1. finance
2. investing
3. stocks

Pricing of Idiosyncratic Equity and Variance Risks∗
Elise Gourier†
Princeton University
October 24, 2014
Abstract
This paper develops a parametric approach to study the equity and variance risk premia
of individual stocks, based on the information contained in their underlying returns at different
frequencies, and option prices. I introduce a continuous-time variation of the Capital Asset Pricing
Model with stochastic volatility and jumps and estimate it using Bayesian filtering techniques
over a time period including the recent financial crisis. I show that idiosyncratic equity risk is
priced on short term horizons, but not on longer terms. In contrast, idiosyncratic variance risk is
substantially priced in the cross-section of returns, for all terms considered. An analysis of sector
Exchange Traded Funds reveals that most of them provide an interesting investment alternative
for investors who are not willing to be exposed to non diversifiable risk. However, two sectors
exhibit high priced idiosyncratic risk, namely the Energy and Financials sectors.
JEL Classification: C38, C51, G12, G13.
Keywords: Risk Premia, Idiosyncratic Risk.
∗
This paper was previously presented with the title ”Information Contents of Option Prices on Idiosyncratic and Systematic Equity Risks”.
Special thanks go to Pierre Collin-Dufresne, Darell Duffie, Damir Filipovi´
c, Genevi`
eve Gauthier, Jakub Jurek, Markus Leippold, Loriano Mancini
and Norman Sch¨
urhoff for their helpful feedback. I also thank the participants of the Brown Bag Seminar at EPFL Lausanne, the eighth OxfordPrinceton Workshop, the Mathematical Finance Days at HEC Montreal, the eighth World Congress of the Bachelier Society and the ORFE Financial
Mathematics Seminar at Princeton University for valuable comments. Financial support from the Swiss National Science Foundation (SNSF) is
gratefully acknowledged.
†
Email: [email protected].
1
1
Introduction
The strong factor structure in the dynamics of asset returns has been widely documented since
the introduction of the Capital Asset Pricing Model (CAPM) in the early sixties. The baseline
CAPM predicts that the risk that drives a stock’s returns and does not come from the market
factor, so-called idiosyncratic equity risk, can be diversified. Therefore it should not be priced in a
perfect capital market. The associated risk premium, which reflects the compensation an investor
requires to bear that risk, should thus be zero. However, empirical studies have suggested that
this may not be the case in the presence of market imperfections, see, a.o., Malkiel and Xu (2001),
Goyal and Santa-Clara (2003), Spiegel and Wang (2005), Fu (2009) and Garcia, Mantilla-Garcia,
and Martellini (2011). Furthermore, recent papers, e.g., Herskovic, Kelly, and Lustig (2014) have
highlighted a similar factor structure among the variances of the idiosyncratic components of returns,
so-called idiosyncratic variances. Sch¨
urhoff and Ziegler (2011) further decomposed the idiosyncratic
variance into a component that is linear in the variance of the market factor, referred to as common
idiosyncratic variance, and a component that is specific to the firm, so-called true idiosyncratic
variance. Measuring the risk premium attached to each source of risk is crucial to understanding the
exposure of a portfolio, and to select appropriate investment and hedging strategies. Quantifying
the idiosyncratic part of risk premia is, however, a difficult task. On the one hand, measuring the
evolution of total risk premia requires knowledge on the conditional distributions of the asset returns
under the empirical measure and an appropriate risk-neutral measure, which is challenging to acquire.
On the other hand, filtering the idiosyncratic component of risk premia relies on assumptions on the
dependence structure between stocks. Such assumptions often give rise to high-dimensional problems
that are neither tractable nor easy to calibrate to market data.
This paper develops a parametric approach to study the equity and variance risk premia of individual
stocks. I introduce a continuous-time extension of the CAPM and estimate it using individual stocks’
returns, realized variances and option prices on every stock. I evaluate its performance in terms of
pricing errors and forecasting accuracy. An analysis of the filtered idiosyncratic variance trajectory
suggests that it contains valuable information on future returns. I show that idiosyncratic equity
risk is priced on short time horizons, and idiosyncratic variance risk for all terms considered. In
particular, the common idiosyncratic variance carries a risk premium which is significantly larger
than zero whereas the risk premium attached to the true idiosyncratic variance is largely negative.
2
Finally, I relate the risk profile of single stocks to the one of the sectors they belong to, and show
that certain sectors present are particularly appealing for investors.
My framework assumes that individual stock returns’ dynamics consist of a systematic component
which is function of the movements of a market factor, and an idiosyncratic component that solely
depends on the firm’ characteristics. I introduce a continuous-time model in the spirit of CollinDufresne, Goldstein, and Yang (2012) and Christoffersen, Fournier, and Jakobs (2013). As risk
premia strongly depend on higher order moments of the underlying returns, using jumps in the
returns and variance process accommodates for the rich stylized facts of the empirical and risk-neutral
distributions of the underlying stocks and market factor. High-frequency returns and options with
short times-to-maturity are used to identify the jumps’ distributions. Options with wide ranges of
maturities allow me to make inferences on the properties of the term structure of risk premia. Based
on empirical evidence, my model accounts for idiosyncratic and systematic diffusive and jump drivers
of the returns and allows for disentangling and analyzing their respective impacts. Furthermore, the
model allows movements of the stochastic variance of single stocks to be correlated with movements
of the market variance. The market factor is proxied by the S&P 500 and VIX indices. As my model
belongs to the affine class of Duffie, Pan, and Singleton (2000), it is tractable and enables estimation
to a cross-section of stocks’ returns and option prices over a relatively long time-series of data.
Based on Bayesian filtering techniques, the estimation reconciles the vast information present in the
data in order to quantify the idiosyncratic component of the asset returns’ dynamics, and better
understand which risks are priced. The estimation procedure is based on the Auxiliary Particle
Filter of Pitt and Shephard (1999) which, combined with a Maximum Likelihood estimation, yields
time-consistent parameters as well as estimates of the distributions of unobservable processes and
their jumps over time. I estimate the model over a time period of data that ranges from 2006 until
the end of 2009, i.e., covering the financial crisis. This allows me to obtain accurate information
on jumps, as these typically do not occur frequently and are hard to detect empirically. The model
performance is examined over the subsequent time period spanning 2010 until the end of 2012.
I show that the model is able to capture stylized facts that have been non-parametrically highlighted
in the literature. In particular, estimation results confirm the relationships between the beta parameters and the risk-neutral variance and skewness of returns. Furthermore, I test the performance of
the model in reproducing variance swap rates. I synthesize swaps from available options and regress
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the resulting rates on model-implied rates for all stocks considered. Cross-sectional (Fama-MacBeth)
and panel regressions indicate that the model accurately represents the synthesized data. I further
challenge its reliability by evaluating how well it forecasts future variance swap rates. I compare the
predictions it delivers to martingale predictions, whose best guess of the future rate is the current
rate. Panel regressions reveal that the model significantly outperforms the martingale predictions
for all future times considered.
I perform a thorough analysis of the different risk sources in the variances of stock returns. Based on
the estimated trajectories of the variance processes, I evaluate the contribution to total variance of,
one the one hand, the variance of the market factor and, on the other hand, the idiosyncratic variance.
I show that, for the stocks considered, the total variance is mainly governed by its idiosyncratic
component in times of market calm, and that the market impact becomes substantial during turmoil
periods. Idiosyncratic variances exhibit a strong factor structure that is similar to the one of the
total variance, and justifies the decomposition of their dynamics into common idiosyncratic and true
idiosyncratic movements.
Furthermore, I perform a comparative study of the behaviour of the variances of single stocks in the
Dow Jones with the sectors they belong to. This analysis uses returns and options’ data on SPDR
Exchange Traded Funds. I find that most sectors have idiosyncratic variances that are smaller than
those of the single stocks. Notable exceptions are, however, the Energy and Materials sectors, which
have idiosyncratic variances that are close to 10% during times of low market volatility. This is also
the case of the Financials sector after the beginning of the financial crisis. The level of idiosyncratic
variance never returned to its pre-crisis level. In turn, the Consumer Staples, Industrials and Health
Care sectors exhibit very low levels of idiosyncratic variances: close to zero outside the market
volatility peaks, and around 10% during market stress.
Knowledge of idiosyncratic variance is important because it can guide portfolio allocation. I point
out that an investment over the period 2006-2012 in a dynamically rebalanced portfolio of stocks
which have an idiosyncratic variance level in the highest quartile yields a higher Sharpe ratio of
6.37%, against 1.21% when one buys the stocks whose idiosyncratic variance is in the lowest quartile.
This indicates that idiosyncratic variance contains information on future returns, suggesting that
idiosyncratic variance risk may be priced in the cross-section of returns.
My main empirical contribution is to provide an investigation of the equity and variance risk premia
4
of individual stocks. The use of filtering techniques allows me to reconcile the information present
in the different data sources and quantify the priced idiosyncratic risk throughout the time period
considered. I show that idiosyncratic equity risk is priced on a time horizon of one month, but
not on terms that exceed six months. In contrast, idiosyncratic variance risk is substantially priced
for all stocks and terms considered. A comparison to results obtained for sector Exchange Traded
Funds reveals some interesting features. As a result to their small level of idiosyncratic variance,
the Consumer Staples, Industrials and Health Care sectors carry very little idiosyncratic variance
risk. Therefore they provide an interesting investment alternative for agents who are not willing
to be exposed to non diversifiable risk. In constrast, the Energy and Financials sectors provide
higher exposure to idiosyncratic risks than most of their components. Decomposing the price of
idiosyncratic equity and variance risks for these sectors further reveals that a substantial part is due
to tail risk.
I base my analysis on a large body of literature that examines the equity and variance risk premia.
Since the introduction of the Capital Asset Pricing Model (CAPM), many empirical studies have tried
to explain expected returns of stocks and indices. Recent contributions include Ang, Hodrick, Xing,
and Zhang (2006), Chang, Christoffersen, and Jacobs (2013), Bollerslev and Todorov (2011), Yan
(2011) and Cremers, Halling, and Weinbaum (2013). The study of the variance risk premium (VRP)
has also been the subject of a number of papers in the last decade, most of which focus on index
returns. Among others, Bakshi and Kapadia (2003), Bakshi and Madan (2006), Amengual (2009),
Adrian and Rosenberg (2008) and Bollerslev and Todorov (2011). Carr and Wu (2009), Driessen,
Maenhout, and Vilkov (2009), Buraschi, Trojani, and Vedolin (2014), Han and Zhou (2013) extend
the analysis of VRP to individual stocks using non-parametric analyses and emphasize its relation
with the volatility level. The link between VRP and equity risk premium (ERP) has been further
explored in the context of the search for return predictability. See, e.g., Bollerslev, Tauchen, and
Zhou (2009), Drechsler and Yaron (2011) and Bollerslev, Morrone, Xu, and Zhou (2014).
Many of the papers investigating the equity and variance risk premia rely on non-parametric approaches. The main advantage of these methods is that their results are model independent. However they must overcome the fact that risk premia are not observable using approximations and
proxies. For example, the expectation of the future integrated variance under the empirical measure
is typically approximated by the realized variance over the past days or weeks. The model I propose
5
aims to represent the stylized facts highlighted empirically, and enables computing risk premia and
their components analytically.
The remainder of this paper is structured as follows. Section 2 presents a preliminary data analysis.
The model description and its properties are given in Section 3. Section 4 details the estimation
procedure. Section 5 investigates the behaviour of idiosyncratic variance for individual stocks and
sectors. Finally, Section 6 examines the estimated ERP and VRP and presents inferences made on
the pricing of idiosyncratic risks.
2
Preliminary data analysis
2.1
Data
The single stocks examined in this paper are the components of the Dow Jones Industrial Average
index. Visa was excluded from the sample as it completed its initial public offering in March 2008. For
every stock in the sample, the dataset includes daily dividend-adjusted prices obtained from CSRP,
five-minute trading prices from the TAQ database as well option prices from OptionMetrics. The
sampling period of high-frequency prices has been chosen to avoid adverse effects of microstructure
noise while keeping a frequency which is high enough to make inference on the properties of jumps. All
data span the time period from the 1st of March 2006 until the end of 2012. I eliminate options with
missing bid/ask prices or zero bid prices, as well as options which have a negative bid-ask spread.
I further remove options with zero open interest and those that violate the arbitrage conditions:
C(t, τ, K) < S(t) and C(t, τ, K) > S(t) − Ke−r(T −t) . To account for the American feature of options,
I follow Broadie, Chernov, and Johannes (2007) and calculate American volatilities implied from a
binomial tree for all mid prices, and then treat the options as European, with an implied volatility
equal to the one found. The error due to this adjustment is negligible for short-maturity options
which have an early exercise premium close to zero, and increases with the maturity of options. To
avoid beginning- and end-of-week effects, I follow standard practice and only keep Wednesday option
prices.
The S&P 500 combines the market capitalizations of the 500 largest US stocks, and therefore is
a natural proxy for the market factor in the US market. It is complemented by the VIX index,
6
which represents the market expectation of the future volatility of S&P 500 returns over the next
thirty days, and is often considered as a fear gauge of the markets. Options on both indices are
liquidly traded. The dataset of underlying levels and options as well as its preliminary treatment is
as described in Bardgett, Gourier, and Leippold (2014), extended to cover the period up to the end
of 2012.
2.2
Factor structure of implied volatilities
In addition to providing useful information on the individual stock returns’ distributions, I show in
this section that options also contain information on the factor structure of stocks.
I analyze the factor structure of implied volatility (IV) surfaces using Functional Principal Component
Analysis (FPCA). While standard PCA decomposes the covariance matrix of a set of time-series to
identify the main vectors of variation, FPCA performs a similar task with time-series of surfaces.1
The detailed procedure is described in Appendix A.
I construct for every Wednesday in the sample a smooth estimator of the implied volatility surface
IVt (m, τ ) over a grid of moneynesses m ∈ [mmin , mmax ] and times-to-maturity τ ∈ [τmin , τmax ] using
available data, and a non-parametric Nadaraya-Watson estimator as detailed in H¨ardle (1992) and
A¨ıt-Sahalia and Lo (1998). The outputs of the FPCA are a set of eigenmodes that represent the
main directions of variation of the surface, and principal component processes that are the projection
of the surface at every point in time onto the eigenmodes.
The eigenmodes and principal component processes that result from the FPCA of the S&P 500
index are represented in Appendix A, in Figures 14 and 15. The first eigenmode represents the
variation coming from the level of the IV surface, i.e. the upwards and downwards shifts of the entire
surface. This mode is not completely flat, there is a slight upward tilt towards short maturities and
smaller moneynesses and a slight downward tilt towards longer maturities and larger moneynesses.
Furthermore, the mode is slightly decreasing as maturity and moneyness increase. This reflects the
fact that deep out-of-the-money (OTM) put IVs tend to move more and deep OTM call IVs less
1
Similar analyses have been applied to indices by Skiadopoulos, Hodges, and Clewlow (2000), Cont and da Fonsecca
(2002), Cont, da Fonsecca, and Durrleman (2002), Daglish, Hull, and Suo (2007). Furthermore, Christoffersen, Fournier,
and Jakobs (2013) decompose the IV surface of single stocks into an affine function of moneyness and time-to-maturity.
Standard PCA is then applied to the time-series of loadings. This procedure is a sub-case of the one I apply, in which
the first direction of variation of the IV is affine in the moneyness and the second is proportional to the time-to-maturity.
I allow eigenmodes to be polynomial functions of both moneyness and maturity.
7
than other IVs. The second mode is a skew mode. OTM put IVs move in opposite directions from
OTM call IVs. The mode changes sign at the ATM level. Overall magnitudes tend to decrease as the
time-to-maturity increases. The third mode is a term structure mode. Short-term IVs move in the
opposite direction from long-term IVs. The magnitude tends to decrease as the moneyness increases.
