Model-Free Risk-Neutral Moments and Proxies

Zhangxin (Frank) Liua,1

aBusiness School, The University of Western Australia, Perth, WA, Australia

Abstract

Estimation of risk-neutral (RN) moments is of great interest to both academics and practi-tioners. 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 BoardOptions Exchange; 3) nonparametric RN moments that are calculated as the difference of im-plied volatilities across moneyness levels; and 4) the level, slope and curvature of the impliedvolatility smirk. More specifically, we investigate the estimation procedure by examining theconsequence of directly using raw option data versus applying various smoothing methods tothe option data. In the simulation study, we study estimation errors arise from integrationtruncation, discreteness of strike prices and asymmetric truncation. We show that applyingsmoothing methods reduces the estimation errors of true moments but the size and directionof estimation errors are largely unquantifiable. In the empirical study, we find that applyingsmoothing methods increases the Kendall and Spearman rank correlations among RN momentestimates. We conduct a case study that examines the relationship between RN skewness andfuture realised stock returns from 1996 to 2014. We show that a strategy that is long the quin-tile portfolio with the lowest RN skewness stocks yields a negative and significant Fama-FrenchFive-Factor alpha. This finding is robust across all RN skewness measures.

Keywords: Risk-Neutral Moments, Skewness, Kurtosis, Implied Volatility Smirk, Skew,Curvature, VIX

Email address: frank.liu@uwa.edu.au (Zhangxin (Frank) Liu )1First 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 ofrisk-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) andCarr and Madan (2001), BKM’s approach supplies a new tool to estimate RN moments andhas received increasing popularity in empirical studies. The primary goal of this paper is toinvestigate the implementation issues in applying their methods. We compare the accuracy inusing the raw and different smooth methods to interpolate option prices in implementing BKMmethod, alongside with several other nonparametric RN moment estimates.

BKM’s approach to compute moments of the RN distribution relies on three sets of con-ditions: 1) the existence of a continuum of strike prices for the underlying security in a givenmaturity; 2) the strike price range spans from zero to infinity; and 3) the option is a Euro-pean option. There are several difficulties with inferring model-free RN moments using thisapproach. From the traded options in the market, we do not observe a continuum of strikeprices. In particular, we often see an unequal range of out-of-the-money (OTM) put strikes andOTM call strikes and the difference can be substantial following a large price moment in theunderlying security. The second condition is also not met because there only exist discretelyspaced strike prices. For the third condition, it does not raise any issues if the main subjectof study is on European options. In the case of American options, which are common amongequity 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 todeal with the first two conditions. Our study is largely motivated by the disagreement in howobserved option prices should be treated when implementing the BKM method. We summarisea subset of studies that have used BKM method and their corresponding approach in treat-ing 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 op-tions 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 priceshave been interpolated and extrapolated using some particular method before been applied inBKM formulas.

[Table 1 about here.]

Dennis and Mayhew (2002) is among the first to apply BKM method to study RN skewnessfrom stock options. In their study, they discuss biases from the discreteness of the strike priceinterval and asymmetry in the domain of integration. Leaning on their simulation study usingBlack-Scholes option prices, their approach to combat issues from the first two conditions is tofilter 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 ruleof 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 smoothapproach. 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 mainsteps in a smooth approach, 1) interpolation between the OTM put with the lowest strike andthe OTM call with the highest strike; and 2) extrapolation beyond the highest and lowest strikeprice to recover both tails.

The literature in BKM applications also seems divided in how to interpolate and extrap-olate. We first discuss the interpolation procedure. Christoffersen et al. (2008) interpolateimplied volatilities (IV) using a cubic spline across moneyness level, defines as K/S, to obtaina continuum of IVs. They then convert IVs back to corresponding option prices. It is impor-tant to point out that the use of IV does not assume the validity of Black-Scholes model. TheIV 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 fitwith a cubic spline interpolation. Jiang and Tian (2007) study how to minimise discretisationand truncation errors in the Chicago Board Options Exchange (CBOE) VIX calculation2. Theauthors propose a solution by interpolating implied volatilities of OTM puts and calls usinga natural cubic spline across strike prices (K) from the lowest OTM put to the highest OTMcall. 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 momentsfrom S&P 500 options. In their study, a different interpolation is done by fitting a cubic splineacross 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 ensuresan IV function in delta is smooth up to second order in terms of the partial derivatives ofoption 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 thethird order in option price itself, while the approach by Christoffersen et al. (2008) ensures theRN density is smooth up to the second order.

Engle and Mistry (2013) study skewness in priced risk factors and individual stocks. They fita quadratic spline with a knot at 0 of moneyness in IV-moenyness space, where the moneynessis defines as ln(K/S)−rT

σ√T

and σ is measured from the historical monthly realised volatility. A

more recent study by Stilger et al. (2015) considers yet another way and interpolates IV usinga 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 high-est and the lowest strike is less subject to deviation. A common approach is to assume a flatstructure 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 studieslisted in Table 1 which have considered a smooth approach.

Jiang and Tian (2007) discuss two drawbacks with this flat extrapolation scheme. The firstone 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 withnegative local RN density and thus violates no-arbitrage conditions. They propose a smooth

2Jiang and Tian (2007) is not included in Table 1 as technically speaking their study does not directlyimplement 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 acertain proportion of the tail density matches with that of the main RN density. However, asthe proportion of RN density on each end is arbitrarily set and lacks a theoretical ground onhow to be calibrated using market data, we find this is challenging to implement if the mainsubject is individual stock option3.

From the discussion above, it is clear to see a divergence exists in choosing the raw or asmooth approach when implementing BKM method. When RN moment is an important factorin an empirical study, however, the consequence from choosing either approach and how thatwould have impact to the empirical findings remains largely undiscussed.

As an example, the disagreement in the relationship between the RN skewness and futurerealised returns may shed some light on this matter. Conrad et al. (2013) implement a rawBKM approach in estimating RN moments. They find a negative relationship between quarterlyaverages of daily RN skewness estimates and subsequent realised quarterly stock returns. Baliand Murray (2013) also adopt a raw BKM approach and create a portfolio of options that onlyexposes to skewness effect. They find a negative relationship between RN skewness and optionportfolios’ returns. On the other hand, Rehman and Vilkov (2002) implement a smooth BKMapproach and document the ex ante skewness is positively related to future stock returns. Thisfinding 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 stocksand 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 thedifference in their findings to the fact that the underperformance in the most negative skewnessstocks is driven by stocks that are too costly to short sell.

Our study is largely motivated by the disagreement in how observed option prices shouldbe treated when implementing the BKM method. We extend our analysis to investigate otherRN moment estimates and proxies, including 1) CBOE moments which are based on CBOE’smethodology in calculating the VIX and SKEW index5; 2) nonparametric RN moments thatare estimated by taking differences of IVs of options at different moneyness levels, including avariation 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, skewnessand excess kurtosis, respectively.

Our motivation of including the aforementioned estimates is twofold. First, there is asubstantial amount of literature that shows the shape of the volatility smirk carries predictive

3In 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 unpublishednote, we have experimented with S&P 500 options by matching different segments on tails across an 18-yearperiod from 1996 to 2014. We find that results of the tail shape can be distinctly different if the range ofavailable strike prices becomes narrow.

4The authors find similar results by using the raw BKM approach as a robustness check.5Note 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 explainedin 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 IVof 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-anteexpected returns. The authors’ primary estimates are BKM raw moments and they also usenonparametric 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 tothe data availability issues6, therefore it is important to investigate the difference in outcomesbetween applying the raw or a smooth method.

The first and the most obvious question to be asked is: what is the difference betweenimplementing the raw and smooth approaches? In other words, hypothesizing these RN momentestimates can theoretically recover the true moments, how large will the estimation errors bewhen the availability of option prices vary and how will smooth approaches improve on theresult? 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 optionprices - we first conduct a control simulation experiment, where the true RN moments can beestimated. 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 mainapplication 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 RNmoment estimates. Furthermore, we investigate the percentage of matching items in top andbottom quintiles between RN moment estimates and the true moments.

Jiang and Tian (2007) and Chang et al. (2012) conduct similar studies to our first researchquestion. Comparing to Jiang and Tian, this paper extends the analysis to RN skewness andkurtosis, as well as including investigations in other nonparametric RN moment estimates.Chang et al. examine the accuracy of the BKM volatility and skewness computations. Theirsimulation design is limited to using Heston (1993)’s stochastic volatility model with one setof parameters as option price generation process. Our study extends the analysis to includethree other models as option price generation processes as well as nine sets of parameters torepresent various market conditions. Moreover, we investigate further in the accuracy issue andour analysis in Kendall rank correlations provides an extension in studying the usefulness ofthese RN estimates as sorting mechanisms.

The second research question is to investigate how the implementation of the raw andsmooth approaches empirically differ using traded option prices. With the absence of true

6Strictly speaking, apart from the CBOE moments, even though the other measures do not require a con-tinuum of strike prices as a necessary condition, the estimation still confronts with the availability issue. Forexample, when calculating a nonparametric RN skewness as the difference between the IV of the 0.25 deltacall 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 optionwith the exact delta does not exist. A few approaches can be considered: 1) replace the missing call with 0.25delta with the closest call available; 2) a linear interpolation between two adjacent calls; and 3) a cubic splineinterpolation of the entire IV smirk to fill the missing call with 0.25 delta. The first approach can be viewed asthe 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. Wecompare the Kendall and Spearman rank correlations and present the differences in the rawmeasures and smooth measures.

We also conduct an empirical case study, where we study the relationship between the RNskewness 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 fiveusing a smooth approach), we investigate the excess return performance in skewness-quintileportfolios. From January 1996 to August 2014, we sort all stocks available from OptionMetricsby each RN skewness estimate in ascending order on the last trading day of each month. Wecompare three skewness-quintile portfolio strategies: 1) a long strategy in quintile 1 stockswith RN skewness in the bottom 20th percentile; 2) a long strategy in quintile 5 stocks withRN skewness in the 80th percentile; and 3) a long strategy in quintile 5 and a short strategy inquintile 1 portfolio.

Our main findings can be easily summarised. First, in the simulation study, we showthat regardless of using the raw or smooth approaches, the point estimate of the true RNmoment is unstable under different conditions. More importantly, the estimation error doesnot follow any particular patterns. The problem is more pronounced in skewness and excesskurtosis. Despite the poor performance in point estimate, the design of the simulation studyallows us to show that smooth approaches increase the Kendall and Spearman rank correlationsbetween 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 asmooth approach to trade option prices, the Kendall and Spearman correlations among RNestimates increase. In other words, if RN estimate is used as a sorting mechanism for portfolioconstruction, our result implies using a smooth approach increases the likelihood that a similarportfolio composition is found across portfolios based on different RN estimates. Third, in theempirical case study that examines the RN skewness and future realised returns, we show thatonly the monthly excess return of the first strategy consistently yields a negative Fama-FrenchFive-Factor (Fama and French, 2015) alpha across all RN skewness estimates. Furthermore, westudy 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 ineach portfolio. We illustrate that although the RN volatility differs numerically between rawand smooth approach, the time-series behavior is similar across different portfolios. A similarbut weaker finding is presented in RN kurtosis.

Our paper is related to the discussion of higher-moments risk in asset pricing and invest-ment management, and contributes to the literature in several ways. First, to the best of ourknowledge, this is the first study to examine the consequence of using raw and smooth approachin calculating model-free RN estimates. Our results may provide an alternative explanation insome mixing empirical findings regarding RN moments. Second, our empirical study is specialin terms of underlying data that we are able to use exchange-traded options data of more than8,000 securities from OptionMetrics in the period between 1996 to 2014. This coverage enablesus to examine the strike price availability issue across different issue types, including stockoptions, index options, options on exchange-traded funds (ETF), among others. Third, ourfindings in the skewness-quintile portfolio study documents a consistent underperformance inquintile 1 skewness portfolios, regardless of how the RN skewness is estimated. This may shedsome light on portfolio management with RN skewness.

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The remainder of the paper is organized as follows. Section 2 describes the method to con-struct each RN measure. Section 3 conducts simulation studies to investigate the relationshipamong RN estimates and true measures. Section 4 shows the data and presents the empiricalresults. Section 5 concludes.

2. Methodology

2.1. BKM Risk-Neutral Moments

Bakshi and Madan (2000) articulate that any payouff function can be spanned and pricedusing an explicit positioning across a continuum of option strikes. BKM demonstrate that theRN annualised τ -period volatility, skewness and excess kurtosis of a security’s log return canbe obtained as7:

VolBKM ≡√EQ(R2)− E2

Q(R)

τ

=

√erτV − µ2

τ(1)

SkewBKM ≡EQ(R3)− 3EQ(R)EQ(R2) + 2E3

Q(R)

(EQ(R2)− E2Q(R))3/2

=erτW − 3erτµV + 2µ3

(erτV − µ2)3/2(2)

KurtBKM ≡EQ(R4)− 4EQ(R)EQ(R3) + 6E2

Q(R)EQ(R2)− E4Q

(EQ(R2)− E2Q(R))2

− 3

=erτX − 4erτµW + 6erτµ2V − 3µ4

(erτV − µ2)2− 3 (3)

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 measuresin 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:

V =

∫ ∞S∗

2(1− ln

(KS∗

))K2

C(K) dK

+

∫ S∗

0

2(1 + ln

(S∗

K

))K2

P (K) dK (4)

W =

∫ ∞S∗

3 ln(KS∗

) (1− 2 ln

(KS∗

))K2

C(K) dK

−∫ S∗

0

3 ln(S∗

K

) (1 + 2 ln

(S∗

K

))K2

P (K) dK (5)

7In BKM, the notation kurt represents the risk-neutral kurtosis. As we are interested in excess kurtosisthroughout the text, we drop out the word excess in the notation for clarity.

