Common Factors in Equity Option Returns

Common Factors in Equity Option Returns
Alex Horenstein1 , Aurelio Vasquez∗2 , and Xiao Xiao3,4
1
Department of Economics, University of Miami
2
ITAM, School of Business
3
Erasmus School of Economics, Erasmus University Rotterdam
4
Erasmus Research Institute of Management - ERIM
May 29, 2019
Abstract
This paper studies the factor structure of the cross-section of delta-hedged equity
option returns. Using latent factor techniques, we find strong evidence supporting the
existence of a factor structure in equity options returns. We find that a four-factor model
captures relevant latent factors and explains the time series and the cross-section of equity
option returns. The factors are the market volatility risk factor and three characteristic-
based factors related to firm size, idiosyncratic volatility, and the difference between
implied and historical volatilities. Stock return factors cannot price the cross-section of
equity option returns.
JEL Classification: C14, G13, G17

Keywords: Cross Section of Option Returns, Latent Factors, Rank Estimation
∗
Corresponding to: Aurelio Vasquez, ITAM, Rio Hondo 1, Alvaro Obregón, Mexico City, Mexico. Tel:
(52) 55 5628 4000 x.6518; Email: aurelio.vasquez@itam.mx.
1
1 Introduction
The identification of factors that drive the comovements of asset returns is a central question
in empirical asset pricing. Existing papers on multi-factor asset pricing models mainly focus
on common factors in stock returns1 . However the factor structure of the cross-section and
time-series of equity option returns is less understood. Options are often viewed as merely
leveraged positions in the underlying stocks. However Bakshi and Kapadia (2003) show that
delta-hedged option returns contain risk premiums beyond the equity premium such as the
variance risk premium. To study the factor structure of option returns provides information
on the factor that drives the cross-section of variance risk premiums. This is the main goal
of our study.
Using a multifactor stochastic volatility model, we show that the expected delta-hedged
equity option returns are driven by the volatility risk of the priced factors. Since no options
are traded on the stock return factors, we work with option portfolios constructed from firm
characteristics that predict option returns. We include eleven characteristics that predict
option returns: size, reversal, momentum, profitability, cash holding and analyst forecast
dispersion in Cao et al. (2017), credit rating in Vasquez and Xiao (2018), the deviation of
realized volatility from implied volatility in Goyal and Saretto (2009), idiosyncratic volatility
in Cao and Han (2013), and the volatility term structure in Vasquez (2017). Empirically, we
construct 105 option portfolios sorted by the eleven characteristics using monthly portfolios of
delta-hedged options from January 1996 to December 2015. The eleven characteristic-based
factors are constructed based on long-short strategies of decile or quintile returns. We also
include two additional market factors: the delta-hedged return of the S&P 500 index option
and the value-weighted delta-hedged return on the individual stocks that are components of
the S&P 500 index. We consider 13 candidate factors in total.
Our identification procedure follows the factor identification protocol suggested by Puk-
thuanthong et al. (2018). We first employ a latent variable analysis of the covariance matrix
of the delta-hedged option returns to understand the factor structure for the cross-section of
option returns. Latent variables are not directly observed but are econometrically inferred
1
Recent studies include Barillas and Shanken (2018), Ahn et al. (2018), Hou et al. (2018), Feng et al.
(2017) among others.
2
from observed variables. We estimate the number of latent factors using six identification
methods: 1) the Eigenvalue Ratio (ER) and 2) the Growth Ratio (GR) estimators by Ahn
and Horenstein (2013), 3) the Edge Distribution (ED) estimator of Onatski (2010), 4) the
BIC3 and 5) IC1 estimators of Bai and Ng (2002), and 6) the Modified Information Criterion
estimator (ABC) of Alessi et al. (2010). We find that there is one strong factor and possibly
up to five weak factors that can explain the cross-section of delta-hedged option returns.
Next we estimate the common latent factors using principal component analysis (PCA) as
suggested in Connor and Korajczyk (1986) and study how well they explain the cross-section
of delta-hedged option returns. We find that the six latent factors explain on average 31.8%
of the time series variation of the 105 portfolios. The correlation between the average return
of the 105 portfolios and the model’s predicted return is 96%.
Latent factors estimated with PCA are difficult to interpret economically since they are
not directly observable. To circumvent this problem, we test if the 13 (observed) candidate
factors are related to the covariance matrix of delta-hedged option returns. Using the rank-
estimation method suggested by Ahn et al. (2018) on the 13 candidate factors, we find that
out of the 462 different combinations of 6 factors, only one set can generate a full rank beta
matrix. The unique set contains the long-short factors constructed from size, cash holding,
analyst dispersion, idiosyncratic volatility, volatility deviation, and credit rating. These six
factors contain most of the relevant information on the cross-section of delta-hedged option
returns. Canonical correlation analysis further confirms this result. Linear combinations of
the six characteristic-based factor candidates are highly correlated with linear combinations
of the six latent factors estimated from PCA.
Next, we find that three out of the six factor candidates suffice to explain the cross-section
of delta-hedged option returns: size, idiosyncratic volatility, and volatility deviation. The
correlation between the average return of the 105 portfolios and the three factor’s predicted
returns is 0.95. The added explanatory power of the three remaining factors: cash holding,
analyst dispersion, and credit rating is negligible. The fourth factor we propose is the market
volatility risk factor. This factor’s loadings are almost constant and as such it is useful to
explain the time-series of option returns but not the cross-section.2
2
Fama and French (1992) and more recently Ahn and Horenstein (2018) show that the stock returns’
3
Given the empirical results, we propose a four-factor model that captures the time-series
and cross-sectional variation in delta-hedged option returns. The factors include the delta-
hedged return of the S&P 500 index options, size, idiosyncratic volatility, and volatility
deviation. The market volatility risk factor mainly explains the time-series variation of the
delta-hedged option returns. The latter three factors capture the cross-sectional comovements
in option returns. Using the 105 option portfolios as test assets in Fama-MacBeth regressions,
we find that the exposures to the three factors are statistically significant in explaining average
delta-hedged returns of the 105 portfolios. The adjusted R2 ranges from 84% to 90% when
using the full sample or subsamples. To address the critique in Lewellen et al. (2010), we
also test the proposed model on a set of delta-hedge option portfolios constructed based on
industrial classification instead of firm characteristics and find that the model explains their
cross-section quite well.
Lastly, we investigate how much information about the cross-section of delta-hedged
option returns are contained in commonly used factors to price the cross-section of stock
returns. We find that stock factors have little to no explanatory power over the 6 latent
factors necessary to explain the cross-section of delta-hedged option returns. The only stock
factor that is sometimes significant at the 1% or less when regressed onto option latent factors
is the BAB factor (for the case of the 2nd and 3rd most important factors). As expected,
the information contained in stock factors seems not relevant for pricing the cross-section of
delta-hedged option returns.
This paper differs from the existing literature in option factors in two important ways.
First, different from Christoffersen et al. (2017a) who study the factor structure of equity
volatility levels, skews, and term structures, we focus on the factor structure of delta-hedged
equity option returns, which reflect the risk premium required by bearing the unfavorable
variance risk. While the implications of their paper are mainly on equity option valuation,
our analysis uncovers the factor structure of equity option returns and the embedded variance
risk premium. Second, in the spirit of Ahn et al. (2018) and Feng et al. (2017), this paper
Market portfolio shows little to no variability in factor loadings and is mostly useful to explain the time-series
dimension of stock returns. We find an analogous relationship between the market volatility risk factor and
option returns. The market volatility risk factor explains the time-series but not the cross-section of option
returns
4
attempts to avoid the proliferation of option returns’ factors capturing similar information.
Like in the stock return literature, multiple predictive characteristics are proposed but not
necessarily all of them capture different information. We are the first to address this issue in
the option market and find that four factors summarize all the information about the cross-
section and time-series among the 13 factors proposed so far in the option return literature.
In Section 2 we present our main analytical results motivating the factor structure in
delta-hedged option returns. Section 3 explains the data used for our empirical analysis. In
Section 4 we perform our quantitative studies: In Section 4.1 we analyze the factor structure
in option returns using latent variable techniques, in Section 4.2 we study which option
returns predictors that best explain the factor structure in option returns, and in Section
4.3 we study the common information between factors in stock returns and factor in option
returns. We conclude in Section 5.
2 Theoretical motivation: Delta-hedged equity option gains

in a multi-factor framework
In this section, we show expected delta-hedged equity option gains in a multi-factor frame-
work, in which stock return and variance are driven by multiple factors. The results show
that the delta-hedged option gains have no sensitivity to standard factors in stock returns,
just to those related to volatility risk.
We denote the stock price and the variance of stock return for firm i as Sti and Vti . The
j
variance of stock i is driven by a factor structure: Vf,t , j = 1, ..., n, where the factors are
independent with each other. The stock price evolves according to the process:
dSti
q
i i i i
= µ (S
t t , Vt )dt + Vti dW1t ,
Sti
n
j
X
i
Vt = β j Vf,t + Zti ,
j=1
j j j i,j
dVf,t = θj (Vf,t )dt + η j (Vf,t )dW2t .
To simplify the analysis, we assume that the correlations among the standard Brownian
5
i and W i,j are all 0. Relaxing this assumption and allowing leverage effect does
motions W1t 2t
not change the main result of the model. Note that if the stock variance is only driven by
the market index variance, the factor structure of the stock variance is based on CAPM that
stock returns have a market component and an idiosyncratic component: Rti = α̂i + β̂ i Rtm +ˆ
it ,
where ˆit is uncorrelated with Rtm . βi = (β̂ i )2 is the sensitivity of individual variance with
1 corresponds to the variance of the market
respect to the variance of the market index and Vf,t
1 ,
index. Similarly, if the stock return is driven by the Fama-French three factor model, Vf,t
2 and V 3 corresponds to the variance of the market index, variance of the SMB factor
Vf,t f,t
and variance of the HML factor.

