Model-Free Risk-Neutral Moments and Proxies
Zhangxin (Frank) Liua,1
a
Business School, The University of Western Australia, Perth, WA, Australia
Abstract
Estimation of risk-neutral (RN) moments is of great interest to both academics and practitioners. We study 1) the model-free measure of RN moments by Bakshi, Kapadia and Madan
(2003); 2) RN moments that are used in the VIX and SKEW index by the Chicago Board
Options Exchange; 3) nonparametric RN moments that are calculated as the difference of implied volatilities across moneyness levels; and 4) the level, slope and curvature of the implied
volatility smirk. More specifically, we investigate the estimation procedure by examining the
consequence of directly using raw option data versus applying various smoothing methods to
the option data. In the simulation study, we study estimation errors arise from integration
truncation, discreteness of strike prices and asymmetric truncation. We show that applying
smoothing methods reduces the estimation errors of true moments but the size and direction
of estimation errors are largely unquantifiable. In the empirical study, we find that applying
smoothing methods increases the Kendall and Spearman rank correlations among RN moment
estimates. We conduct a case study that examines the relationship between RN skewness and
future realised stock returns from 1996 to 2014. We show that a strategy that is long the quintile portfolio with the lowest RN skewness stocks yields a negative and significant Fama-French
Five-Factor alpha. This finding is robust across all RN skewness measures.
Keywords: Risk-Neutral Moments, Skewness, Kurtosis, Implied Volatility Smirk, Skew,
Curvature, VIX
1
Email address: [email protected] (Zhangxin (Frank) Liu )
First Version: August 10, 2015. Work in progress and incomplete. Comments are welcome.
Preprint submitted to SSRN
August 10, 2015
1. Introduction
Bakshi, Kapadia and Madan (2003, hereinafter, BKM) provide a model-free measure of
risk-neutral (RN) volatility, skewness and kurtosis that can be inferred from traded options.
Building on the work by Breeden and Litzenberger (1978), Bakshi and Madan (2000) and
Carr and Madan (2001), BKM’s approach supplies a new tool to estimate RN moments and
has received increasing popularity in empirical studies. The primary goal of this paper is to
investigate the implementation issues in applying their methods. We compare the accuracy in
using the raw and different smooth methods to interpolate option prices in implementing BKM
method, alongside with several other nonparametric RN moment estimates.
BKM’s approach to compute moments of the RN distribution relies on three sets of conditions: 1) the existence of a continuum of strike prices for the underlying security in a given
maturity; 2) the strike price range spans from zero to infinity; and 3) the option is a European option. There are several difficulties with inferring model-free RN moments using this
approach. From the traded options in the market, we do not observe a continuum of strike
prices. In particular, we often see an unequal range of out-of-the-money (OTM) put strikes and
OTM call strikes and the difference can be substantial following a large price moment in the
underlying security. The second condition is also not met because there only exist discretely
spaced strike prices. For the third condition, it does not raise any issues if the main subject
of study is on European options. In the case of American options, which are common among
equity options, the issue may be mitigated if the early exercise premium could be estimated.
In this paper, we limit our discussion to the first two conditions.
The literature in BKM application does not seem to have reached consensus on how to
deal with the first two conditions. Our study is largely motivated by the disagreement in how
observed option prices should be treated when implementing the BKM method. We summarise
a subset of studies that have used BKM method and their corresponding approach in treating the traded option prices in Table 1. In this table, the column “Stock/Index Options”
shows the main type of options that are used to implement BKM method. Most stock options are American style and the majority of index options are European style. The column
“Raw/Smooth” refers to whether the traded option prices are directly used, or the option prices
have been interpolated and extrapolated using some particular method before been applied in
BKM formulas.
[Table 1 about here.]
Dennis and Mayhew (2002) is among the first to apply BKM method to study RN skewness
from stock options. In their study, they discuss biases from the discreteness of the strike price
interval and asymmetry in the domain of integration. Leaning on their simulation study using
Black-Scholes option prices, their approach to combat issues from the first two conditions is to
filter out options without a minimum of two OTM puts and two OTM calls in each maturity.
Because they use the market option prices directly without any interpolation and extrapolation,
we refer this as a raw approach. A number of studies follow this raw approach and the rule
of thumb by Dennis and Mayhew, including Han (2008), Duan and Wei (2009), Conrad et al.
(2013) and most recently Bali et al. (2015).
We refer an approach that interpolates and extrapolates market option prices as a smooth
approach. Since the first two conditions also challenge the RN density recovery from observed
2
option prices, there is a rich literature (e.g. Shimko, 1993; Jackwerth and Rubinstein, 1996;
Figlewski, 2008) that can be borrowed when implementing BKM method. There are two main
steps in a smooth approach, 1) interpolation between the OTM put with the lowest strike and
the OTM call with the highest strike; and 2) extrapolation beyond the highest and lowest strike
price to recover both tails.
The literature in BKM applications also seems divided in how to interpolate and extrapolate. We first discuss the interpolation procedure. Christoffersen et al. (2008) interpolate
implied volatilities (IV) using a cubic spline across moneyness level, defines as K/S, to obtain
a continuum of IVs. They then convert IVs back to corresponding option prices. It is important to point out that the use of IV does not assume the validity of Black-Scholes model. The
IV is used as a transformation process to avoid arbitrage possibilities. Ait-Sahlia and Duarte
(2003) show that the volatility surface is corrected for arbitrage possibilities after being fit
with a cubic spline interpolation. Jiang and Tian (2007) study how to minimise discretisation
and truncation errors in the Chicago Board Options Exchange (CBOE) VIX calculation2 . The
authors propose a solution by interpolating implied volatilities of OTM puts and calls using
a natural cubic spline across strike prices (K) from the lowest OTM put to the highest OTM
call. Similar approach is adopted in Hansis et al. (2010), Buss and Vilkov (2012), Chang et al.
(2012), DeMiguel et al. (2013), among others.
Neumann and Skiadopoulos (2013) study the predictability in the dynamics of RN moments
from S&P 500 options. In their study, a different interpolation is done by fitting a cubic spline
across a delta grid with 1,000 points, where each delta is calculated using the at-the-money
(ATM) IV. As discussed in Figlewski (2008), applying a cubic spline in delta-IV space ensures
an IV function in delta is smooth up to second order in terms of the partial derivatives of
option prices, which is equivalent to fitting a fourth-degree spline in strike-IV space. That is,
Neumann and Skiadopoulos’ approach ensures a corresponding RN density is smooth up to the
third order in option price itself, while the approach by Christoffersen et al. (2008) ensures the
RN density is smooth up to the second order.
Engle and Mistry (2013) study skewness in priced risk factors and individual stocks. They fit
a quadratic spline with a knot at 0 of moneyness in IV-moenyness space, where the moneyness
√
and σ is measured from the historical monthly realised volatility. A
is defines as ln(K/S)−rT
σ T
more recent study by Stilger et al. (2015) considers yet another way and interpolates IV using
a piecewise Hermite polynomial separately for calls and puts across moneyness levels (K/S).
Contrary to variations seen in the interpolation process, the extrapolation beyond the highest and the lowest strike is less subject to deviation. A common approach is to assume a flat
structure in IV function (of different definitions of moneyness) beyond each boundary. That is,
the last known IV on each end is used to fill the rest of grids. This is adopted by all studies
listed in Table 1 which have considered a smooth approach.
Jiang and Tian (2007) discuss two drawbacks with this flat extrapolation scheme. The first
one is that it tends to underestimate the true IV given the observed volatility smile. Second,
the change in slope of the IV function leads to a kink at each end, which is associated with
negative local RN density and thus violates no-arbitrage conditions. They propose a smooth
2
Jiang and Tian (2007) is not included in Table 1 as technically speaking their study does not directly
implement BKM method.
3
pasting condition by matching the slopes of the extrapolated and interpolated segments.
Another interesting extrapolation technique is proposed by Figlewski (2008) . The author
uses a generalised extreme value distribution to extrapolate tails such that the shape of a
certain proportion of the tail density matches with that of the main RN density. However, as
the proportion of RN density on each end is arbitrarily set and lacks a theoretical ground on
how to be calibrated using market data, we find this is challenging to implement if the main
subject is individual stock option3 .
From the discussion above, it is clear to see a divergence exists in choosing the raw or a
smooth approach when implementing BKM method. When RN moment is an important factor
in an empirical study, however, the consequence from choosing either approach and how that
would have impact to the empirical findings remains largely undiscussed.
As an example, the disagreement in the relationship between the RN skewness and future
realised returns may shed some light on this matter. Conrad et al. (2013) implement a raw
BKM approach in estimating RN moments. They find a negative relationship between quarterly
averages of daily RN skewness estimates and subsequent realised quarterly stock returns. Bali
and Murray (2013) also adopt a raw BKM approach and create a portfolio of options that only
exposes to skewness effect. They find a negative relationship between RN skewness and option
portfolios’ returns. On the other hand, Rehman and Vilkov (2002) implement a smooth BKM
approach and document the ex ante skewness is positively related to future stock returns. This
finding is further supported by Stilger et al. (2015). The authors use a smooth BKM approach4
and document that a strategy to long the quintile portfolio with the highest RN skewness stocks
and short the quintile portfolio with the lowest RN skewness stocks on average yields a FamaFrench-Carhart alpha of 55 bps per month. As point out in Stilger et al., they attribute the
difference in their findings to the fact that the underperformance in the most negative skewness
stocks is driven by stocks that are too costly to short sell.
Our study is largely motivated by the disagreement in how observed option prices should
be treated when implementing the BKM method. We extend our analysis to investigate other
RN moment estimates and proxies, including 1) CBOE moments which are based on CBOE’s
methodology in calculating the VIX and SKEW index5 ; 2) nonparametric RN moments that
are estimated by taking differences of IVs of options at different moneyness levels, including a
variation discussed in Mixon (2011) that is superior to other nonparametric skewness measures;
and 3) the level, slope and curvature of the IV smirk as proxies for the RN volatility, skewness
and excess kurtosis, respectively.
Our motivation of including the aforementioned estimates is twofold. First, there is a
substantial amount of literature that shows the shape of the volatility smirk carries predictive
3
In Figlewski (2008) and Birru and Figlewski (2012), the authors recover RN densities from S&P 500 options.
They set generalise extreme value functions to match the proportion of a RN density for the moneyness levels
(K/S) between 0.02 and 0.05 on the left end, and between 0.92 and 0.95 on the right end. In the unpublished
note, we have experimented with S&P 500 options by matching different segments on tails across an 18-year
period from 1996 to 2014. We find that results of the tail shape can be distinctly different if the range of
available strike prices becomes narrow.
4
The authors find similar results by using the raw BKM approach as a robustness check.
5
Note that even though CBOE SKEW index is based on BKM method, the implementation in estimating one
of key parameters is slightly different from the main stream BKM applications. This will be further explained
in Section 2.
4
power for future equity returns and volatilities (e.g., see Mixon, 2009; Cremers and Weibaum,
2010). Going back to the discussion on the RN skewness and future return above, Xing et al.
(2010) document a positive relationship between skewness and future returns. In their study,
the main estimate of daily option implied skew is calculated as the difference between the IV
of OTM puts and ATM calls. The weekly skew is then obtained as an average of daily values.
Bali et al. (2015) demonstrate that ex-ante measure of skewness is positively related to ex-ante
expected returns. The authors’ primary estimates are BKM raw moments and they also use
nonparametric RN moments from differences of IVs at different strikes as a robustness check.
Our second rational is that as the estimation of these alternative measures is also subject to
the data availability issues6 , therefore it is important to investigate the difference in outcomes
between applying the raw or a smooth method.
The first and the most obvious question to be asked is: what is the difference between
implementing the raw and smooth approaches? In other words, hypothesizing these RN moment
estimates can theoretically recover the true moments, how large will the estimation errors be
when the availability of option prices vary and how will smooth approaches improve on the
result? To answer this question, as we do not observe the true moments from the market
- neither physical moments from return distribution nor RN moments implied from option
prices - we first conduct a control simulation experiment, where the true RN moments can be
estimated. Our candidates of RN moments include volatility, skewness and excess kurtosis.
We estimate and analyse the estimation errors against true moments. As one of the main
application with RN moment is to be used as a sorting mechanism (e.g. Conrad et al., 2013;
Stilger et al., 2015), we also calculate the Kendall and Spearman rank correlations among RN
moment estimates. Furthermore, we investigate the percentage of matching items in top and
bottom quintiles between RN moment estimates and the true moments.
Jiang and Tian (2007) and Chang et al. (2012) conduct similar studies to our first research
question. Comparing to Jiang and Tian, this paper extends the analysis to RN skewness and
kurtosis, as well as including investigations in other nonparametric RN moment estimates.
Chang et al. examine the accuracy of the BKM volatility and skewness computations. Their
simulation design is limited to using Heston (1993)’s stochastic volatility model with one set
of parameters as option price generation process. Our study extends the analysis to include
three other models as option price generation processes as well as nine sets of parameters to
represent various market conditions. Moreover, we investigate further in the accuracy issue and
our analysis in Kendall rank correlations provides an extension in studying the usefulness of
these RN estimates as sorting mechanisms.
The second research question is to investigate how the implementation of the raw and
smooth approaches empirically differ using traded option prices. With the absence of true
6
Strictly speaking, apart from the CBOE moments, even though the other measures do not require a continuum of strike prices as a necessary condition, the estimation still confronts with the availability issue. For
example, when calculating a nonparametric RN skewness as the difference between the IV of the 0.25 delta
call and that of the -0.25 delta put, it is common to see one needs a proxy for a 0.25 delta call as such option
with the exact delta does not exist. A few approaches can be considered: 1) replace the missing call with 0.25
delta with the closest call available; 2) a linear interpolation between two adjacent calls; and 3) a cubic spline
interpolation of the entire IV smirk to fill the missing call with 0.25 delta. The first approach can be viewed as
the raw approach, whereas the latter two can be treated as smooth approaches.
5
moments, we focus the investigation on the information content from the RN estimates. We
compare the Kendall and Spearman rank correlations and present the differences in the raw
measures and smooth measures.
We also conduct an empirical case study, where we study the relationship between the RN
skewness and future realised returns. We follow the research design in Stilger et al. (2015).
Using ten RN estimates (five of which are constructed using the raw approach and another five
using a smooth approach), we investigate the excess return performance in skewness-quintile
portfolios. From January 1996 to August 2014, we sort all stocks available from OptionMetrics
by each RN skewness estimate in ascending order on the last trading day of each month. We
compare three skewness-quintile portfolio strategies: 1) a long strategy in quintile 1 stocks
with RN skewness in the bottom 20th percentile; 2) a long strategy in quintile 5 stocks with
RN skewness in the 80th percentile; and 3) a long strategy in quintile 5 and a short strategy in
quintile 1 portfolio.
Our main findings can be easily summarised. First, in the simulation study, we show
that regardless of using the raw or smooth approaches, the point estimate of the true RN
moment is unstable under different conditions. More importantly, the estimation error does
not follow any particular patterns. The problem is more pronounced in skewness and excess
kurtosis. Despite the poor performance in point estimate, the design of the simulation study
allows us to show that smooth approaches increase the Kendall and Spearman rank correlations
between the RN estimates and true moments. The improvement is less for the higher moments.
Second, the finding in simulation study is confirmed by the empirical results. By applying a
smooth approach to trade option prices, the Kendall and Spearman correlations among RN
estimates increase. In other words, if RN estimate is used as a sorting mechanism for portfolio
construction, our result implies using a smooth approach increases the likelihood that a similar
portfolio composition is found across portfolios based on different RN estimates. Third, in the
empirical case study that examines the RN skewness and future realised returns, we show that
only the monthly excess return of the first strategy consistently yields a negative Fama-French
Five-Factor (Fama and French, 2015) alpha across all RN skewness estimates. Furthermore, we
study the monthly average RN volatility and kurtosis in these RN skewness-quintile portfolios.
We use the raw and a smooth approach in estimating the average RN volatility and kurtosis in
each portfolio. We illustrate that although the RN volatility differs numerically between raw
and smooth approach, the time-series behavior is similar across different portfolios. A similar
but weaker finding is presented in RN kurtosis.
Our paper is related to the discussion of higher-moments risk in asset pricing and investment management, and contributes to the literature in several ways. First, to the best of our
knowledge, this is the first study to examine the consequence of using raw and smooth approach
in calculating model-free RN estimates. Our results may provide an alternative explanation in
some mixing empirical findings regarding RN moments. Second, our empirical study is special
in terms of underlying data that we are able to use exchange-traded options data of more than
8,000 securities from OptionMetrics in the period between 1996 to 2014. This coverage enables
us to examine the strike price availability issue across different issue types, including stock
options, index options, options on exchange-traded funds (ETF), among others. Third, our
findings in the skewness-quintile portfolio study documents a consistent underperformance in
quintile 1 skewness portfolios, regardless of how the RN skewness is estimated. This may shed
some light on portfolio management with RN skewness.
6
The remainder of the paper is organized as follows. Section 2 describes the method to construct each RN measure. Section 3 conducts simulation studies to investigate the relationship
among RN estimates and true measures. Section 4 shows the data and presents the empirical
results. Section 5 concludes.
