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IMPLIED VOLATILITY
AROUND THE WORLD:
GEOGRAPHICAL MARKETS AND
ASSET CLASSES
Julian P. Velev
Brian C. Payne
Jiří Trešl
Wilfredo Toledo
Charles University
Center for Economic Research and Graduate Education
Academy of Sciences of the Czech Republic
Economics Institute
CERGE
EI
WORKING PAPER SERIES (ISSN 1211-3298)
Electronic Version
562
Working Paper Series
(ISSN 1211-3298)
562
Implied Volatility Around the World:
Geographical Markets and Asset Classes
Julian P. Velev
Brian C. Payne
Jiří Trešl
Wilfredo Toledo
CERGE-EI
Prague, April 2016
ISBN 978-80-7343-369-7 (Univerzita Karlova v Praze, Centrum pro ekonomický výzkum
a doktorské studium)
ISBN 978-80-7344-375-7 (Národohospodářský ústav AV ČR, v. v. i.)
Implied Volatility Around the World:
Geographical Markets and Asset Classes
Julian P. Velev
University of Puerto Rico and University of Nebraska, Lincoln
Brian C. Payne
U.S. Air Force Academy
Jiří Trešl*
University of Nebraska, Lincoln and CERGE-EI
Wilfredo Toledo
University of Puerto Rico
April 2016
ABSTRACT
This study analyzes the implied volatility-return relationship across asset classes, geographical
regions, and time, which extends efforts documenting the instantaneous relation between
implied volatility changes and index returns. Modeling the relationships as a GARCH process
with lagged terms, we confirm that implied volatility depends on the immediate index changes.
However, contemporaneous volatility changes are also explained by lagged index returns and
past volatility moves. While this short-term volatility behavior is heavily asymmetric on the
side of negative moves, in the long-term there is indifference between positive and negative
moves. Volatility also appears to transfer from larger, primary markets to smaller, secondary
markets, as price moves in larger markets explain a large portion of volatility in smaller
markets. Volatility in larger markets also transfers to the commodity and currency markets.
Keywords: Implied volatility, GARCH, Risk transfer, International asset classes
JEL Classification: C22, C58, F37, G15
* Corresponding author: [email protected]
We would like to thank Jan Hanousek, Jan Novotny, Demian Berchtold, Anastasiya Shamshur, Jeff Bredthauer,
George McCabe, and Emre Unlu. We also thank seminar participants at the 2015 European FMA Venice, the
2015 AFS, and the University of Nebraska. This work was partially supported by the National Science Foundation
(Grant Nos. DMR-1105474 and EPS-1010094) and by GAČR grant No.14-27047S. The views expressed in this
paper are those of the authors and do not necessarily reflect the official policy or position of the Air Force, the
Department of Defense or the US Government. The usual disclaimer applies.
1
I. INTRODUCTION
Even though risk aversion is theoretically defined in asset pricing models (Campbell et
al, 1997; Cochrane, 2001), its empirical identification spans a series of risk aversion indicators,
one of them being the VIX (Coudert and Gex, 2008). The VIX is called the “investors’ fear
gauge,” as it expresses the consensus view about the expected future stock market volatility
(Whaley 2000). 1 In relation to the stock market, it spikes during high impact political and
economic events (e.g. the Gulf wars, 9/11, recent financial crisis), and general market
nervousness such as in 1998 (Whaley,2000; Bloom, 2009). In this study, we confirm that
implied volatility depends on the immediate index changes and find that contemporaneous
volatility changes are also explained by lagged index returns and past volatility moves, shortterm volatility behavior is heavily asymmetric on the side of negative moves, long-term
behavior is indifference between positive and negative moves, and that volatility also appears
to transfer from larger, primary markets to smaller, secondary markets. Lastly, implied
volatility in larger markets also transfers to the commodity and currency markets.
As a proxy for global risk perceptions or investors’ fear, the VIX has become an
explanatory variable not only for the stock price changes (Blair, Poon and Taylor, 2001;
Corrado and Miller, 2005; Giot, 2006) and economic activity (Bloom, 2009; Bekaert and
Hoerova, 2013) but also for other prices such as fixed income securities, commodity, and
currency prices (Stivers and Sun , 2002; Soytas, and Hacihasanoglu, 2011. In a related recent
development, Durand, Lim, and Zumwalt (2011) study the relation not only between the total
market returns and the VIX but also the relation of the three Fama and French (1993) factors
on VIX. They found that the risk premium and the value premium are especially sensitive to
changes in VIX. As well, they show that an increase in volatility is related to a movement to
high-quality stocks and an increase in the required risk premium. Exploiting the relation
between VIX and SPX has led Hilal, Poon, and Tawn (2011) to propose using VIX futures to
hedge the fat tails of the SPX distribution. They use a generalized autoregressive conditional
heteroscedasticity (GARCH) model to filter the SPX data for heteroscedasticity. Further,
Mencia and Sentana (2013) conduct an analysis of VIX valuation models while Li and Zhang
(2013) the diffusion term of the state variables in affine jump-diffusion models.
The terms “implied volatility” and “volatility” are used interchangeably in this study. Implied volatility is the
measure used in all quantitative analysis. Please see Whaley (2009) for a comprehensive discussion and
description of the VIX.
1
2
Despite the fair amount of work on this topic, there are some key points left unaddressed
in the existing literature. First, most studies concentrate on only one index (predominantly SPX
and VIX), thus it is not possible to make an inference of the general validity of the finding.
Second, in most cases the problem of data heteroscedasticity has not been addressed in a
consistent manner. Importantly, most available models relating volatility changes to index
returns consider only coincident changes of the relevant variables. The underlying assumption
of these models is that the only information available to and considered by the investor are the
current spot prices of a single index, which leads to an asymmetric response of volatility to
positive and negative index changes (e.g. Simon, 2003, Giot, 2006, and Whaley, 2000 and
2009). Such models predict that if an index fluctuates around the same value, volatility will
increase indefinitely, which is contrary to empirical evidence and intuition. The more natural
assumption is that the investors have information about past and present index values and other
important market indicators, and they account for this memory when generating expectations
about the future.
We show that the wealth of the available implied volatility data allows one to extract