By Ito’s lemma, we can write the call option price as,
t+τ t+τ t+τ

∂C i i
Z Z Z
i
Ct+τ = Cti + ∆iu dSui + dV + biu du (1)
t t ∂V i u t
∂Cui
where ∆iu = ∂Sui
is the delta of the call option and
n
∂C i 1 i i 2 ∂ 2 C i 1 X j 2 j 2 ∂2C i
biu = + V (S ) + (β ) (η ) .
∂u 2 ∂(S i )2 2 ∂(V i )2
j=1
The no-arbitrage assumption implies that the valuation equation that determines the call
option price is:
n
1 i i 2 ∂2C i 1 X j 2 j 2 ∂2C i ∂C i
V (S ) + (β ) (η ) + rSi +
2 ∂(S i )2 2 ∂(V i )2 ∂Si
j=1
n
X j j ∂C i ∂C i
[ β j (θj (Vf,t ) − λj (Vf,t ))] + − rC i = 0, (2)
∂V i ∂t
j=1
j j
where λj (Vf,t ) = −covt ( dm
mt , dVf,t ) is the variance risk premium for factor j given a pricing
t
kernel mt .
6
Combining Equation (1) and (2), we have:
t+τ t+τ
∂C i
Z Z
i
Ct+τ − Cti = ∆iu dSui + r(C i − Si )du+
t t ∂Si
t+τ n t+τ n
∂C i ∂C i
Z Z
j j
dW2i,j ].
X X
[ β j λj (Vf,t )] i
du + [ β j θj (Vf,t ) i
(3)
t ∂V t ∂V
j=1 j=1
With a delta-hedged portfolio, we buy the call option and dynamically delta-hedge the option
position with time-varying ∆iu . The delta-hedged gain Πit,t+τ is defined as the gain or loss
on a delta-hedged option portfolio in excess of the risk-free rate earned by this portfolio:
t+τ t+τ
∂C i
Z Z
Πit,t+τ = i
Ct+τ − Cti − ∆iu dSui − r(C i − Si )du.
t t ∂Si
From the definition of delta-hedged gain and Equation (3), we obtain the expectation of the
delta-hedged gain for stock option i:
n t+τ Z t+τ X n
∂C i i
Z
j j ∂C
dW2i,j ]]
X
E[Πit,t+τ ] = E[ [ j j
β λ (Vf,t )] i
du + [ β j j
θ (V f,t ) i
t ∂V t ∂V
j=1 j=1
n t+τ
∂C i
Z
j
X
= β j E[ λj (Vf,t )] du] (4)
t ∂V i
j=1
1 , V 2 , ..., V n ) are priced, the

The result shows that, if variance risks of the factors (Vf,t f,t f,t
expected delta-hedged gain of a equity option is driven by the exposure to the variance risk,
∂C i
βj , the price of variance risk, λj , and the vega of the option ∂V i
.
Bakshi and Kapadia (2003) show that the expected delta-hedged gain of the index option
is closely related to the price of volatility risk. If we consider the following price process for
the market index S m with stochastic volatility,
dStm
= µm m m m
p
i t (St , Vt )dt + Vtm dW1t ,
St
dVtm = θm (Vtm )dt + η m (Vtm )dW2t
m
,
and a call option written on the market index is Ctm , according to Bakshi and Kapadia
7
(2003), the expected delta-hedged gain for the index option is:
t+τ
∂C m
Z
E[Πm
t,t+τ ] = E[ λm (Vtm ) du].
t ∂V m
∂C m ∂C i
It follows that, if ∂V m and ∂V i
are at the similar level, we can use the delta-hedged gain of
the index options to replicate the price of volatility risk of the index return. However, there
are no traded options on other stock return factors. In this paper, following the literature
on stock return factors, we consider some long-short characteristic-based option portfolios as
potential candidate of factors. The details of the characteristics and factors are provided in
Section 3.
3 Data and variables description
3.1 Data and sample coverage
We obtain option data on individual stocks from the OptionMetrics Ivy DB database. Sample
period is from January 1996 to December 2015. Implied volatility and Greeks are calculated
by OptionMetrics using the binomial tree from Cox et al. (1979). We obtain stock returns,
prices and credit ratings from the Center for Research on Security Prices (CRSP); balance
sheet data from Compustat and analyst coverage and forecast data from I/B/E/S.
We apply several filters to select the options in our sample. First, to avoid illiquid options,
we exclude options if the trading volume is zero, the bid quote is zero, the bid quote is smaller
than the ask quote, or the average of the bid and ask price is lower than 0.125 dollars. Second,
to remove the effect of early exercise premium in American options, we discard options whose
underlying stock pays a dividend during the remaining life of the option. Therefore, options
in our sample are very close to European style options. Third, we exclude all options that
violate no-arbitrage restrictions. Fourth, we only keep options with moneyness between 0.8
and 1.2. At the end of each month and for each stock with options, we select a call option
that is the closest to being at-the-money with the shortest maturity among those options
with more than one month to maturity. We drop options whose maturity is different from
the majority of options. Our final sample contains 327,016 option-month observations for
8
calls. The time to maturity ranges from 47 to 50 days.
3.2 Construction of the delta-hedged option returns
Since option is a derivative written on a stock, option returns are highly sensitive to stock
returns. In this paper, following the literature, we study the gain of delta-hedged options, such
that the portfolio gain is not sensitive to the movement of the underlying stock. Empirical
studies find that the average gain of the delta-hedged option portfolios is negative for both
indexes and individual stocks (Bakshi and Kapadia (2003), Carr and Wu (2009), and Cao
and Han (2013)). Bakshi and Kapadia (2003) show that the sign and the magnitude of
delta-hedged gain are related to the variance risk premium and jump risk premium. The
delta-hedged option position is constructed by holding a long position in an option, hedged
by a short position of delta shares on the underlying stock. The definition of delta-hedged
option gain follows Bakshi and Kapadia (2003) and is given by
Z t+τ Z t+τ
Πt,t+τ = Ot+τ − Ot − ∆u dSu − ru (Ou − ∆u Su )du,
t t
∂Cu
where Ct represents the price of an European option at time t, ∆u = ∂St is the option
delta at time u, and ru is the annualized risk-free rate at time u. We consider a portfolio
of an option that is hedged discretely N times over the period [t, t + τ ], where the hedge is
rebalanced at each date tn , n = 0, 1, ...N − 1. As shown by Bakshi and Kapadia (2003) in a
simulation setting, the use of the Black-Scholes hedge ratio has a negligible bias in calculating
delta-hedged gains. The discrete delta-hedged option gain up to maturity t + τ is defined as
N −1 N −1
X X an rtn
Πt,t+τ = Ot+τ − Ot − ∆tn [Stn+1 − Stn ] − (Otn − ∆tn Stn ), (5)
365
n=0 n=0
where Ot is the price of the option, ∆tn is the delta of the option at time tn , rtn is the
annualized risk free rate, and an is the number of calendar days between tn and tn + 1. This
definition is used to compute the delta-hedged gain for call and put options by using the
corresponding price and delta. To make the delta-hedged gains comparable across stocks we
use delta-hedged option returns defined as the delta-hedged option gain Πt,t+τ scaled by the
9
absolute value of the securities involved, i.e. ∆t St − Ot for call options. We start the position
at the beginning of each month and close the position at the end of each month. We work
with monthly returns through the empirical analysis.
3.3 Test portfolios and factor candidates in the equity option market
In the literature on cross-section of stock returns, long-short factors are commonly used to
describe stock returns. These factors are constructed with portfolios composed by ranked
stocks by certain characteristics, such as the size factor, the value factor and the momentum
factor. In the equity option market, following the similar logic, we consider the following
predictors which has been shown to have strong predictability in the literature. The predictors
are then used to sort portfolios and construct factors.
We consider six equity characteristics in Cao et al. (2017), which have been found to be
significant predictors of delta-hedged equity option return in the next month. The equity
characteristics are as follows, with notation of the long-short factors in the brackets.
(1) Size (LSsize ): The natural logarithm of the market value of the firm’s equity (e.g.Banz
(1981) and Fama and French (1992)).
(2) Stock return reversal (LSreversal ): The lagged one-month return. (Jegadeesh (1990))
(3) Stock return momentum (LSmom ): The cumulative return on the stock over the 11
months ending at the beginning of the previous month (Jegadeesh and Titman (1993))
(4)CH (LSch ): Cash-to-assets ratio, as in Palazzo (2012), defined as the value of corporate
cash holdings over the value of the firm’s total assets.
(5) Profit (LSprof it ): Profitability, calculated as earnings divided by book equity in which
earnings are defined as income before extraordinary items, as in Fama and French (2006).
(6) Disp (LSdisp ): Analyst earnings forecast dispersion, computed as the standard de-
viation of annual earnings-per-share forecasts scaled by the absolute value of the average
outstanding forecast (Diether et al. (2002)). Cao et al. (2017) find that delta-hedged op-
tion gains increase with size, momentum, reversal, and profitability and decrease with cash
holding and analyst forecast dispersion.
We also consider other option predictors in the literature related to volatility.
(7) Ivol (LSivol ): Stock return idiosyncratic volatility, as in Ang et al. (2006). Cao and
10
Han (2013) find that delta-hedged equity option return decreases monotonically with an
increase in the idiosyncratic volatility of the underlying stock.
(8) Voldev (LSvoldev ): The log difference between the realized volatility and the Black-
Scholes implied volatility for at-the-money options. Goyal and Saretto (2009) find that the
higher the difference, the higher the future straddle return of the equity option.
(9) Vts slope (LSvts ): the difference between long-term and short-term implied volatility.
Vasquez (2017) finds that straddle portfolios with high slopes of the volatility term structure
outperform straddle portfolios with low slopes by a significant amount.
(10) BidAsk (LSbidask ): The ratio of the difference between the bid and ask quotes of
option to the midpoint of the bid and ask quotes at the end of previous month. Christoffersen
et al. (2017b) find that option illiquidity has strong risk premia in the equity option market.
We use bid-ask spread as a proxy of option illiquidity due to data availability.
(11) Rating (LSrating ): Credit ratings are provided by Standard & Poor’s and are mapped
to 22 numerical values, where 1 corresponds to the highest rating (AAA) and 22 corresponds
to the lowest rating (D). Vasquez and Xiao (2018) find that credit rating is a strong predictor
of future option returns. Options with lower credit rating have more negative delta-hedged
returns in the future.
At the end of each month, we sort all stock options into 10 deciles based on the first 10
characteristics described above We sort stock options into 5 quintiles by credit rating because
there are less than 10 different ratings in some months, which leads to missing data in the
portfolio returns. We then start the position at the beginning of the next month and hold
the position until the end of that month. Their corresponding delta-hedged option returns
are calculated according to Section 3.2. We consider the 105 portfolios sorted by 11 different
characteristics as test assets, such that they have enough heterogeneity and the underlying
risk premium associated factors can be detected. The 11 candidate factors are the 10-1
(5-1 for credit rating) return spreads based on the 11 characteristics. We also consider two
candidate factors related to common volatility risk.
(12) Delta-hedged return of the S&P 500 index option (DHidx ): DHidx is a proxy for the
market volatility risk in Coval and Shumway (2001) and Carr and Wu (2009).
(13) Delta-hedged return of the stock options (DHstk ): DHstk is the value-weighted delta-
11
hedged returns on the individual stocks that are components of the S&P 500 index. It is
used as a measure of common individual stock variance risk.
Table 1 below shows the summary statistics for the average returns of the decile (quintile)
delta-hedged option portfolio sorted by the 11 predictors. The table shows that the long-
short returns constructed by buying the top decile (quintile) and selling the bottom decile
(quintile) are all significantly different from zero. The average return spreads range from
−1.48% to 2.31% with t-statistics ranging from −10.49 to 16.05. The delta-hedged equity
option returns increase with size, reversal, momentum, profitability, volatility deviation and
the slope of volatility term structure, while they decrease with cash holding, analyst disper-
sion, idiosyncratic volatility, bid ask spread and credit rating.
[ Table 1 around here]
Since the delta-hedged return of the S&P 500 index options is on average negative, which
represents the negative price of variance risk, we construct the long-short factors based on
the return spreads such that they are all on average negative. The long portfolio is the one
with the highest payoff in bad states of nature as a hedge and the short portfolio is the one
with the lowest payoff in bad states of nature. The summary statistics of the long-short
factors including mean, standard deviation, skewness, kurtosis, 10th, 25th, 50th, 75th and
90th percentiles are reported in Table 2.
To conclude this section, Table 3 shows the correlation coefficients between the option re-
turns predictors. The table shows that the correlation coefficients between the strategies
are mostly below 0.5 with the exception that the correlation between LSdisp and LSivol is
0.57, the correlation between LSdisp and LSprof it is 0.53 and the correlation between LSvoldev
and LScredit is 0.53. The low correlation among the long-short factors suggests that these
variables might capture different information about the cross-section of delta-hedged option
returns. How many different factors do these 13 factors capture? How many of them are
12
relevant for explaining cross-section of option returns? Do they capture similar information
to stock returns factors? We answer these questions in the next section.
4 Empirical Procedure and Results
Following the suggestions in Pukthuanthong et al. (2018), we first perform a latent variable
analysis of the covariance matrix of delta-hedged option returns to uncover its factor struc-
ture. For that purpose, we test the number of common factors in delta-hedged option returns.
Then we estimate those factors using principle component analysis (PCA) as suggested in
Connor and Korajczyk (1986) and study how well they explain the cross-section of delta-
hedged option returns. Factors estimated by PCA are difficult to interpret economically.
Therefore, we test if some factor candidates with economic interpretation are related to the
covariance matrix of delta-hedged option returns using rank-estimation method as suggested
by Ahn et al. (2018). Third, we perform several tests to assess how much of the common
variation captured by the PCA factors is captured by the selected factor candidates. For
this purpose, we use standard regression analysis as well as a canonical correlation analysis
as suggested in Pukthuanthong et al. (2018). Finally, we test whether the factors selected
from the previous steps command risk premium and propose a four factor model to explain
the cross-section of delta-hedged option returns.
4.1 Number of latent factors that drive the comovement of delta-hedged