2. Methodology
2.1. BKM Risk-Neutral Moments
Bakshi and Madan (2000) articulate that any payouff function can be spanned and priced
using an explicit positioning across a continuum of option strikes. BKM demonstrate that the
RN annualised τ -period volatility, skewness and excess kurtosis of a security’s log return can
be obtained as7 :
r
2
EQ (R2 ) − EQ
(R)
VolBKM ≡
τ
r
erτ V − µ2
=
(1)
τ
3
(R)
EQ (R3 ) − 3EQ (R)EQ (R2 ) + 2EQ
SkewBKM ≡
2
2
3/2
(EQ (R ) − EQ (R))
rτ
e W − 3erτ µV + 2µ3
=
(2)
(erτ V − µ2 )3/2
2
4
EQ (R4 ) − 4EQ (R)EQ (R3 ) + 6EQ
(R)EQ (R2 ) − EQ
KurtBKM ≡
−3
2
(EQ (R2 ) − EQ
(R))2
erτ X − 4erτ µW + 6erτ µ2 V − 3µ4
=
−3
(3)
(erτ V − µ2 )2
where r represents the continuously compounded risk-free rate for the τ -period. Note that,
VolBKM is annualised as a standard convention. This is followed in the other volatility measures
in this paper. The risk-neutral expectation of the squared contract (V ), the cubed contract
(W ), the quartic contract (X), and µ can be calculated as:
Z ∞
2 1 − ln SK∗
V =
C(K) dK
K2
S∗
∗ Z S∗
2 1 + ln SK
+
P (K) dK
(4)
K2
0
Z ∞
3 ln SK∗ 1 − 2 ln SK∗
W =
C(K) dK
K2
S∗
∗ ∗
Z S∗
3 ln SK 1 + 2 ln SK
−
P (K) dK
(5)
K2
0
7
In BKM, the notation kurt represents the risk-neutral kurtosis. As we are interested in excess kurtosis
throughout the text, we drop out the word excess in the notation for clarity.
7
∞
4 ln2
3 − ln SK∗
X=
C(K) dK
K2
S∗
∗ ∗
Z S∗
4 ln2 SK 3 + ln SK
P (K) dK
−
K2
0
S(τ )
µ ≡ EQ ln
S
0
V
W
X
−rτ
rτ
1−e
− −
≈e
−
2
6
24
Z
K
S∗
(6)
(7)
where S ∗ is an arbitrary strike price that sets the OTM boundary, C(K) and P (K) represents
the price of an OTM call and put option with strike K, respectively.
In the original model derivation in BKM, each contract (V , W or X) requires the existence
of a continuum of options with strike spanning from 0 to infinity. To approximate the integrals
in eqs. (4) to (6), it is common to implement a trapezoidal approach to discretize and truncate
with available strikes (e.g. see Dennis and Mayhew, 2002; Bali and Murray, 2013; Conrad et
al., 2013):
X 2∆Ki Ki
1 − ln
Q(Ki )
(8)
V ≈
2
Ki
F0
i
X 3∆Ki Ki
Ki
2
W ≈
2 ln
− ln
Q(Ki )
(9)
2
Ki
F0
F0
i
X 4∆Ki Ki
Ki
3
2
− ln
Q(Ki )
(10)
3 ln
X≈
2
Ki
F0
F0
i
where ∆K1 = K2 − K1 , ∆KN = KN − KN −1 and ∆Ki = (Ki+1 − Ki−1 )/2 for i ∈ {2, . . . , N − 1}
where strike price is indexed from low to high. Q(Ki ) is the price of an OTM put (call) option
if Ki is smaller (larger) than the forward level F0 . That is, S ∗ is chosen to be the forward level
F0 = S0 e(r−q)τ with an estimated dividend yield q.
Researchers have considered different ways to approximate the value of a definite integral.
For example, Stilger et al. (2015) apply Simpson’s rule to compute integrals in eqs. (4) to (6),
which uses quadratic polynomials and it is able to converge to the true value of the definite
integral at faster rates comparing to the trapezoidal rule (Atkinson, 1989). Given
2.2. CBOE BKM-Equivalents
CBOE introduced a volatility index (original ticker: VIX; current ticker: VXO) in 1993 by
interpolating ATM implied volatilities of OEX options to construct a 30-day forward-looking
volatility measure. The VIX methodology was updated in 2003 with a reference to a model-free
approach first introduced in Demeterfi, Derman, Kamal and Zou (1999). The principle of the
new VIX is based on a principle that the fair value of future volatility can be captured by the
dynamic hedging of a log contract ln(ST /S0 ). Jiang and Tian (2007) show that this is equivalent
to the model-free implied variance developed in Britten-Jones and Neuberger (2000). Due to
its popularity and well establishment as a market volatility risk proxy, we adopt the majority
of CBOE VIX methodology in constructing VolCBOE but do not consider an interpolation in
term-structure to yield a fixed 30-day measure.
8
Although less popular in both finance industry and academic, CBOE also started publishing
a skewness index (current ticker: SKEW) in 2011. SKEW is designed to become the benchmark
measure for perceived future tail risk of the SPX return distribution. More specifically, the
algorithm of CBOE SKEW is to measure the negative skewness that SKEW = 100 − 10 ∗ S,
where S is the RN skewness. In our study, we consider S rather than the actual SKEW.
Strictly speaking, there does not exist a BKM-equivalent RN kurtosis from CBOE. We lean
the CBOE SKEW method to make an extension. The CBOE moments are given as follows
(CBOE, 2009, 2010):
(
2 )1/2
X
1 F0
∆Ki rτ
2
e Q(Ki ) −
−1
(11)
VolCBOE ≡
2
τ i Ki
τ K0
P3 − 3P1 P2 + 2P13
(P2 − P12 )3/2
P4 − 4P1 P3 + 6P12 P2 − P14
−3
≡
(P2 − P12 )2
SkewCBOE ≡
(12)
KurtCBOE
(13)
where the approximation on each component is performed as:
!
X ∆Ki
P1 ≈ erT −
Q(Ki ) + 1
Ki2
i
!
X 2∆Ki K
i
P2 ≈ erT
1 − ln
Q(Ki ) + 2
Ki2
F0
i
!
X 3∆Ki K
K
i
i
2 ln
− ln2
Q(Ki ) + 3
P3 ≈ erT
2
K
F
F0
0
i
i
!
X 4∆Ki K
K
i
i
P4 ≈ erT
3 ln2
− ln3
Q(Ki ) + 4
2
K
F
F0
0
i
i
(14)
(15)
(16)
(17)
where the terms at the end are adjustments made to compensate the difference between the
forward level F0 and the strike price K0 that is immediately below F0 . They can be computed
as:
F0
F0
1 = − 1 + ln
−
(18)
K0
K0
K0
1 2 K0
F0
2 = 2 ln
− 1 + ln
(19)
F0
K0
2
F0
1
K0
F0
K0
2
3 = 3 ln
ln
−1+
(20)
F0
3
F0
K0
K0
1
K0
K0
F0
3
4 = 4 ln
− ln
+
(21)
ln
F0
4
F0
F0
K0
We present a simple derivation of 1 in Appendix A8 . It is important to note that, V , W and
8
Other terms can be derived following a similar analogy. Exact derivation manuscript is available upon
request.
9
X in eqs. (8) to (10) can be seen as their corresponding counterpart P2 , P3 and P4 in eqs. (19)
to (21) without the terms.
A close examination on eq. (7) and eq. (14) reveals the major difference between the BKM
formulas and the CBOE
eq. (7) is derived in the Appendix in BKM by applying Taylor
P ones.
n
series of exp(R) = 4n=0 Rn! + o(R4 ). In comparison, the method of CBOE is more similar
to the pricing of a log contract (Neuberger, 1994) in the framework set by Bakshi and Madan
(2000). Due to this difference, we do expect to see slight deviations between BKM RN skewness
and kurtosis from those of CBOE.
2.3. Nonparametric Measures
Xing et al. (2010) examine individual stock options in the US market and argue that the
shape of the volatility smirk has predictive power for future equity returns. In their paper, they
estimate skew measure as the difference between the implied volatilities of OTM puts and ATM
calls. Xing et al. base the use of their skew measure on the demand-based option pricing model
of Gârleanu et al. (2007), which documents that the positive relationship between demand for
index options and option expensiveness, measured by implied volatility, can consequently affect
the steepness of the implied volatility skew. Bali et al. (2015) use nonparametric RN estimates
as a robustness check to their raw BKM estimates. We refer interested readers to the summary
provided in Mixon (2011). The nonparametric (NP) moments can be estimated as follows9 :
CIV50 + PIV50
2
≡ CIV25 − PIV25
≡ CIV25 + PIV25 − CIV50 − PIV50
SkewNP
CIV25 − PIV25
=
≡
50 Delta Volatility
VolNP
VolNP ≡
SkewNP
KurtNP
SkewMixon
(22)
(23)
(24)
(25)
where Cn represents the IV of an OTM call with delta n/100, and Pn represents the IV of an
OTM put with delta −n/100. For ease of convenience, we refer these as NP moments.
It is worthwhile to discuss the inclusion of SkewMixon and its difference comparing to SkewNP .
We reproduce some important discussion presented in Mixon (2011). Groeneveld and Meeden
(1984) define four properties to qualify a valid skewness function γ: 1) a scale or location change
for a random variable does not alter γ; 2) γ = 0 for a symmetric distribution; 3) if Y = −X
then γ(Y ) = −γ(X); and 4) if F and G are cumulative distribution functions for X and Y ,
respectively, and F c-proceeds G, then γ(X) ≤ γ(Y ). The first point is particular valid to the
above nonparametric skew measures. For example, the skewness measure should have minimal
dependence on the level of volatility. Mixon (2011) shows that SkewMixon subjects to the least
variations across a range of changes in volatility.
2.4. Measures from Implied Volatility Smirk
IV, as a function of the strike price for a given maturity, has been empirically studied in
Rubinstein (1994), Ait-Sahalia and Lo (1998), Foresi and Wu (2004), among others. There is a
25 −CIV25
Note that in Mixon (2011), the formula is specified as 50PIV
Delta Volatility , which measures the negative skewness.
We implement a necessary transformation to fit in this study.
9
10
rich literature that investigates the information content from the IV smirk. Zhang and Xiang
(2008) use a second-order polynomial to describe the IV-moneyness function. They show that
the level, slope and curvature of the IV smirk can be linked to RN volatility, skewness and
excess kurtosis, respectively. We follow their approach and estiamte these measures as follows:
VolSmirk ≡ γ0
SkewSmirk ≡ γ1
KurtSmirk ≡ γ2
(26)
(27)
(28)
where γ0 , γ1 , and γ2 are referred to as the level, slope and curvature of the IV smirk, respectively.
They are obtained by regressing the IVs with a quadratic function of moneyness:
IV(ξi ) = γ0 (1 + γ1 ξi + γ2 ξi2 ) + i
(29)
where the moneyness measure ξ is chosen to be:
ξi ≡
ln(Ki /F0 )
√
σ̄τ τ
(30)
and where σ̄τ denotes a measure of the average volatility of the underlying asset price. For
ease of convenience, we refer these as Smirk moments. We proxy σ̄τ by the realised volatility
of the underlying asset in the past τ −period. For example, for an option that has 9 days to
maturity, τ9/365 is the annualised standard deviation on the logarithm of the close-to-close daily
total return of the underlying asset in the past 9 days.
Our approach differs from Zhang and Xiang (2008) in several ways. They use a quadratic
function to fit the IV data by minimising the volume-weighted mean square error. We do
not weight the mean squared error by the option volume due to two reasons. First, we do
not have option trade volume in the simulation study. Second, Zhang and Xiang study the
implied volatility smirk from S&P 500 options. Our empirical study covers all issue types from
OptionMetrics and trade volume data is more noisy cross-sectionally. Another deviation from
their approach is that they use VIX value as the proxy for σ̄τ in the moneyness equation,
whereas the realised volatility is chosen in this study.
2.5. Raw Measures and Smoothing Method
Researchers are divided in how to interpolate and extrapolate observed option prices when
implementing BKM method. This is discussed in Section 1. Table 1 provides a list of studies
that have used BKM method and their corresponding treatment in treating the option data. To
cover a wide range of smooth methods, we implement the following approaches in the simulation
study. We limit our discussion to raw and s1 in the empirical study.
Raw We only use the observed option price data.
Smooth1 (s1) The interpolation is done by fitting a natural cubic spline to IV against deltas
between the highest and lowest known option deltas. The extrapolation follows
Jiang and Tian (2007) to match the slopes of the extrapolated and interpolated
segments.
11
Smooth2 (s2) The interpolation is done by fitting a natural cubic spline in IV against moneyness (K/S). The extrapolation step is the same as s1.
Smooth3 (s3) We linearly interpolate IV against option deltas. The extrapolation step is the
same as s1.
More specifically, in s1 and s3, we interpolate and extrapolate the observed IVs to fill in
a total of 1,000 grid points in the delta range from 0.001 to 1. In s2, the interpolation and
extrapolation is done to fill the moneyness-delta space on a total of 1,001 grid points in the
moneyness range from 1/3 to 3. We then calculate the option prices from the fitted IV using
the known interest rate and the adjusted dividend yield (recovered from comparing the security
price and the corresponding forward price provided by OptionMetrics) for a given maturity.
The variable naming convention follows this way: we put the estimation method in the
superscript and data interpolation approach in the subscript. For example, for the BKM
volatility that is constructed using raw data, we name it as VolBKM
raw . For the CBOE skewness
that is constructed using s1 smoothing interpolation, we name it as SkewCBOE
.
s1
3. Simulation Study
3.1. Simulation Design
We conduct Monte-Carlo (MC) simulations to examine various biases arise from the lack
of continuum of strike prices spanning from 0 to infinity. We need two important inputs, 1)
option prices that can be used to calculate various RN estimates presented in Section 2; 2) true
moments that are set as a benchmark target to examine estimation errors. With these inputs,
we can illustrate how the estimation error from each RN moment estimate can be shaped by
altering the availability of option prices. Furthermore, with multiple parameter settings, we
can further investigate the ranking correlations among the RN moment estimates.
Jiang and Tian (2007) study various estimation errors from the implementation of CBOE
VIX method. Hansis, Schlag and Vilkov (2010) discuss the effectiveness of using cubic splines
to interpolate the implied volatilities against moneyness and the importance of smoothing.
The authors use Black and Scholes model, the Heston model, the stochastic volatility and
jump model developed in Bates (1996) and Bakshi, Cao and Chen (1997) as well as SVCJ
model. Their design is meant to be directly comparable to that of Dennis and Mayhew (2002).
However, they do not investigate all three types of approximation errors as outlined in Chang
et al. (2012). Furthermore, as their results from the simulation study are not included in the
paper, it makes difficult to draw any inference.
Appendix B in Chang et al. (2012) discuss the approximation errors in skewness using
simulation option prices with Heston model. They only look at one set of parameters in one
model, in which we will show you the essence of using multiple sets of parameters in different
models. The authors conclude that “it is difficult to estimate skewness accurately when the
width of the integration domain is small” and “. . . we choose a sample of stocks with liquid
option data”. This motivates us to further include an analysis of RN skewness in this section.
We extend the simulation design outlined in Appendix B in Chang et al. (2012) to perform
MC simulations to generate option prices from the Black-Scholes-Merton (BSM) model (Black
and Scholes, 1973; Merton, 1973); Heston stochastic volatility model (Heston, 1993); Merton
jump-diffusion model (Merton, 1976); and Bates stochastic volatility jump-diffusion model
12
(Bates, 1996)10 . It is important to note that a standard MC estimation usually requires a large
number of trials to achieve some reasonable accuracy, at an expense of extra computational
resource usage. A typical procedure is to apply variance reduction techniques, such as applying
control variate technique and using discrete versions of martingale control variate. Provided
the goal of this simulation exercise is to draw direct comparisons with corresponding sections
in Dennis and Mayhew (2002), Jiang and Tian (2007), Chang et al. (2012), we do not adopt
any variance reduction techniques in improving the accuracy of option prices generated in
simulations.
We first outline the MC simulation procedure for each model and then show how the true
moment is estimated. We run MCS in BSM model, which is based on the Geometric Brownian
Motion. Given there is an exact solution to its stochastic differential equation (SDE), we have:
1 2
f
(31)
St = S0 exp
r − σ t + σ Wt
2
for t ∈ [0, T ], which means we could approximate the process (Si )i∈{1,...,N } by:
√
1 2
St+1 = St exp
r − σ ∆t + σ ∆tZt
2
for Zt ∼ N (0, 1) and t ∈ {0, 1, . . . , T − 1}.
In Heston model, the risk-neutral dynamics is governed by the system of SDEs:
√
ft1
dSt = rSt dt + νt St dW
√ f1 p
2
2
f
dνt = κ(θ̃ − νt )dt + ξ νt ρdWt + 1 − ρ dWt
for t ∈ [0, T ]. To simulate the process, we apply the Euler approximation:
p
√
√ √
νt+1 = κ(θ̃ − νt )∆t + ξ νt ρ ∆tZ1,t + 1 − ρ2 ∆tZ2,t
√
√
St+1 = St + rSt ∆t + νt St ∆tZ1,t
(32)
(33)
(34)
(35)
(36)
for Z1,t , Z2,t ∼ N (0, 1) and t ∈ {0, 1, . . . , T − 1}.
In Merton model, the solution to the SDE of Merton under the risk-neutral measure is given
as:
Nt
Y
ft
(r−λk− 12 σ 2 )t+σ W
St = S0 e
Yi
(37)
i=1
for t ∈ [0, T ], where Nt ∼ Pois(λ) and independent jumps Y with ln(Yi ) ∼ N (µJ , vJ2 ). We
apply the Euler simulation:
p
Ut = exp(Pt µJ + Pt vJ Z2,t )
(38)
10
It is interesting to point out that it is possible to calculate option prices in closed form using Fourier
inversion for these models, however, the convergence could fail given some extreme parameter choice (e.g. at
extreme far end of moneyness level). In order to achieve consistency in results, we follow Chang et al. (2012)
and opt to use simulations in this section.
13
St+1 = St exp
µJ + 12 vJ2
r − λ(e
√
1 2
− 1) − σ ∆t + ∆tσZ2,t Ut
2
where Pt ∼ Pois(λ∆t) and t ∈ {0, 1, . . . , T − 1}.