Finally, the fourth mode is a convexity mode. Deep OTM put and call IVs tend to fluctuate more
than ATM option IVs. These findings are consistent with those of Cont and da Fonsecca (2002).
They find that the third mode is the convexity mode, instead of the term structure mode. But this
is likely due to the fact that they use a shorter time-series of data.
The same analysis is conducted on IV surfaces of single stocks. The resulting modes as well as
the percentages of variance they explain are described in Table 1. The first three eigenmodes are
usually the same as those found for the S&P 500. In some cases the term structure mode explains
more variation than the skewness mode. Overall, the first mode explains a large percentage of the
variation of the surfaces, on average 97.5%. The corresponding principal component process is highly
correlated with the one of the S&P 500 surface decomposition, with a correlation of 91.6% on average.
The second and third eigenmodes are also significantly correlated with the corresponding eigenmodes
of the S&P 500 decomposition, with respective average coefficients of 51.6% and 42.6%.
Functional PCAs therefore indicate a strong factor structure of the main driver of IV surfaces,
i.e., the level of the surface. The other drivers, which roughly represent the skewness and term
structure of the surfaces, also exhibit a large positive correlation with the corresponding market
principal components. However, I interpret the fact that this correlation is always substantially
lower than 1 as an indication of firm specific variation in IV. These results are consistent with those
of Christoffersen, Fournier, and Jakobs (2013).
2.3
Factor structure of high-frequency returns
High-frequency data contain valuable information on the fine structure of data. In particular, diverse
measures of variation of log-price increments have been introduced that enable disentangling the
contribution of the diffusive and jump components of the dynamics of returns.2 In this section, I use
2
See for example Andersen, Bollerslev, Diebold, and Labys (2003), Barndorff-Nielsen and Shephard (2004) and
Barndorff-Nielsen and Shephard (2006)
8
high-frequency data to analyze the potential factor structure of some of these measures.3
For each stock I calculate the daily Realized Variance (RV), which captures the total variation of
high-frequency returns over a day, and can be decomposed into the Continuous Variation (CV), which
represents the variation due to continuous movements of the returns, and the Jump Variation (JV)
which captures discontinuous movements of returns. For definitions and convergence properties, see,
e.g., Bollerslev, Todorov, and Li (2013). I apply a PCA to these measures.4 The same analysis is
performed after taking their n-day moving averages, with n ∈ [3 180].
The percentage of variance explained by the first principal component as a function of n is displayed
in Figure 1. Using a larger n leads to the first component explaining a larger percentage of the
variance of all measures, with that percentage being very close to 100% when the 6-month moving
average is considered, except for the jump variation measure. Along the same line, a smaller number
of factors are needed to explain the total amount of variation when n increases. I interpret this
result as an indication that the factor structure explains the overall tendencies of variation measures,
whereas the daily fluctations are mainly caused by firm-specific events. The much smaller percentage
of variation of the jump measure explained by the first components indicates that most jumps are
idiosyncratic, and that the factor structure of the diffusive movements is much stronger than the
one of jumps. These results are in line with Bollerslev, Todorov, and Li (2013), who find that the
number of filtered idiosyncratic jumps in stocks that belong to the S&P 500 exceeds the number of
systematic jumps for all the stocks contained in their sample.
I propose to represent the factor structure of asset returns using a parametric modeling approach.
Most studies using high-frequency data or option prices use non-parametric or semi-parametric approaches, which present more flexibility in analyzing the data and do not face the challenge of
estimation. However, with such approaches it is only possible to draw inferences, e.g., they cannot
be used for out-of-sample exercises, which are important parts of this paper.
3
Luciani and Veredas (2012) and Barigozzi, Brownlees, Gallo, and Veredas (2014) introduce a dynamic factor model
for realized volatilities.
4
A log-transformation is applied to the RV.
9
3
Model
3.1
Model setup
Let (Ω, F, {Ft }t≥0 , P) be a filtered probability space satisfying the usual assumptions, where P denotes the empirical measure. I denote by M = (Mt )t≥0 the value process of the market factor. Its
dynamics under P are specified as follows.
q
dMt
M
M (P)
M (P)
P ZtM
= (rt − λM (vtM
− 1] + γtM (vtM
vtM
+ (eZt − 1)dNt
− , mt )E [e
− , mt ))dt +
− dWt
Mt−
!
q
κM
M (P)
M (P)
M (P)
v
M
M (P)
M
m
dvt = κv
dt
+
σ
m
−
v
vtM
+ yt
dNt
−
− dBt
v
t
M (P) t
κv
(P) M (P)
M√
¯ M (P)
dmt = κM
(θ
− mt )dt + σm
mt dB
m
t
(1)
(2)
(3)
The risk-free rate rt is assumed to be a deterministic function of time. Price returns are driven by
M (P)
a Brownian motion Wt
M (P)
and a Poisson process Nt
. The continuous part of the returns’ vari-
ance is governed by the process (vtM )t≥0 , which mean-reverts towards a stochastic central tendency
M (P)
(mt )t≥0 , up to a scaling factor. Its mean-reversion speed is controlled by κv
. The parameter κM
v
corresponds to the speed of mean-reversion of the variance under an equivalent martingale measure Q.
The stochastic central tendency mean-reverts itself towards a fixed long-run mean θM (P) , with speed
M (P)
κm
. Jumps in the returns process occur simulateneously with jumps in the first variance factor vtM ,
M
M M
M
5
with an intensity that is affine in the two variance processes: λM (vtM
− , mt ) = λ0 + λ1 vt− + λ2 mt .
The sizes of jumps in the returns are assumed to be independent and identically distributed following
a normal distribution. Jumps in the variance are assumed to be exponentially distributed. This implies that the variance can only jump upwards and prevents it from becoming negative. The leverage
effect that impacts the market factor is represented by the correlation coefficient ρM defined such
that dhW M (P) , B M (P) it = ρM dt.
The model for individual stocks is a continuous-time extension of the CAPM.6 I separate the impact of
5
Based on a non-parametric study, Bollerslev and Todorov (2011) argue in favor of time-varying jump intensities
and a tendency for fatter tails in period of high overall volatility.
6
The model is related to the one of Collin-Dufresne, Goldstein, and Yang (2012), but additionally features idiosyncratic jumps and stochastic volatility. The framework which is closest to mine is the one of Christoffersen, Fournier,
and Jakobs (2013), who calibrate a diffusive model to a cross-section of stock returns and option prices. There are
10
the market factor into two contributions. On one side the diffusive random variation of single stocks’
j
returns is dependent on the Brownian motion that drives market returns through the coefficient βdif
f.
On the other side discontinuous movements in the market returns can also translate into jumps in
j
single stocks’ returns, though the coefficient βjump
. A jump of size ZtM in the market log level triggers
a jump of βjump ZtM in the individual log stock price.7 This allows disentangling the effects of both
market drivers on single stocks’ returns. Under P, the dynamics of firm j’s dividend-adjusted stock
prices Stj are given by:
dStj
Stj−
= (r +
γtj )dt
j
+
j
βdif
f
q
q
j
j
M (P)
j(P)
j(P)
M
vt− dWt
+ vtj− dWt
+ (eZt − 1)dNt
− λj (vtj− )EP [eZt − 1]dt
M
M (P)
+ (eβjump Zt − 1)dNt
dvtj = βvj dvtM + κj(P)
v
κjv
j(P)
κv
j
M
P βjump Zt
− λM (vtM
− 1]dt
− , mt )E [e
!
q
θj − vtj
−
dt + σvj
−
j(P)
vtj dBt
(4)
j(P)
+ ytj dNt
.
(5)
I account for the idiosyncratic leverage effect: dhW j(P) , B j(P) it = ρj dt. In line with the preliminary
data analysis, I assume that idiosyncratic returns and variance can jump. As for the market factor,
jumps occur in both processes simultaneously, with an intensity that is affine in the variance process:
λj (vtj− ) = λj0 + λj1 vtj− . The sizes of jumps in the returns are assumed to be normally distributed
with mean µj(P) and variance (σ j(P) )2 . Jumps in the variance follow an exponential distribution with
mean ν j(P) .
The model allows for two new features. On the one hand, it disentangles the respective impacts of
the diffusive and jump components of the market factor on individual stocks. Such differentiation is
justified by the notion of crash aversion introduced by Bates (2008), who argues that the investors
some major differences however between their work and mine. First, they assume that the idiosyncratic variance is
independent from the market variance. An examination of the idiosyncratic variances of stocks suggests that a richer
factor structure would be needed, as further advocated by Serban, Lehoczky, and Seppi (2008), Sch¨
urhoff and Ziegler
(2011) and Herskovic, Kelly, and Lustig (2014). My model accounts for this and decomposes idiosyncratic variance
movements into a common and a true idiosyncratic component. Second, they assume that idiosyncratic variance risk
is not priced. In other words, the variance risk premium of single stocks is assumed to be zero. Although it has been
shown to be smaller in magnitude than the variance risk premium of the index, there is empirical evidence for a non-zero
variance risk premium in individual stocks. Third, they estimate the trajectory of the unobservable spot variance of
the index and of single stocks using a least-square algorithm, i.e., its dynamics are not ensured to be consistent with
the specified model. The use of Bayesian methods allows me to have an estimation which guarantees that the filtered
trajectory of the spot variance is consistent with the model specification. Last, their model does not include jumps,
which have been shown to be important, especially during the financial crisis, to fit option prices, see, e.g., Bakshi,
Cao, and Zhong (2012).
7
Based on a model selection exercise, Bakshi, Cao, and Zhong (2012) propose to use a model with jumps in the
returns and in the volatility of individual stocks.
11
treat jump and volatility risks differently. For example, an investor may want to hedge tail risk, in
j
which case he will find stocks with a larger βjump
more attractive. Both volatility and jump risks
are priced in the cross-section of returns, as has been shown by Cremers, Halling, and Weinbaum
(2013).
On the other hand, my model allows the idiosyncratic variance of individual stocks to be correlated
with the market variance. Sch¨
urhoff and Ziegler (2011) and Herskovic, Kelly, and Lustig (2014)
show that the idiosyncratic variance of individual stocks can be decomposed into a true idiosyncratic
component that is firm-specific and a common idiosyncratic component, which depends on the market
variance. Therefore I divide the movements of the variance into two parts. The first one measures
the impact of the market variance on the idiosyncratic variance. It is linear in βvj . Therefore, βvj
represents the exposure of the idiosyncratic variance to the market variance. This first part is referred
to as the common idiosyncratic variance. The second component of the idioyncratic variance is driven
j(P)
by a firm-specific Brownian motion Bt
j(P)
and a Poisson process Nt
. It is referred to as the true
idiosyncratic variance, as its drivers are assumed to be independent from the drivers of the market
variance. Such representation preserves the affine structure of the model.
The model can be easily extended to multiple factors driving underlying stocks’ returns. Several
factors have been proposed in the asset pricing literature. The size and value factors of the popular
Fama-French three-factor model account for the extra risk present in small companies’ stocks and
value stocks. However, a regression of the returns on the three Fama-French factors over the time
period considered reveals that the betas associated with the SMB (Small market capitalization Minus
Big) and the HML (High book-to-market ratio Minus Low) are close to zero. Histograms of the point
estimates of the two betas are provided in Figure 2. Furthermore, adding factors in the proposed
setup requires additional parametric assumptions on their dynamics, and involves further model and
estimation error. Finally, traded derivatives’ prices would be needed to estimate the risk premia of
these factors. Such generalization would therefore be challenging to apply in practice.
One limitation of my model is that the beta coefficients are assumed to be constant in order to
preserve the affine structure of the setup. This assumption is, however, justified by the time-series of
data considered. Adrian and Franzoni (2009) show that the beta in its traditional sense varies over
time when looking at a long time-series of data, typically including several decades. The advantage
of such exercise is to dispose of a sufficient amount of data points. However, various issues such
12
as the non-stationarity of the data may arise in this case. As I work with options, the amount of
available data is already very large and is sufficient to draw inferences for the recent years covering
the financial crisis, while keeping the betas constant over this time horizon.
3.2
Risk premia specification
Due to incompleteness of the market, the risk-neutral measure is not unique. Following standard
practice, I parameterize the pricing kernel Lt defined in equation (6), which is then calibrated to the
market. The corresponding risk-neutral measure is denoted by Q. Lt can be decomposed into two
components which respectively contain the market prices of risk of the diffusive and jump processes
that play a role in the dynamics of assets returns:
−rt
Lt = e
dQ f jump
= e−rt Ldif
Lt
.
t
dP Ft
(6)
The diffusive component of the pricing kernel is equal to
Z
f
Ldif
t
= exp
t
Λ0s dWs
0
1
−
2
Z
t
Λ0s Λs ds
0
where Wt is the vector of diffusive random processes that drive asset returns at time t and Λt is
the vector of premia attached to every element of Wt , assumed to be proportional to the volatility
levels:
Wt0 =
Λ0t =
M (P)
Wt
ηM
¯ M (P) , W j(P) , B j(P)
, B
t
t
t
q
p
p
M
(P)
M
(P)
κM
κM
j
M
M
v −κv
m −κm
vt ,
mt , η vtj ,
σM
σM
M (P)
, Bt
p
vtM ,
v
m
j(P)
κjv −κv
σvj
q vtj .
(7)
The parameters without superscript (P) in equation (7) either correspond to the Q-counterpart of
the P-parameters, or are invariant under both measures. This specification allows for non-zero equity
and variance risk premia in the market factor and in individual stock returns. The affine structure
of the model is preserved under Q, so that options can be efficiently priced using Fourier methods.
13
The jump component Ljump
accounts for different mean and variance of jumps under P and Q. For
t
the sake of identification, the intensity of jumps is assumed to be equal under both measures:
M (P)
Z t Z
Y φM (Q) (Z M )
n
M M
M (P)
M (Q)
=
exp
λ (vs , ms ) φ
(z) − φ
(z) dz ds
φM (P) (ZnM )
0
R
n=1
Nt
Ljump
t
j(P)
Nt
Y
Y φj(Q) (Znj )
j∈{1,...,J} n=1
φj(P) (Znj )
Z t Z
exp
0
λj (vsj ) φj(P) (z) − φj(Q) (z) dz ds
R
where J is the number of individual stocks considered. The functions φM (ZnM ) and φj (Znj ) refer to
the normal density function, with parameters equal to the mean and variance of jump sizes in the
market factor and single stock j, under the specified measure.
I make the assumption that the leverage and the beta coefficients are the same under the riskneutral and objective measures. The latter assumption is consistent with Serban, Lehoczky, and
Seppi (2008), who find estimates which are statistically close under both measures.
Under this specification of the pricing kernel, the market equity risk premium (ERP) coefficient γtM
present in equation (1) is equal to the sum of a diffusive contribution that is proportional to the
market variance level, and a jump contribution that reflects the different jump size distributions
under the measures P and Q:
M
M
M M
γtM = η M vtM
(vt− , mt ) EP [eZt − 1] − EQ [eZt − 1] .
− + λ
Similarly, the single stock’s ERP coefficient γtj present in equation (4) can be decomposed into a
diffusive and a jump contribution. Each of them is itself the sum of an idiosyncratic component and
j
j
a market component, whose impacts are quantified by βdif
f and βjump :
j
M M
γtj =η j vtj + βdif
f η vt
j
j
M
P βjump ZtM
− 1] − EQ [eβjump Zt − 1] .
+ λj (vtj− ) EP [eZt − 1] − EQ [eZt − 1] + λM (vtM
− , mt ) E [e
Under this specification of the change of measure, the dynamics of the market factor and individual
returns are given in Appendix C.
