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Retrieving Risk Neutral Moments and Expected Quadratic Variation from
Option Prices
by
Leonidas S. Rompolis∗and Elias Tzavalis†
Abstract
This paper derives exact formulas for retrieving risk neutral moments of future payoffs of any
order from generic European-style option prices. It also provides an exact formula for retrieving
the expected quadratic variation of the stock market implied by European option prices, which
nowadays is used as an estimate of the implied volatility, and a formula approximating the jump
component of this measure of variation. To implement the above formulas to discrete sets of option
prices, the paper suggests a numerical procedure and provides upper bounds of its approximation
errors. The performance of this procedure is evaluated through a Monte Carlo exercise. The paper
provides clear cut evidence that ignoring the jump component of the underlying asset can lead
to seriously biased estimates of the new volatility index suggested by the Chicago Board Options
Exchange (CBOE). This is also confirmed by an empirical exercise based on market option prices
written on the S&P 500 index, which shows that the jump component of quadratic variation is
significant and varies substantially during financial crises.
Keywords: Risk neutral moments, characteristic function, expected quadratic variation.
JEL: C14, G11, G12
The authors would like to thank Karim Abadir and Jens Carsten Jackwerth for useful comments
on an early version of the paper.
∗Department of Accounting and Finance, Athens University of Economics and Business. 76 Patissionstreet, 10434 Athens, Greece. Email: [email protected]†Department of Economics, Athens University of Economics and Business. 76 Patission street, 10434
Athens, Greece. Email: [email protected].
1
1 Introduction
There is recently growing interest in the option pricing literature to retrieve model-free estimates of
risk neutral moments (RNM) of the future payoff or return of an underlying asset (stock) implied by
cross-sectional sets of generic European-style call/put options. These moments do not rely on any
option pricing model and thus, have a number of interesting applications in practice. For instance,
they can be used to study the information content of option prices about future stock market
volatility or any other higher-order risk neutral moment (see, e.g. Canina and Figlewski (1993),
Christensen and Prabhala (1998), Dennis and Mayhew (2002), Jiang and Tian (2005) and Bollerslev
and Zhou (2006)). Secondly, they can be employed to examine the relationship between higher-
order risk neutral moments of asset returns and their physical counterparts, and/or to estimate
the implied risk aversion coefficient of the stock and option markets participants (see, e.g. Bakshi,
Kapadia and Madan (2003), Bakshi and Madan (2006a,b) and Rompolis and Tzavalis (2010)).
Thirdly, they can be applied to estimate the risk neutral density (RND) of the underlying asset
return based on an approximation density method (see, e.g. Corrado and Su (1996), Rompolis and
Tzavalis (2008) and Rompolis (2010)).
There are two generally accepted methods of retrieving model-free estimates of risk neutral
moments (RNM) of future asset returns from option prices. As shown in this paper, these are
theoretically consistent with each other. The first relies on nonparametric estimates of the risk
neutral density (RND) of the future payoff of the underlying asset from cross-sectional option prices
in a first step, exploiting the Breeden-Litzenberger (1978) relationship.1 From this density, we can
then obtain estimates of RNM of any order. However, as is well-known in the literature, in order
to be successful nonparametric density estimation methods used to retrieve RNDs require large
sets of cross-section option prices, which are not often available in practice (see Pagan and Ullah
(1999), for a survey).2 The second method of retrieving model-free estimates of RNM of future
asset returns from option prices relies on the work of Bakshi and Madan (2000) showing that
risk neutral asset price payoffs are spanned by a continuum of out-of-the-money (OTM) generic
European-style options. Based on this work, Bakshi, Kapadia and Madan (2003) have provided
1See, e.g. Ait-Sahalia and Lo (1998), Bliss and Panigirtzoglou (2002), Ait-Sahalia and Duarte (2003).2To improve over their performance, they have been suggested semi-parametric methods, but these depend on the
correct specification of the underlying asset price process, see, e.g. Corrado and Su (1996) and Melick and Thomas
(1997).
2
model-free formulas for retrieving the first four non-central RNM of the underlying asset return
from OTM cross-sectional sets of European call and put prices directly, without having first to
recover the correct RND. Using an analogous methodology, Demeterfi et al. (1999) and Britten-
Jones and Neuberger (2000) have derived a formula of implied volatility based on OTM options.
This formula relies on the assumption that the underlying asset price follows a pure continuous
process.
In this paper, we extent the above works and derive exact, closed-form formulas for retrieving
RNM from OTM generic European-style option prices. In particular, the paper extents the work
of Bakshi, Kapadia and Madan (2003) in two main dimensions. First, it provides RNM formulas of
any order and for any random variable taken as a function of the asset payoff. These higher-order
RNM are necessary to better approximate the RND. Second, it derives formulas which can be
applied to retrieve RNM of more complicated payoffs using generic European-style options, such
as Asian or basket, which can be used in applied work. To this end, we first derive the risk neutral
characteristic function of options payoffs. From this function, we can directly obtain RNM formulas
of any order.
The RNM formulas derived by the paper are also used to obtain an exact analytic formula of the
expected quadratic variation of the stock market implied by European options, referred to as the
new-VIX index.3 This measure of volatility does no longer rely of the Black-Scholes (BS) model.
It has been introduced by the Chicago Board Options Exchange (CBOE) to provide model-free
estimates of the stock market volatility and facilitate the trading of volatility derivatives. The
exact formula that the paper derives for the VIX index indicates that, if the underlying asset price
process includes random jumps, then the CBOE formula used to calculate the VIX is only an
approximation of the exact formula. The latter depends on the jump component of the underlying
asset price. As shown in the paper, ignoring this component will lead to biased estimates of the new-
VIX. To estimate this component of expected quadratic variation, the paper derives a model-free
approximation formula based on higher-order RNM.
To implement the above formulas to finite, discrete sets of option prices, the paper suggests
a numerical method which relies on an interpolation-extrapolation smoothing procedure. We use
cubic spline functions to interpolate implied volatilities between available strike prices. To extrap-
3The original VIX now is known as the VXO.
3
olate these volatilities beyond the minimum and maximum strike prices, we employ a linear or,
alternatively, a constant function. We also consider the case of not extrapolating them. To control
for the approximation errors of the above numerical method, the paper provides theoretical upper
bounds of them and, more importantly, it shows how these bounds vary at the endpoints of the
extrapolation scheme. To assess the performance of this method, the paper conducts an extensive
simulation study. This study examines the degree of accuracy of the method with respect to strike
price intervals met in practice, the number of observations of the call or put prices available and,
finally, alternative extrapolation schemes. Our simulation study is also interested in examining if
the model-free formula of the jump component of the expected quadratic variation suggested by
the paper provides accurate estimates of it.
To carry out our simulation study, we generated data from the stochastic volatility with jumps
(SVJ) model suggested in the literature to improve over the performance of the BS and stochastic
volatility models (see, e.g. Bakshi, Cao and Chen (1997)). The results of our simulation study
provide a number of interesting results with important portfolio management implications. They
show that the suggested by the paper numerical method for implementing the RNM formulas to
option price data sets works very satisfactory even for small intervals of strike prices, often available
in practice. This is true independently on the number of the call and put option prices available.
As was expected, the magnitude of the approximation errors reduce as the strike price interval
increases. This is more apparent for higher than second-order moments. Another interesting result
of our simulation study is that ignoring the jump component of the underlying asset price process
clearly leads to biased estimates of the VIX index. This component is adequately approximated
by our model-free suggested formula. Applying this formula to actual option price data, the paper
provides clear cut evidence that the bias of the VIX index due to the omission of the jump com-
ponent of asset price process is significant and it varies substantially, especially during financial
crises.
The paper is organized as follows. Section 2 derives the RNM formulas for different categories of
European-style option prices, while Section 3 derives the exact formula for retrieving the expected
quadratic variation from option price data, taking into account possible jumps in the underlying
asset price process. This section also derives the approximation formula of the jump component
of the expected quadratic variation. Section 4 presents the numerical method for implementing
4
the above formulas to option price data and provides upper bounds of their approximation errors.
Section 5 conducts our simulation exercise. Section 6 carries out our empirical exercise. Section 7
concludes the paper. All of the derivations are given in a technical appendix.
2 Risk neutral moment formulas
Consider a generic European-style option contract with expiration date and strike price .
Denote the random variable upon which the future payoff of the contract is written as (0 ).
This variable is a function of a stochastic process ()∈[0 ] ∈ R, defined over the time interval[0 ], with 0 ≤ 0 . For a plain vanilla option contract, (0 ) is defined as (0 )
≡ , where can be the payoff of any asset, e.g. stock, bond, index, futures or swap, at date
. The random variable (0 ) can also represent the payoff of an Asian option contract with
payoff (0 ) ≡ 1−0
R 0.
Under no arbitrage conditions, the price of a generic European call option contract at time ,
with maturity interval = − , defined as ( ), can be written as follows:
( ) = − [ (0 )−]+ =
= −Z ∞
[ (0 )−][ (0 )], (1)
where is the instantaneous riskless interest rate, denotes the risk neutral probability measure,
[ (0 )−]+ defined as [ (0 )−]+ ≡ max 0 (0 )− is the payoff of the optionat the expiration date , [ (0 )] is the conditional risk neutral density function (RND) of
(0 ), at time . The price of a generic European put option contract at time , with maturity
interval , denoted as ( ), is defined as
( ) = − [ − (0 )]+ =
= −Z
0
[ − (0 )][ (0 )] (2)
where [ − (0 )]+ ≡ max 0 − (0 ) is the payoff of this option at date .Let () be defined as the following general function of payoff (0 ): () ≡ [ (0 )]−
(0), where : R → R is a twice continuously differentiable function under measure and 0
5
is a constant. Then, based on Bakshi and Madan’s (2000) fundamental option pricing theorem,
we can derive an analytic formula of the risk neutral characteristic function (RNCF) of function
() implied by generic European-style option prices, at time This formula is given in the next
proposition.
