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Discussion Papers
Joint Modeling of Call and Put Implied Volatility
Katja Ahoniemi
Helsinki School of Economics, FDPE, and HECER
and
Markku Lanne
University of Helsinki, RUESG, and HECER
Discussion Paper No. 198
November 2007
ISSN 1795-0562
HECER Helsinki Center of Economic Research, P.O. Box 17 (Arkadiankatu 7), FI-00014
University of Helsinki, FINLAND, Tel +358-9-191-28780, Fax +358-9-191-28781,
E-mail info-hecer@helsinki.fi, Internet www.hecer.fi
HECER
Discussion Paper No. 198
Joint Modeling of Call and Put Implied Volatility*
Abstract
This paper exploits the fact that implied volatilities calculated from identical call and put
options have often been empirically found to differ, although they should be equal in
theory. We propose a new bivariate mixture multiplicative error model and show that it is a
good fit to Nikkei 225 index call and put option implied volatility (IV). A good model fit
requires two mixture components in the model, allowing for different mean equations and
error distributions for calmer and more volatile days. Forecast evaluation indicates that in
addition to jointly modeling the time series of call and put IV, cross effects should be added
to the model: put-side implied volatility helps forecast call-side IV, and vice versa. Impulse
response functions show that the IV derived from put options recovers faster from shocks,
and the effect of shocks lasts for up to six weeks.
JEL Classification: C32, C53, G13
Keywords: Implied Volatility, Option Markets, Multiplicative Error Models, Forecasting.
Katja Ahoniemi Markku Lanne
Department of Economics Department of Economics
Helsinki School of Economics University of Helsinki
P.O. Box 1210 P.O. Box 17 (Arkadiankatu 7)
FI-00101 Helsinki FI-00014 University of Helsinki
FINLAND FINLAND
e-mail: katja.ahoniemi@hse.fi e-mail: markku.lanne@helsinki.fi
* This research has been supported by the Okobank Group Research Foundation. Katja
Ahoniemi thanks the Finnish Doctoral Programme in Economics, the Finnish Foundation
for Advancement of Securities Markets, and the Yrj Jahnsson Foundation for financial
support.
1 Introduction
In theory, the implied volatilities derived from a call option and a put option with the
same underlying asset, strike price, and expiration date should be equal - both reect the
markets expectation of the volatility of the returns of the underlying asset during the
remaining life of the two options. However, it has been empirically observed that when
call and put implied volatilities (IV) are backed out of option prices using an option
pricing formula, they often deviate from each other.
The reason behind the inequality of put and call implied volatilities may lie in the
dierent demand structure for calls and puts. There is an inherent demand for put
options that does not exist for similar calls, as institutional investors buy puts regularly
for purposes of portfolio insurance. There are often no market participants looking to sell
the same options to oset this demand, meaning that prices may need to be bid up high
enough for market makers to be willing to become counterparties to the deals. With no
market imperfections such as transaction costs or other frictions present, option prices
should always be determined by no-arbitrage conditions, making implied volatilities of
identical call and put options the same. However, in real-world markets the presence
of imperfections may allow option prices to depart from no-arbitrage bounds if there is,
for example, an imbalance between supply and demand in the market. References to
existing literature and more details on this topic are provided in Section 2.
Despite the fact that call and put-side implied volatilities dier, they must be tightly
linked to one another at all times - after all, they both represent the same market
expectation, and the driving forces behind their values are common. Therefore, it can
be argued that there is potential value added in jointly modeling time series of implied
volatilities, one derived from call option prices and the other from put option prices.
Further, the interactions between the two variables can be studied with cross eects, i.e.
allowing call IV to depend on lagged values of put IV, and vice versa.
The modeling of IV provides a valuable addition to the extensive literature on volatil-
ity modeling. IV is truly a forward-looking measure: implied volatility is the markets
expectation of the volatility in the returns of an options underlying asset during the
remaining life of the option in question. In contrast, other volatility estimates are based
on historical prices. Examples of IV modeling literature include Ahoniemi (2006), who
nds that there is some predictability in the direction of change of the VIX Volatility
Index, an index of the IV of S&P 500 index options. Dennis et al. (2006) nd that
daily innovations in the VIX Volatility Index contain very reliable incremental informa-
tion about the future volatility of the S&P 100 index.
1
Other studies that attempt to
forecast IV or utilize the information contained in IV to trade in option markets include
Harvey and Whaley (1992), Noh et al. (1994), and Poon and Pope (2000). Reliable
forecasts of implied volatility can benet option traders, but many other market partic-
ipants as well: all investors with risk management concerns can benet from accurate
forecasts of future volatility.
The implied volatility data used in this study are calculated separately from call and
put options on the Japanese Nikkei 225 index. Separate time series for call and put-side
IV oer a natural application for the bivariate multiplicative model presented below.
In their analysis of implied volatilities of options on the S&P 500 index, the FTSE 100
1
The data set in Dennis et al. (2006) ends at the end of 1995, when options on the S&P 100 index
were used to calculate the value of the VIX. The Chicago Board Options Exchange has since switched
to S&P 500 options.
1
index, and the Nikkei 225 index, Mo and Wu (2007) nd that U.S. and UK implied
volatilities are more correlated with each other than with Japanese implied volatilities,
indicating that the Japanese market exhibits more country-specic movements. There-
fore, it is interesting to analyze the Japanese option market and its implied volatility
in this context, as investors may be presented with possibilities in the Japanese index
option market that are not available elsewhere. Mo and Wu (2007) also report that the
implied volatility skew is atter in Japan than in the U.S. or UK markets. They con-
clude that in Japan, the risk premium for global return risks is smaller than in the other
two countries. The developments in the Japanese stock market during the late 1990s in
particular are very dierent from Western markets, with prices declining persistently in
Japan. This characteristic also makes the Japanese market unique. Mo and Wu (2007)
observe that out-of-the-money calls have relatively higher IVs in Japan, as investors
there expect a recovery after many years of economic downturn. Investors in Japan
seem to price more heavily against volatility increases than against market crashes.
In this paper, we introduce a new bivariate multiplicative error model (MEM). MEM
models have gained ground in recent years due to the increasing interest in modeling
non-negative time series in nancial market research.
2
The use of MEM models does not
require logarithms to be taken of the data, allowing for the direct modeling of variables
such as the duration between trades, the bid-ask spread, volume, and volatility. Recent
papers that successfully employ multiplicative error modeling in volatility applications
include Engle and Gallo (2006), Lanne (2006, 2007), and Ahoniemi (2007). Lanne (2006)
nds that the gamma distribution is well suited for the multiplicative modeling of the
realized volatility of two exchange rate series, and Ahoniemi (2007), using the same