option returns
As stated in Pukthuanthong et al. (2018), a necessary condition for any empirical factor can-
didate to be a factor is that it should be related to the principal components of the covariance
matrix. In this section, we aim to identify factors that drive option return systematically.
In Section 2, we show that, under a stochastic volatility model, the delta-hedged return of
an equity option portfolio is consistent with a linear factor model if the stock variance follows
a multi-factor structure. In this section we use several techniques developed to estimate
13
the number of factors in approximate linear factor models as defined in Chamberlain and
Rothschild (1983)3 . More precisely, let xit be the response variable for the ith cross-section
unit at time t (i = 1, 2, ..., N , and t = 1, 2, ..., T ). Explicitly, xit can be the return on a
delta-hedged option portfolio i at time t. The response variables xit depend on K empirical
factors ft = (f1t , ..., fKt )0 . That is,
xit = α + Bft + t ,
where xit = (x1t , ..., xKt )0 ; α is the N-vector of individual intercepts; B is the N ×K matrix of
factor loadings (beta matrix); and t is the N-vector of idiosyncratic components of individual
returns at time t. The entries in t can be cross-sectionally correlated. We denote the ith
row of α and B by αi and βt = (β1i , ..., βKi )0 , respectively.
To estimate the number of factors K in delta-hedged option returns, we use as response
variables data on all the delta-hedged portfolio in Section 3.3 (N = 105 portfolios) during the
entire sample period from January 1996 to December 2015 (T = 240 months). As a prelimi-
nary step, we plot in Figure 1 the largest fifteen eigenvalues from the sample second-moment
matrix of the “doubly demeaned” delta-hedged portfolio returns4 . The figure, known as a
“scree plot”, indicates that there are about six common factors, and one of them has much
stronger explanatory power than the other five factors.
[Figure 1 around here]
Pukthuanthong et al. (2018) suggest that the number of factors should be designated
in advance; for example, the number of factors could be chosen such that the cumulative
variance explained by the principal components is at least ninety percent. However, the
“ninety-percent” is an arbitrary cut-off point and a scree plot is not a formal statistical tool
to estimate the number of factors either. Therefore, we also estimate the number of factors
3
The advantage of working with approximate factor models as opposed to the classic exact factor models
(e.g. Ross (1976)) is that the former allows for a certain degree of correlation across idiosyncratic terms while
the later impose an orthogonality condition on the covariance matrix of the idiosyncratic component
4
Let xit be the VW-excess return on PN demeaned” excess return equals xit − x̄i −
Passet i. Then, the “doubly
x̄t + x̄, where x̄i = Tt=1 xit /T , x̄t = N
P
i=1 xit /N , and x̄ = i=1 xi /N . See Ahn and Horenstein (2013) for
the justification of using these doubly demeaned excess returns for estimation of the number of factors.
14
using several consistent methods. More precisely, in Table 4 we present the results obtained
from estimating the number of factors using the Eigenvalue Ratio (ER) and Growth Ratio
(GR) estimators of Ahn and Horenstein (2013), the Edge Distribution (ED) estimator of
Onatski (2010), the BIC3 and IC1 estimators of Bai and Ng (2002), and the Modified Infor-
mation Criterion estimator (ABC) of Alessi et al. (2010). A brief explanation about these
methods is provided in Appendix A.1. We apply the estimators to doubly-demeaned data
(Column 1) and to the raw data (Column 2) in Table 4. ER and GR have been applied only
to doubly-demeaned data.
First, the first column confirms our preliminary results from the scree test that there are
between 1 and 6 common factors in doubly-demeaned delta-hedged option returns. While
ER, GR, and ED capture 1 common factor, BIC3 captures 3 common factors, IC1 captures 6
common factors as well as ABC5 . The factor structure is consistent with having 1 strong factor
and possibly up to 5 weak factors. Second, all estimators capture an additional factor when
we use raw data. This is consistent with the existence of an additional factor having constant
factor loadings. More precisely, this factor with constant factor loadings, corresponding to
the first principal component (PC) extracted from raw data, is the equally-weighted portfolio
(EWP) of the response variables (correlation between the first PC extracted from raw data
and EWP is 0.99). This result has led to many researchers to argue that the most important
factor is the market portfolio (e.g. Brown (1989), Ferson and Korajczyk (1995)), and that
this factor is captured by the first PC from raw returns (or excess-returns over the risk free
rate). While this factor has good power to explain the average time series variation in returns,
given that it produces constant betas, it can barely explain the cross-section of returns. As
Ahn and Horenstein (2018) show, to better capture the factors with variation in betas and
the relevant factors to explain the cross-section of option returns, we can extract common
factors from excess returns over the equally-weighted portfolio and use as response variables
5
For all these estimators we set the parameter kmax, the maximum number of factors to test for, equal to
15.
15
for testing assets pricing models the excess returns over EWP or any other well-diversified
portfolios. In the Appendix A.3, we show that EWP contains unitary loadings. Therefore,
our base latent model consists of the following equation:
6
X
rit − rEW P,t = αi + βik fkt + it ,
k=1
where ri is the return on the ith delta-hedged portfolio, rEW P is the return on the equally-
weighted portfolio (EWP) constructed using the 105 characteristic-based delta hedged-portfolios,
fk corresponds to the kth PC factor6 (where k = 1, . . . , 6), βik is the sensitivity of portfolio
i to factor k, αi is the pricing error of the model with respect to portfolio i, and i is the
idiosyncratic component of portfolio i. The PC factors have been extracted using the same
rotation as Bai and Ng (2002)).
Panel (a) of Table 5 shows the performance of the proposed model with 6 latent variables
as factors. For comparison purpose, we present on Panel (b) the performance of a model
using the 13 candidate factors. The performance metrics we analyze are the percentage of
pricing errors statistically different from 5% generated by each model, the average adjusted
R2 across option portfolios generated by each model, the correlation between the portfolios
expected (excess) returns and the betas of each factor, the correlation between the portfo-
lios expected (excess) returns and the predicted returns from the model (betas times factor
premiums).
The performance metrics of the model with latent variables improve over those of a model
using the 13 candidate factors used to predict option returns. It produces less pricing errors
statistically different than zero, slightly higher adjusted R2 , and lower annualized average
absolute pricing errors. This result is not surprising in principle, since the latent factors are
the most correlated factors with the second-moment matrix of our sample of delta-hedged
option returns. The correlation between expected returns and predicted returns is quite high
6
The kth PC factor is the one corresponding to the kth largest eigenvalue.
16
for both models: 0.97 for the model with 6 latent variables and 0.96 for the model with 13
candidate factors. What is striking from these results is the correlation between expected
returns and the beta of the first latent factor, which amounts to 0.96. Figure 2 plots the
scatter diagram between average returns and the betas of the first latent factor, showing a
clear linear relationship between the two variables.
[Figure 2 around here]
Consistent with the estimation for the number of factors in the previous section, the first
factor seems to suffice for explaining the co-movement in delta-hedged returns. However, the
other 5 weak factors might contain some relevant information. We now analyze how many,
if any, of those weak factors are necessary to improve the model performance.
To answer these question, Panel (c) Table 5 shows the performance metrics of 6 factor
models, in which we increase the number of factors used as independent variables sequentially
from 1 to 6. This panel contains 3 metrics: (i) the number of pricing errors statistically
significant at the 5% or less, (ii) the average adjusted R2 generated by the model, and (iii)
the annualized average absolute pricing error (AAAPE). We do not show the correlation
between expected returns and realized returns because we already know that in this case the
first factor generates a correlation 0.96 of the total of 0.97 generated by all the six latent
factors.
A model with one latent factor generates 54% of alphas different from zero. After adding
a second factor to the model, the percentage decreases to 22% and the minimum is obtained
with 3 factors at 18% of pricing errors different than zero. According to this metric, additional
factors beyond the third do not add relevant pricing information. The average adjusted R2
metric shows that the first factor explains an average of 14% of the returns variation, the
second factor explains 6%, and after that each additional factor only explains 3%. Finally,
the third metric (AAAPE) decreases with the number of factors. However, the change is
quite small after the third factor. Overall, this table suggests that at most three of the latent
factors suffice to have the best fit to price our set of delta-hedged option returns.
It is important to note that we work with “doubly-demeaned” delta-hedged returns. This
17
implies that the factors we analyze are the ones that do not have constant loadings and are
useful for explaining the cross-section of delta-hedged returns. As previously discussed, and
further developed in Appendix A.3, EWP, which corresponds to the first PC estimated from
raw data, is an additional factor that has explanatory power for the time series of returns
but not over their cross-section since it has unitary loadings. We use this factor later on to
add the market portfolio to the proposed benchmark pricing model.
Finally, the problem with statistical and latent factors is that it is hard to understand
them with economic interpretation. As such, it is important to study which factors in the 13
candidate factors are important to explain the latent factors, especially the first one. At the
same time, given that there are at most 6 common factors in total, and at most 3 common
factors relevant for pricing, many of the 13 variables proposed as predictors of option returns
might contain redundant information. Hence, it is important to separate the most relevant
variables for pricing, which is the focus of the next section.
4.2 Relevant candidate factors of the cross-section of delta-hedged option

