices change, which questions whether the effect is linked to financial leverage at all. As always, correlation does not iply causality. Alternative econoic interpretations have been suggested: an anticipated increase in volatility requires a higher rate of return fro the asset, which can only be produced by a fall in the asset price (see, e.g., French et al. (1987) and Capbell and Hentschel (1992)). The leverage explanation suggests that a negative return should ake the fir ore levered, hence riskier and therefore lead to higher volatility; the volatility feedback effect is consistent with the sae correlation but reverses the causality: increases in volatility lead to future negative returns. These different interpretations have been investigated and copared (see Bekaert and Wu (2000)), although at the daily and lower frequencies the direction of the causality ay be difficult to ascertain since they both appear to be instantaneous at the level of daily data (see Bollerslev et al. (2006)). Using higher frequency data, naely five-inute absolute returns to construct a realized volatility proxy over longer horizons, Bollerslev et al. (2006) find a negative correlation between the volatility and the current and lagged returns, which lasts for several days, low correlations between the returns and the lagged volatility and strong correlation between the 2

4 high-frequency returns and their absolute values. Their findings support the dual presence of a prolonged leverage effect at the intradaily level, and an alost instantaneous volatility feedback effect. Differences between the correlation easured using stock-level data and index-level data have been investigated by Duffee (1995). Bollerslev et al. (2011) develop a representative agent odel based on recursive preferences in order to generate a volatility process which exhibits clustering, fractional integration, and has a risk preiu and a leverage effect. Whatever the source(s) or explanation(s) for the presence of the leverage effect correlation, there is broad agreeent in the literature that the effect should be present. So why is there a puzzle, as suggested by the title of this paper? As we will see, using high frequency data and standard estiation techniques, the data stubbornly refuse to confor to these otherwise appealing explanations. We find that, at high frequency and over short horizons, the estiated correlation ρ between the asset returns and changes in its volatility is close to zero, instead of the strong negative value that we have coe to expect. At longer horizons, or especially using option-iplied volatilities, the effect is present. If we accept that the true correlation is indeed negative, then this is especially striking since a correlation estiator relies on second oent, or quadratic (co)variation, quantities and as such should be estiated particularly well at high frequency, or instantaneously, using standard probability liit results. We call this disconnect the leverage effect puzzle, and the purpose of this paper is to exaine the reasons for it. At first read, this behavior of the estiated correlation at high frequency can be reiniscent of the Epps Effect. Starting with Epps (1979), it has indeed been recognized that the epirical correlation between the returns of two assets tends to decrease as the sapling frequency of observation increases. One essential issue that arises in the context of high frequency estiation of the correlation coefficient between two assets is the asynchronicity of their trading, since two assets will generally trade, hence generate high frequency observations, at different ties. Asynchronicity of the observations has been shown to have the potential to generate the Epps Effect. 1 However, the asynchronicity proble is not an issue here since we are focusing on the estiation of the correlation between an asset s returns and its (own) volatility. Because the 1 As a result, various data synchronization ethods have been developed to address this issue: for instance, Hayashi and Yoshida (2005) have proposed a odification of the realized covariance which corrects for this effect; see also Large (2007), Griffin and Ooen (2008), Voev and Lunde (2007), Zhang (2011), Barndorff-Nielsen et al. (2008b), Kinnebrock and Podolskij (2008) and Aït-Sahalia et al. (2010). 3

5 volatility estiator is constructed fro the asset returns theselves, the two sets of observations are by construction synchrone. On the other hand, while asynchronicity is not a concern, one issue that is gerane to the proble we consider in this paper is the fact that one of two variables entering the correlation calculation is latent, naely the volatility of the asset returns. Relative to the Epps Effect, this gives rise to a different set of issues, specifically the need to eploy preliinary estiators or proxies for the volatility variable, such as realized volatility () for exaple, in order to copute its correlation with asset returns. We will show that the latency of the volatility variable is partly responsible for the observed puzzle. One further issue, which is in coon at high frequency between the estiation of the correlation between two asset returns and the estiation of the correlation between an asset s return and its volatility, is that of arket icrostructure noise. When sapled at sufficiently high frequency, asset prices tend to incorporate noise that reflects the echanics of the trading process, such as bid/ask bounces, the different price ipact of different types of trades, liited liquidity, or other types of frictions. To address this issue, we will analyze the effect of using noise-robust high frequency volatility estiators for the purpose of estiating the leverage effect. 2 Our ain results are the following. We provide theoretical results to disentangle the biases involved in estiating the correlation between the returns and volatilities with a sequence of progressively ore realistic estiators. We proceed increentally, in such a way that we can isolate the sources of the bias one by one. Starting with the spot volatility, an ideal but unavailable estiator since volatility is unobservable, we will see that the leverage effect paraeter ρ is 2 In the univariate volatility case, any estiators have been developed to produce consistent estiators despite the presence of the noise. These include the Two Scales Realized Volatility (TS) of Zhang et al. (2005), Multi-Scale Realized Volatility (MS), a odification of TS which achieves the best possible rate of convergence proposed by Zhang (2006), Realized Kernels (RK) by Barndorff-Nielsen et al. (2008a), the Pre-Averaging volatility estiator (PAV) by Jacod et al. (2009), and the Quasi-Maxiu Likelihood Estiator (QMLE) of Xiu (2010) which extends the paraetric Maxiu-Likelihood Estiator of Aït-Sahalia et al. (2005) to the setting of stochastic volatility. Related work include Bandi and Russell (2006), Delattre and Jacod (1997), Fan and Wang (2007), Gatheral and Ooen (2010), Hansen and Lunde (2006), Kalnina and Linton (2008), Li and Mykland (2007), Aït-Sahalia et al. (2011) and Li et al. (2010). To estiate the correlation between two assets, or any two variables that are observable, Zhang (2011) proposed a consistent Two Scales Realized Covariance estiator (TSCV), Barndorff-Nielsen et al. (2008b) a Multivariate Realized Kernel (MRK), Kinnebrock and Podolskij (2008) a ultivariate Pre-Averaging estiator and Aït-Sahalia et al. (2010) a ultivariate Quasi- Maxiu Likelihood Estiator. 4

