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Essays in Volatility Modeling and Option Pricing
by
Mathieu Fournier
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
« Joseph L. Rotman School of Management »
University of Toronto
© Copyright by Mathieu Fournier 2014
Abstracts
Essays in Volatility Modeling and Option Pricing
Mathieu Fournier
Doctor of Philosophy
Graduate Department of Joseph L. Rotman School of Management
University of Toronto
2014
The common thread that runs through my research is the implication of volatility
dynamics for option pricing. In the �rst chapter of this thesis, my co-authors and I study
the presence of a factor structure in equity options. A principal component analysis of
equity options reveals a strong factor structure. Guided by this �nding, we develop an
equity option valuation model that captures this factor structure. The model allows for
stochastic volatility in the market return and also in the idiosyncratic component of equity
returns. The model has a series of predictions relating individual �rm�s beta to patterns
in implied volatilities. The equity option data support the cross-sectional implications
of the estimated model. In the second chapter, I focus on the impact of intermediaries�
risk sharing capacity on index options. I examine how inventory risk and market-maker
wealth jointly determine the value of index options through their e¤ects on the variance
risk premium. My analysis shows that part of the variance risk premium compensates
option market-makers for their exposure to market variance. Based on these �ndings, I
develop a theoretical model in which market variance is stochastic and the representative
market-maker accumulates inventory over time by absorbing end-users� net demand for
index options. Starting from the market-maker�s optimal trading strategy, I derive an
explicit formula linking the variance risk premium to inventory risk and market-maker
wealth. The model predicts that the variance risk premium is in�uenced by the ratio of
inventory risk to the market-maker�s wealth. Estimating the model on S&P 500 index
returns and options, I �nd that it performs well.
ii
Dedication
To my parents and my family.
.
iii
Acknowledgements
I would like to express my gratitude to my supervisor, Peter Christo¤ersen for his continual
guidance. I am also grateful to my committee members Redouane Elkamhi, Christian
Gouriéroux, Chayawat Ornthanalai, and Jason Wei for their many insights and advices.
This work was supported by the Rotman School of Management.
.
iv
Contents
1 The Factor Structure in Equity Options 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Common Factors in Equity Option Prices . . . . . . . . . . . . . . . . . . . 4
1.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Other Stylized Facts in the Cross-Section of Equity Option Prices . 8
1.3 Equity Option Valuation Using a Single-Factor Structure . . . . . . . . . . 9
1.4 Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Factor Structure and Equity Option Sensitivities . . . . . . . . . . . 14
1.4.2 Factor Structure and Equity Option Returns . . . . . . . . . . . . . 16
1.4.3 The Relative Pricing of Index and Equity Options . . . . . . . . . . 17
1.4.4 The Level of Equity Option Volatility . . . . . . . . . . . . . . . . . 18
1.4.5 Equity Option Skews . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.6 The Term Structure of Equity Volatility . . . . . . . . . . . . . . . . 21
1.5 Estimation and Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.1 Estimation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.2 Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5.3 Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.4 Equity Betas and Equity Option IVs . . . . . . . . . . . . . . . . . . 30
1.5.5 Option-Implied and Regression Betas . . . . . . . . . . . . . . . . . 31
1.5.6 The Cross-Section of Idiosyncratic Risk . . . . . . . . . . . . . . . . 32
1.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
v
2 Inventory Risk, Market-Maker Wealth, and the Variance Risk Premium 67
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.2 Empirical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.2.2 De�nition of Main Variables . . . . . . . . . . . . . . . . . . . . . . . 73
2.2.3 Methodology and Predictions . . . . . . . . . . . . . . . . . . . . . . 77
2.2.4 Control Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.2.6 The Financial Crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.4 Model Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.4.1 The Structure of the Variance Risk Premium . . . . . . . . . . . . . 86
2.4.2 Market-Maker Optimal Wealth . . . . . . . . . . . . . . . . . . . . . 88
2.4.3 Risk-Neutral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.5 Estimation and Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.5.1 Estimation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.5.2 Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.5.3 Index Option Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.5.4 Economic Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
vi
List of Tables
2.1 Companies, Tickers and Option Contracts p 60
2.2 Summary Statistics on Implied Volatility 1996-2010 p 61
2.3 Principal Component from Principal Component Analysis of Equity Implied p 62
Volatility
2.4 Model Parameters and Properties. Index and Equity Options p 63
2.5 Model Fit for Index and Equity Put Options p 64
2.6 Idiosyncratic Variance Correlation Matrix p 65
2.7 (A1) Principal Component (PC) Loadings for IV Level, Moneyness Slope, p 66
and Maturity Slope
3.1 Descriptive Statistics p 117
3.2 Implied Volatility, Market-Makers�Inventory, and Delta-Hedged Gains p 118
and Losses by Moneyness and Maturity for SPX Options
3.3 Time-Series Regressions of the One-month of Log-Variance Risk Premium p 119
3.4A Time-Series Regressions of the Term Structure of Log-Variance Risk p 120
Premia. 1997-2011
3.4B Time-Series Regressions of the Term Structure of Variance Risk p 121
Premia. 1997-2011
3.5 Statistics of the Return Distribution of Delta-Hedged Near-the-Money Options p 122
3.6 Return Statistics and Parameter Coe¢ cients for CEV and Heston (1993) p 123
from MLIS. 1997-2011
3.7 Inventory Risk and Market-Maker0s Wealth Parameters, Heston(1993)0s p 124
Variance Risk Premium Coe¢ cient, and SIVSE. 1997-2011
vii
3.8 Model Fit Based on 131,838 SPX Put Options. 1997-2011 p 125
viii
List of Figures
2.1 At-the-MoneyImpliedVolatility.Six Firms and the S&P 500 Index p 49
2.2 Implied Volatility Level, Moneyness and Term Slope. S&P 500 Index and p 50
the First Principal Component from 29 Firms
2.3 Market Delta and Vega of Equity Call Options p 51
2.4 Expected Excess Returns on Equity Options p 52
2.5 Beta and Implied Volatility Across Moneyness and Maturity p 53
2.6A Average Market- (solid) and Model-Implied (dashed) Volatility Smile.High p 54
Volatility (black) and Low Volatility (grey) Days (Panel A)
2.6B Average Market- (solid) and Model-Implied (dashed) Volatility Smile.High p 55
Volatility (black) and Low Volatility (grey) Days (Panel B)
2.7A Market- (solid) and Model-Implied (dashed) Term Structures for At-the- p 56
Money Implied Volatility. Upward-Sloping (grey) and Downward-Sloping
(black) Days (Panel A)
2.7B Market- (solid) and Model-Implied (dashed) Term Structures for At-the- p 57
Money Implied Volatility. Upward-Sloping (grey) and Downward-Sloping
(black) Days (Panel B)
2.8 Implied Volatility Levels, Moneyness Slopes, and Term Structure Slopes p 58
Scatter Plotted Against Beta. 29 Firms
2.9 Regression-based Betas versus Option-Implied Beta. 29 Firms p 59
3.1 S&P 500 Index, Variance Risk Premium, and Delta-Hedged Gains and p 111
Losses of Near-the-Money Options
3.2 CBOE VIX Index and Market-Makers�Inventory Risk p 112
ix
3.3 Market-Makers�Daily and Cumulative Pro�ts and Losses, and p 113
Market-Makers�Bid-Ask Spread Revenue
3.4 Filtered Spot Volatilities Using Daily S&P 500 Returns p 114
3.5 One-Month Variance Risk Premium, IVRMSE, and Implied p 115
Volatility Smile by Model
3.6 The Dollar Response of Index Options to Inventory Risk and p 116
Market-Maker0s Wealth
x
1
Chapter 1
The Factor Structure in Equity Options
1.1 Introduction
In their path-breaking study, Black and Scholes (1973) show that when valuing equity
options in a constant volatility CAPM setting, the beta of the stock does not matter.
Consequently, standard equity option valuation models make no attempt at modeling the
factor structure in the equity market. Typically, a stochastic process is assumed for each
underlying equity price and the option is priced on this stochastic process, ignoring any links
the underlying equity price may have with other equity prices through common factors.
Seminal papers in this vein include Wiggins (1987), Hull and White (1987), and Heston
(1993), Bakshi, Cao and Chen (1997), and Bates (2000, 2008). We show that in a CAPM
setting with stochastic volatility, the beta does indeed matter for equity option prices. We
�nd strong support for this factor structure in a large-scale empirical investigation using
equity option prices.
When considering a single stock option, ignoring an underlying factor structure may
be relatively harmless. However, in portfolio applications it is crucial to understand links
between the underlying stocks. Risk managers need to understand the total exposure to the
underlying risk factors in a portfolio of stocks and stock options. Equity portfolio managers
who use equity options to hedge large downside moves in individual stocks need to know
their overall market exposure. Dispersion traders who sell (expensive) index options and
buy (cheaper) equity options need to understand the market exposure of individual equity
options. See for example Driessen, Maenhout, and Vilkov (2009) for evidence on the market
2
exposure of equity options.
Our empirical analysis of more than three quarters of a million index option prices
and 11 million equity option prices reveals a very strong factor structure. We study three
characteristics of option prices: short-term implied volatility (IV) levels, the slope of IV
curves across option moneyness, and the slope of IV curves across option maturity.
First, we compute the daily time series of implied volatility levels (IV) on the stocks
in the Dow Jones Industrials Average and extract their principal components. The �rst
common component explains 77% of the cross-sectional variation in IV levels and the com-
mon component has an 92% correlation with the short-term implied volatility constructed
from S&P 500 index options. The level of short-term equity option IV appears to be
characterized by a common factor.
Second, a principal component analysis of equity option IV moneyness, known as the
option skew, reveals a signi�cant common component as well. 77% of the variation in
the skew across equities is captured by the �rst principal component. Furthermore, this
common component has a correlation of 64% with the skew of market index options. Third,
60% of the variation in the term structure of equity IV is explained by the �rst principal
component. This component has a correlation of 80% with the IV term slope from S&P
500 index options.
We use the �ndings from the principal component analysis as guidance to develop a
structural model of equity option prices that incorporates a market factor structure. In
line with well-known empirical facts in the literature on index options (see for example
Bakshi, Cao and Chen, 1997; Heston and Nandi, 2000; Bates, 2000; and Jones, 2003),
the model allows for mean-reverting stochastic volatility and correlated shocks to returns
and volatility. Motivated by the principal component analysis, we allow for idiosyncratic
shocks to equity returns which also have mean-reverting stochastic volatility and a separate
leverage e¤ect.
Individual equity returns are linked to the market index using a standard linear factor
model with a constant factor loading. In the resulting two-factor stochastic volatility
3
model, the equity option price is driven by the beta of the stock, the market variance risk
premium, and the market and idiosyncratic variance dynamics.1 The model belongs to the
a¢ ne class, which yields closed-form option pricing formulas. It can be extended to allowing
for market-wide and idiosyncratic jumps.2 Our model shows that equity option sensitivities
and expected returns will depend on beta. It also explains why at-the-money index options
appear to be relatively more expensive than at-the-money options on individual stocks.
The model has three important cross-sectional implications. First, it predicts that
�rms with higher betas have higher implied volatilities. Second, it predicts that �rms
with higher betas have steeper moneyness slopes. Third, higher beta �rms are expected to
have a greater positive (negative) slope when the market variance term-structure is upward
(downward) sloping.
We develop a convenient approach to estimating the model using option data. When
estimating the model on the �rms in the Dow-Jones index, we �nd that it provides a good
�t to observed equity option prices, and the cross-sectional implications of the model are
supported by the data. While it is not the main focus of this paper, our model provides
option-implied estimates of market betas, which is a topic of recent interest, studied by for
example Chang, Christo¤ersen, Jacobs, and Vainberg (2012), and Buss and Vilkov (2012).
Multiple applications in asset pricing and corporate �nance require estimates of beta, such
as cost of capital estimation, performance evaluation, portfolio selection, and abnormal
return measurement.
Our paper is also related to the recent empirical literature on equity options. Dennis and
Mayhew (2002) investigate the relationship between �rm characteristics and risk-neutral
skewness. Bakshi and Kapadia (2003) investigate the volatility risk premium for equity
options. Bakshi, Kapadia, and Madan (2003) derive a skew law for individual stocks,
decomposing individual return skewness into a systematic and idiosyncratic component.
1Existing studies of two-factor SV models mainly focus on index options and do not model a factorstructure in returns. See Taylor and Xu (1994), Bates (2000), and Christo¤ersen, Heston and Jacobs(2009) among others.
2Pan (2002), Broadie, Chernov, and Johannes (2007), and Bates (2008) among others have documentedthe importance of modeling jumps in index options.
4
They �nd that individual �rms display much less (negative) option-implied skewness than
the market index. Duan and Wei (2009) use a version of the model in Bakshi, Kapadia,
and Madan (2003) to relate individual stocks�implied volatilities to systematic risk, and
empirically relate systematic risk to implied volatilities. Bakshi, Cao, and Zhong (2012) in-
vestigate the performance of jump models for equity option valuation. Engle and Figlewski
(2012) develop time series models of implied volatilities and study their correlation dynam-
ics. Kelly, Lustig, and Van Nieuwerburgh (2013) use the model in our paper to study the
pricing of implicit government guarantees to the banking sector. Finally, Carr and Madan
(2012) develop a Levy-based model with factor structure.
Our paper is also related to recent theoretical advances. Mo and Wu (2007) develop
an international CAPM model which has features similar to our model. Elkamhi and
Ornthanalai (2010) develop a bivariate discrete-time GARCH model to extract the market
jump risk premia implicit in individual equity option prices. Finally, Serban, Lehoczky,
and Seppi (2008) develop a non-a¢ ne model to investigate the relative pricing of index and
equity options.
The reminder of the paper is organized as follows. In Section 2 we describe the data
set and present the principal components analysis. In Section 3 we develop the theoretical
model. Section 4 highlights a number of cross-sectional implications of the model. In
Section 5 we estimate the model and investigate its �t to observed index and equity option
prices. Section 6 concludes. The appendix contains the proofs of the propositions.