returns
In the previous section we find that there are at most 6 factors in option returns capable
to explain their cross-sectional variation. However, we do not know how many of the 6
factors are captured by the 13 variables used to predict option returns. This can be solved
by estimating the rank of the beta matrix produced by the 13 variables when regressed onto
the 105 delta-hedged option portfolios. As Ahn et al. (2018) point out, “the rank of the
beta matrix corresponding to a set of factors equals the number of factors whose prices are
identifiable.” In other words, the rank of the beta matrix tells us the number of different
sources of stock returns’ comovement captured by a set of factors. When we apply the
RBIC estimator developed in Ahn et al. (2018), we find that the rank of the beta matrix
generated by the 13 variables proposed as predictors of option returns equals 6. Therefore,
the 13 variables are capturing all the relevant factors containing information about the co-
movement of delta-hedged option returns. This result is consistent with our analysis in
Section 4.1.
Table 6 below shows the correlation coefficients between the 13 candidate factors and
18
the six latent factors. Since the EWP can be considered a factor, although it has unitary
loadings, we add it to the table.
Many characteristic-based factors are relatively highly correlated with the latent factors,
suggesting that some candidate factors might be capturing similar information. As in the
stock returns case, it seems that the variable used to capture the market factor has unitary
loadings (e.g. Ahn et al. (2018)). This means that the delta-hedged index option returns
might be good for capturing the average time-series variation in delta-hedged option returns
but not the cross-sectional variation. We further study this result in Appendix A.3.
Since many variables seem to be capturing similar information, we would like to know
the minimum set of variables necessary to capture the 6 common latent factors. We use rank
estimation methods to answer this question. More precisely, we generate all combinations of
different sets containing 6 factors from the 13 candidate factors and check if any set generates
a full-rank beta matrix. From the set of 13 candidate factors, we can create 462 different sets
of 6 factors. It turns out that only 1 of these 462 sets generates a full rank beta matrix when
using the returns of the delta-hedged portfolios over the EWP as response variables. The
unique set generating a full rank beta matrix contains the following factors: LSsize , LSch ,
LSdisp , LSivol , LSvoldev , and LSrating . This means that these six factors suffice to capture
most (if not all) the relevant information about the cross-section of delta-hedged option re-
turns. Figure 3 shows the relationship between the average returns of the option portfolios
and the predicted returns by a model containing the six relevant factors. The two variables
have a correlation coefficient of 0.95. As shown in Panel (B) of Table 5, this correlation
increases to 0.96 if we use all of the 13 candidate factors. Overall, these six variables contain
all relevant information about the cross-section of delta-hedged option returns.
[ Figure 3 around here]
Previous results show that the six latent factors are captured by the six factor candi-
19
dates. However, we do not know how well the candidate factors capture the latent ones. To
study this, following Pukthuanthong et al. (2018) we use canonical correlation analysis. A
brief explanation on this procedure is on Appendix A.2. Table 7 below shows the canonical
correlations for different pairs of variables.
The first column shows the canonical correlation corresponding using the 6 latent factors
and the 13 factor candidates. It shows that the first five canonical correlations are quite
high. This result indicates that five linear combinations of the 13 factor candidates are highly
correlated with five linear combinations of the 6 latent factors. It appears that the candidate
factors almost perfectly capture five dimensions spanned by the 6 latent factors. However,
we do not know which dimension is missed by just looking at the canonical correlation
coefficients. If it were the one spanned by the first latent factor, the implication would be
that the candidate factors are missing the most important dimension. Therefore, in the
second column we compare the 3 most relevant latent factors with the 13 factor candidates.
We find that the 13 candidate factors capture almost perfectly the three most important
latent factors for pricing, meaning that the 13 factor candidates contain the most important
information for pricing delta-hedged option returns. Columns three and four compare the
latent factors with the subset of the 6 factor candidates that are most relevant with our rank
estimation exercise. We find that the 6 most relevant factor candidates also capture almost
perfectly 3 of the latent factors, and that those latent factors are the 3 most relevant ones.
This further confirms our rank estimation exercise.
We next select the most relevant factors among the six relevant factor candidates, which
are able to capture the most relevant three latent factors. For this purpose, we rank by
importance the six variables which are sufficient to explain the cross-section of delta-hedged
option returns. We pick the most relevant variable as the one that produces by itself the
largest correlation coefficient between beta and average return. The second variable in our
rank is the one that increases the correlation between average return and predicted returns
the most in a model already containing the first variable as regressor. We find that we only
20
need 3 out of the 6 variables to have a correlation between average returns and predicted
returns of 0.95. LSsize is the variable that produces the highest correlation, 0.83. LSivol
increases that correlation further to 0.89 and LSvoldev increases it to 0.95. This does not
mean that the other variables are uninformative, however, the added explanatory power
to the cross-section of option returns by LSch , LSdisp1 , and LSrating is negligible once we
control for the selected three factors. In fact, the three variables chosen generate quite high
canonical correlation with respect to the first two latent factors (0.93 and 0.89), arguably the
most important ones. The third latent factor is explained with lesser accuracy (canonical
correlation is 0.58).
Overall, all our results are consistent with our previous estimation of the number of factors
showing that there is one strong factors and 5 weak factors, where the weakest 3 factors are
much weaker than the other two (see Figure 1). Given these results, we now construct
an asset pricing model that can capture well the time-series and cross-sectional variation
in delta-hedged returns. In the previous analysis, we find that the three most important
factors to capture the cross-sectional comovement in delta-hedged option returns are LSsize ,
LSivol , and LSvoldev . We also show and further investigate in Appendix A.3 that a variable
that captures well the factor with unitary loadings is the delta-hedged return of S&P 500
index options DHidx , which is commonly considered the market factor in the option returns
literature such as in Goyal and Saretto (2009) and Cao and Han (2013). Thus, we propose
the following four-factor model for pricing the delta-hedged equity option returns7 :
rit = αi + βidx,i DHidx,t + βsize,i LSsize,t + βivol,i LSivol,t + βvoldev,i LSvoldev,t
Table 8 below shows the performance metrics for the four-factor model in Panel (a) and
the four-factor model augmented with LSch , LSdisp1 , and LSrating in Panel (b).
We observe in Table 8 that performance of the four-factor model does not increase when
7
Note that the delta-hedged returns are in excess of the risk free rate as in Equation 5.
21
augmented by LSch , LSdisp , and LSrating . We conclude that the four-factor model captures
all the necessary information to price the delta-hedged option returns. However, the factor
structure might change through time and it is important to evaluate the performance of the
proposed factor model in different sub-samples. For this purpose, we run monthly rolling
regressions using 60-month of data in each iteration. Since our data comprise the period
January 1996 to December 2015, we have 181 regressions in total. For each regression we
calculate the correlation between the average returns of the delta-hedged portfolios and the
predicted returns by our four factor model. The average correlation is 0.86, with a standard
deviation of 0.04, a maximum value of 0.93 and a minimum of 0.72. Figure 4 below shows
the value of this parameter through time. This figure shows that the predictive power of the
four factor model we propose is quite stable throughout the time span analyzed.
[ Figure 4 around here]
We further check if the factors proposed in our model are priced in the cross-section
by running Fama-MacBeth regressions for the whole sample and two 10-year sub-samples.
Regression results are reported in Table 9. To avoid multicollinearity problems arising from
having a factor with near constant loadings (see Ahn et al. (2013)), we only test the three
factors that present variability in their loadings: LSsize , LSivol , and LSvoldev . Table 9 shows
that the proposed factors are priced since all the coefficients are statistically significant in
the full sample and two subsamples. As such, the factors in our model passes all the criteria
in the protocol established by Pukthuanthong et al. (2018) to find relevant factors.
As a further robustness check, we evaluate the performance of the four-factor and the
seven-factor models for industry portfolios. We construct the industry portfolios by cate-
gorizing each firm into the industry groups with two-digit code provided by OptionMetrics.
After removing portfolios with missing data, we have 26 industry portfolios in total. We
report performance matrix for the industry portfolios the same as for the 105 characteristic-
22
sorted portfolios in Table 10.
Note that only 2 of the 26 industry portfolios have non-zero alphas in the four-factor
model and the correlation between expected returns and realized returns is 0.83. Again, the
model with four factors works as well as the model with seven factors. The main difference
between using industry portfolios and characteristic-based portfolios is that LSsize does not
seem to be relevant for industry portfolios while LSch does. However, LSch does not add
additional explanation power once we use the four factor model with LSsize , LSivol , and
LSvoldev .
4.3 Are stock factors important in explaining the cross section of option
returns?
We now analyze how much information about the cross-section of delta-hedged option returns
are contained in commonly used factors to price the cross-section of stock returns. We include
the following stock market factors as potential factors to explain the variation in equity option
returns: the five factors in Fama and French (2016) (MKT, SMB, HML, RMW, CMA),
momentum factor in Carhart (1997) (MOM), stock market liquidity risk factors in Pástor
and Stambaugh (2003) (P Sinnov , P Slevel , P Svwf ), betting-against-beta factor in Frazzini and
Pedersen (2014) (BAB), two mispricing factors in Stambaugh and Yuan (2016) (MGMT and
PERF).
Table 11 reports the correlation coefficients between the 12 stock factors and the esti-
mated 6 latent option factors plus EWP. The table shows that stock factors are not highly
correlated with the latent option factors. The first PC factor, which is the most important
one, reports correlations below (absolute) 19% with the stock factors.
To further study the relationship between stock return factors and option return factors,
23
we regress each latent option factor onto all the stock factors. Table 12 reports the regression
results. The table shows that the market factor from stock returns is highly significant for
explaining the variation of EWP. This suggests that the factor with unitary loading in stocks
is related to that of option returns, which possibly comes from the leverage effect that stock
return and stock volatility are correlated.
Stock factors explain around 32% of the variation of EWP, but very little of that of
the 6 latent factors necessary to explain the cross-section of delta-hedged option returns.
The only stock factor that is sometimes significant at the 1% or less when regressed onto
option PC factors is the BAB factor (for the case of the 2nd and 3rd factor). As expected,
the information contained in stock factors seems negligible when pricing the cross-section of
delta-hedged option returns.
5 Conclusion
Despite the very large and still growing literature on common factors in stock returns, there
is limited understanding about the factor structure in delta-hedged equity option returns.
In this paper, we motivate our empirical analysis by showing that in a stochastic volatility
model, the expected delta-hedged option returns are driven by factors related to volatility
risks and not by stock returns’ factors.
In the empirical analysis, we construct 105 option portfolios sorted by 11 characteristics
using monthly portfolios of delta-hedged options from January 1996 to December 2015. The
characteristic-based factors are constructed based on long-short strategies of decile portfolios.
We also consider delta-hedged return of the S&P 500 and average delta-hedged return of the
equity options as two additional candidate factors. Using latent factor techniques on the 105
portfolios, we find strong evidence for the existence of at most six factors in equity options
returns, where three suffice to explain its cross-section variation. Using the 11 characteristic-
based option factors as candidate factors, we find three of them suffice to capture the relevant
24
latent factors and to explain the time series and cross-section of equity option returns. The
three factors are long-short factors constructed based on size, idiosyncratic volatility and
volatility deviation.
Using these three factors in the Fama-MacBeth regression, we find that the exposures to
the three factors are statistically significant in explaining average delta-hedged return of the
105 portfolios in full sample and two subsamples with adjusted R2 higher than 80%.
Finally, we propose a four-factor pricing model for delta-hedged option returns that
adds the delta-hedged return of the S&P 500 index options to the three aforementioned
characteristic-based factors. This factor does not add explanatory power to the cross-section
of option returns but it improves the explanatory power of the model over the time-series
dimension. As expected from the theoretical model, stock factors seems negligible to price
the cross-section of delta-hedged option returns.
25
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A Appendix
A.1 Summary of the number of factors tests used in the paper
One of the most popular methods used to determine the number of common factors is the
Scree Test developed by Cattell (1966). The test is a visual way to find out the number of
factors from the eigenvalues of XX 0 , where X is the T × N matrix of response variables.
Cattell describes his method as follows. “If I plotted the principal components in their sizes,
as a diminishing series, and then joined up the points all through the number of variables
concerned, a relatively sharp break appeared where the true number of factors ended and the
‘detritus’, presumably due to error factors, appeared. From the analogy of the steep descent
of a mountain till one comes to the scree of rubble at the foot of it, I decided to call this the
Scree Test.”
The Scree Test is an eye-ball test (not a formal estimator with known asymptotic proper-
ties). Bai and Ng (2002) (BN, 2002) propose the first consistent estimator for the number of
factors. Their method can be interpreted as finding a consistent threshold to divide the eigen-
values corresponding to common factors from those corresponding to noise. To be specific,
denote ψk (A) as the kth largest eigenvalue of a positive semi-definite matrix A. Define
1 1
µ̃N T,k = ψk ( XX 0 ) = ψk ( XX 0 ),
NT NT
where k = 1, ..., m, and m = min(T, N ). Then, T µ̃N T,k (N µ̃N T,k ) is the kth largest eigenvalue
of the sample covariance matrix of xi· (x·i ) if the means of xit are all zeros. Ahn and
Horenstein (2013) show that if r is the true number of factors, then µ̃N T,k = Op (1) for k < r
while µ̃N T,k = Op (m−1 ) for k > r.
Next we describe all the estimators used in the paper. Let F̃ k be the T × k matrix of the
eigenvectors corresponding to the largest k eigenvalues of XX 0 /(N T ), which is normalized
so that (F̃ k )0 (F̃ k )/T = Ik . Let V (k) be the mean of squared residuals from the regressions
of xi· on F̃ k and with σ̂ 2 = V (kmax), where kmax is the maximum number of factors to be
tested. BN proposed minimizing two types of criteria to consistently estimate the number of
factors:
30
P C(k) = V (k) + σ̂ 2 kg(N, T ),
IC(k) = ln(V (k)) + kg(N, T ),
where g(N, T ) is a penalty function such that g(N, T ) → 0 and g(N, T )m → ∞ as N, T → ∞.

