- Email: [email protected]
On the relation between implied and realized volatility indices: Evidence from the BRIC countries
Accepted Manuscript On the relation between implied and realized volatility indices: Evidence from the BRIC countries S´onia R. Bentes PII: DOI: Reference:
S0378-4371(17)30381-3 http://dx.doi.org/10.1016/j.physa.2017.04.071 PHYSA 18173
To appear in:
Physica A
Received date: 20 January 2017 Revised date: 26 February 2017 Please cite this article as: S.R. Bentes, On the relation between implied and realized volatility indices: Evidence from the BRIC countries, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.04.071 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlights:
This paper studies the relation between implied and realized volatility
We focus on the informational content of implied volatility
We test whether implied volatility is unbiased and efficient
We employ a static OLS regression and an ADL and EC model
Results differ according to the methodology used
*Manuscript Click here to view linked References
On the relation between implied and realized volatility indices: Evidence from the BRIC countries
1, 2
Sónia R. Bentes
1
ISCAL, Av. Miguel Bombarda 20, 1069-035 Lisbon, Portugal, [email protected]
2
BRU-IUL, Av. das Forças Armadas, 1649-025, Lisbon, Portugal
ABSTRACT
This paper investigates the relation between implied (IV) and realized volatility (RV). Using monthly data from the BRIC countries, we assess the informational content of IV in explaining future RV as well as its unbiasedness and efficiency. We employ an ADL (Autoregressive Distributed Lag) and the corresponding EC (Error Correction) model and compare the results with the ones obtained from the OLS regression. Our goal is to assess the fully dynamical relations between these variables and to separate the short from the long-run effects. We found different results for the informational content of IV according to the methodologies used. However, both methods show that IV is an unbiased estimate of RV for India and that IV was not found to be efficient in any of the BRIC countries. Further, EC results reveal the presence of short and long-run effects for India, whereas Russia exhibits only short-run adjustments.
1
Keywords: implied volatility; realized volatility, ADL model, EC model
* Corresponding author. E-mail address: [email protected]
2
1. Introduction
In view of the increasing turbulence and instability of stock markets, volatility has become a very active field of research. An important issue in this debate is how well implied volatility predicts future realized volatility. The former bases on the theory of options and consists in solving the BS model in order to determine the corresponding implied volatility. Sometimes called the “the investor fear gauge” [1], IV is widely regarded as the options markets’ forecast of future volatility. Thus, if option markets are efficient, market implied volatility should be an efficient forecast of future returns volatility. In other words, IV should include the information contained in all the variables in the market information set [2]. Despite the number of studies on this subject, no consensus exists about the predictive power of IV. Besides, another shortcoming in the literature that motivates our research is that in general, studies on this topic are mainly devoted to developed markets, with only very few focusing on the emerging countries. In fact, there are a plethora of studies about the behavior of IV of financial assets and commodities in the developed countries ([3,4], inter alia), but very little regarding implied volatility of the emerging markets (e.g., [5,6]). A possible explanation for this lies in the fact that only recently IVs indices started to be available for these countries (e.g., BRIC).
On the other hand, the bulk of studies on implied volatility only considers that RV is contemporaneously related to IV and dynamically related to its own previous values, at the most. However, lagged values of IV may also influence the current value of RV. In this case, we should consider a fully dynamic model with lags on all the variables, rather than partial dynamic models as used by other authors. Therefore, we propose an alternative approach to test for the predictive power of IV based on the ADL model 3
and in a reparameterization of it – the EC model in order to account for the fully dynamical relations between the variables and to disentangle both the short and the long-run effects.
The goal of this paper is to reexamine the relation between implied and subsequent realized volatility for the BRIC countries. Briefly, we want to test (i) if IV contains some information about future realized volatility (H1), (ii) if it is an unbiased estimate of RV (H2) and, (iii) if IV is efficient (H3). We found that though the results differ according to the methodology used for the first hypothesis they remain the same for the last two. Thus, while the dynamic and static models provide different conclusions about the informational content of IV, both approaches reveal that IV is an unbiased estimate of RV for India and that it is inefficient for all of the analyzed countries. Finally, long and short-run adjustments were found for India whereas Russia only exhibits long-run adjustments to equilibrium.
The format for this paper is as follows: Section 2 outlines the methodology used, Section 3 presents the data, Section 4 for provides the estimation results and Section 5 summarizes the findings and conclude.
2. Methodological approach
We based on the OLS regression of the form [2]:
RVt 0 i IVt ut ,
(1)
where RVt and IVt denote realized and implied volatility at time t, respectively, and test three hypothesis: 4
(H1) IV contains at least some information about the future realized volatility. In this case i in Eq. (1) should be nonzero. (H2) IV is an unbiased estimate of realized volatility. If this holds, then in the expression above 0 0 and i 1 . (H3) Implied volatility is efficient, which implies that in Eq. (1) the residuals ut should be white noise, non-autocorrelated, and uncorrelated with any other variable.