7

X =

∫ ∞S∗

4 ln2(KS∗

) (3− ln

(KS∗

))K2

C(K) dK

−∫ S∗

0

4 ln2(S∗

K

) (3 + ln

(S∗

K

))K2

P (K) dK (6)

µ ≡ EQ ln

(S(τ)

S0

)≈ erτ

(1− e−rτ − V

2− W

6− X

24

)(7)

where S∗ is an arbitrary strike price that sets the OTM boundary, C(K) and P (K) representsthe 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 existenceof a continuum of options with strike spanning from 0 to infinity. To approximate the integralsin eqs. (4) to (6), it is common to implement a trapezoidal approach to discretize and truncatewith available strikes (e.g. see Dennis and Mayhew, 2002; Bali and Murray, 2013; Conrad etal., 2013):

V ≈∑i

2∆Ki

K2i

(1− ln

(Ki

F0

))Q(Ki) (8)

W ≈∑i

3∆Ki

K2i

(2 ln

(Ki

F0

)− ln2

(Ki

F0

))Q(Ki) (9)

X ≈∑i

4∆Ki

K2i

(3 ln2

(Ki

F0

)− ln3

(Ki

F0

))Q(Ki) (10)

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) optionif Ki is smaller (larger) than the forward level F0. That is, S∗ is chosen to be the forward levelF0 = S0e

(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 definiteintegral 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 byinterpolating ATM implied volatilities of OEX options to construct a 30-day forward-lookingvolatility measure. The VIX methodology was updated in 2003 with a reference to a model-freeapproach first introduced in Demeterfi, Derman, Kamal and Zou (1999). The principle of thenew VIX is based on a principle that the fair value of future volatility can be captured by thedynamic hedging of a log contract ln(ST/S0). Jiang and Tian (2007) show that this is equivalentto the model-free implied variance developed in Britten-Jones and Neuberger (2000). Due toits popularity and well establishment as a market volatility risk proxy, we adopt the majorityof CBOE VIX methodology in constructing VolCBOE but do not consider an interpolation interm-structure to yield a fixed 30-day measure.

8

Although less popular in both finance industry and academic, CBOE also started publishinga skewness index (current ticker: SKEW) in 2011. SKEW is designed to become the benchmarkmeasure for perceived future tail risk of the SPX return distribution. More specifically, thealgorithm 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 leanthe CBOE SKEW method to make an extension. The CBOE moments are given as follows(CBOE, 2009, 2010):

VolCBOE ≡

{2

τ

∑i

∆Ki

K2i

erτQ(Ki)−1

τ

[F0

K0

− 1

]2}1/2

(11)

SkewCBOE ≡ P3 − 3P1P2 + 2P 31

(P2 − P 21 )3/2

(12)

KurtCBOE ≡ P4 − 4P1P3 + 6P 21P2 − P 4

1

(P2 − P 21 )2

− 3 (13)

where the approximation on each component is performed as:

P1 ≈ erT

(−∑i

∆Ki

K2i

Q(Ki)

)+ ε1 (14)

P2 ≈ erT

(∑i

2∆Ki

K2i

(1− ln

(Ki

F0

))Q(Ki)

)+ ε2 (15)

P3 ≈ erT

(∑i

3∆Ki

K2i

(2 ln

(Ki

F0

)− ln2

(Ki

F0

))Q(Ki)

)+ ε3 (16)

P4 ≈ erT

(∑i

4∆Ki

K2i

(3 ln2

(Ki

F0

)− ln3

(Ki

F0

))Q(Ki)

)+ ε4 (17)

where the ε terms at the end are adjustments made to compensate the difference between theforward level F0 and the strike price K0 that is immediately below F0. They can be computedas:

ε1 = −(

1 + ln

(F0

K0

)− F0

K0

)(18)

ε2 = 2 ln

(K0

F0

)(F0

K0

− 1

)+

1

2ln2

(K0

F0

)(19)

ε3 = 3 ln2

(K0

F0

)(1

3ln

(K0

F0

)− 1 +

F0

K0

)(20)

ε4 = 4 ln3

(K0

F0

)(1

4ln

(K0

F0

)− ln

(K0

F0

)+F0

K0

)(21)

We present a simple derivation of ε1 in Appendix A8. It is important to note that, V , W and

8Other ε terms can be derived following a similar analogy. Exact derivation manuscript is available uponrequest.

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 ε terms.

A close examination on eq. (7) and eq. (14) reveals the major difference between the BKMformulas and the CBOE ones. eq. (7) is derived in the Appendix in BKM by applying Taylorseries of exp(R) =

∑4n=0

Rn

n!+ 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 skewnessand kurtosis from those of CBOE.

2.3. Nonparametric Measures

Xing et al. (2010) examine individual stock options in the US market and argue that theshape of the volatility smirk has predictive power for future equity returns. In their paper, theyestimate skew measure as the difference between the implied volatilities of OTM puts and ATMcalls. Xing et al. base the use of their skew measure on the demand-based option pricing modelof Garleanu et al. (2007), which documents that the positive relationship between demand forindex options and option expensiveness, measured by implied volatility, can consequently affectthe steepness of the implied volatility skew. Bali et al. (2015) use nonparametric RN estimatesas a robustness check to their raw BKM estimates. We refer interested readers to the summaryprovided in Mixon (2011). The nonparametric (NP) moments can be estimated as follows9:

VolNP ≡ CIV50 + PIV50

2(22)

SkewNP ≡ CIV25 − PIV25 (23)

KurtNP ≡ CIV25 + PIV25 − CIV50 − PIV50 (24)

SkewMixon ≡ CIV25 − PIV25

50 Delta Volatility=

SkewNP

VolNP(25)

where Cn represents the IV of an OTM call with delta n/100, and Pn represents the IV of anOTM 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 changefor a random variable does not alter γ; 2) γ = 0 for a symmetric distribution; 3) if Y = −Xthen 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 theabove nonparametric skew measures. For example, the skewness measure should have minimaldependence on the level of volatility. Mixon (2011) shows that SkewMixon subjects to the leastvariations 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 inRubinstein (1994), Ait-Sahalia and Lo (1998), Foresi and Wu (2004), among others. There is a

9Note that in Mixon (2011), the formula is specified as PIV25−CIV25

50 Delta Volatility , which measures the negative skewness.We implement a necessary transformation to fit in this study.

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 thatthe level, slope and curvature of the IV smirk can be linked to RN volatility, skewness andexcess kurtosis, respectively. We follow their approach and estiamte these measures as follows:

VolSmirk ≡ γ0 (26)

SkewSmirk ≡ γ1 (27)

KurtSmirk ≡ γ2 (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 + γ1ξi + γ2ξ2i ) + εi (29)

where the moneyness measure ξ is chosen to be:

ξi ≡ln(Ki/F0)

στ√τ

(30)

and where στ denotes a measure of the average volatility of the underlying asset price. Forease of convenience, we refer these as Smirk moments. We proxy στ by the realised volatilityof the underlying asset in the past τ−period. For example, for an option that has 9 days tomaturity, τ9/365 is the annualised standard deviation on the logarithm of the close-to-close dailytotal return of the underlying asset in the past 9 days.

Our approach differs from Zhang and Xiang (2008) in several ways. They use a quadraticfunction to fit the IV data by minimising the volume-weighted mean square error. We donot weight the mean squared error by the option volume due to two reasons. First, we donot have option trade volume in the simulation study. Second, Zhang and Xiang study theimplied volatility smirk from S&P 500 options. Our empirical study covers all issue types fromOptionMetrics and trade volume data is more noisy cross-sectionally. Another deviation fromtheir approach is that they use VIX value as the proxy for στ 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 whenimplementing BKM method. This is discussed in Section 1. Table 1 provides a list of studiesthat have used BKM method and their corresponding treatment in treating the option data. Tocover a wide range of smooth methods, we implement the following approaches in the simulationstudy. 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 deltasbetween the highest and lowest known option deltas. The extrapolation followsJiang and Tian (2007) to match the slopes of the extrapolated and interpolatedsegments.

11

Smooth2 (s2) The interpolation is done by fitting a natural cubic spline in IV against mon-eyness (K/S). The extrapolation step is the same as s1.

Smooth3 (s3) We linearly interpolate IV against option deltas. The extrapolation step is thesame as s1.

More specifically, in s1 and s3, we interpolate and extrapolate the observed IVs to fill ina total of 1,000 grid points in the delta range from 0.001 to 1. In s2, the interpolation andextrapolation is done to fill the moneyness-delta space on a total of 1,001 grid points in themoneyness range from 1/3 to 3. We then calculate the option prices from the fitted IV usingthe known interest rate and the adjusted dividend yield (recovered from comparing the securityprice 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 thesuperscript and data interpolation approach in the subscript. For example, for the BKMvolatility that is constructed using raw data, we name it as VolBKM

raw . For the CBOE skewnessthat 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 lackof 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) truemoments 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 byaltering the availability of option prices. Furthermore, with multiple parameter settings, wecan further investigate the ranking correlations among the RN moment estimates.

Jiang and Tian (2007) study various estimation errors from the implementation of CBOEVIX method. Hansis, Schlag and Vilkov (2010) discuss the effectiveness of using cubic splinesto interpolate the implied volatilities against moneyness and the importance of smoothing.The authors use Black and Scholes model, the Heston model, the stochastic volatility andjump model developed in Bates (1996) and Bakshi, Cao and Chen (1997) as well as SVCJmodel. 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 Changet al. (2012). Furthermore, as their results from the simulation study are not included in thepaper, it makes difficult to draw any inference.

Appendix B in Chang et al. (2012) discuss the approximation errors in skewness usingsimulation option prices with Heston model. They only look at one set of parameters in onemodel, in which we will show you the essence of using multiple sets of parameters in differentmodels. The authors conclude that “it is difficult to estimate skewness accurately when thewidth of the integration domain is small” and “. . . we choose a sample of stocks with liquidoption 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 performMC simulations to generate option prices from the Black-Scholes-Merton (BSM) model (Blackand Scholes, 1973; Merton, 1973); Heston stochastic volatility model (Heston, 1993); Mertonjump-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 largenumber of trials to achieve some reasonable accuracy, at an expense of extra computationalresource usage. A typical procedure is to apply variance reduction techniques, such as applyingcontrol variate technique and using discrete versions of martingale control variate. Providedthe goal of this simulation exercise is to draw direct comparisons with corresponding sectionsin Dennis and Mayhew (2002), Jiang and Tian (2007), Chang et al. (2012), we do not adoptany variance reduction techniques in improving the accuracy of option prices generated insimulations.

We first outline the MC simulation procedure for each model and then show how the truemoment is estimated. We run MCS in BSM model, which is based on the Geometric BrownianMotion. Given there is an exact solution to its stochastic differential equation (SDE), we have:

St = S0 exp

((r − 1

2σ2

)t+ σWt

)(31)

for t ∈ [0, T ], which means we could approximate the process (Si)i∈{1,...,N} by:

St+1 = St exp

((r − 1

2σ2

)∆t+ σ

√∆tZt

)(32)

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:

dSt = rStdt+√νtStdW

1t (33)

dνt = κ(θ − νt)dt+ ξ√νt

(ρdW 1

t +√

1− ρ2dW 2t

)(34)

for t ∈ [0, T ]. To simulate the process, we apply the Euler approximation:

νt+1 = κ(θ − νt)∆t+ ξ√νt

(ρ√

∆tZ1,t +√

1− ρ2√

∆tZ2,t

)(35)

St+1 = St + rSt∆t+√νtSt√

∆tZ1,t (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:

St = S0e(r−λk− 1

2σ2)t+σWt

Nt∏i=1

Yi (37)

for t ∈ [0, T ], where Nt ∼ Pois(λ) and independent jumps Y with ln(Yi) ∼ N (µJ , v2J). We

apply the Euler simulation:

Ut = exp(PtµJ +√PtvJZ2,t) (38)

10It is interesting to point out that it is possible to calculate option prices in closed form using Fourierinversion for these models, however, the convergence could fail given some extreme parameter choice (e.g. atextreme 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

((r − λ(eµJ+ 1

2v2J − 1)− 1

2σ2

)∆t+

√∆tσZ2,t

)Ut (39)

where Pt ∼ Pois(λ∆t) and t ∈ {0, 1, . . . , T − 1}.Lastly, Bates model combines Merton and Heston settings with SDEs as:

dSt/St = rdt+√νtStdW

1t + dNt (40)

dνt = κ(θ − νt)dt+ ξ√νt

(ρdW 1

t +√

1− ρ2dW 2t

)(41)

for t ∈ [0, T ], where Nt ∼ Pois(λ) and independent jumps Y with ln(Yi) ∼ N (µJ , v2J). We

apply the Euler simulation:

Ut = exp(PtµJ +√PtvJZ3,t) (42)

νt+1 = κ(θ − νt)∆t+ ξ√νt

(ρ√

∆tZ1,t +√

1− ρ2√

∆tZ2,t

)(43)

St+1 = St exp

((r − λ(eµJ+ 1

2v2J − 1)− 1

2νt

)∆t+

√∆tνtZ1,t

)Ut (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-lookingmeasures. We are also interested in the 180-day measure to draw some comparison with Changet al. (2012). For each model, there are a total of 18 sets of parameters: 9 sets of parametersfor each of the 2 maturities. In the BSM model, we vary the volatility parameter σ. In theHeston model and Bates model, we vary the correlation parameter ρ of Wiener processes ofsecurity price and volatility. In the Merton jump-diffusion model, we vary the intensity ofjumps parameter λ. The numerical choice of the parameters in Table 2 follow that of Jiangand Tian (2007) and Chang et al. (2012). The exact MC simulation procedure is outlines asfollows.

1. Assuming that there are T (T ∈ {22, 124}) trading days for the 30- and 180-day measure,respectively. The iteration for each simulation is T times with an interval ∆t = 1

252.

2. For each model and each parameter choice, we perform a T-iteration for 1 million times.

We calculate the log return ln(Si,T

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:

VolTrue =

√∑106

i=1(Ri − R)2

106 × T/252(45)

14

SkewTrue =1

106

∑106

i=1(Ri − R)3(1

106−1

∑106

i=1(Ri − R)2)3/2

(46)

KurtTrue =1

106

∑106

i=1(Ri − R)4(1

106

∑106

i=1(Ri − R)2)2 − 3 (47)

where Ri ≡ ln(Si,T

S0

)and R = 1

106

∑106

i=1Ri.