Lastly, Bates model combines Merton and Heston settings with SDEs as:
√
ft1 + dNt
dSt /St = rdt + νt St dW
√ f1 p
2
2
f
dνt = κ(θ̃ − νt )dt + ξ νt ρdWt + 1 − ρ dWt
for t ∈ [0, T ], where Nt ∼ Pois(λ) and independent jumps Y with ln(Yi ) ∼ N (µJ , vJ2 ).
apply the Euler simulation:
p
Ut = exp(Pt µJ + Pt vJ Z3,t )
p
√
√ √
νt+1 = κ(θ̃ − νt )∆t + ξ νt ρ ∆tZ1,t + 1 − ρ2 ∆tZ2,t
p
1
µJ + 12 vJ2
St+1 = St exp
r − λ(e
− 1) − νt ∆t + ∆tνt Z1,t Ut
2
(39)
(40)
(41)
We
(42)
(43)
(44)
where Pt ∼ Pois(λ∆t) and t ∈ {0, 1, . . . , T − 1}.
To generate prices for European options, we focus on 30- and 180-day measure. For each
maturity, we consider 9 pairs of parameters to capture a variety of outcomes in volatility,
skewness and kurtosis. This is presented in Table 2.
[Table 2 about here.]
The one month measure is considered due to the popularity concept of monthly portfolio,
as well as the monthly horizon seen in VIX and SKEW, which are both 30-day forward-looking
measures. We are also interested in the 180-day measure to draw some comparison with Chang
et al. (2012). For each model, there are a total of 18 sets of parameters: 9 sets of parameters
for each of the 2 maturities. In the BSM model, we vary the volatility parameter σ. In the
Heston model and Bates model, we vary the correlation parameter ρ of Wiener processes of
security price and volatility. In the Merton jump-diffusion model, we vary the intensity of
jumps parameter λ. The numerical choice of the parameters in Table 2 follow that of Jiang
and Tian (2007) and Chang et al. (2012). The exact MC simulation procedure is outlines as
follows.
1. Assuming that there are T (T ∈ {22, 124}) trading days for the 30- and 180-day measure,
1
.
respectively. The iteration for each simulation is T times with an interval ∆t = 252
2. For each model and each parameter
we perform a T-iteration for 1 million times.
choice,
Si,T
We calculate the log return ln S0 for each of these 1 million trajectories that i ∈
{1, 2, . . . , 106 }.
3. Compute the true volatility, skewness and kurtosis of these 1 million returns as the sample
moment:
sP
106
2
True
i=1 (Ri − R̄)
Vol
=
(45)
106 × T /252
14
Skew
=
KurtTrue
where Ri ≡ ln
Si,T
S0
1
106
and R̄ =
1
106
P106
− R̄)3
3/2
P106
1
2
(R
−
R̄)
i
6
i=1
10 −1
P
106
1
(Ri − R̄)4
106
= P i=1
2 − 3
106
1
2
i=1 (Ri − R̄)
106
True
P106
i=1
i=1 (Ri
(46)
(47)
Ri .
4. Approximate the European call and put option price as:
e−rT /252
P106
max(ST,i − K, 0)
106
P
6
e−rT /252 10
i=1 max(K − ST,i , 0)
P =
106
C=
i=1
(48)
(49)
3.2. Various Types of Approximation Error
Chang et al. (2012) specify three types of errors in implementing a typical trapezoidal
approach in the BKM moments construction. The first one is an integration domain truncation
error that arises from the missing strike prices beyond the range of observed strike prices. The
second one is a discretisation error that is induced by the discreteness of observed strike price.
The third one is an asymmetric integration domain truncation error, as the name suggests, that
the truncation is not symmetric around the the mean/mode/median. They are best presented
in the symbolic forms as follows.
1. Truncation errors:
∞
Z
Z
Kmax
. . . dK →
. . . dK
as K ∈ (0, ∞) → K ∈ [Kmin , Kmax ]
2. Discretization errors:
Z
(50)
Kmin
0
Kmax
. . . dK →
K
max
X
(51)
. . . ∆Ki
(52)
[Kmin , Kmax ] 6= [S0 × a, S0 /a]
(53)
Kmin
Kmin
3. Asymmetric truncation errors:
where a ∈ (0, 1] and S0 is the current spot level.
For every option model, the spot price for the underlying security S0 is set to be 1000. In the
base case (i.e. the ideal case scenario), strike price range is [1000*0.5, 1000/0.5] with a strike
interval ∆K = 1. In the simulation study, we fix ∆K and vary the strike price range to study
the integration domain truncation type of error. In particular, we vary the integration domain
from 0.50 to 0.99 with a step size of 0.01. That is, the strike range goes from [S0 ∗ 0.50, S0 /0.50]
to [S0 ∗ 0.99, S0 /0.99]. That is, we have 50 variations in examining truncation error.
15
In studying the discretisation of strike price type of error, we fix the integration domain to
be [S0 ∗ 0.50, S0 /0.50] and vary the strike interval as ∆K ∈ {1, 2, . . . , 25}. That is, we have 25
variations in examining discretisation error.
The design in studying the asymmetric truncation is worthwhile to elaborate. We fix the
strike interval to be 1 and vary the downside boundary as S0 ∗ uL , where uL = 0.7 + δu; and
upside boundary as S0 ∗ uH , where uH = 0.7 − δu. We set δu to vary from -0.2 to 0.2 with a
step size of 0.01. That is, we have 41 variations in examining asymmetric truncation error. To
see this more clearly, when δu = −0.2, the strike range is [500, 1111.11]; and when δu = 0.2, the
strike range is [900, 2000]. As δu varies from -0.2 to 0.2, the strike range moves from being more
negatively skewed to more positively skewed. The centre is at du = 0, where the asymmetry
is at its minimal. It is important to note that for each pair of asymmetric truncation, the
amount of available strikes are not too different; whereas in the truncation type, the higher the
truncation factor, the smaller amount of strikes available.
As a summary, we have a total of 116 (116 = 50 from truncation + 25 from discretisation
+ 41 from asymmetric truncation) variations from all three errors. Within each variation, we
have a total of 72 true value in each moment category (72 = 9 sets of parameters x 2 maturity
terms x 4 option models). This set up is particularly important when we discuss the ranking
correlations in Section 3.4.
3.3. Estimation Accuracy
We first investigate the estimation accuracy in the truncation error. We illustrate the
approximation errors of volatility, skewness and kurtosis in Figures 2 to 4. The approximation
error is calculated as
Estimation Error =
Estimated Moment − True Moment
True Moment
(54)
It is important to note that the NP and Smirk moments are only proxies for the true moments
and thus the value should differ numerically from the true ones. That is, given our definition
of estimation error, we will not directly interpret the size of estimation errors but focus on the
trajectory and trend across variations in each type of error study. Due to the slight complexity
in the iilustration, we explain the layout and content of these figures in Figure 1.
[Figures 1 to 4 about here.]
In each figure, the 1st column of plots illustrates approximation errors using the raw data
from simulations. The 2nd column applies s1 approach by fitting a natural cubic spline in
interpolating implied volatilities against deltas. The 3rd column applies s2 approach by fitting
a natural cubic spline in interpolating implied volatilities against strike prices. The 4th column
applies s3 approach by linearly interpolating implied volatilities against deltas. Each moment
is calculated using: 1) BKM method in the 1st row; 2) CBOE method in the 2nd row; 3)
non-parametric method in the 3rd row; and 4) implied volatility smirk in the 4th row. For
skewness, moment in the additional 5th row is calculated using Mixon’s method. Within each
small panel, the 1st (2nd ) column reports approximation errors using options with expiration of
22 (124) trading days. Within each panel, options prices are simulated using 1) Black-Scholes
model in the 1st row; 2) Bates stochastic volatility and jump diffusion model in the 2nd row; 3)
16
Heston stochastic volatility model in the 3rd row; and 4) Merton jump-diffusion model in the
4th row. In each plot, different shades of colour represents results from different parameters
used to generate option prices.
The truncation error in volatility estimation is illustrated in Figure 2. For VolBKM
and
raw
CBOE
Volraw , the underestimation of VolTrue is higher when the truncation is larger (i.e. smaller
integration domain range). We also see an increase in the size of errors as the maturity increases.
CBOE
All the smooth methods reduce the size of errors of VolBKM
raw and Volraw , from as large as −80%
Smirk
to less than 0.8%. In terms of the trend of errors, s1 is similar to s3. For VolNP
raw and Volraw ,
the improvement using smooth methods is minimal.
Examining the truncation error in skewness estimation from Figure 3, we find that as the
truncation becomes larger, it is possible to observe both under- and over-estimation of true
skewness in raw and smooth approaches, depending on the parameter choice. For SkewBKM ,
apply smooth methods flattens the trend of errors and reduce the absolute value of errors,
however, the errors of SkewBKM are a lot larger than those of SkewCBOE in each raw and smooth
approach. For SkewNP , SkewSmirk and SkewMixon , it is unclear to see how smooth approaches
improve on the accuracy as the trend look similar to the raw ones.
From Figure 4, the shape of error structures in kurtosis looks similar to what we find in
volatility, albeit the size of errors are much larger in the former. For KurtBKM and KurtCBOE ,
apply smooth methods reduce the magnitude of errors significantly, however, there is no particular pattern in the trend of errors in each smooth method.
[Figures 5 to 7 about here.]
We now move to discuss discretisation errors, as shown in Figures 5 to 7. In Figure 5,
we see that applying smooth methods significantly reduce the estimation errors for BKM and
CBOE volatility estimated from the longer maturity BSM and Merton options, but not for
Bates and Heston options. There is no clear improvement from applying smooth methods in
Smirk volatility. More specifically, s2 significantly increases the size of errors in VolSmirk if the
∆K is relatively small. In Figure 6, we see a similar improvement from smooth approaches
in estimation for BKM and CBOE skewness, but not for NP, Smirk or Mixon skewness. In
Figure 7, it is unclear to see the structure of errors for each measure as the size of errors is
dominated by two parameter sets (ones with darker colour). The only pattern can be found is
that implement smooth approaches reduce the size of errors for options with a longer maturity.
Overall, the discretisation errors are less of a concern than truncation errors for BKM and
CBOE moments.
[Figures 8 to 10 about here.]
Last, we discuss the asymmetric truncation errors, as presented in Figures 8 to 10. Figure 8 shows that estimation errors of BKM and CBOE volatility are significantly reduced by
implementing smooth approaches, particularly for shorter-maturity options. There is little improvement for NP volatility. When apply s2 smooth approach to NP and Smirk volatility, the
estimation errors are actually larger than the raw approach. For skewness estimates in Figure 9,
it is possible to see both under- and over-estimation in errors depending on the choice of parameters. Applying smooth approaches improves the estimation of BKM and CBOE skewness by
17
reducing the size of errors as well as flatten the error pattern. There is no clear improvement to
the error patterns of NP, Smirk and Mixon skewness. In Figure 10, s2 seems to work better for
BKM and CBOE kurtosis for shorter-maturity options, but no clear improvement over s1 and
s3 for longer-maturity options. For NP kurtosis, s2 can potentially create outliers in errors, as
compared to s1 and s3. For Smirk kurtosis, there is no clear benefits from applying smooth
approaches.
Having discussed all three types of approximation errors, it is important to point out that
in reality, it is impossible to disentangle the option data into separate analyses of these three
types of errors. We do observe improvements in the size of estimation errors for BKM and
CBOE moments but the improvement significant drops as we move to higher moments. Our
conclusion for the estimation accuracy is that the exact error for a true moment estimate is, at
its best, unquantifiable.
3.4. Ranking Correlations
We now turn to a different angle in looking at the usage of these moment estimates. One
popular application of RN moments is to use them as a sorting mechanism. As an example,
suppose we have 100 securities and their RN moments can be estimated from traded options. If
the goal is to rank these securities by a RN moment and form portfolios according to a specific
rule, then it is more interesting to find out which of the RN moment estimates gives the more
correct ranking. This question is less challenging than looking for a point estimator.
In the simulation set up, we have a total of 116 (116 = 50 from truncation + 25 from
discretisation + 41 from asymmetric truncation) variations from all three errors. Within each
variation, we have a total of 72 true value in each moment category (72 = 9 sets of parameters
x 2 maturity terms x 4 option models). If we view each variation of error study from one
particular set of parameters, one maturity and one particular option model as one ‘security’,
then we have 8, 352 ‘securities’ in total. For example, we can set security A as a stock that
follows BSM model with σ = 0.1, τ = 22/252 with a strike range from [500, 2000] and ∆K = 1;
and another security B as a stock that follows Bates model with ρ = −0.75, τ = 22/252 with a
strike range from [500, 2000] and ∆K = 4. A good RN estimate should give the same ranking
of A and B as that from a true estimate.
A widely accepted rank correlation coefficient is Kendall’s τ (Kendall, 1948) that essentially
measures the probability of two elements being in the same order in the two ranked lists. In
particular, as we have ties in the ranking (because the true value is the same across different variations in the error study), we calculate the Kendall τ -b statistic. We also estimate the
Spearman’s rank correlation. There is a lengthy discussion in statistic literature on the comparison of Kendall and Spearman’s rank correlations. A general understanding is that Spearman’s
rank correlation is usually larger than Kendall’s τ .
[Tables 3 to 5 about here.]
We present the Kendall and Spearman correlations in Tables 3 to 5 for volatility, skewness
and kurtosis, respectively. Each correlation estimate is calculated based on 8, 352 pairs of true
moment and moment estimate. Kendall correlations are presented in the highlighted cells in
the bottom left part of each table. Spearman correlations are presented in the top right part
of each table.
18
NP
In Table 3, it is interesting to learn that VolNP
raw VolSmirk have a higher Kendall correlations
CBOE
True
BKM
with Vol
than Volraw and Volraw . Applying smooth methods significantly increases the
Kendall correlations for BKM and CBOE volatility. For example, Kendall correlation between
True
VolBKM
increases from 0.55 to 0.95 after applying for s1 smooth method. We do not
raw and Vol
see any improvement for NP and Smirk volatility. Similar findings are presented with Spearman
correlation.
In Table 4, we also see a two-fold increase in Kendall correlations for BKM and CBOE
skewness with all smooth methods. Furthermore, the improvement in CBOE skewness is much
larger, from 0.34 to 0.88 with s1 and s3. Even though there is no improvement from applying
smooth methods to NP, Smirk and Mixon skewness, the Kendall correlations between those
skewness estimates with true skewness are all around 0.9.
In Table 5, we find the Kendall correlation between BKM kurtosis and true kurtosis increases
from 0.06 for raw approach to at least 0.16 for smooth approaches. Larger increases are found
between CBOE kurtosis and true kurtosis. It is interesting to note that applying smooth
approaches reduces the Kendall correlation between NP kurtosis from 0.09 to as low as 0 using
s2. There is no impact to Smirk kurtosis. Similar findings are found with Spearman correlation.
These findings confirm that applying smooth methods increases the usefulness of BKM and
CBOE volatility and skewness, however, the improvements in kurtosis are minimal.
3.5. Portfolio Composition Comparison
A potential downside of using the ranking correlations is that the whole population of ranks
are evaluated. If only the composition of a certain proportion is interested, then the ranking
correlations may overestimate the problem. Stilger et al. (2015) sort equities by the RN
skewness at the end of each month in ascending order. They take a long strategy in stocks
with the highest quintile RN skewness and a short strategy in stocks with the lowest quintile
RN skewness. In this case, the rank order is not as important within each quintile. Inspired on
this design, we perform a matching test of the top and bottom quintile with the true moments.
The test design for the top quintile portfolio is as follows.
For each volatility estimate that is computed using BKM, CBOE, NP and Smirk, there are 72
volatility proxies with 72 corresponding true volatilities from each of 116 variations that study
approximation errors (where 116 = 50 variations in truncation error study + 25 variations
in discretisation + 41 variations in asymmetric truncation and 72 = 9 sets of parameters
x 2 maturity terms x 4 option-price-generation models). In each of 116 variations, we sort
volatility estimates from one of 4 methods (BKM, CBOE etc) in ascending order. We extract
estimates that are above the 80th percentile (a quintile of 72 items is roughly 15). As each
volatility estimate has a corresponding true volatility, we then calculate the percentage of
these corresponding true volatilities are also above the 80th percentile of true volatilities. We
illustrate the distribution of these 116 percentages in a boxplot. This is repeated for skewness
and kurtosis estimates. We replicate these procedures for the bottom quintile by changing the
threshold to be 20th percentile.
[Figures 11 and 12 about here.]
We illustrate the matching comparison of the top (highest) quintile in Figure 11 and that
of the bottom quintile in Figure 12. A close examination of these two figures confirm with
19
our previous findings in twofold. First, applying smooth methods improve BKM and CBOE
moments but the improvement is smaller for higher moments. Second, the improvement is
larger for CBOE moments than BKM ones. Third, there is no improvements to other moment
estimates from implement smooth methods.
4. Empirical Results
4.1. Data
We obtain data from the Ivy DB US OptionMetrics provided through Wharton Research
Data Services. We download the entire database that contains all securities traded from 4
January 1996 to 29 August 2014. We extract the security ID, issue types, date, expiration
date, put and call identifier, strike price, best bid, best offer, implied volatility and delta from
the option price file. We use the average of the bid and ask quotes for each option contract.
We filter out options with zero bids. We further filter out options with non-zero bids but are
beyond two consecutive strike prices with zero bid prices11 .
Interest rates are taken from the CRSP Zero Curve file. We apply a cubic spline to the
interest rate term-structure data to match the length of risk-free rate with the corresponding
option maturity.
