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

The Leverage Eect Puzzle

Disentangling Sources of Bias in High Frequency Inference Yacine A• t-Sahalia Jianqing Fan Yingying Li

Princeton University Princeton University HK UST

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1 THE LEVERAGE EFFECT PUZZLE

1. The Leverage Eect 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

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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

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1 THE LEVERAGE EFFECT PUZZLE

And it appears to be there over long horizons (e.g., S&P 500, 1997- 2011)

800 1000 1200 1400 1600 2000 2005 2010 −0.04 −0.02 0.00 0.02 0.04 Close Returns

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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

• f nancial leverage (Figlewski and Wang, 2000)

{ Daily frequency (Bekaert and Wu, 2000); Higher frequency (Boller- slev et al., 2006)

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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 eect

−0.04 −0.02 0.00 0.02 −400 −200 100 200

S&P 500

Return Change in Volatility (VIX^2) −0.03 −0.02 −0.01 0.00 0.01 0.02 −100 100 200

Dow Jones

Return Change in Volatility (VXD^2)

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1 THE LEVERAGE EFFECT PUZZLE

The puzzle { Despite the economic rationale for a leverage eect 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 eect { Despite the fact that high frequency data should help better iden- tify the quadratic (co)variation of the process

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1 THE LEVERAGE EFFECT PUZZLE

Daily returns and daily changes of integrated volatilities, 2004-2007

−0.04 −0.02 0.00 0.02 −3e−04 −1e−04 1e−04 3e−04

S&P 500 Futures

Return Change in Volatility (TSRV) −0.04 −0.02 0.00 0.02 −3e−04 −1e−04 1e−04 3e−04

E−mini S&P 500

Return Change in Volatility (TSRV) −0.04 −0.02 0.00 0.02 −3e−04 −1e−04 1e−04 3e−04

S&P 500 Futures

Return Change in Volatility (TSRV) −0.10 −0.05 0.00 0.05 −2e−04 0e+00 2e−04 4e−04

MSFT

Return Change in Volatility (TSRV)

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1 THE LEVERAGE EFFECT PUZZLE

With leads and lags

−0.04 −0.02 0.00 0.02 −3e−04 −1e−04 1e−04 3e−04

S&P 500 Futures

Return Lagged Change in Volatility (TSRV) −0.04 −0.02 0.00 0.02 −3e−04 −1e−04 1e−04 3e−04

S&P 500 Futures

Lagged Return Change in Volatility (TSRV) −0.10 −0.05 0.00 0.05 −2e−04 0e+00 2e−04 4e−04

MSFT

Return Lagged Change in Volatility (TSRV) −0.10 −0.05 0.00 0.05 −2e−04 0e+00 2e−04 4e−04

MSFT

Lagged Return Change in Volatility (TSRV)

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1 THE LEVERAGE EFFECT PUZZLE

At longer time horizons

−0.06 −0.04 −0.02 0.00 0.02 0.04 −4e−04 0e+00 2e−04 4e−04

S&P 500 Futures, 1 week

Return Change in Volatility (TSRV) −0.10 −0.05 0.00 0.05 −4e−04 0e+00 4e−04

S&P 500 Futures, 1 month

Return Change in Volatility (TSRV) −0.2 −0.1 0.0 0.1 0.2 −2e−04 0e+00 2e−04 4e−04

MSFT, 3 months

Return Change in Volatility (TSRV) −0.2 −0.1 0.0 0.1 0.2 0.3 −2e−04 0e+00 2e−04 4e−04

MSFT, 6 months

Return Change in Volatility (TSRV)

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1 THE LEVERAGE EFFECT PUZZLE

The Epps Eect (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 Eect { 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

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1 THE LEVERAGE EFFECT PUZZLE

Dierent 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 dierent nonparametric volatility estimators and show that they lead to dierent types of biases when employed to esti- mate . { We disentangle the dierent sources of the biases, and propose adjustments to correct for these biases.