1.2 Common Factors in Equity Option Prices
In this section we �rst introduce the data set used in our study. We then look for evidence
of commonality in three crucial features of the cross-section of equity options: Implied
volatility levels, moneyness slopes (or skews), and volatility term structures. We rely on a
principal component analysis (PCA) of the �rm-speci�c levels of short-term at-the-money
implied volatility (IV), the slope of IV with respect to option moneyness, and the slope of
IV with respect to option maturity. The results of this model-free investigation will help
5
identify desirable features of a factor model of equity option prices.
1.2.1 Data
We rely on end-of-day implied volatility surface data from OptionMetrics starting on Jan-
uary 4, 1996 and ending on October 29, 2010. We use the S&P 500 index to proxy for
the market factor. For our sample of individual equities, we choose the �rms in the Dow
Jones Industrial Average index at the end of the sample. Of the 30 �rms in the index we
excluded Kraft Foods for which data are not available throughout the sample.
The implied volatility surfaces contain options with more than 30 days and less than
365 days to maturity (DTM). We �lter out options that have moneyness (spot price over
strike price) less than 0:7 and larger than 1:3, those that do not satisfy the usual arbitrage
conditions, those with implied volatility less than 5% and greater than 150%, and those for
which the present value of dividends is larger than 4% of the stock price. For each option
maturity, interest rates are estimated by linear interpolation using zero coupon Treasury
yields. Dividends are obtained from OptionMetrics and are assumed to be known during
the life of each option. For each option we discount future dividends from the current spot
price.
The S&P 500 index options are European, but the individual equity options are Amer-
ican style, and their prices may be in�uenced by early exercise premiums. OptionMetrics
therefore uses binomial trees to compute implied volatility for equity options. Using these
implied volatilities, we can treat all options as European-style in the analysis below.
Table 1 presents the number of option contracts, the number of calls and puts, the aver-
age days-to-maturity, and the average implied volatility. We have a total of 775; 670 index
options and on average 758; 976 equity options per �rm. The average implied volatility
for the market is 20:65% during the sample period. Cisco has the highest average implied
volatility (40:68%) while Johnson & Johnson has the lowest average implied volatility
(22:79%). Table 1 also shows that the data set is balanced with respect to the number of
calls and puts.
6
Table 2 reports the average, minimum, and maximum implied volatility, as well as the
average option vega. Note that the index option vega is much higher than the equity vegas
simply because the S&P500 index values are much larger than the typical stock price.
Figure 1 plots the daily average short-term (30 < DTM < 60) at-the-money (0:95 <
S=K < 1:05) implied volatility (IV) for six �rms (black lines) as well as for the S&P 500
index (grey lines). Figure 1 shows that the variation in the short-term at-the-money (ATM)
equity volatility for each �rm is highly related to S&P 500 volatility.
1.2.2 Methodology
We want to assess the extent to which the time-varying volatilities of equities share one or
more common components. In order to gauge the degree of commonality in risk-neutral
volatilities, we need daily estimates of the level and slope of the implied volatility curve,
and of the slope of the term structure of implied volatility for all �rms and the index. To
obtain these estimates, we run the following regression for �rm j for each day t;
IVj;l;t = aj;t + bj;t ��Sjt =Kj;l
�+ cj;t � (DTMj;l) + �j;l;t; (1.1)
where l denotes an option available for �rm j on day t. The regressors are standardized
each day by subtracting the mean and dividing by the standard deviation. We run the
same regression on index option IVs. We interpret aj;t as a measure of the level of implied
volatilities of �rm j on day t. Similarly, bj;t captures the slope of implied volatility curve,
while cj;t proxies for the slope of the term structure of implied volatility.
Once the regression coe¢ cients have been estimated on each day and for each �rm, we
run a PCA analysis on each of the matrices faj;tg, fbj;tg, and fcj;tg. Table 3 contains the
results from the PCA analysis and Figure 2 plots the �rst principal component as well as
the time series of the corresponding index option coe¢ cients, aI;t, bI;t, and cI;t.
7
Common Factors in the Level of Implied Equity Volatility
Panel A in Table 3 contains the results for implied volatility levels. We show the average,
minimum, and maximum loading across �rms for each component. We also report the
total variation captured as well as the correlation of each component with S&P 500 IV.
The results in Panel A of Table 3 are quite striking. The �rst component captures 77%
of the total cross-sectional variation in the level of IV and it has a 92% correlation with
the S&P 500 index IV. This suggests that the equity IVs have a very strong common
component that is highly correlated with index option IVs.
Panel A in Figure 2 shows the time series of IV levels for index options. Panel B plots
the time series of the �rst PCA component of equity IV. The strong relationship between
the two series is readily apparent.
The second PCA component in Panel A of Table 3 explains 13% of the total variation
and the third component explains 2%. The average loadings on these two components are
close to zero. The explanatory power of the second PCA component suggests the need for
a second, �rm-speci�c, source of variation in equity volatility.
Table A.1 in the Appendix reports the individual loadings for each �rm on each factor.
Table A.1 shows that the loadings on the �rst component are positive for all 29 �rms, illus-
trating the pervasive nature of the common factor. The loadings on the second component
take on a wide range of positive and negative values.
Common Factors in the Moneyness Slope
Panel B in Table 3 contains the results for IV moneyness slopes. The moneyness slopes
contain a signi�cant degree of co-movement. The �rst principal component explains 77% of
cross-sectional variation in the moneyness slope. The second and third components explain
6% and 4% respectively and their average loadings are close to zero.
Panel B also shows that the �rst principal component has a 64% correlation with the
moneyness slope of S&P 500 implied volatility. Equity moneyness slope dynamics clearly
seem driven to a non-trivial extent by the market moneyness slope.
8
Panel C in Figure 2 plots the S&P 500 index IV moneyness slope in the top panel
and the �rst principal component from the equity moneyness slopes is shown in Panel D.
The relationship between the �rst principal component and the market moneyness slope
is readily apparent, but not as strong as for the volatility level in Panels A and B. Table
A.1 in the appendix shows that the �rst component has positive loadings on all 29 �rms
whereas the second and third components have positive and negative loadings for di¤erent
�rms.
Common Factors in the Term Structure Slope
Panel C in Table 3 contains the results for IV term structure slopes. The variation in the
term structure slope captured by the �rst principal component is 60%, which is lower than
for spot volatility (Panel A) and the moneyness slope (Panel B). The correlation between
the �rst component and the term slope of S&P 500 index option IV is 80%, which is higher
than for the moneyness slope in Panel B but lower than for the variance level in Panel A.
The second and third components capture 14% and 5% of the variation respectively and
the wide range of loadings on this factor suggest a scope for �rm-speci�c variation in the
IV term structure for equity options.
Panel E in Figure 2 plots the S&P 500 index IV term structure slope in the top panel
and the �rst principal component from the equity term slopes is shown in Panel F. Most
of the spikes in the S&P 500 term structure slope are clearly evident in the �rst princi-
pal component as well. Table A.1 in the Appendix show that the loadings on the �rst
component are positive for all 29 �rms.
We conclude that the market volatility term structure captures a substantial share of
the variation in equity volatility term structures.
1.2.3 Other Stylized Facts in the Cross-Section of Equity Option Prices
The literature on equity options has documented a number of important cross-sectional
stylized facts. Bakshi, Kapadia, and Madan (2003) derive a skew law for individual stocks,
9
decomposing individual return skewness into a systematic and idiosyncratic component.
They theoretically investigate and empirically document the relationship between risk-
neutral market and equity skewness, which a¤ects the relationship between the moneyness
slope for equity and index options. They �nd that the volatility smile for the market
index is on average more negatively sloped than volatility smiles for individual �rms. They
also show that the more negatively skewed the risk-neutral distribution, the steeper the
volatility smile. Finally, they �nd that the risk-neutral equity distributions are on average
less skewed to the left than index distributions. Duan and Wei (2009) build on Bakshi,
Kapadia, and Madan (2003) to relate implied volatilities to systematic risk.
Some studies document cross-sectional relationships between either betas or systematic
risk, estimated using historical data, and characteristics of the equity IVs. Dennis and
Mayhew (2002) �nd that option-implied skewness tends to be more negative for stocks
with larger betas. Duan and Wei (2009) �nd that the level of implied equity volatility
is related to the systematic risk of the �rm and that the slope of the implied volatility
curve is related to systematic risk as well. Finally, Driessen, Maenhout, and Vilkov (2009)
�nd a large negative index variance risk premium, but �nd no evidence of a negative risk
premium on individual variance risk.
These �ndings are at �rst blush not directly related to the �ndings of the PCA analysis
above, which merely documents a strong factor structure of various aspects of implied
equity volatilities. We next outline a structural equity option modeling approach with a
factor structure that captures the results from the PCA analysis outlined above, but is
also able to match the cross-sectional relationships between betas and implied volatilities
documented by these studies.
1.3 Equity Option Valuation Using a Single-Factor Structure
We model an equity market consisting of n �rms driven by a single market factor, It. The
individual stock prices are denoted by Sjt , for j = 1; 2; :::; n. Investors also have access to
a risk-free bond which pays a return of r.
10
The market factor evolves according to the process
dItIt= (r + �I)dt+ �I;tdW
(I;1)t ; (1.2)
where �I is the instantaneous market risk premium and where volatility is stochastic and
follows the standard square root process
d�2I;t = �I(�I � �2I;t)dt+ �I�I;tdW(I;2)t : (1.3)
As in Heston (1993), �I denotes the long-run variance, �I captures the speed of mean
reversion of �2I;t to �I , and �I measures volatility of volatility. The innovations to the market
factor return and volatility are correlated with coe¢ cient �I . Conventional estimates of �I
are negative and large capturing the so-called leverage e¤ect in aggregate market returns.
Individual equity prices are driven by the market factor as well as an idiosyncratic term
which also has stochastic volatility
dSjt
Sjt� rdt = �jdt+ �j
�dItIt� rdt
�+ �j;tdW
(j;1)t (1.4)
d�2j;t = �j(�j � �2j;t)dt+ �j�j;tdW(j;2)t ; (1.5)
where �j denotes the excess return and �j is the market beta of �rm j.
The innovations to idiosyncratic returns and volatility are correlated with coe¢ cient �j .
As suggested by the skew laws derived in Bakshi, Kapadia, and Madan (2003), asymmetry
of the idiosyncratic return component is required to explain the di¤erences in the price
structure of individual equity and index options. Note that this model of the equity market
has a total of 2(n+ 1) innovations.
To develop more intuition for the model, de�ne total spot variance for �rm j at time t
by
Vj;t � �2j�2I;t + �
2j;t: (1.6)
11
Note that our single-factor stochastic volatility (SV) model for the market index in (1.3)
along with the single-factor SV model for idiosyncratic volatility in (1.5) implies a two-
factor SV model for total equity volatility in (1.6). Two-factor SV models have been stud-
ied in Taylor and Xu (1994), Bates (2000), and Christo¤ersen, Heston and Jacobs (2009),
among others. None of these papers models the factor structure in returns that we capture
through the �j , which allows us to separate total equity variance into a systematic and
an idiosyncratic part. When the market and idiosyncratic variances have di¤erent physi-
cal dynamics and/or carry di¤erent risk premia, beta will have important cross-sectional
implications for the pricing of equity options, which is the central focus of our work.
In order to use our stochastic volatility model of the equity market for option valuation
we need to de�ne a stochastic discount factor (SDF). We assume a standard SDF of the
linear form
dMt
Mt= �rdt�
� (I;1)t dW
(I;1)t +
(I;2)t dW
(I;2)t
��Xj
� (j;1)t dW
(j;1)t +
(j;2)t dW
(j;2)t
�; (1.7)
where t �h (I;1)t ;
(I;2)t ; :::;
(j;1)t ;
(j;2)t
icontains the market prices of risk. Given the SDF
in (2.14), the change-of-measure from the physical (P ) distribution to the risk-neutral (Q)
distribution has the exponential form
dQ
dP(t) �Mt exp(rt) = exp
��tR0
udWu �1
2
tR0
0udDW;W
0Eu u
�; (1.8)
where the vector Wu contains the 2(n + 1) innovations for each of the n �rms and the
market, and d h:; :i is the covariance operator.
Note that the correlation between W (I;1)t and W (I;2)
t and between W (j;1)t and W (j;2)
t
imply that the prices of market and idiosyncratic variance risk are (I;2)t + �I (I;1)t and
(j;2)t + �j
(j;1)t ; respectively. In the spirit of Cox, Ingersoll, and Ross (1985) and Heston
12
(1993), among others, we assume a price of market variance risk of the form
(I;2)t + �I
(I;1)t = �I�I;t:
We also assume that the idiosyncratic variance risk is not priced, so that
(j;2)t + �j
(j;1)t = 0;
although this restriction could be relaxed while still obtaining closed-form option prices.
Together the above assumptions yield the following result.
Proposition 1 Given the change-of-measure in (1.8) the processes governing the market
factor and equity returns under the Q-measure are given by
dItIt
= rdt+ �I;td ~W(I;1)t (1.9)
dSjt
Sjt= rdt+ �j
�dItIt� rdt
�+ �j;td ~W
(j;1)t ; (1.10)
and the Q-processes governing the market and idiosyncratic variances are
d�2I;t = ~�I
�~�I � �2I;t
�dt+ �I�I;td ~W
(I;2)t (1.11)
d�2j;t = �j��j � �2j;t
�dt+ �j�j;td ~W
(j;2)t ; (1.12)
with ~�I = �I +�I�I ; ~�I =�I�I~�I, and where d ~Wt denotes the risk-neutral counterpart of dWt
satisfying d ~Wt = dWt + dDW;W
0Et t. The prices of risk are given by
(I;1)t =
�I � �I�I�2I;t�I;t(1� �2I)
and (I;2)t =�I�
2I;t � �I�I
�I;t(1� �2I)(1.13)
(j;1)t =
�j�j;t(1� �2j )
and (j;2)t = ��j�j
�j;t(1� �2j ): (1.14)
The risk-neutral conditional characteristic function ~�j(� ; u; �j) for the equity price, S
jT , is
13
given by
~�j(� ; u; �j) � EQt
hexp
�iu ln
�SjT
��i(1.15)
=�Sjt
�iuexp
�iur� �
�A(� ; u; �j) +B(� ; u)
�� C(� ; u; �j)�2I;t �D(� ; u)�2j;t
�;
where � = T � t and the expressions for A�� ; u; �j
�, B (� ; u), C
�� ; u; �j
�, and D (� ; u) are
provided in Appendix A.