4. Approximate the European call and put option price as:

C =e−rT/252

∑106

i=1 max(ST,i −K, 0)

106(48)

P =e−rT/252

∑106

i=1 max(K − ST,i, 0)

106(49)

3.2. Various Types of Approximation Error

Chang et al. (2012) specify three types of errors in implementing a typical trapezoidalapproach in the BKM moments construction. The first one is an integration domain truncationerror that arises from the missing strike prices beyond the range of observed strike prices. Thesecond 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, thatthe truncation is not symmetric around the the mean/mode/median. They are best presentedin the symbolic forms as follows.

1. Truncation errors: ∫ ∞0

. . . dK →∫ Kmax

Kmin

. . . dK (50)

as K ∈ (0,∞)→ K ∈ [Kmin, Kmax] (51)

2. Discretization errors: ∫ Kmax

Kmin

. . . dK →Kmax∑Kmin

. . . ∆Ki (52)

3. Asymmetric truncation errors:

[Kmin, Kmax] 6= [S0 × a, S0/a] (53)

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 thebase case (i.e. the ideal case scenario), strike price range is [1000*0.5, 1000/0.5] with a strikeinterval ∆K = 1. In the simulation study, we fix ∆K and vary the strike price range to studythe integration domain truncation type of error. In particular, we vary the integration domainfrom 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 tobe [S0 ∗ 0.50, S0/0.50] and vary the strike interval as ∆K ∈ {1, 2, . . . , 25}. That is, we have 25variations in examining discretisation error.

The design in studying the asymmetric truncation is worthwhile to elaborate. We fix thestrike interval to be 1 and vary the downside boundary as S0 ∗ uL, where uL = 0.7 + δu; andupside boundary as S0 ∗ uH , where uH = 0.7 − δu. We set δu to vary from -0.2 to 0.2 with astep size of 0.01. That is, we have 41 variations in examining asymmetric truncation error. Tosee this more clearly, when δu = −0.2, the strike range is [500, 1111.11]; and when δu = 0.2, thestrike range is [900, 2000]. As δu varies from -0.2 to 0.2, the strike range moves from being morenegatively skewed to more positively skewed. The centre is at du = 0, where the asymmetryis at its minimal. It is important to note that for each pair of asymmetric truncation, theamount of available strikes are not too different; whereas in the truncation type, the higher thetruncation 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, wehave a total of 72 true value in each moment category (72 = 9 sets of parameters x 2 maturityterms x 4 option models). This set up is particularly important when we discuss the rankingcorrelations in Section 3.4.

3.3. Estimation Accuracy

We first investigate the estimation accuracy in the truncation error. We illustrate theapproximation errors of volatility, skewness and kurtosis in Figures 2 to 4. The approximationerror 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 momentsand thus the value should differ numerically from the true ones. That is, given our definitionof estimation error, we will not directly interpret the size of estimation errors but focus on thetrajectory and trend across variations in each type of error study. Due to the slight complexityin 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 datafrom simulations. The 2nd column applies s1 approach by fitting a natural cubic spline ininterpolating implied volatilities against deltas. The 3rd column applies s2 approach by fittinga natural cubic spline in interpolating implied volatilities against strike prices. The 4th columnapplies s3 approach by linearly interpolating implied volatilities against deltas. Each momentis 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. Forskewness, moment in the additional 5th row is calculated using Mixon’s method. Within eachsmall panel, the 1st (2nd) column reports approximation errors using options with expiration of22 (124) trading days. Within each panel, options prices are simulated using 1) Black-Scholesmodel 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 the4th row. In each plot, different shades of colour represents results from different parametersused to generate option prices.

The truncation error in volatility estimation is illustrated in Figure 2. For VolBKMraw and

VolCBOEraw , 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.All the smooth methods reduce the size of errors of VolBKM

raw and VolCBOEraw , from as large as −80%

to less than 0.8%. In terms of the trend of errors, s1 is similar to s3. For VolNPraw and VolSmirk

raw ,the improvement using smooth methods is minimal.

Examining the truncation error in skewness estimation from Figure 3, we find that as thetruncation becomes larger, it is possible to observe both under- and over-estimation of trueskewness 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 smoothapproach. For SkewNP, SkewSmirk and SkewMixon, it is unclear to see how smooth approachesimprove 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 involatility, 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 par-ticular 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 andCBOE volatility estimated from the longer maturity BSM and Merton options, but not forBates and Heston options. There is no clear improvement from applying smooth methods inSmirk 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 approachesin estimation for BKM and CBOE skewness, but not for NP, Smirk or Mixon skewness. InFigure 7, it is unclear to see the structure of errors for each measure as the size of errors isdominated by two parameter sets (ones with darker colour). The only pattern can be found isthat 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 andCBOE moments.

[Figures 8 to 10 about here.]

Last, we discuss the asymmetric truncation errors, as presented in Figures 8 to 10. Fig-ure 8 shows that estimation errors of BKM and CBOE volatility are significantly reduced byimplementing smooth approaches, particularly for shorter-maturity options. There is little im-provement for NP volatility. When apply s2 smooth approach to NP and Smirk volatility, theestimation 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 param-eters. 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 tothe error patterns of NP, Smirk and Mixon skewness. In Figure 10, s2 seems to work better forBKM and CBOE kurtosis for shorter-maturity options, but no clear improvement over s1 ands3 for longer-maturity options. For NP kurtosis, s2 can potentially create outliers in errors, ascompared to s1 and s3. For Smirk kurtosis, there is no clear benefits from applying smoothapproaches.

Having discussed all three types of approximation errors, it is important to point out thatin reality, it is impossible to disentangle the option data into separate analyses of these threetypes of errors. We do observe improvements in the size of estimation errors for BKM andCBOE moments but the improvement significant drops as we move to higher moments. Ourconclusion for the estimation accuracy is that the exact error for a true moment estimate is, atits best, unquantifiable.

3.4. Ranking Correlations

We now turn to a different angle in looking at the usage of these moment estimates. Onepopular 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. Ifthe goal is to rank these securities by a RN moment and form portfolios according to a specificrule, then it is more interesting to find out which of the RN moment estimates gives the morecorrect 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 fromdiscretisation + 41 from asymmetric truncation) variations from all three errors. Within eachvariation, we have a total of 72 true value in each moment category (72 = 9 sets of parametersx 2 maturity terms x 4 option models). If we view each variation of error study from oneparticular 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 thatfollows 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 astrike range from [500, 2000] and ∆K = 4. A good RN estimate should give the same rankingof A and B as that from a true estimate.

A widely accepted rank correlation coefficient is Kendall’s τ (Kendall, 1948) that essentiallymeasures the probability of two elements being in the same order in the two ranked lists. Inparticular, as we have ties in the ranking (because the true value is the same across differ-ent variations in the error study), we calculate the Kendall τ -b statistic. We also estimate theSpearman’s rank correlation. There is a lengthy discussion in statistic literature on the compar-ison of Kendall and Spearman’s rank correlations. A general understanding is that Spearman’srank 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, skewnessand kurtosis, respectively. Each correlation estimate is calculated based on 8, 352 pairs of truemoment and moment estimate. Kendall correlations are presented in the highlighted cells inthe bottom left part of each table. Spearman correlations are presented in the top right partof each table.

18

In Table 3, it is interesting to learn that VolNPraw VolNP

Smirk have a higher Kendall correlationswith VolTrue than VolBKM

raw and VolCBOEraw . Applying smooth methods significantly increases the

Kendall correlations for BKM and CBOE volatility. For example, Kendall correlation betweenVolBKM

raw and VolTrue increases from 0.55 to 0.95 after applying for s1 smooth method. We do notsee any improvement for NP and Smirk volatility. Similar findings are presented with Spearmancorrelation.

In Table 4, we also see a two-fold increase in Kendall correlations for BKM and CBOEskewness with all smooth methods. Furthermore, the improvement in CBOE skewness is muchlarger, from 0.34 to 0.88 with s1 and s3. Even though there is no improvement from applyingsmooth methods to NP, Smirk and Mixon skewness, the Kendall correlations between thoseskewness estimates with true skewness are all around 0.9.

In Table 5, we find the Kendall correlation between BKM kurtosis and true kurtosis increasesfrom 0.06 for raw approach to at least 0.16 for smooth approaches. Larger increases are foundbetween CBOE kurtosis and true kurtosis. It is interesting to note that applying smoothapproaches reduces the Kendall correlation between NP kurtosis from 0.09 to as low as 0 usings2. 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 andCBOE 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 ranksare evaluated. If only the composition of a certain proportion is interested, then the rankingcorrelations may overestimate the problem. Stilger et al. (2015) sort equities by the RNskewness at the end of each month in ascending order. They take a long strategy in stockswith the highest quintile RN skewness and a short strategy in stocks with the lowest quintileRN skewness. In this case, the rank order is not as important within each quintile. Inspired onthis 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 72volatility proxies with 72 corresponding true volatilities from each of 116 variations that studyapproximation errors (where 116 = 50 variations in truncation error study + 25 variationsin discretisation + 41 variations in asymmetric truncation and 72 = 9 sets of parametersx 2 maturity terms x 4 option-price-generation models). In each of 116 variations, we sortvolatility estimates from one of 4 methods (BKM, CBOE etc) in ascending order. We extractestimates that are above the 80th percentile (a quintile of 72 items is roughly 15). As eachvolatility estimate has a corresponding true volatility, we then calculate the percentage ofthese corresponding true volatilities are also above the 80th percentile of true volatilities. Weillustrate the distribution of these 116 percentages in a boxplot. This is repeated for skewnessand kurtosis estimates. We replicate these procedures for the bottom quintile by changing thethreshold to be 20th percentile.

[Figures 11 and 12 about here.]

We illustrate the matching comparison of the top (highest) quintile in Figure 11 and thatof 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 CBOEmoments but the improvement is smaller for higher moments. Second, the improvement islarger for CBOE moments than BKM ones. Third, there is no improvements to other momentestimates from implement smooth methods.

4. Empirical Results

4.1. Data

We obtain data from the Ivy DB US OptionMetrics provided through Wharton ResearchData Services. We download the entire database that contains all securities traded from 4January 1996 to 29 August 2014. We extract the security ID, issue types, date, expirationdate, put and call identifier, strike price, best bid, best offer, implied volatility and delta fromthe 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 arebeyond two consecutive strike prices with zero bid prices11.

Interest rates are taken from the CRSP Zero Curve file. We apply a cubic spline to theinterest rate term-structure data to match the length of risk-free rate with the correspondingoption maturity.

We consider OTM options only. We define a put (call) option is OTM if its strike price islower (greater) than the forward price of the underlying asset. We convert the OTM put deltasinto the corresponding call deltas as 1 + put delta = call delta. Underlying security pricesare 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 itsclose price after adjusting for dividends from ‘Distribution file’.

In estimating the moneyness level ξ ≡ ln(K/F0)σ√τ

, where στ in the Smirk moments, we needan input for σ. We obtain realised volatilities for the underlying assets from OptionMetrics‘Historical Volatility’ files. According to its reference manual, realized volatility is calculatedover a list of standard date ranges from 10 to 730 calendar days. The calculation is performedusing a standard deviation on the natural logarithm of the close-to-close daily total return. Weproxy στ by the realised volatility of the underlying asset from this file.

We examine various estimation errors from integration truncation, discretisation and asym-metric truncation in Section 3. Although we cannot investigate these estimation errors empir-ically, it is interesting to present summary statistics to document how these estimation errorsmay have a role with observed data.

[Table 6 about here.]

Table 6 reports the summary statistics of ∆KF0

in the filtered raw data set from January 1996to August 2014. This is a subset of our data that only includes options with five groups ofmaturity 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 370days as 12m. We calculate ∆K1 = K2−K1, ∆KN = KN −KN−1 and ∆Ki = (Ki+1−Ki−1)/2for i ∈ {2, . . . , N − 1} where strike price is indexed from low to high. Issue type is defined

11This 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 than60% of options are written on common stocks, which is followed by options on ETF and indexoptions.

In the simulation, we set the spot level to be 1000 and vary ∆K from 1 to 25, which meanswe vary ∆K

F0roughly from 0.001 to 0.025. Examining the summary statistics in Table 6, we find

that on average the strike step size is 0.087 for stock options with 1 month to maturity. Thisincrease to 0.129 for stocks options with 1 year to maturity. The stirke step size is much smallerfor index options, where on average it is 0.018 for index options with 1 month to maturity and0.025 for those with 1 year to maturity. The concern comes from the maximum step size. Forexample, out of the 18-year period, there is one stock option with 2 months to maturity on oneday that has a strike step size as large as 8.9 times its underlying forward level. It is importantto note that this is based on the filtered data.

[Tables 7 and 8 about here.]

Table 7 shows the summary statistics of Kmin

F0of the lowest OTM put option in the filtered

raw data set12. We find that on average, the mean of lower boundary is around 0.85 for allissue types with 1 month to maturity. Consistent with the common understanding, the lowerboundary decreases as the option maturity increases. Table 8 shows the summary statistics ofKmax

F0of the highest OTM call option in the filtered raw data set. We find that on average, the

mean of upper boundary is around 1.15 for common stocks with 1 month to maturity, which ishigher than that of index options. Similar to the lower boundary, the upper boundary increasesas the option maturity increases.

4.2. Rank Correlations

In this section, we estimate rank correlations among RN moments to study their usefulnessas a sorting mechanism. If a security has a high RN moment measured with BKM and is rankedamong the top 20% when all securities are sorted in ascending order, will this be captured bythe high RN moment measured by other methods? We present the average and the standarddeviation of daily Kendall and Spearman rank correlations of various volatility, skewness andexcess 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 isfirst 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 inparentheses) are then calculated based on daily correlations across the whole sample period. Itis clear to see that applying s1 smooth method increases the rank correlations between BKMand 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

12Note that, the proportion values should be interpreted differently to those found in Table 6. There may bemultiple entries of ∆K

F0from each security on any day with any maturity term, whereas there is only one entry

of Kmin

F0from each security on that day with the same maturity.