We consider OTM options only. We define a put (call) option is OTM if its strike price is
lower (greater) than the forward price of the underlying asset. We convert the OTM put deltas
into the corresponding call deltas as 1 + put delta = call delta. Underlying security prices
are obtained through CRSP. We obtain forward price of each security from OptionMetrics
‘Std Option Price file’. If the forward price is missing, we calculate the present value of its
close price after adjusting for dividends from ‘Distribution file’.
)
√ 0 , where σ̄τ in the Smirk moments, we need
In estimating the moneyness level ξ ≡ ln(K/F
σ τ
an input for σ. We obtain realised volatilities for the underlying assets from OptionMetrics
‘Historical Volatility’ files. According to its reference manual, realized volatility is calculated
over a list of standard date ranges from 10 to 730 calendar days. The calculation is performed
using a standard deviation on the natural logarithm of the close-to-close daily total return. We
proxy σ̄τ by the realised volatility of the underlying asset from this file.
We examine various estimation errors from integration truncation, discretisation and asymmetric truncation in Section 3. Although we cannot investigate these estimation errors empirically, it is interesting to present summary statistics to document how these estimation errors
may have a role with observed data.
[Table 6 about here.]
in the filtered raw data set from January 1996
Table 6 reports the summary statistics of ∆K
F0
to August 2014. This is a subset of our data that only includes options with five groups of
maturity terms: 1) between 28 to 32 days are labeled as 1m; 2) between 58 to 62 days as 2m; 3)
between 88 and 92 days as 3m; 4) between 180 and 185 days as 6m; and 5) between 360 and 370
days as 12m. We calculate ∆K1 = K2 − K1 , ∆KN = KN − KN −1 and ∆Ki = (Ki+1 − Ki−1 )/2
for i ∈ {2, . . . , N − 1} where strike price is indexed from low to high. Issue type is defined
11
This is similar to the filtration standard by CBOE in VIX and SKEW calculation.
20
according to the OptionMetrics Ivy DB reference manual. The figure shows that more than
60% of options are written on common stocks, which is followed by options on ETF and index
options.
In the simulation, we set the spot level to be 1000 and vary ∆K from 1 to 25, which means
roughly from 0.001 to 0.025. Examining the summary statistics in Table 6, we find
we vary ∆K
F0
that on average the strike step size is 0.087 for stock options with 1 month to maturity. This
increase to 0.129 for stocks options with 1 year to maturity. The stirke step size is much smaller
for index options, where on average it is 0.018 for index options with 1 month to maturity and
0.025 for those with 1 year to maturity. The concern comes from the maximum step size. For
example, out of the 18-year period, there is one stock option with 2 months to maturity on one
day that has a strike step size as large as 8.9 times its underlying forward level. It is important
to note that this is based on the filtered data.
[Tables 7 and 8 about here.]
of the lowest OTM put option in the filtered
Table 7 shows the summary statistics of KFmin
0
12
raw data set . We find that on average, the mean of lower boundary is around 0.85 for all
issue types with 1 month to maturity. Consistent with the common understanding, the lower
boundary decreases as the option maturity increases. Table 8 shows the summary statistics of
Kmax
of the highest OTM call option in the filtered raw data set. We find that on average, the
F0
mean of upper boundary is around 1.15 for common stocks with 1 month to maturity, which is
higher than that of index options. Similar to the lower boundary, the upper boundary increases
as the option maturity increases.
4.2. Rank Correlations
In this section, we estimate rank correlations among RN moments to study their usefulness
as a sorting mechanism. If a security has a high RN moment measured with BKM and is ranked
among the top 20% when all securities are sorted in ascending order, will this be captured by
the high RN moment measured by other methods? We present the average and the standard
deviation of daily Kendall and Spearman rank correlations of various volatility, skewness and
excess kurtosis estimates in Tables 9 to 11, respectively.
[Tables 9 to 11 about here.]
We first examine the volatility ones in Table 9. In this table, each pair of correlation is
first estimated for all options with maturities of 1-month, 2-month, 3-month, 6-month and 12month of all issue types on the daily basis. The average and the standard deviation (shown in
parentheses) are then calculated based on daily correlations across the whole sample period. It
is clear to see that applying s1 smooth method increases the rank correlations between BKM
and CBOE volatility, from 0.8 to 0.97 on average, with a reduction in standard deviation,
from 0.11 to 0.04. We also see an increase among other volatility estimates after implementing
12
Note that, the proportion values should be interpreted differently to those found in Table 6. There may be
multiple entries of ∆K
F0 from each security on any day with any maturity term, whereas there is only one entry
of KFmin
from
each
security
on that day with the same maturity.
0
21
s1 approach. This implies that volatility estimates with s1 more or less capture the same
information. In Table 10, the average Kendall and Spearman correlations are smaller than those
seen in volatility ones. We see an increase among rank correlations after applying s1 method.
In Table 11, the Kendall correlation between BKM and CBOE almost doubles from 0.52 to 0.94
after implement s1 approach. It is interesting to learn that both NP and Smirk kurtosis have
low rank correlations, even after implementing the smooth approach. This suggests that NP
and Smirk kurtosis have different information content from those of BKM and CBOE kurtosis.
4.3. Skewness Portfolio Composition and Future Returns
This last section is motivated by the mix findings in the relationship between RN skewness
and future realised returns. Conrad et al. (2013) implement a raw BKM approach in estimating RN moments. They find a negative relationship between quarterly averages of daily RN
skewness estimates and subsequent realised quarterly stock returns. Bali and Murray (2013)
also adopt a raw BKM approach and create a portfolio of options that only exposes to skewness
effect. They find a negative relationship between RN skewness and option portfolios’ returns.
On the other hand, Rehman and Vilkov (2002) implement a smooth BKM approach and document the ex ante skewness is positively related to future stock returns. This finding is further
supported by Stilger et al. (2015). The authors use a smooth BKM approach13 and document
that a strategy to long the quintile portfolio with the highest RN skewness stocks and short the
quintile portfolio with the lowest RN skewness stocks on average yields a Fama-French-Carhart
alpha of 55 bps per month. As point out in Stilger et al., they attribute the difference in their
findings to the fact that the underperformance in the most negative skewness stocks is driven
by stocks that are too costly to short sell.
We follow Stilger et al. (2015) to study skewness-quintile portfolios and the realised returns.
Given we have 10 RN skewness measures (5 of the raw ones and 5 of the s1 ones), this portfolio
study allows us to further investigate the information content carried in these RN skewness
measures. Portfolios are constructed as follows. We only consider equity options. On the last
trading day of each month t, stocks are sorted in ascending order by the corresponding skewness
measure. Each skewness measure is calculated from its options with the shortest maturity (with
at least 10 days to maturity) on that day. Quintile 5 (1) includes stocks with skewness measure
that is above the 80th percentile (below the 20th percentile).
[Table 12 about here.]
We first compare the portfolio composition, as shown in Table 12. This is similar to the
analysis presented in Section 3.5. For each month from January 1996 to August 2014, we first
count the number of matching stocks from each pairwise skewness portfolios and divide this
number by the total number of stocks in each portfolio to estimate the percentage. The average
and the standard deviation (shown in parentheses) of percentages are then calculated for the
whole period. In quintile 5 portfolios, which are presented in the bottom-left part of the table,
we find that applying s1 method significantly increases the percentage of matched securities
among these portfolios. A similar finding is found in quintile 1 portfolios.
13
The authors find similar results by using the raw BKM approach as a robustness check.
22
[Figure 13 about here.]
In Figure 13, we use a heat map to illustrate the monthly excess returns of all skewnessquintile portfolios from 1996 to 2014. We form the skewness-quintile portfolio at the end of
each month t, and calculate the equally-weighted returns of these portfolios at the end of the
following month t + 1. The excess return is obtained by subtracting the monthly risk-free
return from the portfolio return. The adjusted close prices (for dividend splits etc) at time t
and t + 1 are used to calculate the return. The top panel shows the colour key used to represent
excess returns. The top panel also presents the histogram of all monthly excess returns of all
skewness-quintile portfolios. The bottom panel presents the heat map, the time is shown on
the horizontal axis where each skewness-quintile portfolio is illustrated along the vertical axis.
A close examination of this figure reveals that quintile 1 skewness portfolios behave quite
similar, as show by the similar colour intensity vertically. There are some big losses in mid-1998,
around the dot-com bubble from mid-2000 to mid-2001, as well as in GFC. In contrast, we do
not see any similarity in returns across quintile 5 portfolios. In addition, no significant losses
or gains are found in quintile 5 portfolios.
[Figures 14 and 15 about here.]
Figure 14 illustrates the average RN volatility of skewness-quintile portfolios, where the plot
BKM
with VolBKM
is found in the bottom. Examining the
raw is provided in the top and that of Vols1
horizontal axis of colour key (smaller box) of these two plots, we see that the average VolBKM
is
s1
higher than VolBKM
in
the
whole
period.
Although
they
differ
numerically,
the
colour
intensity
raw
in these two plots suggests that they do not make any qualitative difference across time.
Time-series average RN excess kurtosis of these portfolios are shown in Figure 15, where
the plot with KurtBKM
is provided in the top and that of KurtBKM
is found in the bottom.
raw
s1
Similar to what we find in volatility, the colour key shows that the average KurtBKM
is higher
s1
BKM
than KurtBKM
in
the
whole
period.
Furthermore,
in
quintile-1
portoflios
formed
by
Skew
raw ,
raw
CBOE
BKM
CBOE
Skewraw , Skews1 and Skews1 , the average RN excess kurtosis is much higher than the
other portfolios in the whole sample period. This is an interesting finding that may attract
some further investigation in the future.
[Tables 13 and 14 about here.]
Having examined the time-series behaviour of RN skewness-quintile portfolios, we now study
the excess return using the Fama-French Five-Factor model (Fama and French, 2015; hereinafter, FF5). Table 13 shows the excess return performance, measured by ln(Pt+1 /Pt ) − Rf ,
of stock portfolios as well as their FF5 alphas and other factor loadings, including the portfolio
loadings β’s with respect to the market (MKT), size (SMB), value (HML), profitability (RMW)
and investment patterns (CMA) are also reported as well as the explanatory power of the model
(adjusted R2 ). It is clear to see that a long strategy in quintile 1 and a long strategy in quintile
5 portfolios consistently generate significantly negative αFF5 across all skewness measures. We
do not consistently find a 5-1 (long 5 and short 1) strategy yielding a positive and significant
αFF5 across all measures. This is different from the finding by Stilger et al. (2015). It is
important to point out that the difference can be related to a few reasons. First, we cover a
23
longer time period to 2014, as comparing to 2012. Second, we need to remove missing values
across all skewness measures. That is, our universe of stocks may differ from theirs.
If we measure excess return as (Pt+1 − Pt )/Pt − Rf , as shown in Table 14 , regression results
are slightly different. Quintile 1 portfolios still yields significant and negative αFF5 across all
skewness measures. Quintile 5 portfolios do not yield any significant αFF5 for most skewness
measures.
5. Conclusion
RN moments are important sources to study the information embedded in market option
prices. BKM provide a model-free measure of volatility, skewness and kurtosis that can be directly inferred from traded options. In this paper, we study different treatments of option data
before they are input to the BKM formulas. Using MC simulations, we examine the integration
truncation error, discretisation of strike price error and asymmetric truncation error arise from
the lack of a continuum of strike price ranging from zero to infinity. We extend the analysis
to include several other RN moment proxies, including the CBOE moments, nonparametric
moments that are calculated as differences of IV across different moneyness, and the intercept,
slope and curvature of the IV smirk. In the simulation study, we show that the errors of point
estimates of true moments are larger for higher moments, and are largely unquantifiable. Examining the Kendall and Spearman rank correlations, we show that applying smooth methods
significantly improve the information content of RN moments with the true moment.
In the empirical study, we document that truncation errors, discretisation errors and asymmetric truncation errors play a role in estimating the BKM and CBOE moments. Applying the
smooth method increases the rank correlation among these RN moments. In that case study
that examines the RN skewness-quintile portfolios and future realised returns, we find that the
portfolio with the lowest skewness significantly underperform the market, after adjusting for
the Fama-French Five-Factors.
References
The reference list is currently incomplete.
Agarwal, V., Bakshi, G. Huij, J., 2009, Do higher-moment equity risks explain hedge fund
returns. Working paper.
Bakshi, G., Kapadia, N., Madan, D., 2003. Stock Return Characteristics, Skew Laws, and
the Differential Pricing of Individual Equity Options. The Review of Financial Studies 16,
101–143.
Bakshi, G., Madan, D., 2000. Spanning and derivative-security valuation. Journal of Financial
Economics 55, 205–238.
Bali., T. G., Hu, J., Murray, S., 2015. Option implied volatility, skewness, and kurtosis and the
cross-section of expected stock returns. Working paper.
24
Bali., T. G., Murray, S., 2013, Does risk-neutral skewness predict the cross-section of equity
option portfolio returns? Journal of Financial and Quantitative Analysis 48, No. 4, 1145–
1171.
Breeden, D. T., Litzenberger, R. H., 1978. Prices of state-contingent claims implicit in option
prices. Journal of Business 51, 621–651.
Carr, P., Madan, D., 2001. Optimal positioning in derivative securities. Quantitative Finance
1, 19–37.
Chang, B., Christoffersen, P., Jacobs, K., Vainberg, G., 2012, Option-implied measures of
equity risk. Review of Finance 16, 385–428.
Christoffersen, P., Jacobs, K., Vainberg, G., 2008, Forward-looking betas. Working paper.
Conrad. J., Dittmar, R. F., Ghysels, E., 2013, Ex Ante Skewness and Expected Stock Returns.
The Journal of Finance 68, 85–124.
Dennis, P., Mayhew, S., 2002, Risk-Neutral Skewness: Evidence from Stock Options. Journal
of Financial and Quantitative Analysis 37, 471–493.
Duan, J. C., Wei, J., 2009, Systematic risk and the price structure of individual equity options.
The Review of Financial Studies 22, 1981–2006.
Engle, R., Mistry, A., 2014, Priced risk and asymmetric volatility in the cross section of skewness. Journal of Econometrics 182, 135–144.
Figlewski, S., 2008, Estimating the Implied Risk Neutral Density for the U.S. Market Portfolio.
Volatility and Time Series Econometrics: Essarys in Honor of Robert F. Engle, 323–353.
Han, B., 2008, Investor Sentiment and Option Prices. The Review of Financial Studies 21,
387–414.
Hansis, A., Schlag, C., Vilkov, G., 2010, The dynamics of risk-neutral implied moments: evidence from individual options. Working paper.
Jiang, G. J., Tian, Y. S., 2007, Extracting Model-free volatility from option prices: an examination of the VIX index. Journal of Derivatives.
Mixon, S., 2011, What does implied volatility skew measure? Journal of Derivatives, 9–25.
Neumann, M., Skiadopoulos, G. 2013, Predictable dynamics in higher order risk-neutral moments: evidence from the S&P 500 options. Journal of Financial and Quantitative Analysis
48, No. 3, 947–977.
Shimko, D., 1993, Bounds of Probability. Risk 6, 33–37.
Stilger, P. S., Kostakis, A. and Poon, S., 2015. What does risk-neutral skewness tell us about
future stock returns, Working paper.
25
Taylor, S. J., Yadav, P. K., Zhang, Y., 2010, The information content of implied volatilities
and model-free volatility expectations: Evidence from options written on individual stocks.
Journal of Banking & Finance 34, 871–881.
Xing, Y., Zhang, X., Zhao, R., 2010, What does the individual option volatility smirk tell us
about future equity returns? Journal of Financial and Quantitative Analysis 45, 641–662.
Zhang, J. E., Xiang, Y., 2008, The implied volatility smirk. Quantitative Finance 8, 263–284.
26
27
Stock/Index Options
Stock
Stock and Index
Stock
Index
Stock
Stock
Stock
Index
Stock
Stock and Index
Stock
Stock
Stock
Stock
Index
Index
Stock
Stock
Index
Stock
Stock
Stock
Futures Options
Index
Stock
Authors
Dennis and Mayhew (2002)
Christoffersen, Jacobs and Vainberg (2008)
Han (2008)
Agarwal, Bakshi and Huij (2009)
Duan and Wei (2009)
Taylor, Yadav and Zhang (2009)
Hansis, Schlag and Vilkov (2010)
Mixon (2011)
Buss and Vilkov (2012)
Chang, Christoffersen, Jacobs and Vainberg (2012)
Diavatopoulos, Doran, Fodor and Peterson (2012)
Friesen, Zhang and Zorn (2012)
Rehman and Vilkov (2012)
Bali and Murray (2013)
Byun and Kim (2013)
Chang, Christoffersen and Jacobs (2013)
Conrad, Dittmar and Ghysels (2013)
DeMiguel, Plyakha, Uppal and Vilkov (2013)
Neumann and Skiadopoulos (2013)
Engle and Mistry (2013)
An, Ang, Bali and Cakici (2014)
Bali, Hu and Murray (2015)
Chatrath, Miao, Ramchander and Wang (2015)
Gagnon, Power and Toupin (2015)
Stilger, Kostakis and Poon (2015)
Raw
Smooth
Raw
Raw
Raw
Raw
Smooth
Raw
Smooth
Smooth
Raw
Raw
Smooth
Raw
Raw
Smooth
Raw
Smooth
Smooth
Smooth
Raw
Raw
Smooth
Smooth
Raw and Smooth
Raw/Smooth
Interpolate
Interpolate
Interpolate
Interpolate
Interpolate
Interpolate
Interpolate
Interpolate
Interpolate
Interpolate
Interpolate
Interpolate
IV using a cubic spline across moneyness (K/S)
IV using a cubic spline across moneyness (K/S)
IV using a piecewise Hermite polynomial across moenyess (K/S)
IV using a cubic spline across moneyness (K/S)
IV using a cubic spline across deltas
√
IV using a quadratic spline across moneyess ln(K/S)−rT
σ T
IV using a cubic spline across moneyness (K/S)
IV using a cubic spline across moneyness (K/S)
IV using a cubic spline across moneyness (K/S)
IV using a cubic spline across moneyness (K/S)
IV using a cubic spline across moneyness (K/S)
IV using a cubic spline across moneyness (K/S)
Smooth Method
Table 1: This table provides a subset of studies that have applied Bakshi, Kapadia and Madan (2003, BKM)’s method to
calculate risk-neutral moments from option prices. “Stock/Index Options” shows the main type of options that are used to
implement BKM method. “Raw/Smooth” refers to whether the traded option prices (i.e. raw) are directly used, or the option
prices have been interpolated and extrapolated using some particular method before been applied in BKM’s formulas. IV stands
for implied volatility.