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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

• f noise can have unexpected eects when that volatility estimator

is employed in a leverage eect computation

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1 THE LEVERAGE EFFECT PUZZLE

Biases { We proceed incrementally, isolate the sources of the bias one by

• ne

{ 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 eect parameter : all is well

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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

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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. { Robustication bias eect of using noise-robust estimators of the integrated volatility the additional bias term may go either way

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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

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2 THE MODEL AND ESTIMATORS

2. The Model and Estimators

2.1. Data Generating Process

Xt = log-price, Heston model dXt = ( t=2)dt + 1=2

t

dBt dt = ( t)dt + 1=2

t

dWt: { B and W are two BMs with E(dBtdWt) = dt. { parameters , , , and with = =2 Calibrated to = 0:8

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2.1 Data Generating Process 2 THE MODEL AND ESTIMATORS

Leverage eect parameter = lim

s&0 Corr(t+s t; Xt+s Xt):

Integrated volatility Vt; =

Z t

t sds

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2.2 Nonparametric Volatility Estimators 2 THE MODEL AND ESTIMATORS

2.2. Nonparametric Volatility Estimators

RV ^ V RV

t; = =1

X

i=0

(Xt+(i+1) Xt+i)2 Ecient in the absence of noise, biased otherwise

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2.2 Nonparametric Volatility Estimators 2 THE MODEL AND ESTIMATORS

TSRV, MSRV { Data: with additive market microstruture noise, Zt+i = Xt+i + t+i is what's observed { n = =, TSRV a constant, L =

h

TSRVn2=3i the number of grids

• ver which the subsampling is performed and

n = (n L + 1)=n ^ V TSRV

t;

= 1 L

nL

X

i=0

(Zt+(i+L) Zt+i)2 n n

n1

X

i=0

(Zt+(i+1) Zt+i)2

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2.2 Nonparametric Volatility Estimators 2 THE MODEL AND ESTIMATORS

PAV { Zt+i = Xt+i + t+i is what's observed { g(x) = x ^ (1 x), PAV a constant, kn = [PAV=1=2]

^ V PAV

t; =121=2

PAV

nkn+1

X

i=0

( 1 kn

kn1

X

j=bkn=2c

Zt+(i+j) 1 kn

bkn=2c1

X

j=0

Zt+(i+j))2 6 2

PAV n1

X

i=0

(Zt+(i+1) Zt+i)2:

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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 : Corr(t+mt; Xt+mXt) =

• 2 4 + 42

16 m+o(m) For < 0, bias is always positive (unfavorable). But we obtain the desired limit, as ! 0 : this works as intended

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3.1 Discretization bias 3 DISENTENGLING THE BIASES

50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Cor Cor_nu Cor_V Cor_RV Cor_PAV Cor_TSRV 24

3.2 Smoothing bias 3 DISENTENGLING THE BIASES

3.2. Smoothing bias

No spot volatility available, use the integrated volatility Vt; =

R t

t sds

Corr(Vt+m; Vt;; Xt+m Xt) = Corr(t+m t; Xt+m Xt)

| {z }

! as !0

(2m 1) 2

• m2 m=3

1=2 + o(m)

m = 1 gives an estimated correlation of 0:5 So now things start to go wrong as ! 0; the high frequency limit is no longer

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3.2 Smoothing bias 3 DISENTENGLING THE BIASES

50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Cor Cor_nu Cor_V Cor_RV Cor_PAV Cor_TSRV 26

3.3 Estimation error bias 3 DISENTENGLING THE BIASES

3.3. Estimation error bias

True IV is not available, use RV This will shrink the correlation towards zero: Corr( ^ V RV

t+m; ^

V RV

t;; Xt+m Xt)

= Corr(t+m t; Xt+m Xt) (2m 1) 2

• m2 m=3

1=2

• @1 +

12 + 62 (32m 2)C 3

222CCm

1 A

1=2

+ o(m) C = n = 390=252, Cm = m2

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3.3 Estimation error bias 3 DISENTENGLING THE BIASES

m = 1 gives 0; as we found empirically. So things get bad as ! 0!