information about the decision process through which investors and markets form expectations
about future volatility. For the first time to our knowledge, we enhance the prior volatility
modeling technique and analyze volatility behavior across asset classes and across
geographically separate markets. Our model assumes that investors have complete information
about the particular local market (i.e., present and past index and volatility values) as well as
information about the proxy for global volatility. The assumption of semi-form market
efficiency is not far-fetched given the abundance of publicly-available data about prices and
volatility measures. By comparing model estimations between different index-volatility pairs,
across geographical regions and asset classes, we identify transmission mechanisms which are
universal across markets. Knowing how global markets and asset classes import volatility can
lend insight into the transmission patterns of systematic risk throughout the markets. This study
generates key findings on how global markets and asset classes import volatility and helps
explain the transmission patterns of systematic risk.
First, we find that the implied volatility in all markets shows a distinct short-term and
long-term pattern. In the short term the increased volatility results chiefly from a very strong
reaction to price drops. Our result is consistent with previous studies, which consider only the
effect of immediate price changes on the volatility. By expanding the information considered,
we find that these “instantaneous” models in the prior literature artificially constrain the
asymmetric response of volatility to price changes , as it is in fact much larger than previously
3
estimated. Another important contribution we find is that in the long term the idea of market
equilibrium effects dominate, leading to the market’s interpretation of past negative moves as
a sign that the market will start stabilizing.
Second, we find that there is an imbalance between fear and optimism. Volatility is
driven largely by fear, with reaction to negative influences figuring much more prominently
than reaction to positive influences. Here, by negative influences we consider not only
immediate negative index returns considered in previous studies, but also past negative index
returns and past increases in volatility. Generally, we find that the reaction to positive moves
is much more subdued and varied. Positive returns can be interpreted as a good sign (most
stocks), an indifferent sign (Asian markets and currencies), or a bad sign (commodities).
Finally, the results of this study help identify primary and secondary volatility markets.
On the primary markets, price moves generate the expected volatility; the secondary markets
import a large portion of the volatility from the primary markets. The data indicates that the
dependent markets follow the global volatility without discriminating between positive and
negative news.
This study extends the volatility literature in myriad ways. First, it extends the scope of
previous studies by simultaneously analyzing 12 stock indices from multiple global regions,
two commodities, and one exchange traded fund involving currencies. Second, it implements
the GARCH techniques to account for the empirical features of the data. Third, it accounts for
market by including lagged terms in the model and global volatility effects, respectively. We
find that the volatility changes depend not only on the immediate price changes of the
underlying index; they are also strongly related to past index returns and volatility changes as
well as to the perception of global volatility
The paper is organized as follows. Section II summarizes the relevant literature. Section
III explains the methodology used in the estimations, followed by a description of the data in
Section IV. Section V presents the results of the empirical estimations. The final section
discusses the implications and concludes.
II. LITERATURE OVERVIEW
The volume of the options and derivative securities markets greatly exceeds that of the
underlying assets. Valuation of derivatives depends crucially on the market participants’
expectation of future volatility (Wilmott, 2006). Thus implied volatility often garners as much
attention as the option price in option listings, and estimation of the future volatility is a critical
4
aspect of market research. Scholars and practitioners have used various methods of forecasting
volatility, such as historical volatility, implied volatilities from options data (Day and Lewis,
1992), forecasting based on ARCH models (Bollerslev, 1994), and forecasting based on
implied volatility indices. These prior studies have analyzed the contemporaneous relationship
between the US stock market volatility index and returns to its underlying assets. However, to
our knowledge no work has explored volatility in the context of time-variant relationships,
connections among asset classes, or transmission among different international markets. By
filling this void, this study sheds light on the patterns of systematic risk around the world and
through asset classes.
A short history of volatility literature highlights the growing relevance of volatility
measures. In a study commissioned by the CBOE, Whaley (1993) introduced the idea of a
volatility index, designed to measure the 30-day market volatility implied by at-the-money
options written on the S&P 100 (OEX) stock index. The first volatility index, the CBOE
Volatility Index (VIX), was introduced in 1993 and calculated retroactively until 1986. In 2003,
CBOE revamped the VIX methodology to use near- and next-term out-of-the-money options
written on the S&P 500 (SPX) index (CBOE, 2009). It was calculated retroactively until 1990.
Simultaneously, the volatility of the OEX index was renamed VXO and is available from 1986
to the present. A number of other volatility indices have appeared globally since, all calculated
using options on stock indices and commodity and currency exchange traded funds (ETFs).
Meanwhile, studies have shown VIX and other volatility indices outperform other methods as
predictors of near-term market volatility. For example, in a comparison of the realized volatility
of S&P 100 with the prediction of several forecast methods, including an ARCH model using
daily returns, Blair, Poon and Taylor (2001) showed VIX provides the most accurate forecast
for all forecasting horizons. Similarly, Corrado and Miller (2005) examined the forecast quality
of the volatility indices of the S&P 100 (VXO), S&P 500 (VIX), and the NASDAQ 100 (VXN).
They concluded that the volatility indices provide high quality forecasts of future volatility. In
addition, Hsu and Murray (2007) showed very high correlations between VIX and the 30-day
realized volatility even though they found no such correlation between SPX and the realized