Proof. See Appendix A.
This proposition provides several insights. Note that the market factor structure is
preserved under Q. Furthermore, the market beta is the same under the risk-neutral and
physical distributions. This result makes betas estimated from option data appropriate for
applications such as capital budgeting. Duan and Wei (2009) argue that the risk-neutral
and objective betas should be the same. The risk-neutral betas in the model in Serban,
Lehoczky and Seppi (2008) are also identical to the physical betas, and they document
empirically that risk-neutral and physical betas are economically and statistically close for
most stocks.
It is also worth noting that in our modeling framework, higher moments and their
premiums, as de�ned by the di¤erence between the moment under P and Q, are a¤ected
by the drift adjustment in the variance processes. We will discuss this further in the next
section.
Our model is developed within an a¢ ne framework, which implies that the characteristic
function for the logarithm of the index level and the logarithm of the equity price can both
be derived analytically. The characteristic function for the index is identical to that in
Heston (1993). Given the characteristic function in (1.15) for the equity price, the price of
a European equity call option with strike price K and maturity � = T � t is
Cjt (Sjt ;K; � ; �j) = Sjt�
j1(�j)�Ke�r��
j2(�j); (1.16)
14
where the risk-neutral probabilities �j1(�j) and �j2(�j) are de�ned by
�j1(�j) =1
2+e�r�
�Sjt
1Z0
Re
"e�iu lnK~�
j(� ; u� i; �j)iu
#du (1.17)
�j2(�j) =1
2+1
�
1Z0
Re
"e�iu lnK~�
j �� ; u; �j
�iu
#du: (1.18)
While these integrals must be evaluated numerically, they are well-behaved and can be
computed quickly.
1.4 Model Properties
In this section we derive a number of important cross-sectional implications from the model
and investigate if the model captures the stylized facts documented in Section ??. We will
also draw some key implications of the model for equity option sensitivities and expected
returns, as well as for the relative pricing of equity and index options. For convenience we
assume that beta is positive for all �rms below. This is not required by the model but it
simpli�es the interpretation of certain expressions.
1.4.1 Factor Structure and Equity Option Sensitivities
The principal component analysis in Section ?? has revealed a strong factor structure
in equity options, and indicated that the most important factor is closely related to the
dynamics of market index options. In our model, equity option prices are partly driven by
changes in the market index price and in market variance. Classic equity option valuation
models do not provide guidance on how equity option prices respond to changes in market
index variables, but our model has clear implications for the sensitivity of equity option
prices with respect to market level and market variance. To better understand how market
variables a¤ect individual equity options, the following result is useful.
Proposition 2 For a derivative contract f j written on the stock price, Sjt , the sensitivity
15
of f j with respect to the index level, It (the market delta), is given by
@f j
@It=@f j
@Sjt
SjtIt�j :
The sensitivity of f j with respect to the market variance (the market vega) is given by
@f j
@�2I;t=
@f j
@Vj;t�2j :
Proof. See Appendix B.
This proposition shows that the beta of the �rm in a straightforward way provides the
link between the usual stock price delta @fj
@Sjtand the market delta, @f
j
@It, and the link between
the usual equity vega, @fj
@Vj;t, and the market vega @fj
@�2I;t. Note of course that high-beta �rms
will have equity option prices that are more sensitive to changes in the index level and in
market variance.
This result allows market participants with portfolios of equity options on di¤erent
�rms to measure and manage their total exposure to the index level and to the market
variance. It also allows investors engaged in dispersion trading, who sell index options
and buy equity options, to measure and manage their overall exposure to market risk and
market variance risk.
Figure 3 plots the market sensitivities from model option prices. Each line has a
di¤erent beta but the same amount of unconditional total equity variance de�ned by ~Vj �
�2j~�I + �j = 0:1. We set the current spot variance to �2I;t = 0:01 and Vj;t = 0:05, and de�ne
the idiosyncratic variance as the residual �2j;t = Vj;t��2j�2I;t. The market index parameters
are ~�I = 5; ~�I = 0:04; �I = 0:5; �I = �0:8; and the individual equity parameters are
�j = 1; �j = 0:4; and �j = 0. The risk-free rate is 4% per year and option maturity is
3 months. Additionally, we set Sjt =It = 0:1. We plot the market delta (top panel) and
the market vega (bottom panel) against moneyness for �rms with di¤erent betas. The top
panel of Figure 3 shows that the di¤erences in market deltas across �rms with di¤erent
16
betas can be substantial for ATM and ITM call options. The bottom panel of Figure 3
shows that the di¤erences in market vega are also substantial�particularly for ATM calls
where the option exposure to total variance is the largest.
1.4.2 Factor Structure and Equity Option Returns
So far we have focused on option price levels and their sensitivities. It is also instructive to
see the model�s implications for expected equity option returns. The following proposition
provides an expression for the expected (P -measure) equity option return as a function of
the expected market return.3
Proposition 3 For a derivative f j written on the stock price, Sjt , the expected excess
return on the derivative contract is given by:
1
dtEPt
�df j
f j� rdt
�=
@f j
@Sjt
Sjtf j��j + �j�I
�dt
=@f j
@Sjt
Sjtf j�j +
@f j
@Sjt
Sjtf j�j�I
=@f j
@Sjt
Sjtf j�j +
@f j
@It
Itf j�I ;
where @fj
@Itis given in Proposition 2.
Proof. See Appendix C.
The model thus decomposes the excess return on the option into two parts: The delta
of the equity option and the beta of the stock. Put di¤erently, equity options provide
investors with two sources of leverage: First, the beta with respect to the market, and
second, the elasticity of the option price with respect to changes in the stock price.
In Figure 4 we use the parameter values from Figure 3 and set the equity market risk
premium �I equal to 0:075:We plot the expected excess return on equity call options (top
3Recent empirical work on equity and index option returns includes Broadie, Chernov, and Johannes(2009), Goyal and Saretto (2009), Constantinides, Czerwonko, Jackwerth, and Perrakis (2011), Vasquez(2011), and Jones and Wang (2012).
17
panel) and on put options (bottom panel) in percent per day against moneyness for �rms
with di¤erent betas. The top panel of Figure 4 shows that the di¤erences in expected call
returns across �rms with di¤erent betas can be substantial for OTM calls where option
leverage in general is highest. The bottom panel of Figure 4 shows that put option expected
excess returns (which are always negative) also vary most across �rms with di¤erent betas
when the put options are OTM. In general the di¤erences in expected excess returns across
betas are smaller for put options (bottom panel) than for call options (top panel).
1.4.3 The Relative Pricing of Index and Equity Options
The factor structure in our model combined with the market variance premium also has
implications for the relative pricing of index and equity options. A negative variance risk
premium implies that EQt [�2I;t:T ] > EPt [�
2I;t:T ], where the expected integrated market index
variance under P and Q is de�ned by
EPt [�2I;t:T ] � EPt
24 TZt
�2I;sds
35 and EQt [�2I;t:T ] � EQt
24 TZt
�2I;sds
35 : (1.19)
This combined with our assumption that idiosyncratic volatility is not priced, so that
EQt [�2j;t:T ] = EPt [�
2j;t:T ], implies we can write
�2j +EPt [�
2j;t:T ]
EPt [�2I;t:T ]
> �2j +EQt [�
2j;t:T ]
EQt [�2I;t:T ]
:
Multiplying by both denominators yields
��2jE
Pt [�
2I;t:T ] + E
Pt [�
2j;t:T ]
�EQt [�
2I;t:T ] >
h�2jE
Qt [�
2I;t:T ] + E
Qt [�
2j;t:T ]
iEPt [�
2I;t:T ]
, EPt [Vj;t:T ]EQt [�
2I;t:T ] > EQt [Vj;t:T ]E
Pt [�
2I;t:T ];
18
which in turn enables us to write
EQt [�2I;t:T ]
EPt [�2I;t:T ]
>EQt [Vj;t:T ]
EPt [Vj;t:T ]; (1.20)
where EQt [Vj;t:T ] and EPt [Vj;t:T ] denote expected integrated �rm variance under P and Q.
Our model therefore implies that the variance risk premium will be smaller in equity
options than in market index options so that a variance swap (or an at-the-money option)
on the market index will be relatively more expensive than a variance swap (or an at-
the-money option) on the individual equity. This is in line with the empirical evidence in
Bakshi, Kapadia and Madan (2003) among others.
1.4.4 The Level of Equity Option Volatility
Duan and Wei (2009) show empirically that �rms with higher systematic risk have a higher
level of risk-neutral variance. We now investigate if our model is consistent with this
empirical �nding.
Recall from above that the total spot variance for �rm j at time t is de�ned by
Vj;t � �2j�2I;t + �
2j;t:
By decomposing the P -expectation into integrated market variance and idiosyncratic vari-
ance, we have
EPt [Vj;t:T ] = �2jEPt [�
2I;t:T ] + E
Pt [�
2j;t:T ]:
Given our model, the expectation of the integrated total variance for equity j under Q
is
EQt [Vj;t:T ] = �2jEQt [�
2I;t:T ] + E
Qt [�
2j;t:T ] = �2jE
Qt [�
2I;t:T ] + E
Pt [�
2j;t:T ]:
where the second equation holds when idiosyncratic risk is not priced.
For any two �rms having the same level of expected total variance under the P -measure
19
(EPt [V1;t:T ] = EPt [V2;t:T ]) we have
EPt [�21;t:T ]� EPt [�22;t:T ] = �(�21 � �22)EPt [�2I;t:T ]:
Therefore
EQt [V1;t:T ]� EQt [V2;t:T ] = (�21 � �22)E
Qt [�
2I;t:T ] +
�EQt [�
21;t:T ]� E
Qt [�
22;t:T ]
�= (�21 � �22)E
Qt [�
2I;t:T ] +
�EPt [�
21;t:T ]� EPt [�22;t:T ]
�= (�21 � �22)
�EQt [�
2I;t:T ]� EPt [�2I;t:T ]
�:
When the market variance premium is negative, we have ~�I > �I which implies that
EQt [�2I;t:T ] > EPt [�
2I;t:T ]. We therefore have that
�1 > �2 , EQt [V1;t:T ] > EQt [V2;t:T ]:
We conclude that the model is consistent with the �nding in Duan and Wei (2009) that
�rms with high betas tend to have a high level of risk-neutral variance.
1.4.5 Equity Option Skews
To help understand the slope of the equity option implied volatility moneyness curve in our
model, the next proposition shows how beta and index skewness impact equity skewness.
Proposition 4 The conditional total skewness of the integrated returns of �rm j under
P , denoted by TSkPj , is given by
TSkPj;t:T � SkP
Z T
t
dSju
Sju
!= �3j �
EPt [�
2I;t:T ]
EPt [Vj;t:T ]
!3=2� SkPI;t:T +
EPt [�
2j;t:T ]
EPt [Vj;t:T ]
!3=2� SkPj;t:T :
(1.21)
The conditional total skewness of the integrated returns of �rm j under Q, denoted by
20
TSkQj , is given by
TSkQj;t:T � SkQ
Z T
t
dSju
Sju
!= �3j �
EQt [�
2I;t:T ]
EQt [Vj;t:T ]
!3=2� SkQI;t:T +
EQt [�
2j;t:T ]
EQt [Vj;t:T ]
!3=2� SkQj;t:T ;
(1.22)
where SkI;t:T = Sk�R TtdIsIs
�and Skj;t:T = Sk
�R Tt �j;sdW
(j;1)s
�is the market and idiosyn-
cratic �rm j skewness, respectively.
Proof. See Appendix D.
This result shows that �j matters for determining �rm j�s conditional total skewness
and thus the moneyness slope of its implied volatility curve.
Equation (1.22) shows that under the risk neutral measure, �j a¤ects the slope of the
equity implied volatility curve through TSkQj;t:T by a¤ecting the loading on the market
skewness. To see this, consider two �rms with the same expected total variance under Q so
that EQt [V1;t:T ] = EQt [V2;t:T ] which implies EQt [�
2I;t:T ]=E
Qt [V1;t:T ] = EQt [�
2I;t:T ]=E
Qt [V2;t:T ].
When �1 > �2, Firm 1 has a greater loading on index risk-neutral skewness and �rm 2
has a greater loading on idiosyncratic skewness. When the index Q-distribution is more
negatively skewed than the idiosyncratic equity distribution, as found empirically in Bakshi,
Kapadia, and Madan (2003), then our model predicts that higher-beta �rms will have
more negatively skewed Q-distributions. Note that this prediction is in line with the cross-
sectional empirical �ndings of Duan and Wei (2009) and Dennis and Mayhew (2002).
The top panel of Figure 5 plots the implied Black-Scholes volatility from model option
prices based on the parameter values from Figure 3. The top panel in Figure 5 shows that
beta has a substantial impact on the moneyness slope of equity IV even when keeping the
total variance constant: The higher the beta, the larger the moneyness slope.
The factor structure in our model combined with the market variance premium also
has implications for the relative importance for total skewness of the market factor under
21
the Q measure and under the P measure. We can rewrite (1.20) to get
EQt [�2I;t:T ]
EQt [Vj;t:T ]>EPt [�
2I;t:T ]
EPt [Vj;t:T ]: (1.23)
Our model therefore implies that the loading of total equity skewness, TSkj;t:T , on
market skewness, SkI;t:T , is higher under the Qmeasure in (1.22) than under the P measure
in (1.21). This in turn shows that the model can potentially capture the strong commonality
in moneyness slopes across �rms documented in Section ??.