A BN estimator is obtained by minimizing these functions. BIC3 is a PC criterion where
(N +T −k) ln(N T ) N +T
g(N, T ) = NT , while IC1 is an IC criterion where g(N, T ) = NT ln( NN+T
T
). Note
that the threshold values used for the estimators are not unique. Specifically, any finite
multiple of a valid threshold value is also asymptotically valid. To show the relationship
between BN and the Scree Test, note that
N T
1 X 0 X
V (k) = (xi· xi· − x0i· P (F̃ k )xi· ) = µ̃N T,j (6)
NT
i=1 j=k+1
PT
See Ahn and Horenstein (2013) for more details. Then, σ̂ 2 = V (kmax) = j=kmax+1 µ̃N T,j
and the two BN criteria can be written as
T
X T
X
P C(k) = µ̃N T,j + kg(N, T ) µ̃N T,j ,
j=k+1 j=kmax+1
T
X
IC(k) = ln( µ̃N T,j ) + kg(N, T ).
j=k+1
Let k̃P C = mink≤kmax P C(k) and k̃IC = mink≤kmax IC(k), using Equation 6 and the mono-
tonicity of the eigenvalues, we can show that k̃P C = mink≤kmax {k|µ̃N T,j ≥ σ̂ 2 g(N, T )}.
Thus, the PC estimation can be viewed as a Scree Test using σ̂ 2 g(N, T ) as a threshold value.
Overall, the BN estimator can be roughly understood as a formalization of the Scree
Test. The ER (eigenvalue ratio) estimator of Ahn and Horenstein (2013) can be viewed as a
modified version of the PC estimators that does not require the use of threshold values. The
ER estimator is defined by maximizing the following criterion function:
µ̃N T,k V (k − 1) − V (k)

ER(k) = =
µ̃N T,k+1 V (k) − V (k + 1)
31
Thus, the ER estimator is the value of k that maximizes the ratio of the changes in V (k) at
k-1 and k. A similar interpretation can be given to the GR (growth ratio) developed also in
Ahn and Horenstein (2013), which is defined by maximizing the following criterion function:
ln(1 + µ̃N T,k ) ln(V (k − 1)/V (k))