volatility. Thus, analyzing the behavior of VIX and other similar measures is of paramount
importance for derivative securities modeling.
As VIX and similar volatility measures gained traction, a new branch of economic
literature has emerged using VIX as an explanatory variable to proxy for market volatility. In
an early comprehensive study Fleming, Ostdiek, and Whaley (1995) found a relation between
the VXO (then VIX) changes and OEX returns. They used an ordinary least squares (OLS)
5
model with heteroscedasticity-corrected standard errors. They found a negative relation
between the volatility changes and stock index returns—that is, drops (hikes) in the volatility
are associated with increases (decreases) in the stock prices. Also, they identified asymmetry
in the volatility response: it responds stronger to drops in the prices than to hikes. In a later
study, Whaley (2000) investigated the relation between the OEX index returns and VXO
changes. This effort obtained a similar negative and asymmetric relation, which justified the
reference to the VXO as an “investor fear gauge.” Most recently, Whaley (2009) studied the
relationship between VIX changes and SPX returns. The results from his OLS model indicated
that the returns of the SPX explain approximately 55% of the VIX change. Once again, price
drops (increases) increased (reduced) the volatility; the asymmetry in the VIX response to SPX
price moves held, with negative moves feared more. He found an insignificant intercept term,
which is consistent with the intuition that prices should remain steady when volatility does not
change.
Other authors have obtained similar results. Simon (2003) studied the relation between
the NASDAQ 100 (NDX) returns and its volatility VXN. This OLS model corrected for
heteroscedasticity and autocorrelation using the first lag, obtaining similar results to the others:
a negative relation between returns and volatility changes, with asymmetric behavior between
negative and positive moves. The estimation showed that VXN is mean-reverting around the
dot-com crash (i.e., no structural change). Giot (2006) studied the relation between both the
S&P 100 and the NASDAQ 100 and their respective volatilities using OLS with
heteroscedastic-consistent standard errors. The VIX model with the positive and negative
returns of the log of OEX his results are similar to those of Whaley (2009). This study found
analogous results from the OLS regression of the VXN returns with the positive and negative
NDX returns, however, the asymmetry was rather weak. Giot reported structural changes
across time, since the OLS regression parameters take different values in several different time
periods. This finding implies that there is no universal relation between returns and volatility
and that investors may behave differently in different environments.
III. MODEL AND METHODOLOGY
We posit a model where investors follow the present development of a particular asset
index and have access to the index historical returns and implied volatility. In addition, they
are aware of all major economic indicators. Therefore, we take the volatility changes to depend
on the immediate price changes of the underlying index, on the past index returns and volatility
6
changes, and on the global perception of volatility. This general form of the model, which we
will refer to as a global model, is found in equation (1),
∆𝜎𝑡 = 𝛼 + 𝛽0− ∆𝑥𝑡− + 𝛽0+ ∆𝑥𝑡+ + ∑
+∑
𝑗=1
+
−
𝛽𝑖− ∆𝑥𝑡−𝑖
+ 𝛽𝑖+ ∆𝑥𝑡−𝑖
+
𝑖=1
+
−
𝛾𝑗− ∆𝜎𝑡−𝑗
+ 𝛾𝑗+ ∆𝜎𝑡−𝑗
+∑
+
−
𝛿𝑘− ∆𝜍𝑡−𝑘
+ 𝛿𝑘+ ∆𝜍𝑡−𝑘
+ 𝜀𝑡 ,
(1)
𝑘=0
2
ℎ𝑡 = 𝜆 + 𝜇1 𝜀𝑡−1
+ 𝜈1 ℎ𝑡−1
where ∆𝜎𝑡 = 𝜎𝑡 − 𝜎𝑡−1 is the change in volatility index, and ∆𝑥𝑡 = (𝑥𝑡 − 𝑥𝑡−1 )/𝑥𝑡−1
represents the percentage change in the underlying index (i.e., the return). Using this
convention, the index returns are measured in percentage points as volatility, and we interpret
the slope coefficients as the percentage change in the volatility in response to one-percent
change in the index. An additional benefit is that coefficients for different indices are directly
comparable. Furthermore, ∆𝑥𝑡−𝑖 and ∆𝜎𝑡−𝑖 are the lagged terms of the index returns and the
volatility changes, respectively. The ∆𝜍𝑡−𝑖 terms represent the changes in the global volatility,
where the sum includes the instantaneous value and lagged terms. The following model
estimations include two weeks (ten trading days, or lags) of the returns and own volatility
changes and one week (five lags) of the global volatility changes. Finally, because the volatility
indices do not exhibit constant variances, the variance ℎ𝑡 is estimated simultaneously as a
GARCH(1,1) process. The plus and minus superscripts delineate positive and negative changes
of the index returns and the volatility. This separation, which is standard in this literature,
allows us to differentiate the investors’ response to improving or worsening markets.
There are several levels of approximation to the general model given in equation (1).
On the first level, one can ignore the global volatility influence by setting the coefficients 𝛿𝑖± =
0. This model, which we refer to as a local model, accounts for only the present and past values
of the index itself and its volatility. One can partially justify this approximation from the point
of view that the global volatility change is difficult to quantify. Estimation of this model would
yield new coefficients 𝛼 for the intercept, 𝛽𝑖− , 𝛽𝑖+ for the present and past lagged returns, and
𝛾𝑗− , 𝛾𝑖+ for the past volatility changes. The simplest possible model is when we further set the
coefficients of the lagged returns to zero 𝛽𝑖± = 0 for 𝑖 ≥ 1 as well as the lagged volatility
coefficients, 𝛾𝑗± = 0. This model, which we will call instantaneous price model, takes into
account only instantaneous moves in the index price. Most prior literature implements this
model. However, we contend it is difficult to justify relative to the more general models
because it assumes investors act only based on the present information. They are presumed
7
unaware of the history of the traded asset. Estimation of this model will yield yet another set
of coefficients 𝛼, 𝛽0− , 𝛽0+ , which we also estimate in this paper for benchmarking purposes.
Before estimating the models with empirical data, we want to establish some general
properties of the models. First, if all variables which affect the volatility are included in the
model, the intercept 𝛼 must be zero.2 This property must hold because if the model is wellspecified (i.e., includes the correct explanatory variables), and they do not change, then the
market should not expect any change in volatility, which leads to a zero intercept term. The
second property is that if the market is in equilibrium, such as the prices and the volatility
fluctuate around a constant value, the asymmetry of the coefficients is zero. 3 This zero
asymmetry occurs because in market equilibrium the local and global asymmetry cancel any