Bakshi, Kapadia, and Madan (2003) discuss how market skewness in�uences total eq-
uity skewness relative to idiosyncratic skewness. Duan and Wei (2009) highlight the impor-
tance of systematic risk for explaining the cross-section of equity options. We contribute by
showing the role of beta in determining total equity skewness in a fully-speci�ed dynamic
stochastic volatility model. Our results highlight that the negative market variance risk
premium, a robust stylized fact, is critical for explaining the cross-sectional impact of beta.
1.4.6 The Term Structure of Equity Volatility
Our model implies the following two-component term-structure of equity variance
EQt [Vj;t:T ] =��2j~�I + �j
�+ �2j
��2I;t � ~�I
�e�~�I(T�t) +
��2j;t � �j
�e��j(T�t): (1.24)
This expression shows how the term structure of market variance a¤ects the term structure
of variance for �rm j. Given di¤erent systematic and idiosyncratic mean reversion speeds
(~�I 6= �j), �j has important implications for the term-structure of volatilities. In the
empirical work below, we �nd that the risk-neutral idiosyncratic variance process is more
persistent than the market variance. When the idiosyncratic variance process is more
persistent (~�I > �j), higher values of beta imply a faster reversion toward the �rst term
in (1.24), the unconditional total variance ( ~Vj = �2j~�I + �j). As a result, when the market
variance process is less persistent than the idiosyncratic variance, �rms with higher betas
are likely to have steeper volatility term-structures. In other words, higher beta �rms
22
are expected to have a greater positive (negative) term structure slope when the market
variance term-structure is upward (downward) sloping.
The bottom panel of Figure 5 plots the implied Black-Scholes volatility from model
prices against option maturity. Each line has a di¤erent beta but the same amount of
unconditional total equity variance ~Vj = �2j~�I + �j = 0:1. We set the current spot variance
to �2I;t = 0:01 and Vj;t = 0:05, and de�ne the idiosyncratic variance as the residual �2j;t =
Vj;t � �2j�2I;t. The parameter values are as in Figure 3. The bottom panel of Figure 5
shows that beta has a non-trivial e¤ect on the IV term structure: The higher the beta, the
steeper the term structure when the term structure is upward sloping.
In summary, our model makes a number of qualitative predictions for the role of the
market variance risk premium and beta for the pricing of individual equity options: First,
equity option sensitivities and expected returns will depend on beta. Second, at-the-money
index options will be relatively more expensive than at-the-money options on individual
stocks. Third, higher beta stocks will have higher option values even when keeping the
physical volatility �xed. Fourth, stocks with higher beta will have implied volatility curves
with steeper moneyness slopes. Finally, stocks with higher betas will revert more quickly
to the long-run implied volatility.
We now estimate our model on index and equity options in order to quantify the role
of beta in equity option valuation.
1.5 Estimation and Fit
In this section, we �rst describe our estimation methodology. Subsequently we report on
parameter estimates and model �t. Finally we relate the estimated betas to patterns in
observed equity option IVs.
1.5.1 Estimation Methodology
Several approaches have been proposed in the literature for estimating stochastic volatility
models. Jacquier, Polson, and Rossi (1994) use Markov Chain Monte Carlo to estimate
23
a discrete-time stochastic volatility model. Pan (2002) uses GMM to estimate the objec-
tive and risk neutral parameters from returns and option prices. Serban, Lehoczky, and
Seppi�s (2008) estimation strategy is based on simulated maximum likelihood using the
EM algorithm and a particle �lter.
Another approach treats the latent volatility states as parameters to be estimated and
thus avoids �ltering the latent volatility factor. This strategy has been adopted by Bates
(2000) and Santa-Clara and Yan (2010) among others. We follow this strand of literature.
Recall that we need to estimate two vectors of latent variables f�2I;t, �2j;tg and two sets
of structural parameters f�I , �jg, where �I � f~�I ; ~�I ; �I ; �Ig and �j � f�j ; �j ; �j ; �j ; �jg.
Our methodology involves two main steps.
In the �rst step, we estimate the market index dynamicn�I ; �
2I;t
obased on S&P 500
option prices alone. In the second step, we use equity options for �rm j only, we take the
market index dynamic as given, and we estimate the �rm-speci�c dynamicsn�j ; �
2j;t
ofor
each �rm conditional on estimates ofn�I ; �
2I;t
o. This step-wise estimation procedure is
not fully econometrically e¢ cient but it enables us to estimate our model for 29 equities
while ensuring that the same dynamic is imposed for the market-wide index for each of
the 29 �rms. We have con�rmed that this estimating technique has good �nite sample
properties in a Monte Carlo study which is available from the authors upon request.
Each of the two main steps contains an iterative procedure which we now describe in
detail.
Step 1: Parameter Estimation for the Index
Given a set of starting values, �0I , for the structural parameters characterizing the index,
we �rst estimate the spot market variance each day by solving
�2I;t = argmin�2I;t
NI;tXm=1
(CI;t;m � Cm(�0I ; �2I;t))2=V ega2I;t;m, for t = 1; 2; :::; T; (1.25)
24
where CI;t;m is the market price of index option contract m on day t, Cm(�I ; �2I;t) is
the model index option price, NI;t is the number of index contracts available on day t;
and V egaI;t;m is the Black-Scholes sensitivity of the index option price with respect to
volatility evaluated at the implied volatility. These vega-weighted dollar price errors are
a good approximation to implied volatility errors and the computational cost involved is
much lower.4
Once the set of T market spot variances is obtained, we solve for the set of parameters
characterizing the index dynamic as follows
�I = argmin�I
NIXm;t
(CI;t;m � Cm(�I ; �2I;t))2=V ega2I;t;m, (1.26)
where NI �PTt NI;t represents the total number of index option contracts available.
We iterate between (1.25) and (1.26) until the improvement in �t is negligible, which
typically requires 5-10 iterations.
Step 2: Parameter Estimation for Individual Equities
Given an initial value �0j and the estimated �2I;t and �I we can estimate the spot equity
variance each day by solving
�2j;t = argmin�2j;t
Nj;tXm=1
(Cj;t;m � Cm(�0j ; �I ; �2I;t; �2j;t))2=V ega2j;t;m, for t = 1; 2; :::T; (1.27)
where Cj;t;m is the price of equity option m for �rm j with price t, Cm(�j ;�I ; �2I;t; �2j;t)
is the model equity option price, Nj;t is the number of equity contracts available on day t;
and V egaj;t;m is the Black-Scholes Vega of the equity option.
Once the set of T market spot variances is obtained, we solve for the set of parameters
4This approximation has been used in Carr and Wu (2007) and Trolle and Schwartz (2009) among others.
25
characterizing the equity dynamic as follows
�j = argmin�j
NjXm;t
(Cj;t;m � Cm(�j ; �I ; �2I;t; �2j;t))=V ega2j;t;m; (1.28)
where Nj �PTt Nj;t is the total number of contracts available for security j.
We again iterate between (1.27) and (1.28) until the improvement in �t is negligible.
We repeat this estimation procedure for each of the 29 �rms in our data set.
1.5.2 Parameter Estimates
This section presents estimation results for the market index and the 29 �rms for the
1996-2010 period. In order to speed up estimation, we restrict attention to put options
with moneyness in the range 0:9 � S=K � 1:1 and maturities of 2, 4, and 6 months.
We estimate the structural parameters in the model on a panel data set consisting of the
options available on the �rst Wednesday of each month. We end up using a total of 150; 455
equity options and 6; 147 index options when estimating the structural parameters. We
estimate the spot variances on each trading day thus using more than 3:1 million equity
options and 128; 532 index options.
Table 4 reports estimates of the structural parameters that characterize the dynamics
of the systematic variance and the idiosyncratic variance, as well as estimates of the betas.
The top row shows estimates for the S&P 500 index.
The unconditional risk-neutral market index variance ~�I = 0:0610 corresponds to
24:70% volatility per year. Based on the average index spot variance path for the sample,
1T
PTt=1 �
2I;t, we obtain a volatility of 22:23%. The idiosyncratic �j estimates range from
0:0018 for American Express to 0:0586 for Cisco.
The estimate of the mean-reversion parameter for the market index variance ~�I is equal
to 1:13, which corresponds to a daily variance persistence of 1� 1:13=365 = 0:9969 which
is very high, consistent with the existing literature. The idiosyncratic �j range from 0:15
for Bank of America to 1:29 for Merck, indicating that idiosyncratic volatility is highly
26
persistent as well. Only �ve �rms in the sample (JP Morgan, Hewlett-Packard, Intel, IBM,
and Merck) have an idiosyncratic variance process that is less persistent than the market
variance.
Our estimate of the volatility of variance parameter, �I , for the market is 0:371 which
is in line with the estimates found by Bakshi, Cao and Chen (1997), Bates (2000) and Pan
(2002). The volatility of the idiosyncratic variance, �j , is on average 0:180 and ranges from
0:054 for American Express to 0:333 for Cisco.
The estimate of �I is strongly negative (�0:855), capturing the so-called leverage e¤ect
in the index. The idiosyncratic �j are negative for all �rms except one, ranging from �0:724
for Bank of America to +0:297 for Exxon Mobil.
The estimates of beta are reasonable and vary from 0:70 for Johnson & Johnson to 1:24
for American Express. The average beta across the 29 �rms is 0:99. We will analyze the
beta estimates in more detail below.
The average total spot volatility (ATSV) for �rm j is computed as
ATSV =
vuut 1
T
TXt=1
Vj;t =
vuut 1
T
TXt=1
��2j�
2I;t + �
2j;t
�:
Comparing the beta column with the ATSV column in Table 4 shows that ATSV is generally
high when beta is high.
The �nal column of Table 4 reports the systematic risk ratio (SSR) for each �rm. It is
computed from the spot variances as follows
SSR =
PTt=1 �
2j�2I;tPT
t=1
��2j�
2I;t + �
2j;t
� :Table 4 shows that the systematic risk ratio varies from close to 32% for Hewlett-Packard to
above 70% for Exxon Mobile. The systematic risk ratio is 46% on average, indicating that
the estimated factor structure is strongly present in the equity option data. Comparison
of the beta column with the SSR column in Table 4 shows that �rms with similar betas
27
can have radically di¤erent SSR and, vice versa, �rms with very di¤erent betas can have
roughly similar SSRs. This �nding of course suggests a key role for the idiosyncratic
variance dynamic in the model.
1.5.3 Model Fit
We measure model �t using the root mean squared error (RMSE) based on the vegas,
which is consistent with the criterion function used in estimation
Vega RMSE �r1
N
XN
m;t(Cm;t � Cm;t(�))2=V ega2m;t:
We also report the implied volatility RMSE de�ned as
IVRMSE �r1
N
XN
m;t(IVm;t � IV (Cm;t(�)))2;
where IVm;t denotes market IV for option m on day t and IV (Cm;t(�)) denotes model IV.
We use Black-Scholes to compute IV for both model and market prices.
Table 5 reports model �t for the market index and for each of the 29 �rms. We report
results for all contracts, as well as separate results for in- and out-of-the-money puts, and
for 2-month and 6-month at-the-money (ATM) contracts. We also report the IVRMSE
divided by the average market IV in order to assess relative IV �t. Several interesting
�ndings emerge from Table 5.
� First, the Vega RMSE approximates the IVMRSE closely for the index and for all
�rms. This suggests that using Vega RMSE in estimation does not bias the IVRMSE
results.
� Second, the average IVRMSE across �rms is 1:20% and the relative IV (IVRMSE/Average
IV) is 4:05% on average. The �t is very similar across �rms. Overall the �t of the
model is thus quite good across �rms. The best pricing performance for equity op-
tions is obtained for Coca Cola with an IVRMSE of 0:95%. The worst �t is for
28
General Electric with an IVRMSE of 1:64%. Based on the relative IVRMSE, the
best �t is for Intel with 2:90% and the worst is again for GE with 5:66%.
� Third, the average IVRMSE �t across �rms for ITM puts is 1:17% and for OTM puts
it is 1:23%. Using this metric the model �ts ITM and OTM puts roughly equally
well.
� Fourth, the average IVRMSE �t across �rms for 2-month ATM options is 1:10% and
for 6-month ATM options it is 1:08%. The model thus �ts 2-month and 6-month
ATM options equally well on average.
Figure 6 reports the market IV (solid) and model IV (dashed) averaged over time for
di¤erent moneyness categories for each �rm. The black lines (left axis) show the average
on days with above-average IV and the grey lines (right axis) show the average for days
with below-average IV. Moneyness is on the horizontal axis, measured by S=K, with ITM
puts shown on the left side and OTM puts on the right side. Figure 6.A reports on the �rst
15 �rms and Figure 6.B reports on the remaining 14 �rms as well as the index. Note that
in order to properly see the di¤erent patterns across �rms, the vertical axis scale di¤ers in
each subplot, but the range of implied volatility values is kept �xed at 10% across �rms to
facilitate comparisons.
Figure 6 shows that the smiles computed using market prices vary considerably across
�rms, both in terms of level and shape. It is noteworthy that for many of these large
�rms, the smile looks more like an asymmetric smirk�especially on low-volatility days (grey
lines). The IV bias by moneyness are small in general across �rms and no large outliers
are apparent. The model tends to slightly underprice OTM equity puts when volatility is
high (black lines). This is not the case when volatility is low (grey lines).
The smirk is of course a strong stylized fact for index options and it is evident in the
bottom-right panel of Figure 6.B. Comparing with the other panels con�rms the �nding in
Bakshi, Kapadia and Madan (2003) that the market index is generally more (negatively)
skewed than individual �rms. Note that when allowing for a large negative �I ; the Heston
29
(1993) model is able to �t the relatively expensive OTM index put options quite well, but
it still has a small bias and requires additional negative skewness. This could be achieved
by including jumps in returns (Bates, 2000).