Proposition 1 Let (ΩF ) be the probability space restricted to the time interval [0 ], withits filtration F = F; ∈ [0 ], () be the conditional on F characteristic function of therandom payoff () under measure , where : R → R is a twice continuously differentiable
function. Then, the analytic form of () is given as
()
= 1 + 0(0)³ [ (0 )]−0
´+
+½Z +∞
0
£00()− 20()2
¤[()−(0)]( ) +
+
Z 0
0
£00()− 20()2
¤[()−(0)]( )
¾ (3)
where i denotes the imaginary number.
The proof of the proposition is given in the Appendix.
As expected by Bakshi and Madan’s theorem, the RNCF of future payoff () given by Propo-
sition 1 is spanned by a continuum of out-of-the-money (OTM) generic European-style option call
and put prices over different strike prices. From this RNCF, we can derive the well known Breeden
and Litzenberger (1978) formula often used in practice to derive the RND of () from option
prices. This theoretical result implies that retrieving risk neutral moments (RNM) directly from
OTM generic European-style option prices is theoretically consistent with deriving them from the
well known Breeden and Litzenberger formula. This result is established in the next corollary.
Corollary 1 Proposition 1 implies that the conditional on F risk neutral density of (0 ),
denoted [ (0 )], can be calculated based on the Breeden-Litzenberger formula:
[ (0 )] =
½
2()
2 , for (0 ) 6 1
2()
2 , for (0 ) 1,(4)
where 1 = [ (0 )].
6
The proof of the corollary is given in the Appendix.
The RNCF () given by Proposition 1 has a theoretical interest. It can be employed to
obtain analytic, exact formulas of the non-central RNM of the future payoff () of any order
based on generic European-style option prices. These formulas are given in the next proposition.
Proposition 2 Given the results of Proposition 1, the conditional on filtration F th-order non-
central risk neutral moment of payoff () is given as
1
= 0(0)³ [ (0 )]−0
´+
+∙Z +∞
0
00()( ) +
Z 0
0
00()( )¸, (5)
for = 1 and
(6)
= ½Z +∞
0
(;0)( ) +
Z 0
0
(;0)( )
¾,
for ≥ 2, where
(;0) = [()− (0)]−2 [00()(()− (0)) + (− 1)0()2]
The proof of this proposition is given in the Appendix.
As a corollary of Proposition 2, next we give the th-order central RNM of future payoff
(0 ), for ≥ 2.
Corollary 2 Proposition 2 implies that the conditional on filtration F th-order central risk neu-
tral moment of (0 ) is given by the following formula:
e (7)
= (− 1)"Z +∞
1
(;)( ) +
Z 1
0
(;)( )
#,
for ≥ 2, where 1 = [ (0 )] and (;) = ( − 1)
−2.
7
The RNM formulas given by Corollary 2 can be obtained by assuming [ (0 )] = (0 )
and 0 = [ (0 )] in relationships (5)-(6). They indicate some distributional features of
the RND of the future payoff (0 ), implied by OTM call or put prices, which have both
theoretical and practical interest. In particular, for = 2 and = 4 they show that the values
of both the conditional variance (referred to as the volatility) and kurtosis of (0 ), at time ,
increase with the prices of OTM calls and puts. This can explain the very high degree of positive
correlation between these two central RNM of future payoffs implied by European option prices,
which has been found in many empirical studies (see, e.g. Corrado and Su (1997)). For = 3,
formula (7) implies that the third-order central moment determining the skewness of the RND,
denoted as e3, has negative sign when the OTM puts are more expensive that the OTM calls.
This result was first noticed by Bates (1991), based on a different framework of analysis.
Based on Proposition 2, in next corollary we provide exact RNM formulas of the future asset
log-return, defined as ( ) = ln³
´, which are functions of plain vanilla European option prices.
As mentioned in the introduction, these formulas have many interesting applications in the option
pricing literature.
Corollary 3 Proposition 2 implies that the conditional on filtration F th-order non-central RNM
of the asset log-return ( ) is given as
1
=
µ
∙
¸− 1¶−
−∙Z +∞
1
2( ) +
Z
0
1
2( )
¸, (8)
for = 1, and
=
(Z +∞
1
2
∙ln
µ
¶¸−2 ∙− 1− ln
µ
¶¸( )
+
Z
0
1
2
∙ln
µ
¶¸−2 ∙− 1− ln
µ
¶¸( )
), (9)
for ≥ 2.
8
To obtain the RNM formulas of the above corollary, given by (8)-(9), we have set ( ) = ln( )
and 0 = in equations (5)-(6), since (0 ) ≡ for plain vanilla European option prices.
Some interesting remarks on these formulas are given bellow.
Remark 1 The RNM formulas (8)-(9) clearly indicate that only the higher than second-order RNM
of the log-return ( ) can be solely derived from OTM plain vanilla European call and put prices.
For = 2 3 4, the formulas of the RNM given by (9) for the log-return ( ) are consistent
with those derived by Bakshi, Kapadia and Madan (2003).
Remark 2 As relationship (8) clearly indicates, the first order RNM 1 depends also on the
risk neutral expectation
h
iof the underlying asset price considered. Below, we present two
different cases of assets where this expectation can be analytically derived. The first is when the
underlying asset is a stock, or a stock index, which pays a constant dividend yield . Then, we have
h
i= (−) and thus, equation (8) becomes
1
= (−) − 1− ∙Z +∞
1
2( ) +
Z
0
1
2( )
¸ (10)
The second case is when the underlying asset is a futures or a swap contract. Then, we will have
h
i= 1 which implies that formula (8) becomes
1 = −∙Z +∞
1
2( ) +
Z
0
1
2( )
¸. (11)
3 Retrieving implied volatility from option prices
Recently, there is a growing interest in deriving a model-free measure of the asset price process’
()∈[ ] volatility implied from plain vanilla European options. This measure of volatility is
referred to as the annualized expected quadratic variation (see, e.g., Demeterfi et al. (1999), Britten-
Jones and Neuberger (2000), Jiang and Tian (2005) and Carr and Wu (2009)) and it is defined
as 1
hhi
i, where hi is the quadratic variation of log-price process = ln over
time interval [ ]. It nowadays has become the benchmark for measuring stock market volatility.
On September 22, 2003, the Chicago Board Options Exchange (CBOE) started calculating a
new stock market volatility index from option prices based on it. This index is known as the VIX
9
and is widely accepted by market participants. It also constitutes the underlying asset of volatility
derivatives. In the next proposition, we derive an exact formula of the above volatility measure
1
hhi
ibased on the RNM formulas provided in the previous section (see Appendix).
This is done under the more general assumption that the underlying asset price process contains
discontinuous components which are nowadays considered very important in capturing the price
dynamics of many financial assets (see, e.g. Eraker (2004), and Barndorff-Nielsen and Shepard
(2006)).
Proposition 3 Let the stochastic process of the underlying asset price ()∈[ ] be a semimartin-
gale. Then, the annualized expected −period quadratic variation 1
hhi
i, denoted as ,
can be analytically written as follows:
≡ 1
hhi
i=
2
∙Z +∞
1
2( ) +
Z
0
1
2( )
¸+
+2
∙Z
+
−− −
¸+
+2
⎧⎨⎩ X6
µ∆ +
1
2(∆)
2 − − −−
¶⎫⎬⎭ (12)
where = ln is the log-price process.
The proof of this proposition is given in the Appendix.
The analytic formula of , given by equation (12), provides a direct link between RNM and
the expected quadratic variation, which have been studied separately in the literature. According
to this formula, the annualized expected -period quadratic variation, , depends on three terms.
The first, as shown in the proof of Proposition 3, comes along the risk neutral mean. The second
term, given as
hR +
− −
−
i, depends on the underlying asset considered. If, for example,
10
this is a stock (or a stock market index) that pays a constant dividend yield , then it becomes4
∙Z
+
−− −
¸= 1 + ( − ) − (−) .
Examples of assets where term
hR +
− −
−
idoes not have any impact on formula (12)
are futures or swap contracts. In this case, we can easily show the following result:
∙Z
+
−− −
¸= 0.
The third term of equation (12), defined as
≡ 2
⎧⎨⎩ X6
∙∆ +
1
2(∆)
2 − − −−
¸⎫⎬⎭ ,is due to the discontinuous component of the asset price process ()∈[ ].
As is explained in the CBOE white page (see CBOE (2003)), the CBOE procedure calculates
the expected quadratic variation based on the following formula:5
∗ ≡ 2
∙Z +∞
1
2( ) +
Z
0
1
2( )
¸+
+2
h1 + ( − ) − (−)
i (13)
Once an estimate of the expected quadratic variation is obtained, the VIX index is calculated as
the square root of ∗ multiplied by 100. Proposition 3 implies that if the asset price has no
discontinuous component, then formula (13) coincides with the exact formula (12). In any other
case, it constitutes an approximation formula, which ignores the discontinuous component of the
4Note that if = = 0, then
+
−
− −
= 1. In this case, formula (12) reduces to that provided by
Britten-Jones and Neuberger (2000) under the assumption that the underlying asset price follows a diffusion process,
without discontinuous component.