21

s1 approach. This implies that volatility estimates with s1 more or less capture the sameinformation. In Table 10, the average Kendall and Spearman correlations are smaller than thoseseen 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 0.94after implement s1 approach. It is interesting to learn that both NP and Smirk kurtosis havelow rank correlations, even after implementing the smooth approach. This suggests that NPand 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 skewnessand future realised returns. Conrad et al. (2013) implement a raw BKM approach in estimat-ing RN moments. They find a negative relationship between quarterly averages of daily RNskewness 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 skewnesseffect. 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 docu-ment the ex ante skewness is positively related to future stock returns. This finding is furthersupported by Stilger et al. (2015). The authors use a smooth BKM approach13 and documentthat a strategy to long the quintile portfolio with the highest RN skewness stocks and short thequintile portfolio with the lowest RN skewness stocks on average yields a Fama-French-Carhartalpha of 55 bps per month. As point out in Stilger et al., they attribute the difference in theirfindings to the fact that the underperformance in the most negative skewness stocks is drivenby 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 portfoliostudy allows us to further investigate the information content carried in these RN skewnessmeasures. Portfolios are constructed as follows. We only consider equity options. On the lasttrading day of each month t, stocks are sorted in ascending order by the corresponding skewnessmeasure. Each skewness measure is calculated from its options with the shortest maturity (withat least 10 days to maturity) on that day. Quintile 5 (1) includes stocks with skewness measurethat 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 theanalysis presented in Section 3.5. For each month from January 1996 to August 2014, we firstcount the number of matching stocks from each pairwise skewness portfolios and divide thisnumber by the total number of stocks in each portfolio to estimate the percentage. The averageand the standard deviation (shown in parentheses) of percentages are then calculated for thewhole 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 securitiesamong these portfolios. A similar finding is found in quintile 1 portfolios.

13The 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 skewness-quintile portfolios from 1996 to 2014. We form the skewness-quintile portfolio at the end ofeach month t, and calculate the equally-weighted returns of these portfolios at the end of thefollowing month t + 1. The excess return is obtained by subtracting the monthly risk-freereturn from the portfolio return. The adjusted close prices (for dividend splits etc) at time tand t+1 are used to calculate the return. The top panel shows the colour key used to representexcess returns. The top panel also presents the histogram of all monthly excess returns of allskewness-quintile portfolios. The bottom panel presents the heat map, the time is shown onthe 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 quitesimilar, 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 donot see any similarity in returns across quintile 5 portfolios. In addition, no significant lossesor 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 plotwith VolBKM

raw is provided in the top and that of VolBKMs1 is found in the bottom. Examining the

horizontal axis of colour key (smaller box) of these two plots, we see that the average VolBKMs1 is

higher than VolBKMraw in the whole period. Although they differ numerically, the colour intensity

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 KurtBKMraw is provided in the top and that of KurtBKM

s1 is found in the bottom.Similar to what we find in volatility, the colour key shows that the average KurtBKM

s1 is higherthan KurtBKM

raw in the whole period. Furthermore, in quintile-1 portoflios formed by SkewBKMraw ,

SkewCBOEraw , SkewBKM

s1 and SkewCBOEs1 , the average RN excess kurtosis is much higher than the

other portfolios in the whole sample period. This is an interesting finding that may attractsome further investigation in the future.

[Tables 13 and 14 about here.]

Having examined the time-series behaviour of RN skewness-quintile portfolios, we now studythe excess return using the Fama-French Five-Factor model (Fama and French, 2015; here-inafter, 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 portfolioloadings 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 quintile5 portfolios consistently generate significantly negative αFF5 across all skewness measures. Wedo not consistently find a 5-1 (long 5 and short 1) strategy yielding a positive and significantαFF5 across all measures. This is different from the finding by Stilger et al. (2015). It isimportant 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 valuesacross 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 resultsare slightly different. Quintile 1 portfolios still yields significant and negative αFF5 across allskewness measures. Quintile 5 portfolios do not yield any significant αFF5 for most skewnessmeasures.

5. Conclusion

RN moments are important sources to study the information embedded in market optionprices. BKM provide a model-free measure of volatility, skewness and kurtosis that can be di-rectly inferred from traded options. In this paper, we study different treatments of option databefore they are input to the BKM formulas. Using MC simulations, we examine the integrationtruncation error, discretisation of strike price error and asymmetric truncation error arise fromthe lack of a continuum of strike price ranging from zero to infinity. We extend the analysisto include several other RN moment proxies, including the CBOE moments, nonparametricmoments 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 pointestimates of true moments are larger for higher moments, and are largely unquantifiable. Ex-amining the Kendall and Spearman rank correlations, we show that applying smooth methodssignificantly improve the information content of RN moments with the true moment.

In the empirical study, we document that truncation errors, discretisation errors and asym-metric truncation errors play a role in estimating the BKM and CBOE moments. Applying thesmooth method increases the rank correlation among these RN moments. In that case studythat examines the RN skewness-quintile portfolios and future realised returns, we find that theportfolio with the lowest skewness significantly underperform the market, after adjusting forthe 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 fundreturns. Working paper.

Bakshi, G., Kapadia, N., Madan, D., 2003. Stock Return Characteristics, Skew Laws, andthe 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 FinancialEconomics 55, 205–238.

Bali., T. G., Hu, J., Murray, S., 2015. Option implied volatility, skewness, and kurtosis and thecross-section of expected stock returns. Working paper.

24

Bali., T. G., Murray, S., 2013, Does risk-neutral skewness predict the cross-section of equityoption 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 optionprices. Journal of Business 51, 621–651.

Carr, P., Madan, D., 2001. Optimal positioning in derivative securities. Quantitative Finance1, 19–37.

Chang, B., Christoffersen, P., Jacobs, K., Vainberg, G., 2012, Option-implied measures ofequity 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. Journalof 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 skew-ness. 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: evi-dence from individual options. Working paper.

Jiang, G. J., Tian, Y. S., 2007, Extracting Model-free volatility from option prices: an exami-nation 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 mo-ments: evidence from the S&P 500 options. Journal of Financial and Quantitative Analysis48, 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 aboutfuture stock returns, Working paper.

25

Taylor, S. J., Yadav, P. K., Zhang, Y., 2010, The information content of implied volatilitiesand 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 usabout 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

Tab

le1:

This

table

pro

vid

esa

subse

tof

studie

sth

athav

eap

plied

Bak

shi,

Kap

adia

and

Mad

an(2

003,

BK

M)’

sm

ethod

toca

lcula

teri

sk-n

eutr

alm

omen

tsfr

omop

tion

pri

ces.

“Sto

ck/I

ndex

Opti

ons”

show

sth

em

ain

typ

eof

opti

ons

that

are

use

dto

imple

men

tB

KM

met

hod.

“Raw

/Sm

oot

h”

refe

rsto

whet

her

the

trad

edop

tion

pri

ces

(i.e

.ra

w)

are

dir

ectl

yuse

d,

orth

eop

tion

pri

ces

hav

eb

een

inte

rpol

ated

and

extr

apol

ated

usi

ng

som

epar

ticu

lar

met

hod

bef

ore

bee

nap

plied

inB

KM

’sfo

rmula

s.IV

stan

ds

for

implied

vola

tility

.

Auth

ors

Sto

ck/I

ndex

Opti

ons

Raw

/Sm

oot

hSm

oot

hM

ethod

Den

nis

and

May

hew

(200

2)Sto

ckR

aw-

Chri

stoff

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n,

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obs

and

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nb

erg

(200

8)Sto

ckan

dIn

dex

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oot

hIn

terp

olat

eIV

usi

ng

acu

bic

spline

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oney

nes

s(K

/S)

Han

(200

8)Sto

ckR

aw-

Aga

rwal

,B

aksh

ian

dH

uij

(200

9)In

dex

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-D

uan

and

Wei

(200

9)Sto

ckR

aw-

Tay

lor,

Yad

avan

dZ

han

g(2

009)

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ckR

aw-

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sis,

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lag

and

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v(2

010)

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ckSm

oot

hIn

terp

olat

eIV

usi

ng

acu

bic

spline

acro

ssm

oney

nes

s(K

/S)

Mix

on(2

011)

Index

Raw

-B

uss

and

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v(2

012)

Sto

ckSm

oot

hIn

terp

olat

eIV

usi

ng

acu

bic

spline

acro

ssm

oney

nes

s(K

/S)

Chan

g,C

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n,

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obs

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(201

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(201

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/S)

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oot

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dex

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/S)

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lger

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ng

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ise

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mit

ep

olynom

ial

acro

ssm

oen

yess

(K/S

)

27

Table 2: This table describes parameters used in the base case models of our simulation studyin Section 3 and parameters used in examining each type of errors. For each model, there area total of 18 sets of parameters: 9 sets of parameters for each of the 2 maturities. In the BSMmodel, we vary the volatility parameter σ. In the Heston model and Bates model, we varythe correlation parameter ρ of Wiener processes of security price and volatility. In the Mertonjump-diffusion model, we vary the intensity of jumps parameter λ. For every model, the spotprice for the underlying security S0 is set to be 1000. The forward price F0 is then calculatedas S0 exp ((r − q)τ). In the base case, strike price range is [1000*0.5, 1000/0.5] with a strikeinterval ∆K = 1. In the simulation study, we fix ∆K and vary the strike price range to studythe integration domain truncation type and the asymmetric integration domain truncation typeof errors. We fix the strike price range and vary ∆K to study the discretisation of strike pricetype of error.

Panel A

Name Symbols BSM Heston Merton Bates

Spot S0 1000 1000 1000 1000Strike Range [Kmin, Kmax] [S0*0.5, S0/0.5] [S0*0.5, S0/0.5] [S0*0.5, S0/0.5] [S0*0.5, S0/0.5]Strike Interval ∆K 1 1 1 1Time to Maturity τ 22

252, 124

25222252

, 124252

22252

, 124252

22252

, 124252

Interest r 0.05 0.05 0.05 0.05Dividend q 0 0 0 0

Volatility σ 0.1, 0.2, . . . , 0.9 -√

0.05 -Initial Variance ν0 - 0.05 - 0.05

Long-Run Variance θ - 0.05 - 0.05Vol of Vol ξ - 0.15 - 0.15Speed of Mean Reversion κ - 2.00 - 2.00Correlation of S and V ρ - -1, -0.75, . . . , 1 - -1, -0.75, . . . , 1Mean of Jumps µJ - - −0.15σ −0.15σVolatility of Jumps vJ - - 0.152σ2 0.152σ2

Intensity of Jumps λ - - 0.5, 1.0,. . . , 4.5 1.00

Panel B

Type of Errors Parameter Strike Range Strike Interval

Truncation u ∈ {1, 2, . . . , 50} [S0 ∗ (u ∗ 0.01 + 0.49), S0

u∗0.01+0.49] 1

Discretisation ∆K [S0 ∗ 0.5, S0/0.5] ∆K ∈ {1, 2, . . . , 25}Asymmetric Truncation δu ∈ {1, 2, . . . , 41} [S0 ∗ (0.49 + δu/100), S0

0.91−δu/100] 1

28

Tab

le3:

This

table

show

sth

eK

endal

lra

nk

corr

elat

ion

(τ-b

)an

dSp

earm

an’s

rank

(ρ)

corr

elat

ion

coeffi

cien

tsam

ong

vari

ous

vola

tility

mea

sure

sin

the

sim

ula

tion

study

inSec

tion

3.E

ach

corr

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esti

mat

eis

calc

ula

ted

bas

edon

8,35

2pai

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tility

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mat

es(8,3

52=

116×

9×

2×

4,w

her

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6=

50va

riat

ions

intr

unca

tion

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rst

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+25

vari

atio

ns

indis

cret

isat

ion

+41

vari

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inas

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met

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trunca

tion

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sets

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table

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earm

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rrel

atio

ns

are

pre

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din

the

top

righ

tpar

tof

the

table

.

Sp

earm

anN

ame

Vol

Tru

eV

olB

KM

raw

Vol

CB

OE

raw

Vol

NP

raw

Vol

Sm

irk

raw

Vol

BK

Ms1

Vol

CB

OE

s1V

olN

Ps1

Vol

Sm

irk

s1V

olB

KM

s2V

olC

BO

Es2

Vol

NP

s2V

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mir

ks2

Vol

BK

Ms3

Vol

CB

OE

s3V

olN

Ps3

Vol

Sm

irk

s3

Vol

Tru

e0.

680.

660.

930.

940.

990.

970.

930.

940.

990.

960.

930.

710.

990.

970.

930.

94

Vol

BK

Mra

w0.

550.

990.

660.

660.

690.

700.

660.

670.

690.

710.

660.

610.

690.

700.

660.

67V

olC

BO

Era

w0.

520.

930.

660.

650.

660.

690.

660.

660.

670.

700.

650.

620.

660.

690.

660.

66V

olN

Pra

w0.

800.

520.

520.

990.

950.

981.

000.

990.

950.

981.

000.

740.

950.

981.

000.

99V

olS

mir

kra

w0.

820.

520.

520.

920.

960.

990.

991.

000.

960.

980.

990.

760.

960.

990.

991.

00

Vol

BK

Ms1

0.95

0.56

0.53

0.83

0.84

0.98

0.95

0.96

1.00

0.97

0.95

0.72

1.00

0.98

0.95

0.96

Vol

CB

OE

s10.

870.

560.

560.

900.

910.

910.

980.

990.

981.

000.

980.

760.

981.

000.

980.

99V

olN

Ps1

0.79

0.52

0.52

0.97

0.93

0.82

0.90

0.99

0.95

0.98

1.00

0.74

0.95

0.98

1.00

0.99

Vol

Sm

irk

s10.

810.

520.

530.

930.

960.

840.

920.

940.

960.

990.

990.

760.

960.

990.

991.

00

Vol

BK

Ms2

0.95

0.56

0.53

0.82

0.83

0.97

0.90

0.82

0.83

0.98

0.95

0.72

1.00

0.98

0.95

0.96

Vol

CB

OE

s20.

860.

560.

570.

900.

900.

880.

960.

900.

910.

900.

980.

760.

971.

000.

980.

99V

olN

Ps2

0.79

0.52

0.52

0.97

0.92

0.82

0.89

0.99

0.93

0.81

0.89

0.73

0.95

0.98

1.00

0.99

Vol

Sm

irk

s20.

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680.

670.

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760.

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Vol

BK

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0.95

0.56

0.53

0.83

0.84

1.00

0.91

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0.82

0.61

0.98

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0.96

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CB

OE

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810.

520.

530.

930.

960.

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920.

941.

000.

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910.

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690.

840.

920.