Figure 7 reports for each �rm the average (over time) implied volatility as a func-
tion of time to maturity (in years). We split the data set into two groups: Days with
upward-sloping IV term structure and days with downward-sloping IV term structure. We
then compute the median slope on the upward-sloping days and the median slope on the
downward-sloping days. In Figure 7 we report the average market IVs (solid lines) as
well as the average model IVs (dashed lines) on the days with higher-than-median upward-
sloping term structure (grey lines) and on the days with lower-than-median downward-
sloping term structure (black lines). This is done because on many days the term structure
is roughly �at and thus uninteresting. The downward-sloping black lines use the left axis
and the upward-sloping grey lines use the right axis. In order to facilitate comparison
between model and market IVs the level of IVs di¤er between the left and right axis and
they di¤er across �rms. For ease of comparison between term structures the di¤erence
between the minimum and maximum on each axis is �xed at 10% across all �rms.
Figure 7 shows that the term structure of IV di¤ers considerably across �rms. Some
�rms such as Hewlett-Packard tend to mean-revert rather quickly, whereas other �rms such
as 3M have much more persistent term structures. Generally, across �rms, the downward
sloping black lines appear to be steeper than the upward sloping grey lines. This pattern
is matched by the model. It is also worth noting that the model is able to capture the
strong persistence in IV quite well: Figure 7 does not reveal any systematic model biases
in the term structure of IVs. The two-factor stochastic volatility structure of our equity
model is clearly helpful in this regard.
We conclude from Table 5 and Figures 6 and 7 that the model �ts the observed equity
option data quite well. Encouraged by this �nding, we next analyze in some detail how
the estimated betas are related to observed patterns in equity option IVs.
30
1.5.4 Equity Betas and Equity Option IVs
The three main cross-sectional predictions of our model, as discussed in Section ??, are as
follows:
1. Firms with higher betas have higher risk-neutral variance.
2. Firms with higher betas have steeper moneyness slopes. This is equivalent to stating
that �rms with higher betas are characterized by more negative skewness.
3. Firms with higher betas have steeper positive volatility term structures when the term
structure is upward sloping, and steeper negative volatility term structures when the
term structure is downward sloping.
We now document if these theoretical model implications are supported by the estimates
for the 29 Dow-Jones �rms. Consider �rst the level of option-implied volatility. In the top
panel of Figure 8, we scatter plot the time-averaged intercepts from the implied volatility
regression in (1.1), 1TPTt=1 aj;t against the beta estimates from Table 4 for each �rm j. We
then run a regression on the 29 points in the scatter and assess the signi�cance and �t.
The slope has a t-statistic of 6:81 and the regression �t (R2) is quite high at 63%. The
regression line shows the positive relationship between the estimated betas and the average
implied volatility observed in the market prices of equity options.
In the middle panel of Figure 8, we scatter plot the moneyness slope coe¢ cients from
the IV regression in (1.1), 1TPTt=1 bj;t against the beta estimate from Table 4 for each �rm
j. In the moneyness slope regression, the sensitivity to beta has a t-statistic of 4:66 and
an R2 of 45%. The middle panel of Figure 8 clearly shows that higher beta estimates are
associated with steeper slopes of the IV moneyness smile.
Finally, in the bottom panel of Figure 8 we scatter plot the absolute value of the term
structure slope coe¢ cients from (1.1), 1TPTt=1 cj;t against the beta estimate from Table 4
for each �rm. In the term slope regression, the sensitivity to beta has a t-statistic of 4:90
and the R2 is 47%. Panel C shows that higher betas are associated with higher absolute
31
slopes of the term structure in equity IVs: Firms with high betas will tend to have a term
structure of implied volatility curve that decays more quickly to the unconditional level of
volatility compared with �rms with low betas.
We conclude that the cross-sectional relation between our estimates of beta and the
model-free measures of IV level, slope, and term structure is consistent with the three main
model predictions from Section ??.
1.5.5 Option-Implied and Regression Betas
As discussed in section ??, the betas estimated from options seem reasonable. They vary
from 0:70 for Johnson & Johnson to 1:24 for American Express and the average beta across
the 29 �rms is 0:99. To provide additional perspective we also compute regression-based
OLS betas for the same 29 �rms. To be consistent with the option-based estimate, we
estimate a constant beta using daily return data for the entire sample from 1996 to 2010.
The OLS beta is 0:97 on average across �rms.
The top panel of Figure 9 provides a scatter plot of OLS betas versus option-implied
betas. The black line shows the result of a cross-sectional regression of the OLS betas on
the option-implied betas. A number of important conclusions obtain. First, the option-
implied betas are positively correlated with the OLS betas. In fact, the relation between
the two beta estimates is very strong, which is evidenced by the high R-square of the
regression (84%) and the fact that the top panel of Figure 9 contains very few outliers.
Second, option-implied betas have a smaller dispersion (15%) than historical betas (31%).
Note that this larger dispersion of the historical betas yields a regression slope larger than
one and a negative regression intercept when regressing historical beta on option implied
beta.
The pattern of betas in the top panel of Figure 9 is also interesting in light of the well-
known �nite sample biases in estimating OLS betas, and the common practice of shrinking
the betas toward one to account for this bias. To reduce the potential small-sample bias in
OLS beta, we follow Vasicek (1973) and shrink the raw time-series estimate of beta toward
32
the cross-sectional mean according to
�SHj = ! � �OLSj + (1� !) � 1;
where �OLSj is the OLS beta estimate. Following Vasicek (1973) and Frazzini and Pedersen
(2013), among others, we set ! = 0:6.
When we plot the betas with shrinkage versus the option-implied betas in the bottom
panel of Figure 9, the results are perhaps even more striking. Note how the 45� degree line
(in grey) now almost perfectly lines up with the �tted line (in black) obtained by regressing
the betas with shrinkage on the option-implied betas.
We conclude that overall the relationship between regression-based and option-implied
beta is surprisingly strong. It may prove interesting to see if this relationship also holds
for betas computed over shorter windows of options and returns. We leave that for future
work.
1.5.6 The Cross-Section of Idiosyncratic Risk
A number of recent studies investigate co-movements between �rm-level volatilities. Engle
and Figlewski (2012) model the dynamics of correlations between implied volatilities, and
investigate the role of VIX as a factor in explaining �rm-level implied volatilities. Schürho¤
and Ziegler (2010) study the relative pricing of equity and index variance swaps. Kelly,
Lustig, and Van Nieuwerburgh (2012) show that there is a strong factor structure in �rm-
level historical volatility, distinct from the common variation in returns. Surprisingly, they
�nd that idiosyncratic volatility contains a factor structure similar to the one in total
volatility.
Motivated by these �ndings, Table 6 presents the correlation matrix between the idio-
syncratic variances for the 29 �rms estimated from the model. Clearly, Table 6 con�rms the
results of Kelly, Lustig, and Van Nieuwerburgh (2012), which are obtained using historical
returns data.
33
While Table 6 may be interpreted as suggesting the need for a richer factor model, note
that the results of Kelly, Lustig, and Van Nieuwerburgh (2012) are robust to the inclusion
of additional known factors. Note also that while correlated idiosyncratic variances signal
the presence of additional factors, it is not necessarily the case that these additional factors
are priced in equity options.
1.6 Summary and Conclusions
Principal component analysis reveals a strong factor structure in equity options. The �rst
common component explains 77% of the cross-sectional variation in IV and the common
component has a 92% correlation with the short-term implied volatility constructed from
S&P 500 index options. Furthermore, 77% of the variation in the equity skew is captured
by the �rst principal component. This common component has a correlation of 64% with
the skew of market index options. Also, 60% of the variation in the term structure of
equity IV is explained by the �rst principal component. This component has a correlation
of 80% with the term slope of the option IV from S&P500 index options.
Motivated by the �ndings from the principal component analysis, we develop a struc-
tural model of equity option prices that incorporates a market factor. The model allows
for mean-reverting stochastic volatility and correlated shocks to returns and volatility. Mo-
tivated by the principal components analysis, we allow for idiosyncratic shocks to equity
prices which also have mean-reverting stochastic volatility and a separate leverage e¤ect.
Individual equity returns are linked to the market index using a standard linear factor
model with a constant beta factor loading. We derive closed-form option pricing formulas
as well as results for option hedging and option expected returns. The model has three
important cross-sectional implications: higher beta stocks have higher option values for a
given physical volatility �xed, higher beta stocks have implied volatility curves with steeper
moneyness slopes, and higher beta stocks revert more quickly to the mean long-run implied
volatility.
We develop a convenient estimation method for estimation and �ltering based on option
34
prices. When estimating the model on the �rms in the Dow-Jones index, we �nd that it
provides a good �t to observed equity option prices. Moreover, we show that the estimates
strongly con�rm the three main cross-sectional model implications.
Several issues are left for future research. First, it would be interesting to empirically
study the implications of our models for option price sensitivities and option returns.
Second, it may be useful to extend the model, for instance by allowing for two stochastic
volatility factors in the market price process, as in Bates (2000), or by allowing for jumps
in the market price (Bates, 2008; Bollerslev and Todorov, 2011). Third, combining option
information with high-frequency returns when estimating the parameters in our model
following Andersen, Fusari and Todorov (2013) may lead to better estimates of the spot
volatility process and of beta (see also Patton and Verardo, 2012; Hansen, Lunde, and
Voev, 2012). Joint estimation on returns and options will also allow us to identify the
variance risk premium. Finally, characterizing the time-variation in option-implied betas
would be of signi�cant interest.
Appendix
This appendix collects proofs of the propositions.
A.1 Proof of Proposition 1. Part 1: Risk-Neutral Distribution
First, de�ne the stochastic exponential �(:)
�
�tR0
!0udWu
�� exp
�tR0
!0udWu �
1
2
tR0
!0udDW;W
0Eu!u
�; (1.29)
where !u is a 2(n+1) real or complex valued vector adapted to the Brownian �ltration (see
Protter (1990) p. 85). Given the de�nition of �(:), we can express the change-of-measure
(1.8) asdQ
dP(t) = �
��tR0
0udWu
�; (1.30)
35
where Wu �hW(I;1)u ;W
(I;2)u ; :::;W
(j;1)u ;W
(j;2)u
i0and u �
h (I;1)u ;
(I;2)u ; :::;
(j;1)u ;
(j;2)u
i0are
the vectors of innovations and market prices of risk, respectively. Given (1.4) and the
de�nition of �(:), we can write
Sjt
Sj0= �
�tR0
�j�I;udW(I;1)u +
tR0
�j;udW(j;1)u
�exp((r + �j + �j�I)t): (1.31)
By imposing the no-arbitrage condition on the individual equity prices, Sjt , we must have
EPs
"Sjt
Sjs
dQdP (t)dQdP (s)
exp(�r(t� s))#= 1, R(t) � Sjt
Sj0
dQ
dP(t) exp(�rt) is a P -martingale.
Therefore, Sjt �s no-arbitrage condition constrain the process R(t) to be a P -martingale.
Note that we have omitted the superscript j in the de�nition of R(t). Given (1.30) and
(1.31), R(t) can be orthogonalized in the following manner R(t) = F (t)G(t) where
F (t) � exp((�j + �j�I)t)�
�tR0
�j�I;udW(I;1)u
��
��tR0
(I;1)u dW (I;1)u �
tR0
(I;2)u dW (I;2)u
�(1.32)
�
�tR0
�j;udW(j;1)u
��
��tR0
(j;1)u dW (j;1)u �
tR0
(j;2)u dW (j;2)u
�;
and
G(t) � �
0@�Xk=2j;I
�tR0
(k;1)u dW (k;1)u +
tR0
(k;2)u dW (k;2)u
�1A :
where in order to decompose dQdP (t) and Sjt =S
j0 we have used �(Xt + Yt) = �(Xt)�(Yt)
for orthogonal processes. By properties of stochastic exponentials, we know that �(:)
are P -martingales which implies that G(t) is a P -martingale. Since F (t) and G(t) are
independent, R(t) will be a P -martingale if and only if F (t) is a P -martingale. Using the
result �(Xt)�(Yt) = �(Xt + Yt) exp(hX;Y it) to rewrite (1.32), we have
�
�tR0
�j�I;udW(I;1)u
��
��tR0
(I;1)u dW (I;1)u �
tR0
(I;2)u dW (I;2)u
�= �
�tR0
��j�I;u � (I;1)u
�dW (I;1)
u �tR0
(I;2)u dW (I;2)u
�exp
��tR0
�j�I;u
� (I;1)u + �I
(I;2)u
�du
�;
36
and
�
�tR0
�j;udW(j;1)u
��
��tR0
(j;1)u dW (j;1)u �
tR0
(j;2)u dW (j;2)u
�= �
�tR0
��j;u � (j;1)u
�dW (j;1)
u �tR0
(j;2)u dW (j;2)u
�exp
��tR0
�j;u
� (j;1)u + �j
(j;2)u
�du
�:
Combining the previous expressions with (1.32), we see that F (t) will be a P -martingale
whenever
exp
��tR0
�j�I;u
� (I;1)u + �I
(I;2)u
�du
�exp
��tR0
�j;u
� (j;1)u + �j
(j;2)u
�du
�exp((�j+�j�I)t) = 1;
which is satis�ed when
�I � �I;t( (I;1)t + �I
(I;2)t ) = 0 dP dt a:s: (1.33)
�j � �j;t( (j;1)t + �j (j;2)t ) = 0 dP dt a:s:; (1.34)
where denotes the product of the two measures and where (1.33) is the no-arbitrage
condition for the market index obtained by imposing EPs
�ItIs
dQdP(t)
dQdP(s)exp(�r(t� s))
�= 1.