14
3.3
Properties of the model
Due to the affine structure of the model, option prices are available in closed-form and involve the
characteristic functions of returns, which are known up to the resolution of Ordinary Differential
Equations (ODEs). Assuming the existence of derivatives on the market factor, the characteristic
function of market returns is given in Bardgett, Gourier, and Leippold (2014). The one of single
stocks’ returns can be derived similarly.
Proposition 3.1. The Laplace transform of the single stocks’ log returns S˜tj = log Stj is exponential
affine in the factor processes:
h
i
˜
ΨS˜T (t, s˜t , vtj , vtM , mt ; ω) := EQ eωST |Ft
j
˜
M +χ(T −t)·m
= eα(T −t)·St +β(T −t)·vt +γ(T −t)·vt
t +η(T −t)
where the coefficients α, β, γ, χ and η are functions defined on [0, T ] by ODEs presented in Appendix
D and ω ∈ C is chosen so that the conditional expectations are well defined.
The availability and tractability of the Laplace transforms are desirable features, as they allow using
Fourier methods to calculate option prices in a fast and efficient manner.
Another important property of the model is that risk premia are available in closed-form. The
integrated equity risk premium (IERP) is defined as follows8 :
1
IERP (t, T ) =
T −t
j
"
EPt
log
STj
Stj
"
#
−
EQ
t
log
STj
Stj
#!
.
(8)
It represents the difference between the expected return made by an investor who buys stock j at
time t and keeps it until time T , under the empirical and risk-neutral measures. It is composed of
two parts. On the one hand, the diffusive part depends on the market price of idiosyncratic risk η j
and on the price of risk of the market factor, η M . On the other hand, the jump component depends
j j
j j S −S
ST −St
This definition is equivalent to the usual definition IERP j (t, T ) = T 1−t EPt T j t − EQ
up to the
j
t
St
St
"
#
"
#!
2
j j 2
j
j
ST −St
ST −St
convexity adjustment O EPt
− EQ
. In the following, the notation Et [.] is used in place of
j
j
t
8
St
St
E[.|Ft ].
15
on premia attached to the mean and variance of the jump sizes.
The integrated variance risk premium (IVRP) reflects the amount that an investor would be willing
to pay to be hedged against the fluctuations in an asset’s variance. It is defined as follows:
IV RP j (t, T ) =
i
h
i
1 Ph
j
j
Q
Et QV[t,T
−
E
QV
t
]
[t,T ]
T −t
(9)
where QV[t,T ] denotes the quadratic variation of the log price process, i.e., the sum of the integrated
variance of returns and the squared jumps in the time interval considered.
As a result of the affine property of the model, the IERP and IVRP are available in closed-form and
can be expressed as linear functions of the latent variance processes. Details on the calculation of
the IERP and IVRP are available in Appendix E.
Finally, the model is consistent with empirical facts highlighted in the literature on the relationship
between the beta and risk-neutral moments of the conditional distribution of log-returns. Duan and
Wei (2009) find that firms with higher betas tend to have a high level of risk-neutral variance. In my
j
2 M
model, the risk-neutral variance rate can be decomposed into a diffusion part vtj− + (βdif
f ) vt− that
j 2
j j
M 2
M M
j
j
j 2
2
M 2
increases with βdif
f , and a jump part (σ ) + (µ ) λ (vt− )+(βjump ) (σ ) + (µ ) λ (vt− , mt )
j
that increases with βjump
. Appendix F shows that the total risk-neutral variance of stock j over
[t, T ] also increases in both betas.
Furthermore, Dennis and Mayhew (2002) and Duan and Wei (2009) find that higher beta firms have
more negatively skewed risk-neutral distributions. In my model and as detailed in Appendix F, the
skewness rate can be decomposed into a market effect and an idiosyncratic component. The market
component increases in absolute value proportionally to (βdif f )3 , and is negative. The idiosyncratic
component is closer to zero. Therefore my model reflects the findings of Dennis and Mayhew (2002)
and Duan and Wei (2009).
4
Estimation
I estimate the model using data from 2006 until to 2009, covering the financial crisis. This period
is subsequently referred to as the in-sample period. The model performance is examined on the
16
subsequent time period (out-of-sample period) spanning 2010 up to the end of 2012. Table 2 reports
the amount of options included in the in- and out-of-sample periods for every stock. As the number
of traded options tends to increase over time, for many stocks it is higher in the out-of-sample period
than in the in-sample period, which is a challenge for the model.
4.1
Estimation method
??
I estimate the model in two steps. The first step consists in estimating the parameters of the
market equations (1) - (3). I reconcile the dynamics of the S&P 500 and VIX indices following the
estimation procedure outlined in Bardgett, Gourier, and Leippold (2014). The affine characteristics
of the model allows me to use the Fourier Cosine method introduced by Fang and Oosterlee (2008)
to calculate the prices of VIX options. My procedure is based on the Auxiliary Particle Filter of Pitt
and Shephard (1999) which, combined with a Maximum Likelihood estimation, yields time-consistent
parameters as well as estimates of the distributions of unobservable processes over time and their
jumps.9 Given the parameters and latent processes driving the market factor 10 , the second step uses
a similar algorithm to estimate the parameters and filter the processes that drive every individual
stock. In this second step, all estimations can be run in parallel, making the procedure particularly
efficient. I cast the problem in a state-space form, which consists of a transition equation and three
measurement equations. The transition equation is an Euler discretization of the SDEs satisfied by
the variance process of the individual stock prices under P, on a uniform time grid of M + 1 points
t ∈ {t0 = 0, t1 = ∆t, ..., tM = M ∆t}, M ∈ N? so that M ∆t corresponds to the end of the period
considered. For t ∈ {0, ..., M − 1}:
∆vtj
=
βvj ∆vtM
+
κj(P)
v
κjv
j(P)
κv
!
j
θ −
vtj
∆t +
σvj
q
j(P)
vtj ∆Bt
j(P)
+ yt
j(P)
∆Nt
.
The measurements consist of log-returns, option prices and daily values of the RV. The first measurement equation is obtained using the Euler discretization of the dynamics of log-returns. The second
9
This procedure allows obtaining at each point in time an estimate for the density of the latent factors, consistent
with the specified stochastic differential equation (SDE). In that sense it provides a significant improvement over the
typically used least-square estimation, see e.g, Christoffersen, Fournier, and Jakobs (2013), where the estimation treats
the unobservable factors as parameters.
10
At every time t, I use the empirical mean of the filtered density of these factors as point estimate.
17
matches the logarithm of model-implied option prices to market data. It is a vector equation. Its
dimension is equal to the number of options that are traded at time t. The error term is assumed to
follow a normal distribution centered at 0, with a heteroskedastic variance that is exponential affine
in the bid-ask spread, the log-moneyness of the option and its time-to-maturity. The last measurement equation expresses the variance rate as an affine function of the realized variance, following
Wu (2011). The innovations are assumed to be normally distributed centered at zero, with variance
proportional to vtj to account for the fact that the error made when estimating the variance tends to
increase with the variance level.
The parameters driving the jump distribution under the empirical measure were found hard to
estimate precisely. Therefore, they were estimated in a preliminary investigation of high-frequency
returns and fixed throughout the particle filtering exercise. The algorithm used to estimate them is
outlined in Appendix B. Estimated intra-day jumps were aggregated on a daily basis, so that the
empirical mean and standard deviation of daily jumps could be calculated.
4.2
Estimation results
The estimation performance is evaluated by examining root mean-square errors (RMSEs) and relative
errors, as well as the evolution of sequential likelihoods in- and out-of-sample. RMSEs are reported
in Table 3. For the sake of conciseness, the other performance measures are not reported but are
available upon request. There is a slight loss of quality in the estimation of deep OTM options.
This is not surprising as the tails of the risk-neutral distribution are typically hard to estimate.
In addition, short-term options tend to be better fitted than longer-term options. This could be
improved by making the mean reversion level of the idiosyncratic variance stochastic.
Parameter estimates as well as their standard errors are displayed in Table 4. The mean size of return
jumps under P is very close to zero, which is in line with Yan (2011). Under the risk-neutral measure,
it tends to be more negative, consistent with a positive jump equity risk premium. Furthermore,
the risk-neutral speed of mean reversion of the variance process is always significantly lower than its
empirical counterpart, which pulls the variance risk premium towards negative values.
j
The beta coefficients are reported in Table 5. Stocks are grouped by industry. The diffusive beta βdif
f
varies between 0.450 for Procter & Gamble and 1.221 for Goldman Sachs. Companies specialized in
18
j
financial services have the highest values of βdif
f . As those firms were at the core of the financial
crisis, this result was expected. Note that they are not the companies with the largest market
capitalizations, as could be expected because the S&P 500 is capitalization-weighted. For example,
j
Johnson & Johnson (JNJ) has a low βdif
f = 0.497 but one of the largest market capitalizations.
Companies specialized in consumer staples, such as The Coca-Cola Company (KO), tend to have a
relatively low exposure to diffusive market movements.
The exposure of a stock to the market is directly linked to its risk-neutral variance, as Figure 3
illustrates. As a model-free measure of the risk-neutral variance, I use the two-month at-the-money
(ATM) IV. Its evolution is displayed for three stocks on the left panel of the figure, over the time
period March 2006 - December 2012. The estimated diffusive betas of these stocks are respectively:
0.497 for JNJ, 0.670 for IBM and 1.221 for GS. The graph clearly indicates that the overall level of
j
IV increases with βdif
f . In the absence of jumps, my model specification implies that the risk-neutral
j
variance is proportional to the square of βdif
f . The right graph of Figure 3 represents the median
j
of the two-month ATM IV versus the estimated value of βdif
f squared across stocks. As expected,
the graph indicates a monotonic relationship between the two quantities. I do not find any evidence
j
that the risk-neutral variance is increasing in the jump exposure coefficient βjump
, which is justified
by the fact that the variance is mainly driven by its continuous part.
Similarly, I examine the relationship between the returns’ implied skewness and the betas. The left
graph of Figure 3 suggests that the magnitude of the IV peaks in times of crisis also increases with
the coefficient βdif f . As a model-free proxy of the risk-neutral skewness, I use the average difference
between the 95% and the 5% quantiles of the two-month ATM IV. Figure 4 represents this value for
3
every stock versus the point estimate of βdif
f . This graph indicates a monotonic relation between
the two quantities, in agreement with theory.
4.3
Out-of-sample performance: test on variance swaps
As a consequence of the affine structure of the model, variance swap (VS) rates are available in closed
form. The rate with term τ for company j is given by:
VSj,model
t,τ
=
EQ
t
t+τ
Z
VarRate (u)du
j
t
19
(10)
where the variance rate is given by:
j
2 M
VarRatej (t) =vtj− + (βdif
f ) vt−
j
)2 (σ M )2 + (µM )2 λM (vtM
+ (σ j )2 + (µj )2 λj (vtj− ) + (βjump
− , mt ).
4.3.1
(11)
Out-of-sample pricing
To evaluate the out-of-sample performance of the model, I compare synthesized VS rates with maturity one month to model-implied rates. Every week I synthesize VS using a portfolio of OTM
options as described in Carr and Wu (2009). The resulting VS rates are denoted by VSj,data . The
replication procedure assumes that the underlying stock does not jump. If this assumption is not
satisfied, the error is dependent on the compensator of the jump component. Therefore, to make
the two quantities comparable, I only consider the continuous part of the model-implied VS rate.
I perform a linear regression with the synthesized one-month VS rate on the left hand-side and
its model-implied counterpart on the right hand-side. The error term includes different potential
sources of discrepancies between the two quantities. First, the replication involves interpolation and
extrapolation procedures that take as input a finite number of options with different maturities and
moneynesses. The amount of available options varies from week to week, and yields as output the
implied volatilities of options defined on a grid of moneynesses, with maturity one month. The
quality of the replication depends on the number of options with maturity close to one month that
are available on a specific date, and on the range of their moneynesses. Second, there is inherent
noise in the data due, for example, to the bid-ask spread, that is further reflected in the error term.
Finally, the model is calibrated to a large amount of datapoints, and the estimation is clearly subject
to model specification error as well as estimation error. The intercept should be zero if there is no
systematic bias in the estimation. The slope parameter reflects the ability of the model to capture
the amplitude of the movements of VS rates, it should be one if the model perfectly replicates the
magnitude of the data on average. Non-systematic errors will be reflected in the error term.
I consider two types of regressions in Table 6: repeated cross-sectional regressions (Fama-MacBeth)
and a pooled panel regression. For the cross-sectional regressions, I report the time-series mean
of the coefficient estimates and the R2 . The standard errors are calculated using the Newey-West
procedure to account for autocorrelation, with eight lags. Cluster-robust standard errors are used in
20
the panel regression following the method of Cameron, Gelbach, and Miller (2011), to account for
cross-sectional correlation and time-series auto- and cross-correlations.
Both cross-sectional and panel regressions yield estimates of the intercept that are close to zero,
which indicates that the model does not present a bias in its representation of one-month VS rates.
Estimates of the slope parameter are both slightly smaller than 1. Therefore the model tends to
underestimate the magnitude of VS rates. The hypothesis that the model correctly prices VS, i.e.,
that the intercept is zero and the slope coefficient one, has a p-value of 96% when the regression
is run every week, and 87% when it is run for the whole panel at once. The average R2 from the
cross-sectional regressions is 82%, it goes down to 70% in the panel regression. In summary, the
model fits suynthesized VS rates remarkably well.
4.3.2
Forecasting
I further evaluate the out-of-sample performance of the model in forecasting VS rates. The modelimplied conditional expectation of a time-u VS rate is EPt [VSu ]. I regress the ex ante VS rate at time
u on this expectation, using cross-sectional and panel regressions. I further challenge the model by
comparing the predictions it delivers to a prediction based on the current VS rate at time t. The
latter method is referred to a as martingale prediction, it assumes that the VS rates follow a random
walk, in which case the best guess of future rate that it gives is the rate of today. Given the high
persistence in VS rates dynamics, the martingale prediction is a rather challenging benchmark. If
the forecast is accurate, the intercept of the regressions should be zero and the slope parameter one.
I test this hypothesis and compare the p-values delivered by both prediction methods, for different
future times u.
Table 7 reports the results of the regressions for three values of u − t: one week, one month and one
month and a half. The R2 of the regression using the model prediction is comparable to the one
using the martingale prediction, for all values of u − t. As expected, the R2 decreases with u − t
and is higher for the cross-sectional regressions than for the panel regressions. The p-value of the
null hypothesis is close to 1 for u − t equal to one week with the model prediction. The martingale
prediction, in turn, yields a p-value of only 0.10 in the panel regression. For u−t equal to a month and
six weeks. the p-values of the cross-sectional regressions remain high, around 0.8, for both the model
and the martingale prediction. The panel regressions, in turn, highlight the better performance of
21
the model predictions, with a p-value of 0.19 against 0.05 for u − t equal to one month and of 0.24
against 0.03 for u − t equal to six weeks. To summarize, the model accurately predicts VS rates and
outperforms a martingale model.
5
5.1
Idiosyncratic variance
Descriptive analysis
The total variance swap rate at time t, given by equation (11), decomposes into a diffusive part vtj− +
j
j
2 M
j 2
j 2 j j
2
M 2
(βdif
f ) vt , which is measure-independent, and a jump part [(σ ) +(µ ) ]λ (vt− )+(βjump ) [(σ ) +
(µM )2 ]λM (vtM
− , mt ), which is different under P and Q because of the jump risk premia. Under P, the
diffusive variance represents on average 86% of the total variance rate, and 85% under Q.