Table 2: This table describes parameters used in the base case models of our simulation study
in Section 3 and parameters used in examining each type of errors. For each model, there are
a total of 18 sets of parameters: 9 sets of parameters for each of the 2 maturities. In the BSM
model, we vary the volatility parameter σ. In the Heston model and Bates model, we vary
the correlation parameter ρ of Wiener processes of security price and volatility. In the Merton
jump-diffusion model, we vary the intensity of jumps parameter λ. For every model, the spot
price for the underlying security S0 is set to be 1000. The forward price F0 is then calculated
as S0 exp ((r − q)τ ). In the base case, strike price range is [1000*0.5, 1000/0.5] with a strike
interval ∆K = 1. In the simulation study, we fix ∆K and vary the strike price range to study
the integration domain truncation type and the asymmetric integration domain truncation type
of errors. We fix the strike price range and vary ∆K to study the discretisation of strike price
type of error.
Panel A
Name
Spot
Strike Range
Strike Interval
Time to Maturity
Interest
Dividend
Volatility
Initial Variance
Long-Run Variance
Vol of Vol
Speed of Mean Reversion
Correlation of S and V
Mean of Jumps
Volatility of Jumps
Intensity of Jumps
Symbols
S0
[Kmin , Kmax ]
∆K
τ
r
q
σ
ν0
θ̃
ξ
κ
ρ
µJ
vJ
λ
BSM
Heston
Merton
Bates
1000
[S0 *0.5, S0 /0.5]
1
22 124
,
252 252
0.05
0
0.1, 0.2, . . . , 0.9
-
1000
[S0 *0.5, S0 /0.5]
1
22 124
,
252 252
0.05
0
0.05
0.05
0.15
2.00
-1, -0.75, . . . , 1
-
1000
[S0 *0.5, S0 /0.5]
1
22 124
,
252 252
0.05
0√
0.05
−0.15σ
0.152 σ 2
0.5, 1.0,. . . , 4.5
1000
[S0 *0.5, S0 /0.5]
1
22 124
,
252 252
0.05
0
0.05
0.05
0.15
2.00
-1, -0.75, . . . , 1
−0.15σ
0.152 σ 2
1.00
Panel B
Type of Errors
Parameter
Strike Range
Strike Interval
S0
]
u∗0.01+0.49
Truncation
u ∈ {1, 2, . . . , 50} [S0 ∗ (u ∗ 0.01 + 0.49),
Discretisation
∆K
[S0 ∗ 0.5, S0 /0.5]
S0
Asymmetric Truncation δu ∈ {1, 2, . . . , 41} [S0 ∗ (0.49 + δu/100), 0.91−δu/100
]
28
1
∆K ∈ {1, 2, . . . , 25}
1
29
0.95
0.87
0.79
0.81
0.95
0.86
0.79
0.59
0.95
0.87
0.79
0.81
VolBKM
s1
VolCBOE
s1
VolNP
s1
VolSmirk
s1
VolBKM
s2
VolCBOE
s2
VolNP
s2
VolSmirk
s2
VolBKM
s3
VolCBOE
s3
VolNP
s3
VolSmirk
s3
Kendall
0.93
0.52
0.52
0.55
0.52
0.80
0.82
0.56
0.56
0.52
0.52
0.56
0.56
0.52
0.46
0.56
0.56
0.52
0.52
0.68
VolBKM
raw
VolBKM
raw
VolCBOE
raw
VolNP
raw
VolSmirk
raw
VolTrue
VolTrue
Name
0.53
0.56
0.52
0.53
0.53
0.57
0.52
0.47
0.53
0.56
0.52
0.53
0.52
0.52
0.99
0.66
VolCBOE
raw
0.83
0.90
0.97
0.93
0.82
0.90
0.97
0.66
0.83
0.90
0.97
0.93
0.92
0.66
0.66
0.93
VolNP
raw
0.84
0.91
0.93
0.96
0.83
0.90
0.92
0.69
0.84
0.91
0.93
0.96
0.66
0.65
0.99
0.94
VolSmirk
raw
1.00
0.91
0.82
0.84
0.97
0.88
0.82
0.61
0.91
0.82
0.84
0.69
0.66
0.95
0.96
0.99
VolBKM
s1
0.91
1.00
0.90
0.92
0.90
0.96
0.89
0.68
0.90
0.92
0.98
0.70
0.69
0.98
0.99
0.97
VolCBOE
s1
0.82
0.90
0.99
0.94
0.82
0.90
0.99
0.67
0.94
0.95
0.98
0.66
0.66
1.00
0.99
0.93
VolNP
s1
0.84
0.92
0.94
1.00
0.83
0.91
0.93
0.69
0.96
0.99
0.99
0.67
0.66
0.99
1.00
0.94
VolSmirk
s1
0.97
0.90
0.82
0.83
0.90
0.81
0.60
1.00
0.98
0.95
0.96
0.69
0.67
0.95
0.96
0.99
VolBKM
s2
0.88
0.96
0.90
0.91
0.89
0.67
0.98
0.97
1.00
0.98
0.99
0.71
0.70
0.98
0.98
0.96
VolCBOE
s2
0.82
0.89
0.99
0.93
0.66
0.95
0.98
0.95
0.98
1.00
0.99
0.66
0.65
1.00
0.99
0.93
VolNP
s2
0.61
0.68
0.67
0.69
0.72
0.76
0.73
0.72
0.76
0.74
0.76
0.61
0.62
0.74
0.76
0.71
VolSmirk
s2
0.91
0.82
0.84
1.00
0.97
0.95
0.72
1.00
0.98
0.95
0.96
0.69
0.66
0.95
0.96
0.99
VolBKM
s3
0.90
0.92
0.98
0.98
1.00
0.98
0.76
0.98
1.00
0.98
0.99
0.70
0.69
0.98
0.99
0.97
VolCBOE
s3
0.94
0.95
0.98
0.95
0.98
1.00
0.74
0.95
0.98
1.00
0.99
0.66
0.66
1.00
0.99
0.93
VolNP
s3
0.96
0.99
0.99
0.96
0.99
0.99
0.76
0.96
0.99
0.99
1.00
0.67
0.66
0.99
1.00
0.94
Spearman
VolSmirk
s3
Table 3: This table shows the Kendall rank correlation (τ -b) and Spearman’s rank (ρ) correlation coefficients among various
volatility measures in the simulation study in Section 3. Each correlation estimate is calculated based on 8,352 pairs of volatility
estimates (8, 352 = 116 × 9 × 2 × 4, where 116 = 50 variations in truncation error study + 25 variations in discretisation +
41 variations in asymmetric truncation, 9 sets of parameters with 2 maturity terms from 4 option-price-generation models).
Kendall correlations are presented in the highlighted cells in the bottom left part of the table. Spearman correlations are
presented in the top right part of the table.
30
0.22
0.34
0.89
0.86
0.49
0.88
0.90
0.91
0.51
0.83
0.90
0.70
0.49
0.88
0.90
0.91
0.87
0.88
0.88
0.88
SkewBKM
s1
SkewCBOE
s1
SkewNP
s1
Smirk
Skews1
SkewBKM
s2
SkewCBOE
s2
SkewNP
s2
SkewSmirk
s2
SkewBKM
s3
SkewCBOE
s3
SkewNP
s3
SkewSmirk
s3
SkewMixon
raw
SkewMixon
s1
SkewMixon
s2
SkewMixon
s3
Kendall
SkewTrue
SkewBKM
raw
SkewCBOE
raw
SkewNP
raw
SkewSmirk
raw
Skew
True
Name
0.24
0.25
0.26
0.25
0.48
0.25
0.23
0.23
0.49
0.26
0.24
0.16
0.48
0.25
0.23
0.23
0.67
0.21
0.23
0.30
SkewBKM
raw
0.34
0.35
0.35
0.35
0.24
0.39
0.33
0.35
0.27
0.41
0.34
0.26
0.24
0.39
0.33
0.35
0.32
0.34
0.82
0.44
SkewCBOE
raw
0.95
0.91
0.89
0.91
0.50
0.88
0.95
0.90
0.49
0.81
0.93
0.66
0.50
0.88
0.95
0.90
0.85
0.30
0.43
0.97
SkewNP
raw
0.83
0.84
0.84
0.84
0.47
0.82
0.86
0.89
0.48
0.77
0.85
0.66
0.47
0.82
0.85
0.89
0.32
0.45
0.96
0.96
SkewSmirk
raw
0.53
0.53
0.53
0.53
1.00
0.52
0.50
0.50
0.90
0.47
0.50
0.38
0.52
0.50
0.50
0.60
0.33
0.65
0.63
0.63
SkewBKM
s1
0.89
0.89
0.87
0.89
0.52
1.00
0.87
0.91
0.52
0.89
0.86
0.68
0.87
0.91
0.67
0.36
0.50
0.98
0.95
0.97
SkewCBOE
s1
0.92
0.96
0.93
0.96
0.50
0.87
1.00
0.90
0.51
0.81
0.97
0.67
0.90
0.67
0.97
0.32
0.44
0.99
0.96
0.98
SkewNP
s1
0.89
0.89
0.87
0.89
0.50
0.91
0.90
1.00
0.50
0.84
0.89
0.69
0.65
0.98
0.98
0.32
0.46
0.98
0.97
0.98
SkewSmirk
s1
0.53
0.54
0.55
0.54
0.90
0.52
0.51
0.50
0.56
0.52
0.40
0.97
0.70
0.68
0.66
0.60
0.36
0.66
0.64
0.65
SkewBKM
s2
0.82
0.83
0.82
0.83
0.47
0.90
0.81
0.84
0.81
0.66
0.70
0.63
0.98
0.94
0.95
0.36
0.51
0.93
0.91
0.95
SkewCBOE
s2
0.90
0.94
0.95
0.94
0.50
0.86
0.97
0.89
0.67
0.69
0.94
0.67
0.97
1.00
0.98
0.33
0.44
0.98
0.96
0.98
SkewNP
s2
0.66
0.67
0.67
0.67
0.38
0.68
0.67
0.69
0.55
0.81
0.80
0.52
0.82
0.80
0.81
0.23
0.35
0.79
0.78
0.80
SkewSmirk
s2
0.53
0.53
0.53
0.53
0.52
0.50
0.50
0.97
0.64
0.67
0.52
1.00
0.67
0.67
0.65
0.60
0.33
0.65
0.63
0.63
SkewBKM
s3
0.89
0.89
0.87
0.89
0.87
0.91
0.67
0.70
0.98
0.97
0.82
0.67
1.00
0.97
0.98
0.36
0.50
0.98
0.95
0.97
SkewCBOE
s3
0.92
0.95
0.93
0.96
0.90
0.67
0.97
0.68
0.94
1.00
0.80
0.67
0.97
1.00
0.98
0.32
0.44
0.99
0.96
0.98
SkewNP
s3
0.89
0.89
0.87
0.89
0.65
0.98
0.98
0.66
0.95
0.98
0.81
0.65
0.98
0.98
1.00
0.32
0.46
0.98
0.97
0.98
SkewSmirk
s3
0.95
0.92
0.95
0.71
0.98
0.98
0.97
0.71
0.94
0.97
0.80
0.71
0.98
0.98
0.97
0.35
0.44
0.99
0.94
0.96
SkewMixon
raw
0.97
1.00
0.99
0.71
0.98
0.99
0.98
0.73
0.95
0.99
0.81
0.71
0.98
0.99
0.98
0.36
0.45
0.98
0.95
0.97
SkewMixon
s1
0.97
0.98
0.99
0.72
0.97
0.99
0.97
0.73
0.94
0.99
0.80
0.71
0.97
0.99
0.97
0.38
0.46
0.97
0.95
0.97
SkewMixon
s2
0.99
1.00
0.99
0.71
0.98
0.99
0.98
0.73
0.95
0.99
0.81
0.71
0.98
0.99
0.98
0.36
0.45
0.98
0.95
0.97
Spearman
SkewMixon
s3
Table 4: This table shows the Kendall rank correlation (τ -b) and Spearman’s rank (ρ) correlation coefficients among various
skewness measures in the simulation study in Section 3. Each correlation estimate is calculated based on 8,352 pairs of skewness
estimates (8, 352 = 116 × 9 × 2 × 4, where 116 = 50 variations in truncation error study + 25 variations in discretisation +
41 variations in asymmetric truncation, 9 sets of parameters with 2 maturity terms from 4 models). Kendall correlations are
presented in the highlighted cells in the bottom left part of the table. Spearman correlations are presented in the top right part
of the table.
31
0.16
0.31
0.01
0.18
0.17
0.26
0.00
0.02
0.16
0.31
0.02
0.18
KurtBKM
s1
KurtCBOE
s1
KurtNP
s1
KurtSmirk
s1
KurtBKM
s2
KurtCBOE
s2
KurtNP
s2
KurtSmirk
s2
KurtBKM
s3
KurtCBOE
s3
KurtNP
s3
KurtSmirk
s3
Kendall
0.89
0.13
0.16
0.52
0.45
0.12
0.26
0.49
0.44
0.13
-0.07
0.52
0.45
0.13
0.26
0.08
0.06
0.08
0.09
0.17
KurtBKM
raw
KurtBKM
raw
KurtCBOE
raw
KurtNP
raw
KurtSmirk
raw
KurtTrue
KurtTrue
Name
0.45
0.46
0.14
0.26
0.42
0.43
0.15
-0.06
0.45
0.46
0.15
0.26
0.15
0.17
0.97
0.12
KurtCBOE
raw
0.29
0.36
0.79
0.36
0.27
0.30
0.72
0.19
0.29
0.36
0.79
0.36
0.33
0.20
0.21
0.13
KurtNP
raw
0.31
0.38
0.33
0.71
0.30
0.33
0.32
0.36
0.31
0.38
0.32
0.71
0.23
0.24
0.47
0.21
KurtSmirk
raw
0.99
0.67
0.24
0.45
0.82
0.70
0.22
0.00
0.67
0.24
0.45
0.70
0.63
0.42
0.44
0.20
KurtBKM
s1
0.67
0.99
0.35
0.51
0.58
0.66
0.35
0.14
0.36
0.51
0.83
0.64
0.64
0.51
0.52
0.39
KurtCBOE
s1
0.24
0.35
0.98
0.32
0.25
0.31
0.92
0.23
0.32
0.35
0.50
0.19
0.21
0.91
0.46
0.03
KurtNP
s1
0.45
0.51
0.32
0.99
0.37
0.38
0.29
0.34
0.61
0.67
0.45
0.38
0.38
0.51
0.84
0.22
KurtSmirk
s1
0.82
0.58
0.24
0.37
0.84
0.25
0.00
0.95
0.76
0.37
0.53
0.64
0.58
0.39
0.43
0.21
KurtBKM
s2
0.70
0.66
0.31
0.38
0.32
0.04
0.95
0.87
0.82
0.44
0.55
0.60
0.58
0.44
0.46
0.32
KurtCBOE
s2
0.22
0.35
0.91
0.29
0.26
0.36
0.45
0.33
0.49
0.96
0.41
0.20
0.22
0.86
0.44
0.00
KurtNP
s2
0.00
0.15
0.23
0.34
-0.02
0.04
0.35
-0.02
0.16
0.31
0.40
-0.13
-0.11
0.27
0.45
-0.01
KurtSmirk
s2
0.67
0.24
0.45
0.95
0.87
0.33
-0.02
1.00
0.82
0.35
0.61
0.70
0.63
0.42
0.44
0.20
KurtBKM
s3
0.35
0.51
0.82
0.76
0.82
0.49
0.17
0.83
1.00
0.50
0.67
0.64
0.64
0.51
0.52
0.40
KurtCBOE
s3
0.32
0.35
0.49
0.36
0.44
0.96
0.32
0.35
0.50
1.00
0.46
0.18
0.20
0.91
0.46
0.03
KurtNP
s3
0.61
0.67
0.45
0.53
0.55
0.41
0.40
0.61
0.67
0.45
1.00
0.38
0.38
0.51
0.84
0.22
Spearman
KurtSmirk
s3
Table 5: This table shows the Kendall rank correlation (τ -b) and Spearman’s rank (ρ) correlation coefficients among various
kurtosis measures in the simulation study in Section 3. Each correlation estimate is calculated based on 8,352 pairs of kurtosis
estimates (8, 352 = 116 × 9 × 2 × 4, where 116 = 50 variations in truncation error study + 25 variations in discretisation +
41 variations in asymmetric truncation, 9 sets of parameters with 2 maturity terms from 4 option-price-generation models).
Kendall correlations are presented in the highlighted cells in the bottom left part of the table. Spearman correlations are
presented in the top right part of the table.