data set as in the present study, nds that MEM models together with a gamma error
distribution are a good t to data on Nikkei 225 index implied volatility. All the above-
mentioned MEM applications consider univariate models, but Cipollini et al. (2006)
build a multivariate multiplicative error model using copula functions instead of directly
employing a multivariate distribution. In our application, we use a bivariate gamma
distribution to model the residuals.
Our results show that it is indeed useful to jointly model call and put implied volatil-
ities. The chosen mixture bivariate model with a gamma error distribution is a good
t to the data, as shown by coecient signicance and diagnostic checks. The addition
of lagged cross eects turns out to be important for one-step-ahead daily forecast per-
formance. Our model correctly forecasts the direction of change in IV on over 70% of
trading days in an out-of-sample analysis. Impulse response functions are also calcu-
lated, and they reveal that there is considerable persistence in the data: shocks do not
fully disappear until thirty trading days elapse. Also, put-side IV recovers more quickly
from shocks than call-side IV, indicating that the market for put options may price more
eciently due to larger demand and trading volumes.
This paper proceeds as follows. Section 2 discusses the dierences in the markets for
call and put options in more detail. Section 3 describes the bivariate mixture multiplica-
tive error model estimated in this paper. Section 4 presents the data, model estimation
results, and diagnostic checks of the chosen model specication. Impulse response func-
tions are discussed in Section 5, and forecasts are evaluated in Section 6. Section 7
concludes.
2
A special case of multiplicative error models is the autoregressive conditional duration (ACD) model,
for which an abundant literature has emerged over the past ten years.
2
2 The Markets for Call and Put Options
There is an abundance of literature investigating the dierences in the markets for call
and put options. Bollen and Whaley (2004) have documented that put options account
for 55 % of trades in S&P 500 index options, and that the level of implied volatility
calculated from at-the-money (ATM) options on the S&P 500 index is largely driven
by the demand for ATM index puts. Buraschi and Jackwerth (2001), using an earlier
data set of S&P 500 index options, report that put volumes are around three times
higher than call volumes. There is also evidence that out-of-the-money (OTM) puts
in particular can be overpriced, at least part of the time (Bates (1991), Dumas et al.
(1998), Bollen and Whaley (2004)). Garleanu et al. (2006) document that end users
(non-market makers) of options have a net long position in S&P 500 index puts, and
that net demand for low-strike options (such as OTM puts) is higher than the demand
for high-strike options. The results of Chan et al. (2004) from Hang Seng Index options
in Hong Kong are similar to those of Bollen and Whaley (2004) in that net buying
pressure is more correlated with the change in implied volatility of OTM put options
than in-the-money put options. Also, trading in Hang Seng Index puts determines the
shape of the volatility smile to a greater degree than trading in calls. If OTM puts
are consistently overpriced, investors who write such options could earn excess returns
(empirical evidence in support of this is provided in e.g. Bollen and Whaley (2004)).
On the other hand, Jackwerth (2000) nds that it is more protable to sell ATM puts
than OTM puts in the S&P 500 index option market. Fleming (1999) compares ATM
S&P 100 index calls and puts, and nds that selling puts is more protable than selling
calls.
Further evidence on dierent market mechanisms for calls and puts is provided by
Rubinstein (1994), who notes that after the stock market crash of October 1987, prices
of OTM puts were driven upwards, changing the volatility smile into the now-observed
volatility skew. He hypothesizes that the crash led to OTM puts being more highly val-
ued in the eyes of investors. Fleming (1999) observes that institutional buying pressure
rose dramatically after the 87 crash. Ederington and Guan (2002) also remark that the
volatility smile may be caused in part by hedging pressures which drive up the prices of
puts with low strike prices. They point out that this notion is supported by both trad-
ing volume evidence and the fact that in equity markets, implied volatilities calculated
from options with low strike prices have been found to be higher than actual volatilities.
Das and Uppal (2004) note that downside jumps in international equity markets tend to
occur at the same time. Mo and Wu (2007) also report that large downside moves are
more likely to be global rather than country-specic movements. As a consequence of
this, investors cannot avoid drops in portfolio value by diversifying internationally. This
then creates additional pressure to acquire portfolio insurance from put options, driving
up their prices.
Even if the demand for a put option causes its price (and implied volatility) to rise,
no-arbitrage conditions should ensure that the price of a call option with the same strike
price and maturity date yields an implied volatility that is equal to the one derived from
the put counterpart. But as Fleming (1999) writes,
..., transaction costs and other market imperfections can allow option prices
to deviate from their true values without signaling arbitrage opportunities.
The possibility that option prices can depart from no-arbitrage bounds, thus allowing
3
call and put IV to dier, has been documented numerous times in earlier work. Hentschel
(2003) points out that noise and errors in option prices stemming from xed tick sizes,
bid-ask spreads, and non-synchronous trading can contribute to miscalculated implied
volatilities, and to the volatility smile. Garleanu et al. (2006) develop a model for op-
tion prices that allows for departures from no-arbitrage bounds. These arise from the
inability of market makers to perfectly hedge their positions at all times, which in turn
allows option demand to aect option prices. Empirical evidence lends support to this
theory: market makers require a premium for delivering index options. Even market
makers cannot fully hedge their exposures due to issues such as transaction costs, the
indivisibility of securities, and the impossibility of executing rebalancing trades contin-
uously (Figlewski (1989)), and capital requirements and sensitivity to risk (Shleifer and
Vishny (1997)). When market makers face unhedgeable risk, they must be compensated
through option prices for bearing this risk. In fact, Garleanu et al. (2006) nd that
after periods of dealer losses, the prices of options are even more sensitive to demand.
Other impediments to arbitrage include the fact that a stock index portfolio is dicult
and costly to trade, but if an investor uses futures, she must bear basis and possibly
tracking risk (Fleming (1999)): spot and futures prices may not move hand-in-hand at
all times, and the underlying asset of the futures contracts may not be identical to the
asset being hedged. Liu and Longsta (2004) demonstrate that it can often be optimal
to underinvest in arbitrage opportunities, as mark-to-market losses can be considerable
before the values of the assets involved in the trade converge to the values that eventu-
ally produce prots to the arbitrageur. When it is suboptimal to fully take advantage of
an arbitrage opportunity, there is no reason why the arbitrage could not persist for even
a lengthy amount of time. Bollen and Whaley (2004), in their analysis of the S&P 500
option market, nd support for the hypothesis that limits to arbitrage allow the demand
for options to aect implied volatility.
3 The Model
In this section, we present the bivariate mixture multiplicative error model (BVMEM)
that will be used to model the two time series of implied volatilities described in Section
4. Consider the following bivariate model
v
t
=
t

