The Leverage Eect Puzzle Disentangling Sources of Bias in High - PowerPoint PPT Presentation

The Leverage E�ect Puzzle Disentangling Sources of Bias in High Frequency Inference Yacine A• �t-Sahalia Jianqing Fan Yingying Li Princeton University Princeton University HK UST 1

1 THE LEVERAGE EFFECT PUZZLE 1. The Leverage E�ect Puzzle � Expect to �nd a negative correlation between returns and changes in their volatility { empirically, rising asset prices are accompanied by declining volatil- ity { and vice versa 2

1 THE LEVERAGE EFFECT PUZZLE � Economic interpretations { Firm becomes more leveraged as its stock price goes down, hence riskier (Black, 1976; Christie, 1982). { Risk premia (French et al., 1987, Campbell and Hentschel, 1992) � same negative correlation between returns and volatility changes, but reverses the direction of the causation � an increase in volatility raises the risk inherent in holding an asset and so expect that the asset should then provide a higher future expected return � which in turn necessitates a decline in price � Either way, we expect to observe a negative correlation 3

1 THE LEVERAGE EFFECT PUZZLE � And it appears to be there over long horizons (e.g., S&P 500, 1997- 2011) 1600 Close Returns 0.04 1400 0.02 1200 0.00 1000 −0.02 800 −0.04 2000 2005 2010 4

1 THE LEVERAGE EFFECT PUZZLE � Empirical �ndings { Asymmetry: declining prices have a higher impact on volatility (Nelson, 1991; Engle and Ng, 1993; Yu, 2005). { Magnitude seems too large to be attributed solely to an increase of �nancial leverage (Figlewski and Wang, 2000) { Daily frequency (Bekaert and Wu, 2000); Higher frequency (Boller- slev et al., 2006) 5

1 THE LEVERAGE EFFECT PUZZLE � Evidence from the estimation of parametric stochastic volatility models suggests � � � 0 : 7 or � 0 : 8 for the S&P500, a �nding fairly robust across models and time periods � This is corroborated when using VIX or VXD as proxies for volatility, for example at the daily frequency: �nd a strong e�ect S&P 500 Dow Jones 200 200 Change in Volatility (VXD^2) Change in Volatility (VIX^2) 100 100 0 0 −200 −100 −400 −0.04 −0.02 0.00 0.02 −0.03 −0.02 −0.01 0.00 0.01 0.02 Return Return 6

1 THE LEVERAGE EFFECT PUZZLE � The puzzle { Despite the economic rationale for a leverage e�ect and the pres- ence detected in the data using volatility proxies { Volatilities estimated from the sample price path at moderate to high frequencies show very little evidence of the leverage e�ect { Despite the fact that high frequency data should help better iden- tify the quadratic (co)variation of the process 7

1 THE LEVERAGE EFFECT PUZZLE Daily returns and daily changes of integrated volatilities, 2004-2007 S&P 500 Futures E−mini S&P 500 3e−04 3e−04 Change in Volatility (TSRV) Change in Volatility (TSRV) 1e−04 1e−04 −1e−04 −1e−04 −3e−04 −3e−04 −0.04 −0.02 0.00 0.02 −0.04 −0.02 0.00 0.02 Return Return S&P 500 Futures MSFT 4e−04 3e−04 Change in Volatility (TSRV) Change in Volatility (TSRV) 2e−04 1e−04 0e+00 −1e−04 −2e−04 −3e−04 −0.04 −0.02 0.00 0.02 −0.10 −0.05 0.00 0.05 Return Return 8

Lagged Change in Volatility (TSRV) Lagged Change in Volatility (TSRV) −2e−04 0e+00 2e−04 4e−04 −3e−04 −1e−04 1e−04 3e−04 −0.04 −0.10 −0.02 S&P 500 Futures −0.05 MSFT Return Return With leads and lags 0.00 0.00 0.02 0.05 9 Change in Volatility (TSRV) Change in Volatility (TSRV) −2e−04 0e+00 2e−04 4e−04 −3e−04 −1e−04 1e−04 3e−04 −0.04 −0.10 −0.02 S&P 500 Futures Lagged Return Lagged Return −0.05 1 MSFT 0.00 THE LEVERAGE EFFECT PUZZLE 0.00 0.02 0.05

1 THE LEVERAGE EFFECT PUZZLE At longer time horizons S&P 500 Futures, 1 week S&P 500 Futures, 1 month 4e−04 4e−04 Change in Volatility (TSRV) Change in Volatility (TSRV) 2e−04 0e+00 0e+00 −4e−04 −4e−04 −0.06 −0.04 −0.02 0.00 0.02 0.04 −0.10 −0.05 0.00 0.05 Return Return MSFT, 3 months MSFT, 6 months 4e−04 4e−04 Change in Volatility (TSRV) Change in Volatility (TSRV) 2e−04 2e−04 0e+00 0e+00 −2e−04 −2e−04 −0.2 −0.1 0.0 0.1 0.2 −0.2 −0.1 0.0 0.1 0.2 0.3 Return Return 10