Table 6: This table shows the summary statistics of ∆K
in the filtered raw data set from
F0
OptionMetrics, from January 1996 to August 2014. ∆K1 = K2 − K1 , ∆KN = KN − KN −1
and ∆Ki = (Ki+1 − Ki−1 )/2 for i ∈ {2, . . . , N − 1} where strike price is indexed from low to
high. F0 is the forward spot price. In the maturity column, options with time to maturity 1)
between 28 to 32 days are labeled as 1m; 2) between 58 to 62 days as 2m; 3) between 88 and
92 days as 3m; 4) between 180 and 185 days as 6m; and 5) between 360 and 370 days as 12m.
Issue type is defined according to the OptionMetrics Ivy DB reference manual.
Maturity
Issue Type
Proportion
Min
Q1
Mean
Median
Q3
Max
1m
1m
1m
1m
1m
1m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
4.5%
66.1%
15.2%
0.1%
9.6%
4.5%
0.005
0.002
0.001
0.009
0.001
0.007
0.037
0.035
0.011
0.062
0.007
0.087
0.082
0.087
0.022
0.108
0.018
0.130
0.067
0.075
0.016
0.103
0.012
0.123
0.116
0.125
0.026
0.142
0.021
0.162
1.506
2.041
1.638
0.450
0.463
2.099
2m
2m
2m
2m
2m
2m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
4.7%
66.4%
14.1%
0.1%
10.4%
4.3%
0.008
0.004
0.001
0.008
0.001
0.008
0.047
0.051
0.012
0.065
0.008
0.094
0.098
0.107
0.025
0.120
0.020
0.146
0.083
0.094
0.018
0.115
0.014
0.132
0.131
0.141
0.030
0.153
0.024
0.177
1.373
8.887
8.112
0.463
10.922
3.824
3m
3m
3m
3m
3m
3m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
4.3%
61.0%
15.4%
0.1%
15.2%
4.0%
0.008
0.004
0.001
0.008
0.001
0.009
0.045
0.048
0.011
0.067
0.009
0.088
0.097
0.106
0.025
0.133
0.024
0.145
0.079
0.090
0.017
0.122
0.015
0.128
0.130 0.946
0.140 22.270
0.029 0.858
0.168 0.746
0.028 3.205
0.178 2.121
6m
6m
6m
6m
6m
6m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.1%
71.7%
14.1%
0.1%
5.3%
3.6%
0.006
0.004
0.002
0.009
0.001
0.014
0.050
0.053
0.012
0.071
0.011
0.096
0.110
0.118
0.029
0.150
0.028
0.154
0.087
0.096
0.020
0.128
0.018
0.134
0.139 2.553
0.148 24.161
0.035 4.465
0.182 1.384
0.035 1.591
0.184 2.999
12m
12m
12m
12m
12m
12m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
4.9%
71.2%
13.5%
0.1%
9.1%
1.2%
0.009
0.004
0.002
0.030
0.002
0.023
0.054
0.057
0.011
0.110
0.010
0.085
0.128
0.129
0.039
0.155
0.025
0.154
0.095 0.157
0.098 0.159
0.026 0.049
0.130 0.179
0.017 0.035
0.121 0.184
32
3.084
5.602
1.656
0.613
1.988
1.176
Table 7: This table shows the summary statistics of KFmin
of the lowest OTM put option in
0
the filtered raw data set from OptionMetrics, from January 1996 to August 2014. F0 is the
forward spot price. In the maturity column, options with time to maturity 1) between 28 to
32 days are labeled as 1m; 2) between 58 to 62 days as 2m; 3) between 88 and 92 days as 3m;
4) between 180 and 185 days as 6m; and 5) between 360 and 370 days as 12m. Issue type is
defined according to the OptionMetrics Ivy DB reference manual.
Maturity
Issue Type
Proportion
Min
Q1
Mean
Median
Q3
Max
1m
1m
1m
1m
1m
1m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.2%
76.9%
7.4%
0.1%
3.3%
7.0%
0.310
0.169
0.231
0.593
0.248
0.193
0.811
0.804
0.845
0.865
0.803
0.803
0.857
0.851
0.884
0.898
0.854
0.853
0.875
0.867
0.909
0.910
0.867
0.871
0.922
0.916
0.950
0.946
0.921
0.923
1.000
1.000
1.000
1.000
1.000
1.000
2m
2m
2m
2m
2m
2m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.3%
78.0%
6.5%
0.2%
3.3%
6.6%
0.248
0.120
0.154
0.488
0.229
0.162
0.764
0.752
0.807
0.823
0.757
0.775
0.821
0.812
0.857
0.870
0.819
0.830
0.841
0.830
0.887
0.884
0.836
0.851
0.899
0.892
0.941
0.929
0.900
0.907
1.000
1.000
1.000
1.000
1.000
1.000
3m
3m
3m
3m
3m
3m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.2%
75.2%
6.6%
0.2%
6.4%
6.3%
0.234
0.082
0.123
0.525
0.234
0.207
0.671
0.664
0.710
0.779
0.722
0.693
0.755
0.748
0.791
0.837
0.795
0.770
0.775
0.763
0.825
0.856
0.813
0.790
0.854
0.846
0.897
0.906
0.888
0.868
1.000
1.000
1.000
0.993
1.000
0.999
6m
6m
6m
6m
6m
6m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.4%
79.7%
6.1%
0.3%
2.2%
6.3%
0.135
0.051
0.098
0.386
0.087
0.212
0.611
0.605
0.677
0.747
0.668
0.701
0.714
0.708
0.774
0.812
0.751
0.771
0.735
0.725
0.816
0.824
0.774
0.794
0.833
0.824
0.898
0.890
0.861
0.861
1.000
1.000
1.000
0.995
1.000
1.000
12m
12m
12m
12m
12m
12m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.2%
81.1%
6.3%
0.2%
5.0%
2.2%
0.090
0.034
0.059
0.300
0.075
0.180
0.407
0.397
0.463
0.543
0.546
0.513
0.522
0.521
0.589
0.631
0.699
0.636
0.517
0.505
0.579
0.617
0.755
0.632
0.633
0.636
0.716
0.744
0.873
0.759
0.996
1.000
1.000
0.992
0.997
0.999
33
Table 8: This table shows the summary statistics of KFmax
of the highest OTM call option in
0
the filtered raw data set from OptionMetrics. F0 is the forward spot price. In the maturity
column, options with time to maturity 1) between 28 to 32 days are labeled as 1m; 2) between
58 to 62 days as 2m; 3) between 88 and 92 days as 3m; 4) between 180 and 185 days as 6m;
and 5) between 360 and 370 days as 12m. Issue type is defined according to the OptionMetrics
Ivy DB reference manual.
Maturity
Issue Type
Proportion
Min
Q1
Mean
Median
Q3
Max
1m
1m
1m
1m
1m
1m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.2%
76.9%
7.4%
0.1%
3.3%
7.0%
1.000
1.000
1.000
1.001
1.000
1.000
1.067
1.070
1.030
1.053
1.048
1.078
1.145
1.153
1.111
1.107
1.107
1.182
1.113
1.120
1.060
1.090
1.079
1.137
1.187
1.194
1.128
1.143
1.129
1.235
3.455
4.635
6.121
1.682
3.158
3.598
2m
2m
2m
2m
2m
2m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.3%
78.0%
6.5%
0.2%
3.3%
6.6%
1.000
1.000
1.000
1.000
1.000
1.000
1.089
1.095
1.039
1.062
1.065
1.090
1.194
1.206
1.141
1.136
1.143
1.227
1.149
1.160
1.077
1.111
1.108
1.158
1.243 5.552
1.256 10.102
1.157 9.160
1.177 1.967
1.175 12.114
1.274 5.560
3m
3m
3m
3m
3m
3m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.2%
75.2%
6.6%
0.2%
6.4%
6.3%
1.000
1.000
1.000
1.001
1.000
1.000
1.123
1.130
1.058
1.083
1.077
1.116
1.285
1.299
1.213
1.184
1.215
1.345
1.215
1.225
1.116
1.150
1.131
1.228
1.363 4.758
1.375 24.051
1.229 12.015
1.238 1.990
1.218 4.962
1.439 6.261
6m
6m
6m
6m
6m
6m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.4%
79.7%
6.1%
0.3%
2.2%
6.3%
1.000
1.000
1.000
1.000
1.000
1.000
1.146
1.149
1.066
1.115
1.082
1.094
1.364
1.363
1.246
1.231
1.224
1.286
1.269
1.274
1.138
1.184
1.156
1.182
1.471 6.230
1.466 26.094
1.288 22.323
1.293 2.768
1.268 5.271
1.349 6.499
12m
12m
12m
12m
12m
12m
ADR/ADS
Common Stock
ETF
Fund
Market Index
Not Specified
5.2%
81.1%
6.3%
0.2%
5.0%
2.2%
1.001
1.000
1.000
1.028
1.000
1.000
1.339
1.312
1.217
1.191
1.080
1.124
1.831
1.759
1.567
1.418
1.239
1.519
1.593
1.535
1.367
1.356
1.183
1.300
2.016 9.693
1.939 14.005
1.586 14.355
1.490 2.821
1.323 3.854
1.693 5.879
34
Table 9: This table shows the average and the standard deviation of daily Kendall τ -b and
Spearman ρ correlations of various volatility estimates in the empirical study in Section 4. Definition and calculation of each risk-neutral volatility measure is provided in Section 2. Kendall
τ -b correlations are presented in the highlighted cells. Each pair of correlation is first estimated
for all options with maturities of 1-month, 2-month, 3-month, 6-month and 12-month of all
issue types on the daily basis. The average and the standard deviation (shown in parentheses)
are then calculated based on daily correlations from 1996 to 2014. The definition of maturity
is provided in Table 6. The subscript raw refers to a measure based on the raw data set from
OptionMetrics. The subscript s1 refers to a measure based on the smoothing method 1 that
fits a natural cubic spline in interpolating implied volatilities against deltas.
Spearman
CBOE
NP
Smirk
BKM
CBOE
NP
Name
VolBKM
Vol
Vol
Vol
Vol
Vol
Vol
VolSmirk
raw
raw
raw
raw
s1
s1
s1
s1
VolBKM
raw
VolCBOE
raw
VolNP
raw
VolSmirk
raw
VolBKM
s1
VolCBOE
s1
VolNP
s1
VolSmirk
s1
0.91
(0.07)
0.93
(0.17)
0.84
(0.15)
0.80
(0.11)
0.83
(0.17)
0.84
(0.18)
0.70
(0.16)
0.72
(0.17)
0.95
(0.07)
0.87
(0.15)
0.87
(0.15)
0.82
(0.17)
0.84
(0.17)
0.73
(0.15)
0.73
(0.15)
0.71
(0.16)
0.71
(0.16)
0.88
(0.11)
0.90
(0.11)
0.94
(0.07)
0.94
(0.09)
0.94
(0.18)
0.85
(0.16)
0.99
(0.06)
0.89
(0.10)
0.91
(0.09)
0.96
(0.05)
0.96
(0.06)
Kendall
35
0.95
(0.15)
0.86
(0.14)
0.96
(0.11)
0.96
(0.09)
0.97
(0.04)
0.87
(0.10)
0.90
(0.09)
0.95
(0.15)
0.86
(0.14)
0.97
(0.11)
0.97
(0.08)
0.93
(0.18)
0.85
(0.16)
0.99
(0.06)
0.99
(0.03)
0.99
(0.03)
0.96
0.97
(0.10) (0.08)
0.97
0.98
(0.09) (0.07)
0.98
(0.07)
0.94
(0.07)
0.89
(0.10)
0.93
(0.08)
0.94
(0.17)
0.85
(0.16)
0.98
(0.08)
0.99
(0.05)
Table 10: This table shows the average and the standard deviation of daily Kendall τ -b and
Spearman ρ correlations of skewness estimates in the empirical study in Section 4. Definition
and calculation of each risk-neutral skewness measure is provided in Section 2. Kendall τ -b
correlations are presented in the highlighted cells. Each pair of correlation is first estimated for
all options with maturities of 1-month, 2-month, 3-month, 6-month and 12-month of all issue
types on the daily basis. The average and the standard deviation (shown in parentheses) are
then calculated based on daily correlations from 1996 to 2014. The definition of maturity is
provided in Table 6. The subscript raw refers to a measure based on the raw data set from
OptionMetrics. The subscript s1 refers to a measure based on the smoothing method 1 that
fits a natural cubic spline in interpolating implied volatilities against deltas.
Name
SkewBKM
raw
SkewBKM
raw
SkewCBOE
raw
SkewNP
raw
SkewSmirk
raw
SkewMixon
raw
SkewBKM
s1
SkewCBOE
s1
SkewNP
s1
SkewSmirk
s1
SkewMixon
s1
SkewCBOE
raw
SkewNP
raw
SkewSmirk
raw
SkewMixon
raw
SkewBKM
s1
SkewCBOE
s1
SkewNP
s1
SkewSmirk
s1
Spearman
SkewMixon
s1
0.89
(0.10)
0.31
(0.24)
0.29
(0.23)
0.43
(0.24)
0.4
(0.24)
0.68
(0.22)
0.50
(0.22)
0.46
(0.23)
0.83
(0.17)
0.78
(0.18)
0.75
(0.17)
0.68
(0.19)
0.52
(0.23)
0.65
(0.23)
0.68
(0.19)
0.71
(0.18)
0.68
(0.19)
0.52
(0.25)
0.66
(0.24)
0.70
(0.21)
0.29
(0.23)
0.28
(0.23)
0.85
(0.15)
0.70
(0.21)
0.69
(0.21)
0.55
(0.23)
0.53
(0.22)
0.63
(0.21)
0.85
(0.19)
0.76
(0.17)
0.48
(0.22)
0.46
(0.23)
0.71
(0.20)
0.82
(0.16)
0.86
(0.12)
0.97
(0.07)
0.51
(0.24)
0.52
(0.25)
0.77
(0.22)
0.79
(0.22)
0.66
(0.22)
0.70
(0.20)
0.71
(0.21)
0.82
(0.18)
0.81
(0.17)
0.77
(0.11)
0.22
(0.20)
0.31
(0.21)
0.36
(0.19)
0.20
(0.20)
0.29
(0.21)
0.33
(0.20)
0.52
(0.19)
0.68
(0.17)
0.62
(0.16)
0.59
(0.17)
0.55
(0.18)
0.21
(0.20)
0.40
(0.20)
0.36
(0.20)
0.52
(0.18)
0.53
(0.18)
0.19
(0.19)
0.38
(0.19)
0.33
(0.20)
0.38
(0.20)
0.38
(0.22)
0.72
(0.15)
0.48
(0.19)
0.55
(0.18)
0.50
(0.21)
0.51
(0.21)
0.55
(0.19)
0.73
(0.17)
0.67
(0.14)
0.53
(0.18)
0.54
(0.19)
0.54
(0.19)
0.60
(0.16)
0.73
(0.13)
Kendall
36
0.89
(0.08)
0.38
(0.21)
0.61
(0.20)
0.55
(0.19)
0.39
(0.22)
0.64
(0.20)
0.57
(0.19)
0.51
(0.20)
0.67
(0.18)
0.66
(0.16)
Table 11: This table shows the average and the standard deviation of daily Kendall τ -b and
Spearman ρ correlations of excess kurtosis estimates in the empirical study in Section 4. Definition and calculation of each risk-neutral kurtosis measure is provided in Section 2. Kendall
τ -b correlations are presented in the highlighted cells. Each pair of correlation is first estimated
for all options with maturities of 1-month, 2-month, 3-month, 6-month and 12-month of all
issue types on the daily basis. The average and the standard deviation (shown in parentheses)
are then calculated based on daily correlations from 1996 to 2014. The definition of maturity
is provided in Table 6. The subscript raw refers to a measure based on the raw data set from
OptionMetrics. The subscript s1 refers to a measure based on the smoothing method 1 that
fits a natural cubic spline in interpolating implied volatilities against deltas.
Spearman
CBOE
NP
Smirk
BKM
CBOE
NP
Name
KurtBKM
Kurt
Kurt
Kurt
Kurt
Kurt
Kurt
KurtSmirk
raw
raw
raw
raw
s1
s1
s1
s1
KurtBKM
raw
KurtCBOE
raw
KurtNP
raw
KurtSmirk
raw
KurtBKM
s1
KurtCBOE
s1
KurtNP
s1
KurtSmirk
s1
0.61
(0.22)
0.05
(0.22)
-0.14
(0.23)
0.52
(0.24)
0.04
(0.19)
0.05
(0.24)
-0.10
(0.19)
0.03
(0.22)
0.14
(0.21)
0.49
(0.18)
0.48
(0.19)
0.07
(0.17)
0.17
(0.22)
0.30
(0.24)
0.31
(0.24)
0.05
(0.18)
0.11
(0.20)
0.14
(0.20)
0.14
(0.21)
0.25
(0.21)
0.14
(0.20)
0.09
(0.30)
0.06
(0.27)
0.19
(0.25)
0.25
(0.31)
0.27
(0.31)
0.33
(0.25)
0.59
(0.22)
Kendall
37
0.64
(0.19)
0.39
(0.26)
0.2
(0.24)
0.34
(0.40)
0.94
(0.06)
0.21
(0.23)
0.28
(0.26)
0.62
(0.19)
0.4
(0.26)
0.2
(0.25)
0.36
(0.40)
0.1
(0.21)
0.08
(0.22)
0.33
(0.26)
0.43
(0.30)
0.24
(0.27)
0.16
(0.24)
0.2
(0.24)
0.73
(0.26)
0.98
(0.04)
0.28
(0.28)
0.3
(0.29)
0.39
(0.33)
0.4
(0.33)
0.44
(0.30)
0.22
(0.24)
0.29
(0.26)
0.33
(0.25)
Table 12: This table shows the average and the standard deviation of percentage of matched
securities in the top and bottom skewness quintile portfolios in the empirical study in Section 4.