t
, t = 1, 2, ..., T,
where the conditional mean

t
=
_

1t

2t
_
=
_

1
+

q
1
i=1

1i
v
1,ti
+

p
1
j=1

1j

1,tj

2
+

q
2
i=1

2i
v
2,ti
+

p
2
j=1

2j

2,tj
_
and
t
is a stochastic positive-valued error term such that E (
t
|F
t1
) = 1 with F
t1
=
{v
tj
, j 1}. In what follows, this specication will be called the BVMEM(p
1
, q
1
; p
2
, q
2
)
model. As the conditional mean equations of the model are essentially the same as the
conditional variance equations in the GARCH model in structure, the constraints on
parameter values that guarantee positivity in GARCH models also apply to each of the
equations of the BVMEM model. As outlined in Nelson and Cao (1992), the parameter
values in a rst-order model must all be non-negative. In a higher-order model, positivity
of all parameters is not necessarily required. For example, in a model with p
i
= 1 and
q
i
= 2, i = 1, 2, the constraints are
i
0,
i1
0, 0
i
< 1, and
1

i1
+
i2
0. It
4
should be noted that this basic conditional mean specication must often be augmented
with elements such as cross eects between the variables and seasonality eects. In these
cases, one must ensure that positivity continues to be guaranteed. For example, if the
coecients for lagged cross terms are positive, no problems in achieving positivity arise.
The multiplicative structure of the model was suggested for volatility modeling in
the univariate case by Engle (2002), who proposed using the exponential distribution.
However, the gamma distribution nests, among others, the exponential distribution, and
is therefore more general. Also, the ndings of Lanne (2006, 2007) and Ahoniemi (2007)
lend support to the gamma distribution.
The error term
t
is assumed to follow a bivariate gamma distribution, which is
a natural extension of the univariate gamma distribution used in previous literature
(Lanne (2006, 2007) and Ahoniemi (2007)). Of the numerous bivariate distributions
having gamma marginals, the specication suggested by Nagao and Kadoya (1970) is
considered (for a discussion on alternative bivariate gamma densities, see Yue et al.
(2001)). This particular specication is quite tractable and thus well suited for our
purposes. Collecting the parameters into vector = (
1
,
2
, , ), the density function
can be written as
f

1
,
2
(
1t
,
2t
; ) =
(
1

2
)
(+1)/2
(
1t

2t
)
(1)/2
exp
_

1t
+
2

2t
1
_
() (1 )
(1)/2
I
1
_
2

1t

2t
1
_
,
where () is the gamma function, is the Pearson product-moment correlation coe-
cient, and I
1
() is the modied Bessel function of the rst kind. The marginal error
distributions have distinct scale parameters
1
and
2
, but the shape parameter, , is
the same for both. However, since the error term needs to have mean unity, we impose
the restrictions that the shape parameters are the reciprocals of the scale parameters,
i.e., = 1/
1
and = 1/
2
, indicating that
1
=
2
= 1/. In other words, we will also
restrict the scale parameters to be equal. This is not likely to be very restrictive in our
application, as earlier evidence based on univariate models in Ahoniemi (2007) indicates
that the shape and scale parameters for the time series used in this study, the implied
volatilities of Nikkei 225 call and put options, are very similar.
Incorporating the restrictions discussed above and using the change of variable the-
orem, the conditional density function of v
t
= (v
1t
, v
2t
)

is obtained as
f
t1
(v
1t
, v
2t
; ) = f

1
,
2
_
v
1t

1
1t
, v
2t

1
2t
_

1
1t

1
2t
(1)
=

(+1)
_
v
1t
v
2t

1
1t

1
2t

(1)/2
exp
_

(v
1t

1
1t
+v
2t

1
2t
)
1
_
() (1 )
(1)/2

I
1
_
_
2
_
v
1t
v
2t

1
1t

1
2t
1
_
_

1
1t

1
2t
.
5
Consequently, the conditional log-likelihood function can be written as
3
l
T
() =
T

t=1
l
t1
() =
T

t=1
ln [f
t1
(v
1t
, v
2t
; )] ,
and the model can be estimated with the method of maximum likelihood (ML) in a
straightforward manner. Although the gamma distribution is quite exible in describing
the dynamics of implied volatilities, in our empirical application it turned out to be
inadequate as such. In particular, it failed to capture the strong persistence in the
implied volatility time series. As an extension, we consider a mixture specication
that allows for the fact that nancial markets experience dierent types of regimes,
alternating between calm and more volatile periods of time. Dierent parameter values
can be assumed to better describe periods of larger shocks compared with periods of
smaller shocks, and error terms are allowed to come from two gamma distributions
whose shape and scale parameters can dier. Earlier evidence from Lanne (2006) and
Ahoniemi (2007) indicates that the use of a mixture specication improves the t of a
multiplicative model as well as the forecasts obtained from the models.
We will assume that the error term
t
is a mixture of of
(1)
t
and
(2)
t
with mix-
ing probability , and that
(1)
t
and
(2)
t
follow the bivariate gamma distribution with
parameter vectors
1
and
2
, respectively. In other words, the error term is
(1)
t
with
probability and
(2)
t
with probability 1 (0 < < 1). The model based on this
assumption will subsequently be called the mixture-BVMEM model. The conditional
log-likelihood function becomes
l
T
() =
T

t=1
l
t1
() =
T

t=1
ln
_
f
(1)
t1
(v
1t
, v
2t
;
1
) + (1 ) f
(2)
t1
(v
1t
, v
2t
;
2
)
_
,
where f
(1)
t1
(v
1t
, v
2t
;
1
) and f
(2)
t1
(v
1t
, v
2t
;
2
) are given by (1) with replaced by
1
and

2
, respectively.
Assuming that v
t
is stationary and ergodic, it is reasonable to apply standard asymp-
totic results in statistical inference. In particular, approximate standard errors can be
obtained from the diagonal elements of the matrix
_