94K

endal

l

29

Tab

le4:

This

table

show

sth

eK

endal

lra

nk

corr

elat

ion

(τ-b

)an

dSp

earm

an’s

rank

(ρ)

corr

elat

ion

coeffi

cien

tsam

ong

vari

ous

skew

nes

sm

easu

res

inth

esi

mula

tion

study

inSec

tion

3.E

ach

corr

elat

ion

esti

mat

eis

calc

ula

ted

bas

edon

8,35

2pai

rsof

skew

nes

ses

tim

ates

(8,3

52=

116×

9×

2×

4,w

her

e11

6=

50va

riat

ions

intr

unca

tion

erro

rst

udy

+25

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atio

ns

indis

cret

isat

ion

+41

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atio

ns

inas

ym

met

ric

trunca

tion

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ofpar

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from

4m

odel

s).

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dal

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the

table

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pre

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table

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rue

0.30

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0.97

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0.97

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0.63

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0.98

0.96

0.97

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raw

0.22

0.82

0.30

0.32

0.60

0.36

0.32

0.32

0.60

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670.

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0.97

0.82

0.67

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0.97

0.98

0.98

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wN

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0.90

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0.33

0.95

0.85

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0.87

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0.83

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0.47

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0.64

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wC

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0.88

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0.39

0.88

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0.87

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000.

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endal

l

30

Tab

le5:

This

table

show

sth

eK

endal

lra

nk

corr

elat

ion

(τ-b

)an

dSp

earm

an’s

rank

(ρ)

corr

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ion

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cien

tsam

ong

vari

ous

kurt

osis

mea

sure

sin

the

sim

ula

tion

study

inSec

tion

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ach

corr

elat

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esti

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eis

calc

ula

ted

bas

edon

8,35

2pai

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mat

es(8,3

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0.37

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0.18

0.26

0.26

0.36

0.71

0.45

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0.67

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0.43

0.30

0.33

0.70

0.66

0.31

0.38

0.84

0.45

0.04

0.87

0.82

0.44

0.55

Ku

rtN

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0.00

0.13

0.15

0.72

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0.35

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Ku

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Ku

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320.

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260.

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320.

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32K

end

all

31

Table 6: This table shows the summary statistics of ∆KF0

in the filtered raw data set fromOptionMetrics, 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 tohigh. 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 and92 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 ADR/ADS 4.5% 0.005 0.037 0.082 0.067 0.116 1.5061m Common Stock 66.1% 0.002 0.035 0.087 0.075 0.125 2.0411m ETF 15.2% 0.001 0.011 0.022 0.016 0.026 1.6381m Fund 0.1% 0.009 0.062 0.108 0.103 0.142 0.4501m Market Index 9.6% 0.001 0.007 0.018 0.012 0.021 0.4631m Not Specified 4.5% 0.007 0.087 0.130 0.123 0.162 2.099

2m ADR/ADS 4.7% 0.008 0.047 0.098 0.083 0.131 1.3732m Common Stock 66.4% 0.004 0.051 0.107 0.094 0.141 8.8872m ETF 14.1% 0.001 0.012 0.025 0.018 0.030 8.1122m Fund 0.1% 0.008 0.065 0.120 0.115 0.153 0.4632m Market Index 10.4% 0.001 0.008 0.020 0.014 0.024 10.9222m Not Specified 4.3% 0.008 0.094 0.146 0.132 0.177 3.824

3m ADR/ADS 4.3% 0.008 0.045 0.097 0.079 0.130 0.9463m Common Stock 61.0% 0.004 0.048 0.106 0.090 0.140 22.2703m ETF 15.4% 0.001 0.011 0.025 0.017 0.029 0.8583m Fund 0.1% 0.008 0.067 0.133 0.122 0.168 0.7463m Market Index 15.2% 0.001 0.009 0.024 0.015 0.028 3.2053m Not Specified 4.0% 0.009 0.088 0.145 0.128 0.178 2.121

6m ADR/ADS 5.1% 0.006 0.050 0.110 0.087 0.139 2.5536m Common Stock 71.7% 0.004 0.053 0.118 0.096 0.148 24.1616m ETF 14.1% 0.002 0.012 0.029 0.020 0.035 4.4656m Fund 0.1% 0.009 0.071 0.150 0.128 0.182 1.3846m Market Index 5.3% 0.001 0.011 0.028 0.018 0.035 1.5916m Not Specified 3.6% 0.014 0.096 0.154 0.134 0.184 2.999

12m ADR/ADS 4.9% 0.009 0.054 0.128 0.095 0.157 3.08412m Common Stock 71.2% 0.004 0.057 0.129 0.098 0.159 5.60212m ETF 13.5% 0.002 0.011 0.039 0.026 0.049 1.65612m Fund 0.1% 0.030 0.110 0.155 0.130 0.179 0.61312m Market Index 9.1% 0.002 0.010 0.025 0.017 0.035 1.98812m Not Specified 1.2% 0.023 0.085 0.154 0.121 0.184 1.176

32

Table 7: This table shows the summary statistics of Kmin

F0of the lowest OTM put option in

the filtered raw data set from OptionMetrics, from January 1996 to August 2014. F0 is theforward spot price. In the maturity column, options with time to maturity 1) between 28 to32 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 isdefined according to the OptionMetrics Ivy DB reference manual.

Maturity Issue Type Proportion Min Q1 Mean Median Q3 Max

1m ADR/ADS 5.2% 0.310 0.811 0.857 0.875 0.922 1.0001m Common Stock 76.9% 0.169 0.804 0.851 0.867 0.916 1.0001m ETF 7.4% 0.231 0.845 0.884 0.909 0.950 1.0001m Fund 0.1% 0.593 0.865 0.898 0.910 0.946 1.0001m Market Index 3.3% 0.248 0.803 0.854 0.867 0.921 1.0001m Not Specified 7.0% 0.193 0.803 0.853 0.871 0.923 1.000

2m ADR/ADS 5.3% 0.248 0.764 0.821 0.841 0.899 1.0002m Common Stock 78.0% 0.120 0.752 0.812 0.830 0.892 1.0002m ETF 6.5% 0.154 0.807 0.857 0.887 0.941 1.0002m Fund 0.2% 0.488 0.823 0.870 0.884 0.929 1.0002m Market Index 3.3% 0.229 0.757 0.819 0.836 0.900 1.0002m Not Specified 6.6% 0.162 0.775 0.830 0.851 0.907 1.000

3m ADR/ADS 5.2% 0.234 0.671 0.755 0.775 0.854 1.0003m Common Stock 75.2% 0.082 0.664 0.748 0.763 0.846 1.0003m ETF 6.6% 0.123 0.710 0.791 0.825 0.897 1.0003m Fund 0.2% 0.525 0.779 0.837 0.856 0.906 0.9933m Market Index 6.4% 0.234 0.722 0.795 0.813 0.888 1.0003m Not Specified 6.3% 0.207 0.693 0.770 0.790 0.868 0.999

6m ADR/ADS 5.4% 0.135 0.611 0.714 0.735 0.833 1.0006m Common Stock 79.7% 0.051 0.605 0.708 0.725 0.824 1.0006m ETF 6.1% 0.098 0.677 0.774 0.816 0.898 1.0006m Fund 0.3% 0.386 0.747 0.812 0.824 0.890 0.9956m Market Index 2.2% 0.087 0.668 0.751 0.774 0.861 1.0006m Not Specified 6.3% 0.212 0.701 0.771 0.794 0.861 1.000

12m ADR/ADS 5.2% 0.090 0.407 0.522 0.517 0.633 0.99612m Common Stock 81.1% 0.034 0.397 0.521 0.505 0.636 1.00012m ETF 6.3% 0.059 0.463 0.589 0.579 0.716 1.00012m Fund 0.2% 0.300 0.543 0.631 0.617 0.744 0.99212m Market Index 5.0% 0.075 0.546 0.699 0.755 0.873 0.99712m Not Specified 2.2% 0.180 0.513 0.636 0.632 0.759 0.999

33

Table 8: This table shows the summary statistics of Kmax

F0of the highest OTM call option in

the filtered raw data set from OptionMetrics. F0 is the forward spot price. In the maturitycolumn, options with time to maturity 1) between 28 to 32 days are labeled as 1m; 2) between58 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 OptionMetricsIvy DB reference manual.

Maturity Issue Type Proportion Min Q1 Mean Median Q3 Max

1m ADR/ADS 5.2% 1.000 1.067 1.145 1.113 1.187 3.4551m Common Stock 76.9% 1.000 1.070 1.153 1.120 1.194 4.6351m ETF 7.4% 1.000 1.030 1.111 1.060 1.128 6.1211m Fund 0.1% 1.001 1.053 1.107 1.090 1.143 1.6821m Market Index 3.3% 1.000 1.048 1.107 1.079 1.129 3.1581m Not Specified 7.0% 1.000 1.078 1.182 1.137 1.235 3.598

2m ADR/ADS 5.3% 1.000 1.089 1.194 1.149 1.243 5.5522m Common Stock 78.0% 1.000 1.095 1.206 1.160 1.256 10.1022m ETF 6.5% 1.000 1.039 1.141 1.077 1.157 9.1602m Fund 0.2% 1.000 1.062 1.136 1.111 1.177 1.9672m Market Index 3.3% 1.000 1.065 1.143 1.108 1.175 12.1142m Not Specified 6.6% 1.000 1.090 1.227 1.158 1.274 5.560

3m ADR/ADS 5.2% 1.000 1.123 1.285 1.215 1.363 4.7583m Common Stock 75.2% 1.000 1.130 1.299 1.225 1.375 24.0513m ETF 6.6% 1.000 1.058 1.213 1.116 1.229 12.0153m Fund 0.2% 1.001 1.083 1.184 1.150 1.238 1.9903m Market Index 6.4% 1.000 1.077 1.215 1.131 1.218 4.9623m Not Specified 6.3% 1.000 1.116 1.345 1.228 1.439 6.261

6m ADR/ADS 5.4% 1.000 1.146 1.364 1.269 1.471 6.2306m Common Stock 79.7% 1.000 1.149 1.363 1.274 1.466 26.0946m ETF 6.1% 1.000 1.066 1.246 1.138 1.288 22.3236m Fund 0.3% 1.000 1.115 1.231 1.184 1.293 2.7686m Market Index 2.2% 1.000 1.082 1.224 1.156 1.268 5.2716m Not Specified 6.3% 1.000 1.094 1.286 1.182 1.349 6.499

12m ADR/ADS 5.2% 1.001 1.339 1.831 1.593 2.016 9.69312m Common Stock 81.1% 1.000 1.312 1.759 1.535 1.939 14.00512m ETF 6.3% 1.000 1.217 1.567 1.367 1.586 14.35512m Fund 0.2% 1.028 1.191 1.418 1.356 1.490 2.82112m Market Index 5.0% 1.000 1.080 1.239 1.183 1.323 3.85412m Not Specified 2.2% 1.000 1.124 1.519 1.300 1.693 5.879

34

Table 9: This table shows the average and the standard deviation of daily Kendall τ -b andSpearman ρ correlations of various volatility estimates in the empirical study in Section 4. Def-inition and calculation of each risk-neutral volatility measure is provided in Section 2. Kendallτ -b correlations are presented in the highlighted cells. Each pair of correlation is first estimatedfor all options with maturities of 1-month, 2-month, 3-month, 6-month and 12-month of allissue 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 maturityis provided in Table 6. The subscript raw refers to a measure based on the raw data set fromOptionMetrics. The subscript s1 refers to a measure based on the smoothing method 1 thatfits a natural cubic spline in interpolating implied volatilities against deltas.

SpearmanName VolBKM

raw VolCBOEraw VolNP

raw VolSmirkraw VolBKM

s1 VolCBOEs1 VolNP

s1 VolSmirks1

VolBKMraw 0.91 0.93 0.94 0.95 0.95 0.93 0.94

(0.07) (0.17) (0.18) (0.15) (0.15) (0.18) (0.17)VolCBOE

raw 0.80 0.84 0.85 0.86 0.86 0.85 0.85(0.11) (0.15) (0.16) (0.14) (0.14) (0.16) (0.16)

VolNPraw 0.83 0.70 0.99 0.96 0.97 0.99 0.98

(0.17) (0.16) (0.06) (0.11) (0.11) (0.06) (0.08)VolSmirk

raw 0.84 0.72 0.95 0.96 0.97 0.99 0.99(0.18) (0.17) (0.07) (0.09) (0.08) (0.03) (0.05)

VolBKMs1 0.87 0.73 0.88 0.89 0.99 0.96 0.97

(0.15) (0.15) (0.11) (0.10) (0.03) (0.10) (0.08)VolCBOE

s1 0.87 0.73 0.90 0.91 0.97 0.97 0.98(0.15) (0.15) (0.11) (0.09) (0.04) (0.09) (0.07)

VolNPs1 0.82 0.71 0.94 0.96 0.87 0.89 0.98

(0.17) (0.16) (0.07) (0.05) (0.10) (0.10) (0.07)VolSmirk

s1 0.84 0.71 0.94 0.96 0.90 0.93 0.94(0.17) (0.16) (0.09) (0.06) (0.09) (0.08) (0.07)

Kendall

35

Table 10: This table shows the average and the standard deviation of daily Kendall τ -b andSpearman ρ correlations of skewness estimates in the empirical study in Section 4. Definitionand calculation of each risk-neutral skewness measure is provided in Section 2. Kendall τ -bcorrelations are presented in the highlighted cells. Each pair of correlation is first estimated forall options with maturities of 1-month, 2-month, 3-month, 6-month and 12-month of all issuetypes on the daily basis. The average and the standard deviation (shown in parentheses) arethen calculated based on daily correlations from 1996 to 2014. The definition of maturity isprovided in Table 6. The subscript raw refers to a measure based on the raw data set fromOptionMetrics. The subscript s1 refers to a measure based on the smoothing method 1 thatfits a natural cubic spline in interpolating implied volatilities against deltas.