6 already estiated with a bias that is due solely to discretization. The unobservable spot volatility is frequently estiated by a local tie-doain soothing ethod which involves integrating the spot volatility over tie, locally. Replacing the spot volatility by the (also unavailable) true integrated volatility, the bias for estiating ρ is even larger, but reains quantifiable. The increental bias is due to soothing. Replacing the true integrated volatility by an estiated integrated volatility, the bias for estiating ρ becoes so large that, when calibrated on realistic paraeter values, the estiated ρ becoes essentially zero, which is indeed what we find epirically. The increental bias represents the effect of the estiation error. We then exaine the effect of using noise-robust estiators of the integrated volatility, and copute the resulting additional bias ter, which can of course go in the reverse direction. Based on the above results, we propose a regression approach to copute bias-corrected estiators of ρ. We investigate these effects in the context of the Heston stochastic volatility odel, which has the advantage of providing explicit expressions for all these bias ters. The paper is organized as follows. Section 2 docuents the presence of the leverage effect puzzle. The prototypical odel for understanding the puzzle and nonparaetric estiators for spot volatility are described in Section 3. Section 4 presents the ain results of the paper, which unveil the biases of estiating leverage effect paraeter in all steps of approxiations. A novel solution to the puzzle is proposed in Section 5, which is convincingly deonstrated by Monte Carlo siulations in Section 6 and by epirical studies in Section 7 using the highfrequency data fro S&P500 and Microsoft. Section 8 concludes. The appendix contains the atheatical proofs. 2 Motivation: The Leverage Effect Puzzle To otivate the theoretical analysis that follows, we start with a straightforward epirical exercise to illustrate the leverage effect puzzle. A scatter plot of estiated changes of volatilities and returns provides a siple way to exaine graphically the relationship between estiated changes in volatility and changes in log-prices (i.e., log-returns). Figure 1 shows scatter plots of the differences of estiated daily volatilities ˆV t ˆV t against the corresponding returns of horizon days for several assets, where ˆV t is the integrated daily volatility estiated by the noise-robust TS estiator. If we start with long horizons, as shown in Figure 1, we see that the effect is present in the data. 5

7 +++ Insert Figure 1 Here +++ In addition to the evidence that coes fro long horizons, the effect is even stronger epirically if we use a different easureent altogether of the asset volatility, based on arket prices of derivatives. In the case of the S&P 500 index, we eploy VIX, which is the square root of the par variance swap rate with thirty day to aturity; that is, VIX easures the square-root of the risk neutral expectation of the S&P 500 variance over the next thirty calendar days. Using this arket-based volatility easure, the leverage effect is indeed very strong as deonstrated in Figure Insert Figure 2 Here +++ Yet, starting at the daily horizon, even when using high frequency volatility estiates, we see in Figure 3 that the scatter plot of ˆD t = ˆV t ˆV t 1 against daily returns R t shows no apparent leverage effect for the different assets considered. As discussed in the Introduction, different econoic explanations provide for different causation between returns and their volatility. To be robust against the tiing differences that different causality explanations would generate, we next exaine scatter plots of different tie lags and leads such as {( ˆD t 1, R t )} and {( ˆD t, R t 1 )}. The evidence again reveals no leverage effect. Siilar results are obtained if we eploy different tie periods and/or different noise-robust volatility estiators such as QMLE or PAV. +++ Insert Figure 3 Here +++ There are sound econoic rationales to support a prior that a leverage effect is present in the data, and we do indeed find it in Figures 1 and 2. So why are we unable to detect it on short horizon based on high frequency volatility estiates that should provide precise volatility proxies? This is the nature of the leverage effect puzzle that we seek to understand. Can it be the result of eploying estiators that are natural at high frequency for the latent volatility variable, but soehow result in biasing the estiated correlation all the way down to zero? Why does this happen? The goal of this paper is to understand the sources of the puzzle and propose a solution. 6

8 3 Data Generating Process and Estiators In order to study the leverage effect puzzle, we need two ingredients: nonparaetric volatility estiators that are applicable at high frequency, and data generating processes for the logreturns and their volatility in the for of a stochastic volatility odel. Eploying a specific stochastic volatility odel has the advantage that the properties of nonparaetric estiators of the correlation between asset returns and their volatility becoe fully explicit. We can derive theoretically the asyptotic biases of different nonparaetric estiators applied to this odel, and verify their practical relevance via sall saple siulation experients. Put together, these ingredients lead to a novel solution to the leverage puzzle by introducing a tuning paraeter (represented by below) that attepts to iniize the estiation bias. 3.1 Stochastic Volatility Model The specification we eploy for this purpose is the stochastic volatility odel of Heston (1993) for the log-price dynaics: dx t = (µ ν t /2)dt + σ t db t (1) dν t = κ(α ν t )dt + γν 1/2 t dw t, (2) where ν t = σ 2 t, B and W are two standard Brownian otions with E(dB t dw t ) = ρdt, and the paraeters µ, α, κ, γ and ρ are constants. We assue that the initial variance ν 0 > 0 is a realization fro the stationary (invariant) distribution of (2) so that ν t is a stationary process. Under Feller s condition 2κα > γ 2, the process ν t stays positive, a condition that is always assued in what follows. Note that ρ = li s 0 Corr(ν t+s ν t, X t+s X t ) (3) so that the leverage effect is suarized by the paraeter ρ under the Heston odel (1)-(2). Throughout the paper, we refer to the correlation (3) between changes in volatility and changes in asset log-prices, i.e., returns, as the leverage effect. Other papers define it as the correlation between the level of volatility and returns, or the correlation between the level of absolute returns and returns (see, e.g., Bollerslev et al. (2006).) The latter definition, however, would not predict that the paraeter ρ should be identified as the high frequency liit of that correlation; while that alternative definition is appropriate at lower frequencies, it yields 7