Next, in order to compare the efficiency of implied volatility with that of past realized volatility we employ a multiple regression as given by:
RVt 0 i IVt h RVt 1 ut ,
where RVt 1 is the realized volatility at time t-1 and h
(2) represents the
corresponding parameter. We test hypothesis (H4), which asserts that if i IVt is an efficient forecast, h should be not statistically significant and the values of the R 2 and the information criteria of Eq. (2) should be not significantly different from those of Eq. (1).
Although regressions (1) and (2) have proved to be useful to characterize the behavior of implied and realized volatility, they neglect the fully dynamic cross-relations1 between the two variables and do not distinguish between short and long-run effects. In order to overcome these drawbacks we use an ADL(p,q), with p, q > 0, and an EC to model the relation between realized and implied volatility in the BRIC countries.
1
Note that Eq. (2) only accounts for a partial dynamic relation between the variables since it includes
a lagged variable of RV but not of IV. 5
Recall that an ADL(p,q) model can be defined as:
RVt 0 k 0 ik IVt k j 1 hj RVt j t , q
p
(3)
where a change in one or more explanatory variables can cause instantaneously changes in the dependent variable, but also in the lagged relations between variables. Note that in the expression above IVt k and RVt j stand for implied and realized volatility at times
t k
and
t j
, respectively and ik and hj are the estimated
parameters.
On the other hand, an EC(p,q) model can be expressed by:
ln RVt k 0 k1 ln IVt k ih RVt p 0 i IVt p t , p
where
RV
t p
(4)
0 i IVt p denotes the error correction term and ih measures
the adjustment speed, that is, how RV changes in response to disequilibrium.
3. Data
We used monthly data of the BOVESPA (Brazil), RTS Standard Index (Russia), S&P CNX NIFTY (India) and CSI 300 (China) indices as well as the corresponding implied volatility indices: VBOV (Brasil), RTSVX (Russia), INVIXN (India) and IVCSI (China). The sample period covers 11 years from January 2005 to January 2016 totaling 181 observations per indice.
6
Following [2,7] we used the 30-day realized volatility as a proxy for realized volatility:
2
260 22 Pt i RV 100 , ln 22 i 1 Pt i 1
(5)
where Pt i and Pt i 1 are the given prices at times t+i and t+i-1, respectively.
All data were collected from Datastream database. Table 1 shows the descriptive statistics for realized and implied volatilities.
Table 1 Summary statistics of the realized and implied volatilities Brazil
Russia
India
China
Mean
26.66169
32.34871
25.65043
27.93043
Std. Dev.
13.22811
21.84182
14.08291
11.7087
Skewness
3.272464
2.869206
1.778287
0.911113
Kurtosis
18.6815
13.0546
6.238763
2.925273
Jarque-Bera
1275.293
435.579
53.99054
12.47286
Mean
47.21525
40.78711
29.18929
39.16783
Std. Dev.
16.95852
25.41428
10.64901
20.10892
Skewness
-0.262687
3.199907
1.459176
-0.318112
Kurtosis
1.641992
15.03222
5.464257
1.6169
Jarque-Bera
9.36423
603.6286
34.0418
8.691549
Realized volatlity
Implied volatility
We found that implied volatility means exceed realized for all the BRIC countries. This encounters support in [2,8] who observed similar patterns. In addition, except for India, all countries show higher standard deviations for implied volatility than for 7
realized.
In general, statistics reveal positive skewness (except for BOVESPA and CSI implieds) and excess kurtosis. The forth moment is only lower than three for China realized and implied volatilities, which may suggest that these indices might have a distinct volatility behavior when compared with the remainder ones. Finally, all volatility series exhibit significant departures from normality as indicated by the Jarque Bera test.
4. Estimation results
Table 2 presents the estimates of 0 and i (Eq. (1)) for the BRIC countries.
Table 2 OLS Implied volatility estimates (Eq. (1)) Country
Brazil
α0
30.14806
**
(3.8180) Russia
India
China
8.63855
-0.07384
*
0.58132 (0.0726)
-0.34044
0.89043
(4.1299)
(0.1331)
(2.7286)
DW
-
Adjusted-R2
0.74066
-0.00057
1.18215
0.45037
(0.0761)
(3.4832)
27.29916
2(1)
αi
**
0.01612
**
33.2488
**
**
0.67817
2.01243
0.44322
-
0.62792
-0.01059
(0.0620)
Standard error estimates are in brackets ** Statistically significant at the 1% level * Statistically significant at the 5% level
The results show that except for India 0 is not significantly different from zero. On 8
the other hand, i is statistically significant for Russia (0.58132) and for India (0.89043) but not for the other countries. We performed a 2(1) test for the null hypothesis of i 1 , which revealed the non-rejection of the null for India. Hence, H1 (IV contains at least some information about future RV) holds for Russia and India while H2 (IV is an unbiased estimate of the future RV) is only confirmed for India. On the contrary, IV does not appear to contain any useful information to predict future RV for Brazil and China, at least in the long-run equilibrium.