Definition and calculation of each risk-neutral skewness measure is provided in Section 2. On
the last trading day of each month t, stocks are sorted in ascending order by the corresponding
skewness measure. Each skewness measure is calculated from its options with the shortest
maturity (with at least 10 days to maturity) on that day. Quintile 5 (1) includes stocks
with skewness measure that is above the 80th percentile (below the 20th percentile). For each
month from January 1996 to August 2014, we first count the number of matching stocks from
each pairwise skewness portfolios and divide this number by the total number of stocks in
each portfolio to estimate the percentage. The average and the standard deviation (shown
in parentheses) of percentages are then calculated for the whole period. The subscript raw
refers to a measure based on the raw data set from OptionMetrics. The subscript s1 refers to
a measure based on the smoothing method 1 that fits a natural cubic spline in interpolating
implied volatilities against deltas.
Name
SkewBKM
raw
SkewBKM
raw
SkewCBOE
raw
SkewNP
raw
SkewSmirk
raw
SkewMixon
raw
SkewBKM
s1
SkewCBOE
s1
SkewNP
s1
SkewSmirk
s1
SkewMixon
s1
SkewCBOE
raw
SkewNP
raw
SkewSmirk
raw
SkewMixon
raw
SkewBKM
s1
SkewCBOE
s1
SkewNP
s1
SkewSmirk
s1
Quintile 1
SkewMixon
s1
0.82
(0.05)
0.28
(0.07)
0.3
(0.07)
0.37
(0.06)
0.38
(0.05)
0.56
(0.07)
0.37
(0.06)
0.35
(0.06)
0.70
(0.06)
0.61
(0.07)
0.69
(0.05)
0.63
(0.05)
0.42
(0.09)
0.50
(0.09)
0.51
(0.08)
0.66
(0.06)
0.63
(0.05)
0.43
(0.09)
0.51
(0.08)
0.52
(0.07)
0.30
(0.06)
0.33
(0.06)
0.74
(0.04)
0.60
(0.06)
0.55
(0.08)
0.47
(0.07)
0.46
(0.06)
0.52
(0.06)
0.83
(0.03)
0.61
(0.05)
0.43
(0.06)
0.43
(0.04)
0.59
(0.07)
0.69
(0.05)
0.68
(0.06)
0.92
(0.04)
0.43
(0.09)
0.45
(0.09)
0.59
(0.08)
0.60
(0.07)
0.55
(0.07)
0.57
(0.09)
0.59
(0.09)
0.69
(0.07)
0.69
(0.04)
0.49
(0.09)
0.49
(0.04)
0.49
(0.05)
0.49
(0.04)
0.38
(0.05)
0.38
(0.06)
0.38
(0.06)
0.81
(0.05)
0.94
(0.05)
0.83
(0.04)
0.58
(0.04)
0.55
(0.04)
0.47
(0.04)
0.53
(0.04)
0.47
(0.04)
0.42
(0.06)
0.42
(0.06)
0.36
(0.05)
0.41
(0.06)
0.36
(0.06)
0.77
(0.04)
0.79
(0.05)
0.85
(0.03)
0.80
(0.05)
0.82
(0.04)
0.82
(0.04)
0.86
(0.05)
0.81
(0.04)
0.90
(0.03)
0.83
(0.04)
0.79
(0.04)
0.81
(0.04)
0.82
(0.04)
0.82
(0.04)
0.85
(0.03)
Quintile 5
38
0.91
(0.03)
0.76
(0.05)
0.87
(0.04)
0.78
(0.04)
0.78
(0.05)
0.91
(0.04)
0.81
(0.05)
0.81
(0.05)
0.94
(0.05)
0.83
(0.04)
Table 13: This table shows the excess return performance, measured by ln(Pt+1 /Pt ) − Rf , of
stock portfolios sorted on the basis of risk-neutral skewness measures of individual stock, during
the period from January 1996 to August 2014. Definition and calculation of each risk-neutral
skewness measure is provided in Section 2. On the last trading day of each month t, stocks
are sorted in ascending order by each skewness measure. For each stock, skewness measures
are calculated from its options with the shortest maturity (with at least 10 days to maturity)
on that day. Quintile 5 (1) includes stocks with the top (bottom) 20th percentile of skewness
measure. We then calculate the equally-weighted returns of these portfolios at the end of the
following month t + 1. The excess return is then obtained by subtracting the monthly risk-free
return from the portfolio return. The adjusted close prices (for dividend splits etc) at time t
and t + 1 are used to calculate the return. Quintile 5 − 1 is a hypothetical portfolio that takes
long positions in quintile 5 and short positions in quintile 1. We do not consider cost for short
selling or other transaction related costs. Mean return reports the average monthly portfolio
excess return in the sample period. αFF5 stands for the monthly portfolio alpha estimated
from the Fama-French 5-factor model. The portfolio loadings β’s with respect to the market
(MKT), size (SMB), value (HML), profitability (RMW) and investment patterns (CMA) are
also reported as well as the explanatory power of the model (adjusted R2 ). t-values calculated
using Newey-West standard errors with 4 lags are provided in parentheses. ***, **, and *
indicate statistical significance at the 1%, 5% and 10% level, respectively.
The table is presented on the following page.
39
Table 13: Continued
Log Ret
SkewBKM
raw
SkewCBOE
raw
SkewNP
raw
SkewSmirk
raw
SkewMixon
raw
SkewBKM
s1
SkewCBOE
s1
SkewNP
s1
SkewSmirk
s1
SkewMixon
s1
Quintiles
Mean excess return
1 (lowest)
-0.004
5 (highest)
-0.008
5-1
-0.007
1 (lowest)
-0.005
5 (highest)
-0.006
5-1
-0.003
1 (lowest)
-0.016
5 (highest)
-0.003
5-1
0.011
1 (lowest)
-0.010
5 (highest)
-0.002
5-1
0.006
1 (lowest)
-0.008
5 (highest)
-0.003
5-1
0.004
1 (lowest)
-0.007
5 (highest)
-0.003
5-1
0.001
1 (lowest)
-0.007
5 (highest)
-0.003
5-1
0.002
1 (lowest)
-0.016
5 (highest)
-0.003
5-1
0.011
1 (lowest)
-0.009
5 (highest)
-0.004
5-1
0.003
1 (lowest)
-0.010
5 (highest)
-0.003
5-1
0.005
αFF5
βMKT
βSMB
βHML
βRMW
βCMA
Adj-R2
∗∗∗
∗∗∗
∗∗∗
∗
-0.072
(-1.086)
-0.413∗∗
(-2.320)
-0.342
(-1.634)
-0.123
(-1.386)
-0.530∗∗
(-2.472)
-0.407
(-1.513)
0.915
-0.010
(-7.271)
-0.015∗∗∗
(-7.493)
-0.007∗∗∗
(-2.940)
1.026
0.433
(27.679) (7.866)
1.358∗∗∗ 0.580∗∗∗
(20.348) (4.855)
0.333∗∗∗
0.150
(3.729) (0.989)
0.108
(1.886)
0.295∗∗
(2.502)
0.185
(1.276)
-0.012∗∗∗
(-8.475)
-0.014∗∗∗
(-8.783)
-0.004∗∗
(-2.184)
1.071∗∗∗ 0.484∗∗∗ 0.111∗∗
(33.828) (9.214) (2.221)
1.331∗∗∗ 0.572∗∗∗ 0.323∗∗∗
(23.382) (5.374) (3.095)
0.261∗∗∗
0.090 0.211∗∗
(3.850) (0.734) (1.906)
-0.023∗∗∗
(-13.067)
-0.011∗∗∗
(-6.573)
0.010∗∗∗
(6.170)
1.373∗∗∗
(33.020)
1.232∗∗∗
(21.684)
-0.140∗∗
(-2.425)
0.888
0.351
-0.138∗∗ -0.147∗∗
(-2.159) (-2.033)
-0.157 -0.506∗∗∗
(-1.263) (-3.267)
-0.020
-0.359∗
(-0.138) (-1.886)
0.925
0.148
(1.652)
0.159
(1.506)
0.010
(0.093)
-0.408∗∗∗ -0.454∗∗∗
(-3.610) (-3.452)
-0.070 -0.374∗∗∗
(-0.669) (-2.979)
0.337∗∗∗
0.079
(3.527)
(0.543)
0.917
-0.018∗∗∗
(-10.285)
-0.011∗∗∗
(-7.132)
0.005∗∗∗
(2.908)
1.234∗∗∗ 0.531∗∗∗ 0.339∗∗∗
(30.438) (6.578) (4.426)
1.235∗∗∗ 0.625∗∗∗ 0.235∗∗
(23.864) (7.925) (2.149)
0.002
0.097
-0.106
(0.042) (1.493) (-0.970)
-0.186∗ -0.419∗∗∗
(-1.775) (-4.088)
-0.080
0.279∗∗
(-0.810) (-2.400)
0.105
0.140
(1.214)
(1.278)
0.909
-0.016∗∗∗
(-9.540)
-0.011∗∗∗
(-6.952)
0.003∗
(1.906)
1.165∗∗∗ 0.463∗∗∗ 0.237∗∗∗
(28.795) (6.834) (3.303)
1.256∗∗∗ 0.602∗∗∗ 0.274∗∗∗
(24.232) (6.841) (2.599)
0.092∗∗ 0.141∗∗
0.036
(2.247) (2.553) (0.404)
-0.037 -0.278∗∗∗
(-0.395) (-3.509)
-0.095 -0.330∗∗
(-0.925) (-2.495)
-0.058
-0.051
(-0.586) (-0.426)
0.913
-0.014∗∗∗
(-8.676)
-0.012∗∗∗
(-7.058)
-0.000
(-0.156)
1.100∗∗∗ 0.417∗∗∗
(31.840) (7.793)
1.289∗∗∗ 0.587∗∗∗
(23.516) (5.763)
0.190∗∗∗
0.172
(3.137) (1.641)
0.131∗∗
(2.326)
0.291∗∗
(2.565)
0.158
(1.397)
-0.076 -0.275∗∗∗
(-1.052) (-3.378)
-0.127 -0.352∗∗
(-0.975) (-2.324)
-0.052
-0.077
(-0.390) (-0.429)
0.925
-0.014∗∗∗
(-8.578)
-0.011∗∗∗
(-6.917)
0.001
(0.258)
1.112∗∗∗ 0.427∗∗∗
(30.684) (7.748)
1.295∗∗∗ 0.594∗∗∗
(23.706) (5.866)
0.184∗∗∗
0.169∗
(3.118) (1.667)
0.135∗∗
(2.315)
0.294∗∗
(2.534)
0.158
(1.362)
-0.076 -0.303∗∗∗
(-1.016) (-3.680)
-0.112 -0.308∗∗
(-0.889) (-2.084)
-0.037
-0.004
(-0.284) (-0.025)
0.921
0.237∗∗ -0.442∗∗∗ -0.495∗∗∗
(2.449) (-3.736) (-3.603)
0.140
-0.074 -0.312∗∗∗
(1.380) (-0.743) (-2.733)
-0.098 0.366∗∗∗
0.183
(-0.825)
(3.955)
(1.228)
0.916
-0.024∗∗∗ 1.413∗∗∗
(-13.204) (31.352)
-0.011∗∗∗ 1.218∗∗∗
(-7.232) (22.184)
0.010∗∗∗ -0.194∗∗∗
(6.698) (-3.338)
0.651∗∗∗
(6.747)
0.651∗∗∗
(7.189)
0.002
(0.023)
0.648∗∗∗
(6.220)
0.656∗∗∗
(8.151)
0.010
(0.125)
-0.017∗∗∗
(-9.948)
-0.012∗∗∗
(-6.901)
0.003
(1.298)
1.169∗∗∗ 0.556∗∗∗ 0.310∗∗∗
(32.580) (9.249) (4.689)
1.269∗∗∗ 0.592∗∗∗ 0.264∗∗
(21.475) (5.951) (2.125)
0.101
0.038
-0.047
(1.528) (0.382) (-0.323)
-0.143∗ -0.336∗∗∗
(-1.833) (-3.758)
-0.126 -0.363∗∗
(-0.984) (-2.476)
0.016
-0.027
(0.134) (-0.166)
-0.017∗∗∗
(-9.995)
-0.012∗∗∗
(-7.421)
0.004∗∗
(2.226)
1.168∗∗∗ 0.501∗∗∗ 0.285∗∗∗
(28.831) (7.254) (3.824)
1.248∗∗∗ 0.596∗∗∗ 0.255∗∗
(23.160) (6.913) (2.341)
0.081∗
0.097
-0.031
(1.734) (1.534) (-0.292)
-0.077 -0.375∗∗∗
(-0.768) (-3.982)
-0.060 -0.278∗∗
(-0.573) (-2.255)
0.016
0.097
(0.163)
(0.749)
40
0.909
0.216
0.912
0.246
0.913
0.004
0.909
0.106
0.901
0.161
0.902
0.144
0.921
0.297
0.916
0.898
0.021
0.909
0.913
0.03
Table 14: This table shows the excess return performance, measured by (Pt+1 − Pt )/Pt − Rf , of
stock portfolios sorted on the basis of risk-neutral skewness measures of individual stock, during
the period from January 1996 to August 2014. Definition and calculation of each risk-neutral
skewness measure is provided in Section 2. On the last trading day of each month t, stocks
are sorted in ascending order by each skewness measure. For each stock, skewness measures
are calculated from its options with the shortest maturity (with at least 10 days to maturity)
on that day. Quintile 5 (1) includes stocks with the top (bottom) 20th percentile of skewness
measure. We then calculate the equally-weighted returns of these portfolios at the end of the
following month t + 1. The excess return is then obtained by subtracting the monthly risk-free
return from the portfolio return. The adjusted close prices (for dividend splits etc) at time t
and t + 1 are used to calculate the return. Quintile 5 − 1 is a hypothetical portfolio that takes
long positions in quintile 5 and short positions in quintile 1. We do not consider cost for short
selling or other transaction related costs. Mean return reports the average monthly portfolio
excess return in the sample period. αFF5 stands for the monthly portfolio alpha estimated
from the Fama-French 5-factor model. The portfolio loadings β’s with respect to the market
(MKT), size (SMB), value (HML), profitability (RMW) and investment patterns (CMA) are
also reported as well as the explanatory power of the model (adjusted R2 ). t-values calculated
using Newey-West standard errors with 4 lags are provided in parentheses. ***, **, and *
indicate statistical significance at the 1%, 5% and 10% level, respectively.
The table is presented on the following page.