price asymmetry. One corollary results from this second property: the instantaneous price
model has symmetric coefficients.4 In this model the positive and negative moves have to offset
each other. Thus, asymmetry is only possible if the index trends constantly upward.
Proof: If we consider a moment 𝑡 such as during a sufficiently long time before 𝑡 nothing changed for
±
±
considerable amount of time (∆𝑥𝑡−𝑖
= 0, 𝑖 ≥ 0 and ∆𝜎𝑡−𝑗
= 0, 𝑗 ≥ 1) the model predicts ∆𝜎𝑡 = 𝛼. However,
under these conditions, if there are no other external to the model factors influencing the volatility, the volatility
should remain constant ∆𝜎𝑡 = 0. Therefore, the intercept 𝛼 = 0. On the contrary if the intercept was not zero
this would imply a drift of the volatility over time even if the market does not change, which clearly contradicts
empirical evidence. Thus, the presence if the intercept can only be explained by the influence of factors not
included in the model.
3
Proof: Equilibrium would imply that the expectation value of the price and volatility changes is zero over a
sufficiently large interval 〈∆𝑥〉 = (∑𝑡 ∆𝑥𝑡 )/𝑇 = 0 and similarly 〈∆𝜎〉 = 0. We can write 〈∆𝑥〉 = 〈∆𝑥 + 〉 + 〈∆𝑥 − 〉,
̅̅̅̅ , where ̅∆𝑥
̅̅̅ is a constant independent of 𝑡 which measured the
from where it follows that 〈∆𝑥 − 〉 = −〈∆𝑥 + 〉 = ∆𝑥
̅̅̅̅. Then averaging eq. (1) we obtain
average size of the price fluctuations. Similarly 〈∆𝜎 − 〉 = −〈∆𝜎 + 〉 = ∆𝜎
2
〈∆𝜎〉 = 𝛼 + ∑
𝑖=0
(𝛽𝑖− − 𝛽𝑖+ ) 〈∆𝑥 − 〉 + ∑
𝑗=1
(𝛾𝑗− − 𝛾𝑗+ ) 〈∆𝜎 − 〉 + ∑
𝑘=1
(𝛿𝑘− − 𝛿𝑘+ ) 〈∆𝜍 − 〉 = 0
(1)
where we first set the intercept to zero in accordance with the previous preposition. At this point it is convenient
to introduce the following quantities
𝑎𝑥 = ∑
𝑖=0
(𝛽𝑖− − 𝛽𝑖+ ) , 𝑎𝜎 = ∑
𝑖=1
(𝛾𝑗− − 𝛾𝑗+ ) , 𝑎𝜍 = ∑
𝑖=0
(𝛿𝑘− − 𝛿𝑘+ )
(2)
which have the meaning in the asymmetry between the coefficients for the positive and negative moves for the
index prices, the implied index volatility, and the global volatility respectively. With this notation we can write
𝑎𝑥 = −
̅̅̅̅
̅̅̅
∆𝜎
∆𝜍
𝑎𝜎 −
𝑎
̅∆𝑥
̅̅̅
̅∆𝑥
̅̅̅ 𝜍
(3)
which indicates that in equilibrium the price asymmetry is cancelled by the asymmetry of the local and global
volatilities. The coefficients correct for the relative difference in the size of the fluctuations.
4
Proof: If the coefficients of the lagged price terms are zero, then 𝛽0− = 𝛽0+ . The asymmetry in this model can
̅̅̅̅ .
appear only at the cost of an artificial intercept term, in which case 𝛽0− − 𝛽0+ = −𝛼/∆𝑥
8
IV. DATA
We estimate the models on real market data for 15 different index-volatility pairs. Of
these, 12 are stock indices that represent the three most important geographical regions (US,
EU, and Asia Pacific), two are commodity ETFs (gold and oil), and one is a currency ETF (the
US dollar-to-Euro exchange). Considering these data in the aggregate permits us to compare
the models across geographical regions and among asset classes. The data comes directly from
the CBOE, the Euronext, and public sources Yahoo and Google Finance.
The CBOE, which originated the Standard & Poor’s 500 (SPX) Volatility Index (VIX)
in 1993, also compiles volatility indices based on options on all major US indices including the
Dow Jones Industrial Average (DJIA) Volatility Index (VDX); the Nasdaq 100 (NDX)
Volatility Index (VXN); and the Russell 2000 (RUT) Volatility Index (RVX). As demand for
volatility information and volatility-based hedging tools continued to increase, CBOE licensed
the methodology to other exchanges. Volatility indices are now calculated by Euronext on
several of the EU stock indices, including the UK Financial Times Stock Exchange 100 (FTSE)
Volatility Index (VFTSE); the Dutch Amsterdam Exchange (AEX) Volatility Index (VAEX);
the French Cotation Assistée en Continu (CAC) Volatility Index (VCAC); and the Belgium 20
(BEL) Volatility Index (VBEL). These indices were launched in 2007-2008 but have been
calculated retroactively to the beginning to 2000. Some exchanges calculate their volatility
indices based on different methodologies. For example, the Deutsche Börse compiles the
Deutscher Aktienindex (DAX) Volatility Index (VDAX) is based on implied volatility. The
CBOE methodology is also used to compile volatility indices in the Asia Pacific region,
including the Hong Kong Hang Seng Index (HSI) Volatility Index (VHSI) and the Japan Nihon
Keizai Shimbun 225 (NIKKEI) Volatility Index (VJX).
In 2008, due to the growing options market on commodity and currency exchange
traded funds (ETFs), CBOE started using the VIX methodology to compile volatility indices
for these markets as well. The ETFs are securities designed to mimic the prices of certain
commodities or currency exchange rates. They are traded as stocks at exchanges. Thus,
investors trading ETFs get exposure to the commodity or currency price changes without
owning any commodities or currencies. In addition, ETFs allow buying fractional amounts,
buying on a margin, and/or selling short. CBOE calculates and publishes the SPDR Gold Shares
(GLD) Volatility Index (GVZ); the US Oil Fund (USO) Volatility Index (OVX); and Currency
Share Euro Trust (FXE) Volatility Index (EVZ) (see Exhibit 1). Given their global relevance
and practical attention they receive, we use the first two to proxy for commodities’ price9
volatility behavior; we use the last ETF as a proxy to analyze currencies’ price-volatility
behavior. We leave the exploration of other commodities and currency relationships to future
research.
Exhibit 1 defines the volatility-index pairs and shows their sample size. As discussed,
we analyze pairs from around the globe, commodity, and currency asset classes. At the
extremes, there are as many as 5,544 observations (SPX-VIX) and as few as 1,005 (FXEEVZ).5
Exhibit 1: Index and Volatility Overview
This Exhibit shows the index and volatility overview of stock, commodity, and currency indices with
the corresponding volatility indices, including the date of the index origination and the number of
observations until the end of 2011.
Index
Volatility (Ticker)
Inception
Observations (Daily)
US
SPX
VIX
1/1/1990
5,544
DJIA
VXD
10/7/1997
3,582
NDX
VXN
10/10/2000
2,743
RUT
RVX
1/1/2004
2,013
FTSE (UK)
VFTSE
1/4/2000
2,997
CAC 40 (France)
VCAC
1/3/2000
3,056
DAX (Germany)
VDAX
1/2/1992
1,550
AEX (Netherlands)
VAEX
1/3/2000
3,055
BEL 20 (Belgium)
VBEL
1/3/2000
2,731
Asia
N225 (Japan)
VXJ
1/1/1998
3,434
HSI (Hong Kong)
VHSI
1/1/2007
1,234
Commodities
USO (Oil)
OVX
5/10/2007
1,172
GLD (Gold)
GVZ
6/3/2008
904
EVZ
11/1/2007
1,005
(Ticker)
Stocks
European Union
Currency
FXE (USD-EUR)
Since they are not central to the study’s results yet consume much space, we refrain from presenting the
descriptive summary statistics. They are available upon request.
5
10
V. ESTIMATION RESULTS
We use Japan’s NIKKEI Volatility Index (i.e., N225-VJX index-volatility pair) to
illustrate the model estimation described in Section II, noting we follow the same procedure to