Finally,
Table
3
presents
the
residual’s
diagnostics
of
autocorrelation
[Breusch-Godfrey serial correlation LM test - 2(2)] and the Pearson correlation with the explanatory variable for each regression of the four countries. The ADF tests on the residual coefficients rejected the null of a unit root in all cases. Likewise, tests for residual’s normality also rejected the null for all countries. Therefore, apart from stationarity and the evidence of no correlation of the residuals with other variables, H3 is not confirmed in any of the estimated regressions.
Table 3 OLS Implied volatility residual’s diagnostics (Eq. (1)) Country
2(2)
p-value
-coeff
t-stats
ADF
J-B
Brazil
42.38127
0.0000
**
-4.61E-15
-4.70E-14
-4.86770
**
1351.065
**
Russia
20.44255
0.0000
**
-7.61E-16
-6.63E-15
-6.29018
**
426.9766
**
India
0.639823
0.7262
6.40E-16
4.70E-15
-7.39663
**
164.9593
**
China
44.94833
0.0000
-1.87E-15
-1.75E-14
-3.99191
**
12.76808
**
**
** Statistically significant at the 1% level * Statistically significant at the 5% level
Table 4 exhibits the estimates of 0 , i and h (Eq. (2)) for each of the BRIC 9
countries.
Table 4 AR(1) implied volatility estimates (Eq. (2)) Country
Brazil
α0
9.14554
αi
*
0.01205
0.63678
(0.0617)
(0.0778)
-0.31209
1.11221
(3.0113)
(0.1779)
(0.2071)
-1.45660
1.04433
(4.6852)
(0.3139)
(0.2388)
0.01348
0.68743
(0.0464)
(0.0791)
(4.0126) Russia
India
China
9.00710
8.15143 (3.0223)
**
**
2(1)
αh
**
DW
Adjusted-R2
**
10.64169
**
1.93741
0.39050
**
7.1992
**
1.69663
0.59935
1.73402
0.43567
2.33100
0.45566
-0.12938
0.35688
**
10.63752
**
Standard error estimates are in brackets ** Statistically significant at the 1% level * Statistically significant at the 5% level
In this regression we test the hypothesis that i IVt is an efficient forecast, with the null of i 0 . The results show that this hypothesis (H4) only holds for India. We also found that for this country the R 2 coefficient is not significantly different between both regressions (India: 0.44322 vs. 0.43567). Similarly, the Schwarz Information Criterion provided the same conclusions. We performed a 2(1) test for the null hypothesis of i h 1 , where we did not reject the null for India.
The residuals of the estimated regressions are stationary for all countries (ADF tests were performed and the null hypothesis of a unit root was rejected at the 1% level or lower, in all cases). However, the potential presence of residual’s serial correlation may distort the conclusions about the ability of IV to predict future RV, which was evidenced by the low DW statistics observed in Eq. (1) for all countries except for 10
India (2.01243). In an attempt to overcome this problem we estimated four ADL(p,q) models for the countries under consideration. We chose the number of lags of RV (p) and IV (q) in Eq. (3) in order to eliminate the residuals’ autocorrelation. For our purpose, p = q ≤ 2 suffices to eliminate the autocorrelation (note that the number of lags used in each case may differ from country to country). Table 5 provides a summary of the results obtained with this approach.
Table 5 ADL(p,q) [p ≤ 2 and q ≤ 2] implied volatility estimates (Eq. (3)) Country
Brazil
i 0 i1 i 2
α0
9.14554
2(1) -
2(1) -
∑=0
DW
∑=1
*
0.01205
0.03816
-
*
-0.18140
4.89672
*
207.705
0.89043
44.7825
**
0.00442
0.00929
Adjusted-R2
1.93741
0.39050
1.89863
0.63435
0.67817
2.01243
0.44322
-
1.99626
0.48247
(4.0126) Russia
7.75569
**
(3.2211) India
-0.34044 (4.1299)
China
6.37977
*
(3.1241) Standard error estimates are in brackets ** Statistically significant at the 1% level * Statistically significant at the 5% level
In the context of an ADL regression, we should consider the null H0:
i 0
iq 0 while testing the first hypothesis (H1). Hence, a 2(1) test was
performed with q = 0, , 2 lags. Results show the non-rejection of the null for Brazil and China. By the same token, we considered 0 0 and i 0
iq 1 (q = 0,
11
, 2) to test for H2. In this case, we did not reject the joint null hypothesis for India.
Finally, we test H3 by looking at the properties of the residuals ut (Table 6). We observe that serial correlation, correlation with the explanatory variable and stationarity are no longer a problem. Despite this limited evidence, we believe that ADL regressions provide a better framework for the type of analysis under consideration than the static OLS regressions.