41
Table 14: Continued
Simp Ret
SkewBKM
raw
SkewCBOE
raw
SkewNP
raw
SkewSmirk
raw
SkewMixon
raw
SkewBKM
s1
SkewCBOE
s1
SkewNP
s1
SkewSmirk
s1
SkewMixon
s1
Quintiles
Mean excess return
1 (lowest)
0.004
5 (highest)
0.009
5-1
0.003
1 (lowest)
0.004
5 (highest)
0.008
5-1
0.002
1 (lowest)
0.001
5 (highest)
0.010
5-1
0.007
1 (lowest)
0.002
5 (highest)
0.010
5-1
0.006
1 (lowest)
0.002
5 (highest)
0.011
5-1
0.006
1 (lowest)
0.003
5 (highest)
0.010
5-1
0.005
1 (lowest)
0.002
5 (highest)
0.011
5-1
0.006
1 (lowest)
0.001
5 (highest)
0.010
5-1
0.007
1 (lowest)
0.002
5 (highest)
0.010
5-1
0.006
1 (lowest)
0.001
5 (highest)
0.010
5-1
0.006
αFF5
βMKT
βSMB
βHML
βRMW
βCMA
Adj-R2
∗∗
∗∗∗
∗∗∗
0.931
-0.003
(-2.541)
0.001
(0.487)
0.002
(0.556)
0.980
(33.720)
1.301∗∗∗
(17.785)
0.322∗∗∗
(3.363)
0.441
(8.777)
0.646∗∗∗
(5.259)
0.207
(1.308)
0.060
(1.145)
0.197
(1.501)
0.136
(0.823)
-0.078
(-1.164)
-0.322∗∗
(-2.065)
-0.245
(-1.171)
-0.043
(-0.483)
-0.359∗
(-1.685)
-0.316
(-1.124)
-0.003∗∗
(-2.537)
0.000
(0.057)
0.000
(0.312)
1.024∗∗∗ 0.493∗∗∗
(39.335) (10.130)
1.277∗∗∗ 0.611∗∗∗
(22.012) (5.413)
0.254∗∗∗
0.120
(3.566) (0.909)
0.062
(1.327)
0.240∗∗
(2.119)
0.176
(1.391)
-0.150∗∗
(-2.475)
-0.071
(-0.590)
0.078
(0.488)
-0.059
(-0.790)
-0.373∗∗
(-2.347)
-0.314
(-1.524)
-0.007∗∗∗
(-4.392)
0.002
(1.596)
0.007∗∗∗
(4.456)
1.297∗∗∗
(27.035)
1.178∗∗∗
(25.426)
-0.118∗∗
(-2.109)
0.714∗∗∗
(7.615)
0.685∗∗∗
(8.061)
-0.027
(-0.324)
0.055 -0.348∗∗∗
(0.624) (-3.752)
0.081
-0.039
(0.810) (-0.461)
0.024 0.308∗∗∗
(0.229)
(3.554)
-0.301∗∗
(-2.285)
-0.249∗∗
(-2.208)
0.051
(0.342)
0.916
-0.006∗∗∗
(-4.661)
0.002
(1.241)
0.006∗∗∗
(3.780)
1.179∗∗∗
(30.398)
1.185∗∗∗
(26.655)
0.007
(0.148)
0.552∗∗∗ 0.274∗∗∗
(7.185) (3.530)
0.653∗∗∗
0.158
(8.146) (1.459)
0.103
-0.118
(1.566) (-1.138)
-0.136 -0.304∗∗∗
(-1.541) (-3.065)
-0.054
-0.155
(-0.602) (-1.351)
0.081
0.149
(1.021)
(1.368)
0.916
-0.005∗∗∗
(-4.584)
0.002
(1.365)
0.005∗∗∗
(3.510)
1.115∗∗∗
(35.555)
1.204∗∗∗
(26.744)
0.090∗∗
(2.350)
0.476∗∗∗ 0.182∗∗∗
(7.568) (3.159)
0.641∗∗∗
0.183∗
(7.214) (1.803)
0.167∗∗∗
-0.001
(2.997) (-0.006)
-0.007 -0.193∗∗∗
(-0.091) (-2.756)
-0.054
-0.207
(-0.558) (-1.641)
-0.047
-0.014
(-0.506) (-0.120)
0.928
-0.004∗∗∗
(-3.927)
0.002
(0.852)
0.003
(1.613)
1.044∗∗∗ 0.434∗∗∗
(42.364) (10.558)
1.241∗∗∗ 0.618∗∗∗
(23.353) (5.932)
0.198∗∗∗
0.186∗
(3.072) (1.661)
0.073
(1.644)
0.203∗
(1.736)
0.128
(1.013)
-0.072
(-1.311)
-0.070
(-0.581)
0.001
(0.005)
-0.187∗∗
(-2.478)
-0.223
(-1.472)
-0.036
(-0.184)
0.943
-0.004∗∗∗
(-3.851)
0.002
(1.039)
0.004∗
(1.832)
1.053∗∗∗ 0.446∗∗∗
(41.892) (10.424)
1.249∗∗∗ 0.626∗∗∗
(24.028) (6.090)
0.198∗∗∗
0.183∗
(3.208) (1.705)
0.076∗
(1.676)
0.212∗
(1.782)
0.134
(1.064)
-0.071 -0.216∗∗∗
(-1.229) (-2.869)
-0.066
-0.182
(-0.561) (-1.231)
0.004
0.034
(0.029)
(0.181)
-0.007∗∗∗ 1.337∗∗∗
(-4.401) (24.048)
0002 1.162∗∗∗
(1.203) (26.525)
0.006∗∗∗ -0.173∗∗∗
(4.270) (-2.872)
0.706∗∗∗
(6.823)
0.687∗∗∗
(9.341)
-0.017
(-0.194)
0.144 -0.382∗∗∗
(1.389) (-3.772)
0.062
-0.052
(0.656) (-0.651)
-0.084 0.329∗∗∗
(-0.709)
(3.853)
0.935
0.889
0.167
0.912
0.224
0.908
0.011
0.905
0.134
0.889
0.142
0.939
0.892
0.141
0.910
0.928
1.112∗∗∗ 0.576∗∗∗ 0.249∗∗∗
(34.154) (10.095) (3.856)
1.216∗∗∗ 0.625∗∗∗
0.177
(23.449) (6.271) (1.473)
0.106
0.051
-0.073
(1.630) (0.494) (-0.508)
-0.122∗ -0.248∗∗∗
(-1.960) (-2.962)
-0.076
-0.229
(-0.666) (-1.623)
0.044
0.019
(0.369)
(0.113)
-0.006∗∗∗
(-5.204)
0.001
(0.755)
0.005∗∗
(3.551)
1.113∗∗∗
(37.040)
1.195∗∗∗
(26.069)
0.083∗
(1.880)
-0.040 -0.278∗∗∗
(-0.494) (-3.465)
-0.025
-0.148
(-0.267) (-1.272)
0.013
0.129
(0.139)
(1.023)
42
0.282
-0.342∗∗
(-2.399)
-0.185∗
(-1.830)
0.157
(1.025)
-0.006∗∗∗
(-5.105)
0.001
(0.794)
0.005∗∗
(2.489)
0.512∗∗∗ 0.221∗∗∗
(8.363) (3.827)
0.624∗∗∗
0.165
(7.426) (1.562)
0.114∗
-0.058
(1.856) (-0.567)
0.863
0.923
0.259
0.892
0.021
0.925
0.911
0.053
43
st
Estimated Moment−True Moment
.
True Moment
Base
case parameters used in each model are described in Table 2. In each figure, the 1 column of plots illustrates approximation errors using the raw
data from simulations. The 2nd column applies the smoothing method 1 by fitting a natural cubic spline in interpolating implied volatilities against
deltas. The 3rd column applies the smoothing method 2 by fitting a natural cubic spline in interpolating implied volatilities against strike prices. The
4th column applies the smoothing method 3 by linearly interpolating implied volatilities against deltas. In Figures 2, 4, 5, 7, 8 and 10, each moment
is calculated using: 1) BKM method in the 1st row; 2) CBOE method in the 2nd row; 3) non-parametric method in the 3rd row; and 4) implied
volatility smirk in the 4th row. In Figures 3, 6 and 9, moment in the additional 5th row is calculated using Mixon’s method. For example, Panel 3b
refers to CBOE moments calculated from the second smoothing method. Methods to calculate each moment are outlined in Section 2. Within each
panel, the 1st (2nd ) column reports approximation errors using options with expiration of 22 (124) trading days. Within each panel, options prices are
simulated using 1) Black-Scholes model in the 1st row; 2) Bates stochastic volatility and jump diffusion model in the 2nd row; 3) Heston stochastic
volatility model in the 3rd row; and 4) Merton jump-diffusion model in the 4th row. In each plot, different shades of colour represents results from
different parameters used to generate option prices.
Figure 1: This figure describes the layout and content of Figures 2 to 10. Approximation errors are defined as
44
errors on the vertical axis are plotted against integration domain width u on the horizontal axis, which is defined as [S0 ∗ (u ∗ 0.01 + 0.49), S0 /(u ∗
0.01 + 0.49)] where u ∈ {1, 2, . . . , 50}. Different shades of colour represents results from different parameters used to generate option prices.
Figure 2: Volatility Approximation Error - Integration domain truncation. Layout of the figure is explained in Figure 1. In each plot, approximation
45
errors on the vertical axis are plotted against integration domain width u on the horizontal axis, which is defined as [S0 ∗ (u ∗ 0.01 + 0.49), S0 /(u ∗
0.01 + 0.49)] where u ∈ {1, 2, . . . , 50}. Different shades of colour represents results from different parameters used to generate option prices.
Figure 3: Skewness Approximation Error - Integration domain truncation. Layout of the figure is explained in Figure 1. In each plot, approximation
46
on the vertical axis are plotted against integration domain width u on the horizontal axis, which is defined as [S0 ∗ (u ∗ 0.01 + 0.49), S0 /(u ∗ 0.01 + 0.49)]
where u ∈ {1, 2, . . . , 50}. Different shades of colour represents results from different parameters used to generate option prices.
Figure 4: Kurtosis Approximation Error - Integration domain truncation. Layout of figure is explained in Figure 1. In each plot, approximation errors
47
errors on the vertical axis are plotted against strike price interval ∆K on the horizontal axis, which is defined as ∆K ≡ Ki −Ki−1 , ∆K ∈ {1, 2, . . . , 25}.
Different shades of colour represents results from different parameters used to generate option prices.
Figure 5: Volatility Approximation Error - Discretisation of strike prices. Layout of the figure is explained in Figure 1. In each plot, approximation
48
errors on the vertical axis are plotted against strike price interval ∆K on the horizontal axis, which is defined as ∆K ≡ Ki −Ki−1 , ∆K ∈ {1, 2, . . . , 25}.
Different shades of colour represents results from different parameters used to generate option prices.
Figure 6: Skewness Approximation Error - Discretisation of strike prices. Layout of the figure is explained in Figure 1. In each plot, approximation
49
on the vertical axis are plotted against strike price interval ∆K on the horizontal axis, which is defined as ∆K ≡ Ki − Ki−1 , ∆K ∈ {1, 2, . . . , 25}.
Different shades of colour represents results from different parameters used to generate option prices.
Figure 7: Kurtosis Approximation Error - Discretisation of strike prices. Layout of figure is explained in Figure 1. In each plot, approximation errors
50
plot, approximation errors on the vertical axis are plotted against asymmetry in integration domain du on the horizontal axis, which is defined as
[S0 ∗ (0.49 + δu/100), S0 /(0.91 − δu/100)] where δu ∈ {1, 2, . . . , 41}. The asymmetry is at its minimum when δu = 21. Different shades of colour
represents results from different parameters used to generate option prices.
Figure 8: Volatility Approximation Error - Asymmetry in integration domain truncation. Layout of the figure is explained in Figure 1. In each
51
plot, approximation errors on the vertical axis are plotted against asymmetry in integration domain du on the horizontal axis, which is defined as
[S0 ∗ (0.49 + δu/100), S0 /(0.91 − δu/100)] where δu ∈ {1, 2, . . . , 41}. The asymmetry is at its minimum when δu = 21. Different shades of colour
represents results from different parameters used to generate option prices.
Figure 9: Skewness Approximation Error - Asymmetry in integration domain truncation. Layout of figure is explained in Figure 1. In each
52
plot, approximation errors on the vertical axis are plotted against asymmetry in integration domain du on the horizontal axis, which is defined as
[S0 ∗ (0.49 + δu/100), S0 /(0.91 − δu/100)] where δu ∈ {1, 2, . . . , 41}. The asymmetry is at its minimum when δu = 21. Different shades of colour
represents results from different parameters used to generate option prices.
Figure 10: Kurtosis Approximation Error - Asymmetry in integration domain truncation. Layout of figure is explained in Figure 1. In each
53
true measure in Section 3. For each volatility estimate that is computed using BKM, CBOE, NP and Smirk, there are 72 volatility proxies with 72
corresponding true volatilities from each of 116 variations that study approximation errors (where 116 = 50 variations in truncation error study + 25
variations in discretisation + 41 variations in asymmetric truncation and 72 = 9 sets of parameters x 2 maturity terms x 4 option-price-generation
models). In each of 116 variations, we sort volatility estimates from one of 4 methods (BKM, CBOE etc) in ascending order. We extract estimates
that are above the 80th percentile (a quintile of 72 items is roughly 15). As each volatility estimate has a corresponding true volatility, we then
calculate the percentage of these corresponding true volatilities are also above the 80th percentile of true volatilities. Each boxplot illustrates the
distribution of these 116 percentages. We then repeat this for skewness and kurtosis estimates. In the boxplot panel with the title “Raw”, we
perform this process for the raw data from simulations. The title “Smooth1” means we use the smoothing method 1 by fitting a natural cubic spline
in interpolating implied volatilities against deltas. The title “Smooth2” means we use the smoothing method 2 by fitting a natural cubic spline in
interpolating implied volatilities against strike prices. The title “Smooth3” means we use the smoothing method 3 by linearly interpolating implied
volatilities against deltas.
Figure 11: This figure presents boxplots of percentages of matching items in the top quintile between each moment measure and the corresponding
54
true measure in Section 3. For each volatility estimate that is computed using BKM, CBOE, NP and Smirk methods, there are 72 volatility proxies
with 72 corresponding true volatilities from each of 116 variations that study approximation errors (where 116 = 50 variations in truncation error
study + 25 variations in discretisation + 41 variations in asymmetric truncation and 72 = 9 sets of parameters x 2 maturity terms x 4 option-pricegeneration models). In each of 116 variations, we sort volatility estimates from one of 4 methods (BKM, CBOE etc) in ascending order. We extract
estimates that are below the 20th percentile (a quintile of 72 items is roughly 15). As each volatility estimate has a corresponding true volatility, we
then calculate the percentage of these corresponding true volatilities are also below the 20th percentile of true volatilities. Each boxplot illustrates
the distribution of these 116 percentages. We then repeat this for skewness and kurtosis estimates. In the boxplot panel with the title “Raw”, we
perform this process for the raw data from simulations. The title “Smooth1” means we use the smoothing method 1 by fitting a natural cubic spline
in interpolating implied volatilities against deltas. The title “Smooth2” means we use the smoothing method 2 by fitting a natural cubic spline in
interpolating implied volatilities against strike prices. The title “Smooth3” means we use the smoothing method 3 by linearly interpolating implied
volatilities against deltas.
Figure 12: This figure presents boxplots of percentages of matching items in the bottom quintile between each moment measure and the corresponding
55
return is calculated as ln(Pt+1 /Pt ) − Rf . We first form stock portfolios on the basis of each risk-neutral skewness measure. Definition and calculation
of each risk-neutral skewness measure is provided is Section 2. On the last trading day of each month t, stocks are sorted in ascending order by each
skewness measure. For each stock, skewness measures are calculated from its options with the shortest maturity (with at least 10 days to maturity) on
that day. Quintile 5 (1) includes stocks with the top (bottom) 20% skewness measure. We calculate the equally-weighted returns of these portfolios
at the end of the following month t + 1. The excess return is obtained by subtracting the monthly risk-free return from the portfolio return. The
adjusted close prices (for dividend splits etc) at time t and t + 1 are used to calculate the return. The top panel shows the colour key used to represent
excess returns. The top panel also presents the histogram of all monthly excess returns of all skewness-quintile portfolios. The bottom panel presents
the heat map, the time is shown on the horizontal axis where each skewness-quintile portfolio is illustrated along the vertical axis.
Figure 13: This figure present a heat map of monthly excess returns of skewness-quintile portfolios from February 1996 to August 2014. The excess
Figure 14: This figure present a heat map of average risk-neutral volatility of skewness-quintile
portfolios from February 1996 to August 2014. The risk-neutral volatility is calculated as
BKM
. We first form stock portfolios on the basis of each risk-neutral skewness
VolBKM
raw and Vols1
measure. Definition and calculation of each risk-neutral skewness measure is provided is Section 2. On the last trading day of each month t, stocks are sorted in ascending order by each
skewness measure. For each stock, skewness measures are calculated from its options with the
shortest maturity (with at least 10 days to maturity) on that day. Quintile 5 (1) includes stocks
with the top (bottom) 20% skewness measure. We calculate the average risk-neutral volatility
of each portfolio. The panel above the heat map shows the colour key used to represent the
average risk-neutral volatility. It also presents the histogram of all average risk-neutral volatility of all skewness-quintile portfolios. In the heat map, the time is shown on the horizontal
axis where each skewness-quintile portfolio is illustrated along the vertical axis. Results using
BKM
VolBKM
) is included in the top (bottom) panel.
raw (Vols1
Figures are presented on the next page.
56
Figure 14: Continued
VolBKM
raw
VolBKM
s1
57
Figure 15: This figure present a heat map of average risk-neutral excess kurtosis of skewnessquintile portfolios from February 1996 to August 2014. The risk-neutral excess kurtosis is
BKM
calculated as KurtBKM
. We first form stock portfolios on the basis of each riskraw and Kurts1
neutral skewness measure. Definition and calculation of each risk-neutral skewness measure is
provided is Section 2. On the last trading day of each month t, stocks are sorted in ascending
order by each skewness measure. For each stock, skewness measures are calculated from its
options with the shortest maturity (with at least 10 days to maturity) on that day. Quintile
5 (1) includes stocks with the top (bottom) 20% skewness measure. We calculate the average
risk-neutral excess kurtosis of each portfolio. The panel above the heat map shows the colour
key used to represent the average risk-neutral excess kurtosis. It also presents the histogram
of all average risk-neutral excess kurtosis of all skewness-quintile portfolios. In the heat map,
the time is shown on the horizontal axis where each skewness-quintile portfolio is illustrated
along the vertical axis. Results using KurtBKM
(KurtBKM
) is included in the top (bottom)
raw
s1
panel. There are some extreme outliers (where excess kurtosis KurtBKM
exceeds 60) presented
s1
in portfolios that formed in September 2010, July 2013, November 2013 and May 2014. For
illustration purpose, we remove these observations from the heat map.
Figures are presented on the next page.
58
Figure 15: Continued
KurtBKM
raw
KurtBKM
s1
59
Appendix A. Derivation of 1 in eq. (18)
To see how we derive 1 from EQ (ln(Sτ /F0 )) to compensate for the difference between the
forward price F0 and the strike price K0 that is immediate below F0 , we start with valuing
EQ (ln(Sτ /K0 )). It is important to note that, in an idealized world where strike prices are
quoted continuously from 0 to ∞, F0 = K0 .
Rather than deriving it directly, let us suppose we can hold a portfolio of options, Π,
spanning all strikes K ∈ (0, ∞) that will all expire in τ -period and is individually weighted
inversely proportional to K 2 . That is, at time 0, the portfolio is worth:
Z K0
Z ∞
1
1
Π=
max(K − Sτ , 0) dK +
max(Sτ − K, 0) dK
2
2
K
0
K0 K
ST
+ ln K0
= −1 − ln ST +
K0
S T − K0
ST
=
− ln
(A.1)
K0
K0
K0
ST
ST
+ ln
= EQ ln
∴ EQ ln
F0
K0
F0
ST − K0
K0
= EQ
− Π + ln
K0
F0
(A.2)
In the last step in eq. (A.2), we make a substitution from the result in eq. (A.1). It is straightforward to see that EQ (Π) is approximated by the first half of eq. (14). The focus is now on
the other two terms in eq. (A.2):
K0
ST − K0
1 = EQ
+ ln
K0
F0
EQ (ST )
K0
=
− 1 + ln
K0
F0
F0
F0
=
− 1 − ln
K0
K0
F0
F0
= − 1 + ln
−
(A.3)
K 0 K0
as found in eq. (18). For the other terms in eqs. (19) to (21), similar derivations of risk-neutral
expectation of the squared contract (V ), the cubed contract (W ), the quartic contract (X) can
be conducted. The exact derivation manuscript is available upon request.
60