9 a degenerate high frequency liit since it easures the correlation between two variables that are of different orders of agnitude in that liit. High frequency data can be eployed to estiate the correlation between volatility levels and returns, but only over longer horizons, as it is indeed eployed in Bollerslev et al. (2006). We consider a different proble: the nature of the leverage effect puzzle we identify lies in the fact that it is difficult to translate the otherwise straightforward short horizon / high frequency liit (3) into a eaningful estiate of the paraeter ρ. 3.2 Nonparaetric Estiation of Volatility and Sapling Our first statistical task will be to understand why natural approaches to estiate ρ based on (3) do not yield a good estiator when nonparaetric estiates of volatility based on highfrequency data are eployed. With a sall tie horizon (e.g., one day or = 1/252 year), let V t, = t t ν s ds (4) denote the integrated volatility fro tie t to t and ˆV t, be an estiate of it based on the discretely observed log-price process X t, which additionally ay be containated with the arket icrostructure noise. Recall that the quantity of interest is ρ and is based on (3). However, the spot volatility process ν t is not directly observable and has to be estiated by 1 ˆVt,. Thus, corresponding to a given estiator ˆV, a natural and feasible estiator of ρ is ˆρ = Corr( ˆV t+s, ˆV t,, X t+s X t ). (5) With s =, ˆV t+s, and ˆV t, are estiators of integrated volatilities over consecutive intervals. This is a natural choice for paraeter s: changes of daily estiated integrated volatility are correlated with changes of daily prices in two consecutive days. However, as to be deonstrated later, the choice of s = (changes over ultiple days apart) can be ore advantageous. We now specify the different nonparaetric estiators of the integrated volatility that will be used for ˆV t,. We assue that the log-price process X t is observed at higher frequency, corresponding to a tie interval δ (e.g., one observation every 10 seconds). In order for the nonparaetric estiate ˆV t, to be sufficiently accurate, we need δ ; asyptotically, we assue that 0 and δ 0 in such a way that /δ. 8

10 In the absence of icrostructure noise, the log prices X iδ (i = 0, 1,, n) are directly observable, and the ost natural (and asyptotically optial) estiator of V t, is the realized volatility /δ 1 ˆV t, = (X t +(i+1)δ X t +iδ ) 2. (6) i=0 Here, for siplicity of exposition, we assue there is an observation at tie t, and that the ratio /δ is an integer; otherwise /δ should be replaced by its integer part [ /δ], without any asyptotic consequences. In practice, high frequency observations of log-prices are likely to be containated with arket icrostructure noise. Instead of observing the log-prices X t+iδ, we observe the noisy version Z t+iδ = X t+iδ + ϵ t+iδ, (7) where the ϵ t+iδ s are white noise rando variables with ean zero and standard deviation σ ϵ. With this type of observations, we can use noise-robust ethods such as TS, PAV, QMLE or RK to obtain consistent estiates of the integrated volatility. We will first use the TS estiator, as it is relatively siple to analyze. Specifically, letting n = /δ, θ TS be a constant, L = [θ TS n 2/3 ] the nuber of grids over which the subsapling is perfored and n = (n L + 1)/n, the TS estiator is defined as ˆV TS t, = 1 n L (Z t +(i+l)δ Z t +iδ ) 2 n n 1 (Z t +(i+1)δ Z t +iδ ) 2. (8) L n i=0 The TS estiator is siple to analyze but is not rate-optial, converging at rate n 1/6 instead of the optial rate n 1/4. Thus, it is expected to incur a slightly larger estiation error. We therefore consider the rate-efficient pre-averaging volatility estiator (PAV) as proposed by Jacod et al. (2009) with the weight function chosen as g(x) = x (1 x). More specifically, let θ PAV be a constant, k n = [θ PAV n], we consider ˆV PAV t, = 12 n kn+1 θ PAV n i=0 ( 1 k n 1 k n j= k n /2 i=0 Z t +(i+j)δ 1 k n /2 1 k n j=0 Z t +(i+j)δ ) 2 6 n 1 θpav 2 n (Z t +(i+1)δ Z t +iδ ) 2. (9) i=0 A consistent estiator of the variance is provided in Jacod et al. (2009), as well as a consistent estiator of the integrated quarticity t t σ4 sds (see (21)). 9