Table 6 ADL(p,q) [p ≤ 2 and q ≤ 2] implied volatility residual’s diagnostics (Eq. (3)) Country
2(2)
-coeff
t-stats
Brazil
0.87503
0.6456
2.93E-15
2.98E-14
-9.84962
**
450.0783
**
Russia
1.17036
0.5570
-0.04515
-0.38885
-8.21263
**
95.76550
**
India
0.63982
0.7262
6.40E-16
4.70E-15
-7.39663
**
164.9593
**
China
1.38352
0.5007
-2.21E-15
-2.05E-14
-9.26492
**
6.830386
*
p-value
ADF
J-B
** Statistically significant at the 1% level * Statistically significant at the 5% level
Based on the ADL(1,1) model we obtain the corresponding EC for the countries with significant i in the static OLS regression. This leads us to the exclusion of Brazil and China. While 01 provides information about the short-run adjustment of RV on IV, ih indicates the speed of adjustment to deviations in the long-run relation between IV and RV. Table 7 presents the EC implied volatility estimates 01 and
ih for Russia and India.
12
Table 7 EC implied volatility estimates (Eq. (4)) Country
α01
Brazil
Russia
India
αih
-
-
-
-
0.38489
-0.00777
(0.2357)
(0.0038)
0.96187
**
-0.02640
(0.2771)
(0.0058)
-
-
-
-
China
Adjusted-R2
DW
-
-
*
2.20015
0.03913
**
1.89021
0.26885
-
-
Standard error estimates are in brackets ** Statistically significant at the 1% level * Statistically significant at the 5% level
The results point to the presence of both significant short and long-run adjustments in the relation between IV and RV for India, whereas Russia only shows significant long-run adjustments. On the other hand, the deviation from the long-run relation between RV and IV takes a very short time to re-attain equilibrium with the lowest duration occurring for Russia.
5. Conclusions
This paper examined the relation between IV and RV in the context of an ADL and EC models and compared the results with the conventional approach based on the OLS regression. In particular, we wanted to test (i) if IV contained at least some information about future RV (H1), (ii) if it was an unbiased estimate about RV (H2) (iii) and, (iii) its efficiency (H3). While the ADL model allowed us to remove the residual’s autocorrelation evidenced by the static OLS regressions, the EC disentangled the short 13
from the long-run dynamics in the process. This was important insofar it allowed us to understand whether the predictor were able to adapt itself to the actual evolution of realized volatility and how fast did it adapt.
We found that for H1 the results differ across the different methodology used but remain the same for H2 and H3. Thus, while the conventional approach based on the OLS regression indicate that implied volatility contains information about future realized volatility for Russia and India, the ADL model reveals that H1 holds for Brazil and China. However, both methods indicate that IV was an unbiased estimate of future realized volatility for India. Similarly, the third hypothesis was not confirmed in the OLS neither in the ADL regressions. The EC model estimates revealed the existence of short and long-run adjustments for India, while there was only evidence of significant long-run adjustments for Russia. The speed of adjustment to equilibrium was found to be faster for Russia than for India.
This paper provides new evidence on the relation between implied and realized volatilities in the context of an ADL and EC frameworks. It adds to literature by investigating the impact of the dynamical relations between those variables and not only the contemporaneous or partial dynamic relations as proposed by earlier studies on this subject. It also shows that though a cluster, the BRIC countries may exhibit different patterns of behavior in what concerns volatility. Future research could extend our analysis by considering the effect of memory and asymmetry in this relation or even by using an alternative measure of volatility, such as, entropy to reassess this relation.
14
References
[1] R.E. Whaley, The investor fear gauge, Journal of Portfolio Management 23 (2000) 12-26. [2] B.J. Christensen, N.R. Prabhala, The relation between implied and realized volatility, Journal of Financial Economics 50 (1998) 125-150. [3] P. Agnolucci, Volatility in crude oil futures: A comparison of the predictive ability of GARCH and implied volatility models, Energy Economics 31 (2009) 316-321. [4] J.B. Blair, S.-H. Poon, S.J. Taylor, Forecasting S&P 100 volatility: the incremental information content of implied volatilities and high-frequency index returns, Journal of Econometrics 105 (2001) 5-26. [5] S.O. Nam, S.Y. Oh, H.K. Kim, B.C. Kim, An empirical analysis of price discovery and pricing bias in KOSPI 200 stock index derivatives markets, International Review of Financial Analysis 15 (2006) 398-414. [6] E.B. Vrught, US and Japanese macroeconomic news and stock market volatility in Asia-Pacific, Pacific-Basin Finance Journal 17 (2009) 611-627. [7] D.H. Hsu, B.M Murray, On the volatility of volatility, Physica A 380 (2007) 366-376. [8] C.J. Corrado, T.W. Miller Jr, The forecast quality of CBOE implied volatility indexes, Journal of Future Markets 25 (2005) 339-373.