GR(k) = =
ln(1 + µ̃N T,k+1 ) ln(V (k)/V (k + 1))
where k < kmax. An advantage of Ahn and Horenstein (2013) estimators is that they do
not depend on a pre-specified threshold function. This is important because the threshold
function depends mostly on the values of N and T and not too much on the intrinsic char-
acteristics of the data. In addition, there are infinitely many consistent threshold functions
that could be used in Bai and Ng (2002).
Alessi et al. (2010) (ABC, 2010) propose to estimate the number of factors using different
subsamples and different multiples of the BN penalty functions. The final estimators is the
one that is invariant to the subsamples used and the change in the multiplicative constant
of the penalty function in a certain range. Overall, the ABC estimator can be considered as
a refinement of BN. The last estimator we use is the Edge Distribution (ED) estimator of
Onatski (2010) that estimates the number of factors using differenced eigenvalues. A novelty
of his approach is that the threshold value is estimated, not pre-specified like in BN or ABC.
To sum up, we use different estimators but all of them can be linked to the behavior of
the eigenvalues from the second moment matrix of the data. The rationale to all of them is
to divide information from noise, with the known property that eigenvalues corresponding to
eigenvectors having common information do not vanish as the dimension of the panel increases
(N,T) while those corresponding to eigenvectors related to the idiosyncratic component do
vanish. BN separates information from noise using a pre-specified penalty function. ABC
refines BN but still uses a pre-specified penalty function. ED estimates the penalty function
from the data. ER and GR do not use penalty functions.
A.2 Canonical Correlation
In this subsection, we explain the details of the canonical correlation. Suppose there are K
latent factors selected from Section 4.1: F = (f1 , f2 , ..., fL )0 and L candidate factors as return
32
spreads constructed by sorting portfolios based on characteristics: X = (x1 , x2 , ..., xL ). Their
mean factors are µF and µX and their covariance matrices are ΣF and ΣX . The covariance
matrix between F and X is ΣF X = E[(X − µX )(f − µF )0 ].
Define linear combinations of X and F as U and V: U = a0 F and V = b0 X, where a and b
are constant vectors. Note that the variance of the two vectors and the covariance between
the two vectors are:
V ar(U ) = a0 ΣF a, V ar(V ) = b0 ΣX b, and Cov(U, V ) = a0 ΣF X b.
The first pair of canonical variate (U1 , V1 ) is defined via the pair of linear combination vectors
{a1 , b1 } that maximize the following correlation, subject to the condition that U1 and V1 have
unit variance:
Cov(U, V ) a0 ΣF X b
Corr(U, V ) = p =√ 0 √ .
a ΣF a b0 ΣX b
p
V ar(U ) V ar(V )
The remaining canonical variates (Up , Vp ) maximize the above correlation subject to having
unit variance and being uncorrelated with ((Uq , Vq )) for all q < p. The p-th pair of canonical
variates is given by,
−1/2 −1/2
Up = u0p ΣF F, and Vp = vp0 ΣX X
−1/2 −1/2
where up is the p-th eigenvector of ΣF ΣF X Σ−1
X ΣXF ΣF and vp is the p-th eigenvector
−1/2 −1/2
of ΣX ΣXF Σ−1
F ΣF X ΣX . The p-th canonical correlation is given by Corr(Up , Vp ) = ρp ,
−1/2 −1/2 −1/2 −1/2
where 0 ρ2p is the p-th eigenvalue of ΣF ΣF X Σ−1
X ΣXF ΣF and ΣX ΣXF Σ−1
F ΣF X ΣX .
A.3 Equal weighted portfolio (EWP) contains unitary loadings
In the main body of the paper, we argue that EWP is a factor with constant loadings. Ahn
et al. (2013) show that these factors lead to specious conclusions of statistical significance
when using them as explanatory variables in typical asset pricing tests like the Fama-MacBeth
two pass regression method. In addition, Ahn et al. (2018) show that subtracting a factor
with constant loadings from the response variables in asset pricing tests help better identify
33
the relevant factors. Therefore, in the main body of the paper we use as response variable
in our test excess delta-hedged portfolio returns over the EWP. In this Appendix we show
that not only EWP is a factor with unitary loadings but also the Straddleindex factor, which
is usually used as a market factor in the option returns literature (Goyal and Saretto (2009)
and Cao and Han (2013) for example).
To assess the variability of the different factor loadings we use the IB estimator proposed
in Ahn and Horenstein (2013). More precisely, if we have N response variables and K factors,
then IBk = N β̄k2 / N 2
P
i=1 β̂ik where β̂ik is the estimated beta for asset i corresponding to factor
k and β̄k = N
P
i=1 β̂ik /N , for i = 1, . . . , N and k = 1, . . . , K. The IBk estimator is between
0 and 1 and is equivalent to the uncentered R-square from regressing a vector of ones onto
the vector of factor k’s loadings. The closer to 1 the value of IB the closer to a constant the
vector of factor loadings. Table AX below shows the values of the IB estimator for two factor
models. Panel (a) shows the results from regressing the 105 delta-hedged option returns on a
model consisting of the EWP and the three option factors proposed in the main body of the
paper (Size, ivol, and voldev) and Panel (b) show the results from a similar factor model in
which EWP has been replaced by the Straddleindex factor. The table also reports the average
value of the factor loadings as well as the R2 of the models and the correlation between the
predicted returns and the average returns of the models.
The Table shows that the IB estimator is almost equal to 1 for both EWP and the
StraddleIndex factor. As such, these factors might have good explanatory power for the
time series regressions but fail to explain the cross-section of delta-hedged option returns, as
shown by the low correlation between their estimated betas and the average returns of the
delta-hedged portfolios.
34
Figure 1: Eigenvalues from the Second-moment Matrix of the “Doubly-demeaned” Delta-
hedged Portfolio Returns
This figure shows the largest fifteen eigenvalues from the sample second-moment matrix of the “doubly de-
meaned” returns of the 105 delta-hedged portfolios. The sample period is from January 1996 to December
2015.
35
Figure 2: Beta-return Relationship of the First Latent Factor
This figure shows the scatter diagram between the average returns of the 105 delta-hedged option portfolios
and their corresponding betas of the first latent factor. The sample period is from January 1996 to December
2015.
36
Figure 3: Average Returns and Predicted Returns by the Model with Six Latent Factors
This figure shows the relationship between the average returns of the 105 option portfolios and the predicted
returns by the model containing the six latent factors.
37
Figure 4: Correlation of Average Return and Predicted Return by the Three-factor Model
Over Time
We run monthly rolling regressions using 60-month of data in each iteration. Since our data comprise the
period January 1996 – December 2015, we have 181 regressions. For each regression we calculate the correlation
between the average returns of the delta-hedged portfolios and the predicted returns by our three-factor model.
This figure shows the correlation over time.
38
Table 1: Delta-hedged option return sorted by 11 characteristics
1 2 3 4 5 6 7 8 9 10 10 − 1
Size -2.10 -1.26 -0.78 -0.63 -0.55 -0.44 -0.42 -0.31 -0.27 -0.23 1.87***
(-14.50) (-9.22) (-6.06) (-5.66) (-4.87) (-4.17) (-3.97) (-2.96) (-2.44) (-2.26) (16.05)
Reversal -1.31 -0.82 -0.65 -0.67 -0.58 -0.58 -0.53 -0.54 -0.59 -0.72 0.60***
(-7.84) (-6.28) (-5.89) (-6.41) (-5.14) (-5.71) (-5.71) (-5.45) (-5.77) (-5.13) (5.45)
Mom -1.31 -0.82 -0.73 -0.62 -0.56 -0.54 -0.50 -0.54 -0.54 -0.80 0.51***
(-9.17) (-6.71) (-6.50) (-6.02) (-5.36) (-5.30) (-4.85) (-5.13) (-4.50) (-4.90) (3.53)
Ch -0.49 -0.45 -0.46 -0.48 -0.55 -0.60 -0.65 -0.69 -0.81 -1.66 -1.18***
(-4.55) (-4.19) (-4.30) (-4.27) (-4.84) (-5.31) (-5.98) (-5.55) (-6.65) (-10.36) (-8.65)
Profit -1.91 -0.95 -0.66 -0.55 -0.48 -0.46 -0.46 -0.36 -0.46 -0.54 1.37***
(-12.04) (-7.95) (-5.57) (-5.06) (-4.43) (-4.42) (-4.45) (-3.42) (-4.46) (-4.60) (14.29)
Disp -0.46 -0.45 -0.45 -0.42 -0.47 -0.62 -0.61 -0.75 -0.91 -1.13 -0.67***
(-4.76) (-4.52) (-4.32) (-3.69) (-4.32) (-5.61) (-5.03) (-6.20) (-7.10) (-8.34) (-7.77)
Ivol -0.35 -0.34 -0.37 -0.43 -0.51 -0.53 -0.72 -0.80 -1.12 -1.83 -1.48***
(-4.33) (-3.75) (-3.79) (-4.05) (-4.83) (-4.64) (-6.22) (-5.79) (-7.57) (-11.69) (-13.04)
Voldev -2.32 -1.27 -0.93 -0.81 -0.60 -0.44 -0.32 -0.22 -0.08 -0.00 2.31***
39
(-20.32) (-12.17) (-8.77) (-8.27) (-5.37) (-3.75) (-2.61) (-1.58) (-0.61) (-0.02) (14.85)
Vts -2.26 -1.00 -0.69 -0.61 -0.44 -0.38 -0.34 -0.30 -0.34 -0.53 1.72***
(-14.67) (-7.18) (-5.84) (-5.87) (-3.96) (-3.47) (-3.27) (-2.77) (-3.27) (-4.78) (16.80)
BidAsk -0.21 -0.39 -0.41 -0.54 -0.65 -0.79 -0.86 -0.91 -1.02 -1.06 -0.85***
(-1.88) (-3.81) (-3.57) (-5.18) (-5.75) (-7.02) (-7.93) (-8.02) (-8.33) (-7.50) (-10.36)
Credit -0.18 -0.25 -0.31 -0.49 -0.97 -0.78***
(-1.79) (-2.47) (-3.22) (-4.38) (-7.91) (-10.49)
This table reports summary statistics of delta-hedged option returns (in percentage) sorted by various characteristics. The sample period
is January 1996 to December 2015. Size is the natural logarithm of the market value of the firm’s equity. Reversal is the lagged one-
month return. Mom is the cumulative return on the stock over the 11 months ending at the beginning of the previous month. CH is the
Cash-to-assets ratio, defined as the value of corporate cash holdings over the value of the firm’s total assets in Palazzo (2012). Profit is
calculated as earnings divided by book equity in Fama and French (2006). Disp is the analyst earnings forecast dispersion, computed as
the standard deviation of annual earnings-per-share forecasts scaled by the absolute value of the average outstanding forecast in Diether
et al. (2002)). Ivol is the annualized stock return idiosyncratic volatility in Ang et al. (2006). Voldev is the log difference between the
realized volatility and the Black-Scholes implied volatility for at-the-money options in Goyal and Saretto (2009). Vts is the volatility
term structure, defined as the difference between long-term and short-term implied volatility in Vasquez (2017). BidAsk is the difference
between the bid and ask quotes of option divided by the midpoint of the bid and ask quotes at the end of the previous month. Credit is
the credit ratings provided by Standard & Poor’s. The ratings are mapped to 22 numerical values, where 1 corresponds to the highest
rating (AAA) and 22 corresponds to the lowest rating (D). We report equal-weighted returns and 10-1 return spread in the table. The
Newey-West t-statistics are reported in Parenthesis.
Table 2: Summary Statistics of the Long-short Factors
Std. 10th. 25th. 50th. 75th. 90th.