2
l
T
(

)/

_
1
, where

de-
notes the ML estimate of . Similarly, Wald and likelihood ratio (LR) tests for general
hypotheses will have the conventional asymptotic
2
null distributions. Note, however,
that hypotheses restricting the number of mixture components do not have the usual

2
distributions due to the problem of unidentied parameters (see e.g. Davies (1977)).
We will not attempt such tests, but assume throughout that there are two mixture
components. The adequacy of the assumption will be veried by means of diagnostic
procedures (see Section 4.3).
3
Specically, for observation t,
l
t1
() = ( + 1) ln () +
1
2
( 1) [ln (v
1t
) + ln (v
2t
) ln (
1t
) ln(
2t
)]

v
1t

1
1t
+ v
2t

1
2t

1
ln [()] ln (1 )
1
2
( 1) ln ()
+ln

I
1

v
1t
v
2t

1
1t

1
2t
1

ln (
1t
) ln (
2t
) .
6
4 Estimation Results
4.1 Data
The data set in this study covers 3,194 daily closing observations from the period 1.1.1992
- 31.12.2004, and was obtained from Bloomberg Professional Service (see Figure 1). The
rst eleven years of the full sample, or 1.1.1992 - 31.12.2002, comprise the in-sample of
2,708 observations. The nal two years, 2003 and 2004, are left as the 486-day out-of-
sample to be used for forecast evaluation.
The call-side (put-side) implied volatility time series is calculated as an unweighted
average of Black-Scholes implied volatilities from two nearest-to-the-money call (put)
options from the nearest maturity date. Rollover to the next maturity occurs two cal-
endar weeks prior to expiration in order to avoid possibly erratic behavior in IV close
to option expiration. ATM options are typically used to estimate the markets expected
volatility for the remainder of the options maturity, as trading volumes are usually high
for ATM options. Also, ATM options have the highest sensitivity to volatility.
1
0
2
0
3
0
4
0
5
0
6
0
7
0
1992 1994 1996 1998 2000 2002 2004
1
0
2
0
3
0
4
0
5
0
6
0
7
0
1992 1994 1996 1998 2000 2002 2004
Figure 1: Nikkei 225 index call implied volatility (upper panel) and put implied volatility (lower panel)
1.1.1992 - 31.12.2004.
Table 1 provides descriptive statistics on both the call-side IVs (NIKC) and put-side
IVs (NIKP). The average level of put-side implied volatility is higher in the sample of this
study, a phenomenon which has also been documented in the U.S. markets by Harvey
and Whaley (1992).
7
NIKC NIKP
Maximum 70.84 74.87
Minimum 9.26 8.80
Mean 24.68 24.82
Median 23.42 23.84
Standard deviation 7.07 7.41
Skewness 1.10 0.94
Excess kurtosis 2.42 1.79
Table 1: Descriptive statistics for NIKC and NIKP for the full sample of 1.1.1992 - 31.12.2004.
4.2 Model Estimation
Given the clear linkages between the implied volatilities of call and put options on the
same underlying asset outlined above, call-side (put-side) IV can be expected to be a
signicant predictor of future put-side (call-side) IV. Therefore, the model presented
in Section 3 is augmented with lagged cross terms, so that call (put) implied volatility
depends on its own history as well as on the history of put (call) implied volatility. Bollen
and Whaley (2004) nd that in the U.S. market, the demand for ATM index puts drives
both the changes in ATM put implied volatility and the changes in ATM call implied
volatility. Therefore, we expect that for our Japanese implied volatility data, lagged put
IV will be more signicant in explaining call IV than lagged call IV will be in explaining
put IV.
Dummy variables for Friday eects of put-side IV are also added due to the im-
provement in diagnostics achieved after the addition (see Section 4.3 for more details on
diagnostic checks). The level of IV is lowest on Fridays for both call and put options,
4
but trading volumes are highest on Fridays. An analysis of trading volumes of close-
to-the-money, near-term maturity call and put options on the Nikkei 225 index reveals
that during the two-year out-of-sample period used in this study, put options account
for 52.0 % of trading volume (measured with number of contracts traded). The share of
puts is lowest on Mondays (50.4%) and largest on Fridays (53.6%).
Findings similar to ours concerning weekly seasonality have been reported in previous
studies. Pe na et al. (1999) nd that in the Spanish stock index market, the curvature
of the volatility smile at the beginning of the week is statistically signicantly dierent
from the smile at the end of the week. Lehmann and Modest (1994) report that trading
volumes on the Tokyo Stock Exchange are substantially lower on Mondays than on other
days of the week. They hypothesize that this is due to reduced demand by liquidity
traders due to the risk of increased information asymmetry after the weekend. Also,
bid-ask spreads are largest on Mondays, making transaction costs highest at the start
of the week. The signicance of trading volumes is highlighted by Mayhew and Stivers
(2003), who nd that implied volatility performs well when forecasting individual stock
return volatility, but only for those stocks whose options have relatively high trading
volumes.
In order to take cross eects and the observed seasonal variation into account, we
need to modify the basic model presented in Section 3. Let
mt
denote the conditional
mean of mixture component m (m = 1, 2), and
mt
= (
C
mt
,
P
mt
)

, where
C
mt
and
P
mt
4
The level of IV is highest on Mondays. However, dummies for Monday eects were not statistically
signicant.
8
are the conditional means of the call and put implied volatilities, respectively.
The specications of the conditional means are

C
mt
=
C
m
+
q
C

i=1

C
mi
v
C,ti
+
r
C

i=1

C
mi
v
P,ti
+
s
C

i=1

CP
mi
D
i
v
P,ti
+
p
C

j=1

C
mj

C
m,tj
and

P
mt
=
P
m
+
q
P

i=1

P
mi
v
P,ti
+
r
P

i=1

P
mi
v
C,ti
+
s
P

i=1

PP
mi
D
i
v
P,ti
+
p
P

j=1

P
mj

P
m,tj
where the s are the coecients for lagged cross terms, and D
i
receives the value of 1
on Fridays, and zero otherwise. As mentioned above, the dummy variable in both the
call and put mean equations is for put-side Friday eects (coecients
CP
mi
and
PP
mi
).
This specication is later referred to as the unrestricted model.
In order to fully understand the value of including cross eects between NIKC and
NIKP in the model, an alternative specication with no cross terms was also estimated.
In this model, dummies for Friday eects are also included, but due to the elimination
of cross eects, the dummy in the equation for NIKC captures the Friday eect of call-
side, not put-side, implied volatility. In the second model specication, or the restricted
model,