estimate all 15 volatility-index pairs. Before estimating the models it is instructive to explore
some characteristics of the data. Exhibit 2 plots the NIKKEI’s daily time series for both the
stock and the volatility indices from January 1, 1998, to December 31, 2011.6 The correlation
between price drops and volatility spikes becomes clearly visible. Also, periods with steadily
increasing prices are associated with decreasing volatility. However, the graphs also indicate
that the data may be non-stationary.7 Since the results of all stationarity tests indicate the series
are I(1), we construct our model for the first differences as discussed in the previous section
and shown in equation (1). The first differences are also clearly correlated, and large jumps in
the price are associated with large jumps in the volatility (see Exhibit 2). The changes are
centered around zero, yet they show signs of variance clustering (i.e., periods with large
variance are followed by large variance, and vice versa). The situation is essentially identical
with the other index-volatility pairs and implies that it is necessary to model this volatility. For
this reason we implement GARCH models in subsequent estimations.
6
Comparable data for other series is available upon request.
We perform augmented Dickey-Fuller (ADF) unit root tests for both N225 and VXJ. The unit root test for N225
shows clear stochastic trend without deterministic trend or intercept. Similar test for VXJ were not conclusive, 7
therefore we performed two additional tests: Elliott-Rothenberg-Stock test for unit roots and KwiatkowskiPhillips-Schmidt-Shin test for stationary both with heteroscedasticity and autocorrelation consistent variance. The
results of both tests are consistent with the hypothesis that VXJ is 𝐼(1). ADF test without intercept does not reject
the null hypothesis. However, the null hypothesis is rejected if intercept is included, although the estimated 𝜌 ≈
0.99. Since it is known that ADF has very low power against near unit roots (especially when deterministic terms
are included) we consider the test inconclusive.
7
11
VXJ (%)
75
10
VXJ (%)
3
N225 (10 )
20
N225 (%)
Exhibit 2: Japan’s NIKKEI Stock Index: An Example
This Exhibit shows Japan’s NIKKEI stock index and its implied volatility index, the N225 and VXJ,
respectively, and their first differences for the 13-year period from January 1, 1998 to December 31,
2011. ΔN225 represents relative change and ΔVXJ absolute change, both in percentage points.
15
10
5
50
25
0
-10
10
0
-10
98
99
00
01
02
03
04
05
06
07
08
09
10
11
12
Date
1. Instantaneous price model
First, we estimate the simple instantaneous price model, akin to prior studies. A
straightforward OLS estimation immediately reveals problems with heteroscedasticity. The
residuals are not white noise but instead exhibit variance clustering. The ARCH LM
heteroscedasticity test confirms the hypothesis that the variance is not constant. Therefore, the
OLS estimators do not have the regular properties, making it appropriate to adjust them.
Beyond the NIKKEI, this problematic situation also occurs with the other index-volatility pairs,
including SPX-VIX and NDX-VXN. Virtually all prior literature has documented this
heteroscedasticity issue and addressed it using OLS models with heteroscedasticity-corrected
standard errors.
Instead, we take an arguably more robust approach. In order to account for the
heteroscedasticity problem, we treat the variance of the residuals with a GARCH(1,1) model.
Estimation of the instantaneous price model for N225-VXJ generates these parameter
estimates: 𝛼 = −0.394, 𝛽0− = −0.925, 𝛽0+ = −0.138. The model is well-specified, and the
explanatory power is respectable with adjusted 𝑅 2 = 35.9%. The asymmetry between negative
and positive price moves is large (𝑎𝑥 = −0.787). The behavior of the estimators is consistent
12
with what is obtained for all other index-volatility pairs studied in the literature, including for
SPX-VIX (Whaley, 2000, and Whaley, 2009) and NDX-VXN (Simon, 2003 and Giot, 2006).
The variance is found to have a high degree of persistence, with 𝜆 = 0.134, 𝜇1 = 0.155, 𝜈1 =
0.796. As we will see, the model estimation results for the N225-VJX are fairly typical for a
stock index-volatility pair. The main difference is that the asymmetry is larger than that of other
indices. The intercept 𝛼 = −0.394 is statistically significant and very large. A non-zero
negative intercept means that when prices do not change the volatility decreases. The fact that
N225 has decreased over the period and the very large asymmetry between the reaction to
negative and positive moves has led to the large intercept term. This inconsistency with the
model’s first general property reveals the model’s deficiency. This phenomenon is obvious in
the case of N225-VXJ, but it is not so obvious from the estimation of the SPX-VIX model. In
these situations, the intercept is smaller, and some studies even found it not statistically
significant (e.g., Whaley, 2009).
Having thus illustrated the procedure, we proceed with the model estimation for all
index pairs. Unit root tests reveal that all indices have stochastic trends and the first differences
are stationary. Therefore the GARCH(1,1) model works well in all cases. All models show
very strongly autoregressive variance of the residuals (between 80-90%).8 The results of the
model estimation for all index pairs are listed and plotted in Exhibit 3.
8
Untabulated.
13
Exhibit 3: Summary of the Instantaneous Price Model
This Exhibit shows the summary of the instantaneous price model estimation coefficients, coefficient
asymmetry, and predictive power of various volatility indices. GARCH(1,1) model is used for the
estimation. Panel A shows the intercept, the reaction to positive and negative index moves, the relative
asymmetry, and the predictive power for all index pairs. Here relative asymmetry is cited 𝑎′𝑥 =
(𝛽0− − 𝛽0+ )/𝛽0+ in percent..****,***,**,* Indicates significance at 0.1%, 1%, 5%, 10% levels
respectively. Panel B shows a plot of the estimation coefficients and the adjusted R2 from Panel A. The
bars indicate the standard error.
Panel A: Instantaneous Price Model
Coefficient Estimates
STOCKS
Relative
Asymmetry
𝑎′𝑥
Adj.
𝑅2
𝛼
𝛽0−
𝛽0+
VIX
–0.089****
–1.089****
–0.675****
61
0.61
RVX
–0.075****
–0.823****
–0.583****
41
0.62
VXD
–0.094****
–0.996****
–0.646****
54
0.57
VXN
–0.097****
–0.722****
–0.472****
53
0.47
VFTSE
–0.116****
–1.093****
–0.661****
65
0.56
VDAX
–0.012
–0.803****
–0.661****
21
0.46
VCAC
–0.058****
–0.829****
–0.616****
35
0.36
VAEX
–0.145****
–0.929****
–0.528****
76
0.36
VBEL
–0.019
–0.552****
–0.455****
21
0.11
VHSI
–0.493****
–1.092****
–0.148****
636
0.48
VXJ
–0.394****
–0.925****
–0.138****
568
0.36
GVZ
–0.522****
–0.592****
0.698****
–15
0.15
OVX
–0.514****
–0.701****
0.092***
661
0.14
EVZ
–0.140****
–0.538****
–0.010