Mean Skew Kurt
Dev. Pctl. Pctl. Pctl. Pctl. Pctl.
LSsize -1.87 1.65 -3.99 -2.85 -1.82 -1.03 0.07 0.01 0.80
LSreversal -0.60 1.57 -2.33 -1.29 -0.68 0.23 1.09 0.33 3.37
LSmom -0.51 1.95 -2.60 -1.36 -0.35 0.48 1.58 -0.63 3.27
LSch -1.18 1.66 -3.00 -2.25 -1.11 -0.29 0.76 0.06 1.04
LSprof it -1.37 1.46 -3.19 -2.23 -1.31 -0.50 0.17 -0.20 1.27
LSdisp -0.67 1.27 -2.00 -1.32 -0.73 -0.10 0.60 -0.27 5.59
LSivol -1.48 1.69 -3.33 -2.42 -1.62 -0.40 0.47 0.30 2.76
LSvoldev -2.31 1.64 -4.70 -3.26 -1.96 -1.10 -0.48 -0.69 0.57
LSvts -1.72 1.50 -3.57 -2.56 -1.71 -0.91 -0.21 0.29 2.54
LSbidask -0.85 1.26 -2.37 -1.58 -0.91 -0.21 0.68 -0.18 2.93
LScredit -1.03 1.47 -2.81 -1.90 -1.07 -0.23 0.63 -0.08 2.77
DHidx -0.16 0.36 -0.47 -0.38 -0.22 0.02 0.32 1.08 3.84
DHstk -0.06 0.19 -0.22 -0.16 -0.08 0.03 0.13 -0.90 16.19
This table reports summary statistics of the returns on long-short portfolios (in percent-
age) that go long in stock options with high (low) values for a certain characteristic and
short stock options with low (high) values. To make the average sign of the factors returns
consistent with the negative market straddle return, which represents variance risk pre-
mium, we go long in options with low values and short with high values for size, reversal,
momentum, profitability, volatility deviation and slope of the volatility term structure,
such that all factor returns are on average negative. The sample period is from January
1996 to December 2015.
40
Table 3: Correlation Matrix of the Candidate Factors in the Equity Option Market
LSsize LSreversal LSmom LSch LSprof it LSdisp LSivol LSvoldev LSvts LSbidask LScredit DHidx DHstk
LSsize 1.00
LSreversal 0.17 1.00
LSmom 0.33 0.07 1.00
LSch 0.28 0.03 -0.22 1.00
LSprof it 0.51 0.06 0.10 0.50 1.00
LSdisp 0.26 0.04 0.12 0.31 0.43 1.00
LSivol 0.49 0.20 -0.02 0.53 0.57 0.50 1.00
LSvoldev 0.13 -0.01 0.14 0.05 0.08 -0.01 -0.21 1.00
41
LSvts 0.15 0.33 0.03 0.20 0.22 0.30 0.40 0.16 1.00
LSbidask 0.50 0.13 0.18 -0.06 0.19 0.00 0.03 -0.01 0.00 1.00
LScredit 0.47 0.23 0.16 0.23 0.36 0.37 0.53 0.06 0.35 0.16 1.00
DHidx 0.04 0.08 -0.07 0.15 0.11 0.16 0.19 -0.07 0.09 0.01 0.09 1.00
DHstk 0.03 0.11 -0.15 0.24 0.13 0.22 0.27 -0.03 0.20 -0.09 0.12 0.55 1.00
This table reports correlation matrix of the 13 candidate factors in the equity option market. The first 10 factors are 10-1 return
spread sorted by size, reversal, momentum, cash holding, profitability, analyst forecast dispersion, idiosyncratic volatility, deviation of
log realized volatility from log implied volatility, volatility term structure and bid-ask spread. The 11th factor is the 5-1 return spread
sorted by credit rating. The last two factors are the delta-hedged return of the S&P 500 index options (DHidx ) and average delta-hedged
return from the stock options that are components in S&P500 DHstk . Sample period is from January 1996 to December 2015.
Table 4: Estimation for the Number of Factors in the Delta-hedged Option Portfolios
Demeaned Raw
Data Data
Eigenvalue Ratio (ER) estimator in Ahn and Horenstein (2013) 1 NA
Growth Ratio (GR) estimator in Ahn and Horenstein (2013) 1 NA
Edge Distribution (ED) estimator in Onatski (2010) 1 2
Modified Bayesian information criterion (BIC3) estimator in Bai and Ng (2002) 3 4
Information Criterion (IC1) estimator in Bai and Ng (2002) 6 7
Modified Information Criterion estimator (ABC) in Alessi et al. (2010) 6 7
This table presents results obtained from estimating the number of factors using the
Eigenvalue Ratio (ER) and Growth Ratio (GR) estimators of Ahn and Horenstein (2013),
the Edge Distribution (ED) estimator of Onatski (2010), the BIC3 and IC1 estimators
of Bai and Ng (2002), and the Modified Information Criterion estimator (ABC) of Alessi
et al. (2010). The test assets are 105 characteristic-sorted delta-hedged option portfolios
reported in Table 1. The sample period is January 1996 to December 2015. We apply the
estimators to doubly-demeaned data (Column 1) and to the raw data (Column 2). ER
and GR are applied only to doubly-demeaned data.
42
Table 5: Performance of the Model with 6 Latent Factors and the Model with 13 Candidate
Factors
Panel A: Model with 6 Latent Factors Panel B: Model with 13 Candidate Factors
Non-zero Alphas 0.238 Non-zero Alphas 0.295
Average Adj. R2 0.318 Average Adj. R2 0.303
Corr E(R)-Predicted Return 0.967 Corr E(R)-Predicted Return 0.963
Average Abs. Alpha (annualized) 1.625 Average Abs. Alpha (annualized) 1.785
Corr E(R) and Beta of Factor 1 -0.96 Corr E(R) and Beta of LSsize -0.475
Corr E(R) and Beta of Factor 2 0.143 Corr E(R) and Beta of LSreversal -0.154
Corr E(R) and Beta of Factor 3 0.048 Corr E(R) and Beta of LSmom -0.046
Corr E(R) and Beta of Factor 4 -0.008 Corr E(R) and Beta of LSch -0.344
Corr E(R) and Beta of Factor 5 0.009 Corr E(R) and Beta of LSprof it -0.295
Corr E(R) and Beta of Factor 6 -0.005 Corr E(R) and Beta of LSdisp -0.23
Corr E(R) and Beta of LSivol -0.373
Corr E(R) and Beta of LSvoldev -0.429
Corr E(R) and Beta of LSvts -0.38
Corr E(R) and Beta of LSbidask -0.077
Corr E(R) and Beta of LScredit -0.216
Corr E(R) and Beta of ST Ridx -0.01
Corr E(R) and Beta of ST Rstk 0.19
Panel C: Increase the number of factors from 1 to 6

1 Factor 2 Factors 3 Factors 4 Factors 5 Factors 6 Factors
Non-zero alphas (5%) 0.543 0.219 0.181 0.21 0.257 0.238
Average Adj R2 0.138 0.195 0.232 0.26 0.288 0.318
AAAPE 5.461 2.275 1.719 1.681 1.628 1.625
This table shows the performance of the proposed model with 6 latent factors in Panel
A and the performance of a model with 13 candidate factors in Panel B. In Panel A and
B, we report the percentage of pricing errors statistically different than 5% generated by
each model, the average adjusted R2 across option portfolios each model generates, the
correlation between the portfolios expected (excess) returns and the betas of each factor,
the correlation between the portfolios expected (excess) returns and the models’ predicted
returns (betas times factor premiums). Panel C presents the performance metrics of 6
factor models, in which we increase the number of factors used as independent variables
sequentially from 1 to 6. We report the number of pricing errors statistically significant
at the 5% or less, the average adjusted R2 generated by the model, and the annualized
average absolute pricing error (AAAPE).
43
Table 6: Correlation Coefficients between the 13 Candidate Factors and Six Latent Factors
Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6 EWP

LSsize 0.71 0.02 0.48 -0.25 -0.24 -0.15 0.25
LSreversal 0.28 0.11 0.13 0.64 -0.25 -0.09 0.24
LSmom 0.13 -0.43 0.72 0.01 0.35 -0.01 -0.22
LSch 0.64 0.60 -0.23 -0.36 -0.06 0.02 0.33
LSprof it 0.71 0.31 0.16 -0.31 0.03 -0.23 0.30
LSdisp 0.56 0.39 0.23 -0.02 0.40 0.01 0.37
LSivol 0.80 0.72 0.25 -0.04 0.10 -0.15 0.56
LSvoldev 0.20 -0.50 -0.34 0.00 0.23 0.27 -0.18
LSvts 0.53 0.27 -0.01 0.54 0.13 0.13 0.34
LSbidask 0.20 -0.23 0.33 -0.06 -0.56 -0.21 0.10
LScredit 0.63 0.28 0.33 0.13 -0.01 0.19 0.36
DHidx 0.14 0.19 0.04 0.04 -0.02 0.00 0.38
DHstk 0.21 0.30 -0.04 0.08 0.03 0.00 0.50
This table shows the Pearson correlation coefficients between the 13 candidate factors
and the six latent factors. The 13 candidate factors are described in Table 1. The sample
period is from January 1996 to December 2015.
44
Table 7: Canonical Correlation
6 Latent Factors vs 3 Latent Factors vs 6 Latent Factors vs 3 Latent Factors vs

13 Candidate Factors 13 Candidate Factors 6 Candidate Factors 6 Candidate Factors
1 0.99 0.98 0.98 0.96
2 0.98 0.96 0.93 0.90
3 0.95 0.89 0.85 0.75
4 0.91 0.69
45
5 0.91 0.42
6 0.42 0.22
This table shows the canonical correlations for different pairs of factors. The first column shows the canonical correlation
corresponding to the 6 latent factor and the 13 candidate factors. In the second column we compare the 3 most relevant
latent factors with the 13 candidate factors. Columns three and four compare the latent factors with the subset of the 6
candidate factors we find most relevant with our rank estimation exercise, which are LSsize , LSch , LSdisp , LSivol , LSvoldev ,
and LSrating A brief explanation on the canonical correlation is in Appendix A.2.
Table 8: Performance of the Models with 4 Factors and 7 Factors
(105 Characteristic-sorted Portfolios)
Panel A: Model with 4 candidate factors Panel B: Model with 7 candidate factors
Non-zero Alphas 0.05 Non-zero Alphas 0.07
Average Adj. R2 0.34 Average Adj. R2 0.35
Corr E(R) and Beta of DHidx -0.10 Corr E(R) and Beta of DHidx 0.06
Corr E(R) and Beta of LSsize -0.50 Corr E(R) and Beta of LSsize -0.47
Corr E(R) and Beta of LSivol -0.61 Corr E(R) and Beta of LSivol -0.47
Corr E(R) and Beta of LSvoldev -0.63 Corr E(R) and Beta of LSvoldev -0.52
Corr E(R) and Beta of LSch -0.32
Corr E(R) and Beta of LSdisp -0.28
Corr E(R) and Beta of LScredit -0.30
This table shows the performance of the model with 4 option factors DHidx , LSsize ,
LSivol , LSvoldev in Panel A and the performance of the model augmented with LSch ,
LSdisp , and LScredit in Panel B. The test assets are the 105 characteristic-sorted portfolios.
We report the percentage of pricing errors statistically different than 5% generated by
each model, the average Adjusted R2 across option portfolios each model generates, the
returns (betas times factor premiums).
46
Table 9: Fama-Macbeth Regressions for the 105 Portfolios
Intercept LSsize LSivol LSvoldev Adjusted R2