5Note that there is a small difference in the formula employed by the CBOE and equation (13). Instead of ,
the CBOE formula uses the first strike price below the futures price , denoted as ∗, as integration bound. This
means that the second term of ∗ can be written as
2
ln
∗
−
∗− 1
' −1
∗− 12
(see also Jiang and Tian (2007)) so as for the two formulas to be consistent with each other.
11
asset price ()∈[ ], defined by . The latter is not observed in the market. In the next
proposition, we provide a model-free formula which can be used to approximate .
Proposition 4 Let the stochastic process of the underlying asset price ()∈[ ] be a semimartin-
gale. If
⎡⎣ X6
(∆)
⎤⎦ ∞, for all ∈ N+, (14)
where = ln, then ≡ 2
nP6
h∆ +
12(∆)
2 − −−−
iocan be approximated as
' −2
µ1
3!3 +
1
4!4
¶ (15)
where 3 and 4 are respectively the third and fourth-order RNM, at time .
The proof of this proposition is given in the Appendix.
The approximation of , given by formula (15) of Proposition 4, indicates that one can
calculate the jump component of the annualized quadratic variation using higher-order RNM
3 and 4. The latter can be estimated by OTM call and put option prices in a model-free way,
using formula (9). The practical implication of these results is that one can correct the implied
volatility estimate ∗, calculated by the CBOE, for the jump component based on market
data, using formula (15). The accuracy of formula (15) to capture the jump component of the
expected quadratic variation will be examined in a simulation study, presented in Section 5.
4 Implementation of the RNM formulas
To implement the RNM and the expected quadratic variation formulas derived in the previous
sections to observed option prices, we need to rely on an efficient numerical method which fits them
into finite, discrete cross-sectional sets of option prices. Following recent literature,6 a method
which can be used for this purpose relies on an interpolation-extrapolation procedure of option
prices implied volatilities.7 More specifically, this method first interpolates implied volatilities
6See Campa, Chang and Reider (1998), Bliss and Panigirtzoglou (2002), Dennis and Mayhew (2002) and Jiang
and Tian (2005), inter alia.7Note that interpolation and extrapolation of implied volatilities, instead of option prices, has been recently
suggested in many studies (see, e.g. fn 6). This is done in order to avoid numerical difficulties in fitting smooth
12
over the observed closed interval of strike prices [minmax], where min and max denote the
minimum and maximum strike prices observed, respectively. Then, it extrapolates these volatilities
over intervals (0min] and [max+∞), where option prices are not available.The interpolation of the implied volatilities is often carried out by using cubic splines, since
these functions give smooth and accurate values over interval [minmax]. On the other hand,
the extrapolation of the implied volatilities over intervals (0min] and [max+∞) can be doneeither by using a linear function (see, e.g., Bliss and Panigirtzogou (2002)) or a constant (see, e.g.,
Jiang and Tian (2005)). Both of these functions are truncated at the strike prices 0 (which is
considered as an approximation of a zero strike price) and ∞. The latter is considered as an
approximation of a strike price which tends to infinity. The strike prices 0 and ∞ are calculated
so as to correspond to put and call option prices, respectively, which are very close to zero, e.g.,
smaller than 10−3. Instead of these two extrapolation schemes, a third choice would be not to
extrapolate the implied volatility function, i.e. to set 0 = min and ∞ = max (see CBOE
(2003)). If the length of the available strike price interval [minmax] is sufficiently large, then
this can be considered as a natural choice, which can provide accurate estimates of RNM and
expected quadratic variation from observed option prices.
The above suggested numerical methods for implementing the RNM or expected quadratic
variation formulas to option prices lead to an approximation error, whose significance must be
investigated. This error is related to the curve-fitting scheme of implied volatilities over the interval
[0∞]. As is expected, it will depend on the number of option prices available, the length of
the interval [minmax] and the extrapolation scheme chosen. As is shown by equations (6) (or
(9)), the impact of this approximation error on the estimates of the RNM will depend on the
moment-specific function (;0), entering into the integrals of the RNM formulas employed.
Given that this error can not be measured in practice, due to the fact that the true theoretical
implied volatility function is not known, we will appraise its size effects on the estimates of the
RNM and the expected quadratic variation formulas by deriving upper and lower bounds of it. This
is done in the next proposition. The results of this proposition refer to the risk neutral moments of
the asset log-return ( ), given by equations (9) and (10), and the expected quadratic variation
approximation formula (13), used by CBOE.
functions into option prices. To convey the observed option prices into implied volatilities and vise versa, we use the
Black-Scholes (BS) formula. This methodology does not require the BS model to be the true option pricing model.
13
Proposition 5 Let ( ) (or ( )) denote the true call (or put) pricing function with re-
spect to variable = ln(), while b( ) (or b( )) denote the call (or put) pricing functionimplied by our suggested smoothing method for implementing the RNM formulas to finite sets of
European call and put prices. If b and d ∗ denote the approximated values of the true risk neu-tral moments and the approximated expected quadratic variation, ∗, respectively, obtained
by our numerical method, then the following approximation error bounds for b and d ∗ can berespectively derived:
¯b − ¯6
((∞;) + (0;) + ) for ∈ N+ (16)
and ¯d ∗ − ∗¯6 2
((∞; 1) + (0; 1) + 1) , (17)
where = max∈(0∞)
¯( )− b( )
¯and = max
∈(00)
¯( )− b( )¯ constitute up-
per bounds of the call and put option pricing functions approximation errors, respectively, 0 =
ln(0), ∞ = ln(∞), and (;) and (;) are two functions which are evaluated at
endpoints 0 and ∞ and they are defined as follows:
(;) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩1− − for = 1⎧⎨⎩ (2(− 1)−1−(−1) − −1−) if − 1 6
−1− if − 1 for > 2
and
(;) =
⎧⎨⎩ − − 1 for = 1
(−1)−1−1− for > 2,
respectively. is the truncation error of strike prices at endpoints ∞ and 0, respectively,
defined as
=
¯Z +∞
∞ (;0)( ) +
Z 0
0
(;0)( )
¯.
where (;0) =
⎧⎨⎩12 for = 1
2
hln³
´i−2 h− 1− ln
³
´ifor > 2
.
14
The proof of the proposition in given in the Appendix.
The results of Proposition 5 imply that the upper bounds of the approximation errors of the
RNM and the approximated expected quadratic variation, defined as¯b −
¯and
¯d ∗ − ∗¯,
respectively, depend critically on the option pricing function approximation errors and ,
multiplied with functions (;) and (;), respectively. These functions indicate that the
approximation errors¯b −
¯and
¯d ∗ − ∗¯depend on the order of the RNM, . To
investigate more analytically the effects of functions (;) and (;) on¯b −
¯and¯d ∗ − ∗
¯, next we derive the limits of them for → +∞ (or → −∞), while in Figure 1 we
plot their values over , for different RNM orders . Note that this figure gives the values of the
joint function of (;) and (;), which is written as
(;) =
⎧⎨⎩ (;) if 0
(;) if > 0
By taking limits of , we can prove the following result:
lim→+∞(; 1) = 1 for = 1
lim→+∞(;) = 2(− 1)−1−(−1) for > 2
and
lim→−∞ (;) = +∞ for ∈ N+
We can also prove that
(;) 6 () ∀ ∈ R+ and ∈ N+
where () is given as
() =
⎧⎨⎩ 1 for = 1
2(− 1)−1−(−1) for > 2
The above results indicate that function (;), which multiplies the call option pricing approx-
imation error , is bounded by function (). Although the values of () increase with the
15
order of RNM,8 the effects of (;) on¯b −
¯and
¯d ∗ − ∗¯can be controlled. This
can be confirmed by the plots of Figure 1 which clearly indicate that, for the empirically plausible
values of endpoint ∞ ∈ [0 06], the values of function (;) are small and close to each other
independently of .
In contrast to (;), the above results indicate that function (;), which multiplies the
put option pricing approximation error , is not bounded. This function tends to infinity as
→ −∞. Although this result theoretically implies that the size-effects of the error on¯b − ¯and
¯d ∗ − ∗¯may not be controlled, in practice this may not be true. This may
happen because, for empirical plausible values of endpoint 0 ∈ [−1 0], function (;) does nottake very large positive values. This can be confirmed by the inspection of the plots of Figure 1.9
Taking together the results of Figure 1 and Proposition 5 indicate that, when implementing
the RNM or formulas to option prices, the extrapolation scheme, and the endpoints 0 and
∞ chosen lead to a trade-off effect between truncation error and option pricing approximation
errors and . As 0 and ∞ tend to zero and infinity, respectively, decreases,
while and increase, since both intervals (0 0) and (0 ∞) increase. The value of
function (;) controlling the effect of on bounds¯b −
¯and
¯d ∗ − ∗¯will
also increase. If the negative effect caused by 0 → 0 and ∞ → ∞ on dominates its
positive effect on , then extrapolating implied volatilities through linear or constant functions
will increase the magnitude of the above bounds. The effects of , and and of the
alternative extrapolation choices considered on these bounds are investigated by our a Monte Carlo
simulation study, presented in the next section.