0.08
USA
EU
ASIA
COMMODITY
CURRENCY
14
Panel B: Graphical Representation of the Estimation for each Volatility Index
 
,  ,  (%)
0.5
0.0
-0.5
-1.0
40

20

2
R (%)
60


2

R
EVZ
OVX
GVZ
VXJ
VHSI
VBEL
VAEX
VCAC
VDAX
VFTSE
VXN
VDX
RVX
VIX
0
(a) Stock indices
Interpreting the results, we first compare the behavior of the stock indices. Exhibit 3 leads to
some interesting observations. First, in most major markets there is a fairly uniform tendency
for a 1% drop in prices to cause an increase in volatility from 0.72-1.09% (i.e., 𝛽0− ranges from
–0.72 to –1.09). An exception is Belgium, where a 1% drop causes a 0.55% increase in
volatility. This is, however, combined with a relatively low explanatory power of the model,
which indicates that there are external factors influencing the volatility on this secondary
market. The reaction to positive moves is also fairly uniform, at least on the US and EU
markets, where 1% increase in prices decreases the volatility by 0.46-0.68%. The big exception
occurs in the Asian markets, where positive moves do very little to reduce the volatility – about
0.15% in both cases. In the aggregate, the relative asymmetry, 𝑎′𝑥 = (𝛽0− − 𝛽0+ )/𝛽0+ , is within
several tens of percent for all markets, except for the Asian markets, where it is several hundred
percent.
The signs of the coefficients and the asymmetry are consistent with previous studies
(e.g. Whaley, 2009) except that the coefficients are smaller and the asymmetry is larger relative
to the OLS estimation of the same models. The main difference from the previous studies is
that we find that almost all intercepts are significant (and negative) and can be fairly large. The
15
size of the intercept appears proportional to the asymmetry between negative and positive
moves, which could indicate that the intercept is spurious and the model misspecified. It is also
noticeable that on the major markets a very large fraction of the volatility moves are explained
by price changes (i.e., adjusted 𝑅 2 between 35-60%).
Based on these observations, we can conclude that regardless of the geographical
separation the indices demonstrate certain universal behaviors. First, all coefficients are
negative (𝛼, 𝛽0− , 𝛽0+ < 0). This means that negative price moves increase volatility while
positive moves decrease it, which is consistent with previous studies of individual index pairs.
Second, there is an asymmetry between negative and positive price changes. Reductions in
prices cause larger fluctuations in volatility than increases (|𝛽0− | > |𝛽0+ |), again a fact which
has been extensively discussed in the literature, especially in the context of the SPX-VIX pair.
Third, the intercept is not zero (|𝛼| > 0). This means that if price does not change the volatility
diminishes. This is a universal feature of this model which reveals the model deficiency. In the
case of SPX-VIX and the other US indices the intercept is small but nevertheless statistically
significant. Finally, the explanatory power of the models, in the form of adjusted 𝑅 2 , is
generally high. It is greater for the larger markets which means that larger portion of the
volatility change is explained by the price changes of the market itself. In other words larger
markets are more self-sufficient.
(b) Commodity indices
Next we consider the commodity volatility indices results in Exhibit 3. Negative price moves
increasing the volatility and large intercepts are both features shared with the stock indices. A
crucial difference exists, however, between the stock and commodity results. The coefficients
of the positive price moves are positive, which implies that positive commodity price moves
also increase the volatility. While positive stock moves make the investors’ return certain and
relax the volatility or “fear,” with commodities upward moves are also perceived unfavorably.
A reason for this could be that the demand for commodities is driven to a large extent by actual
users, for whom an increase in the price means increased expenses, rather than investors. The
degree of this behavior is quite different for gold and oil, and it is much more pronounced for
gold. Since markets fear all moves for commodities, there is no reason to discuss asymmetry.
However, it is evident that the relative size of the coefficients in gold is reversed, with positive
moves causing more disturbance than negative moves. This could result from the possibility
that markets might perceive an increase in the price of gold as a sign of a general economic
16
meltdown when investors start moving funds from other markets to gold. Finally, we notice
that the explanatory power of the models is fairly small, about 14%, which means that a large
part of the volatility is due to factors other than the price changes themselves (i.e., it is
transferred from other markets).
Thus, we can conclude that commodities share some of the features with the stock
indices. However, they also display behavior which is common between them and markedly
different from that of the stocks. First, the coefficient for positive moves is positive (𝛼, 𝛽0− <
0, 𝛽0+ > 0), which means that any type of price moves increases volatility. Also, the intercept
is very large (|𝛼| > 0). This is a result of the sign of the coefficients; the only times when
volatility drops are when the prices do not change. Thus, the large intercepts are necessary to
keep the volatility within normal bounds. Again, this result suggests poor model specification.
(c) Currency indices
Finally, we consider the currency index in comparison to the stock and commodities behavior.
Here the common features are again that the market perceives negative price moves as bad, and
the large intercept exists. However, the currency index shows yet another unique behavior. The
coefficient for positive moves is zero (𝛼, 𝛽0− < 0, 𝛽0+ ≈ 0). Investors are therefore indifferent
to jumps in the euro price (i.e., cheap dollar) but scared of euro price drops (i.e., expensive
dollar). However, here we note that the explanatory power of the model is even smaller, around
8%, so a great part of the source of the volatility is external.
Overall, the instantaneous price model estimation reveals that negative price moves are
a universal trigger for the volatility. However, perhaps surprisingly, positive price moves can
be considered as good, bad, or indifferent in the different asset markets. Also, interesting
geographical trends are observed. For example, the reaction of investors to price moves in the
same geographical region is similar, but it differs in strength between regions with the Asian
markets exhibiting extreme asymmetry underlined by very weak enthusiasm with regard to
positive price changes.
2. Local model
The instantaneous price model in the prior section has been the choice in most studies of this
subject and provides us with some valuable insights. For example, the asymmetry in the
reaction to short term price changes is present in this model, and as we will see, remains valid
17
even when we consider more general models. In addition, the model is useful to identify general
trends in the behavior of the volatility in different asset classes and across geographical regions.
However, both the small explanatory power of some of the models and the observation that the
asymmetry between negative and positive moves and the intercept are related indicate that the
instantaneous model is not well specified. The graphs of the volatility indices show that the
volatility is mean reverting (e.g., Exhibit 2). This behavior is consistent with intuition that the
volatility must remain within some limits and cannot increase or decrease indefinitely.
However, this behavior contradicts the predictions the models, because if the asymmetry
between the positive and negative moves is large, then the general upward motion of the index
cannot compensate for the coefficient asymmetry. In this case the volatility will tend to increase
indefinitely. Since unbounded variance does not happen in practice, it necessitates an intercept
term. Therefore, the non-zero intercept seems to indicate a problem with the specification of
the model.
We address this issue with the local model, which includes lagged returns [i.e.,
parameters 𝛽𝑖− and 𝛽𝑖+ in equation (1)] and lagged volatility changes as independent variables
(i.e., parameters 𝛾𝑗− and 𝛾𝑗+ ). Specifically, we include two trading weeks’ (i.e., ten days) worth
of lagged daily volatility changes. Exhibit 4 presents these results. When compared to the
instantaneous price model (Exhibit 3), the inclusion of extra explanatory variables increases
the explanatory power (i.e., adjusted R2) of the models, even controlling for the additional
independent variables, which testifies for the importance of memory in the formation of
volatility expectations. The main trends remain the same. In particular, negative price changes
universally lead to an increase in volatility. The market welcomes positive price changes in the
stock indices, the commodity market indices fear them, and the currency index is indifferent to
positive price shocks. These observations partially redeem the instantaneous price model,
which also captures these trends. However, the local model does address some deficiencies in
the instantaneous price model.
First, the most striking observation is that the intercept for almost all indices is
indistinguishable from zero. This is true even for indices with significantly large intercepts in
the instantaneous model, such as VHSI and GVZ. The only index which has a highly
statistically significant intercept term is OVZ, where 𝛼 = −0.326.9 One possible conclusion is
that oil is the only asset in which volatility decreases even when the prices are stagnant.
Alternatively, this could be a sign that there are other variables unaccounted for in our model
9
Note that the EVZ intercept is significant only at the 10% level.
18
which influence the oil’s volatility. In general, the vanishing intercept coincides with our
expectations and indicates that the local model is well specified. The inclusion of a sufficient
number of lagged terms for both the prices and the volatilities is necessary to describe correctly
the volatility changes.
19
Exhibit 4: Summary of the Local Model
This Exhibit shows the summary of the local price model estimation coefficients, relative asymmetry,
and predictive power of various volatility indices. GARCH(1,1) model is used for the estimation, using
10 lags for own volatility and own returns. Panel A shows the estimate coefficients, the relative
asymmetry, and the predictive power for all index pairs. ****,***,**,* Indicates significance at 0.1%,
1%, 5%, 10% levels. Panel B shows a plot of the estimation coefficients and the adjusted R2 from Panel
A. Panel C shows the contemporaneous and lagged term coefficient estimates of the VIX-SPX
regression for the local model. The bars represent the standard error.
Panel A: Local Model
Coefficient Estimates
STOCKS
Relative
Asymmetry
𝑎′𝑥
Adj.
𝑅2
𝛼
𝛽0−
𝛽0+
VIX
0.031
–1.178****
–0.562****
110
0.64
RVX
0.047
–0.877****
–0.503****
74
0.63
VXD
0.014
–1.059****
–0.565****
87
0.59
VXN
0.034
–0.786****
–0.389****
102
0.49
VFTSE
0.033
–1.189****
–0.547****
118
0.58
VDAX
0.051
–0.910****
–0.580****
57
0.48
VCAC
0.046
–0.890****
–0.555****
60
0.38
VAEX
0.023
–1.006****
–0.446****
126
0.35
VBEL
–0.008
–0.591****
–0.406****
46
0.14
VHSI
–0.113
–1.204****
–0.008