SpearmanName SkewBKM

raw SkewCBOEraw SkewNP

raw SkewSmirkraw SkewMixon

raw SkewBKMs1 SkewCBOE

s1 SkewNPs1 SkewSmirk

s1 SkewMixons1

SkewBKMraw 0.89 0.31 0.43 0.50 0.75 0.71 0.29 0.55 0.48

(0.10) (0.24) (0.24) (0.22) (0.17) (0.18) (0.23) (0.23) (0.22)SkewCBOE

raw 0.77 0.29 0.4 0.46 0.68 0.68 0.28 0.53 0.46(0.11) (0.23) (0.24) (0.23) (0.19) (0.19) (0.23) (0.22) (0.23)

SkewNPraw 0.22 0.20 0.68 0.83 0.52 0.52 0.85 0.63 0.71

(0.20) (0.20) (0.22) (0.17) (0.23) (0.25) (0.15) (0.21) (0.20)SkewSmirk

raw 0.31 0.29 0.52 0.78 0.65 0.66 0.70 0.85 0.82(0.21) (0.21) (0.19) (0.18) (0.23) (0.24) (0.21) (0.19) (0.16)

SkewMixonraw 0.36 0.33 0.68 0.62 0.68 0.70 0.69 0.76 0.86

(0.19) (0.20) (0.17) (0.16) (0.19) (0.21) (0.21) (0.17) (0.12)

SkewBKMs1 0.59 0.52 0.38 0.50 0.53 0.97 0.51 0.77 0.70

(0.17) (0.18) (0.20) (0.21) (0.18) (0.07) (0.24) (0.22) (0.20)SkewCBOE

s1 0.55 0.53 0.38 0.51 0.54 0.89 0.52 0.79 0.71(0.18) (0.18) (0.22) (0.21) (0.19) (0.08) (0.25) (0.22) (0.21)

SkewNPs1 0.21 0.19 0.72 0.55 0.54 0.38 0.39 0.66 0.82

(0.20) (0.19) (0.15) (0.19) (0.19) (0.21) (0.22) (0.22) (0.18)SkewSmirk

s1 0.40 0.38 0.48 0.73 0.60 0.61 0.64 0.51 0.81(0.20) (0.19) (0.19) (0.17) (0.16) (0.20) (0.20) (0.20) (0.17)

SkewMixons1 0.36 0.33 0.55 0.67 0.73 0.55 0.57 0.67 0.66

(0.20) (0.20) (0.18) (0.14) (0.13) (0.19) (0.19) (0.18) (0.16)Kendall

36

Table 11: This table shows the average and the standard deviation of daily Kendall τ -b andSpearman ρ correlations of excess kurtosis estimates in the empirical study in Section 4. Def-inition and calculation of each risk-neutral kurtosis measure is provided in Section 2. Kendallτ -b correlations are presented in the highlighted cells. Each pair of correlation is first estimatedfor all options with maturities of 1-month, 2-month, 3-month, 6-month and 12-month of allissue 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 maturityis provided in Table 6. The subscript raw refers to a measure based on the raw data set fromOptionMetrics. The subscript s1 refers to a measure based on the smoothing method 1 thatfits a natural cubic spline in interpolating implied volatilities against deltas.

SpearmanName KurtBKM

raw KurtCBOEraw KurtNP

raw KurtSmirkraw KurtBKM

s1 KurtCBOEs1 KurtNP

s1 KurtSmirks1

KurtBKMraw 0.61 0.05 0.09 0.64 0.62 0.1 0.24

(0.22) (0.22) (0.30) (0.19) (0.19) (0.21) (0.27)KurtCBOE

raw 0.52 -0.14 0.06 0.39 0.4 0.08 0.16(0.24) (0.23) (0.27) (0.26) (0.26) (0.22) (0.24)

KurtNPraw 0.04 -0.10 0.19 0.2 0.2 0.33 0.2

(0.19) (0.19) (0.25) (0.24) (0.25) (0.26) (0.24)KurtSmirk

raw 0.05 0.03 0.14 0.34 0.36 0.43 0.73(0.24) (0.22) (0.21) (0.40) (0.40) (0.30) (0.26)

KurtBKMs1 0.49 0.30 0.14 0.25 0.98 0.28 0.39

(0.18) (0.24) (0.20) (0.31) (0.04) (0.28) (0.33)KurtCBOE

s1 0.48 0.31 0.14 0.27 0.94 0.3 0.4(0.19) (0.24) (0.21) (0.31) (0.06) (0.29) (0.33)

KurtNPs1 0.07 0.05 0.25 0.33 0.21 0.22 0.44

(0.17) (0.18) (0.21) (0.25) (0.23) (0.24) (0.30)KurtSmirk

s1 0.17 0.11 0.14 0.59 0.28 0.29 0.33(0.22) (0.20) (0.20) (0.22) (0.26) (0.26) (0.25)

Kendall

37

Table 12: This table shows the average and the standard deviation of percentage of matchedsecurities 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. Onthe last trading day of each month t, stocks are sorted in ascending order by the correspondingskewness measure. Each skewness measure is calculated from its options with the shortestmaturity (with at least 10 days to maturity) on that day. Quintile 5 (1) includes stockswith skewness measure that is above the 80th percentile (below the 20th percentile). For eachmonth from January 1996 to August 2014, we first count the number of matching stocks fromeach pairwise skewness portfolios and divide this number by the total number of stocks ineach portfolio to estimate the percentage. The average and the standard deviation (shownin parentheses) of percentages are then calculated for the whole period. The subscript rawrefers to a measure based on the raw data set from OptionMetrics. The subscript s1 refers toa measure based on the smoothing method 1 that fits a natural cubic spline in interpolatingimplied volatilities against deltas.

Quintile 1Name SkewBKM

raw SkewCBOEraw SkewNP

raw SkewSmirkraw SkewMixon

raw SkewBKMs1 SkewCBOE

s1 SkewNPs1 SkewSmirk

s1 SkewMixons1

SkewBKMraw 0.82 0.28 0.37 0.37 0.69 0.66 0.30 0.47 0.43

(0.05) (0.07) (0.06) (0.06) (0.05) (0.06) (0.06) (0.07) (0.06)SkewCBOE

raw 0.49 0.3 0.38 0.35 0.63 0.63 0.33 0.46 0.43(0.09) (0.07) (0.05) (0.06) (0.05) (0.05) (0.06) (0.06) (0.04)

SkewNPraw 0.49 0.38 0.56 0.70 0.42 0.43 0.74 0.52 0.59

(0.04) (0.05) (0.07) (0.06) (0.09) (0.09) (0.04) (0.06) (0.07)SkewSmirk

raw 0.49 0.38 0.81 0.61 0.50 0.51 0.60 0.83 0.69(0.05) (0.06) (0.05) (0.07) (0.09) (0.08) (0.06) (0.03) (0.05)

SkewMixonraw 0.49 0.38 0.94 0.83 0.51 0.52 0.55 0.61 0.68

(0.04) (0.06) (0.05) (0.04) (0.08) (0.07) (0.08) (0.05) (0.06)

SkewBKMs1 0.58 0.42 0.77 0.82 0.79 0.92 0.43 0.59 0.57

(0.04) (0.06) (0.04) (0.04) (0.04) (0.04) (0.09) (0.08) (0.09)SkewCBOE

s1 0.55 0.42 0.79 0.86 0.81 0.91 0.45 0.60 0.59(0.04) (0.06) (0.05) (0.05) (0.04) (0.03) (0.09) (0.07) (0.09)

SkewNPs1 0.47 0.36 0.85 0.81 0.82 0.76 0.78 0.55 0.69

(0.04) (0.05) (0.03) (0.04) (0.04) (0.05) (0.05) (0.07) (0.07)SkewSmirk

s1 0.53 0.41 0.80 0.90 0.82 0.87 0.91 0.81 0.69(0.04) (0.06) (0.05) (0.03) (0.04) (0.04) (0.04) (0.05) (0.04)

SkewMixons1 0.47 0.36 0.82 0.83 0.85 0.78 0.81 0.94 0.83

(0.04) (0.06) (0.04) (0.04) (0.03) (0.04) (0.05) (0.05) (0.04)Quintile 5

38

Table 13: This table shows the excess return performance, measured by ln(Pt+1/Pt) − Rf , ofstock portfolios sorted on the basis of risk-neutral skewness measures of individual stock, duringthe period from January 1996 to August 2014. Definition and calculation of each risk-neutralskewness measure is provided in Section 2. On the last trading day of each month t, stocksare sorted in ascending order by each skewness measure. For each stock, skewness measuresare 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 skewnessmeasure. We then calculate the equally-weighted returns of these portfolios at the end of thefollowing month t+ 1. The excess return is then obtained by subtracting the monthly risk-freereturn from the portfolio return. The adjusted close prices (for dividend splits etc) at time tand t+ 1 are used to calculate the return. Quintile 5− 1 is a hypothetical portfolio that takeslong positions in quintile 5 and short positions in quintile 1. We do not consider cost for shortselling or other transaction related costs. Mean return reports the average monthly portfolioexcess return in the sample period. αFF5 stands for the monthly portfolio alpha estimatedfrom 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) arealso reported as well as the explanatory power of the model (adjusted R2). t-values calculatedusing 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: ContinuedLog Ret Quintiles Mean excess return αFF5 βMKT βSMB βHML βRMW βCMA Adj-R2

SkewBKMraw 1 (lowest) -0.004 -0.010∗∗∗ 1.026∗∗∗ 0.433∗∗∗ 0.108∗ -0.072 -0.123 0.915

(-7.271) (27.679) (7.866) (1.886) (-1.086) (-1.386)5 (highest) -0.008 -0.015∗∗∗ 1.358∗∗∗ 0.580∗∗∗ 0.295∗∗ -0.413∗∗ -0.530∗∗ 0.888

(-7.493) (20.348) (4.855) (2.502) (-2.320) (-2.472)5-1 -0.007 -0.007∗∗∗ 0.333∗∗∗ 0.150 0.185 -0.342 -0.407 0.351

(-2.940) (3.729) (0.989) (1.276) (-1.634) (-1.513)

SkewCBOEraw 1 (lowest) -0.005 -0.012∗∗∗ 1.071∗∗∗ 0.484∗∗∗ 0.111∗∗ -0.138∗∗ -0.147∗∗ 0.925

(-8.475) (33.828) (9.214) (2.221) (-2.159) (-2.033)5 (highest) -0.006 -0.014∗∗∗ 1.331∗∗∗ 0.572∗∗∗ 0.323∗∗∗ -0.157 -0.506∗∗∗ 0.909

(-8.783) (23.382) (5.374) (3.095) (-1.263) (-3.267)5-1 -0.003 -0.004∗∗ 0.261∗∗∗ 0.090 0.211∗∗ -0.020 -0.359∗ 0.216

(-2.184) (3.850) (0.734) (1.906) (-0.138) (-1.886)

SkewNPraw 1 (lowest) -0.016 -0.023∗∗∗ 1.373∗∗∗ 0.651∗∗∗ 0.148 -0.408∗∗∗ -0.454∗∗∗ 0.917

(-13.067) (33.020) (6.747) (1.652) (-3.610) (-3.452)5 (highest) -0.003 -0.011∗∗∗ 1.232∗∗∗ 0.651∗∗∗ 0.159 -0.070 -0.374∗∗∗ 0.912

(-6.573) (21.684) (7.189) (1.506) (-0.669) (-2.979)5-1 0.011 0.010∗∗∗ -0.140∗∗ 0.002 0.010 0.337∗∗∗ 0.079 0.246

(6.170) (-2.425) (0.023) (0.093) (3.527) (0.543)

SkewSmirkraw 1 (lowest) -0.010 -0.018∗∗∗ 1.234∗∗∗ 0.531∗∗∗ 0.339∗∗∗ -0.186∗ -0.419∗∗∗ 0.909

(-10.285) (30.438) (6.578) (4.426) (-1.775) (-4.088)5 (highest) -0.002 -0.011∗∗∗ 1.235∗∗∗ 0.625∗∗∗ 0.235∗∗ -0.080 0.279∗∗ 0.913

(-7.132) (23.864) (7.925) (2.149) (-0.810) (-2.400)5-1 0.006 0.005∗∗∗ 0.002 0.097 -0.106 0.105 0.140 0.004

(2.908) (0.042) (1.493) (-0.970) (1.214) (1.278)

SkewMixonraw 1 (lowest) -0.008 -0.016∗∗∗ 1.165∗∗∗ 0.463∗∗∗ 0.237∗∗∗ -0.037 -0.278∗∗∗ 0.913

(-9.540) (28.795) (6.834) (3.303) (-0.395) (-3.509)5 (highest) -0.003 -0.011∗∗∗ 1.256∗∗∗ 0.602∗∗∗ 0.274∗∗∗ -0.095 -0.330∗∗ 0.909

(-6.952) (24.232) (6.841) (2.599) (-0.925) (-2.495)5-1 0.004 0.003∗ 0.092∗∗ 0.141∗∗ 0.036 -0.058 -0.051 0.106

(1.906) (2.247) (2.553) (0.404) (-0.586) (-0.426)

SkewBKMs1 1 (lowest) -0.007 -0.014∗∗∗ 1.100∗∗∗ 0.417∗∗∗ 0.131∗∗ -0.076 -0.275∗∗∗ 0.925

(-8.676) (31.840) (7.793) (2.326) (-1.052) (-3.378)5 (highest) -0.003 -0.012∗∗∗ 1.289∗∗∗ 0.587∗∗∗ 0.291∗∗ -0.127 -0.352∗∗ 0.901

(-7.058) (23.516) (5.763) (2.565) (-0.975) (-2.324)5-1 0.001 -0.000 0.190∗∗∗ 0.172 0.158 -0.052 -0.077 0.161

(-0.156) (3.137) (1.641) (1.397) (-0.390) (-0.429)

SkewCBOEs1 1 (lowest) -0.007 -0.014∗∗∗ 1.112∗∗∗ 0.427∗∗∗ 0.135∗∗ -0.076 -0.303∗∗∗ 0.921

(-8.578) (30.684) (7.748) (2.315) (-1.016) (-3.680)5 (highest) -0.003 -0.011∗∗∗ 1.295∗∗∗ 0.594∗∗∗ 0.294∗∗ -0.112 -0.308∗∗ 0.902

(-6.917) (23.706) (5.866) (2.534) (-0.889) (-2.084)5-1 0.002 0.001 0.184∗∗∗ 0.169∗ 0.158 -0.037 -0.004 0.144

(0.258) (3.118) (1.667) (1.362) (-0.284) (-0.025)