C
mt
=
C
m
+
q
C

i=1

C
mi
v
C,ti
+
s
C

i=1

CC
mi
D
i
v
C,ti
+
p
C

j=1

C
mj

C
m,tj
and

P
mt
=
P
m
+
q
P

i=1

P
mi
v
P,ti
+
s
P

i=1

PP
mi
D
i
v
P,ti
+
p
P

j=1

P
mj

P
m,tj
.
The estimation results for both the unrestricted and the restricted model are pre-
sented in Table 2. The parameter values for all s, s and s meet the Nelson and
Cao (1992) constraints discussed in Section 3. Also, the coecients of cross terms (s)
and dummies (s) are positive, so positivity is guaranteed in the model. Compared with
the full version of the unrestricted model,
C
11
,
P
1
, and
P
12
are constrained to be equal
to zero, which is validated by a likelihood ratio test with p-value 0.273.
5
The probability parameter is quite high for the unrestricted model, close to 0.92.
Therefore, the second regime, which displays larger shocks, occurs on only some eight
percent of the trading days in the in-sample. The estimated shape (and scale) parameters
of the error distribution dier considerably between the two regimes, with residuals more
dispersed in the second regime. Figure 2 shows the joint error density of the unrestricted
model with the parameters estimated for the rst regime, while the error density for the
second regime is depicted in Figure 3. It should be noted that the scale of the z-axis
is dierent in the two gures. The errors are much more tightly concentrated around
unity in the rst, more commonly observed, regime, whereas the tail area is emphasized
in the second regime.
5
The model originally included six dummies: both rst-regime equations had Friday-eect dummies
for the intercept, own lagged value, and the lagged value of the other variable. Only the put-side Friday
eects were statistically signicant, and p-values from likelihood ratio tests validated the constraining
of the other dummies to zero.
9
Unrestricted Model Restricted Model
Log likelihood -12370.0 -12653.4
0.919** (0.012) 0.883** (0.018)

1
126.594** (3.745) 127.3036** (4.614)

1
0.094** (0.027) 0.019 (0.033)

C
1
1.264** (0.164) 0.298** (0.085)

C
11
0.514** (0.020) 0.617** (0.025)

C
12
0.104** (0.019) -0.223** (0.048)

C
11
0.321** (0.017) -

CC
11
- 0.043** (0.006)

CP
11
0.045** (0.005) -

C
11
- 0.584** (0.046)

P
1
- 0.254** (0.077)

P
11
0.529** (0.020) 0.611** (0.023)

P
12
- -0.215** (0.053)

P
11
0.247** (0.018) -

PP
11
0.043** (0.005) 0.046** (0.006)

P
11
0.210** (0.026) 0.581** (0.048)

2
20.043** (2.003) 23.413** (2.288)

2
0.360** (0.063) 0.378** (0.058)

C
2
1.401 (0.772) 0.718 (0.368)

C
21
0.185** (0.070) 0.216** (0.044)

C
21
0.150 (0.101) -

C
21
0.627** (0.108) 0.769** (0.048)

P
2
0.835 (0.688) 0.933* (0.403)

P
21
0.244** (0.087) 0.270** (0.050)