11 4 Biases in Estiation of the Leverage Effect We now present the first results of the paper, consisting of the biases of estiators of the leverage effect paraeter ρ in four progressively ore realistic scenarios, each eploying a different nonparaetric volatility estiator. These progressive scenarios help us docuent an increental source for the bias: discretization, soothing, estiation error and arket icrostructure noise. 4.1 True Spot Volatility: Discretization Bias First, we consider the unrealistic but idealized situation in which the spot volatility process ν s is in fact directly observable. This helps us understand the error in estiating ρ that is due to discretization alone. Theore 1 reports the correlation between asset returns and changes of the instantaneous volatility, fro which the bias can easily be coputed. Theore 1. Changes of the true spot volatility and changes of log-prices have the following correlation: Corr(ν s+t ν t, X s+t X t ) = 1 e ρ κs κ (s ) ( ) + e κs 1 γ 2 γρ κ 4κ 2 κ. (10) + s Let us denote the right hand side of the expression in Theore 1 as C 1 (s, κ, γ, α, ρ). Fro Theore 1, the bias due to the discrete approxiation can be easily coputed, in the for C 1 (s, κ, γ, α, ρ) ρ. In particular, we have the following Proposition expressing the bias as a function of the integration interval and the interval length over which changes are evaluated,, 1, under different asyptotic assuptions on the sapling schee: Proposition 1. When 0, we have Corr(ν t+ ν t, X t+ X t ) = ρ ρ (γ2 4γκρ + 4κ 2 ) + o( ). (11) 16κ Since the value ρ is negative, the first order of the bias is positive, which pulls the function C 1 (s, κ, γ, α, ρ) towards zero, weakening the leverage effect. Figure 4 shows precisely how the function C 1 (, κ, γ, α, ρ) varies with for two sets of paraeter values: (ρ, κ, γ, α) = ( 0.8, 5, 0.5, 0.1) and (ρ, κ, γ, α) = ( 0.3, 5, 0.05, 0.04) when is taken to be 1/252. The forer set of paraeters was adapted fro those in Aït-Sahalia and Kiel (2010) and the latter set 10

12 was taken to weaken the leverage effect but to observe the Feller s condition: 2κα > γ 2. As expected, the saller the, the saller the discretization bias. +++ Insert Figure 4 Here True Integrated Volatility: Soothing Bias The spot volatilities are latent. They can be (and usually are) estiated by a local average of integrated volatility, which is basically a soothing operation, over a sall tie horizon. How big are the biases for estiating ρ even in the idealized situation where the integrated volatility is known precisely? The following theore gives an analytic expression for the resulting soothing bias: Theore 2. Changes of the true integrated volatility and changes of log-prices have the following correlation: Corr(V t+, V t,, X t+ X t ) = A 2 /(B 2 C 2 ) (12) where A 2 = 2γ(1 κ) + 4 κ 2 ρ 2γe κ + e ( κ(+1) e 2 κ (γ 4κρ) 2e κ (γ 2κρ) + γ ), B 2 = 2 e ( κ(+1) 2e κ (e κ 1) 2) + 2 κ 2, and C 2 = γ 2 ( κ + e κ 1) + 4γκρ ( κ e κ + 1) + 4 κ 3. While the expressions in Theore 2 are exact, further insights can be gained when we consider the resulting asyptotic expansion as 0. where is fixed and while still 0. We focus again on both situations Proposition 2. The following asyptotic expansions show the increental bias due to soothing induced by the local integration of spot volatilities: (2 1) Corr(V t+, V t,, X t+ X t ) = Corr(ν t+ ν t, X t+ X t ) 2 2 /3 O( ) when 0 for any + o( ) when, 0. (13) 11

13 The first expression is true when is any fixed integer. For the second expression, note that the asyptote of the correction factor (2 1) 2 2 /3 = 1 + O( 1 ). (14) Hence, when is large, unlike what the initial intuition ight have suggested, the bias of estiated ρ based on integrated volatilities is asyptotically the sae as that of the estiated ρ based on spot volatilities. Figure 4 shows the resulting nuerical values (dotted curves) for the sae sets of paraeters. They are plotted along with the correlations of the other estiators to facilitate coparisons. First, as expected, the bias is larger than that when spot volatilities are eployed. Figure 4 also reveals an interesting shape of biases of the idealized estiate of spot volatility. When is sall, the bias is large and so is when is large. There is an optial choice of that iniizes the bias. For the case = 1/252, with the chosen paraeters as in the left panel of Figure 4 [(ρ, κ, γ, α) = ( 0.8, 5, 0.5, 0.1)], the optiu is 0 = 16 with the optial value 0.74, leading to a bias of On the other hand, using the natural choice = 1, the estiated correlation is about 0.5, eaning that the bias is about 40% of the true value. 4.3 Estiated Integrated Volatility: Shrinkage Bias due to Estiation Error Theores 1 and 2 provide a partial solution to the puzzle. If the spot volatility were observable, the ideal estiate of leverage effect is to use the change of volatility over two consecutive intervals against the changes of the prices over the sae tie interval, i.e. = 1. However, when the spot volatility has to be estiated, even with the ideally estiated integrated volatility V t,, the choice of = 1 is far fro being optial. Indeed the resulting bias is quite large: for ρ = 0.8, with the sae set of paraeters as above, the estiated ρ is about 0.5 even when eploying the idealized true integrated volatility V t,. When the saple version of integrated volatility is used, we should expect that the leverage effect is further asked by estiation error. This is due to the well-known shrinkage bias of coputing correlation when variables are easured with errors. In fact, we already know that it becoes so large that it asks copletely the leverage effect when = 1 is used as in Figures 1 and 2. We now derive the theoretical bias expressions corresponding to this ore realistic case. 12