SkewNPs1 1 (lowest) -0.016 -0.024∗∗∗ 1.413∗∗∗ 0.648∗∗∗ 0.237∗∗ -0.442∗∗∗ -0.495∗∗∗ 0.916

(-13.204) (31.352) (6.220) (2.449) (-3.736) (-3.603)5 (highest) -0.003 -0.011∗∗∗ 1.218∗∗∗ 0.656∗∗∗ 0.140 -0.074 -0.312∗∗∗ 0.921

(-7.232) (22.184) (8.151) (1.380) (-0.743) (-2.733)5-1 0.011 0.010∗∗∗ -0.194∗∗∗ 0.010 -0.098 0.366∗∗∗ 0.183 0.297

(6.698) (-3.338) (0.125) (-0.825) (3.955) (1.228)

SkewSmirks1 1 (lowest) -0.009 -0.017∗∗∗ 1.169∗∗∗ 0.556∗∗∗ 0.310∗∗∗ -0.143∗ -0.336∗∗∗ 0.916

(-9.948) (32.580) (9.249) (4.689) (-1.833) (-3.758)5 (highest) -0.004 -0.012∗∗∗ 1.269∗∗∗ 0.592∗∗∗ 0.264∗∗ -0.126 -0.363∗∗ 0.898

(-6.901) (21.475) (5.951) (2.125) (-0.984) (-2.476)5-1 0.003 0.003 0.101 0.038 -0.047 0.016 -0.027 0.021

(1.298) (1.528) (0.382) (-0.323) (0.134) (-0.166)

SkewMixons1 1 (lowest) -0.010 -0.017∗∗∗ 1.168∗∗∗ 0.501∗∗∗ 0.285∗∗∗ -0.077 -0.375∗∗∗ 0.909

(-9.995) (28.831) (7.254) (3.824) (-0.768) (-3.982)5 (highest) -0.003 -0.012∗∗∗ 1.248∗∗∗ 0.596∗∗∗ 0.255∗∗ -0.060 -0.278∗∗ 0.913

(-7.421) (23.160) (6.913) (2.341) (-0.573) (-2.255)5-1 0.005 0.004∗∗ 0.081∗ 0.097 -0.031 0.016 0.097 0.03

(2.226) (1.734) (1.534) (-0.292) (0.163) (0.749)

40

Table 14: This table shows the excess return performance, measured by (Pt+1−Pt)/Pt−Rf , ofstock portfolios sorted on the basis of risk-neutral skewness measures of individual stock, duringthe period from January 1996 to August 2014. Definition and calculation of each risk-neutralskewness measure is provided in Section 2. On the last trading day of each month t, stocksare sorted in ascending order by each skewness measure. For each stock, skewness measuresare 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 skewnessmeasure. We then calculate the equally-weighted returns of these portfolios at the end of thefollowing month t+ 1. The excess return is then obtained by subtracting the monthly risk-freereturn from the portfolio return. The adjusted close prices (for dividend splits etc) at time tand t+ 1 are used to calculate the return. Quintile 5− 1 is a hypothetical portfolio that takeslong positions in quintile 5 and short positions in quintile 1. We do not consider cost for shortselling or other transaction related costs. Mean return reports the average monthly portfolioexcess return in the sample period. αFF5 stands for the monthly portfolio alpha estimatedfrom 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) arealso reported as well as the explanatory power of the model (adjusted R2). t-values calculatedusing 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: ContinuedSimp Ret Quintiles Mean excess return αFF5 βMKT βSMB βHML βRMW βCMA Adj-R2

SkewBKMraw 1 (lowest) 0.004 -0.003∗∗ 0.980∗∗∗ 0.441∗∗∗ 0.060 -0.078 -0.043 0.931

(-2.541) (33.720) (8.777) (1.145) (-1.164) (-0.483)5 (highest) 0.009 0.001 1.301∗∗∗ 0.646∗∗∗ 0.197 -0.322∗∗ -0.359∗ 0.863

(0.487) (17.785) (5.259) (1.501) (-2.065) (-1.685)5-1 0.003 0.002 0.322∗∗∗ 0.207 0.136 -0.245 -0.316 0.282

(0.556) (3.363) (1.308) (0.823) (-1.171) (-1.124)

SkewCBOEraw 1 (lowest) 0.004 -0.003∗∗ 1.024∗∗∗ 0.493∗∗∗ 0.062 -0.150∗∗ -0.059 0.935

(-2.537) (39.335) (10.130) (1.327) (-2.475) (-0.790)5 (highest) 0.008 0.000 1.277∗∗∗ 0.611∗∗∗ 0.240∗∗ -0.071 -0.373∗∗ 0.889

(0.057) (22.012) (5.413) (2.119) (-0.590) (-2.347)5-1 0.002 0.000 0.254∗∗∗ 0.120 0.176 0.078 -0.314 0.167

(0.312) (3.566) (0.909) (1.391) (0.488) (-1.524)

SkewNPraw 1 (lowest) 0.001 -0.007∗∗∗ 1.297∗∗∗ 0.714∗∗∗ 0.055 -0.348∗∗∗ -0.301∗∗ 0.916

(-4.392) (27.035) (7.615) (0.624) (-3.752) (-2.285)5 (highest) 0.010 0.002 1.178∗∗∗ 0.685∗∗∗ 0.081 -0.039 -0.249∗∗ 0.912

(1.596) (25.426) (8.061) (0.810) (-0.461) (-2.208)5-1 0.007 0.007∗∗∗ -0.118∗∗ -0.027 0.024 0.308∗∗∗ 0.051 0.224

(4.456) (-2.109) (-0.324) (0.229) (3.554) (0.342)

SkewSmirkraw 1 (lowest) 0.002 -0.006∗∗∗ 1.179∗∗∗ 0.552∗∗∗ 0.274∗∗∗ -0.136 -0.304∗∗∗ 0.916

(-4.661) (30.398) (7.185) (3.530) (-1.541) (-3.065)5 (highest) 0.010 0.002 1.185∗∗∗ 0.653∗∗∗ 0.158 -0.054 -0.155 0.908

(1.241) (26.655) (8.146) (1.459) (-0.602) (-1.351)5-1 0.006 0.006∗∗∗ 0.007 0.103 -0.118 0.081 0.149 0.011

(3.780) (0.148) (1.566) (-1.138) (1.021) (1.368)

SkewMixonraw 1 (lowest) 0.002 -0.005∗∗∗ 1.115∗∗∗ 0.476∗∗∗ 0.182∗∗∗ -0.007 -0.193∗∗∗ 0.928

(-4.584) (35.555) (7.568) (3.159) (-0.091) (-2.756)5 (highest) 0.011 0.002 1.204∗∗∗ 0.641∗∗∗ 0.183∗ -0.054 -0.207 0.905

(1.365) (26.744) (7.214) (1.803) (-0.558) (-1.641)5-1 0.006 0.005∗∗∗ 0.090∗∗ 0.167∗∗∗ -0.001 -0.047 -0.014 0.134

(3.510) (2.350) (2.997) (-0.006) (-0.506) (-0.120)

SkewBKMs1 1 (lowest) 0.003 -0.004∗∗∗ 1.044∗∗∗ 0.434∗∗∗ 0.073 -0.072 -0.187∗∗ 0.943

(-3.927) (42.364) (10.558) (1.644) (-1.311) (-2.478)5 (highest) 0.010 0.002 1.241∗∗∗ 0.618∗∗∗ 0.203∗ -0.070 -0.223 0.889

(0.852) (23.353) (5.932) (1.736) (-0.581) (-1.472)5-1 0.005 0.003 0.198∗∗∗ 0.186∗ 0.128 0.001 -0.036 0.142

(1.613) (3.072) (1.661) (1.013) (0.005) (-0.184)

SkewCBOEs1 1 (lowest) 0.002 -0.004∗∗∗ 1.053∗∗∗ 0.446∗∗∗ 0.076∗ -0.071 -0.216∗∗∗ 0.939

(-3.851) (41.892) (10.424) (1.676) (-1.229) (-2.869)5 (highest) 0.011 0.002 1.249∗∗∗ 0.626∗∗∗ 0.212∗ -0.066 -0.182 0.892

(1.039) (24.028) (6.090) (1.782) (-0.561) (-1.231)5-1 0.006 0.004∗ 0.198∗∗∗ 0.183∗ 0.134 0.004 0.034 0.141

(1.832) (3.208) (1.705) (1.064) (0.029) (0.181)

SkewNPs1 1 (lowest) 0.001 -0.007∗∗∗ 1.337∗∗∗ 0.706∗∗∗ 0.144 -0.382∗∗∗ -0.342∗∗ 0.910

(-4.401) (24.048) (6.823) (1.389) (-3.772) (-2.399)5 (highest) 0.010 0002 1.162∗∗∗ 0.687∗∗∗ 0.062 -0.052 -0.185∗ 0.923

(1.203) (26.525) (9.341) (0.656) (-0.651) (-1.830)5-1 0.007 0.006∗∗∗ -0.173∗∗∗ -0.017 -0.084 0.329∗∗∗ 0.157 0.259

(4.270) (-2.872) (-0.194) (-0.709) (3.853) (1.025)

SkewSmirks1 1 (lowest) 0.002 -0.006∗∗∗ 1.112∗∗∗ 0.576∗∗∗ 0.249∗∗∗ -0.122∗ -0.248∗∗∗ 0.928

(-5.105) (34.154) (10.095) (3.856) (-1.960) (-2.962)5 (highest) 0.010 0.001 1.216∗∗∗ 0.625∗∗∗ 0.177 -0.076 -0.229 0.892

(0.794) (23.449) (6.271) (1.473) (-0.666) (-1.623)5-1 0.006 0.005∗∗ 0.106 0.051 -0.073 0.044 0.019 0.021

(2.489) (1.630) (0.494) (-0.508) (0.369) (0.113)

SkewMixons1 1 (lowest) 0.001 -0.006∗∗∗ 1.113∗∗∗ 0.512∗∗∗ 0.221∗∗∗ -0.040 -0.278∗∗∗ 0.925

(-5.204) (37.040) (8.363) (3.827) (-0.494) (-3.465)5 (highest) 0.010 0.001 1.195∗∗∗ 0.624∗∗∗ 0.165 -0.025 -0.148 0.911

(0.755) (26.069) (7.426) (1.562) (-0.267) (-1.272)5-1 0.006 0.005∗∗ 0.083∗ 0.114∗ -0.058 0.013 0.129 0.053

(3.551) (1.880) (1.856) (-0.567) (0.139) (1.023)

42

Fig

ure

1:T

his

figu

red

escr

ibes

the

layo

ut

and

conte

nt

of

Fig

ure

s2

to10.

Ap

pro

xim

ati

on

erro

rsare

defi

ned

as

Est

imate

dM

om

ent−

Tru

eM

om

ent

Tru

eM

om

ent

.B

ase

case

par

amet

ers

use

din

each

mod

elar

ed

escr

ibed

inT

ab

le2.

Inea

chfi

gu

re,

the

1stco

lum

nof

plo

tsil

lust

rate

sap

pro

xim

ati

on

erro

rsu

sin

gth

era

wd

ata

from

sim

ula

tion

s.T

he

2nd

colu

mn

app

lies

the

smooth

ing

met

hod

1by

fitt

ing

an

atu

ral

cub

icsp

lin

ein

inte

rpola

tin

gim

pli

edvo

lati

liti

esagain

std

elta

s.T

he

3rd

colu

mn

app

lies

the

smoot

hin

gm

eth

od

2by

fitt

ing

an

atu

ral

cub

icsp

lin

ein

inte

rpola

tin

gim

pli

edvo

lati

liti

esagain

stst

rike

pri

ces.

Th

e4th

colu

mn

app

lies

the

smoot

hin

gm

eth

od

3by

lin

earl

yin

terp

ola

tin

gim

pli

edvo

lati

liti

esagain

stdel

tas.

InF

igu

res

2,

4,

5,

7,

8an

d10,

each

mom

ent

isca

lcu

late

du

sin

g:1)

BK

Mm

eth

od

inth

e1

stro

w;

2)

CB

OE

met

hod

inth

e2n

dro

w;

3)

non

-para

met

ric

met

hod

inth

e3rd

row

;an

d4)

imp

lied

vola

tili

tysm

irk

inth

e4th

row

.In

Fig

ure

s3,

6an

d9,

mom

ent

inth

ead

dit

ion

al

5th

row

isca

lcula

ted

usi

ng

Mix

on’s

met

hod

.F

or

exam

ple

,P

an

el3b

refe

rsto

CB

OE

mom

ents

calc

ula

ted

from

the

seco

nd

smooth

ing

met

hod

.M

eth

od

sto

calc

ula

teea

chm

om

ent

are

ou

tlin

edin

Sec

tion

2.

Wit

hin

each

pan

el,

the

1st(2

nd)

colu

mn

rep

orts

app

roxim

atio

ner

rors

usi

ng

op

tion

sw

ith

exp

irati

on

of

22

(124)

trad

ing

day

s.W

ith

inea

chp

an

el,

op

tion

sp

rice

sare

sim

ula

ted

usi

ng

1)B

lack

-Sch

oles

mod

elin

the

1stro

w;

2)

Bate

sst

och

ast

icvo

lati

lity

an

dju

mp

diff

usi

on

mod

elin

the

2nd

row

;3)

Hes

ton

stoch

ast

icvo

lati

lity

mod

elin

the

3rdro

w;

and

4)M

erto

nju

mp

-diff

usi

on

mod

elin

the

4thro

w.

Inea

chp

lot,

diff

eren

tsh

ades

of

colo

ur

rep

rese

nts

resu

lts

from

diff

eren

tpar

amet

ers

use

dto

gen

erat

eop

tion

pri

ces.

43

Fig

ure

2:V

olat

ilit

yA

pp

roxim

atio

nE

rror

-In

tegr

atio

nd

om

ain

tru

nca

tion

.L

ayout

of

the

figu

reis

exp

lain

edin

Fig

ure

1.

Inea

chplo

t,ap

pro

xim

ati

on

erro

rson

the

ver

tica

lax

isar

ep

lott

edag

ain

stin

tegr

ati

on

dom

ain

wid

thu

on

the

hori

zonta

laxis

,w

hic

his

defi

ned

as

[S0∗

(u∗

0.01

+0.4

9),S

0/(u∗

0.01

+0.4

9)]

wh

ereu∈{1,2,...,5

0}.