Figure 5 displays the decomposition of the diffusive and jump parts of the variance rate into an
idiosyncratic component and a component that comes from the effect of the market factor, for a
typical stock. The impact of the market factor is very low in times of low market volatility, i.e. at
the beginning of the sample period. The variance rate is then almost solely driven by the idiosyncratic
variance of the firm, which is typically much higher than the market variance due to the averaging
effect in the market portfolio. In turn, during volatility peaks, the market factor has a stronger
j
influence on the assets’ total variance, which is scaled by the βdif
f parameter. For a typical stock,
the influence of the market factor is of the same order of magnitude as the idiosyncratic variance
during these peaks, causing the variance to be multiplied by a factor of approximately 2. The
decomposition of the jump part of variance rates follows a similar pattern. The main difference
however is that the market component usually dominates the idiosyncratic component after the first
volatility peak.
The preliminary data analysis highlighted a strong common component in single stocks’ variances.
This phenomenon is also striking for their idiosyncratic components. Applying PCA to the total
instantaneous variance rate and its idiosyncratic component, I find that the first principal component
explains more than 92% of the variation of the total variance rate, and the first three components
more than 97%. The numbers are only slightly lower for the idiosyncratic variance, as the first
component explains 81% of its variation and the three first components more than 92%. These
22
results are generally in line with Herskovic, Kelly, and Lustig (2014)11 .
The analysis of Herskovic, Kelly, and Lustig (2014) further suggests that idiosyncratic variance varies
with the size of the company and the industry it belongs to. As the methodology I use to estimate
the idiosyncratic variance is completely different from theirs, it is worth revisiting their results. To
analyze the influence of the company’s size on its idiosyncratic variance, I sort the considered firms
by their market capitalization as of end of 2012 and divide them into five buckets. Into each of
these buckets, I calculate the average total diffusive and jump variance rates across time. Taking the
variance rate of the bucket of firms with the smallest market capitalizations as a reference process,
I regress the time-series of the average variance rates in the other size buckets on the reference
time-series. Table 8 summarizes the results of the regressions. They indicate that firms with the
largest market capitalization have a smaller variance rate on average. Exceptions are companies that
suffered more from the financial crisis, such as GE, which presents higher variance rates than the
other companies in the bucket.
I conduct the same analysis on the diffusive idiosyncratic variance rates and find the same pattern,
as shown in Table 9. As parameter estimates do not seem to indicate that companies of smaller size
tend to have a larger exposure to the market factor, I conclude that the idiosyncratic variance is
responsible for the fact that the total variance rate decreases when the size of the company increases.
This ordering does not hold anymore for the true idiosyncratic variance, which pinpoints the common
idiosyncratic variance as the determinant of this phenomenon.
Herskovic, Kelly, and Lustig (2014) suggest that the industry also has an impact on the idiosyncratic
variance. To examine this assertion, I repeat the estimation exercise using data which are industry
related. SPDR Exchange traded funds (ETF) were introduced in 1998 and track the performance of
the Global Industry Classification Standard sectors of the S&P 500. Every fund is a portfolio of the
components of the index in the corresponding sector. American options started trading on the funds
and knew a fast-growing success. A summary of the data available on sector ETFs is provided in
Table 10. All options are of American type; the dataset is treated similarly to the options on single
stocks.
Figure 6 represents the evolution of the total idiosyncratic variance v j , as estimated by the particle
11
Herskovic, Kelly, and Lustig (2014) use a completely different methodology to obtain these results. They calibrate
a factor model to returns and compute the idiosyncratic variance as the variance of the residuals.
23
filter, for each sector considered and the stocks of the Dow Jones that belong to this sector. The
variance levels are compared to the trajectory of the market factor. Until the end of 2007, most
sectors have idiosyncratic variances which are comparable to the one of the market factor, between 0
and 3%, and lower than the one of the considered stocks. The variance of a portfolio is composed of a
weighted average of the individual variances, where the weights are equal to the square of the weights
that form the portfolio, and a term that corresponds to the covariances. If the correlation between
the stocks in the portfolio is not too large, the covariance term is close to zero, and it is common
that the portfolio’s variance be smaller than the individual variances. The Energy and Materials
sectors distinguish themselves from others because they have higher idiosyncratic variances, around
8%. As the variance tends to increase when the market capitalization of the stock decreases, it
is reasonable to assume that most stocks of the S&P 500 that have not been represented because
they do not belong to the Dow Jones, have larger levels of idiosyncratic variance that the stocks
considered which are in the same sector. In the Energy sector, the two stocks considered, Exxon
Mobil Corp (XOM) and Chevron Corp (CVX), represent around 30% of the holdings of the ETF.
DuPont (DD) represents around 10% of the holdings of the Materials sector’s ETF. Therefore, the
large idiosyncratic variance of the Enery and Materials sectors in 2006 and 2007 may be explained
by a larger variance of the smaller stocks in the portfolios. At the end of 2007, the variance of
the Financials sector starts increasing swiftly, and reaches 15% mid 2008. Such early increase is
specific to that sector and does not occur in any other sector. During the market variance peak, all
sectors are exhibit increases in idiosyncratic variance. However, the Consumer Staples, Industrials
and Health Care sectors are only marginally affected. The maximum level they reach is around 10%.
The idiosyncratic variance of the Materials sector peaks around 20%. For the Energy, Technology
and Consumer Discretionary sectors, the maximum is around 30%. The Financial sector stands out
with a maximum idiosyncratic variance that is close to the highest variance of the market factor,
above 50%. This level is also close to the maximum level of idiosyncratic variance reached by the
single stocks considered. Another striking pecularity of the Financials sector is that its idiosyncratic
variance reaches a first peak end of 2008, simultaneously with the market variance, then goes down
to 25% in early 2009, when the market variance goes down below 15%, but then it goes up again
and reaches a peak that is similar to the first one around March 2009. This second peak is specific
to the Financials sector. It occurs at a much smaller scale in the market variance, which goes up to
20%, and is insignificant in the variance trajectories of the other sectors. The only individual stocks
24
that witness this second peak are the stocks in the Financials sector as well as General Electric
(GE) and UnitedHealth Group Inc. (UNH), which belong to the Industrials and Health Care sectors
respectively. An explanation for the peculiar behaviour of GE is that it used to have a consumer
lending division, and was therefore exposed to the risks specific to the financial industry. After the
main market variance peak, the Energy and Materials sectors revert to an idiosyncratic variance
level that is higher than the one of the market, as at the beginning of the times series. It is also the
case for the Financials sector.
Figure 7 illustrates the decomposition of the evolution of idiosyncratic variance into true idiosyncratic
increments and market-induced increments. Two possible cases are observed. In both cases, the
idiosyncratic movements control the evolution of the variance in low volatility periods. The difference
occurs in high volatility times. In the first case, the idiosyncratic movements have the same magnitude
as in calm times, then the variance peak comes from the impact of the market. This is the case of
CAT (Caterpillar Inc.) for example, and for most stocks examined. In the second case, when the
volatility peak does not replicate the one of the market, the true idiosyncratic variance remains
in control of the total idiosyncratic variance. This is the case for GE (General Electric), and for
companies in the financial industry such as JPMorgan Chase and Goldman Sachs. For these stocks,
the volatility peak lasts longer than the one of the market, and therefore the idiosyncratic variance
takes over the entire peak.
5.2
Relation to returns
According to standard asset pricing theory, idiosyncratic volatility should not be priced in the crosssection of returns, as investors should be able to diversify this risk away. However, in the last decade
many papers have found a relation between idiosyncratic volatility and expected returns.12
12
This phenomenon is sometimes referred to as the ”volatility anomaly”. There has been a debate on the sign of this
relation. Ang, Hodrick, Xing, and Zhang (2006), Brockman and Yan (2008), Ang, Hodrick, Xing, and Zhang (2009),
Jiang, Xu, and Yao (2009) and Guo and Savickas (2010) find an anomalous negative relation. In contrast, Malkiel
and Xu (2001), Goyal and Santa-Clara (2003), Spiegel and Wang (2005), Fu (2009) and Garcia, Mantilla-Garcia, and
Martellini (2011) find a positive relation. A number of studies also find that idiosyncratic volatility is not a priced
risk factor, e.g., Fama and MacBeth (1973), Bali, Cakici, Yan, and Zhang (2005), Bali and Cakici (2008), Fink, Fink,
and He (2012), Huang, Liu, Rhee, and Zhang (2010) and Han and Lesmond (2011). A few explanations have been
proposed in the literature to explain the potential pricing of idiosyncratic risk in the cross-section of returns. During
market turmoil, the assumption of perfectly liquid markets that underlies classical asset pricing theories no longer
holds. Furthermore, agents may exhibit specific preferences with respect to the skewness and kurtosis of the returns’
distribution. Langlois (2013) argues that such preferences may incentivise investors to remain underdiversified, leading
to a non zero price of idiosyncratic risk. Berrada and Hugonnier (2013) suggest that idiosyncratic volatility is generated
by aggregated forecast error, in an incomplete information setting.
25
My model permits to analyze the relationship between idiosyncratic volatility and returns. To
illustrate how the knowledge of the idiosyncratic volatility can help a portfolio manager selecting the
assets that are relevant to a specific investor, I examine the performance of four weekly-rebalanced
portfolios. Every week, I sort the values of idiosyncratic variances. Each portfolio contains a quarter
of the stocks. Portfolio 1 contains the seven stocks which have the smallest idiosyncratic variance
level, portfolio 4 contains those that have the largest variance levels. Stocks are given equal weights.
Figure 8 illustrates the evolution of the wealth for the four portfolios. I assume an initial level of
wealth of 100. Table 11 reports the main statistics of the portfolios. An investor who buys a portfolio
of stocks with a higher idiosyncratic variance tends to receive higher returns, and a larger Sharpe
ratio. The results remain unchanged if one sorts portfolios according to the idiosyncratic variance
rate (which also includes the idiosyncratic jumps in returns), the ratio of idiosyncratic variance over
total variance or the level of true idiosyncratic variance. They are also robust to a different weighting
mechanism, where the position in each stock is proportional to the level of its variance.
6
Pricing of idiosyncratic risks
The results of the last section suggest that idiosyncratic variance may contain useful information
on future returns. This section examines whether idiosyncratic risk is priced in the cross-section of
assets, by evaluating the contribution of idiosyncratic equity and variance risks to the equity and
variance risk premia. The risk premia are calculated following equations (8) and (9). They are
available in closed-form at every time t conditionally on the values of the latent variance factors.
These values are estimated following the procedure outlined in section ??.
6.1
Equity Risk Premium
The typical behaviour of the Integrated Equity Risk Premium (IERP) is illustrated in Figure 9,
which represents the case of DD (DuPont). It is positive for all stocks and of smaller magnitude
than the risk premium of the market. Its term structure is similar to the one of the index, i.e., its
magnitude increases when the term increases during market calm, indicating that investors require a
larger compensation when they invest on a longer-term horizon. In contrast, during market turmoil,
the IERP becomes larger for smaller terms, as investors are unsure how markets will behave in the
26
near future and require compensation for that. The decomposition into an idiosyncratic and market
component for a short-term horizon investment varies from stock to stock. For some stocks such
as Boeing, idiosyncratic risk is not priced in the IERP, which suggests that investors think this
risk can be diversified away. For other stocks like DuPont, the idiosyncratic component of the IERP
constitutes an important part of the total IERP. However, for investments with longer time horizons,
results are consistent across stocks and indicate that idiosyncratic risk is no longer priced.
Figure 11 represents the one-month IERPs of sectors and compares them to the IERP of the Dow
Jones components which belong to them. The sector IERP is smaller than the single stocks’ IERPs
for all sectors but Energy and Financials. Furthermore, these two sectors are characterized by a
large IERP even in times of market calm (around 5% in 2006). The computation of the ratio of
idiosyncratic IERP over total IERP shows that the idiosyncratic IERP is responsible for a large
part of the total premium for the Energy and Financials sectors: 86% and 83% respectively. It
is also responsible for a substantial part of the Consumer Discretionary (XLY), Industrials (XLI)
amd Health Care (XLV) sectors: 62%, 38% and 23% respectively. Looking one level below and
separating the jump and diffusion contributions to the idiosyncratic IERP reveals that jumps only
play a non negligible role for the Energy and Financials sectors, where they represent 31% and 51%
of the total idiosyncratic component for the IERP. Investing in ETFs for all sectors except Energy
and Financials is therefore a way for investors to be exposed to less diversifiable risk than what they
would otherwise have to bear if investing in single stocks. Speculators who, in turn, would like to
be exposed to idiosyncratic risk, and particularly to tail risk, can invest in the Energy or Financials
ETFs, whose idiosyncratic IERP is larger than most of their components. These results should be
put in perspective with those of Kelly and Lustig (2011) who show that government guarantees
removed part of the tail risk of the financial sector. They show that there is still substantial tail risk
that remains priced.
6.2
Variance Risk Premium
A typical example of the Integrated Variance Risk Premium (IVRP) is represented in Figure 10. It
tends to be negative, which means that investors are willing to pay to get protection against the
variance fluctuations in most stocks. The amplitude of the IVRP varies very much from stock to
stock. The pattern is similar across stocks, i.e., the IVRP is small in absolute value in times of
27
market calm, and peaks downward during market turmoil. The term structure is similar to the one
of the market factor, i.e., the VRP becomes more negative when the term increases. This indicates
that investors are willing to pay larger amounts to be covered against fluctuations in stock volatility,
when they invest for longer terms. Such term structure can be benefited from in a trading strategy,
as detailed in Filipovi´c, Gourier, and Mancini (2013). Note that my results do not contradict those
of Driessen, Maenhout, and Vilkov (2009) who find no evidence of a negative risk premium on
individual variance risk. Indeed, they are considering relatively short terms, from 14 to 60 days, and
have option data from 1996 until 2003, i.e., prior to the volatility peak. In contrast to the IERP, a
substantial part of the IVRP consists of its idiosyncratic component, for both short and long terms,
as shown on the lower panels of Figure 10.
Figure 12 represents the six-month IVRPs of sectors and compares them to the IERP of the Dow
Jones components which belong to them. The six-month term is chosen as the magnitude of the
IVRP increases with the term. Results are qualitatively similar for the one-month term. This analysis
confirms that the Energy and Financials sectors are particularly risky sectors, because their variance
risk premia are strongly negative. The part that corresponds to idiosyncratic risk is very high: 90
and 91% respectively. The jump component does not play a significant role in the variance risk
premia. It accounts for 7% and 5% of the total variance risk premia of the Energy and Financials
sectors. Two other sectors exhibit significant priced variance risk, namely the Materials and the
Consumer Discretionary Sectors. 84% and 89% of their variance risk premia is idiosyncratic. If I go
one level further and decompose the continuous part of the idiosyncratic variance risk premia into a
component that is due to the common idiosyncratic variance, controlled by βvj , and another due to
the true idiosyncratic variance, I obtain the results displayed on Figure 13. Interestingly, the common
idiosyncratic variance carries a risk premium that tends to be close to zero in times of market calm,
and significantly positive during market turmoil. This result is in line with the results of Sch¨
urhoff
and Ziegler (2011). However, I find that a large part of the continuous idiosyncratic IVRP is driven
by the true idiosyncratic variance: between 60% and 97% for the four sectors mentioned above. This
justifies the resulting negative risk premia that I obtain.
28
7
Conclusion
This paper revisit the question: does idiosyncratic risk matter? It introduces a flexible parametric framework for the study of the risks embedded in individual stocks. Using a particle filtering
algorithm, I estimate the trajectory of idiosyncratic variance, based on the information contents of
option prices and returns at different frequencies. I investigate the decomposition of the idiosyncratic variance movements into common idiosyncratic and true idiosyncratic changes. I show that
idiosyncratic variance is dependent on the size and industry of the stock, and that it is a priced factor
of risk. A thorough analysis of the equity and variance risk premia of individual stocks uncovers
a substantial idiosyncratic component, in contrast to what standard asset pricing models predict.