14 The following theore calculates the bias of using a data driven estiator of the integrated volatility in the absence of icrostructure noise. In other words, we use the realized volatility estiator. Let n be the nuber of observations during each interval. Assue for siplicity that the observation intervals are equally spaced at a distance δ = /n. Theore 3. When n C and 2 C for C, C shows the increental bias due to estiation error induced by the use of : Corr( ˆV t+, ˆV t,, X t+ X t ) = Corr(ν t+ ν t, X t+ X t ) ( ακ + 6γ 2 (3γ 2 γ 2 )κc 3 2 γ2 κ 2 CC (0, ), the following expansion (2 1) 2 (15) 2 /3 ) 1/2 [1 + o( )]. The above theore docuents the bias when there is no arket icrostructure noise. Interestingly, it is decoposed into two factors. The first factor is the soothing bias and the second factor is the shrinkage bias due to the estiation errors. The second factor reflects the cost of estiating the latency of volatility process. The larger the C, the saller the shrinkage bias. Siilarly, the larger the, the saller the shrinkage bias. To appreciate the bias due to the use of, the ain ter in Theore 3 as a function of is depicted in Figure 4 for the sae sets of paraeters as entioned above. The daily sapling frequency is taken to be n = 390 (one observation per inute) so that C = 390/252. In particular, the choice of = 1 corresponds to the natural estiator but it results in a very large bias. Even in the absence of arket icrostructure noise, the estiated correlation based on the natural estiator ˆρ = Corr( ˆV t+, ˆV t,, X t+ X t ) (16) is very close to 0. This provides a atheatical explanation for why the leverage effect cannot be detected epirically using a natural approach. On the other hand, Theore 3 also hints at a solution to the leverage effect puzzle: with an appropriate choice of, there is hope to ake the leverage effect detectable. For the left panel of Figure 4, if the optial = 27 is used, the estiated correlation is now 0.694, when the true value is

15 4.4 Estiated Noise-Robust Integrated Volatility: Shrinkage Bias due to Estiation Error and Noise Correction Error Under the ore realistic case where allowance is ade for the presence of arket icrostructure noise under (7), the integrated volatility V t is estiated based on noisy log-returns, using biascorrected high-frequency volatility estiators such as TS, PAV, QMLE or RK. In this case, as we will see, detecting the leverage effect based on the natural estiator is even harder. It ay in fact even result in an estiated correlation coefficient with the wrong sign. Again, the tuning paraeter can help resolve the issue. We start with TS and then consider PAV as well. Other ethods can be eployed too, but the coputations becoe increasingly tedious ore so than they already are! Recall the definition of θ TS in the TS estiator, which deterines the constant factor of the large scale. Theore 4. When n 1/3 C T S, σ 2 ϵ / C ϵ and 2 C with C T S, C ϵ and C (0, ), the following expansion shows the increental bias due to estiation error and noise correction induced by the use of TS: where TS TS (2 1) Corr( ˆV t+, ˆV t,, Z t+ Z t ) = Corr(ν t+ ν t, X t+ X t ) 2 2 /3 A 4 = B 4 = (1 + A 4 + B 4 ) 1/2 [1 + o( )], 96θ 2 TS C2 ϵ C TS αγ 2 (6 2 3κC ) 8θ TS (2ακ + γ 2 ) κc TS γ 2 (6 2 3κC ). For the sae reasons behind the above theore, using the paraeter helps resolving the leverage effect probles. When θ TS is taken to be 0.5, with = 1 and the sae set of paraeters (ρ, κ, γ, α,, n) = ( 0.8, 5, 0.5, 0.1, 1/252, 390), the leverage effect is barely noticeable whereas using = 73 yields a correlation of Even though the bias is large, the leverage effect is clearly noticeable. Again, the estiating biases can be decoposed into two factors. The first factor is the soothing bias, the sae as that in the and PAV below. The second factor reflects the shrinkage biases due to estiation errors and noise correction errors. The rate of convergence 14

16 of TS is slower than that of. This is reflected in the factor C TS which is of order n 1/3, rather than C = n in. Siilarly, since PAV below has a faster rate of convergence than TS, its corresponding shrinkage bias is saller than TS but larger than. This is reflected in C PAV = n 1/2 in Theore 5 below. A parallel result to Theore 4 for PAV is the following. Theore 5. When n 1/2 C P AV, σ 2 ϵ / C ϵ, and 2 C with C P AV, C ϵ and C (0, ), the following expansion shows the increental bias due to estiation error and noise correction induced by the use of PAV: where PAV PAV (2 1) Corr( ˆV t+, ˆV t,, Z t+ Z t ) = Corr(ν t+ ν t, X t+ X t ) 2 2 /3 (1 + A 5 + B 5 + C 5 ) 1/2 [1 + o( )], A 5 = B 5 = 24Φ 22 θ PAV (2ακ + γ 2 ) ψ2c 2 P AV κγ 2 (6 2 3κC ) 96Φ 12 C ϵ θ PAV ψ2c 2 P AV γ 2 (6 2 3κC ) C 5 = where ψ 2 = 1 12, Φ 11 = 1 6, Φ 12 = 1 96, Φ 22 = Φ 11 C 2 ϵ θ 3 PAV ψ2 2C P AV αγ 2 (6 2 3κC ), Theore 4 and Theore 5 are also illustrated in Figure 4 in which the ain ters of the correlations are plotted. Again when = 1, the correlation is nearly zero, whereas with the ideal choice of = 48, the ideal correlation is 0.599, still significantly saller than the true one of A Solution to the Puzzle: Model-Independent Bias Corrections The previous section docuented the various biases arising when estiating the leverage effect paraeter ρ in four progressively ore realistic scenarios. The essage was decidedly glooy: even in idealized situations, the bias is large, and attepts to correcting for the latency of the 15