5 Simulation study
In this section, we assess the performance of the numerical method suggested in the previous section
for implementing the model-free RNM formulas (9)-(10) and the expected quadratic variation
8The function () takes the following values for = 2 6 : (2) = 147, (3) = 324, (4) = 1075,
(5) = 4688 and (6) = 25267.9Note that, in our simulation experiments presented in the next section, almost all the strike price intervals are
found to be between the values of the endpoints 0 = −1 and ∞ = 057. In 1000 iterations, it was found only one
case where 0 = −13 which lies outside the above interval [0 = −1 ∞ = 057]. This happened when a linear
function was used in the extrapolation scheme, instead of a constant.
16
formula (13) to plain vanilla European option prices. This is done by conducting an extensive
Monte Carlo simulation exercise. This exercise will also show if our numerical method is robust to
two sources of errors. The first is due to the interpolation of implied volatilities by cubic splines
at observed strike price intervals. The second is related to the extrapolation scheme adopted. This
error is associated with truncation error . Apart from these two sources of errors, a third one
can be attributed to the numerical integration technique chosen.10 However, as noted recently by
Jiang and Tian (2005) and it can be confirmed by our simulation exercise, this third type of error
has not significant effects on numerical procedures used for the calculation of integrals of OTM
option prices.
Our simulation study consists of eighteen in total different experiments. These can evaluate
the performance of our numerical method with respect to the length of the strike price interval
[minmax], the number of observed call and put prices (denoted, respectively as and
) and, finally, the extrapolation scheme employed. The latter considers also the choice of not
extrapolating, as a non-extrapolation scheme. The different strike intervals that we consider in our
study are the following: [0867 1046], [0735 1118] and [0596 1279], while the different pairs of
call and put number of prices ( ) assumed are given as: ( = 5 = 7), ( = 11 =
18) and ( = 21 = 34). The above intervals of strike prices and pairs of ( ) are chosen
so as to make our simulation study as close as possible to reality. They constitute representative
strike intervals [minmax] and pairs of ( ) of small, medium and large size taken
from a cross-sectional sets of European option prices written on the S&P 500 index with maturity
interval one month which covers the period from January 1996 to October 2009. This sample of
option and strike prices is used in our empirical exercise, which follows in the next section.
The theoretical European call and put prices used in our simulation study are generated from
the stochastic volatility with jumps (SVJ) model.11 This model is frequently used in practice to
10This source of error is known as discretization error. By choosing a large number of knot points in the numerical
integration technique, this error can be proved negligible. In our simulation study, to calculate the integrals we
employ the Gaussian quadrature numerical procedure.11The full-specification of the diffusion-jump asset price process ()∈[ ] under the risk neutral measure for
the SVJ model is given as follows:
=
( − − )+
√1 +
with = ( − )+
√2
( = 1) = , ln(1 + ) ∼
2
,
17
improve upon the performance of the BS and stochastic volatility models, as it implies high levels of
negative skewness and kurtosis of the RND. These two features of the RND are consistent with the
pattern of the implied volatility across different strike prices observed in reality (see, e.g. Bakshi,
Cao and Chen (1997)). To generate option prices closed to the market ones, we are based on values
of the structural parameters of the SVJ model found in the literature (see Bakshi, Cao and Chen
(1997)). That is, we have assumed the following values for the structural parameters of the model
under the risk neutral measure: = 001, = 04, = 393, = −052, = 061, = −0104and = 014. To be consistent with the intervals of strike prices considered in our simulation
study, the current values of the two state variables of the SVJ model and are taken to be the
mean of the S&P 500 index and the variance of its log-return for the sample period from January
1996 to October 2009. These are given as = 1 132 and = 0026, respectively. For the interest
rate , we used the average level of the three-month US Treasury bill for the above period given
as = 005. For the dividend yield , we used its average estimate over the same period given as
= 0015. This was calculated based on the OptionMetrics data set.
Based on the above values of the SVJ option pricing model parameters and state variables,
we generated European call and put option prices with one-month to maturity interval (i.e. =
0083). We have chosen this short maturity interval, given that the biggest failures of most of
the European option pricing models occur for it. From these sets of option prices, we derived
the implied volatilities which are then used to fit the functions of our interpolation-extrapolation
scheme. Following Ait-Sahalia and Duarte (2003), we added a noise term to our generated set of
implied volatilities drawn from the uniform distribution, with interval support [−0025 0025]. Theabove experiment is repeated 1000 times. The noise term added to the implied volatilities can be
taken to reflect random effects of the bid-ask spread and the different degree of liquidity on call
and put prices.12 These perturbated implied volatilities were then used to obtain estimates of the
RNM using formulas (9)-(10) and of the expected quadratic variation ignoring the jump component
using formula (13) based on the numerical method suggested in the previous section.
where = ln(1 + )− 122 , and 1 and 2 are two correlated Brownian motions with correlation coefficient .
12Note that the perturbated implied volatilities or their associated option prices do not violate the arbitrage
conditions, i.e. monotonicity and convexity.
18
5.1 RNM
To assess the performance of our numerical method in retrieving RNM from option price data, we
will employ the root mean squared error (RMSE) metric. This is defined as the following distance
between the RNM estimates retrieved by our method b and their true values:
RMSE =
rh¡b −
¢2iThe RMSE criterion can be decomposed as follows:
RMSE2 = RSB2 +RV2,
where
RSB =
q¡¡b¢−
¢2and RV =
rh¡b −
¡b¢¢2istand for the root squared bias (RSB) and the root variance (RV) (i.e. the standard deviation) ofb, respectively. The RMSE metric can be thought of as a measure of the overall performanceof a numerical method to retrieve accurate estimates of RNM from option prices. The RSB and
RV metrics can be considered as more appropriate for assessing properties of the method such as
unbiasedness and efficiency, respectively. These two properties are important for good density and
moment estimators. The true values of the RNM and expected quadratic variation used in the
calculation of the above all metrics were derived by differentiating the moment-generating function
of the risk neutral distribution of the log-return ( ) implied by the SVJ model.13
Table 1 presents the results of our Monte Carlo simulation study for the case that the extrapo-
13The moment-generating function of the SVJ model is given as:
() = (−−)+()+()+()
where = +22 − 1 and
() =
2(− + ) − 2 ln
1−
1−
() =
− +
2
1−
1−
() =
+
2
22−1
with
=− +
− − and =
(− )2 − 2(− 1)
19
lation scheme’s function is constant, while Table 2 gives the results for the case that this function
is linear. Finally, Table 3 reports the results for the case that non-extrapolation scheme is chosen,
i.e., 0 = min and ∞ = max. More specifically, the three tables report the theoretical and
approximated by our numerical method values of the first six non-central RNM of the asset return
( ), as well as the values of the following three metrics RMSE, RSB and RV for them. The
approximated values of the RNM reported in the tables constitute average values of their estimates
over the 1000 iterations of our Monte Carlo exercise. In Tables 1 to 3, we also provide average
values of the estimates of the upper bounds of the approximation error given by equation (16) over
the 1000 iterations. These are based on the values of ( ) and ( ) generated by the SVJ
model.
The results of the above tables lead to a number of interesting conclusions which have important
practical implications. First, they clearly show that the theoretical RNM formulas (9)-(10) can be
successfully implemented through our suggested numerical method to discrete sets of option prices.
The values of the RNM provided are very close to their theoretical ones, as the values of the three
metrics reported in the tables are very small. This is true for all different ( )-pairs of call and
put prices considered. The results of the tables indicate that the accuracy of higher-order RNM
estimates (i.e. for 2) increases with the length of the strike interval [minmax]. This
result was expected, as more information of OTM options is required to obtain accurate estimates
of higher-order RNM. As the values of the RSB metric reveal, the increase in the accuracy of
our numerical method with the length of the strike price interval can be attributed to the smaller
magnitude of the bias encountered. Further support for the accuracy of the method can be obtained
see Bates (1996). Based on this moment-generating function, we derived the theoretical values of the th-order RNM
by numerically calculating the th-order derivative of () at = 0, i.e.
=
()
=0
The annualized expected quadratic variation implied by the SVJ can be calculated analytically based on the
following formula:
1
hi
= 2( − )− 2
1 + 2
+
1
22 +
1
22 −
+22 + 1
.
Thus the jump component is given as:
= 2
+
1
22 +
1
22 −
+22 + 1
20
by the inspection of the estimates of the upper bounds of the approximations error¯b −
¯,
reported in the tables. These are found to be very small and, as was expected, they decrease with
the length of the strike price interval.
Regarding the extrapolation schemes chosen, the results of the tables reveal that, with the
exception of the first two moments, our numerical method performs better both in terms of accuracy
and efficiency for strike intervals of smaller length (i.e. for [0867 1046]) when the constant function
is used, instead of the linear. When the length of the strike interval increases, the two functions are
found to perform similarly. The worse performance of the extrapolation scheme employing a linear
function for small strike price intervals can be attributed to the fact that this scheme can lead to
a lower value of the integral bound 0 (or 0), compared to that with a constant function. Such
value of 0 implies that function (0;), multiplying the effects of the put pricing error
on the RNM estimates, will take a large value which, in turn, will lead to a large in magnitude
approximation error of our numerical implementation procedure, as noted in the previous section.