0.56
VXJ
–0.082
–0.903****
–0.053**
0.37
GVZ
–0.105
-0.723****
0.880****

-18
OVX
–0.326****
-0.721****
0.098***
634
0.17
EVZ
–0.122*
–0.550****
0.014

0.11
USA
EU
ASIA
COMMODITY
CURRENCY
20
0.23
Panel B: Graphical Representation of the Estimation for each Volatility Index
 
,  ,  (%)
0.5
0.0
-0.5
-1.0
40

20

2
R (%)
60


2

R
EVZ
OVX
GVZ
VXJ
VHSI
VBEL
VAEX
VCAC
VDAX
VFTSE
VXN
VDX
RVX
VIX
0
Panel C: Contemporaneous and Lagged Term Coefficient Estimates of the VIX-SPX Index Pair
Regression
 
 ,  (%)
0.0
-0.5

l

l

l
 
 ,  (%)

l
-1.0
0.1
0.0
-0.1
0
1
2
3
4
5
6
Lag (days)
21
7
8
9
10
Second, we observe that the degree of asymmetry increases relative to the instantaneous
price model. That is, the coefficients of the negative returns increase and the positive returns
decrease. The asymmetry is now close to hundred percent for most models and becomes
extremely large for the Asian markets. This means that the instantaneous model artificially
constrains the short term coefficients and underestimates the short-term asymmetry. The most
spectacular deviations occur in the Asian markets, where the coefficients of the positive returns
drop to zero, engendering an infinite measure for relative asymmetry.
In addition to having improved specification, the local model provides more
information about the behavior of the volatility through the lagged coefficients.10 Exhibit 4,
Panel C plots the price and volatility lagged coefficients of the VIX-SPX estimation, with the
bars representing the standard errors. Although the zero- and first-lagged negative returns
increase the volatility as expected, longer lags for negative returns decrease it (𝛽𝑖− > 0 for 𝑖 >
2). Furthermore, the lagged volatility coefficients tend to be negative, which means that past
increases in the volatility work to decrease the current volatility (𝛾𝑗+ < 0) – that is, volatility
tends to mean-revert over time. The combined effect of the two is that the asymmetry is
essentially exactly balanced in the long term, which is again consistent with our expectations.
Another noticeable feature is that the negative influences (i.e., price drops and volatility spikes)
produce larger effects than the positive influences (i.e., price increases and volatility drops). In
general, the reaction to positive news is more cautious.
Based on the new lagged data we can further identify elements of universal behavior.
First, short-lag negative returns increase the volatility but the long-lag terms decrease it.
Positive lagged returns generally have less effect on the volatility. Second, lagged volatility
increases actually tend to reduce the current volatility. Lagged volatility drops have less effect.
Finally, overall in the long term positive and negative contributions to the volatility cancel out.
This collection of results suggests that although investors react strongly to immediate threats
to their wealth, they act as if they believe in a long-term equilibrium.
Overall, we contend that the local model is a marked improvement over the
instantaneous price model represented in prior literature. Although the instantaneous model
captures the main thrusts of the short-term volatility reaction to price changes, it underestimates
the asymmetry. Moreover, the instantaneous model completely misses the long-term behavior.
Very importantly, the behavior on the long term is consistent with market equilibrium, which
10
Parameter estimates for all lagged coefficients are available upon request. We do not present them in the spirit
of parsimony.
22
causes past price drops and past increases in the volatility to be treated as positive signs. The
total asymmetry is zero when we account for long-term behavior. Finally, the more general
local model reveals that the market reacts to positive signs a lot less tepidly than to negative
signs over the whole timeframe.
3. Global model
Although the local model is well specified and yields relevant insights, it is conceivable
that including additional information could improve it. The local model assumes that investors
have accounted for all the market information about the instantaneous and past movements of
the local index price and its volatility. At the same time, investors are most certainly aware not
only of the situation in a particular market but also of what happens in other markets. Thus a
possible improvement in the model involves including external factors. We find that for the
major markets a very large part of the volatility change is explained through immediate and
past index returns and the past volatility changes. One can therefore consider these markets as
generators of volatility. On the other, secondary markets price and past volatility changes are
not enough to explain the volatility changes as fully as for the primary markets based on the
adjusted R2 values. Thus, we can posit that some part of the volatility transfers from the primary
markets that generate it to the secondary markets. To test this possibility, we use VIX as a
proxy of global volatility and include it in the estimation by augmenting the model with the
+
−
terms ∑𝑘=0 𝛿𝑘− ∆𝜍𝑡−𝑘
+ 𝛿𝑘+ ∆𝜍𝑡−𝑘
, 𝑘 = 0 to 5, where as ∆𝜍 is a measure of the global volatility
as discussed in the methodology. As a proxy for the global volatility we will use the VIX index,
which is the typical assumption used in the literature (see Stivers and Sun, 2002 and Sari,
Soytas, and Hacihasanoglu, 2011). This assumption obviously precludes us from estimating
the global model for the US indices because of the large degree of redundancy between VIX
and the other US market volatility measures.
Exhibit 5, Panel A presents the estimations of the global model. Panel B shows
graphically the geographic and asset class trends in the parameter estimates and explanatory
power resulting from the global model. First, we observe that the coefficients of the
instantaneous and the first few lagged coefficients for the VIX are statistically significant and
positive (in the range of 0.1-0.3, as illustrated in Panel C). Therefore, the different geographical
markets are indeed influenced by the global volatility. The reaction to the positive and negative
VIX changes is approximately the same, which means that the respective index volatilities
simply follow global volatility. The inclusion of the additional information increases the
23
explanatory power of the model across the board, however, the effect is most pronounced for
the smaller markets. This finding is consistent with the expectation that the smaller markets
import the volatility while the larger markets generate it. Comparing the estimation coefficients
with those of the local model (in Exhibit 4), we observe that all the trends identified in the local
model are present and quantitatively the same.
24
Exhibit 5: Summary of the Global Model
This Exhibit shows the summary of the global price model estimation coefficients, relative asymmetry,
and predictive power of various volatility indices. A GARCH(1,1) model is used for the estimation,
using 10 lags for own volatility and own returns and 5 lags for the global volatility. Panel A shows the
estimate coefficients, the relative asymmetry, and the predictive power.. ****,***,**,* Indicates
significance at 0.1%, 1%, 5%, 10% levels. Panel B shows a plot of the estimation coefficients and the
adjusted R2 from Panel A. Panel C shows a plot of the contemporaneous and lagged term coefficients
for the global volatility. The bars indicate the standard error.
Panel A: Global Model
Coefficient Estimates
STOCKS
Relative
Asymmetry
𝑎′𝑥
Adj.
𝑅2
𝛼
𝛽0−
𝛽0+
VFTSE
0.01
–0.990****
–0.413****
140
0.64
VDAX
–0.002
–0.760****
–0.459****
66
0.55
VCAC
0.032
–0.785****
–0.473****
66
0.44
VAEX
–0.002
–0.830****
–0.327****
154
0.43
VBEL
0.014
–0.486****
–0.389****
25
0.21
VHSI
–0.135*
–0.995****
0.039