17 volatility, or for the presence of arket icrostructure noise, do not iprove atters. In fact, they often ake atters worse. But, fortunately, they also point towards potential solutions to the bias proble. 5.1 Back to the Latent Spot Volatility First, we show that all the additional biases that are introduced by the latency of the spot volatility can be corrected, and the proble is reduced to the discretization bias left in Theore 1. Recall the asyptotic expression given in Theore 2, which can be inverted to yield: Corr(ν t+ ν t, X t+ X t ) = 2 2 /3 (2 1) Corr(V t+, V t,, X t+ X t )+O( ). (17) Thus, up to a ultiplicative correction factor that is independent of the odel s paraeters, the integrated volatility V can work as well as the spot volatility ν. The effectiveness of this siple bias correction is deonstrated in Figure Insert Figure 5 Here +++ In the absence of icrostructure noise, using the realized volatility (6), the asyptotic relative bias in coparison with the use of the true spot volatility is given by Theore 3. Using the expressions given there, we can correct the bias due to the estiate of realized volatility back to that based on the spot volatility. However, such a correction involves unknown paraeters in the Heston odel and depends on the paraetric assuption. However, the ethod is applicable to a wider array of data generating processes. A nonparaetric correction consists in using the following result, deonstrated in the Appendix, in the proof of Theore 3: Proposition 3. When n C and 2 C with C and C (0, ), Corr(ν t+ ν t, X t+ X t ) = c /3 (2 1) where c 3 is given by Corr( ˆV t+, ˆV t,, X t+ X t ) + o( ), (18) ( ) 1/2 4E [σt 4 ] 2 c 3 = 1 n Var( ˆV t+, ˆV t, ). (19) 16

18 Note that in (19), the stationarity of the process of ν t is used so that the correction factor does not depend on t. In practice, we can estiate E [σ 4 t ] nonparaetrically based on the fact that the quarticity satisfies n n 3 (X t+(i+1)δ X t+iδ ) 4 P i=0 t+ t σ 4 sds as n for any fixed. Hence a long run average of scaled quarticity can be used to estiate E [σ 4 t ]. The variance in (19) can be estiated by its saple version. For the TS estiator, the bias correction adits the sae for as (18) with a different correction. Proposition 4. When n 1/3 C T S, σ 2 ϵ / C ϵ and 2 C for constants C T S, C ϵ and C (0, ), Corr(ν t+ ν t, X t+ X t ) = c /3 (2 1) where since c 4 = ( TS Corr( ˆV t+, TS ˆV t,, Z t+ Z t ) + o( ) ) 1/2 1 48θ 2 TS σ4 ϵ + 8θ TS E [σt 4 ] 2 3n 1/3 TS TS Var( ˆV t+, ˆV t, ). (20) Two unknown quantities are involved and can be estiated nonparaetrically here. For σ ϵ, ˆV t, 2nσ2 ϵ + t a long run average of ( t σ2 sds, ˆV t, ˆV TS t, t t σ2 sds for fixed and big n, we can conclude that TS ˆV t, )/2n can be used as a good estiate of σ2 ϵ. This is siilar to the way the average of the subsapled estiators is bias-corrected to construct TS. For E [σ 4 t ], since arket icrostructure noise is involved, the situation is a bit ore coplicated than before. Consistent noise-robust estiators of t t σ4 sds are proposed in Zhang et al. (2005) 17

19 and Jacod et al. (2009); we can use for instance the estiator called ˆQ n t in the latter paper: ˆQ n t = 1 3θ 2 P AV ψ2 2 + P t δ θ 4 P AV ψ2 2 δ 4θ 4 P AV ψ2 2 t σ 4 t dt, n k n +1 i=0 n 2k n+1 i=0 ( 1 k n 1 k n j= k n /2 (( 1 k n 1 k n i+2k n 1 j= k n/2 Z t +(i+j)δ 1 k n /2 1 k n j=0 Z t +(i+j)δ 1 k n/2 1 k n j=0 ) (Z t +(j+1)δ Z t +jδ ) 2 j=i+k n Z t +(i+j)δ ) 4 Z t +(i+j)δ ) 2 n 2 (Z t +(i+1)δ Z t +iδ ) 2 (Z t +(i+3)δ Z t +(i+2)δ ) 2. i=1 (21) where ψ 2 = 1 12, k n = [θ PAV n] for an appropriately chosen θpav. A scaled long run average of this estiator can be used to estiate E [σ 4 t ]. For PAV, we have Proposition 5. When n 1/2 C P AV, σ 2 ϵ / C ϵ, and 2 C for constants C P AV, C ϵ and C (0, ), Corr(ν t+ ν t, X t+ X t ) = c /3 (2 1) PAV Corr( ˆV t+, PAV ˆV t,, Z t+ Z t ) + o( ) where with ( ) 1/2 2(A 5 + B 5 + C c 5 = 1 5) n 1/2 PAV PAV Var( ˆV t+, ˆV t, ), (22) A 5 = 4Φ 22θ PAV E [σt 4 ], ψ2 2 B 5 = 8Φ 12E [σ 2 t ] σ 2 ϵ θ PAV ψ 2 2 C 5 = 4Φ 11σϵ 4, θpav 3 ψ2 2 where ψ 2, Φ 11, Φ 12, Φ 22 are as in Theore 5. A ore direct way is to use the long run average of the quantity Γ n t defined in (3.7) of Jacod et al. (2009) to estiate A 5 + B 5 + C 5. 18,