Following for example Heston (1993), we assume that the market price of variance risks
are proportional to their spot volatilities �I;t and �j;t, that is
(I;2)t + �I
(I;1)t = �I�I;t (1.35)
(j;2)t + �j
(j;1)t = �j�j;t: (1.36)
Solving (1.33), (1.34), (1.35), and (1.36) restricting attention to the subset of solutions
satisfying �j = 0; where idiosyncratic variance risk is not priced, we have
(I;1)t =
�I � �I�I�2I;t�I;t(1� �2I)
and (I;2)t =
�I�2I;t � �I�I
�I;t(1� �2I)(1.37)
37
(j;1)t =
�j�j;t(1� �2j )
and (j;2)t = �
�j�j
�j;t(1� �2j ): (1.38)
Given (1.37) and (1.38), an application of the Girsanov�s theorem delivers the risk-neutral
Brownian motions d ~Wt = dWt + dDW;W
0Et t where
d ~W(I;1)t = dW
(I;1)t +
��I�I;t
�dt (1.39)
d ~W(I;2)t = dW
(I;2)t + (�I�I;t) dt (1.40)
d ~W(j;1)t = dW
(j;1)t +
��j�j;t
�dt (1.41)
d ~W(j;2)t = dW
(j;2)t : (1.42)
The previous results can now be used to risk-neutralize the processes (1.2) to (1.5) and
obtain (1.9), (1.10), (1.11), and (1.12). For the market factor using (1.39) in (1.2) and
(1.40) in (1.3), we have
dItIt
= (r + �I)dt+ �I;tdW(I;1)t
= (r + �I)dt+ �I;t
�d ~W
(I;1)t � �I
�I;tdt
�= rdt+ �I;td ~W
(I;1)t ; (1.43)
and
d�2I;t = �I(�I � �2I;t)dt+ �I�I;tdW(I;2)t
= �I(�I � �2I;t)dt+ �I�I;t�d ~W
(I;2)t � �I�I;tdt
�= ~�I
�~�I � �2I;t
�dt+ �I�I;td ~W
(I;2)t ; (1.44)
where ~�I = �I + �I�I and ~�I =�I�I~�I. Using a similar argument, we can show that
dSjt
Sjt� rdt = �j
�dItIt� rdt
�+ �j;td ~W
(j;1)t (1.45)
d�2j;t = �j(�j � �2j;t)dt+ �j�j;td ~W(j;2)t : (1.46)
38
A.2 Proof of Proposition 1. Part 2: Equity Characteristic Function
For ease of notation, we de�ne the integrated Brownian ~W 1�k;t:T
�TRt
�k;ud ~W(k;1)u and the
integrated variance �2k;t:T �TRt
�2k;udu for k 2 fI; jg. Given the Q-processes, one can apply
Ito�s lemma to ln(Sjt ) and obtain (after integration) the following expression for individual
equity log-returns
ln
SjTSjt
!= r� � 1
2
��2j;t:T + �
2j�2I;t:T
�+ ~W 1
�j;t:T+ �j ~W
1�I;t:T
; (1.47)
where � = T � t. Therefore, the conditional characteristic function of the risk-neutral
log-returns takes the form
~�LR �
� ; u; �j�= EQt
�exp
�iu
�r� � 1
2
��2j;t:T + �
2j�2I;t:T
�+ ~W 1
�j;t:T+ �j ~W
1�I;t:T
���:
(1.48)
Using the de�nition of the stochastic exponential �(�) in (1.29), we have
��� ~W 1
�k;t:T
�= exp
�� ~W 1
�k;t:T� (�)
2
2
D~W 1�k; ~W 1
�k
Et:T
�= exp
�� ~W 1
�k;t:T� 12�2�2k;t:T
�;
(1.49)
for k 2 fj; Ig, which allows us to write (1.48) as
~�LR �
� ; u; �j�= exp(iur�) � (1.50)
EQt
h��iu�i ~W
1�I;t:T
���iu ~W 1
�j;t:T
�exp
���g1(u; �j)�
2I;t:T + g2(u)�
2j;t:T
��i
where g1(u; �j) =iu2 �
2j (1� iu) and g2(u) = iu
2 (1� iu). Following Carr and Wu (2004) and
Detemple and Rindisbacher (2010), we de�ne the following change-of-measure
dC
dQ(t) � �
�iu�j ~W
1�I;0:t
���iu ~W 1
�j;0:t
�: (1.51)
39
Combining (1.50) with the change of measure (1.51), we can write
~�LR �
� ; u; �j�= exp(iur�)EQt
"dCdQ(T )
dCdQ(t)
exp���g1(u; �j)�
2I;t:T + g2(u)�
2j;t:T
��#
) ~�LR �
� ; u; �j�= exp(iur�)ECt
�exp(�g1(u; �j)�2I;t:T )
�ECt
�exp
��g2(u)�2j;t:T
��: (1.52)
Given an extension of the Girsanov theorem to the complex plane, under the C-measure
we have
dWC;(I;2)t = d ~W
(I;2)t � (iu�I�j�I;t)dt
dWC;(j;2)t = d ~W
(j;2)t � (iu�j�j;t)dt:
As a result,
d�2k;t = �Ck (�Ck � �2k;t)dt+ �k�k;tdW
C;(k;2)t (1.53)
where
�CI = ~�I � iu�I�j�I ; �CI =~�I~�I
�CI; �Cj = �j � iu�j�j ; and �Cj =
�j�j
�Cj:
We can now make use of the closed-form solution for the moment generating function of
ECt [exp��g(u)�2t:T
�] to obtain the following expression for ~�
LR(�);
~�LR �
� ; u; �j�= exp
�iur� �
�A(� ; u; �j) +B(� ; u)
�� C(� ; u; �j)�2I;t �D(� ; u)�2j;t
�;
(1.54)
with
A(� ; u; �j) =~�I~�I
�2I
(2 ln
(1�
1(u; �j)� �CI21(u; �j)
�1� e�1(u;�j)�
�!+�1(u; �j)� �CI
��
)(1.55)
40
B(� ; u) =�j�j
�2j
(2 ln
1�
2(u)� �Cj22(u)
�1� e�2(u)�
�!+�2(u)� �Cj
��
)(1.56)
C�� ; u; �j
�=
2(1� e�1(u;�j)� )g1(u; �j)
21(u; �j)��1(u; �j)� �CI
� �1� e�1(u;�j)�
� (1.57)
D(� ; u) =2(1� e�2(u)� )g2(u)
22(u)��2(u)� �Cj
� �1� e�2(u)�
� ; (1.58)
and where
1(u; �j) =q(�CI )
2 + 2�2Ig1(u; �j) and 2(u) =q(�Cj )
2 + 2�2jg2(u);
with
g1(u; �j) =iu
2�2j (1� iu) and g2(u) =
iu
2(1� iu);
and
�CI = ~�I � iu�I�j�I and �Cj = �j � iu�j�j
Using the fact that ~�j �� ; u; �j
�= eiu ln(S
jt )~�
LR �� ; u; �j
�, the previous equations can be
used to compute the price of a European call option written on Sj .
B. Proof of Proposition 2
Within our model, the index price (It) takes the following form under the risk-neutral
measure
It = I0 exp�rt� 1
2�2I;0:t + ~W 1
�I;0:t
�:
Taking the derivative of the index price It with respect to �j ~W1�I;0:t
gives
@It
@�j ~W1�I;0:t
=@It
@ ~W 1�I;0:t
@ ~W 1�I;0:t
@�j ~W1�I;0:t
=@It
@ ~W 1�I;0:t
@�j ~W
1�I;0:t
@ ~W 1�I;0:t
!�1=It�j;
41
where the second equality makes use of the inverse function theorem which holds as long
as �j 6= 0. Moreover, as long as It 6= 0 the inverse function theorem also implies that
@�j ~W1�I;0:t
@It=
@It
@�j ~W1�I;0:t
!�1=�jIt: (1.59)
Furthermore, within our model the equity price is given by
Sjt = Sj0 exp(rt�1
2
��2j;0:t + �
2j�2I;0:t
�+ ~W 1
�j;0:t + �j~W 1�I;0:t
);
which implies@Sjt
@�j ~W1�I;0:t
= Sjt : (1.60)
Combining (1.59) and (1.60) implies
@Sjt@It
=@Sjt
@�j ~W1�I;0:t
@�j ~W1�I;0:t
@It=SjtIt�j :
Therefore, for any derivative f j written on Sj the sensitivity of f j with respect to market
value, It (market delta), is
@f j
@It=@f j
@Sjt
@Sjt@It
=@f j
@Sjt
SjtIt�j : (1.61)
For the sensitivity of f j with respect to market variance (market vega), we have
@f j
@�2I;t=
@f j
@Vj;t
@Vj;t@�2I;t
=@f j
@Vj;t
@(�2j�2I;t + �
2j;t)
@�2I;t=
@f j
@Vj;t�2j :
C. Proof of Proposition 3
The proof of this proposition is adapted from Broadie, Chernov, and Johannes (2009) to
our set-up. By application of Ito�s lemma to the derivative contract f j written on Sj ,
42
combined with the pricing PDE, allows us to write the dynamic of df j under P as
df j =nrf j � f jSrS
jt � f
jVj�2j ~�I(
~�I � �2I;t) + ~�j(~�j � �2j;t)odt
+f jSdSjt + f
jVjdVj;t
, df j =nrf j � f jSrS
jt � f
jVj�2j ~�I(
~�I � �2I;t) + �j(�j � �2j;t)odt (1.62)
+f jSdSjt + f
jVjdVj;t:
where f jx denotes the partial derivative of f j with respect to x. Note that in the previous
equation, we have assumed that idiosyncratic risk is not priced, which is consistent with
Proposition 1 (i.e. ~�j = �j and ~�j = �j). Moreover,
EPt [dSjt ]
dt= (r + �j + �j�I)S
jt
EPt [dVj;t]
dt= �2j�I
��I � �2I;t
�+ �j(�j � �2j;t):
(1.63)
Consequently, combining (1.62) and (1.63) leads to
1
dtEPt
�df j
f j� rdt
�=
f jSf jEPt
hdSjt � rS
jt dti+f jVjf j�2jE
Pt
hd�2I;t � ~�I(~�I � �2I;t)dt
i+f jVjf jEPt
�d�2j;t � �j(�j � �2j;t)dt
�;
which simpli�es to
1
dtEPt
�df j
f j� rdt
�= f jS
Sjtf j��j + �j�I
�+ f jVj
�2jf j(~�I~�I � �I�I): (1.64)
As in Heston (1993), our risk neutralization implies that ~�I~�I = �I�I . Consequently, we
obtain1
dtEPt
�df j
f j� rdt
�= f jS
Sjtf j��j + �j�I
�=@f j
@Sjt
Sjtf j�j +
@f j
@It
Itf j�I ;
43
where the second equation uses the result in Proposition 2.
D. Proof of Proposition 4
The following argument is derived under the P measure; however, a similar argument can
be developed under the Qmeasure. Given the de�nition of skewness, the total (conditional)
skewness of the integrated return of �rm j is
SkP
R Tt
dSju
Sju
!�
EPt
��R TtdSjuSju� EPt
hR TtdSjuSju
i�3��EPt
��R TtdSjuSju� EPt
hR TtdSjuSju
i�2��3=2 : (1.65)
Given that R Tt
dSju
Sju� EPt
"R Tt
dSju
Sju
#= �jW
1�I;t:T
+W 1�j;t:T
;
equation (1.65) can be simpli�ed to
SkP
R Tt
dSju
Sju
!=
EPt
���jW
1�I;t:T
+W 1�j;t:T
�3��EPt
���jW
1�I;t:T
+W 1�j;t:T
�2��3=2 :
By the properties of Ito integrals and the independence of W (I;1) and W (j;1), we have
EPt
���jW
1�I;t:T
+W 1�j;t:T
�2�= �2j � EPt
��2I;t:T
�+ EPt
��2j;t:T
�= EPt [Vj;t:T ];
and
EPt
���jW
1�I;t:T
+W 1�j;t:T
�3�= �3j � EPt
��W 1�I;t:T
�3�+ EPt
��W 1�j;t:T
�3�:
44
Consequently, the total (conditional) skewness of the integrated return of �rm j takes the
form
SkP
R Tt
dSju
Sju
!= �3j
EPt
��W 1�I;t:T
�3��EPt [Vj;t:T ]
�3=2 +
EPt
��W 1�j;t:T
�3��EPt [Vj;t:T ]
�3=2 (1.66)
= sign(�j) � �3j � EPt [�
2I;t:T ]
EPt [Vj;t:T ]
!3=2�EPt
��W 1�I;t:T
�3��EPt [�
2I;t:T ]
�3=2 +0@EPt
h�2j;t:T
iEPt [Vj;t:T ]
1A3=2 � EPt��W 1�j;t:T
�3��EPt
h�2j;t:T
i�3=2 :For positive beta �rms, we obtain
SkP
R Tt
dSju
Sju
!= �3j �
EPt [�
2I;t:T ]
EPt [Vj;t:T ]
!3=2� SkPI;t:T +
EPt [�
2j;t:T ]
EPt [Vj;t:T ]
!3=2� SkPj;t:T ; (1.67)
where
SkPI;t:T = SkP�R T
t
dIsIs
�and SkPj;t:T = SkP
�R Tt �j;sdW
(j;1)s
�are the market and idiosyncratic skewness, respectively.
45
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49
Figure 1: At-the-Money Implied Volatility. Six Firms and the S&P 500 Index
1996 1998 2000 2002 2004 2006 2008 2010
20
40
60
80
100A
TM Im
plie
d V
ol.
Boeing
1996 1998 2000 2002 2004 2006 2008 2010
20
40
60
80
100Coca Cola
1996 1998 2000 2002 2004 2006 2008 2010
20
40
60
80
100
ATM
Impl
ied
Vol
.
Home Depot
1996 1998 2000 2002 2004 2006 2008 2010
20
40
60
80
100Merck
1996 1998 2000 2002 2004 2006 2008 2010
20
40
60
80
100
ATM
Impl
ied
Vol
.