Therefore idiosyncratic risk matters, and must be taken into consideration by investors when forming
their portfolios. Finally, I compare the risk present in individual stocks to the risk that is in the
sectors they belong to. This analysis is based on the information contained in sector ETFs and
derivatives on them. I show that most sector ETFs provide an interesting investment alternative
for agents who are not willing to be exposed to non diversifiable equity risk, as their idiosyncratic
risk premia are close to zero. Two notable exceptions are the Energy and Financials sectors, which
provide significant exposure to idiosyncratic equity risks, and include substantial tail risk. In addition to the two aforementioned sectors, the Materials and Consumer Discretionary sectors contain
substantial priced variance risk. This risk is driven mainly by the true idiosyncratic variance, which
carries a large negative premium. In contrast, the common idiosyncratic variance is shown to carry
a risk premium that is close to zero in times of market calm, and significantly positive during market
turmoil.
29
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33
Figures and Tables
Figure 1: Variance explained by the first principal component in the PCA of high-frequency measures
of returns’ variation of individual stocks’. The measures considered are the Realized Variance (RV),
the Bipower Variation (BV) and the Continuous Variation (CV) on the left panel, and the Jump
Variation on the right panel. They are computed based on 5-minute tick data.
1
1
0.95
0.8
0.9
0.85
0.6
0.8
0.4
0.75
RV
BV
CV
0.7
0.65
0
50
100
Lag (days)
0.2
0
0
150
50
100
Lag (days)
150
7
14
6
12
5
10
beta HML
beta SMB
Figure 2: Histograms of the betas associated with the SMB and HML factors in the Fama-French
three-factor model.
4
3
8
6
2
4
1
2
0
−8
−6
−4
−2
0
2
0
−0.01
4
−3
x 10
34
0
0.01
0.02
0.03
Figure 3: Left panel: Evolution of the two-month at-the-money (ATM) level of the implied volatility
(IV) of single stock options from January 4, 2006 until December 26, 2012. The ATM IV has been
interpolated on the basis of available market data, using the biharmonic spline interpolation method
of T.Sandwell (1987). Right panel: Median level of interpolated two-month ATM IV from March 3,
j
2
2006 until December 26, 2012, versus the point estimate of the parameter (βdif
f ) , which measures
the impact of diffusive market movements on the returns of stock j.
1.4
JNJ
IBM
GS
1.5
GS
AXP
1
j
2
(βdif
f)
2−month ATM IV
1.2
0.8
0.6
1
CAT
GE
DIS
DD BA
JPM
MSFT
NKE
HD
0.5
0.4
0.2
JNJ
Jan−08
Jan−10
0
0.1
Jan−12
TRV
CVX
UTX
XOM
MRK
MMM
PFE
WMT
MCD
IBM
VZ
T
KO
0
INTC
PG
0.15
0.2
0.25
0.3
median of 2-month ATM IV
0.35
Figure 4: Difference between the 95% and the 5% quantile of interpolated two-month ATM IV from
j
March 3, 2006 until December 26, 2012, versus the point estimate of the parameter βdif
f (left panel)
j
j
3
resp. (βdif
f ) (right panel). βdif f measures the impact of diffusive market movements on the returns
of stock j.
1.4
2
GS
1.2
GS
1.5
AXP
CAT
GE
DIS
BADD
0.8
0.6
JPM
MSFT
NKE
HD
INTC
AXP
j
3
(βdif
f)
j
βdif
f
1
1
CAT
GE
DIS
BADD
TRV
0.5
T
CVX
UTX
KOXOM
PFE
MMM MRK
WMT
MCD
IBM
VZ
PG
JNJ
JPM
MSFT
NKE
HD
INTC
T
CVX
UTX
KOXOM
PFE
MMM MRK
WMT
MCD
IBM
VZ
PG
JNJ
0.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7
95% minus 5% quantile of 2-month ATM IV
TRV
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
95% minus 5% quantile of 2-month ATM IV
35
Figure 5: Decomposition of the evolution of the diffusive and jump components of the variance rate
(Total var. rate) of INTC (βdif f = 0.75, βjump = 1.35) into an idiosyncratic component (Idio. part)
and a component that results from the impact of the market factor (Market part).
0.12
Total var. rate
Idio. part
Market part
0.6
0.4
0.2
Total var. rate
Idio. part
Market part
0.1
Jump variance rate
Diffusive variance rate
0.8
0.08
0.06
0.04
0.02
0
0
Jan−08
Jan−10
Jan−12
Jan−08
36
Jan−10
Jan−12
Figure 6: Total idiosyncratic variance of every sector versus the variance of its components that are
part of the Dow Jones. The sector ”Utilities” has not been represented as no constituent of the Dow
Jones is part of it.
Energy
Technology
0.7
0.7
Market
XLE
XOM
CVX
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
2007
2008
2009
2010
2011
Market
XLK
MSFT
VZ
IBM
T
INTC
CSCO
0.6
0
2012
2007
2008
Materials
2012
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
2007
2008
2009
2010
2011
Market
XLP
PG
KO
WMT
0.6
0.5
0
2012
2007
2008
ConsumerDiscretionary
2009
2010
2011
2012
Industrials
0.7
0.7
Market
XLY
DIS
HD
MCD
NKE
0.6
0.5
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
2007
2008
2009
2010
2011
Market
XLI
GE
UTX
MMM
BA
CAT
0.6
0.4
0
2012
2007
2008
HealthCare
2009
2010
2011
2012
Financials
0.7
0.7
Market
XLV
JNJ
PFE
MRK
UNH
0.6
0.5
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
2007
2008
2009
2010
2011
2012
Market
XLF
JPM
AXP
GS
TRV
0.6
0.4
0
2011
0.7
Market
XLB
DD
0.6
0
2010
ConsumerStaples
0.7
0
2009
37
0
2007
2008
2009
2010
2011
2012
Figure 7: Decomposition of the evolution of the idiosyncratic variance of two stocks (CAT on the
left and GE on the right) into true idiosyncratic movements and common movements.
Total id. var
True id. var
Market var
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
Jan−08
Jan−10
Total id. var
True id. var
Market var
0.5
vˆt
vˆt
0.5
0
Jan−12
Jan−08
Jan−10
Jan−12
Figure 8: Evolution of the wealth process of four portfolios that are weekly rebalanced depending
on the level of idiosyncratic variance (Panel A) and of true idiosyncratic variance (Panel B) of each
stock. Portfolio 1 contains every week a combination of the 7 stocks with the smallest idiosyncratic
variance. Portfolio 4 contains the 7 stocks with the largest idiosyncratic variance. Stocks are equally
weighted. The initial wealth level is 100.
Panel A
Panel B
200
1(smallest)
2
3
4 (largest)
1(smallest)
2
3
4 (largest)
Wealth
150
Wealth
150
200
100
50
100
Jan−08
Jan−10
50
Jan−12
38
Jan−08
Jan−10
Jan−12
Figure 9: Left panel: Term structure of the IERP for DD (DuPont), from March 3, 2006 until December 26, 2012. Right panel: Evolution of the instantaneous IERP and decomposition into idiosyncratic
and systematic movements from March 3, 2006 until December 26, 2012, for DD (DuPont).
0.2
∆t
1m
3m
6m
IERP(t,T)
IERP(t,T)
0.15
Total
Idiosyncratic
0.4
0.1
0.05
0.3
0.2
0.1
0
0
Jan−08
Jan−10
Jan−12
Jan−08
Jan−10
Jan−12
Figure 10: Upper panel: Term structure of the IVRP for DD (DuPont), from March 3, 2006 until
December 26, 2012. lower panels: Dcomposition of the instantaneous and 6-month IVRP into
idiosyncratic and systematic movements.
∆t
1m
3m
6m
0.04
IVRP(t,T)
0.02
0
−0.02
−0.04
−0.06
Jan−08
−3
5
x 10
Jan−10
Instantaneous IVRP
Jan−12
6−month IVRP
0.01
0
0
−0.01
−5
−0.02
−10
−0.03
−0.04
−15
Total
Idiosyncratic
−20
Jan−08
Jan−10
Total
Idiosyncratic
−0.05
−0.06
Jan−12
39
Jan−08
Jan−10
Jan−12
Figure 11: One-month total IERP of every sector versus the IERP of its components that are part of
the Dow Jones. The sector ”Utilities” has not been represented as no constituent of the Dow Jones
is part of it.
Energy
Technology
0.45
0.45
XLE
XOM
CVX
0.4
XLK
MSFT
VZ
IBM
T
INTC
CSCO
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
2007
2008
2009
2010
2011
2012
2007
2008
Materials
2009
2010
2011
2012
ConsumerStaples
0.45
0.45
XLB
DD
0.4
XLP
PG
KO
WMT
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
2007
2008
2009
2010
2011
2012
2007
2008
ConsumerDiscretionary
2009
2010
2011
2012
Industrials
0.45
0.45
XLY
DIS
HD
MCD
NKE
0.4
0.35
XLI
GE
UTX
MMM
BA
CAT
0.4
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
2007
2008
2009
2010
2011
2012
2007
2008
HealthCare
2009
2010
2011
2012
Financials
0.45
0.45
XLV
JNJ
PFE
MRK
UNH
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
2007
2008
2009
2010
2011
2012
XLF
JPM
AXP
GS
TRV
0.4
40
0
2007
2008
2009
2010
2011
2012
Figure 12: Six-month total IVRP of every sector versus the IVRP of its components that are part of
the Dow Jones. The sector ”Utilities” has not been represented as no constituent of the Dow Jones
is part of it.
Energy
Technology
−0.02
−0.02
−0.04
−0.04
−0.06
−0.06
−0.08
−0.08
−0.1
−0.1
−0.12
−0.12
−0.14
−0.14
−0.16
−0.16
XLE
XOM
CVX
−0.18
−0.2
XLK
MSFT
VZ
IBM
T
INTC
CSCO
−0.18
−0.2
−0.22
−0.22
2007
2008
2009
2010
2011
2012
2007
2008
Materials
2009
2010
2011
ConsumerStaples
−0.02
−0.02
−0.04
−0.04
−0.06
−0.06
−0.08
−0.08
−0.1
−0.1
−0.12
−0.12
−0.14
−0.14
−0.16
−0.16
−0.18
XLP
PG
KO
WMT
−0.18
−0.2
−0.2
XLB
DD
−0.22
2007
2008
2009
2010
2011
−0.22
2012
2007
2008
ConsumerDiscretionary
2009
2010
2011
−0.02
−0.04
−0.04
−0.06
−0.06
−0.08
−0.08
−0.1
−0.1
−0.12
−0.12
−0.14
−0.14
−0.16
XLI
GE
UTX
MMM
BA
CAT
−0.16
XLY
DIS
HD
MCD
NKE
−0.18
−0.2
−0.22
2008
2009
2010
2011
2012
Industrials
−0.02
2007
2012
−0.18
−0.2
−0.22
2012
2007
2008
HealthCare
2009
2010
2011
2012
Financials
−0.02
−0.02
−0.04
−0.04
−0.06
−0.06
−0.08
−0.08
−0.1
−0.1
−0.12
−0.12
−0.14
−0.14
−0.16
−0.16
XLV
JNJ
PFE
MRK
UNH
−0.18
−0.2
−0.22
2007
2008
2009
2010
2011
2012
XLF
JPM
AXP
GS
TRV
−0.18
−0.2
41
−0.22
2007
2008
2009
2010
2011
2012
Figure 13: Six-month idiosyncratic common IVRP of every sector versus the IVRP of its components
that are part of the Dow Jones. The sector ”Utilities” has not been represented as no constituent of
the Dow Jones is part of it.
Energy
Technology
0.05
0.05
XLE
XOM
CVX
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
−0.01
−0.01
−0.02
2007
2008
2009
2010
2011
XLK
MSFT
VZ
IBM
T
INTC
CSCO
0.04
−0.02
2012
2007
2008
Materials
2009
2010
2011
2012
ConsumerStaples
0.05
0.05
XLP
PG
KO
WMT
XLB
DD
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
−0.01
−0.01
−0.02
2007
2008
2009
2010
2011
−0.02
2012
2007
2008
ConsumerDiscretionary
2011
2012
0.05
XLY
DIS
HD
MCD
NKE
0.04
0.03
0.02
0.02
0.01
0.01
0
0
−0.01
−0.01
2007
2008
2009
2010
2011
XLI
GE
UTX
MMM
BA
CAT
0.04
0.03
−0.02
2012
2007
2008
HealthCare
2009
2010
2011
2012
Financials
0.05
0.05
XLV
JNJ
PFE
MRK
UNH
0.04
0.03
0.02
0.02
0.01
0.01
0
0
−0.01
−0.01
2007
2008
2009
2010
2011
2012
XLF
JPM
AXP
GS
TRV
0.04
0.03
−0.02
2010
Industrials
0.05
−0.02
2009
42
−0.02
2007
2008
2009
2010
2011
2012
Table 1: This table reports the results of the Functional Principal Analysis (FPCA) of the IV
surfaces of single stocks. The stocks considered are the constituents of the Dow Jones. The second
column describes what the top three eigenmodes (EM) of the decomposition represent. The next
three columns provide the percentage of variance explained by each of them. The last three columns
give the correlation between each principal component (PC) process the stocks’ FPCA and the
corresponding principal component process of the S&P 500 FPCA. All numbers are in percentages.