1 THE LEVERAGE EFFECT PUZZLE � The Epps E�ect (1979) { Empirical correlation between the returns of two assets tends to decrease as the sampling frequency increases { Asynchronicity of the observations has been shown to have the potential to generate the Epps E�ect { As a result, various data synchronization methods have been de- veloped to address this issue: see e.g., Hayashi and Yoshida (2005) { The asynchronicity problem is not an issue here { The volatility estimator is constructed from the asset returns them- selves, the two sets of observations are by construction synchrone 11

1 THE LEVERAGE EFFECT PUZZLE � Di�erent issue here: latency { We need to employ preliminary estimators of the volatility variable, such as realized volatility (RV) for example, in order to compute its correlation with asset returns. { We consider the consequences of the latency of the volatility vari- able when estimating � . { We examine di�erent nonparametric volatility estimators and show that they lead to di�erent types of biases when employed to esti- mate � . { We disentangle the di�erent sources of the biases, and propose adjustments to correct for these biases. 12

1 THE LEVERAGE EFFECT PUZZLE � Market microstructure noise { We incorporate noise-robust high frequency volatility estimators, such as TSRV, MSRV, PAV, RK, QMLE { We show that robustifying the volatility estimator for the presence of noise can have unexpected e�ects when that volatility estimator is employed in a leverage e�ect computation 13

1 THE LEVERAGE EFFECT PUZZLE � Biases { We proceed incrementally, isolate the sources of the bias one by one { Discretization bias � starting with the spot volatility, an ideal but unavailable estima- tor since volatility is unobservable � as the sampling frequency increases, the estimator converges to the leverage e�ect parameter � : all is well 14

1 THE LEVERAGE EFFECT PUZZLE { Smoothing bias � the unobservable spot volatility is replaced by a local time-domain smoothing estimator � replacing the spot volatility by the (also unavailable) true inte- grated volatility � the bias for estimating � is now present as we sample more fre- quently 15

1 THE LEVERAGE EFFECT PUZZLE { Estimation error bias � replacing the true IV by an estimated integrated volatility, RV � the bias for estimating � becomes so large that the estimated � becomes essentially zero � which is indeed what we �nd empirically. { Robusti�cation bias � e�ect of using noise-robust estimators of the integrated volatility � the additional bias term may go either way 16

1 THE LEVERAGE EFFECT PUZZLE � Bias correction { We determine the form of the bias terms { We propose a regression approach to compute bias-corrected esti- mators of � 17

2 THE MODEL AND ESTIMATORS 2. The Model and Estimators 2.1. Data Generating Process � X t = log-price, Heston model dX t = ( � � � t = 2) dt + � 1 = 2 dB t t d� t = � ( � � � t ) dt + �� 1 = 2 dW t : t { B and W are two BMs with E ( dB t dW t ) = �dt . { parameters � , � , � , � and � with � = �= 2 � Calibrated to � = � 0 : 8 18

2.1 Data Generating Process 2 THE MODEL AND ESTIMATORS � Leverage e�ect parameter � = lim s & 0 Corr( � t + s � � t ; X t + s � X t ) : � Integrated volatility Z t V t; � = t � � � s ds 19

2.2 Nonparametric Volatility Estimators 2 THE MODEL AND ESTIMATORS 2.2. Nonparametric Volatility Estimators � RV � =� � 1 X V RV ( X t � �+( i +1) � � X t � �+ i� ) 2 ^ t; � = i =0 � E�cient in the absence of noise, biased otherwise 20

2.2 Nonparametric Volatility Estimators 2 THE MODEL AND ESTIMATORS � TSRV, MSRV { Data: with additive market microstruture noise, Z t + i� = X t + i� + � t + i� is what's observed h � TSRV n 2 = 3 i { n = � =� , � TSRV a constant, L = the number of grids over which the subsampling is performed and � n = ( n � L + 1) =n n � L X = 1 V TSRV ( Z t � �+( i + L ) � � Z t � �+ i� ) 2 ^ t; � L i =0 n � 1 X � � n ( Z t � �+( i +1) � � Z t � �+ i� ) 2 n i =0 21

2.2 Nonparametric Volatility Estimators 2 THE MODEL AND ESTIMATORS � PAV { Z t + i� = X t + i� + � t + i� is what's observed { g ( x ) = x ^ (1 � x ), � PAV a constant, k n = [ � PAV =� 1 = 2 ] b k n = 2 c� 1 n � k n +1 X k n � 1 X X t; � =12 � 1 = 2 ( 1 Z t � �+( i + j ) � � 1 V PAV ^ Z t � �+( i + j ) � ) 2 � PAV k n k n i =0 j =0 j = b k n = 2 c n � 1 X � 6 � ( Z t � �+( i +1) � � Z t � �+ i� ) 2 : � 2 PAV i =0 22

3 DISENTENGLING THE BIASES 3. Disentengling the Biases 3.1. Discretization bias � Suppose that the true spot volatility is available, then over a time span s = m � : � � 2 � 4 ��� + 4 � 2 � � Corr( � t + m � � � t ; X t + m � � X t ) = � � m �+ o ( m �) 16 � � For � < 0, bias is always positive (unfavorable). � But we obtain the desired limit, � as � ! 0 : this works as intended 23