15
S0378-4371(17)30381-3 http://dx.doi.org/10.1016/j.physa.2017.04.071 PHYSA 18173
To appear in:
Physica A
Received date: 20 January 2017 Revised date: 26 February 2017 Please cite this article as: S.R. Bentes, On the relation between implied and realized volatility indices: Evidence from the BRIC countries, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.04.071 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlights:
This paper studies the relation between implied and realized volatility
We focus on the informational content of implied volatility
We test whether implied volatility is unbiased and efficient
We employ a static OLS regression and an ADL and EC model
Results differ according to the methodology used
*Manuscript Click here to view linked References
On the relation between implied and realized volatility indices: Evidence from the BRIC countries
1, 2
Sónia R. Bentes
1
ISCAL, Av. Miguel Bombarda 20, 1069-035 Lisbon, Portugal, [email protected]
2
BRU-IUL, Av. das Forças Armadas, 1649-025, Lisbon, Portugal
ABSTRACT
This paper investigates the relation between implied (IV) and realized volatility (RV). Using monthly data from the BRIC countries, we assess the informational content of IV in explaining future RV as well as its unbiasedness and efficiency. We employ an ADL (Autoregressive Distributed Lag) and the corresponding EC (Error Correction) model and compare the results with the ones obtained from the OLS regression. Our goal is to assess the fully dynamical relations between these variables and to separate the short from the long-run effects. We found different results for the informational content of IV according to the methodologies used. However, both methods show that IV is an unbiased estimate of RV for India and that IV was not found to be efficient in any of the BRIC countries. Further, EC results reveal the presence of short and long-run effects for India, whereas Russia exhibits only short-run adjustments.
1
Keywords: implied volatility; realized volatility, ADL model, EC model
* Corresponding author. E-mail address: [email protected]
2
1. Introduction
In view of the increasing turbulence and instability of stock markets, volatility has become a very active field of research. An important issue in this debate is how well implied volatility predicts future realized volatility. The former bases on the theory of options and consists in solving the BS model in order to determine the corresponding implied volatility. Sometimes called the “the investor fear gauge” [1], IV is widely regarded as the options markets’ forecast of future volatility. Thus, if option markets are efficient, market implied volatility should be an efficient forecast of future returns volatility. In other words, IV should include the information contained in all the variables in the market information set [2]. Despite the number of studies on this subject, no consensus exists about the predictive power of IV. Besides, another shortcoming in the literature that motivates our research is that in general, studies on this topic are mainly devoted to developed markets, with only very few focusing on the emerging countries. In fact, there are a plethora of studies about the behavior of IV of financial assets and commodities in the developed countries ([3,4], inter alia), but very little regarding implied volatility of the emerging markets (e.g., [5,6]). A possible explanation for this lies in the fact that only recently IVs indices started to be available for these countries (e.g., BRIC).
On the other hand, the bulk of studies on implied volatility only considers that RV is contemporaneously related to IV and dynamically related to its own previous values, at the most. However, lagged values of IV may also influence the current value of RV. In this case, we should consider a fully dynamic model with lags on all the variables, rather than partial dynamic models as used by other authors. Therefore, we propose an alternative approach to test for the predictive power of IV based on the ADL model 3
and in a reparameterization of it – the EC model in order to account for the fully dynamical relations between the variables and to disentangle both the short and the long-run effects.
The goal of this paper is to reexamine the relation between implied and subsequent realized volatility for the BRIC countries. Briefly, we want to test (i) if IV contains some information about future realized volatility (H1), (ii) if it is an unbiased estimate of RV (H2) and, (iii) if IV is efficient (H3). We found that though the results differ according to the methodology used for the first hypothesis they remain the same for the last two. Thus, while the dynamic and static models provide different conclusions about the informational content of IV, both approaches reveal that IV is an unbiased estimate of RV for India and that it is inefficient for all of the analyzed countries. Finally, long and short-run adjustments were found for India whereas Russia only exhibits long-run adjustments to equilibrium.
The format for this paper is as follows: Section 2 outlines the methodology used, Section 3 presents the data, Section 4 for provides the estimation results and Section 5 summarizes the findings and conclude.
2. Methodological approach
We based on the OLS regression of the form [2]:
RVt 0 i IVt ut ,
(1)
where RVt and IVt denote realized and implied volatility at time t, respectively, and test three hypothesis: 4
(H1) IV contains at least some information about the future realized volatility. In this case i in Eq. (1) should be nonzero. (H2) IV is an unbiased estimate of realized volatility. If this holds, then in the expression above 0 0 and i 1 . (H3) Implied volatility is efficient, which implies that in Eq. (1) the residuals ut should be white noise, non-autocorrelated, and uncorrelated with any other variable.