Top 3 eigenmodes
AXP
BA
CAT
CVX
DD
DIS
GE
GS
JPM
KO
T
XOM
HD
INTC
IBM
JNJ
MCD
MRK
MSFT
MMM
NKE
PFE
PG
TRV
UTX
VZ
WMT
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
level,
skewness, TS
TS, skewness
skewness, convexity
skewness, TS
skewness, TS
skewness, TS
TS, skewness
skewness, TS
TS, skewness
skewness, TS
skewness, TS
skewness, TS
skewness, TS
skewness, TS
skewness, TS
skewness, TS
skewness, TS
skewness, TS
TS, skewness
skewness, TS
TS, skewness
skewness, TS
skewness, TS
TS, skewness
skewness, TS
TS, skewness
skewness, TS
Var. explained by EM
1st EM 2nd EM 3rd EM
99.6
98.8
91.8
96.8
98.8
98.7
99.7
99.1
99.6
98.8
94.8
86.8
98.6
97.6
98
97.7
96
98.1
98.9
97.7
98.5
98.7
97.7
99.6
98.4
97.2
97.5
0.2
0.6
6.3
2.5
0.7
0.6
0.2
0.5
0.2
0.7
3.54
9.4
0.9
1.5
1.1
1.6
2.7
1
0.7
1.4
0.7
0.6
1.4
0.2
0.9
1.9
1.7
43
0.2
0.5
1.0
0.4
0.5
0.5
0.1
0.3
0.1
0.4
1.1
2.3
0.4
0.6
0.7
0.5
0.8
0.6
0.3
0.6
0.6
0.5
0.7
0.1
0.5
0.6
0.5
Correl. with PC of S&P 500 IV FPCA
1st PC 2nd PC
3rd PC
91.6
95.4
92.0
94.5
97.3
96.7
89.6
88.2
91.4
95.3
87.0
85.1
88.6
88.0
94.0
95.4
81.2
87.7
94.1
97.1
95.4
93.8
94.7
90.2
97.8
85.7
85.3
49.8
65.6
24.7
72.0
60.9
44.9
46.1
34.2
37.1
53.7
57.8
45.2
55.6
56.0
59.7
52.2
55.1
60.8
48.3
63.3
55.0
44.8
51.0
47.8
63.2
29.6
58.5
40.3
43.6
5.1
56.5
43.7
32.9
32.3
33.8
40.5
57.8
52.9
24.2
47.8
33.6
39.7
42.5
26.4
44.7
59.1
46.2
57.0
54.3
51.5
41.2
53.0
47.7
48.4
44
13770
2827
4913
2124
2342
1564
4078
5233
4459
All options
m ≤ 0.7
0.7 < m ≤ 0.95
0.95 < m ≤ 1.05
1.05 < m ≤ 1.2
m > 1.2
τ ≤ 2m
2m < τ ≤ 6m
τ > 6m
IBM
11583
1885
3942
1993
2114
1649
3095
4965
3523
9335
1107
3337
1624
1854
1413
2649
3731
2955
10564
1405
4160
2158
1959
882
3731
4213
2620
All options
m ≤ 0.7
0.7 < m ≤ 0.95
0.95 < m ≤ 1.05
1.05 < m ≤ 1.2
m > 1.2
τ ≤ 2m
2m < τ ≤ 6m
τ > 6m
BA
9702
760
3873
1668
1840
1561
2639
4259
2804
All options
m ≤ 0.7
0.7 < m ≤ 0.95
0.95 < m ≤ 1.05
1.05 < m ≤ 1.2
m > 1.2
τ ≤ 2m
2m < τ ≤ 6m
τ > 6m
7969
531
2810
1480
1563
1585
2493
3197
2279
All options
m ≤ 0.7
0.7 < m ≤ 0.95
0.95 < m ≤ 1.05
1.05 < m ≤ 1.2
m > 1.2
τ ≤ 2m
2m < τ ≤ 6m
τ > 6m
AXP
7054
1448
3061
1441
817
287
2056
2848
2150
3742
260
1566
966
642
308
1018
1406
1318
JNJ
10178
2004
4075
2018
1443
638
2767
4350
3061
7858
695
3187
1663
1625
688
2025
3417
2416
10302
1647
4007
2214
1942
492
3383
4312
2607
7089
609
2685
1429
1360
1006
2055
2890
2144
DD
DIS
5357
360
1931
1078
1187
801
1500
2222
1635
7899
456
2412
1368
1614
2049
2204
3452
2243
GE
6129
621
2159
1123
1242
984
1660
2404
2065
13156
1710
4593
2225
2892
1736
3230
6050
3876
8554
1556
3036
1697
1533
732
2511
3335
2708
9765
1619
3728
1719
1482
1217
2765
3926
3074
Out-of-sample period
5889
361
2153
1168
1284
923
1596
2242
2051
In-sample period
7870
705
2501
1359
1682
1623
2522
3086
2262
9790
1426
3848
1739
1886
891
2135
4558
3097
5605
451
1824
986
1134
1210
1333
2422
1850
PFE
10518
1542
3644
1602
2034
1703
2281
4866
3371
NKE
11213
1712
4420
2166
2079
836
3696
4683
2834
MMM
11033
1721
4709
2105
1692
806
3592
4612
2829
Out-of-sample period
6411
347
2345
1447
1495
777
1980
2634
1797
In-sample period
MSFT
8739
1182
3314
1343
1563
1337
2247
3902
2590
6337
657
2249
1035
1151
1245
1970
2410
1957
CSCO
MRK
10745
2141
4233
1540
1593
1238
2726
4545
3474
8070
730
2801
1409
1731
1399
1963
3389
2718
CVX
MCD
13499
2147
4494
1892
2286
2680
3606
5712
4181
12038
1044
4311
2008
2160
2519
3556
5360
3122
CAT
7501
1746
3187
1508
860
200
2211
3063
2227
5772
450
2411
1405
1049
457
1707
2244
1821
PG
14722
2099
4327
1808
2560
3928
4288
5633
4801
16351
1958
5491
2228
2705
3969
4903
6182
5266
GS
3726
553
1562
823
588
200
944
1482
1300
3991
369
1386
713
811
712
1011
1593
1387
TRV
19939
2575
6596
3437
4262
3069
4881
9053
6005
10606
956
3783
1865
1954
2048
2949
4615
3042
JPM
11307
1478
4509
2127
2109
1084
3517
4956
2834
8184
648
2663
1336
1605
1932
2072
3601
2511
UNH
8625
2028
3552
1767
1041
237
2239
3758
2628
6129
322
2443
1667
1156
541
1571
2636
1922
KO
10129
1963
3575
1784
1850
957
2570
4511
3048
6274
584
2215
1138
1304
1033
1474
2693
2107
UTX
9262
1381
3995
1787
1737
362
2406
4272
2584
7009
622
2437
1331
1601
1018
2287
2760
1962
T
VZ
13925
2422
5776
2691
2493
543
3652
6541
3732
6500
600
2293
1280
1466
861
1922
2552
2026
10121
2237
3824
1691
1515
854
2895
3943
3283
8655
829
3059
1432
1633
1702
2469
3378
2808
XOM
8194
1903
3425
1543
980
343
2171
3590
2433
8544
638
3194
1762
1925
1028
2291
3596
2657
WMT
11834
1940
4687
2163
2164
880
3423
5374
3037
7843
639
2531
1282
1646
1745
1997
3429
2417
HD
11458
1370
4061
1816
2501
1710
2814
5307
3337
6550
570
2003
973
1305
1699
2002
2585
1963
INTC
Table 2: This table reports the number of options included in the in-sample (e.g., estimation period) and out-of-sample exercises for
each stock in the Dow Jones. The in-sample period starts in March 2006, until the end of 2009. The out-of-sample period ranges from
the beginning of 2010 until the end of 2012.
45
0.264
0.245
0.267
0.301
0.255
0.193
0.097
0.232
0.426
All options
m ≤ 0.7
0.7 < m ≤ 0.95
0.95 < m ≤ 1.05
1.05 < m ≤ 1.2
m > 1.2
τ ≤ 2m
2m < τ ≤ 6m
τ > 6m
BA
0.500
0.516
0.593
0.529
0.416
0.258
0.251
0.427
0.710
0.520
0.379
0.542
0.690
0.526
0.377
0.214
0.439
0.753
All options
m ≤ 0.7
0.7 < m ≤ 0.95
0.95 < m ≤ 1.05
1.05 < m ≤ 1.2
m > 1.2
τ ≤ 2m
2m < τ ≤ 6m
τ > 6m
All options
m ≤ 0.7
0.7 < m ≤ 0.95
0.95 < m ≤ 1.05
1.05 < m ≤ 1.2
m > 1.2
τ ≤ 2m
2m < τ ≤ 6m
τ > 6m
CAT
0.199
0.136
0.202
0.251
0.210
0.072
0.085
0.172
0.290
0.215
0.245
0.240
0.219
0.174
0.078
0.125
0.193
0.283
CVX
0.230
0.131
0.232
0.295
0.251
0.169
0.092
0.186
0.345
0.375
0.369
0.403
0.410
0.341
0.183
0.138
0.299
0.560
0.168
0.129
0.196
0.187
0.124
0.071
0.061
0.147
0.266
0.204
0.187
0.232
0.220
0.179
0.125
0.104
0.200
0.270
DD
DIS
0.179
0.239
0.216
0.188
0.170
0.100
0.113
0.162
0.246
GE
0.092
0.078
0.096
0.115
0.091
0.056
0.042
0.064
0.145
0.385
0.310
0.486
0.417
0.255
0.100
0.131
0.310
0.578
0.302
0.278
0.324
0.375
0.267
0.148
0.172
0.269
0.412
0.077
0.061
0.085
0.089
0.073
0.035
0.034
0.049
0.120
0.074
0.078
0.087
0.093
0.060
0.031
0.044
0.056
0.104
In-sample period
0.350
0.265
0.398
0.318
0.340
0.236
0.397
0.279
0.400
0.315
0.173
0.190
0.161
0.127
0.308
0.220
0.478
0.372
Out-of-sample period
0.168
0.129
0.188
0.201
0.172
0.105
0.083
0.161
0.236
PG
0.196
0.089
0.184
0.279
0.229
0.117
0.080
0.162
0.294
KO
TRV
0.301
0.224
0.311
0.352
0.294
0.169
0.095
0.244
0.431
T
UNH
0.215
0.197
0.192
0.246
0.256
0.173
0.099
0.182
0.338
UTX
0.330
0.304
0.375
0.391
0.261
0.144
0.121
0.313
0.451
HD
VZ
0.149
0.108
0.181
0.155
0.103
0.065
0.053
0.105
0.248
0.080
0.069
0.073
0.100
0.088
0.065
0.040
0.051
0.128
0.089
0.080
0.085
0.112
0.110
0.059
0.047
0.088
0.118
INTC
0.150
0.090
0.140
0.226
0.142
0.083
0.085
0.135
0.205
0.332
0.344
0.305
0.392
0.365
0.197
0.132
0.257
0.499
WMT
0.150
0.163
0.160
0.163
0.124
0.077
0.062
0.133
0.229
0.189
0.251
0.217
0.201
0.171
0.106
0.078
0.162
0.271
0.211
0.212
0.190
0.240
0.248
0.135
0.105
0.174
0.306
0.791
0.520
1.167
0.478
0.287
0.134
0.432
1.014
0.730
0.485
0.482
0.555
0.530
0.482
0.269
0.221
0.410
0.693
XOM
0.288
0.278
0.230
0.374
0.365
0.160
0.141
0.239
0.401
0.137
0.097
0.162
0.147
0.096
0.059
0.041
0.085
0.232
0.213
0.169
0.219
0.255
0.226
0.124
0.084
0.180
0.330
0.281
0.257
0.289
0.289
0.240
0.304
0.128
0.271
0.373
0.233
0.113
0.197
0.344
0.290
0.162
0.089
0.189
0.346
0.209
0.214
0.224
0.201
0.207
0.150
0.096
0.194
0.282
1.042
0.834
0.989
1.047
1.211
1.026
0.442
0.857
1.462
0.224
0.229
0.221
0.283
0.234
0.109
0.090
0.179
0.335
0.430
0.453
0.349
0.488
0.542
0.375
0.223
0.352
0.639
JPM
0.331
0.264
0.277
0.389
0.419
0.216
0.151
0.306
0.460
0.710
0.611
0.809
1.006
0.732
0.388
0.336
0.577
1.026
PFE
0.095
0.107
0.119
0.099
0.063
0.035
0.042
0.065
0.144
GS
1.418
1.463
1.639
1.616
1.361
0.888
0.816
1.308
1.902
NKE
0.125
0.096
0.120
0.169
0.100
0.124
0.064
0.120
0.180
MMM
0.251
0.213
0.265
0.289
0.239
0.138
0.082
0.213
0.404
Out-of-sample period
0.128
0.062
0.077
0.169
0.180
0.088
0.077
0.116
0.175
In-sample period
0.213
0.267
0.221
0.227
0.198
0.157
0.075
0.186
0.325
MSFT
0.100
0.049
0.097
0.147
0.114
0.059
0.058
0.078
0.148
0.110
0.083
0.099
0.129
0.134
0.102
0.063
0.096
0.155
CSCO
MRK
0.483
0.411
0.602
0.523
0.315
0.170
0.184
0.455
0.653
0.599
0.543
0.615
0.660
0.650
0.442
0.322
0.530
0.799
MCD
0.476
0.403
0.540
0.577
0.476
0.310
0.148
0.440
0.671
0.305
0.338
0.327
0.359
0.309
0.175
0.148
0.295
0.429
JNJ
0.274
0.274
0.282
0.319
0.281
0.167
0.122
0.230
0.400
0.243
0.255
0.201
0.293
0.311
0.173
0.146
0.228
0.325
IBM
0.386
0.418
0.353
0.387
0.429
0.384
0.198
0.352
0.550
All options
m ≤ 0.7
0.7 < m ≤ 0.95
0.95 < m ≤ 1.05
1.05 < m ≤ 1.2
m > 1.2
τ ≤ 2m
2m < τ ≤ 6m
τ > 6m
AXP
Table 3: This table reports the root mean-square errors (RMSEs) which result from the estimation for each stock’s options, in the inand out-of-sample periods. The in-sample period starts in March 2006, until the end of 2009. The out-of-sample period ranges from
the beginning of 2010 until the end of 2012.
Table 4: This table reports the point estimates of the parameters which govern single stock returns’
dynamics. Below each estimate, the corresponding standard error is given in parentheses. The
estimation period starts in March 2006, until the end of 2009.