P
21
0.146 (0.090) -

P
21
0.612** (0.111) 0.717** (0.053)
Table 2: Estimation results for the BVMEM model. Standard errors calculated from the nal Hessian
matrix are given in parentheses. (**) indicates statistical signicance at the one-percent level, and (*)
at the ve-percent level.
10
Figure 2: Density of residuals in the rst regime of the unrestricted model.
Figure 3: Density of residuals in the second regime of the unrestricted model.
11
The correlation of errors, or , is higher in the second regime, making changes in
call and put IV more correlated when volatility is high. This is also clearly visible in
Figures 2 and 3. The coecients of cross terms are signicant at the one-percent level
in the rst, more common regime, and jointly signicant in the second regime (p-value
from LR test equal to 0.007). The coecients of the cross terms are higher in the rst
regime, making the cross eects more pronounced. In other words, the cross eects are
smaller when volatility is high. For both regimes, the eect put-side IV has on call-side
IV is larger than the eect call IV has on put IV, although the dierence in coecients
is quite small in the second regime. Friday dummies for the rst lag of put IV are
also signicant and positive, indicating that the eect of the lagged put IV is larger
on Fridays, when trading volumes are highest. Values of intercepts are higher in the
second regime, consistent with the notion that this regime occurs on days when shocks
are larger. The clearly greater s in the second regime indicate higher persistence in
that regime. This can be interpreted as a sign that once the second regime is entered,
it is likely that large shocks persist, i.e. there is volatility clustering present.
As we are interested in seeing the relevance of cross terms for forecast performance,
we also present the results for the restricted model without these cross eects. It should
be noted that the null hypothesis of all coecients of cross terms equal to zero is rejected
by an LR test at all reasonable signicance levels. In the restricted model, the estimate
of is smaller than in the unrestricted model, but the rst regime remains clearly more
prevalent. The parameters of the error distribution are very similar, but the correlation
of the residuals is lower in the rst regime than it was with the unrestricted model.
One notable dierence to the parameter values of the unrestricted model is that the
coecients of the second lags are both signicant in the rst regime, and have a negative
sign. This suggests that the exclusion of cross eects results in biased estimates of these
parameters. The dummies for Friday eects are signicant, indicating that the data
behaves somewhat dierently when the trading volume is at its highest. As the shape
(and scale) parameters of the error distribution that are estimated for the restricted
model are very close in value to those for the unrestricted model, the graphs for error
densities are qualitatively similar as those in Figures 2 and 3 and are therefore not
displayed.
4.3 Diagnostics
Most standard diagnostic tests are based on a normal error distribution, which renders
these tests unfeasible for our purposes due to the use of the gamma distribution. Also, as
our model specication has two mixture components and switching between the regimes
is random, there is no straightforward way to obtain residuals.
In order to investigate the goodness-of-t of our model, diagnostic evaluations can
nevertheless be conducted by means of so-called probability integral transforms of the
data. This method was suggested by Diebold et al. (1998) and extended to the multivari-
ate case by Diebold et al. (1999). The probability integral transform in the univariate
case (for one IV series) is obtained as
z
t
=
_
y
t
0
f
t1
(u)du (2)
where f
t1
() is the conditional density of the implied volatility with the chosen model
specication. The transforms are independently and identically uniformly distributed
12
in the range [0,1] if the model is correctly specied. Although commonly employed in
the evaluation of density forecasts, this method is also applicable to the evaluation of
in-sample t. In the bivariate case, Diebold et al. (1999) recommend evaluating four
sets of transforms: z
C
t
, z
P
t
, z
C|P
t
, and z
P|C
t
. The transforms z
C
t
and z
P
t
are based on the
marginal densities of the call and put implied volatilities, respectively. Similarly, z
C|P
t
is based on the density of call IV conditional on put IV, and vice versa for z
P|C
t
.
Graphical analyses of the probability integral transforms are commonplace. These
involve both a histogram of the transforms, that allows for determining uniformity, as
well as autocorrelation functions of demeaned probability integral transforms and their
squares. The graphical approach allows for easily identifying where a possible model
misspecication arises. Figure 4 presents the 25-bin histogram and autocorrelations for
z
C|P
t
, and Figure 5 for z
P|C
t
for the unrestricted model. Figures 6 and 7 present the
equivalent graphs for z
C
t
and z
P
t
, respectively.
Most columns of the histograms fall within the 95 % condence interval, which is
based on Pearsons goodness-of-t test. Although there are some departures from the
condence bounds (between zero and four, depending on the case), there is no indication
that the model would not be able to capture the tails of the conditional distribution
properly. It must be noted that Pearsons test statistics and condence interval are not
exactly valid, as their calculation does not take estimation error into account. However,
this omission most likely leads to rejecting too frequently.
The autocorrelations of the demeaned probability integral transforms also provide
encouraging evidence, although some rejections do occur at the ve percent (but not at
the ten percent) level.
6
There clearly seems to be some remaining autocorrelation in the
squares of the demeaned probability integral transforms. This same nding has been
made previously with univariate models for volatility data (see Ahoniemi (2007) and
Lanne (2006, 2007)). A potential explanation is that the model is not quite sucient in
capturing the time-varying volatility of implied volatility.
The removal of dummy variables from the unrestricted model results in a clear de-
terioration in the autocorrelation diagnostics, and consequently, we have deemed the
inclusion of weekly seasonality eects relevant for our model. The diagnostics for the
restricted model, or the model without cross eects, are somewhat better than those
for the unrestricted model, especially where autocorrelations are concerned.
7
The im-
provement in diagnostics due to the removal of cross eects is surprising, as the cross
terms are statistically signicant and improve forecasts (see Section 6 for discussion on
forecasts).
In order to verify that our unrestricted model takes the high persistence in the data
into account, we compare the autocorrelation functions (ACF) estimated from the call
and put IV data to those calculated from data simulated with our model. Figure 8
depicts the autocorrelation functions of NIKC and NIKP, as well as the autocorrelation
functions generated by the unrestricted mixture-BVMEM model after simulating 100,000
data points. A 95% condence band is drawn around the estimated autocorrelation
functions. The band is obtained by simulating 10,000 series of 3,194 data points (equal
to the full sample size), and forming a band that encompasses 95% of the autocorrelations
6
The condence bands of the autocorrelations are also calculated without estimation error accounted
for.
7
To save space, the diagnostic graphs for the unrestricted model without dummy variables and for
the restricted model are not presented in the paper, but are available from the authors upon request.
13
Figure 4: Diagnostic evaluation of z
C|P
t
: NIKC conditional on NIKP. Histograms of probability integral
transforms in the upper panel, and autocorrelation functions of demeaned probability integral transforms
(middle panel) and their squares (lower panel). The dotted lines depict the boundaries of the 95%
condence interval.
Figure 5: Diagnostic evaluation of z
P|C
t
: NIKP conditional on NIKC. Histograms of probability integral
transforms in the upper panel, and autocorrelation functions of demeaned probability integral transforms
(middle panel) and their squares (lower panel). The dotted lines depict the boundaries of the 95%
condence interval.
14
Figure 6: Diagnostic evaluation of z
C
t
: Marginal NIKC. Histograms of probability integral transforms
in the upper panel, and autocorrelation functions of demeaned probability integral transforms (middle
panel) and their squares (lower panel). The dotted lines depict the boundaries of the 95% condence
interval.
Figure 7: Diagnostic evaluation of z
P
t
: Marginal NIKP. Histograms of probability integral transforms
in the upper panel, and autocorrelation functions of demeaned probability integral transforms (middle