This can be confirmed by the results of Table 2, see Subpanel C1, which covers the case where
endpoint 0 takes its lowest value given by 0 = −13.Compared to the non-extrapolation scheme, the scheme based on a constant function performs
better in terms of accuracy, while the opposite is true in terms of efficiency. The linear function
extrapolation scheme outperforms the non-extrapolation one in terms of accuracy only for small
sample sizes and large strike price intervals. In terms of efficiency, again the non-extrapolation
scheme outperforms the linear one. The approximation error bounds reported in the tables reveal
that the non-extrapolation scheme outperforms both the constant and linear ones, for the majority
of sample sizes and strike price intervals examined. As mentioned in the previous section, this
can be explained by the fact that the negative effect caused by 0 → 0 and ∞ → ∞ on
dominates its positive effect on , which implies that extrapolating the implied volatility function
increases the approximation error bounds. This result has important practical implications. As the
true RNM are unknown, it means that a constant or a linear function extrapolation scheme may
increase the approximation error bounds¯b −
¯, for all . Thus, it may be better to choose
not to extrapolate implied volatilities over intervals (0min] and [max+∞). The magnitude ofthe truncation error encountered in this case can be assessed by fitting an option pricing formula
into option price data.
21
5.2 Expected quadratic variation
The results of our simulation exercise evaluating the performance of our numerical method to
retrieve estimates of the expected quadratic variation from option prices are reported in Table
4. The table presents average values of ∗, implied by the formula used by CBOE which ignores the
jump component term , over the 1000 iterations, and average values of term , approximated
through relationship (15). As in Tables 1-3, this is done for different sample sizes and strike price
intervals. The sum of the above values of ∗ and provides estimates of . For reasons of
space, the table presents results only for the case of the non-extrapolation scheme. Instead of values
of the RMSE, RSB and RV metrics, the table reports estimates of the error bounds of , ∗
and . These are obtained based on the values of the call and put prices generated by the SVJ
model.
The results of the table clearly indicate that the estimates of ∗ and , as well as those of
obtained through our numerical procedure are very close to their theoretical values, predicted
by the SVJ model. The latter are given in the first row of the table. The estimates of ∗ and
converge to their true values, as the strike price interval [minmax] and sample size
increase. These results indicate that the model-free formula (15) provides accurate estimates of
. As is predicted by the theory, these can explain the downward bias of ∗, due to the omission
of the jump component from formula (12) used by the CBOE. The theoretical value of term
is given as = 00014, which can be translated into an underestimation of the true VIX index
(given as 100√ ) by 33 basis points. As each index basis points is worth $10 per VIX futures
contract, this means an undervaluation of $330 per contract.
6 Empirical application
Based on the theoretical formulas derived in the previous sections, in this section we retrieve
estimates of the jump component term of expected quadratic variation from option price
data. This can show how important is the jump component of quadratic variation , ignored by
CBOE’s formula calculating the VIX index, i.e., ∗, based on (13).
To carry out this exercise, we rely on call and put option data written on the S&P 500 index.
More specifically, we use the implied volatility surface (IVS) of the S&P 500 index provided by
OptionMetrics Ivy DB database. This volatility surface is constructed from implied volatilities
22
with kernel smoothing technique, which is described in details in OptionMetrics data manual. This
data set contains implied volatilities (of both calls and puts) on a grid of fixed maturities and option
deltas. Using these Black-Scholes delta values OptionMetrics also calculate the implied strike price
of each implied volatility value. We select the IVS of 1-month time-to-maturity options every
third Wednesday of each month from January 1996 to October 2009. As the risk-free interest rate
and dividend yield we use the estimates employed in the OptionMetrics calculations. The interest
rate is derived from British Banker’s Association LIBOR rates and settlement prices of Chicago
Mercantile Exchange Eurodollar futures. The dividend yield is estimated by the put-call parity
relation of at-the-money option contracts.
Table 5 reports average values of ∗ and over our whole sample, based on formulas (13)
and (15), respectively. t-statistics are in parentheses. In addition to these values, the table also
reports average values of RNM 3 and 4, needed by formula (15). The estimates of ∗ and
across all points of our sample are graphically presented in Figures 2 and 3, respectively. To
retrieve the above estimates, we rely on the numerical interpolation-extrapolation procedure which
assumes a non-extrapolation scheme. As shown in the previous section, this leads to smaller error
bounds. Since these mainly depend on the truncation error , Table 5 reports estimates of them
for ∗, 3, 4 and , as well as the percentage of these bounds due to . These bounds are
calculated by fitting the SVJ model into our option price data and using the formulas of Proposition
5 to calculate errors , and .14 Figure 2 presents upper and lower intervals of ∗
adjusted by its error bound¯d ∗ − ∗
¯. The upper interval of ∗ can be considered as an
estimate of it which is net of truncation error , since this is found to explain almost all (about
97.76%) of the error bound.
The results of Table 5, and Figures 2 and 3 clearly indicate that constitutes a significant
component of expected quadratic variation , which implies that ∗ provides downward biased
estimates of . moves closely with ∗ during our sample and it takes its highest values
during the periods of financial crises, i.e., the 1997-1998 Asian crisis, the 2002 US corporate crisis
and Lehman Brothers’ collapse in year 2008, respectively. As was expected, investors’ expectations
for possible random jumps in the market are intensified during these periods. The negative bias of
14Note that we have chosen to calculate errors , and , and, hence, bounds ∗ − ∗
−
and −
, given that this model fits better into our data and is used very frequently in the empirical literature.23
∗ implies an undervaluation of a VIX futures contract written on the implied volatility . For
example, a negative bias of 50 basis points (observed immediately after Lehman Brothers’ default,
according to Figure 3) implies an undervaluation of $500 per VIX futures contract.15
7 Conclusion
This paper provides analytic formulas for retrieving model-free risk neutral moments (RNM) of any
order of future asset payoffs implied by generic out-of-the-money European-style option call/put
prices, such as Asian or plain vanilla. Based on these formulas, next the paper derives an exact
formula of the expected quadratic variation which nowadays is used as a new measure of implied
volatility by the Chicago Board Options Exchange (CBOE), known as the VIX index. It also
derives a formula which enables us to approximate the jump component of the expected quadratic
variation in a model-free way, based on the third and fourth-order RNM. This component is not
taken into account by the formula used by CBOE to calculate the VIX index.
To implement the above formulas to discrete sets of option prices, the paper suggests a numerical
procedure based on an interpolation-extrapolation, curve-fitting scheme of implied by option prices
volatilities. Interpolation is used between the observed strike prices interval, while extrapolation
is carried out for those which are unobserved. To control for the approximation errors of this
numerical method, the paper derives upper bounds of these errors. These may be proved very
useful, in practice, to assess the relative magnitude of approximation errors encountered when
retrieving RNM and/or expected quadratic variation from option price data.
The performance of our numerical method is assessed through an extensive Monte Carlo exercise,
assuming that the stochastic volatility with jumps (SVJ) model characterizes our data. The results
of this exercise clearly indicate that this method can be successfully employed to retrieve efficient
estimates of RNM of any order and expected quadratic variation from option price data. This
is true even for strike price intervals of small length, which are often available in practice, and
independently on the number of call or put prices. They also indicate that calculating the expected
quadratic variation based on the CBOE’s VIX formula can lead to downward biased estimates of it,
if the underlying asset price process includes a jump component. This bias matches the estimates
15Note that this can be thought of as a conservative estimate of , given that a non-extrapolation scheme is
employed. If we add the unspecified integral parts of higher-order RNM, controlling for the truncation error , to
this estimate then the value of will increase.
24
of the model-free approximation function of the jump component term of quadratic variation,
suggested by the paper.
In an empirical application, the paper shows that the jump component term of expected
quadratic variation is significant and varies substantially across time, especially during financial
crises. In such periods, it is shown that the CBOE estimates of the VIX are substantially under-
valued. This finding has important asset pricing implications. It means that the VIX futures
contracts written on this index can be severely undervalued, due to the jump component bias.
A Appendix
In the Appendix we provide proofs of the main theoretical results of the paper.
A.1 Proof of Proposition 1
To prove Proposition 1, we are based on the result that, for a twice continuously differentiable
function (·) of the payoff (0 ), the following result holds:
[ (0 )]
= (0) + 0(0)[ (0 )−0] +
+
Z +∞
0
00()[ (0 )−]+ +
Z 0
0
00()[ − (0 )]+, (18)
for some constant 0 (see Bakshi and Madan (2000) and Carr and Madan (2001)).
Suppose that : R→ C : (0 ) 7−→ [(0 )], ∀ ∈ R. Then, equation (18) implies
[(0 )]
= (0) + 0(0)(0)( (0 )−0) +
+
Z +∞
0
h00()() − 20()2()
i[ (0 )−]+ +
+
Z 0
0
h00()() − 20()2()
i[ − (0 )]+.
Multiplying both sides of the last relationship with −(0)− and taking the conditional expec-
25
tation with respect to measure yields
−
n[((0 ))−(0)]
o= − + −0(0)
[ (0 )−0] +
+−
½Z +∞
0
£00()− 20()2
¤[()−(0)][ (0 )−]+
¾+−
½Z 0
0
£00()− 20()2
¤[()−(0)][ − (0 )]+
¾.
Cancelling out − from both sides of the last relationship proves (3), where the RNCF () is
defined as () =
©[((0 ))−(0)]ª.
A.2 Proof of Corollary 1
To prove this corollary, first notice that from the call and put option price relationships (1) and (2),
as well as the call-put arbitrage parity relationship ( )−( ) = −³ [ (0 )]−
´,
we can obtain the following equilibrium boundary conditions for the call and put prices:
lim→∞
( ) = 0
lim→1
( )− ( ) = 0 (19)
lim→0
( ) = 0
and
lim→∞
( )
= 0
lim→1
( )
− ( )
= −− (20)
lim→0
( )
= 0.