Proctor & Gamble
1996 1998 2000 2002 2004 2006 2008 2010
20
40
60
80
100Walmart
Notes to Figure: We plot the time series of implied volatility for six �rms (black) and the
S&P 500 index (grey). On each day we use contracts with between 30 and 60 days to
maturity and a moneyness (S=K) between 0:95 and 1:05. For every trading day and every
security, we average the available implied volatilities to obtain an estimate of at-the-money
implied volatility.
50
Figure 2: Implied Volatility Level, Moneyness and Term Slope.
S&P 500 Index and the First Principal Component from 29 Firms
1996 1998 2000 2002 2004 2006 2008 201010
20
30
40
50
60
IV L
evel
Panel A: S&P500 IV Level
1996 1998 2000 2002 2004 2006 2008 201010
20
30
40
50
60
IV L
evel
Panel B: 1st PC of Equity IV Level
1996 1998 2000 2002 2004 2006 2008 20100
2
4
6
8
IV M
oney
ness
Slo
pe
Panel C: S&P500 IV Moneyness Slope
1996 1998 2000 2002 2004 2006 2008 20100
2
4
6
8
IV M
oney
ness
Slo
pe
Panel D: 1st PC of Equity IV Moneyness Slope
1996 1998 2000 2002 2004 2006 2008 20108
6
4
2
0
2
IV T
erm
Slo
pe
Panel E: S&P500 IV Term Slope
1996 1998 2000 2002 2004 2006 2008 20108
6
4
2
0
2
IV T
erm
Slo
pe
Panel F: 1st PC of Equity IV Term Slope
Notes to Figure: Panel A plots the implied volatility level from S&P 500 index options.
Panel B plots the �rst principal component (1st PC) of implied volatility levels from options
on 29 equities in the Dow-Jones index. Panel C plots the slope of implied volatility with
respect to moneyness from short-term S&P 500 index options. Panel D plots the 1st PC of
the implied volatility moneyness slopes from options on the 29 equities. Panel E plots the
slope of the implied volatility term structure from S&P 500 index options. Panel F plots
the 1st PC of the implied volatility term structure from options on the 29 equities.
51
Figure 3: Market Delta and Vega of Equity Call Options
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
0.05
0.1
0.15 M
arke
t Del
taMarket Delta of Equity Call Options
Beta = 1.3
Beta = 1
Beta = 0.7
0.4 0.6 0.8 1 1.2 1.4 1.6 1.81.80
4
8
12
Moneyness (S/K)
Mar
ket V
ega
Market Vega of Equity Call Options
Notes to Figure: We plot the sensitivity of equity options to changes in market level (market
delta) and market volatility (market vega). Each line has a di¤erent beta but the same
amount of unconditional total equity variance ~Vj = �2j~�I + �j = 0:1. We set the current
spot variance to �2I;t = 0:01 and Vj;t = 0:05, and de�ne the idiosyncratic variance as the
residual �2j;t = Vj;t � �2j�2I;t. The market index parameters are ~�I = 5; ~�I = 0:04; �I =
0:5; �I = �0:8; and the individual equity parameters are �j = 1; �j = 0:4; and �j = 0.
The risk-free rate is 4% per year, option maturity is 3 months, and we set Sjt =It = 0:1:
52
Figure 4: Expected Excess Returns on Equity Options
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
0.3
0.6
0.9
1.2
1.5D
aily
Exc
ess R
etur
n (%
)Call Option Excess Return
Beta = 1.3
Beta = 1
Beta = 0.7
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40.5
0.4
0.3
0.2
0.1
00
Moneyness (S/K)
Dai
ly E
xces
s Ret
urn
(%)
Put Option Excess Return
Notes to Figure: We plot expected excess returns in percent per day on call and put options
using the model. Each line has a di¤erent beta but the same amount of unconditional total
equity variance ~Vj = �2j~�I + �j = 0:1. We set the current spot variance to �2I;t = 0:01 and
Vj;t = 0:05, and de�ne the idiosyncratic variance as the residual �2j;t = Vj;t � �2j�2I;t. The
market index parameters are ~�I = 5; ~�I = 0:04; �I = 0:5; �I = �0:8; �I = 0:075; and the
individual equity parameters are �j = 1; �j = 0:4; �j = 0, and �j = 0. The risk-free rate
is 4% per year and option maturity is 3 months.
53
Figure 5: Beta and Implied Volatility Across Moneyness and Maturity
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.222
24
26
28
30
32
Moneyness (S/K)
Impl
ied
Vola
tility
Beta and Implied Volatility Across Moneyness
0.5 1 1.5 2 2.5 322
24
26
28
30
32
Years
Impl
ied
Vola
tility
Beta and the AttheMoney Implied Volatility TermStructure
Beta = 1.3
Beta = 1
Beta = 0.7
Notes to Figure: The top panel plots option implied volatility (IV) from model prices with
3 months to maturity against moneyness (S=K). The bottom panel plots model IV for
at-the-money options against time-to-maturity in years. Each line has a di¤erent beta but
the same amount of unconditional total equity variance ~Vj = �2j~�I + �j = 0:1. We set the
current spot variance to �2I;t = 0:01 and Vj;t = 0:05, and de�ne the idiosyncratic variance as
the residual �2j;t = Vj;t��2j�2I;t. The market index parameters are ~�I = 5; ~�I = 0:04; �I =
0:5; �I = �0:8; and the individual equity parameters are �j = 1; �j = 0:4; and �j = 0.
The risk-free rate is 4% per year.
54
Figure 6.A: Average Market- (solid) and Model-Implied (dashed) Volatility Smile.
High Volatility (black) and Low Volatility (grey) Days
0.9 1 1.138
43
48
Impl
ied
Vol
.
26
31
36Alcoa
0.9 1 1.136
41
46
20
25
30American Express
0.9 1 1.134
39
44
17
22
27Bank of America
0.9 1 1.130
35
40
Impl
ied
Vol
.
23
28
33Boeing
0.9 1 1.132
37
42
25
30
35Caterpillar
0.9 1 1.136
41
46
20
25
30JP Morgan
0.9 1 1.122
27
32
Impl
ied
Vol
.
19
24
29Chevron
0.9 1 1.144
49
54
27
32
37Cisco
0.9 1 1.127
32
37
18
23
28AT&T
0.9 1 1.123
28
33
Impl
ied
Vol
.
15
20
25Coca Cola
0.9 1 1.132
37
42
22
27
32Disney
0.9 1 1.127
32
37
20
25
30Dupont
0.9 1 1.121
26
31
Impl
ied
Vol
.
18
23
28Exxon Mobil
Moneyness (S/K)0.9 1 1.1
30
35
40
18
23
28General Electric
Moneyness (S/K)0.9 1 1.1
37
42
47
27
32
37HewlettPackard
Moneyness (S/K)
Notes to Figure: We plot the market IV (solid) and model IV (dashed) averaged over time
for di¤erent moneyness categories for each �rm. The black lines (left axis) show the average
on days with above-average IV and the grey lines (right axis) show the average for days
with below-average IV. Moneyness measured by S=K is on the horizontal axis.
55
Figure 6.B: Average Market- (solid) and Model-Implied (dashed) Volatility Smile.
High Volatility (black) and Low Volatility (grey) Days
0.9 1 1.134
39
44
Impl
ied
Vol
.
23
28
33Home Depot
0.9 1 1.139
44
49
27
32
37Intel
0.9 1 1.128
33
38
18
23
28IBM
0.9 1 1.121
26
31
Impl
ied
Vol
.
15
20
25Johnson & Johnson
0.9 1 1.125
30
35
21
26
31McDonald's
0.9 1 1.126
31
36
21
26
31Merck
0.9 1 1.131
36
41
Impl
ied
Vol
.
20
25
30Microsoft
0.9 1 1.124
29
34
19
24
293M
0.9 1 1.128
33
38
21
26
31Pfizer
0.9 1 1.122
27
32
Impl
ied
Vol
.
16
21
26Procter & Gamble
0.9 1 1.129
34
39
19
24
29Travellers
0.9 1 1.127
32
37
20
25
30United Technologies
0.9 1 1.127
32
37
Impl
ied
Vol
.
18
23
28Verizon
Moneyness (S/K)0.9 1 1.1
27
32
37
19
24
29Walmart
Moneyness (S/K)0.9 1 1.1
20
25
30
13
18
23S&P500
Moneyness (S/K)
Notes to Figure: We plot the market IV (solid) and model IV (dashed) averaged over time
for di¤erent moneyness categories for each �rm. The black lines (left axis) show the average
on days with above-average IV and the grey lines (right axis) show the average for days
with below-average IV. Moneyness measured by S=K is on the horizontal axis.
56
Figure 7.A: Market- (solid) and Model-Implied (dashed) Term Structures for
At-the-Money Implied Volatility. Upward-Sloping (grey) and Downward-Sloping (black)
Days
2 4 638
43
48
Impl
ied
Vol
.
31
36
41Alcoa
2 4 640
45
50
25
30
35American Express
2 4 641
46
51
26
31
36Bank of America
2 4 633
38
43
Impl
ied
Vol
.
27
32
37Boeing
2 4 632
37
42
29
34
39Caterpillar
2 4 639
44
49
24
29
34JP Morgan
2 4 624
29
34
Impl
ied
Vol
.
22
27
32Chevron
2 4 645
50
55
32
37
42Cisco
2 4 629
34
39
22
27
32AT&T
2 4 626
31
36
Impl
ied
Vol
.
19
24
29Coca Cola
2 4 632
37
42
25
30
35Disney
2 4 628
33
38
23
28
33Dupont
2 4 624
29
34
Impl
ied
Vol
.
21
26
31
Months to Maturity
Exxon Mobil
2 4 629
34
39
22
27
32
Months to Maturity
General Electric
2 4 637
42
47
28
33
38
Months to Maturity
HewlettPackard
Notes to Figure: The solid black line (left axis) shows the average market IV on days with
steeper-than-median downward-sloping term structures and the grey line (right axis) shows
the average market IV on days with steeper-than-median upward-sloping term structures.
The dashed lines show the corresponding average model IVs. Moneyness (S=K) is between
0:95 and 1:05.
57
Figure 7.B: Market- (solid) and Model-Implied (dashed) Term Structures for
At-the-Money Implied Volatility. Upward-Sloping (grey) and Downward-Sloping (black)
Days
2 4 633
38
43
Impl
ied
Vol
.
27
32
37Home Depot
2 4 638
43
48
30
35
40Intel
2 4 629
34
39
22
27
32IBM
2 4 622
27
32
Impl
ied
Vol
.
18
23
28Johnson & Johnson
2 4 626
31
36
21
26
31McDonald's
2 4 626
31
36
24
29
34Merck
2 4 632
37
42
Impl
ied
Vol
.
26
31
36Microsoft
2 4 624
29
34
22
27
323M
2 4 626
31
36
25
30
35Pfizer
2 4 625
30
35
Impl
ied
Vol
.
19
24
29Procter & Gamble
2 4 627
32
37
22
27
32Travellers
2 4 628
33
38
23
28
33United Technologies
2 4 629
34
39
Impl
ied
Vol
.
21
26
31
Months to Maturity
Verizon
2 4 629
34
39
22
27
32
Months to Maturity
Walmart
2 4 622
27
32
16
21
26
Months to Maturity
S&P500
Notes to Figure: The solid black line (left axis) shows the average market IV on days with
steeper-than-median downward-sloping term structures and the grey line (right axis) shows
the average market IV on days with steeper-than-median upward-sloping term structures.
The dashed lines show the corresponding average model IVs. Moneyness (S=K) is between
0:95 and 1:05.
58
Figure 8: Implied Volatility Levels, Moneyness Slopes, and Term Structure Slopes Scatter
Plotted Against Beta. 29 Firms
0.7 0.8 0.9 1 1.1 1.2 1.320
30
40
50
Ave
rage
IV L
evel
0.7 0.8 0.9 1 1.1 1.2 1.31
1.5
2
2.5
Ave
rage
Mon
eyne
ss S
lope
0.7 0.8 0.9 1 1.1 1.2 1.30.6
0.8
1
1.2
1.4
OptionImplied BetaAve
rage
Abs
olut
e T
erm
Slo
pe
Notes to Figure: We plot the average implied volatility (IV) levels (top panel), the average
moneyness slopes (middle panel), and the average absolute value of the term-structure
slopes (bottom panel) against the estimated betas from Table 4.
59
Figure 9: Regression-based Betas versus Option-Implied Beta. 29 Firms
0.7 0.9 1.1 1.30.5
0.7
0.9
1.1
1.3
1.5
1.71.7
OL
S B
eta
0.7 0.9 1.1 1.30.5
0.7
0.9
1.1
1.3
1.5
1.71.7
OptionImplied Beta
OL
S B
eta
with
Shr
inka
ge
Notes to Figure: We plot the regression-based OLS beta estimates against option-implied
beta (top panel), and OLS beta with shrinkage against option-implied betas (bottom panel)
for 29 �rms. The OLS betas are obtained from a CAPM regression using daily returns
during January 4, 1996 to October 29, 2010. The betas with shrinkage are de�ned by
0:4 + 0:6 � �OLSj where �OLSj is the OLS beta. In both panels, the �tted line from the
regression is displayed in black while the grey line represents the 45�line.
60
61
62
63
64
65
66
67
Chapter 2
Inventory Risk, Market-Maker Wealth, and the
Variance Risk Premium
2.1 Introduction
The variance of the stock market evolves stochastically through time. Market variance is
thus itself risky. With a daily trading volume of more than 110 billion dollars, S&P 500
index (SPX) options are the most actively traded securities providing exposure to market
variance risk. These options are priced at a signi�cant premium over the realized variance
of the underlying S&P 500 index. This is commonly referred to as the (market) variance
risk premium. Bollerslev, Tauchen, and Zhou (2009) demonstrate that the variance risk
premium predicts future stock market returns. The variance risk premium is thus a poten-
tial state variable for the economy, and a better understanding of the factors impacting it
is of signi�cant interest.