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3.3 Estimation error bias 3 DISENTENGLING THE BIASES

50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Cor Cor_nu Cor_V Cor_RV Cor_PAV Cor_TSRV 29

3.4 Robustication bias 3 DISENTENGLING THE BIASES

3.4. Robustication bias

In the presence of noise, the log-returns data are Z = X + ; let's use a noise-robust volatility estimator instead of RV. TSRV: Corr( ^ V TSRV

t+m; ^

V TSRV

t;

; Zt+m Zt) = Corr(t+m t; Xt+m Xt) (2m 1) 2

• m2 m=3

1=2

(1 + A4 + B4)1=2 + o(m) A4 =

962

TSRVC2

• CTSRV2(6m23Cm)

B4 =

8TSRV(2+2) CTSRV2(6m23Cm)

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3.4 Robustication bias 3 DISENTENGLING THE BIASES

{ n1=3 ! CTSRV and 2

= ! C

{ m = 1 gives 0.

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3.4 Robustication bias 3 DISENTENGLING THE BIASES

PAV Corr( ^ V PAV

t+m; ^

V PAV

t; ; Zt+m Zt)

= Corr(t+m t; Xt+m Xt) (2m 1) 2

• m2 m=3

1=2

(1 + A5 + B5 + C5)1=2 + o(m) A5 =

2422PAV(2+2) 2

2CPAV2(6m23Cm)

B5 =

9612C PAV 2

2CPAV2(6m23Cm)

C5 =

4811C2

• 3

PAV 2 2CPAV2(6m23Cm)

{ n1=2 ! CPAV and 2

= ! C:

{ As for RV and TSRV, m = 1 gives 0

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3.4 Robustication bias 3 DISENTENGLING THE BIASES

50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Cor Cor_nu Cor_V Cor_RV Cor_PAV Cor_TSRV 33

4 ONE POSSIBLE SOLUTION

4. One Possible Solution

4.1. Bias corrections

Model-based discretization and smoothing biases corrected by am = 2

• m2 m=3

1=2 =(2m 1)

for integrated volatility This is the reciprocal of the correction term we calculated for integra- tion of the volatility

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4.1 Bias corrections 4 ONE POSSIBLE SOLUTION

20 40 60 80 100 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Cor Cor_nu Cor_V Cor_V => Cor_nu

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4.1 Bias corrections 4 ONE POSSIBLE SOLUTION

Estimation error bias: RV nonparametrically corrected by the multi- plicative factor c3am: c3 =

@1

4E

h

4

t

i

2 n Var( ^ V RV

t+m; ^

V RV

t;)

1 A

1=2

Robustication error bias: TSRV and PAV corrected by the multiplica- tive factor c4am or c5am respectively c4 =

@1

482

TSRV4 + 8TSRVE

h

4

t

i

2 3n1=3 Var( ^ V TSRV

t+m; ^

V TSRV

t;

)

1 A

1=2

; c5 =

@1

2(A0

5 + B0 5 + C0 5)

n1=2 Var( ^ V PAV

t+m; ^

V PAV

t; )

1 A

1=2

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4.2 Bias correction by regression 4 ONE POSSIBLE SOLUTION

4.2. Bias correction by regression

Compute correlation between changes of estimated volatilities and log- prices over a time span s = m. Conduct a preliminary correction by the appropriate factor ciam as described above. This removes the additional bias terms, leaving only the discretization error bias which is linear in m:

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4.2 Bias correction by regression 4 ONE POSSIBLE SOLUTION

Run the linear regression of the estimated correlation on the time horizon over a range where linearity is appropriate: ^ m = + m + o(m) Take the intercept as the estimate of = leverage eect parameter, i.e. extrapolating to m = 0. Model-free as long as the linear structure is appropriate.

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5 SIMULATIONS

5. Simulations

Include noise, = :0005 min-by-min data over 5 years Simulation results consistent with theoretical results

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5 SIMULATIONS

50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Cor Cor_nu Cor_V Cor_RV Cor_PAV Cor_TSRV

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5 SIMULATIONS

Eectiveness of the regression method: 1 minute data, m = (6; 17) Min. 1st Qu. Median 3rd Qu. Max. Mean SD utilized population parameters in correction