Next, in order to compare the efficiency of implied volatility with that of past realized volatility we employ a multiple regression as given by:
RVt 0 i IVt h RVt 1 ut ,
where RVt 1 is the realized volatility at time t-1 and h
(2) represents the
corresponding parameter. We test hypothesis (H4), which asserts that if i IVt is an efficient forecast, h should be not statistically significant and the values of the R 2 and the information criteria of Eq. (2) should be not significantly different from those of Eq. (1).
Although regressions (1) and (2) have proved to be useful to characterize the behavior of implied and realized volatility, they neglect the fully dynamic cross-relations1 between the two variables and do not distinguish between short and long-run effects. In order to overcome these drawbacks we use an ADL(p,q), with p, q > 0, and an EC to model the relation between realized and implied volatility in the BRIC countries.
1
Note that Eq. (2) only accounts for a partial dynamic relation between the variables since it includes
a lagged variable of RV but not of IV. 5
Recall that an ADL(p,q) model can be defined as:
RVt 0 k 0 ik IVt k j 1 hj RVt j t , q
p
(3)
where a change in one or more explanatory variables can cause instantaneously changes in the dependent variable, but also in the lagged relations between variables. Note that in the expression above IVt k and RVt j stand for implied and realized volatility at times
t k
and
t j
, respectively and ik and hj are the estimated
parameters.
On the other hand, an EC(p,q) model can be expressed by:
ln RVt k 0 k1 ln IVt k ih RVt p 0 i IVt p t , p
where
RV
t p
(4)
0 i IVt p denotes the error correction term and ih measures
the adjustment speed, that is, how RV changes in response to disequilibrium.
3. Data
We used monthly data of the BOVESPA (Brazil), RTS Standard Index (Russia), S&P CNX NIFTY (India) and CSI 300 (China) indices as well as the corresponding implied volatility indices: VBOV (Brasil), RTSVX (Russia), INVIXN (India) and IVCSI (China). The sample period covers 11 years from January 2005 to January 2016 totaling 181 observations per indice.
6
Following [2,7] we used the 30-day realized volatility as a proxy for realized volatility:
2
260 22 Pt i RV 100 , ln 22 i 1 Pt i 1
(5)
where Pt i and Pt i 1 are the given prices at times t+i and t+i-1, respectively.
All data were collected from Datastream database. Table 1 shows the descriptive statistics for realized and implied volatilities.
Table 1 Summary statistics of the realized and implied volatilities Brazil
Russia
India
China
Mean
26.66169
32.34871
25.65043
27.93043
Std. Dev.
13.22811
21.84182
14.08291
11.7087
Skewness
3.272464
2.869206
1.778287
0.911113
Kurtosis
18.6815
13.0546
6.238763
2.925273
Jarque-Bera
1275.293
435.579
53.99054
12.47286
Mean
47.21525
40.78711
29.18929
39.16783
Std. Dev.
16.95852
25.41428
10.64901
20.10892
Skewness
-0.262687
3.199907
1.459176
-0.318112
Kurtosis
1.641992
15.03222
5.464257
1.6169
Jarque-Bera
9.36423
603.6286
34.0418
8.691549
Realized volatlity
Implied volatility
We found that implied volatility means exceed realized for all the BRIC countries. This encounters support in [2,8] who observed similar patterns. In addition, except for India, all countries show higher standard deviations for implied volatility than for 7
realized.
In general, statistics reveal positive skewness (except for BOVESPA and CSI implieds) and excess kurtosis. The forth moment is only lower than three for China realized and implied volatilities, which may suggest that these indices might have a distinct volatility behavior when compared with the remainder ones. Finally, all volatility series exhibit significant departures from normality as indicated by the Jarque Bera test.
4. Estimation results
Table 2 presents the estimates of 0 and i (Eq. (1)) for the BRIC countries.
Table 2 OLS Implied volatility estimates (Eq. (1)) Country
Brazil
α0
30.14806
**
(3.8180) Russia
India
China
8.63855
-0.07384
*
0.58132 (0.0726)
-0.34044
0.89043
(4.1299)
(0.1331)
(2.7286)
DW
-
Adjusted-R2
0.74066
-0.00057
1.18215
0.45037
(0.0761)
(3.4832)
27.29916
2(1)
αi
**
0.01612
**
33.2488
**
**
0.67817
2.01243
0.44322
-
0.62792
-0.01059
(0.0620)
Standard error estimates are in brackets ** Statistically significant at the 1% level * Statistically significant at the 5% level
The results show that except for India 0 is not significantly different from zero. On 8
the other hand, i is statistically significant for Russia (0.58132) and for India (0.89043) but not for the other countries. We performed a 2(1) test for the null hypothesis of i 1 , which revealed the non-rejection of the null for India. Hence, H1 (IV contains at least some information about future RV) holds for Russia and India while H2 (IV is an unbiased estimate of the future RV) is only confirmed for India. On the contrary, IV does not appear to contain any useful information to predict future RV for Brazil and China, at least in the long-run equilibrium.