0.63
VXJ
–0.283****
–0.773****
–0.030
0.40
GVZ
–0.135
–0.612****
0.744****

–18
OVX
–0.381****
–0.559****
0.11****
406
0.25
EVZ
–0.083
–0.459****
0.062

0.16
EU
ASIA
COMMODITY
CURRENCY
25
0.38
Panel B: Graphical Representation of the Estimation for each Volatility Index
 
,  ,  (%)
0.5
0.0
-0.5
-1.0
40

20

2



EVZ
OVX
GVZ
VXJ
VHSI
VBEL
VAEX
VDAX
VFTSE
0
2
R
VCAC
R (%)
60
Panel C: Global Model Regression Coefficients for the Global Volatility
0.35

0

0.30
0
0.20
0.15
0.10
26
EVZ
OVX
GVZ
VXJ
VHSI
VBEL
VAEX
VCAC
VDAX
0.05
VFTSE


0,0
0.25
Overall, the fact that the VIX coefficients are significant means that global risk
perceptions influence all markets. The reaction to positive and negative moves of the global
volatility is not significantly different, which means that all markets to some extent follow the
global trend. Also, the marked increase in explanatory power (i.e., adjusted 𝑅 2 ) for the
secondary markets confirms the hypothesis that much of the volatility is transferred to these
markets. This result could occur from investors taking positions in several markets
simultaneously. These observations help us separate markets into two groups – the first which
are largely self-sufficient and generate their own volatility and the second, which to a large
extent follow the developments in other markets. Not surprisingly, the largest markets are the
most self-sufficient. Also, perhaps not surprisingly, the commodities and currency markets are
much less self-sufficient than the equity markets.
Finally a comparison of the local and global models with the instantaneous price model
is shown in Exhibit 6. We plot the relative change in the asymmetry and the adjusted 𝑅 2 with
respect with the instantaneous price model. It becomes obvious that for the larger markets the
improvement of 𝑅 2 is relatively modest. However, for the small markets the adjusted 𝑅 2 can
increase by up to 75% when we account for memory. Even more significantly, the improvement
in the adjusted 𝑅 2 can reach 100-150% when the global volatility is included in the estimation.
This evidence clearly indicates that a large portion of the volatility transfers to the smaller
markets. On the other hand, the asymmetry of the coefficients of the negative and positive
moves increases by 30-60% for most indices, which indicates that the instantaneous price
model severely constrains the asymmetry.
27
Exhibit 6: General Trends
This Exhibit visualizes the relative change in the asymmetry and the adjusted R2 of the local (blue) and
the global (orange) models with respect with the instantaneous price model.
120
90
60
x
x
x
(a -aipm)/aipm (%)
150
30
150
2
(R -Ripm)/Ripm (%)
0
2
2
100
50
EVZ
OVX
GVZ
VXJ
VHSI
VBEL
VAEX
VCAC
VDAX
VFTSE
VXN
VDX
RVX
VIX
0
V. CONCLUSIONS
This paper enriches the volatility literature by examining these key issues with the
following main findings: (1) the volatility in all markets shows a distinct short-term and longterm pattern. In the short-term, the volatility results mainly from a very strong reaction to price
drops. By expanding the information considered, we find that the asymmetry is much larger
than previously estimated in the prior literature; (2) in the long term the idea of market
equilibrium takes hold, compensating for the asymmetric response to positive and negative
price changes; (3) by incorporating past volatility information across asset classes – not simply
contemporaneous behavior for equity markets – we find fear largely drives volatility, with
reaction to negative influences figuring much more prominently than reaction to positive
influences. The reaction to positive moves is much more subdued and varied. Positive returns
can be interpreted as a good sign (most stocks), an indifferent sign (Asian markets and
currencies), or a bad sign (commodities); and (4) we can identify primary and secondary
volatility markets, with the secondary markets following the primary global volatility trends
without discriminating between positive and negative news.
28
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30
Abstrakt
Tato studie analyzuje vztah mezi výnosy a implikovanou volatilitou napříč třídami aktiv,
geografických oblastí, a času, a zároveň rozšiřuje znalosti mezi okamžitou změnou
implikované volatility a indexových výnosů. Modelování vztahů GARCH procesem
potvrzujeme, že implikovaná volatilita závisí na okamžité změně indexu. Nicméně, současné
změny volatility jsou rovněž vysvětleny předchozími výnosy indexu a volatility. Zatímco
krátkodobá volatilita je silně asymetrická na straně negativních pohybů, v dlouhodobém
horizontu se nevyskytuje rozdíl mezi kladnými a zápornými pohyby. Volatilita také přechází z
větších, primárních trhů na menší, sekundární, a pak i na trhy komoditní a měnové.
31
Working Paper Series
ISSN 1211-3298
Registration No. (Ministry of Culture): E 19443
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Papers (including their dissemination) were supported from institutional support RVO 67985998
from Economics Institute of the ASCR, v. v. i.
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acknowledged at the beginning of the Paper.
(c) Julian P. Velev, Brian C. Payne, Jiří Trešl, and Wilfredo Toledo, 2016
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