Diff

eren

tsh

ades

of

colo

ur

rep

rese

nts

resu

lts

from

diff

eren

tp

ara

met

ers

use

dto

gen

erate

op

tion

pri

ces.

44

Fig

ure

3:S

kew

nes

sA

pp

roxim

atio

nE

rror

-In

tegr

atio

nd

om

ain

tru

nca

tion

.L

ayou

tof

the

figu

reis

exp

lain

edin

Fig

ure

1.

Inea

chp

lot,

ap

pro

xim

ati

on

erro

rson

the

ver

tica

lax

isar

ep

lott

edag

ain

stin

tegr

ati

on

dom

ain

wid

thu

on

the

hori

zonta

laxis

,w

hic

his

defi

ned

as

[S0∗

(u∗

0.01

+0.4

9),S

0/(u∗

0.01

+0.4

9)]

wh

ereu∈{1,2,...,5

0}.

Diff

eren

tsh

ades

of

colo

ur

rep

rese

nts

resu

lts

from

diff

eren

tp

ara

met

ers

use

dto

gen

erate

op

tion

pri

ces.

45

Fig

ure

4:K

urt

osis

Ap

pro

xim

atio

nE

rror

-In

tegr

atio

nd

om

ain

tru

nca

tion.

Lay

ou

toffi

gu

reis

exp

lain

edin

Fig

ure

1.

Inea

chp

lot,

ap

pro

xim

ati

on

erro

rson

the

vert

ical

axis

are

plo

tted

agai

nst

inte

grat

ion

dom

ain

wid

thu

on

the

hori

zonta

laxis

,w

hic

his

defi

ned

as

[S0∗(u∗0.0

1+

0.4

9),S

0/(u∗0.0

1+

0.4

9)]

wh

ereu∈{1,2,...,5

0}.

Diff

eren

tsh

ades

ofco

lou

rre

pre

sents

resu

lts

from

diff

eren

tp

ara

met

ers

use

dto

gen

erate

op

tion

pri

ces.

46

Fig

ure

5:V

olat

ilit

yA

pp

roxim

atio

nE

rror

-D

iscr

etis

ati

on

of

stri

kep

rice

s.L

ayou

tof

the

figu

reis

exp

lain

edin

Fig

ure

1.

Inea

chp

lot,

ap

pro

xim

ati

on

erro

rson

the

vert

ical

axis

are

plo

tted

agai

nst

stri

kep

rice

inte

rval∆K

on

the

hori

zonta

laxis

,w

hic

his

defi

ned

as

∆K≡K

i−K

i−1,

∆K∈{1,2,...,2

5}.

Diff

eren

tsh

ades

ofco

lou

rre

pre

sents

resu

lts

from

diff

eren

tp

ara

met

ers

use

dto

gen

erate

op

tion

pri

ces.

47

Fig

ure

6:S

kew

nes

sA

pp

roxim

atio

nE

rror

-D

iscr

etis

ati

on

of

stri

ke

pri

ces.

Lay

ou

tof

the

figu

reis

exp

lain

edin

Fig

ure

1.

Inea

chplo

t,ap

pro

xim

ati

on

erro

rson

the

vert

ical

axis

are

plo

tted

agai

nst

stri

kep

rice

inte

rval∆K

on

the

hori

zonta

laxis

,w

hic

his

defi

ned

as

∆K≡K

i−K

i−1,

∆K∈{1,2,...,2

5}.

Diff

eren

tsh

ades

ofco

lou

rre

pre

sents

resu

lts

from

diff

eren

tp

ara

met

ers

use

dto

gen

erate

op

tion

pri

ces.

48

Fig

ure

7:K

urt

osis

Ap

pro

xim

atio

nE

rror

-D

iscr

etis

ati

on

of

stri

ke

pri

ces.

Lay

ou

tof

figu

reis

exp

lain

edin

Fig

ure

1.

Inea

chp

lot,

ap

pro

xim

ati

on

erro

rson

the

vert

ical

axis

are

plo

tted

agai

nst

stri

kep

rice

inte

rval

∆K

on

the

hori

zonta

laxis

,w

hic

his

defi

ned

as

∆K≡K

i−K

i−1,

∆K∈{1,2,...,2

5}.

Diff

eren

tsh

ades

ofco

lou

rre

pre

sents

resu

lts

from

diff

eren

tp

ara

met

ers

use

dto

gen

erate

op

tion

pri

ces.

49

Fig

ure

8:V

olat

ilit

yA

pp

roxim

atio

nE

rror

-A

sym

met

ryin

inte

gra

tion

dom

ain

tru

nca

tion

.L

ayou

tof

the

figu

reis

exp

lain

edin

Fig

ure

1.

Inea

chp

lot,

app

roxim

atio

ner

rors

onth

eve

rtic

alax

isar

ep

lott

edagain

stasy

mm

etry

inin

tegra

tion

dom

aindu

on

the

hori

zonta

laxis

,w

hic

his

defi

ned

as

[S0∗

(0.4

9+δu/1

00),S

0/(

0.9

1−δu/1

00)]

wh

ereδu∈{1,2,...,4

1}.

Th

easy

mm

etry

isat

its

min

imu

mw

hen

δu=

21.

Diff

eren

tsh

ad

esof

colo

ur

rep

rese

nts

resu

lts

from

diff

eren

tp

aram

eter

su

sed

togen

erate

op

tion

pri

ces.

50

Fig

ure

9:S

kew

nes

sA

pp

roxim

atio

nE

rror

-A

sym

met

ryin

inte

gra

tion

dom

ain

tru

nca

tion

.L

ayou

tof

figu

reis

exp

lain

edin

Fig

ure

1.

Inea

chp

lot,

app

roxim

atio

ner

rors

onth

eve

rtic

alax

isar

ep

lott

edagain

stasy

mm

etry

inin

tegra

tion

dom

aindu

on

the

hori

zonta

laxis

,w

hic

his

defi

ned

as

[S0∗

(0.4

9+δu/1

00),S

0/(

0.9

1−δu/1

00)]

wh

ereδu∈{1,2,...,4

1}.

Th

easy

mm

etry

isat

its

min

imu

mw

hen

δu=

21.

Diff

eren

tsh

ad

esof

colo

ur

rep

rese

nts

resu

lts

from

diff

eren

tp

aram

eter

su

sed

togen

erate

op

tion

pri

ces.

51

Fig

ure

10:

Ku

rtos

isA

pp

roxim

atio

nE

rror

-A

sym

met

ryin

inte

gra

tion

dom

ain

tru

nca

tion

.L

ayou

tof

figu

reis

exp

lain

edin

Fig

ure

1.

Inea

chp

lot,

app

roxim

atio

ner

rors

onth

eve

rtic

alax

isar

ep

lott

edagain

stasy

mm

etry

inin

tegra

tion

dom

aindu

on

the

hori

zonta

laxis

,w

hic

his

defi

ned

as

[S0∗

(0.4

9+δu/1

00),S

0/(

0.9

1−δu/1

00)]

wh

ereδu∈{1,2,...,4

1}.

Th

easy

mm

etry

isat

its

min

imu

mw

hen

δu=

21.

Diff

eren

tsh

ad

esof

colo

ur

rep

rese

nts

resu

lts

from

diff

eren

tp

aram

eter

su

sed

togen

erate

op

tion

pri

ces.

52

Fig

ure

11:

Th

isfi

gure

pre

sents

box

plo

tsof

per

centa

ges

of

matc

hin

git

ems

inth

eto

pqu

inti

leb

etw

een

each

mom

ent

mea

sure

an

dth

eco

rres

pon

din

gtr

ue

mea

sure

inS

ecti

on3.

For

each

vola

tili

tyes

tim

ate

that

isco

mp

ute

du

sin

gB

KM

,C

BO

E,

NP

an

dS

mir

k,

ther

eare

72

vola

tili

typ

roxie

sw

ith

72

corr

esp

ond

ing

tru

evo

lati

liti

esfr

omea

chof

116

vari

atio

ns

that

stu

dy

ap

pro

xim

ati

on

erro

rs(w

her

e116

=50

vari

ati

on

sin

tru

nca

tion

erro

rst

ud

y+

25

vari

atio

ns

ind

iscr

etis

atio

n+

41va

riat

ion

sin

asym

met

ric

tru

nca

tion

an

d72

=9

sets

of

para

met

ers

x2

matu

rity

term

sx

4op

tion

-pri

ce-g

ener

ati

on

mod

els)

.In

each

of11

6va

riat

ion

s,w

eso

rtvo

lati

lity

esti

mate

sfr

om

on

eof

4m

eth

od

s(B

KM

,C

BO

Eet

c)in

asc

end

ing

ord

er.

We

extr

act

esti

mate

sth

atar

eab

ove

the

80th

per

centi

le(a

qu

inti

leof

72it

ems

isro

ugh

ly15).

As

each

vola

tili

tyes

tim

ate

has

aco

rres

pon

din

gtr

ue

vola

tility

,w

eth

enca

lcu

late

the

per

centa

geof

thes

eco

rres

pon

din

gtr

ue

vola

tili

ties

are

als

oab

ove

the

80

thp

erce

nti

leof

tru

evo

lati

liti

es.

Each

box

plo

til

lust

rate

sth

ed

istr

ibu

tion

ofth

ese

116

per

centa

ges.

We

then

rep

eat

this

for

skew

nes

san

dku

rtosi

ses

tim

ate

s.In

the

box

plo

tp

an

elw

ith

the

titl

e“R

aw”,

we

per

form

this

pro

cess

for

the

raw

dat

afr

omsi

mu

lati

ons.

Th

eti

tle

“S

mooth

1”

mea

ns

we

use

the

smooth

ing

met

hod

1by

fitt

ing

an

atu

ral

cub

icsp

lin

ein

inte

rpol

atin

gim

pli

edvo

lati

liti

esag

ain

std

elta

s.T

he

titl

e“S

mooth

2”

mea

ns

we

use

the

smooth

ing

met

hod

2by

fitt

ing

an

atu

ral

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53

Fig

ure

12:

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54

Fig

ure

13:

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onth

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nes

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from

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ruary

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ugu

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f.

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ock

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5(1

)in

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ith

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(bott

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re.

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ula

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y-w

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turn

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ivid

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eb

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om

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tm

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eis

show

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izon

tal

axis

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ere

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s-qu

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ort

folio

isil

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rate

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ng

the

vert

ical

axis

.

55

Figure 14: This figure present a heat map of average risk-neutral volatility of skewness-quintileportfolios from February 1996 to August 2014. The risk-neutral volatility is calculated asVolBKM

raw and VolBKMs1 . 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 Sec-tion 2. On the last trading day of each month t, stocks are sorted in ascending order by eachskewness measure. For each stock, skewness measures are calculated from its options with theshortest maturity (with at least 10 days to maturity) on that day. Quintile 5 (1) includes stockswith the top (bottom) 20% skewness measure. We calculate the average risk-neutral volatilityof each portfolio. The panel above the heat map shows the colour key used to represent theaverage risk-neutral volatility. It also presents the histogram of all average risk-neutral volatil-ity of all skewness-quintile portfolios. In the heat map, the time is shown on the horizontalaxis where each skewness-quintile portfolio is illustrated along the vertical axis. Results usingVolBKM

raw (VolBKMs1 ) is included in the top (bottom) panel.

Figures are presented on the next page.

56

Figure 14: Continued

VolBKMraw

VolBKMs1

57

Figure 15: This figure present a heat map of average risk-neutral excess kurtosis of skewness-quintile portfolios from February 1996 to August 2014. The risk-neutral excess kurtosis iscalculated as KurtBKM

raw and KurtBKMs1 . We first form stock portfolios on the basis of each risk-

neutral skewness measure. Definition and calculation of each risk-neutral skewness measure isprovided is Section 2. On the last trading day of each month t, stocks are sorted in ascendingorder by each skewness measure. For each stock, skewness measures are calculated from itsoptions with the shortest maturity (with at least 10 days to maturity) on that day. Quintile5 (1) includes stocks with the top (bottom) 20% skewness measure. We calculate the averagerisk-neutral excess kurtosis of each portfolio. The panel above the heat map shows the colourkey used to represent the average risk-neutral excess kurtosis. It also presents the histogramof 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 illustratedalong the vertical axis. Results using KurtBKM

raw (KurtBKMs1 ) is included in the top (bottom)

panel. There are some extreme outliers (where excess kurtosis KurtBKMs1 exceeds 60) presented

in portfolios that formed in September 2010, July 2013, November 2013 and May 2014. Forillustration purpose, we remove these observations from the heat map.

Figures are presented on the next page.

58

Figure 15: Continued

KurtBKMraw

KurtBKMs1

59

Appendix A. Derivation of ε1 in eq. (18)

To see how we derive ε1 from EQ(ln(Sτ/F0)) to compensate for the difference between theforward price F0 and the strike price K0 that is immediate below F0, we start with valuingEQ(ln(Sτ/K0)). It is important to note that, in an idealized world where strike prices arequoted 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 weightedinversely proportional to K2. That is, at time 0, the portfolio is worth:

Π =

∫ K0

0

1

K2max(K − Sτ , 0) dK +

∫ ∞K0

1

K2max(Sτ −K, 0) dK

= −1− lnST +STK0

+ lnK0

=ST −K0

K0

− lnSTK0

(A.1)

∴ EQ

(lnSTF0

)= EQ

(lnSTK0

+ lnK0

F0

)= EQ

(ST −K0

K0

− Π + lnK0

F0

)(A.2)

In the last step in eq. (A.2), we make a substitution from the result in eq. (A.1). It is straight-forward to see that EQ(Π) is approximated by the first half of eq. (14). The focus is now onthe other two terms in eq. (A.2):

ε1 = EQ

(ST −K0

K0

+ lnK0

F0

)=EQ(ST )

K0

− 1 + lnK0

F0

=F0

K0

− 1− lnF0

K0

= −(

1 + lnF0

K0

− F0

K0

)(A.3)

as found in eq. (18). For the other ε terms in eqs. (19) to (21), similar derivations of risk-neutralexpectation of the squared contract (V ), the cubed contract (W ), the quartic contract (X) canbe conducted. The exact derivation manuscript is available upon request.

60