Finally,
Table
3
presents
the
residual’s
diagnostics
of
autocorrelation
[Breusch-Godfrey serial correlation LM test - 2(2)] and the Pearson correlation with the explanatory variable for each regression of the four countries. The ADF tests on the residual coefficients rejected the null of a unit root in all cases. Likewise, tests for residual’s normality also rejected the null for all countries. Therefore, apart from stationarity and the evidence of no correlation of the residuals with other variables, H3 is not confirmed in any of the estimated regressions.
Table 3 OLS Implied volatility residual’s diagnostics (Eq. (1)) Country
2(2)
p-value
-coeff
t-stats
ADF
J-B
Brazil
42.38127
0.0000
**
-4.61E-15
-4.70E-14
-4.86770
**
1351.065
**
Russia
20.44255
0.0000
**
-7.61E-16
-6.63E-15
-6.29018
**
426.9766
**
India
0.639823
0.7262
6.40E-16
4.70E-15
-7.39663
**
164.9593
**
China
44.94833
0.0000
-1.87E-15
-1.75E-14
-3.99191
**
12.76808
**
**
** Statistically significant at the 1% level * Statistically significant at the 5% level
Table 4 exhibits the estimates of 0 , i and h (Eq. (2)) for each of the BRIC 9
countries.
Table 4 AR(1) implied volatility estimates (Eq. (2)) Country
Brazil
α0
9.14554
αi
*
0.01205
0.63678
(0.0617)
(0.0778)
-0.31209
1.11221
(3.0113)
(0.1779)
(0.2071)
-1.45660
1.04433
(4.6852)
(0.3139)
(0.2388)
0.01348
0.68743
(0.0464)
(0.0791)
(4.0126) Russia
India
China
9.00710
8.15143 (3.0223)
**
**
2(1)
αh
**
DW
Adjusted-R2
**
10.64169
**
1.93741
0.39050
**
7.1992
**
1.69663
0.59935
1.73402
0.43567
2.33100
0.45566
-0.12938
0.35688
**
10.63752
**
Standard error estimates are in brackets ** Statistically significant at the 1% level * Statistically significant at the 5% level
In this regression we test the hypothesis that i IVt is an efficient forecast, with the null of i 0 . The results show that this hypothesis (H4) only holds for India. We also found that for this country the R 2 coefficient is not significantly different between both regressions (India: 0.44322 vs. 0.43567). Similarly, the Schwarz Information Criterion provided the same conclusions. We performed a 2(1) test for the null hypothesis of i h 1 , where we did not reject the null for India.
The residuals of the estimated regressions are stationary for all countries (ADF tests were performed and the null hypothesis of a unit root was rejected at the 1% level or lower, in all cases). However, the potential presence of residual’s serial correlation may distort the conclusions about the ability of IV to predict future RV, which was evidenced by the low DW statistics observed in Eq. (1) for all countries except for 10
India (2.01243). In an attempt to overcome this problem we estimated four ADL(p,q) models for the countries under consideration. We chose the number of lags of RV (p) and IV (q) in Eq. (3) in order to eliminate the residuals’ autocorrelation. For our purpose, p = q ≤ 2 suffices to eliminate the autocorrelation (note that the number of lags used in each case may differ from country to country). Table 5 provides a summary of the results obtained with this approach.
Table 5 ADL(p,q) [p ≤ 2 and q ≤ 2] implied volatility estimates (Eq. (3)) Country
Brazil
i 0 i1 i 2
α0
9.14554
2(1) -
2(1) -
∑=0
DW
∑=1
*
0.01205
0.03816
-
*
-0.18140
4.89672
*
207.705
0.89043
44.7825
**
0.00442
0.00929
Adjusted-R2
1.93741
0.39050
1.89863
0.63435
0.67817
2.01243
0.44322
-
1.99626
0.48247
(4.0126) Russia
7.75569
**
(3.2211) India
-0.34044 (4.1299)
China
6.37977
*
(3.1241) Standard error estimates are in brackets ** Statistically significant at the 1% level * Statistically significant at the 5% level
In the context of an ADL regression, we should consider the null H0:
i 0
iq 0 while testing the first hypothesis (H1). Hence, a 2(1) test was
performed with q = 0, , 2 lags. Results show the non-rejection of the null for Brazil and China. By the same token, we considered 0 0 and i 0
iq 1 (q = 0,
11
, 2) to test for H2. In this case, we did not reject the joint null hypothesis for India.
Finally, we test H3 by looking at the properties of the residuals ut (Table 6). We observe that serial correlation, correlation with the explanatory variable and stationarity are no longer a problem. Despite this limited evidence, we believe that ADL regressions provide a better framework for the type of analysis under consideration than the static OLS regressions.