26
Next, write the RNCF given by (3) for () = and 0 = [ (0 )] as follows:
()
= 1− 2Z +∞
1
(−1)( ) − 2Z 1
0
(−1)( )
= 1 + Z +∞
1
2(−1)
2( ) +
Z 1
0
2(−1)
2( )
= 1 +
⎧⎨⎩"(−1)
( )
#+∞1
−Z +∞
1
(−1)
( )
⎫⎬⎭+
("(−1)
( )
#10
−Z 1
0
(−1)
( )
)
= 1 + ∙lim
→∞(−1)( )− lim
→1
(−1)(( )− ( ))
− lim→0
(−1)( )¸−
"Z +∞
1
(−1)
( )
+
Z 1
0
(−1)
( )
#.
Using the set of arbitrage boundary conditions (19) and the following equality¯(−1)
¯= 1,
the above relationship for () can be written as follows:
()
= 1−
(∙(−1)
( )
¸+∞1
−Z +∞
1
(−1)2( )
2
)−
−½∙
(−1)( )
¸10
−Z 1
0
(−1)2( )
2
¾= 1−
½lim
→∞(−1)
( )
− lim→1
(−1)∙( )
− ( )
¸− lim
→0(−1)
( )
¾+
Z +∞
1
(−1)2( )
2 + (−)
Z 1
0
(−1)2( )
2.
27
Given (20), the last relationship can be written in a more compact way as
() = Z +∞
1
(−1)2( )
2 +
Z 1
0
(−1)2( )
2, (21)
where () =
n[(0 )−1]
o. Based on (21), we can derive the RNCF of (0 ) as
() =
Z +∞
0
∙11
2( )
2+ 161
2( )
2
¸.
The last relationship implies Breeden-Litzenberger’s (1978) RND formula of payoff (0 ), given
by equation (4).
A.3 Proof of Proposition 2
To derive the non-central RNM formulas given by Proposition 2, first write the RNCF () in
the form of a Taylor’s series expansion as
()
= 1 + 0(0)³ [ (0 )]−0
´+
+Z +∞
0
£00()− 20()2
¤ ∞X=0
[(()− (0))]
!( )
+Z 0
0
£00()− 20()2
¤ ∞X=0
[(()− (0))]
!( ),
where [()−(0)] =P∞
=0([()−(0)])
!. After algebraic manipulations, the last equation can
be rewritten as follows:
()
= 1 + n0(0)
h ( (0 ))−0
i+
Z +∞
0
00()( ) +
Z 0
0
00()( )¾
+
∞X=2
½Z +∞
0
(;0)( ) +
Z 0
0
(;0)( )
¾()
!,
28
with (;0) = [()− (0)]−2 [00()(()− (0)) + (− 1)0()2].
Based on the last relationship, we can derive equations (5) and (6) of Proposition 2. This
can be done based on the moment representation of a characteristic function, i.e. () =P∞=0
()
!, where is the th-order non-central moment of payoff ().
A.4 Proof of Proposition 3
To prove Proposition 3, first note that applying the Ito lemma for semimartingales (see Protter
(1990)) to function = we have:
= − +1
2− hi +
µ − − − −∆ − 1
2− (∆)
2
¶,
which implies
1
2 hi =
−− +
µ∆ +
1
2(∆)
2 − − −−
¶. (22)
Integrating the last formula yields
1
2hi =
=
Z
+
−− ln
µ
¶+
X6
µ∆ +
1
2(∆)
2 − − −−
¶. (23)
Taking the conditional risk neutral expectations of equation (23) and, then, multiplying both sides
of the resulting equation with 2 yields
1
hhi
i=
2
∙Z
+
−
¸− 2
∙ln
µ
¶¸+
+2
⎧⎨⎩ X6
µ∆ +
1
2(∆)
2 − − −−
¶⎫⎬⎭ .
29
Substituting the first-order risk-neutral moment, given by formula (8), into the last formula and
rearranging terms yields
1
hhi
i=
2
∙Z +∞
1
2( ) +
Z
0
1
2( )
¸+
+2e
∙Z
+
−− −
¸+
+2
⎧⎨⎩ X6
µ∆ +
1
2(∆)
2 − − −−
¶⎫⎬⎭ ,which proves formula (12).
A.5 Proof of Proposition 4
To prove Proposition 4, first note that applying the Taylor’s series expansion we have the following
result:
∆ +1
2(∆)
2 − − −−
= ∆ +1
2(∆)
2 + 1− ∆ = −∞X=3
(∆)
!,
Integrating the last formula and taking risk neutral expectations, given that condition (14) holds,
yields
= −2
∞X=3
1
!
⎡⎣ X6
(∆)
⎤⎦ '' −2
⎛⎝ 13!
⎡⎣ X6
(∆)3
⎤⎦+ 1
4!
⎡⎣ X6
(∆)4
⎤⎦⎞⎠ , (24)
after multiplying with 2 . Applying Ito’s lemma to functions 3 and 4 and, then, taking risk
neutral expectations yields:
⎡⎣ X6
(∆)3
⎤⎦ = 3 − 3
∙Z
+
( −)2
¸− 3
∙Z
+
( −) hi¸
30
and
⎡⎣ X6
(∆)4
⎤⎦ = 4 − 4
∙Z
+
( −)3
¸− 6
∙Z
+
( −)2 hi
¸
−4
⎡⎣ X6
( −) (∆)3
⎤⎦ respectively. Thus, terms
hP6 (∆)
3iand
hP6 (∆)
4ican be approximated
by third and fourth-order RNM 3 and 4, respectively. Substituting the last two approximated
formulas into (24) yields formula (15).
A.6 Proof of Proposition 5
To prove this proposition, first notice that, for 1 2 ∈ R and ∈ N+, we haveZ 2
1
−2−(− 1− ) = −12 −2 − −11 −1 . (25)
We will first derive the bounds of the approximation error of our numerical method for implementing
the risk neutral moment 1 and the approximated expected quadratic variation, denoted as ∗.
This error can be written as
¯b1 − 1¯
= ¯Z ∞
1
2
³ b( )− ( )´ +
Z
0
1
2
³ b( )− ( )´−
−Z +∞
∞
1
2( ) −
Z 0
0
1
2( )
¯6
∙Z ∞
1
2
¯ b( )− ( )¯ +
Z
0
1
2
¯ b( )− ( )¯ + 1
¸,
where
1 =
¯Z +∞
∞
1
2( ) +
Z 0
0
1
2( )
¯
31
is the truncation error. The above relationship can be rewritten as follows:
¯b1 − 1¯
6 ∙Z ∞
1
2
¯ b( )− ( )¯ +
Z
0
1
2
¯ b( )− ( )¯ + 1
¸6
∙
Z ∞
1
2 +
Z
0
1
2 + 1
¸=
=
((∞; 1) + (0; 1) + 1) , (26)
where
= max∈(0∞)
¯( )− b( )
¯and = max
∈(00)
¯( )− b( )¯
and
(; 1) = 1− − and (; 1) = − − 1,
with 0 = ln(0) and ∞ = ln(∞). Based on the last inequality, it is straightforward to
show that the approximation error of the estimated by our method expected quadratic variation is
given as ¯d ∗ − ∗¯6 2
((∞; 1) + (0; 1) + 1) .
To derive upper bounds for the approximation errors of our method to retrieve higher than the
first-order RNM, first notice that the moment-specific function (;0) can be written as
(;0) =
2
∙ln
µ
¶¸−2 ∙− 1− ln
µ
¶¸
and
¯b − ¯
= ¯Z ∞
(;0)³ b( )− ( )
´+
+
Z
0
(;0)³ b( )− ( )
´ −
−Z +∞
∞ (;0)( ) −
Z 0
0
(;0)( )
¯
32
6 ∙Z ∞
| (;0)|¯ b( )−( )
¯+
+
Z
0
| (;0)|¯ b( )− ( )
¯ +
¸
where
=
¯Z +∞
∞ (;0)( ) +
Z 0
0
(;0)( )
¯
The above relationship for¯b −
¯becomes
¯b − ¯
6 ∙Z ∞
| (;0)|¯ b( )− ( )
¯+
+
Z
0
| (;0)|¯ b( )− ( )
¯ +
¸
=
∙Z ∞
0
| (;)|¯ b( )− ( )
¯ +
Z 0
0
| (;)|¯ b( )− ( )
¯ +
¸,
where = ln() and
(;) = −2−(− 1− ),
which implies the following relationship:
¯b − ¯6
∙
Z ∞
0
| (;)| +
Z 0
0
| (;)| +
¸(27)
To complete the proof of the proposition, we need to derive analytic formulas for the two integrals
appeared in the last formula. For the first integral, if we assume that − 1 6 ∞, then we haveZ ∞
0
| (;)| =
Z −1
0
| (;)| +Z ∞
−1| (;)|
=
Z −1
0
−2−(− 1− ) −
Z ∞
−1−2−(− 1− )
= (− 1)−1−(−1) −h−1∞ −(−1) − (− 1)−1−(−1)
i= 2(− 1)−1−(−1) −−1∞ −∞
33
by equation (25). If − 1 ∞, then we haveZ ∞
0
| (;)| =
Z ∞
0
−2−(− 1− ) = −1∞ −∞
also by equation (25). The above results implies that the analytic formula ofR ∞0| (;)| is
given as Z ∞
0
| (;)| = (∞;) (28)
with
(;) =
⎧⎨⎩ (2(− 1)−1−(−1) − −1−) if − 1 6
−1− if − 1 for > 2
The analytic formula of the second integral in equation (27) is given as
Z 0
0
| (;)| = (−1)−2Z 0
0
−2−(− 1− ) = (−1)−1−10 −0
or Z 0
0
| (;)| = (0;) (29)
where
(;) = (−1)−1−1− for > 2.