panel) and their squares (lower panel). The dotted lines depict the boundaries of the 95% condence
interval.
15
pointwise at each lag. As the ACFs generated by our model fall within the band at each
lag, it can be concluded that the observed ACFs could have been generated by our
mixture-BVMEM model.
Figure 8: NIKC (upper panel) and NIKP (lower panel) autocorrelation functions. The solid lines depict
the ACFs estimated from the full sample of the data, the lines with long dashes are the ACFs implied by
the mixture-BVMEM model with 100,000 simulated data points, and the lines with short dashes draw
95% condence bands around the ACFs.
The diagnostics underscore the necessity for using a mixture model in this case. We
also estimated a BVMEM model with only one regime, and the diagnostic checks clearly
reveal its inadequacy. In particular, the four histograms for that model show that this
specication fails to account for the tails of the conditional distribution, giving too little
weight to values close to zero and unity and too much weight to the mid-range of the
distribution.
8
However, the imbalance in the histograms is not as severe as with the
univariate models in Ahoniemi (2007), thus indicating that even for models without a
mixture structure, joint modeling improves the t to the Nikkei 225 IV data somewhat.
5 Impulse Response Analysis
The bivariate nature of our model allows for a further analysis of how the variables
adjust dynamically to shocks. In order to investigate this issue, impulse responses of
various types are calculated with both model specications presented above. This anal-
ysis should also uncover more evidence pertaining to the persistence of the data. The
more interesting specication for this purpose is naturally the unrestricted model, which
includes lagged cross terms in the rst regime. Also, as the coecients of the cross
terms are signicant, the unrestricted model specication is favored over the restricted
8
The estimation results and diagnostic evaluation for the no-mixture model are available from the
authors upon request.
16
version.
9
We generate the impulse responses by simulating data according to the conditional
mean proles method proposed by Gallant et al. (1993). It turns out that after approx-
imately 40 periods, the eects of all considered shocks go to zero. Therefore, we present
impulse responses up to 40 periods (trading days) ahead. The calculation of the impulse
response functions proceeds as follows: we generate 1,000 series of 40 random error terms
from gamma distributions with the shape and scale parameters estimated above. Also,
we generate 1,000 series of 40 random numbers that are uniformly distributed on the
interval [0,1]. These series are used in each period to determine which regime the model
is in: if the value of the random number exceeds the value of , the mean equation
for the second regime is used. To get initial values, a starting point in the data set is
chosen, and then 1,000 paths, forty days ahead into the future, are simulated from that
point onwards with the random error terms, random regime indicators, and estimated
parameter values. Another set of 1,000 paths are also simulated, this time with a shock
added to the values of NIKC, NIKP, or both in time period 0. The baseline value and the
value aected by the shock are calculated simultaneously, so that the same random error
terms and regime indicators are used for both. The averages of the 1,000 realizations are
taken for each of the forty days, and the impulse response function is then obtained as
the dierence between the series aected by the shocks and the baseline series without
the shocks.
In order to select a realistic magnitude for the shocks, we follow Gallant et al. (1993)
and study a scatter plot of demeaned NIKC and NIKP. The scatter plot, shown in Figure
9, helps to identify perturbations to NIKC and NIKP that are consistent with the actual
data. As expected, the scatter plot reveals a strong correlation in the two time series.
On the basis of the graphical analysis, six dierent plausible shock combinations are
selected: (10,10), (10,0), (0,10), (-10,-10), (-10,0) and (0,-10). In other words, the shock
is introduced directly into the value of NIKC or NIKP (or both), rather than into the
error terms of the model.
The impulse responses for the rst three shock combinations are presented in Figures
10, 11, and 12, respectively. The starting point in the data was July 11, 1996, a time
when both NIKC and NIKP were historically quite low. Four noteworthy conclusions can
be drawn from the analysis. Most importantly, put-side IV recovers from shocks more
rapidly than call-side IV, which is evident in all three gures - even when the shock
aects only put-side IV (Figure 12). This result could be based on the phenomenon
documented by Bollen and Whaley (2004): the demand for ATM index puts drives the
level of ATM implied volatility (in U.S. markets). Also, trading volumes for puts are
higher (measured with number of contracts). Therefore, the pricing of puts may be
somewhat more ecient, allowing shocks to persist for shorter periods of time than in a
less ecient market.
Second, the eects of shocks take a relatively long time to disappear entirely: some
thirty trading days, or six weeks, seem to elapse before the eect of a shock is completely
wiped out. This nding gives further support to the existence of considerable persistence
in the data.
Third, the impulse responses are similar regardless of the starting point that is se-
lected from the data. Four dierent starting points were in fact considered: a moment
9
Dummy variables are removed from the mean equations in the impulse response analysis. The
removal of weekly seasonality does not aect the general shape of the impulse response functions, but
makes results more easily readable from graphs.
17
Figure 9: Scatter plot of demeaned NIKC (x-axis) and demeaned NIKP (y-axis) for in-sample period
(1.1.1992 - 31.12.2002).
when both IVs were low, a moment when both were high, a moment when NIKC was
considerably higher than NIKP, and a moment when NIKP was considerably higher than
NIKC. This third result is to be expected as the nonlinearity in the mixture-BVMEM
model arises primarily through the mixture of two regimes, with the selection of the mix-
ture component being random rather than dependent on the past values of the implied
volatilities. The fourth and nal observation is that our model does not allow positive
and negative shocks to have eects of diering magnitude. Therefore, only the impulse
responses to positive shocks are presented.
Without the evidence on historical values provided by the scatter plot, it could be
argued that a shock of the type (10,0), or any shock with a clearly dierent magnitude
for call and put IVs, is not realistic. As outlined above, both IVs represent the markets
expectation of future volatility, and should thus be equal. However, empirical analyses
again lend support to the fact that call and put IV can dier even considerably at
times, due to market imperfections and demand shocks. As an example, the dierence
between NIKP and NIKC is greatest on Sept. 12, 2001, or immediately after the 9/11
terrorist attacks, when the demand for put options was extremely high. On that day,
the dierence between the put and call implied volatilities was 33.6.
With the restricted model, or the model without cross eects, the impulse responses
look very dierent. The eect of a shock lasts for less than ten days, or less than two
weeks. As there are no lagged cross terms in this model specication, if a shock aects
only one variable, the other is (naturally) entirely unaected and the impulse response
is at.
6 Forecasts
In this section, we turn our attention to the forecasting ability of the two models outlined
in Section 4.2. Forecast evaluation is based on two separate criteria: the direction of
18
Figure 10: Impulse response function for a shock of (10,10) with the unrestricted model.
Figure 11: Impulse response function for a shock of (10,0) with the unrestricted model.
Figure 12: Impulse response function for a shock of (0,10) with the unrestricted model.
19
change in IV as well as the traditional forecast accuracy measure mean squared error
(MSE). It is of particular interest whether the inclusion of cross eects between the two
time series can improve the earlier forecast performance of univariate models for NIKC
and NIKP investigated by Ahoniemi (2007).
Both daily (one-step-ahead) and ve-step-ahead forecasts were calculated with the
mixture-BVMEM model specications outlined above for the 486-day out-of-sample pe-
riod of 1.1.2003 - 31.12.2004. Days when public holidays fall on weekdays and the
observed value of implied volatility does not change were omitted from the data set.
Parameter values are treated in two ways: they are either estimated once using the data
from the in-sample period and then kept xed, or re-estimated each day. If parameter