• corNutorho
• 0.856
• .817
• .801
• .781
• .751
• .800

.026 corVtorho

• 0.883
• .824
• .802
• .781
• .742
• .804

.031 corRVtorho

• 0.928
• .832
• .799
• .771
• .708
• .801

.043 corPAVtorho

• 1.033
• .832
• .792
• .717
• .614
• .779

.079 corTSRVtorho

• 1.076
• .835
• .778
• .702
• .477
• .774

.117 Estimated parameters in correction

• corRVtorhoE
• 0.892
• .825
• .801
• .778
• .731
• .803

.034 corPAVtorhoE

• 0.953
• .838
• .794
• .751
• .655
• .794

.062 corTSRVtorhoE

• 1.923
• .965
• .824
• .745
• .559
• .874

.211

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5 SIMULATIONS

15 second data, m = (6; 17) Min. Q1 Q2 Q3 Max. Mean SD corNutorho

• 0.853
• .814
• .802
• .781
• .732
• .799

.024 corVtorho

• 0.873
• .825
• .810
• .787
• .722
• .805

.029 corRVtorho

• 0.878
• .829
• .810
• .788
• .694
• .805

.033 corPAVtorho

• 0.897
• .831
• .799
• .759
• .612
• .794

.058 corTSRVtorho

• 0.992
• .871
• .795
• .731
• .536
• .800

.094 corRVtorhoE

• 0.875
• .827
• .809
• .787
• .707
• .806

.030 corPAtorhoE

• 0.896
• .829
• .797
• .768
• .674
• .797

.045 corTSRVtorhoE

• 1.780
• .966
• .885
• .818
• .603
• .915

.175 Improved results at higher frequency

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6 EMPIRICAL DATA: ESTIMATING

6. Empirical Data: Estimating

Biased plug-in methods, 1mn data, 2004-2007

50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Cor Cor_VIX Cor_PAV Cor_TSRV 43

6 EMPIRICAL DATA: ESTIMATING

Bias-corrected methods

50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Correction from PA 55 60 65 70 75 80 85 −0.46 −0.42 −0.38 −0.34 m Correction from PA 20 40 60 80 100 −0.9 −0.7 −0.5 −0.3 m Correction from PA 50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Correction from TSRV 60 70 80 90 100 −0.9 −0.8 −0.7 −0.6 −0.5 m Correction from TSRV 20 40 60 80 100 −0.9 −0.7 −0.5 −0.3 m Correction from TSRV

44

6 EMPIRICAL DATA: ESTIMATING

Biased plug-in methods, MSFT, Jan 2005 - Jun 2007 m 1 2 5 10 21 63 126 252 One observation per minute TSRV .087 .049 .016

• .027
• .120
• .290
• .363
• .221

PAV .030

• .002
• .017
• .039
• .169
• .339
• .405
• .280

One observation per 5 seconds TSRV

• .006
• .034
• .047
• .069
• .207
• .364
• .362
• .345

PAV

• .051
• .074
• .102
• .115
• .257
• .412
• .400
• .404

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6 EMPIRICAL DATA: ESTIMATING

Bias-corrected methods, MSFT, Jan 2005 - Jun 2007, 1mn data

50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Correction from PA 130 140 150 160 170 −0.38 −0.34 −0.30 −0.26 m Correction from PA 50 100 150 −0.8 −0.6 −0.4 −0.2 0.0 m Correction from PA 50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Correction from TSRV 130 140 150 160 170 −0.45 −0.40 −0.35 −0.30 m Correction from TSRV 50 100 150 −0.8 −0.6 −0.4 −0.2 0.0 m Correction from TSRV

46

6 EMPIRICAL DATA: ESTIMATING

Bias-corrected methods, MSFT, Jan 2005 - Jun 2007, 5sec data

50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Correction from PA 130 140 150 160 170 −0.42 −0.38 −0.34 −0.30 m Correction from PA 50 100 150 −0.8 −0.6 −0.4 −0.2 0.0 m Correction from PA 50 100 150 200 250 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 m Correction from TSRV 130 140 150 160 170 −0.44 −0.40 −0.36 −0.32 m Correction from TSRV 50 100 150 −0.8 −0.6 −0.4 −0.2 0.0 m Correction from TSRV

47

7 CONCLUSIONS

7. Conclusions

Leverage eect puzzle: why do high-frequency data not reveal a strong leverage eect? Sources of bias: discretization, smoothing, estimation error (shrink- age), robustication. Proposed bias correction and linear extrapolation Empirically, nd agreement with values for from the estimation of stochastic volatility models or the use of VIX proxies.

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