Substituting equations (28) and equations (29) into equation (27), we obtain:
¯b − ¯6
((∞;) + (0;) + ) for ∈ N+
which proves Proposition 5.
34
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37
Table 1: Risk neutral moments estimates
(extrapolation scheme with constant function)
1 2 3 4 5 6
Theoretical 0.0012 0.0035 -0.0004 0.00016 -0.000056 0.000026
Panel A: = 5 = 7
Panel A1: [minmax] = [0867 1046]
Estimates 0.0013 0.0033 -0.00024 0.00007 -0.000013 318× 10−6
RMSE 0.00011 0.00028 0.00015 0.000096 0.000043 0.000023
RSB 0.00011 0.00028 0.00015 0.000096 0.000043 0.000023
RV 0.00001 0.00002 311× 10−6 105× 10−6 295× 10−7 85× 10−8
BD 0.000313 0.000732 0.000404 0.000212 0.000103 0.000051
Panel A2: [minmax] = [0735 1118]
Estimates 0.0012 0.0035 -0.00039 0.00014 -0.000043 0.000015
RMSE 0.00002 0.00004 0.000014 0.000024 0.000012 0.00001
RSB 0.000017 0.000037 0.000014 0.000024 0.000012 0.00001
RV 0.00001 0.00002 349× 10−6 165× 10−6 623× 10−7 279× 10−7
BD 0.000076 0.000174 0.000116 0.000081 0.000052 0.000035
(Continued)
38
Panel A3: [minmax] = [0596 1279]
Estimated 0.0012 0.0035 -0.0004 0.00016 -0.000054 0.000024
RMSE 0.000014 0.00003 515× 10−6 323× 10−6 173× 10−6 206× 10−6
RSB 757× 10−6 0.000015 321× 10−7 248× 10−6 157× 10−6 203× 10−6
RV 0.000012 0.000026 514× 10−6 207× 10−6 707× 10−7 363× 10−7
Bound 0.000179 0.000467 0.000447 0.000386 0.000315 0.000248
(Continued)
39
Panel B: = 11 = 18
Panel B1: [minmax] = [0867 1046]
Estimates 0.0013 0.0033 -0.00024 0.00007 -0.000013 318× 10−6
RMSE 0.00011 0.00028 0.00015 0.000096 0.000043 0.000023
RSB 0.00011 0.00028 0.00015 0.000096 0.000043 0.000023
RV 698× 10−6 0.000014 283× 10−6 925× 10−7 276× 10−7 81× 10−8
BD 0.000313 0.000732 0.000404 0.000212 0.000103 0.000051
Panel B2: [minmax] = [0735 1118]
Estimates 0.0012 0.0035 -0.00039 0.00014 -0.000043 0.000015
RMSE 0.00002 0.00004 0.000014 0.000024 0.000012 0.00001
RSB 0.000019 0.000041 0.000014 0.000024 0.000012 0.00001
RV 712× 10−6 0.000014 225× 10−6 111× 10−6 467× 10−7 223× 10−7
BD 0.000060 0.000133 0.000082 0.000056 0.000036 0.000024
Panel B3: [minmax] = [0596 1279]
Estimates 0.0012 0.0035 -0.0004 0.00016 -0.000054 0.000024
RMSE 856× 10−6 0.000017 321× 10−6 279× 10−6 163× 10−6 204× 10−6
RSB 455× 10−7 946× 10−7 668× 10−7 249× 10−6 157× 10−6 203× 10−6
RV 855× 10−6 0.000017 313× 10−6 124× 10−6 434× 10−7 228× 10−7
BD 0.000022 0.000056 0.000053 0.000046 0.000038 0.000030
(Continued)
40
Panel C: = 21 = 34
Panel C1: [minmax] = [0867 1046]
Estimates 0.0013 0.0033 -0.00024 0.00007 -0.000013 318× 10−6
RMSE 0.00011 0.00028 0.00015 0.000096 0.000043 0.000023
RSB 0.00011 0.00028 0.00015 0.000096 0.000043 0.000023
RV 591× 10−6 0.000011 269× 10−6 899× 10−7 269× 10−7 80× 10−8
Bound 0.000313 0.000732 0.000404 0.000212 0.000103 0.000051
Panel C2: [minmax] = [0735 1118]
Estimates 0.0012 0.0035 -0.00039 0.00014 -0.000043 0.000015
RMSE 0.00002 0.00004 0.000014 0.000024 0.000012 0.00001
RSB 0.000019 0.000041 0.000014 0.000024 0.000012 0.00001
RV 516× 10−6 0.00001 166× 10−6 902× 10−7 411× 10−7 205× 10−7
BD 0.000060 0.000133 0.000082 0.000056 0.000036 0.000024
(Continued)
41
Panel C3: [minmax] = [0596 1279]
Estimates 0.0012 0.0035 -0.0004 0.00016 -0.000054 0.000024
RMSE 612× 10−6 0.000012 227× 10−6 267× 10−6 161× 10−6 205× 10−6
RSB 674× 10−7 136× 10−6 619× 10−7 252× 10−6 158× 10−6 204× 10−6
RV 609× 10−6 0.000012 218× 10−6 887× 10−7 313× 10−7 169× 10−7
BD 817× 10−6 0.000019 0.000017 0.000014 0.000011 912× 10−6Notes: This table presents the average values of estimates of the first six non-central risk neutral moments of ( ), denoted as for
= 1 2 6, based on our suggested method for implementing these formulas over 1000 Monte Carlo iterations. It also gives the RMSE
(root mean square error), RSB (root square bias) and RV (root variance) values of these estimates. BD denotes the average value of the
approximation error bounds given by equations (16) over the 1000 iterations. The extrapolation scheme used by our method is based on a
constant function. The theoretical values of the risk neutral moments reported in the table are implied by the SVJ. This model is used to
generate option prices in our simulation study.
42
Table 2: Risk neutral moments estimates
(extrapolation scheme with linear function)
1 2 3 4 5 6
Theoretical 0.0012 0.0035 -0.0004 0.00016 -0.000056 0.000026
Panel A: = 5 = 7
Panel A1: [minmax] = [0867 1046]
Estimates 0.0011 0.0037 -0.00068 0.0004 -0.00028 0.00025
RMSE 0.000067 0.00021 0.00031 0.00027 0.00027 0.00027
RSB 0.00005 0.00017 0.00027 0.00023 0.00023 0.00022
RV 0.00004 0.00012 0.00014 0.00014 0.00014 0.00015
BD 0.000460 0.001543 0.002823 0.004820 0.007769 0.012036
Panel A2: [minmax] = [0735 1118]
Estimates 0.0012 0.0035 -0.00041 0.00018 -0.000065 0.000034
RMSE 0.000012 0.000027 0.000016 0.000014 0.00001 944× 10−6
RSB 398× 10−6 0.000011 0.000014 0.000012 954× 10−6 867× 10−6
RV 0.000012 0.000024 807× 10−6 607× 10−6 489× 10−6 375× 10−6
BD 0.000106 0.000298 0.000376 0.000431 0.000469 0.000492
(Continued)
43
Panel A3: [minmax] = [0596 1279]
Estimates 0.0012 0.0035 -0.0004 0.00016 -0.000054 0.000026
RMSE 0.000015 0.000032 532× 10−6 246× 10−6 174× 10−6 483× 10−7
RSB 941× 10−7 0.000018 170× 10−6 118× 10−6 153× 10−6 20× 10−7
RV 0.000012 0.000026 514× 10−6 207× 10−6 707× 10−7 363× 10−7
BD 0.000196 0.000556 0.000632 0.000640 0.000607 0.000553
(Continued)
44
Panel B: = 11 = 18
Panel B1: [minmax] = [0867 1046]
Estimates 0.0011 0.0038 -0.00085 0.00063 -0.00056 0.00059
RMSE 0.00018 0.00054 0.00073 0.00081 0.00094 0.0011
RSB 0.000093 0.0003 0.00045 0.00046 0.00051 0.00057
RV 0.00015 0.00045 0.00057 0.00067 0.00079 0.00095
BD 0.000436 0.001454 0.002638 0.004466 0.007139 0.010969
Panel B2: [minmax] = [0735 1118]
Estimates 0.0012 0.0035 -0.0004 0.00018 -0.000062 0.000035
RMSE 0.000018 0.000032 0.000023 0.000024 0.000013 0.000013
RSB 864× 10−6 0.000016 239× 10−6 0.000016 577× 10−6 898× 10−6
RV 0.000015 0.000027 0.000023 0.000017 0.000012 962× 10−6
BD 0.000020 0.000049 0.000053 0.000055 0.000055 0.000055
Panel B3: [minmax] = [0596 1279]
Estimates 0.0012 0.0035 -0.0004 0.00016 -0.000054 0.000026
RMSE 864× 10−6 0.000017 389× 10−6 231× 10−6 202× 10−6 999× 10−7
RSB 126× 10−6 179× 10−6 185× 10−6 121× 10−6 154× 10−6 374× 10−7
RV 855× 10−6 0.000017 342× 10−6 197× 10−6 130× 10−6 926× 10−7
BD 0.000035 0.000083 0.000089 0.000087 0.000080 0.000071
(Continued)
45
Panel C: = 21 = 34
Panel C1: [minmax] = [0867 1046]
Estimates 0.00091 0.0043 -0.0016 0.0017 -0.0021 0.0026
RMSE 0.00056 0.00179 0.00276 0.00376 0.00504 0.00676
RSB 0.00027 0.00084 0.00124 0.00160 0.00205 0.00264
RV 0.00049 0.00158 0.00246 0.00341 0.00461 0.00622
BD 0.000426 0.001420 0.002567 0.004329 0.006898 0.010564
Panel C2: [minmax] = [0735 1118]
Estimates 0.0011 0.0035 -0.00039 0.00020 -0.000061 0.000044
RMSE 0.000051 0.000081 0.000071 0.000059 0.000034 0.000031
RSB 0.000026 0.000044 0.000014 0.000036 533× 10−6 0.000018
RV 0.000044 0.000068 0.000069 0.000047 0.000034 0.000025
BD 0.000019 0.000042 0.000041 0.000040 0.000039 0.000039
Panel C3: [minmax] = [0596 1279]
Estimates 0.0012 0.0035 -0.0004 0.00017 -0.000053 0.000027
RMSE 670× 10−6 0.000012 609× 10−6 568× 10−6 492× 10−6 341× 10−6
RSB 172× 10−6 212× 10−6 310× 10−6 281× 10−6 273× 10−6 151× 10−6
RV 648× 10−6 0.000012 524× 10−6 493× 10−6 409× 10−6 306× 10−6
BD 948× 10−6 0.000023 0.000025 0.000024 0.000022 0.000020
Notes: The reported values in this table correspond to those in Table 1 using a linear function in the extrapolation scheme.