values are not stable over time, there can be added value in updating them before cal-
culating each new forecast. When parameter values are updated daily, the forecasts are
calculated from rolling samples. In other words, the rst observation is dropped and a
new one added each day, in order to include information that is as relevant as possible.
The one-step-ahead forecasts are evaluated in terms of both directional accuracy
and MSE. Although an accurate forecast of the future level of IV can be valuable to all
market participants with risk management concerns, a correct forecast of the direction
of change in implied volatility can be useful for option traders. Various option spreads,
such as the straddle, can yield prots for the trader if the view on direction of change
(up or down) is correct, ceteris paribus.
The forecast results are summarized in Table 3. The directional accuracy of the
bivariate model in the two-year out-of-sample is superior to the performance of univariate
models. The BVMEM model predicts the direction of change correctly on 348 days out
of 486 for NIKC, and on 351 days for NIKP. This is in contrast to the results in Ahoniemi
(2007), where the best gures from multiplicative models were 336 and 321 for NIKC and
NIKP, respectively. There would appear to be some value to updating parameter values
each day. This improves directional accuracy for NIKC clearly, and yields a lower mean
squared error for both series of forecasts. However, the direction of change is predicted
correctly for NIKP on one day more when daily updating is not employed. Both the
unrestricted and the restricted model make more upward mistakes for NIKC, i.e. the
models make a prediction of an upward move too often. When predicting the direction
of change of NIKP, the unrestricted model forecasts a move downwards too often, but
the restricted model a move upwards too often.
A useful statistical test of the sign forecasting ability of the BVMEM models is the
test statistic presented in Pesaran & Timmermann (1992). This market timing test can
help conrm that the percentage of correct sign forecasts is statistically signicant. The
p-values from the test are below 0.00001 for all the series of forecasts in Table 3, so the
null hypothesis of predictive failure can be rejected at the one-percent level for all four
forecast series.
The values for MSE in Table 3 indicate that more accurate forecasts can be obtained
for NIKC. Mean squared errors are lower than with the univariate models in Ahoniemi
(2007), even with the restricted model. The Diebold-Mariano (1995) test (henceforth
the DM test) conrms that the improvement upon the equivalent univariate models
is statistically signicant for the unrestricted model, with the null hypothesis of equal
predictive accuracy rejected at the ve-percent level for NIKC and the one-percent level
for NIKP. Again, this lends support to the joint modeling of the time series, and the
inclusion of cross eects. The forecast accuracy of NIKC with updating coecients is
signicantly better than that with xed coecients. For NIKP, the dierence between
20
the two alternative treatments of parameter values is not statistically signicant.
10
NIKC NIKP
Correct sign % MSE Correct sign % MSE
Unrest., updating 348 71.6% 4.21 350 72.0% 5.24
Unrest., xed 341 70.2% 4.31 351 72.2% 5.26
Rest., updating 332 68.3% 4.34 336 69.1% 5.49
Rest., xed 332 68.3% 4.39 332 68.3% 5.55
Table 3: Correct sign predictions (out of 486 trading days) and mean squared errors for forecasts from
the BVMEM model with both updating and xed parameter values. The best values within each column
are in boldface.
Overall, the results obtained for the Nikkei 225 index option market are superior to
those obtained for e.g. the U.S. market. Ahoniemi (2006) nds that ARIMA models can
predict the correct direction of change in the VIX index on 62 percent of trading days at
best with an identical out-of-sample period. Brooks and Oozeer (2002) model the implied
volatility of options on Long Gilt Futures that are traded in London. Their model has
a directional accuracy of 52.5 %. Pesaran and Timmermann (1995) predict the sign of
excess returns in the U.S. stock market, with results falling within the range of 58% to
60.5%. Gencay (1998) uses a technical trading strategy for the Dow Jones Industrial
Average and achieves the correct directional forecast on 57-61 percent of trading days.
Our earlier discussion on the eects of limits to arbitrage could perhaps explain why
Japanese IV is more predictable in sign than the IV in other markets. If arbitrage is
more dicult to carry out in Japan, option prices can depart from their true values to
a greater degree, making the market more forecastable.
Table 4 presents the MSEs for the 482 ve-step-ahead forecasts that could be cal-
culated within the chosen out-of-sample. The unrestricted model continues to be the
better forecaster for NIKP, but surprisingly, the simpler model, or the specication with-
out cross eects, yields lower MSEs for NIKC. The Diebold-Mariano test also rejects the
null of equal forecast accuracy at the ten-percent level when comparing the restricted
and unrestricted models with updating coecients for NIKC, but not at the ve-percent
level. For the corresponding NIKP values (8.79 and 9.36), the null is not rejected. The
results for NIKP are better than in Ahoniemi (2007), but for NIKC, the univariate
models provide lower mean squared errors. The DM test does not reject the null when
the best MSEs from univariate and bivariate models are compared (this applies to both
NIKC and NIKP). Therefore, no conclusive evidence is provided regarding the best fore-
cast model for a ve-day horizon, but in statistical terms, the BVMEM model is at least
as good as univariate models.
10
A bivariate model specication without cross eects and dummy terms is not a better forecaster
than univariate models, regardless of whether directional accuracy or MSE is used as the measure of
forecast performance. For example, the directional accuracy of this model is 332 out of 486 at best for
NIKC, and 318 for NIKP. The detailed results for this third model specication are available from the
authors.
21
NIKC NIKP
Unrestricted Model, updating 7.21 8.79
Unrestricted Model, xed 7.65 8.78
Restricted Model, updating 6.48 9.36
Restricted Model, xed 6.60 9.43
Table 4: Mean squared errors for 482 ve-step-ahead forecasts from the BVMEM model with both
updating and xed parameter values. The best values within each column are in boldface.
7 Conclusions
It has often been empirically observed that implied volatilities calculated from otherwise
identical call and put options are not equal. Market imperfections and demand pressures
can make this phenomenon allowable, and this paper seeks to answer the question of
whether call and put IVs can be jointly modeled, and whether joint modeling has any
value for forecasters.
We show that the implied volatilities of Nikkei 225 index call and put options can be
successfully jointly modeled with a mixture bivariate multiplicative error model, using a
bivariate gamma error distribution. Diagnostics show that the joint model specication
is a good t to the data, and coecients are statistically signicant. Two mixture
components are necessary to fully capture the characteristics of the data set, so that
days of large and small shocks are modeled separately. There are clear linkages between
the implied volatilities calculated from call and put option prices, as lagged cross terms
are statistically signicant. The IV derived from put options is a more important driving
factor in our model than the IV from calls, as dummy variables for Friday eects of put-
side IV are revealed to be signicant and to improve the diagnostics of the joint model.
Impulse response analysis indicates that put-side IV recovers more quickly from
shocks than call-side IV. Shocks persist for a relatively lengthy period of time (thirty
trading days), which is consistent with good forecastability. Also, as the nonlinear
feature of our model is primarily the random switching between regimes, the point of
time in which a shock is introduced does not aect the behavior of the impulse response
functions.
The BVMEM model provides better one-step-ahead forecasts than its univariate
counterparts. Both directional accuracy and mean squared errors improve when jointly
modeling call and put implied volatility. The direction of change in implied volatility is
correctly forecast on over 70% of the trading days in our two-year out-of-sample period.
When forecasting ve trading days ahead, the BVMEM model is at least as good as
univariate models in statistical terms. Based on the combined evidence from all forecast
evaluations, we conclude that joint modeling and the inclusion of cross eects improves
the forecastability of Nikkei 225 index option implied volatility, and can provide added
value to all investors interested in forecasting future Japanese market volatility.
22
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