20 5.2 Correcting the Discretization Bias Fro Spot Volatilities The above results reveal that the biases due to the various estiates are correctable back to the case where the spot volatility can be viewed as observable. However, Theore 1 iplies that the estiate of ρ based on ν t itself is also biased. If the odel were known, then the bias (11) can be coputed and corrected. However, this depends on the Heston odel and its unknown paraeters. A paraeter-independent ethod is as follows. Let ˆρ = Corr(ν t+ ν t, X t+ X t ). Then, by Theore 1 we see that ˆρ = ρ + b + o( ). (23) This suggests that the paraeter of interest ρ (as well as the slope b but this is not needed) can be estiated by running a linear regression of the data {(, ˆρ )}. The bias-corrected estiate of ρ is siply the intercept of that linear regression. The scatter plot of {(, ˆρ )} can also suggest a region of to run the above siple linear regression (23). The above discussion suggests a rather general strategy for bias correction. The ethod does not depend on the Heston odel paraeters. First, copute the siple correlation between estiated changes of volatilities and estiated changes of prices. Second, conduct a preliinary bias correction according to (17) (22), depending on which estiated volatilities are used. Third, run the siple regression equation (17) for the preliinary bias corrected estiated correlations. Fourth, take the intercept of the siple linear regression as the final estiate. The ethod turns out to be very effective in pratice, as we will now see. 6 Monte-Carlo Siulations In this section, we use siulation studies to reproduce the leverage effect puzzle and its proposed solution and to verify the practical validity of the results presented in the previous section. 6.1 The data generating process The true log-price is siulated fro the Heston odel (1)-(2) with broadly realistic paraeter values: α = 0.1 γ = 0.5, κ = 5, ρ = 0.8 and µ = 0.05 over trading days in five years ( = 1/252). Each day, the sapling frequency is one inute per saple, giving an 19

21 intra-day nuber of observations of n = 390. Therefore the total nuber of observations over 5 years is N = = 491, 400. The true price is latent. Instead, the observed data {Z iδ } 491,400 i=1 are containated with the arket icrostructure as in (7) with i.i.d N (0, σ 2 ϵ ) noise, and σ ϵ = (The case when the observation frequency is higher, n = 1560 is also studied. The results are collected at the end of this section.) 6.2 Vizualizing the leverage effect puzzle With the latent spot volatility ν t and latent price X t known in siulated data, we can easily exaine the correlation of {(X t X (t 1), ν t ν (t 1) )} over N observations. As expected, the leverage effect is strong, with the saple correlation being for a given realization. This is in line with the result of Theore 1. Next, consider the ore realistic situation that the spot volatility needs to be estiated by a soothing ethod such as a local integrated average V t, = t t σ2 t dt. A natural estiate is the average of daily spot volatility ˆV t, = n 1 n j=1 ˆσ2 t +j/n. In this ideal situation, σ2 t +j/n is known, resulting in V t, = n 1 n j=1 σ2 t +j/n. The correlation of {(X (t+1) X t, V (t+1), V t, )} 1259 t=1 is for the given realization. This is in line with the result of Theore 2. The agnitude of the leverage effect paraeter ρ is significantly under-estiated. To appreciate the effect of the tuning paraeter, the upper panels of Figure 6 plots the correlation {(X (t+) X t, ν (t+), ν t, )} 1260 t=1 and {(X (t+) X t, V (t+), V t, )} 1260 t=1 against. To exaine the sapling variabilities, the siulation is conducted 100 ties. The averages of the saple correlations are plotted along with its standard deviation in the figure. The ipact of can easily be seen and the natural estiate based on V t, with = 1 is far fro optial. +++ Insert Figure 6 Here +++ In practice, the integrated volatility is not observable. It has to be estiated using the discretely observed data. In absence of arket icrostructure noise, the realized volatility provides a good estiate of the integrated volatility. Using based on the siulated latent prices Xi n, we have a saple correlation of 0.25 for the sae realization discussed above. More generally, the correlation of {{(X (t+) X t, ˆV (t+), ˆV t, )}1260 t=1 as a function of is depicted in the lower left panel of Figure 6. As above, this is repeated 100 ties so that the average correlations along with their errors at each are coputed. 20

22 For a ore realistic situation, the integrated volatility has to be estiated based on the containated log prices Z t in (7). The volatility paraeter is now estiated by the correlation {(Z (t+) Z t, PAV PAV ˆV (t+), ˆV t, )}1260 TS t=1 or {(Z (t+) Z t, ˆV (t+), TS ˆV t, )}1260 t=1 with a suitable choice of. The lower iddel and right panels of Figure 6 shows the correlation as a function of. In particular, when = 1, the saple correlation is erely for PAV and for TS for the sae siulated path as entioned above, which would be interpreted in practice as showing little support for the leverage effect. But we know that this is due to the statistical bias of the procedure as deonstrated in Theore 4. Using PAV with = 26, the saple correlation is 0.682; and using TS with = 62, the saple correlation is While this is still a biased estiate, the leverage effect is now clearly seen. The averages of these correlations against are plotted together in Figure 7. These are in line with what the theory predicts (see the left panel of Figure 4). +++ Insert Figure 7 Here Effectiveness of the Bias Correction Method We now illustrate the effectiveness of the bias correction ethod proposed in Section 5. We siulate saple paths with the sae paraeters as above. θ TS and θ PAV are both taken to be 0.5. For the linear regression ethod, we use the set of values of as {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}. Let us denote the bias corrected estiate of ρ as cornutorho with the correlation based on the spot volatility as the input. Siilarly, corvtorho, cortorho, corpavtorho and cortstorho are the bias corrected ρ using the linear regression respectively based on the corrected curves fro V, ˆV ˆV PAV and ˆV TS, using equations (17), (18), (22) and (20). The true values of Eσ 4 t, Eσ 2 t and σ ϵ are plugged in. Table 1 suarizes the results based on 100 siulations of (T = 5) inute-by-inute (n=390) data over a five-year period. +++ Insert Table 1 Here +++ The ean of all of these corrected estiates are all close to the true value ρ =-0.8. This eans that these estiates are unbiased. The progressive harder of the probles can easily be seen fro the SD of the estiates. As deonstrated in Figure 8, the TS with the available saple size incurs large estiation errors. This translates into the large error for estiating 21