AXP
BA
CAT
CVX
DD
DIS
GE
GS
JPM
KO
T
XOM
HD
INTC
IBM
JNJ
MCD
MRK
MSFT
MMM
NKE
PFE
PG
TRV
UTX
VZ
WMT
CSCO
UNH
κjP
v
θ jP
y jP
µjP
σ jP
κjv
θj
yj
µj
σj
λ0
j
λ1
j
j
σv
ρj
ηj
5.900
1.491
7.000
1.666
3.700
2.280
7.700
1.840
5.700
1.214
6.100
1.460
8.500
2.837
8.166
1.922
4.600
1.572
7.389
1.696
6.300
1.194
6.360
1.243
5.474
1.129
6.500
1.825
4.700
0.000
8.200
2.341
6.700
0.964
6.686
1.163
6.100
1.351
8.200
1.679
7.029
5.285
6.680
4.934
9.000
2.757
5.700
0.900
5.200
1.481
5.700
1.267
5.700
1.540
7.600
1.858
4.700
1.912
0.001
0.040
0.067
0.030
0.078
0.070
0.042
0.050
0.060
0.150
0.060
0.040
0.066
0.030
0.065
0.061
0.024
0.110
0.063
0.027
0.013
0.090
0.047
0.074
0.027
0.067
0.028
0.100
0.033
0.150
0.000
0.110
0.048
0.070
0.032
0.037
0.054
0.070
0.043
0.050
0.021
0.052
2.979
0.030
2.978
0.100
0.030
0.100
0.015
0.100
0.045
0.100
0.049
0.100
0.040
0.130
0.035
0.140
0.039
-0.030
0.028
0.010
0.040
-0.030
0.026
-0.050
0.048
-0.030
0.011
0.000
0.076
-0.010
0.038
-0.040
0.013
-0.010
0.021
-0.030
0.040
0.001
0.049
0.000
0.037
0.000
0.016
-0.060
0.058
-0.010
0.000
-0.010
0.028
-0.010
0.024
-0.030
0.028
0.010
0.042
0.030
0.052
-0.060
0.049
0.000
0.028
-0.100
0.102
-0.040
0.036
-0.020
0.043
-0.020
0.037
-0.020
0.034
-0.010
0.037
0.010
0.051
0.050
0.103
0.050
0.028
0.050
0.028
0.050
0.030
0.050
0.014
0.014
0.039
0.100
0.030
0.100
0.012
0.020
0.054
0.150
0.054
0.011
0.065
0.111
0.042
0.013
0.083
0.150
0.073
0.020
0.000
0.050
0.026
0.080
0.027
0.122
0.064
0.050
0.026
0.160
0.031
0.111
2.214
0.120
2.208
0.160
0.011
0.130
0.027
0.150
0.040
0.190
0.037
0.190
0.128
0.050
0.025
0.110
0.046
2.900
1.183
2.900
1.390
4.500
1.328
2.500
1.543
5.500
1.630
1.200
1.561
3.500
1.186
2.460
1.504
2.400
1.314
1.649
1.124
2.600
1.626
2.460
1.939
2.460
1.410
2.400
0.840
2.300
0.000
2.300
1.211
4.500
1.622
1.649
2.097
3.500
1.147
1.900
0.456
2.117
1.446
1.640
0.995
3.300
0.595
3.700
0.128
3.200
0.997
3.200
0.792
3.600
1.170
4.900
1.999
2.400
1.428
0.002
0.016
0.020
0.027
0.023
0.024
0.011
0.023
0.002
0.014
0.008
0.030
0.003
0.016
0.007
0.022
0.002
0.016
0.005
0.011
0.006
0.014
0.007
0.017
0.004
0.022
0.029
0.028
0.003
0.000
0.000
0.013
0.017
0.016
0.006
0.027
0.014
0.016
0.002
0.022
0.006
5.059
0.005
5.072
0.011
0.010
0.027
0.017
0.017
0.017
0.017
0.012
0.010
0.022
0.020
0.032
0.030
0.021
0.080
0.042
0.120
0.064
0.040
0.052
0.040
0.033
0.060
0.018
0.060
0.028
0.020
0.029
0.067
0.010
0.040
0.026
0.030
0.015
0.040
0.028
0.033
0.034
0.055
0.019
0.060
0.029
0.100
0.000
0.020
0.038
0.040
0.055
0.033
0.045
0.020
0.047
0.080
0.037
0.041
3.132
0.070
3.172
0.040
0.040
0.060
0.030
0.040
0.053
0.040
0.036
0.040
0.038
0.060
0.024
0.020
0.053
-0.010
0.049
-0.020
0.048
-0.040
0.036
-0.020
0.039
-0.060
0.017
-0.040
0.058
-0.060
0.021
-0.080
0.013
-0.020
0.037
-0.070
0.042
-0.020
0.037
-0.090
0.033
-0.050
0.017
-0.030
0.051
-0.020
0.000
-0.080
0.043
-0.020
0.037
-0.070
0.031
-0.060
0.030
-0.050
0.027
-0.050
0.034
-0.090
0.036
-0.010
0.045
-0.060
0.040
0.010
0.044
0.010
0.028
0.020
0.081
-0.060
0.027
-0.100
0.054
0.110
0.063
0.090
0.060
0.050
0.102
0.170
0.037
0.130
0.048
0.090
0.058
0.160
0.051
0.111
0.017
0.090
0.058
0.091
0.053
0.110
0.063
0.122
0.040
0.060
0.023
0.190
0.060
0.170
0.000
0.090
0.036
0.070
0.100
0.122
0.055
0.070
0.087
0.130
0.026
0.165
2.138
0.135
2.119
0.150
0.049
0.130
0.030
0.070
0.106
0.070
0.122
0.070
0.081
0.150
0.056
0.130
0.036
0.080
0.042
0.120
0.064
0.040
0.052
0.040
0.033
0.060
0.018
0.060
0.028
0.020
0.029
0.067
0.010
0.040
0.026
0.030
0.015
0.040
0.028
0.033
0.034
0.055
0.019
0.060
0.029
0.100
0.000
0.020
0.038
0.040
0.055
0.033
0.045
0.020
0.047
0.080
0.037
0.041
3.132
0.070
3.172
0.040
0.040
0.060
0.030
0.040
0.053
0.040
0.036
0.040
0.038
0.060
0.024
0.020
0.053
4.700
2.194
6.300
1.804
5.500
1.767
5.100
2.427
6.700
1.721
5.900
1.677
7.600
1.806
4.482
1.876
9.500
2.840
7.389
1.877
9.300
3.443
6.050
1.749
9.400
2.650
4.900
2.407
7.500
0.000
5.500
2.473
6.500
1.682
9.974
4.369
8.100
2.204
4.400
2.371
5.474
3.499
9.770
7.787
7.900
2.112
6.600
0.782
6.500
2.131
6.500
1.631
6.500
1.955
3.700
2.530
4.600
3.698
0.300
0.113
0.300
0.103
0.200
0.091
0.200
0.115
0.250
0.055
0.450
0.186
0.500
0.194
0.333
0.057
0.300
0.075
0.202
0.105
0.330
0.061
0.273
0.070
0.273
0.070
0.300
0.118
0.300
0.000
0.350
0.127
0.400
0.131
0.223
0.106
0.200
0.108
0.300
0.061
0.273
1.812
0.310
1.824
0.400
0.118
0.300
0.105
0.350
0.106
0.350
0.100
0.350
0.121
0.250
0.100
0.250
0.078
-0.550
0.105
-0.600
0.062
-0.550
0.092
-0.600
0.050
-0.600
0.029
-0.450
0.126
-0.500
0.137
-0.589
0.020
-0.600
0.047
-0.589
0.116
-0.600
0.059
-0.666
0.062
-0.589
0.022
-0.550
0.096
-0.500
0.000
-0.650
0.097
-0.600
0.059
-0.589
0.041
-0.500
0.101
-0.550
0.039
-0.547
2.682
-0.590
2.732
-0.550
0.051
-0.550
0.049
-0.500
0.100
-0.500
0.124
-0.500
0.130
-0.550
0.052
-0.550
0.077
0.395
0.208
0.400
0.341
0.663
0.162
0.901
0.189
-0.050
0.159
0.750
0.348
0.750
0.255
0.095
0.152
0.200
0.213
0.658
0.164
0.390
0.176
0.650
0.168
0.850
0.133
0.350
0.243
-0.100
0.000
-0.050
0.117
0.850
0.154
-0.100
0.249
0.715
0.136
0.453
0.123
0.893
0.163
0.425
0.225
0.298
0.101
0.280
0.108
0.400
0.184
0.650
0.255
0.400
0.219
0.900
0.318
0.200
0.122
0.008
0.028
0.004
0.002
0.000
0.001
0.002
0.001
0.001
0.000
0.003
0.002
0.011
0.001
0.000
0.011
0.002
0.008
0.000
0.002
0.001
0.004
0.018
0.010
0.001
0.000
0.000
0.000
46
Table 5: This table reports the point estimates and standard errors (in italic) of the beta coefficients.
The estimation period starts in March 2006, until the end of 2009.
j
Est.
βdif f
Std. err.
j
Est.
βjump
Std. err.
j
βv
Est.
Std. err.
Consumer staples
KO
PG
WMT
0.607
0.450
0.560
0.130
0.189
0.144
0.607
9.000
0.650
0.129
0.408
0.283
0.301
0.300
0.300
0.375
0.209
0.412
Pharmaceutical
JNJ
MRK
UNH
PFE
0.410
0.607
0.810
0.600
0.094
0.110
0.258
0.878
8.200
6.686
1.250
6.680
0.229
0.164
0.228
2.264
0.200
0.301
0.000
0.490
0.486
0.446
0.618
0.000
Energy
CVX
XOM
0.660
0.607
0.086
0.119
1.100
1.105
0.221
0.192
0.200
0.259
0.520
0.438
Technology
IBM
MSFT
VZ
T
CSCO
INTC
0.530
0.800
0.500
0.560
1.140
0.750
0.000
0.149
0.215
0.000
0.268
0.103
4.700
6.100
5.700
0.819
1.000
6.500
0.000
0.545
0.411
0.000
0.001
0.215
0.150
0.250
0.150
0.368
0.000
0.250
0.000
0.246
0.216
0.000
0.463
0.628
Industrials
MMM
UTX
GE
BA
CAT
0.600
0.650
0.920
0.890
0.990
0.162
0.209
0.163
0.163
0.150
8.200
0.002
0.400
1.450
0.650
0.480
0.382
0.593
0.738
0.407
0.200
0.150
0.000
0.150
0.450
0.222
0.250
0.310
0.283
0.278
Materials
DD
0.770
0.080
0.300
0.225
0.550
0.690
Consumer discretionary
MCD
HD
NKE
DIS
0.540
0.779
0.790
0.900
0.210
0.073
1.053
0.216
0.600
5.474
7.029
1.100
0.232
0.109
1.942
0.244
0.150
0.273
0.157
0.100
0.409
0.349
1.314
0.234
Financials
AXP
JPM
TRV
GS
1.120
0.840
0.720
1.221
0.384
0.219
0.160
0.199
0.550
0.950
0.900
0.497
0.301
0.228
0.000
0.107
0.050
0.250
0.600
0.273
0.171
0.439
0.113
0.404
Table 6: This table reports the results of a regression comparing synthesized VS rates following the
procedure described in Carr and Wu (2009) with model-implied VS rates, following equation (10).
The sampling frequency is weekly. The first column is based on repeated cross-sectional regressions
(Fama-MacBeth). The reported coefficients are averages of the weekly coefficient estimates. Standard
errors are calculated using the method of Newey-West with 8 lags. The standard errors of the
intercept coefficients are multiplied by 100. The second column reports the results of a pooled panel
regression. The last line reports the p-value when testing a hypothesis test with null H0 : the model
perfectly represents VS rates, i.e., the intercept is zero and the slope is one.
Cross-sectional
Panel
0.98
(0.20)
0.88
(0.02)
-0.27
(0.27)
0.96
(0.03)
0.82
357
0.82
0.70
9639
0.79
Intercept ×100
Slope
R2
N
H0 p-value [%]
47
Table 7: This table reports the results of a regression comparing ex ante synthesized VS rates
at time t + τ , where VS rates are computed following the procedure described in Carr and Wu
(2009), with model-implied predictions at time t, given by EP [VSt+τ ]. The model predictions are
compared to predictions given by the martingale method, whose prediction of future VS rate is the
current VS rate. Cross-sectional regressions (Fama-MacBeth) as well as pooled panel regressions
are conducted, for three values of τ : one week (1w), one month (1m) and 6 weeks (5w). For
cross-sectional regressions, the reported coefficients are averages of the weekly coefficient estimates.
Standard errors are calculated using the method of Newey-West with 8 lags. The standard errors
of the intercept coefficients are multiplied by 100. The last line reports the p-value when testing a
hypothesis test with null H0 : the prediction perfectly matches future VS rates, i.e., the intercept is
zero and the slope is one.
τ = 1w
Model
Martingale
Cr. sec.
Panel
Cr. sec.
Panel
Intercept ×100
Slope
R2
N
H0 p-value [%]
τ = 1m
Model
Martingale
Cr. sec.
Panel
Cr. sec.
Panel
τ = 6w
Model
Martingale
Cr. sec.
Panel
Cr. sec.
Panel
1.03
(0.21)
0.86
(0.02)
0.34
(0.32)
0.99
(0.05)
0.58
(0.16)
0.95
(0.01)
2.38
(1.27)
0.71
(0.15)
1.59
(0.30)
0.94
(0.06)
-1.21
(0.57)
1.44
(0.10)
1.27
(0.28)
0.87
(0.03)
3.06
(1.19)
0.63
(0.14)
2.04
(0.39)
0.99
(0.08)
-0.23
(0.74)
1.42
(0.15)
1.51
(0.35)
0.85
(0.05)
3.50
(1.13)
0.58
(0.13)
0.78
356
0.93
0.63
10324
0.96
0.85
356
0.97
0.51
10324
0.10
0.68
353
0.84
0.42
10237
0.19
0.67
353
0.84
0.40
10237
0.05
0.63
351
0.80
0.26
10179
0.24
0.62
351
0.79
0.34
10179
0.03
Table 8: This table reports the results of a regression comparing the average total diffusive variance
rate of the bucket of firms with the smallest market capitalizations at the end of 2012 with the
average variance rates in other size buckets. The notation ’k/1’ is used to refer to the regression
of the average diffusive variance rate in bucket k on the rate in bucket 1. Bucket 1 contains the
firms with the smallest market capitalizations, bucket 5 those with the largest. In the last column,
the variance of GE was removed from the calculation of the average. Standard errors are given in
parentheses and are in percentages.
Intercept×100
Slope
R2
2/1
3/1
4/1
5/1
5/1 (w/o GE)
0.30
(0.09)
0.87
(0.56)
0.34
(0.09)
0.78
(0.60)
-0.94
(0.11)
0.59
(0.65)
0.05
(0.08)
0.66
(0.50)
0.25
(0.09)
0.55
(0.56)
0.98
0.98
0.97
0.98
0.97
48
Table 9: This table reports the results of a regression comparing the average idiosyncratic diffusive
variance rate of the bucket of firms with the smallest market capitalizations at the end of 2012 with
the average variance rates in other size buckets. The notation ’k/1’ is used to refer to the regression
of the average diffusive variance rate in bucket k on the rate in bucket 1. Bucket 1 contains the
firms with the smallest market capitalizations, bucket 5 those with the largest. In the last column,
the variance of GE was removed from the calculation of the average. Standard errors are given in
parentheses and are in percentages.
Intercept×100
Slope
R2
2/1
3/1
4/1
5/1
5/1 (w/o GE)
0.34
(0.11)
0.93
(1.35)
0.18
(0.11)
0.86
(1.32)
-0.66
(0.14)
0.57
(1.49)
0.09
(0.09)
0.74
(1.14)
0.24
(0.10)
0.59
(1.25)
0.93
0.92
0.85
0.98
0.86
49
50
All options
m ≤ 0.7
0.7 < m ≤ 0.95
0.95 < m ≤ 1.05
1.05 < m ≤ 1.2
m > 1.2
τ ≤ 2m
2m < τ ≤ 6m
τ > 6m
All options
m ≤ 0.7
0.7 < m ≤ 0.95
0.95 < m ≤ 1.05
1.05 < m ≤ 1.2
m > 1.2
τ ≤ 2m
2m < τ ≤ 6m
τ > 6m
45988
5038
18475
8508
9344
4630
10311
19570
16107
48568
3519
16933
9162
9669
9285
10511
20987
17070
XLE
7698
875
3928
2000
860
35
2003
3675
2020
7341
149
2928
2546
1488
234
1887
3588
1866
XLU
7869
1130
3306
1801
1357
275
1807
3585
2477
11617
1401
4678
2352
2371
815
3261
5525
2831
11033
1721
4709
2105
1692
806
3592
4612
2829
11017
1428
4804
2540
1913
332
2835
5195
2987
8251
365
3158
2399
1758
574
1932
3864
2455
Out-of-sample period
XLY
XLB
XLP
XLK
In-sample period
5609
10959
6411
98
631
347
2176
4252
2345
1725
2690
1447
1284
2202
1495
326
1184
777
1186
2833
1980
2665
5171
2634
1758
2955
1797
11666
1842
4664
2347
2131
682
2846
5368
3452
8323
328
3092
2535
1746
622
1822
3918
2583
XLI
8100
905
3947
2237
980
31
2013
3844
2243
5611
66
2005
2319
1082
139
1314
2744
1553
XLV
19939
2575
6596
3437
4262
3069
4881
9053
6005
10606
956
3783
1865
1954
2048
2949
4615
3042
XLF
Table 10: Number of options included in the in-sample (e.g., estimation period) and out-of-sample exercises for each stock.
Table 11: This table reports the statistics of the returns of four portfolios that are weekly rebalanced
depending on the level of idiosyncratic variance (Panel A) and true idiosyncratic variance (Panel
B) of each stock, from the beginning of March 2006 until the end of December 2012. Portfolio 1
contains every week a combination of the 7 stocks with the smallest idiosyncratic variance. Portfolio
4 contains the 7 stocks with the largest idiosyncratic variance. The initial wealth level is 100. The
average return, its standard deviation (Std. dev.) and its Sharpe ratio are multiplied by 100.
Average return
Std. dev.
Sharpe ratio
1
2
3
4
0.02
0.07
0.22
0.23
2.02
2.38
2.69
3.76
Panel A
1.21
3.20
8.28
6.37
1
2
3
4
0.03
0.11
0.16
0.25
1.95
2.36
2.72
3.85
Panel B
1.54
4.87
6.05
5.53
51
Skewness
Kurtosis
-1.16
-0.44
-0.67
-0.16
10.60
6.92
7.46
7.86
-0.81
-0.31
-0.79
-0.25
9.48
5.30
9.03
7.51
Technical Appendix
A
A.1
Functional Principal Component Analysis
Procedure
I first construct for every Wednesday in my sample a smooth estimator of the implied volatility surface
IVt (m, τ ) over a grid of moneynesses m ∈ [mmin , mmax ] and times-to-maturity τ ∈ [τmin , τmax ], using
a non-parametric Nadaraya-Watson estimator as detailed in H¨ardle (1992) and A¨ıt-Sahalia and Lo
(1998). The boundaries of the grid are chosen such that enough data points are available over time
to accurately intrapolate the IV surface. Following Cont and da Fonsecca (2002), I calculate the
weekly variations ∆Xt (m, τ ) = ln IVt (m, τ ) − ln IVt−1 (m, τ ) of the logarithm of the implied volatility
at every grid point and consider the random field Ut (m, τ ) = ∆Xt (m, τ ). I assume stationarity and
apply a Karhunen-Lo`eve (K-L) decomposition to Ut (m, τ ), which is a generalization of the classic
PCA to higher dimensional random fields. Intuitively, the K-L expansion provides a second-moment
characterization of random surface in terms of orthogonal functions fk and uncorrelated random
variables Uk as follows:
U (m, τ ) =
N
KL
X
Uk fk (m, τ ).
k=1
The fk are the eigenfunctions of the covariance function K(m, τ, m0 , τ 0 ) for (m, τ ) and (m0 , τ 0 ) ∈ A =
[mmin , mmax ] × [τmin , τmax ], defined as follows:
K(m, τ, m0 , τ 0 ) = cov(U (m, τ ), U (m0 , τ 0 )).