Full sample 0.49 -2.11 -1.35 -2.31 0.90
(9.19) (-13.47) (-10.66) (-9.16)
Jan. 1996 - Dec. 2005 -0.00 -2.35 -1.38 -2.39 0.84

(-0.00) (-18.63) (-9.12) (-9.73)
Jan. 2006 - Dec. 2015 -0.00 -1.59 -1.23 -2.10 0.84

(-0.01) (-14.25) (-10.28) (-13.14)
This table reports Fama-MacBeth regression results for the 105 portfolios for the full
sample from January 1996 to December 2015 and for the two sub-samples from January
1996 to December 2005 and from January 2006 to December 2015. We first run the time-
series regression of the return of the 105 portfolios on the three factors and get the betas.
We then run the cross-section regression of average return of the 105 portfolios on the
estimated betas from the first step. The table reports regression coefficients, adjusted R2
and the t-statistics from the second step of regression. The standard errors are corrected
by Shanken (1992).
47
Table 10: Performance of the Models with 4 Factors and 7 Factors
(26 Industry Portfolios)
Panel A: Model with 4 candidate factors Panel B: Model with 7 candidate factors
Non-zero alphas 0.08 Non-zero alphas 0.08
Average Adj R2 0.22 Average Adj R2 0.25
Corr E(R) and Beta of DHidx -0.29 Corr E(R) and Beta of DHidx -0.43
Corr E(R) and Beta of LSsize -0.10 Corr E(R) and Beta of LSsize -0.09
Corr E(R) and Beta of LSivol 0.70 Corr E(R) and Beta of LSivol 0.44
Corr E(R) and Beta of LSvoldev 0.74 Corr E(R) and Beta of LSvoldev 0.54
Corr E(R) and Beta of LSch 0.71
Corr E(R) and Beta of LSdisp -0.14
Corr E(R) and Beta of LScredit 0.03
This table shows the performance of the model with 4 option factors DHidx , LSsize , LSivol ,
LSvoldev , in Panel A and the performance of the model augmented with LSch , LSdisp , and
LScredit in Panel B. The test assets are the 26 industry characteristic-sorted portfolios.
We report the percentage of pricing errors statistically different than 5% generated by
each model, the average Adjusted R2 across option portfolios each model generates, the
returns (betas times factor premiums).
48
Table 11: Correlation Coefficients of the Stock Factors and the Six Latent Option Factors
Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6 EWP

MKT 0.10 0.23 -0.12 -0.01 -0.08 -0.10 0.36
SMB 0.19 0.16 -0.08 -0.02 -0.05 -0.08 0.29
HML -0.13 -0.14 0.09 -0.07 -0.15 -0.14 -0.05
RMW -0.12 -0.21 0.15 -0.07 0.06 -0.07 -0.20
CMA -0.14 -0.18 0.04 0.05 -0.06 -0.04 -0.11
MOM 0.18 0.12 0.21 0.06 0.12 0.00 0.02
BAB 0.09 0.10 0.31 -0.04 -0.12 -0.10 0.16
MGMT -0.10 -0.19 0.09 0.00 0.02 0.00 -0.20
PERF 0.03 -0.10 0.22 0.03 0.14 0.09 -0.11
LIQDLEV EL 0.05 0.14 0.09 0.04 -0.33 0.09 -0.25
LIQDIN N OV 0.03 0.09 0.11 -0.02 -0.21 -0.11 -0.21
LIQDV W F 0.06 -0.05 0.10 0.03 -0.12 0.00 -0.18
This table shows the correlation coefficients between the stock factors and the estimated 6
latent option factors plus EWP. Stock factors include the five factors in Fama and French
(2016) (MKT, SMB, HML, RMW, CMA), momentum factor in Carhart (1997) (MOM),
stock market liquidity risk factors in Pástor and Stambaugh (2003) (P Sinnov , P Slevel ,
P SV W F ), betting-against-beta factor in Frazzini and Pedersen (2014) (BAB) and the two
mispricing factors in Stambaugh and Yuan (2016) (MGMT and PERF).
49
Table 12: Regression of the Latent Option Factors and EWP on Stock Factors
Latent Latent Latent Latent Latent Latent

EWP
Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6
MKT 1.20 3.83** -0.80 -0.13 1.19 -2.73 11.58***
SMB 3.47*** 2.58 -2.36 -2.74 0.93 -3.89 10.50***
HML -2.20 -4.02 7.06* -4.14 -8.39* -1.94 -1.56
RMW 0.05 -2.44 -7.01* -1.83 8.51* -10.22** -2.42
CMA -5.23* -5.79 -7.75 8.24 0.77 -3.95 -3.78
MOM 2.47** 4.53** 0.37 1.37 3.80* -4.40** -0.28
BAB 2.72** 7.66*** 6.73 -1.13 -3.48* -1.01 12.66***
MGMT 3.15 0.68 0.49 -0.78 3.23 5.46 0.50
PERF -2.09 -5.33** 5.42** -0.94 -2.90 5.98** 0.74
LIQDlevel 0.00 1.36 -1.08 1.67 -5.06*** 0.00 2.65
LIQDinnov -0.22 -1.25 2.91** -1.44 1.55 -0.83 0.63
LIQDvwf 0.29 -2.91* 1.59 0.68 -2.13 0.74 -2.61
Adjusted R2 0.13 0.21 0.16 0.04 0.19 0.09 0.32
This table reports the regression result of each option latent factor and EWP (equal-
weighted portfolio of all 105 portfolios) on the stock factors. Stock factors include the five
factors in Fama and French (2016) (MKT, SMB, HML, RMW, CMA), momentum factor
in Carhart (1997) (MOM), stock market liquidity risk factors in Pástor and Stambaugh
(2003) (P Sinnov , P Slevel , P SV W F ), betting-against-beta factor in Frazzini and Pedersen
(2014) (BAB) and the two mispricing factors in Stambaugh and Yuan (2016) (MGMT
and PERF).
50
Table A1: Values of the IB Estimator for Two Factor Models
Panel (a)
EWP LSsize LSivol LSvoldev
Average Adj R2 0.87
Corr E(R)-Beta -0.10 0.50 0.61 0.63
IB 0.995 0.00 0.00 0.00
Corr E(R)-Predicted Return 0.95
Panel (b)
DHidx LSsize LSivol LSvoldev
Average Adj R2 0.34
Corr E(R)-Beta 0.10 0.50 0.61 0.63
IB 0.991 0.00 0.87 0.19
Corr E(R)-Predicted Return 0.95
This table shows the values of the IB estimator proposed in Ahn and Horenstein (2013)
for two factor models. Panel (a) shows the results from regressing the 105 delta-hedged
option returns on a model consisting of the EWP (equal-weighted portfolio) and the three
option factors proposed in the main body of the paper (Size, ivol, and voldev) and Panel
(b) show the results from a similar factor model in which EWP has been replaced by the
DHidx factor (delta-hedged return of the S&P 500 index options). The table also reports
the average value of the factor loadings as well as the R2 of the models and the correlation
between the predicted returns and the average returns of the models.
51

Common Factors in Equity Option Returns

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Common Factors in Equity Option Returns

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Common Factors in Equity Option Returns

Alex Horenstein1 , Aurelio Vasquez∗2 , and Xiao Xiao3,4

May 29, 2019

JEL Classification: C14, G13, G17

2 Theoretical motivation: Delta-hedged equity option gains

and variance of the HML factor.

t+τ t+τ t+τ

1 , V 2 , ..., V n ) are priced, the

3 Data and variables description

3.1 Data and sample coverage

3.2 Construction of the delta-hedged option returns

[ Table 1 around here]

[ Table 2 around here]

[ Table 3 around here]

4 Empirical Procedure and Results

4.1 Number of latent factors that drive the comovement of delta-hedged

[Figure 1 around here]

[ Table 4 around here]

[ Table 5 around here]

[Figure 2 around here]

4.2 Relevant candidate factors of the cross-section of delta-hedged option

[ Table 6 around here]

[ Figure 3 around here]

[ Table 7 around here]

rit = αi + βidx,i DHidx,t + βsize,i LSsize,t + βivol,i LSivol,t + βvoldev,i LSvoldev,t

[ Table 8 around here]

[ Figure 4 around here]

[ Table 9 around here]

[ Table 10 around here]

[ Table 11 around here]

[ Table 12 around here]

Mark M Carhart. On persistence in mutual fund performance. The Journal of finance, 52

Narasimhan Jegadeesh. Evidence of predictable behavior of security returns. The Journal of

A.1 Summary of the number of factors tests used in the paper

IC(k) = ln(V (k)) + kg(N, T ),

where g(N, T ) is a penalty function such that g(N, T ) → 0 and g(N, T )m → ∞ as N, T → ∞.

and the two BN criteria can be written as

µ̃N T,k V (k − 1) − V (k)

ln(1 + µ̃N T,k ) ln(V (k − 1)/V (k))

A.2 Canonical Correlation

V ar(U ) = a0 ΣF a, V ar(V ) = b0 ΣX b, and Cov(U, V ) = a0 ΣF X b.

A.3 Equal weighted portfolio (EWP) contains unitary loadings

Std. 10th. 25th. 50th. 75th. 90th.

Panel C: Increase the number of factors from 1 to 6

Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6 EWP

6 Latent Factors vs 3 Latent Factors vs 6 Latent Factors vs 3 Latent Factors vs

(105 Characteristic-sorted Portfolios)

Intercept LSsize LSivol LSvoldev Adjusted R2

Jan. 1996 - Dec. 2005 -0.00 -2.35 -1.38 -2.39 0.84

Jan. 2006 - Dec. 2015 -0.00 -1.59 -1.23 -2.10 0.84

(26 Industry Portfolios)

Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6 EWP

Latent Latent Latent Latent Latent Latent

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