23 ρ. When the sapling frequency is ore frequent than one saple per inute, the estiation error can be reduced. In suary, Table 1 provides a stark evidence that the ethods in Section 5 solve the leverage effect puzzle. It also quantifies the extent to which the proble gets progressively harder. In practice, the paraeters are unknown and hence the correction based directly on the odel paraeters as in Table 1 is not feasible. In the following, we deonstrate the effectiveness of the non-paraetric ethods as described in section 5.1 to obtain corrections. cortorhoe, corpavtorhoe and cortstorhoe are the bias corrected ρ using the linear regression based on the non-paraetrically corrected curves fro ˆV, ˆV PAV and ˆV TS respectively. Estiates are based on the sae siulated saple paths as above. The results are collected in Table Insert Table 2 Here +++ In this section, we used a fixed range of to run the regression regardless of the ethod and realizations. As discussed in Section 5, the data-dependent and estiator-dependent choice of the range can iprove the results further. The scatterplot helps us epirically deterine an appropriate range. To illustrate the point, we randoly select a saple path fro the siulation, and then plot the corrected correlations up to the curve corresponding to Theore 1: see Figure 8. After that, we identify the range of values such that the corrected curve is roughly linear. For this saple path, a range of = (30, 31,, 70) gives us a corpavtorhoe of -0.83, and a range of = (40, 41,, 80) gives us a cortstorhoe of These are better than the corrections based on a generic choice of = (6, 7,, 17), which yields an estiated correlation of and respectively. This tailor-ade ethod will be used in the epirical study below in order to iprove the estiation. +++ Insert Figure 8 Here +++ Note that as shown in the upper right panel of Figure 8, occasionally we ay obtain extree results that the corrected correlation has an absolute value bigger than 1, especially when is sall. One can truncate these observations back to the correct range as one likes. Indeed, typically these values won t affect our final results because we typically choose reasonably big values of in the final step of correction using regression. 22

24 Without bias correction, as deonstrated in Section 2, sapling data at higher frequency does not give us a better assessent of the leverage effect. With the bias correction, we would expect better results. Table 3 and Figure 9 deonstrate this further by the results with the sapling frequency of one observation every 15 seconds, fro which we see siilar results but with reduced estiation errors. +++ Insert Table 3 Here Insert Figure 9 Here Epirical Evidence on the Leverage Effect at High Frequency We now apply our bias corrected ethods to exaine the presence of the leverage effect using high-frequency data. We have seen in Section 2 that, due to the latency of the volatility process, it is nearly ipossible to use only returns data and no extraneous volatility proxy to get as nice a plot as what was shown in Figure 3. Nevertheless, we will deonstrate that the new tool is able to reveal the presence of a strong leverage effect contained in high-frequency data. We only focus on the S&P 500 and MSFT returns; we have applied the ethods to various data sets and the conclusions are siilar. 7.1 S&P 500 data Based on the high-frequency returns (1 inute per saple) on S&P 500 futures fro January 2004 to Deceber 2007, the naive or natural estiates give the results reported in Table 4. The leverage effect at the natural choice of = 1 is nearly 0. Even with the data-optiized choice of, the correlation with TS is estiated to be around 0.44 and that with the PAV is around -0.50, uch saller than that coputed based on the VIX. The upper panel of Figure 10 suarizes the saple correlations based on TS, PAV and VIX respectively, versus horizon. +++ Insert Table 4 Here

25 We now apply our bias corrected ethods. First, we copute the preliinarily biascorrected estiates using both TS and PAV for a wide range of choice of. Their scatter plots are presented in Figure 10, which are quite curly. We then took the first increasing region (because the biases are expected to get larger with after correction so that the slope should be positive) that appears approxiately linear. For TS, we take = (55, 56, 100) to run the regression and obtain the intercept of This is our bias-corrected estiated leverage effect paraeter iplied in the high frequency data. Siilarly, for PAV, we selected the range = (55, 56,, 85), the bias-corrected correlation based on PAV is estiated as Insert Figure 10 Here Microsoft We now use our ethod to exaine how strong the leverage effect for the Microsoft corporation. The high-frequency returns at saple frequencies of one data point per inute and one per 5 seconds fro January 2005 to June 2007 are used for estiating the leverage effect paraeter. Again, we apply both the naive ethod, the siple saple correlation, and ore sophisticated volatility estiation ethods, based on preliinary correction and linear regression. Table 5 suarizes the results. Again, the leverage effects are barely noticeable at both sapling frequencies for natural choices of (sall values of ). +++ Insert Table 5 Here +++ For the data sapled at one observation per inute, the preliinary corrections are suarized in the left panel of Figure 11. This helps us deterine a region where the linear regression should be run. Based on regression (taking = (130, 131,, 170)), the leverage effect paraeter is estiated as 0.72 with TS and 0.68 with PAV. Again, we find that the leverage effect is uch stronger than one would obtain without bias-correction. +++ Insert Figure 11 Here +++ The analysis for the data sapling at 5-second frequency produces siilar results. The range = (130, 131,, 170) is deterined based on preliinary corrections depicted in Figure 12 (left panel). Using this range of estiates to run a siple linear regression yields an estiated leverage effect paraeter. It is 0.76 based on TS and 0.78 based on PAV. +++ Insert Figure 12 Here