The covariance function is numerically computed from the time-series of smooth implied volatility
surfaces. Its eigenfunctions result from a spectral decomposition of the function, done by solving the
homogeneous Fredholm integral equation of the second kind:
Z
K(x1 , x2 )fk (x1 )dx1 = λk fk (x2 ),
A
52
where the λk are the eigenvalues of the covariance function, x1 and x2 ∈ A. A difficulty of solving
the Fredholm equation lies in the availability of eigenfunctions. To reduce the dimensionality of
the problem I use as in Cont and da Fonsecca (2002) a Garlekin procedure13 , which expresses the
eigenfunctions as a linear combination of basis functions hj :
fk (m, τ ) =
X
aij hj (m, τ ) + .
(12)
j
The coefficients aij define a matrix A. Let scalar product of two surfaces u and v : A → R is defined
as:
Z
hu, vi =
u(x)v(x)dx.
A
The error term is set to be orthogonal to each basis function hj so that one ultimately has to solve
for a finite number of equations. The symmetric matrices B and C are defined as:
Bij = hhi , hj i,
Z
Cij =
Z
dm0 dτ 0 hi (m, τ )K(m, τ, m0 , τ 0 )hj (m0 , τ 0 )
dmdτ
A
A
In practice, each coefficient is calculated numerically using Simpson’s rule of integration. The final
problem is summarized in the following matrix eigenvalue problem, where one has to solve for D
and A, D being a diagonal matrix with the eigenvalues of the covariance function λk on its main
diagonal: CA = DBA.
The eigenfunctions are recovered from equation (12) and the principal components processes xk (t)
obtained by projecting the random surface Ut on the eigenfunctions fk :
xk (t) = hXt − X0 , fk i.
13
Alternative methods involving, e.g., wavelets have been proposed to deal with potentially non-smooth surfaces and
processes that exhibit non-stationarity or autocorrelation. See for example Phoon, Huang, and Quek (2002)
53
A.2
Results for the S&P 500
Figure 14: Eigenmodes resulting from the FPCA of weekly S&P 500 options’ implied volatilities
from March 1st, 2006 until December 31, 2012. The first eigenmode explains 98%, and the sum of
the first four eigenmode explain 99.88% of the variation of the IV surface over time.
First eigenmode
Second eigenmode
0.04
0.06
0.02
0
0.04
−0.02
0.02
0.5
−0.04
0.5
T
0
0.7
m0.8
0.9
1.1
1
0
T
Third eigenmode
0.7
−3
x 10
m0.8
0.9
1
1.1
1
1.1
Fourth eigenmode
15
10
0.01
5
0.005
0
0
−5
0.5
−0.005
−0.01
0.5
1.5
0.4
1
0.3
T
0.2
0.1 0.5
T
m
54
0
0.7
m0.8
0.9
Figure 15: Principal component processes resulting from the FPCA of weekly S&P 500 options’
implied volatilities from March 1st, 2006 until December 31, 2012.
14
−2
12
−3
x2 (t)
x1 (t)
10
8
−4
−5
6
−6
4
Jan−08
Jan−10
−7
Jan−12
10
2
5
1
0
0
x4 (t)
x3 (t)
2
−5
−2
−15
−3
Jan−08
Jan−10
−4
Jan−12
55
Jan−10
Jan−12
Jan−08
Jan−10
Jan−12
−1
−10
−20
Jan−08
B
High-frequency data analysis
For each stock I denote by ∆ni p = p i − p i−1 the variation of log-price increments over time grid
n
n
i−1 i every
day
t
=
1,
...,
T
.
The
following
measures of variation of log price returns are calculated
,
n n
based on 5-minute tick data. These measures have been introduced for generic semimartingale
processes, see Andersen, Bollerslev, Diebold, and Labys (2003), Bollerslev, Todorov, and Li (2013),
Mancini (2009), and used extensively since.
The realized variance RVt is defined as the sum of intra-day squared returns, it converges in
probability towards the quadratic variation of the underlying return process:
RVt =
tn+n
X
|∆ni p|2 →P QV[t,t+1] ,
i=tn+1
The realized variance can be decomposed into the continuous variation CVt that captures the
”small” movements of the returns, which are likely to be generated by a continuous process, and the
jump variation JVt , which captures the larger movements, better represented by a jump process:
CVt =
tn+n
X
|∆ni p|2 1{|∆ni p|≤αi n−ω¯ } .
i=tn+1
Z
t+1 Z
P
JVt = RVt − CVt →
t
x2 µ(ds, dx).
R
µ(., .) refers to the jump measure of the underlying returns. The threshold αi is set following Bollerslev and Todorov (2011) and Bollerslev, Todorov, and Li (2013). The estimated mean and standard
deviation of jump sizes under the empirical measure are set to the empirical moments of aggregated
intra-day returns that exceed the level αi .
56
C
Change of measure
The market dynamics are as follows under a risk-neutral measure Q, where the parameters without
superscript are either Q-parameters or parameters that are invariant under both measures:
q
dMt
M
M
Q ZtM
+ (eZt − 1)dNtM
= (r − λM (vtM
,
m
)E
[e
−
1])dt
+
vtM
−
t
− dWt
Mt−
q
M
M
M
dvtM = κM
(m
−
v
)dt
+
σ
vtM
+ ytM dNtM
t
− dBt
v
v
t−
M
M√
¯M
dmt = κM
− mt )dt + σm
mt dB
m (θ
t
M
with EQ [eZt − 1] = eµ
M + 1 (σ M )2
2
− 1 and
M (P)
dWt
M (P)
dBt
q
vtM
=
−η
− dt
M (P)
M (Q)
q
κv
− κv
M (Q)
dt
= dBt
+ vtM
−
σvM
M (Q)
dWt
M
M (P)
¯ M (Q) + √mt κm
= dB
t
M (P)
¯
dB
t
θM (P) =
M (Q)
− κm
M
σm
dt
M (Q) M (Q)
θ
κm
M (P)
κm
Individual stocks have the following dynamics:
dStj
Stj−
=rdt +
j
βdif
f
q
q
j
j
M
M
vt− dWt + vtj− dWtj + (eZt − 1)dNtj − λj (vtj− )EQ [eZt − 1]dt
j
M
j
M
Q βjump Zt
+ (eβjump Zt − 1)dNtM − λM (vtM
− 1]dt
− , mt )E [e
q
dvtj =βvj dvtM + κjv (θj − vtj− )dt + σvj vtj− dBtj + ytj dNtj .
j
j + 1 (σ j )2
2
with EQ [eZt − 1] = eµ
− 1 and
j(P)
dWt
j(P)
dBt
q
− η vtj− dt
q
j(P)
κv − κjv
= dBtj + vtj−
dt.
σvj
=
dWtj
j
57
D
Option pricing
Assume that the characteristic functions of the process S˜ takes the following exponential form:
h
i
˜
ΨS˜T (t, s˜t , vtj , vtM , mt ; ω) := EQ eωST |Ft
j
˜
M +χ(T −t)·m +η(T −t)
t
= eα(T −t)·St +β(T −t)·vt +γ(T −t)·vt
,
. with ω ∈ C. Let us define the joint Laplace transforms of jumps in returns and variances:
j Z j +φM Z M
j j
1
j j 2
M
M
1
M
] = eφ µ + 2 (φ σ ) eφ µ + 2 (φ σ
1
1
j j
M M
θy (ψ j , ψ M ) = EQ [eψ y +ψ y ] =
.
1 − ψj yj
1 − ψM yM
θZ (φj , φM ) = EQ [eφ
M )2
Coefficients are functions of the time-to-maturity and satisfy a system of ODEs, ∀t ∈ (0, T ]. The
first equation reduces to α(T − t) = α(0) = ω. The remaining ODEs are as follows:
0
β (T − t) = ω
−λj1 (θZ (1, 0)
2
1
1
1
− 1) −
− β(T − t)κjv + ω 2 +
β(T − t)σvj + ωβ(T − t)σvj ρj +
2
2
2
λj1 [θZ (ω, 0)θy (β(T − t), 0) − 1]
1 j
1 j 2
j
2
j M
M
γ 0 (T − t) = ω −λM
(θ
(0,
β
)
−
1)
−
(β
)
−
β(T
−
t)β
κ
−
γ(T
−
t)κ
+
ωβdif f +
Z
1
v v
v
jump
2 dif f
2
2 1
2
1
j
j
M M
βvj σvM ρM + ωγ(T − t)βdif
β(T − t)βvj σvM +
γ(T − t)σvM + ωβ(T − t)βdif
f σv ρ +
f
2
2
h
i
j
j
β(T − t)γ(T − t)βvj (σvM )2 + λM
θ
(0,
ωβ
)θ
(0,
β(T
−
t)β
+
γ(T
−
t))
−
1
y
Z
1
v
jump
h
i
1
j
j
χ0 (T − t) = − ωλM
θZ (0, βjump
) − 1 + κM
(χ(T − t)σm )2 +
2
v β(T − t)βv + γ(T − t) − χ(T − t)κm +
2
h
i
j
j
λM
θ
(0,
ωβ
)θ
(0,
β(T
−
t)β
+
γ(T
−
t))
−
1
y
Z
2
v
jump
h
i
j
η 0 (T − t) = ω r − λj0 (θZ (1, 0) − 1) − λM
(θ
(0,
β
)
−
1)
+ β(T − t)κjv θj + χ(T − t)κm θM +
Z
0
jump
h
i
j
j
M
j
λ0 [θZ (ω, 0)θy (β(T − t), 0) − 1] + λ0 θZ (0, ωβjump )θy (0, β(T − t)βv + γ(T − t)) − 1 .
Boundary conditions are given by α(0) = ω, β(0) = γ(0) = χ(0) = η(0) = 0.
58
E
Integrated Risk Premia
Under the definition given in equation (8), the IERP can be decomposed into an idiosyncratic
component IERP Idio (t, T ) and a market component IERP M arket (t, T ), with:
IERP
Idio
(t, T ) =η
j
Z
EPt
T
t
1
−
2
IERP
M arket
(t, T )
Z T
Z T
j j
j j
Q
P
λ (vs )ds
λ (vs )ds − E
− E [e − 1] E
t
t
Z
Z T
Z T
j j
j Q
j
j(P) P
j
Q
λ (vs )ds − µ E
vs ds + µ E
vs ds − E
vsj ds
T
Z
EP
Q
Zj
j
M P
=βdif
f η Et
Z
T
λ
j
(vsj )ds
t
t
t
t
T
vsM ds
t
Z T
Z T
P
M M
Q
M M
− E [e
− 1] E
λ (vs , ms )ds − E
λ (vs , ms )ds
t
t
Z T
Z T
1 j
M
Q
M
2
P
vs ds − E
vs ds
− (βdif f ) E
2
t
t
Z T
Z T
j
M (P) P
M M
M Q
M j
+ βjump µ
E
λ (vs , ms )ds − µ E
λ (vs , ms )ds .
Q
j
βjump
ZM
t
t
The calculation of the IERP therefore amounts to computing the expectation of the integrated
variances under the empirical and risk-neutral measures. Due to the affine property of the model,
these expectations can be written as linear functions of vtj , vtM and mt . Coefficients are derived by
solving the ODE satisfied by f (t, T ) = E[vT ], under P and Q, for vT = vTj and vT = vTM . The ODE
is obtained directly from the stochastic differential equation satisfied by the variance processes. For
conciseness, coefficients are not provided but available upon request.
Similarly, the IVRP can be decomposed into an idiosyncratic component IV RP Idio (t, T ) and a
market component IV RP M arket (t, T ), such that
59
.
IV RP
Idio
Z T
Z T
Q
j
P
j
vs ds
Et
vs ds − Et

t

 t
NTj
NTj
1  P X


 X
+
(Znj )2 
(Znj )2  − EQ
Et 
t 
T −t
j
j
1
(t, T ) =
T −t
n=Nt +1
n=Nt +1
and
IV RP
M arket
j
2
(βdif
f)
Z T
Z T
Q
M
P
M
vs ds
(t, T ) =
Et
vs ds − Et
T −t
t
t
 



NTj
NTj
j
2
(βjump )  P  X

 X

+
(ZnM )2  − EQ
(ZnM )2  .
Et 
t 
T −t
j
j
n=Nt +1
n=Nt +1
The expectation of the squared jumps is obtained using the independence between the jump sizes
and the Poisson process driving their occurrence time. For example in the case of the individual
stock:


NTj
Z
X

j 2
j 2
j 2
Et 
(Zn )  = [(µ ) + (σ ) ]
E[λj (vsj )]ds.
t
n=Ntj +1
F
T
Model properties
In this section I examine the relationship between the betas and the risk-neutral moments of the
returns. Let t = t0 < t1 < ... < tn = T be a uniform time grid over a given time period [0, T ], with
∆t = ti − ti−1 , i > 0. The total risk-neutral variance of stock j over [t, T ] is given by14 :
"Z
j
TotalVar (t, T ) = Et
T
d log Suj
−
Et [d log Suj ]
2 #
t
Z
=
T
j
Z
VarRate (u)du + o
t
T
VarRate (u)du
j
t
where VarRate is the instantaneous variance rate defined in equation (11) and the rest is a function
of the characteristics of the jump distributions. As the variance swap rate, the total variance is
14
All expectations are taken under Q.
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therefore increasing in the betas.
I define the risk-neutral skewness rate of stock j as follows:
Et
1
SkewRate (t) = lim
∆t→0 ∆t1/2
∆ log Stj
−
3 Et [∆ log Stj ]
j
Et
M
= SkewRate
∆ log Stj
j
(t)(βjump
)3
−
2 3/2
Et [∆ log Stj ]
VarRateM (t)
VarRatej (t)
3/2
+ SkewRate
Idio(t)
VarRateIdio (t)
VarRatej (t)
3/2
where VarRateM (t) and SkewRateM (t) are the spot variance and skewness of the market factor,
j
j
and VarRateIdio (t) and SkewRateIdio (t) are those of stock j obtained by setting βjump
= βdif
f = 0.
As µM is typically negative, the market skewness is negative, and pulls the total skewness towards
negative values. The idiosyncratic skewness is for typical parameter values close to zero. Note that
the instantaneous skewness rate is only non zero because of the jump term. In a diffusion setup,
as the third moment of Brownian motion increments is zero, there is no instantaneous skewness.
Following Christoffersen, Fournier, and Jakobs (2013), the total skewness of stock j over the time
interval [t, T ] is defined as:
Et
TotalSkewj (t) =
Et
R
R
T
t
T
t
d log Suj
d log Suj
−
−
3 Et [d log Suj ]
2 3/2
Et [d log Suj ]
The total skew can be decomposed similarly to the instantaneous skew rate into an idiosyncratic
skew and a market skew, whose diffusive and jump parts are respectively proportional to (βdif f )3
and (βjump )3 . A larger (βdif f )3 pushes the skew to be more negative. The impact of (βjump )3 is,
however, not as clear as for the spot skew, because of an additional cross-term that is close to zero
for small idiosyncratic jumps.
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Online Appendix
62