Table 6 ADL(p,q) [p ≤ 2 and q ≤ 2] implied volatility residual’s diagnostics (Eq. (3)) Country
2(2)
-coeff
t-stats
Brazil
0.87503
0.6456
2.93E-15
2.98E-14
-9.84962
**
450.0783
**
Russia
1.17036
0.5570
-0.04515
-0.38885
-8.21263
**
95.76550
**
India
0.63982
0.7262
6.40E-16
4.70E-15
-7.39663
**
164.9593
**
China
1.38352
0.5007
-2.21E-15
-2.05E-14
-9.26492
**
6.830386
*
p-value
ADF
J-B
** Statistically significant at the 1% level * Statistically significant at the 5% level
Based on the ADL(1,1) model we obtain the corresponding EC for the countries with significant i in the static OLS regression. This leads us to the exclusion of Brazil and China. While 01 provides information about the short-run adjustment of RV on IV, ih indicates the speed of adjustment to deviations in the long-run relation between IV and RV. Table 7 presents the EC implied volatility estimates 01 and
ih for Russia and India.
12
Table 7 EC implied volatility estimates (Eq. (4)) Country
α01
Brazil
Russia
India
αih
-
-
-
-
0.38489
-0.00777
(0.2357)
(0.0038)
0.96187
**
-0.02640
(0.2771)
(0.0058)
-
-
-
-
China
Adjusted-R2
DW
-
-
*
2.20015
0.03913
**
1.89021
0.26885
-
-
Standard error estimates are in brackets ** Statistically significant at the 1% level * Statistically significant at the 5% level
The results point to the presence of both significant short and long-run adjustments in the relation between IV and RV for India, whereas Russia only shows significant long-run adjustments. On the other hand, the deviation from the long-run relation between RV and IV takes a very short time to re-attain equilibrium with the lowest duration occurring for Russia.
5. Conclusions
This paper examined the relation between IV and RV in the context of an ADL and EC models and compared the results with the conventional approach based on the OLS regression. In particular, we wanted to test (i) if IV contained at least some information about future RV (H1), (ii) if it was an unbiased estimate about RV (H2) (iii) and, (iii) its efficiency (H3). While the ADL model allowed us to remove the residual’s autocorrelation evidenced by the static OLS regressions, the EC disentangled the short 13
from the long-run dynamics in the process. This was important insofar it allowed us to understand whether the predictor were able to adapt itself to the actual evolution of realized volatility and how fast did it adapt.
We found that for H1 the results differ across the different methodology used but remain the same for H2 and H3. Thus, while the conventional approach based on the OLS regression indicate that implied volatility contains information about future realized volatility for Russia and India, the ADL model reveals that H1 holds for Brazil and China. However, both methods indicate that IV was an unbiased estimate of future realized volatility for India. Similarly, the third hypothesis was not confirmed in the OLS neither in the ADL regressions. The EC model estimates revealed the existence of short and long-run adjustments for India, while there was only evidence of significant long-run adjustments for Russia. The speed of adjustment to equilibrium was found to be faster for Russia than for India.
This paper provides new evidence on the relation between implied and realized volatilities in the context of an ADL and EC frameworks. It adds to literature by investigating the impact of the dynamical relations between those variables and not only the contemporaneous or partial dynamic relations as proposed by earlier studies on this subject. It also shows that though a cluster, the BRIC countries may exhibit different patterns of behavior in what concerns volatility. Future research could extend our analysis by considering the effect of memory and asymmetry in this relation or even by using an alternative measure of volatility, such as, entropy to reassess this relation.
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References
[1] R.E. Whaley, The investor fear gauge, Journal of Portfolio Management 23 (2000) 12-26. [2] B.J. Christensen, N.R. Prabhala, The relation between implied and realized volatility, Journal of Financial Economics 50 (1998) 125-150. [3] P. Agnolucci, Volatility in crude oil futures: A comparison of the predictive ability of GARCH and implied volatility models, Energy Economics 31 (2009) 316-321. [4] J.B. Blair, S.-H. Poon, S.J. Taylor, Forecasting S&P 100 volatility: the incremental information content of implied volatilities and high-frequency index returns, Journal of Econometrics 105 (2001) 5-26. [5] S.O. Nam, S.Y. Oh, H.K. Kim, B.C. Kim, An empirical analysis of price discovery and pricing bias in KOSPI 200 stock index derivatives markets, International Review of Financial Analysis 15 (2006) 398-414. [6] E.B. Vrught, US and Japanese macroeconomic news and stock market volatility in Asia-Pacific, Pacific-Basin Finance Journal 17 (2009) 611-627. [7] D.H. Hsu, B.M Murray, On the volatility of volatility, Physica A 380 (2007) 366-376. [8] C.J. Corrado, T.W. Miller Jr, The forecast quality of CBOE implied volatility indexes, Journal of Future Markets 25 (2005) 339-373.
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