46
Table 3: Risk neutral moments estimates
(non-extrapolation scheme)
1 2 3 4 5 6
Theoretical 0.0012 0.0035 -0.0004 0.00016 -0.000056 0.000026
Panel A: = 5 = 7
Panel A1: [minmax] = [0867 1046]
Estimates 0.001460 0.002936 -0.000206 0.000041 −592× 10−6 962× 10−7
RMSE 0.000271 0.000599 0.000199 0.000126 0.000050 0.000025
RSB 0.000271 0.000599 0.000199 0.000126 0.000050 0.000025
RV 839× 10−6 0.000017 177× 10−6 405× 10−7 697× 10−8 123× 10−8
BD 0.000274 0.000606 0.000200 0.000127 0.000050 0.000025
Panel A2: [minmax] = [0735 1118]
Estimates 0.001222 0.003461 -0.000369 0.000126 -0.000034 0.000011
RMSE 0.000034 0.000078 0.000036 0.000042 0.000022 0.000015
RSB 0.000033 0.000074 0.000036 0.000042 0.000022 0.000015
RV 0.000011 0.000023 345× 10−6 143× 10−6 437× 10−7 156× 10−6
BD 0.000061 0.000138 0.000062 0.000052 0.000026 0.000017
(Continued)
47
Panel A3: [minmax] = [0596 1279]
Estimates 0.001183 0.003548 -0.000403 0.000163 -0.000053 0.000023
RMSE 0.000014 0.000029 535× 10−6 505× 10−6 301× 10−6 337× 10−6
RSB 666× 10−6 0.000013 130× 10−6 461× 10−6 293× 10−6 335× 10−6
RV 0.000013 0.000026 519× 10−6 206× 10−6 704× 10−7 350× 10−7
BD 0.000132 0.000325 0.000240 0.000166 0.000106 0.000067
(Continued)
48
Panel B: = 11 = 18
Panel B1: [minmax] = [0867 1046]
Estimates 0.001461 0.002936 -0.000206 0.000041 −592× 10−6 962× 10−7
RMSE 0.000272 0.000600 0.000199 0.000126 0.000050 0.000025
RSB 0.000271 0.000599 0.000199 0.000126 0.000050 0.000025
RV 515× 10−6 0.000011 107× 10−6 254× 10−7 438× 10−8 789× 10−9
BD 0.000272 0.000600 0.000199 0.000126 0.000050 0.000025
Panel B2: [minmax] = [0735 1118]
Estimates 0.001224 0.003457 -0.000369 0.000126 -0.000034 0.000011
RMSE 0.000035 0.000079 0.000036 0.000042 0.000022 0.000015
RSB 0.000035 0.000078 0.000036 0.000042 0.000022 0.000015
RV 0.000007 0.000015 209× 10−6 878× 10−7 270× 10−7 982× 10−8
BD 0.000038 0.000087 0.000040 0.000043 0.000023 0.000015
Panel B3: [minmax] = [0596 1279]
Estimates 0.001191 0.003532 -0.000403 0.000163 -0.000053 0.000023
RMSE 866× 10−6 0.000018 356× 10−6 479× 10−6 296× 10−6 336× 10−6
RSB 137× 10−6 294× 10−6 164× 10−6 463× 10−6 292× 10−6 336× 10−6
RV 855× 10−6 0.000018 316× 10−6 123× 10−6 428× 10−7 217× 10−7
BD 0.000021 0.000049 0.000033 0.000025 0.000015 0.000011
(Continued)
49
Panel C: = 21 = 34
Panel C1: [minmax] = [0867 1046]
Estimates 0.001461 0.002936 -0.000206 0.000041 −592× 10−6 962× 10−7
RMSE 0.000272 0.000600 0.000199 0.000126 0.000050 0.000025
RSB 0.000272 0.000599 0.000199 0.000126 0.000050 0.000025
RV 366× 10−6 754× 10−6 762× 10−7 181× 10−7 319× 10−8 577× 10−9
BD 0.000272 0.000600 0.000199 0.000126 0.000050 0.000025
Panel C2: [minmax] = [0735 1118]
Estimates 0.001224 0.003457 -0.000369 0.000126 -0.000034 0.000011
RMSE 0.000035 0.000079 0.000036 0.000042 0.000022 0.000015
RSB 0.000035 0.000078 0.000036 0.000042 0.000022 0.000015
RV 505× 10−6 0.000010 0.000001 624× 10−6 195× 10−6 711× 10−6
BD 0.000036 0.000081 0.000037 0.000042 0.000022 0.000015
(Continued)
50
Panel C3: [minmax] = [0596 1279]
Estimates 0.001191 0.003532 -0.000403 0.000163 -0.000053 0.000023
RMSE 629× 10−6 0.000013 2.72E-06 473× 10−6 295× 10−6 337× 10−6
RSB 159× 10−6 337× 10−6 159× 10−6 465× 10−6 294× 10−6 336× 10−6
RV 608× 10−6 0.000012 220× 10−6 875× 10−7 306× 10−7 155× 10−7
BD 718× 10−6 0.000016 0.000010 0.000010 636× 10−6 546× 10−6Notes: The reported values in this table correspond to those in Table 1 using a non-extrapolation scheme.
51
Table 4: Estimates of the expected quadratic variation and its jump component (Simulation study)
∗ ∗ ∗
Theoretical 0.0422 0.0408 0.0014 0.0422 0.0408 0.0014 0.0422 0.0408 0.0014
[minmax] = 5 = 7 = 11 = 18 = 21 = 34
[0867 1046] Estimates 0.0351 0.0344 0.0008 0.0351 0.0343 0.0008 0.0351 0.0343 0.0008
BD 0.0075 0.0066 0.0009 0.0074 0.0065 0.0009 0.0074 0.0065 0.0009
[0735 1118] Estimates 0.0414 0.0401 0.0013 0.0414 0.0400 0.0013 0.0414 0.0400 0.0013
BD 0.0018 0.0015 0.0003 0.0011 0.0009 0.0002 0.0010 0.0009 0.0002
[0596 1279] Estimates 0.0425 0.0410 0.0014 0.0423 0.0408 0.0014 0.0423 0.0408 0.0014
BD 0.0043 0.0032 0.0011 0.0007 0.0005 0.0002 0.0002 0.0002 0.0001
Notes: This table presents average values of estimates of the expected quadratic variation ignoring the jump component of the underlying
stock price (denoted as ∗) over 1000 iterations, the jump component of this variation approximated by formula (15), as well as the sum
of ∗ and defined as = ∗+ . BD denotes the average value of the approximation error bounds of , ∗ and implied by
the results of Proposition 5. The theoretical values of , ∗ and reported in the table are implied by the SVJ, used to generate option
prices in our simulation study. The numerical method employed to implement the above formulas to the data assumes a non-extrapolation
scheme.
52
Table 5: Estimates of expected quadratic variation and its jump component
∗ 3 4
00397
(1149)
−000009(−501)
000003
(334)
000033
(524)
BD 0.0159 0.00052 0.00029 0.0024
Truncation error (%) 94.76% 97.73% 99.60% 97.92%
Notes: This table presents average values of estimates of the expected quadratic variation ignoring the
jump component of the underlying stock price (denoted as ∗), the third and fourth-order RNM, and the
jump component of quadratic variation, approximated by formula (15). These estimates rely on our
numerical method, assuming a non-extrapolation scheme. BD denotes the average value of the error bounds
of ∗ and , implied by the formulas of Proposition 5. The reported values of BD are obtained by
fitting the SVJ into our option price data. Truncation error (%) constitutes the percentage of BD due
to truncation error . The sample period is January 1996 to October 2009.
53
Figure 1: The function (;) for different values of .
54
02101995 0281999 0262003 0242007
0.0
0.1
0.2
0.3
0.4
0.5
IV*`IV*` ≤ bound
Figure 2: Estimates of d ∗ and d ∗± bound, January 1996-October 2009.
55
02101995 0281999 0262003 0242007
0.0
0.1
0.2
0.3
0.4
0.5
Figure 3: Estimates of the jump component , January 1996-October 2009.
56