On average, the net demand of SPX option by end-users is positive (see, among others,
Bollen and Whaley, 2004, and Gârleanu, Pedersen, and Poteshman, 2009). Market-makers
in these options thus act as net sellers and build up large negative inventories over time.
Through their inventory, intermediaries are exposed to market variance risk in addition to
the conventional equity market risk. While market-makers can e¢ ciently hedged equity
market risk using index future contracts, frictions impair their ability to eliminate their
billion dollars exposure to market variance.1 Consequently, market-makers bear large vari-
1Bates (2003) discusses the constraints faced by market-makers to hedge their aggregate exposure tovariance risk. Bakshi, Cao, and Chen (1997), Buraschi and Jackwerth (2001), and Bakshi and Kapadia
68
ance risk and in�uence the variance risk premium. Focusing on the intermediation of SPX
options, our contribution is to demonstrate, empirically and theoretically, the joint impact
of market-makers�inventory risk and wealth on the variance risk premium.
We consider two factors in�uencing the required compensation of market-makers that
should help explain the variance risk premium. The �rst is the risk exposure of index
option market-makers to market volatility. We refer to it as inventory risk. The second
is market-maker wealth. Inventory risk measures the loading of market-makers� delta-
hedged inventory on the variance risk premium. A positive link between inventory risk and
the variance risk premium implies that a higher risk exposure results in higher expected
compensation for intermediaries. Moreover, wealth should also in�uence the variance risk
premium. As they become more constrained, risk-averse market-makers should require a
higher compensation. Surprisingly, little is known about the extent to which these variables
jointly explain the premium for market variance.
Our empirical analysis uses �fteen years of market-making activity for SPX options
and more than one million quotes. We establish our main empirical results using time-
series regressions of daily changes in variance risk premium controlling for lagged daily
changes. First, we show that both inventory risk and changes in market-maker wealth
are statistically signi�cant to explain the variance risk premium. The inclusion of both
variables leads to a 9% increase in the adjusted R-Squared at the daily frequency relative
to various controls.
When inventory risk decreases (i.e. becomes more negative) by one standard deviation,
it leads to a 1:2% decrease in the variance risk premium. Moreover, the e¤ect of inventory
on the variance risk premium is magni�ed when market-makers experience dramatic wealth
losses. When market-makers� loss is within its 90th percentile, a one standard deviation
decrease in inventory risk causes about a 9% decrease in next day�s variance risk premium.
These results are robust to subsample tests. They are also robust to various controls
(2003) argue that index options are non-redundant securities. Thus, in addition to illiquidity issues, hedgingSPX options using other assets introduces basis risk.
69
including S&P 500 return, S&P 500 jump, investors�disagreement, and net buying pressure
for SPX options.
Bollen and Whaley (2004) demonstrate that net buying pressures are positively related
to changes in option implied volatilities. However, the channels by which net demand
impacts option implied volatilies and prices remain unclear. Our empirical analysis com-
plements their work by identifying and quantifying the e¤ect of one of these channels. When
SPX option net buying pressures are large and positive, market-makers�aggregate exposure
to market variance decreases (becomes more negative) on average. Because market-makers
exposure cannot be fully hedged, they command a higher compensation, which implies in a
more negative variance risk premium. Ultimately, a more negative variance risk premium
implies a higher level of risk-neutral volatility, and thus more expensive options.
Our work is related to the literature on the variance risk premium. For instance, Bakshi
and Kapadia (2003), Vilkov (2008), and Carr and Wu (2009) study the properties of the
variance risk premium for index and equity options. Eglo¤, Leippold, and Wu (2010), and
Todorov (2010) investigate the ability of a stochastic volatility-jump model to explain the
variance risk premium. Bollerslev, Tauchen, and Zhou (2009) derive the implications of
a long-run risk model for the variance risk premium. More recently, Buraschi, Trojani,
and Vedolin (2014) relate investors�disagreement and the market price of volatility.2 This
paper contributes to this literature by documenting and modeling the impact of option
intermediation on the variance risk premium.
To further assess the robustness of our results, we investigate the ability of inventory
risk and changes in market-maker wealth to explain the daily changes in the variance
risk premium measured for various horizons. While robust to the entire term structure
of premia, the impact of inventory risk and market-maker wealth is prominent at short-
horizons. This �nding relates to Aït-Sahalia, Karaman, and Mancini (2014), who conclude
that investors�fear of a market crash is captured in short-term variance risk premia.
2See also Bakshi and Kapadia (2006) who relate the variance risk premium to investor risk aversionand the higher-order moments. Moreover, Driessen, Maenhout, and Vilkov (2009) explain the variance riskpremium by the correlation risk of individual equity.
70
Theoretically, Gârleanu, Pedersen, and Poteshman (2009) set the stage for our analysis.
The authors establish that when perfect replication is not possible, net demand exerts pres-
sure on option prices. Their study demonstrates how demand impacts option prices under
various sources of incompleteness while leaving open the question of how intermediaries�
wealth and risk exposure interact to in�uence risk premia. Our work complements their
study by deriving the explicit relation between market-maker�s wealth, inventory risk, and
the variance risk premium in a fully speci�ed stochastic volatility (SV) framework.
We model a continuous-time economy in which the market variance has its own dynamic
and a risk-averse representative market-maker quotes index options. The time-varying
market variance causes part of the market-maker inventory to be a¤ected by unhedgeable
variance risk. To account for �uctuations in the market-maker�s variance risk exposure, we
allow inventory risk to vary over time. We solve for the endogenous variance risk premium,
that is, the utility-maximizing variance risk premium that clears the index option market.
Not only does the model show that inventory risk and the variance risk premium should
be positively related but it also demonstrates that the impact of inventory on the premium
is non-linearly decreasing in the market-maker�s wealth.
In standard SV models, researchers assume that the variance risk premium is the prod-
uct of a time-invariant parameter with the latent spot variance (see, among others, Heston,
1993, Bates, 2000, and Pan, 2002). This speci�cation attributes any discrepancy between
the objective properties of index returns and the risk-neutral probability measure implied
by option prices to the marginal investor�s preference over market variance. Our model
departs from this strand of literature by capturing the e¤ect of inventory risk and market-
maker wealth on the variance risk premium. Since inventory risk has its own dynamic and
the utility-maximizing wealth process is known, unlike Gârleanu, Pedersen, and Potesh-
man (2009) our model is straightforward to use for pricing. We estimate the model on
S&P 500 returns and a large panel of SPX puts. The model outperforms Heston (1993).
Interestingly and relative to Heston (1993), the model performs particularly well during
the crisis and when pricing out-of-the-money puts, which challenge traditional models.
71
Surveying the empirical performance of various option pricing models, Bates (2003)
concludes that a focus on the explicit �nancial intermediation of the underlying risks by
option market-makers is needed. To the best of our knowledge, our model is the �rst to
address Bates (2003)�s challenge.
Most importantly, the model can be used to uncover the economic impact of inventory
risk and wealth on SPX option prices. During turbulent times, when market-maker loss is
within its 90th percentile, it causes up to 2:31% daily price increase.
This paper is related to the strand of literature that study option bid-ask spreads
and returns. Stoikov and Saglam (2009) propose a mean variance framework to ana-
lyze the optimal option bid-ask spread policy when markets are incomplete. Muravyev
(2013) demonstrates that selling pressure in�uences equity option returns. Christo¤ersen,
Goyenko, Jacobs, and Karoui (2014) study the e¤ect of stock and equity option illiquidity
on equity option returns. We contribute to this strand of the literature by showing the
impact of variations in market-makers�risk sharing capacity on the premium for market
variance, which is an important determinant of index option prices and returns.
Also related to our work is the literature that studies the e¤ect of intermediary con-
straints on option prices. Leippold and Su (2011) examine the in�uence of margin re-
quirements and funding costs on the implied-volatility smile in a constant volatility model.
Chen, Joslin, and Ni (2013) investigate the jump premium embedded in index options and
its predictive ability for stock market returns. To motivate their �ndings, the authors build
an equilibrium model in which the consumption process has a constant di¤usive volatility
and time-varying jumps. Moreover, intermediary constraints are captured in reduced form
through time-varying risk-aversion tied to the jump intensity. Our study contributes to
both of these papers by designing a model in which market variance is stochastic, and
the market-maker wealth is endogenous. Both of these features are needed to obtain ex-
plicit implications of intermediaries�risk exposure and wealth on the premium for market
variance.
The rest of the paper is organized as follows. In Section 2, we present the data and the
72
main variables used in this study. We also proceed to our main empirical tests. Section
3 formulates the theoretical analysis. In Section 4, we discuss the model implications. In
Section 5, we estimate the model and investigate its �t. Finally, we conclude in Section 6.
2.2 Empirical Analysis
2.2.1 Data
SPX options trade only at the Chicago Board Option Exchange (CBOE). To construct
the aggregate inventory of CBOE market-makers, we rely on the Market Data Express
Open/Close database for obtaining the daily non market-makers (end-users) order �ows
for SPX options starting on January 1, 1996 and ending on December 30, 2011. The data
divides the daily open/close buy and sell origins into two groups: �rm and customer. Cus-
tomer origin is further subdivided in three categories depending on order size. Since we are
interested in measuring SPX market-makers�aggregate inventory, we disregard individual
origins and group orders into buy and sell. More precisely, on each day we aggregate the
total buys and sells for each contract and take the di¤erence which corresponds to end-
users�net demand for that contract on that day. Because SPX options are European, the
time-series of market-makers�inventory for each contract can be reconstructed by summing
up the negative of end-users�net demand over time starting from the �rst day the contract
is quoted. We �lter out Leaps (options with more than one year to maturity), and restrict
the sample to start on January 1, 1997 to avoid biases in inventory measurement.3
For SPX option prices, we rely on end-of-day data from OptionMetrics starting on
January 1, 1997 and ending on December 30, 2011. From all the quotes provided, we
�lter out options that have moneyness (spot price over strike price) less than 0:8 and
larger than 1:2, those with a mid-quote< 3=8, those that do not satisfy the usual arbitrage
conditions, those with implied volatility less than 5% and greater than 150%, and those
3To correctly measure inventory for a given option, the full time-series of end-users�order �ows for thatoption must be observed. Because some options quoted at the beginning of 1997 have been issued at thebeginning of 1996, accurate measurement of inventories for these contracts constrains the empirical analysisto start on January 1, 1997.
73
with less than ten days to maturity. For each option maturity, interest rates are estimated
by linear interpolation using zero coupon Treasury yields. The dividend yield is obtained
from OptionMetrics. We then merge the inventory data with the OptionMetrics database.
The �nal sample contains more than one million quotes for SPX puts and calls over the
1997-2011 period.
To construct the variance risk premium, we need daily estimates of realized variance. To
this end, we obtain high frequency data for S&P 500 index futures from Tickdata starting
on January 1, 1997 and ending on September 30, 2012.4 Based on multiple grids of intraday
squared returns, we construct daily measures of average realized variance following Zhang,
Mykland, and Aït-Sahalia (2005).
2.2.2 De�nition of Main Variables
Variance Risk Premium
The variance risk premium captures the di¤erence between physical and risk-neutral market
variances. At time t the variance risk premium with a horizon T is
V RP (t; T ) � RV (t; T )�RNV (t; T ) ; (2.1)
where RV (t; T ) � EPt [R t+Tt Vsds] denotes the expected integrated physical variance and
RNV (t; T ) � EQt [R t+Tt Vsds] is the expected integrated risk-neutral variance. Suppose
that we want to obtain a model-free estimate of the one-month variance risk premium on
day t, that is V RP (t; 30). Following Carr and Wu (2009), our �rst step is to compute
annualized RV (t; 30) from realized variances as
RV (t; 30) =365
30�
30Xi=1
RVt+i�1
!: (2.2)
4For some of our empirical tests, we need estimates of realized variance up to September 30, 2012 toconstruct a measure of the 9-month ex-post realized variance on December 30; 2011.
74
As in Carr and Wu (2009), we follow the equity premium literature and proxy expected
physical variance by ex-post realized variances.5 To measure expected integrated risk-
neutral variance, we follow Britten-Jones and Neuberger (2000) and Bollerslev, Tauchen,
and Zhou (2009) among others. We compute RNV (t; 30) from a portfolio of SPX call
options as
RNV (t; 30) = 2 �1Z0
C(t; 30;Ke�r�30=365)� C(t; 0;K)K2
� dK; (2.3)
where C(t; T;K) is the price of a call observed at time t with time to maturity T and
strike price K, and r is the risk-free rate. We evaluate (2.3) using the trapezoidal rule.
Armed with the two measures of expected integrated variance, the one-month variance risk
premium on day t is V RP (t; 30) = RV (t; 30)�RNV (t; 30).
Table 1 presents the daily average of implied volatility, vega, days to maturity, number
of quotes, and volume.6 We also report the yearly averages of the one-month variance
risk premium. Market implied volatility is 22:28% on average. As found in earlier studies
(see, among others, Bakshi and Kapadia, 2003, Vilkov, 2008, and Carr and Wu, 2009), the
yearly averages of the variance risk premium are consistently negative. In our sample, the
risk-neutral variance exceeds the realized variance by 2:13%.
Figure 1 plots the time-series of the S&P 500 index level in the top panel, and of the
one-month variance risk premium expressed in percentage in middle panel. The variance
risk premium is mostly negative before 2008. Interestingly, during the �nancial crisis it
spikes to large positive and negative values contemporaneously to the abrupt price drop in
the S&P 500 index. To investigate if the large variations in variance risk premium during
the �nancial crisis are induced by measurement errors, we plot the weekly averages of daily
gains and losses of delta-hedged near-the-money options in the bottom panel of Figure
1. This exercise is motivated by Bakshi and Kapadia (2003), who show that the gains
and losses from delta-hedged positions in options are informative about the variance risk
5Note that on average ex-post realized variances are good proxies for expected realized variances.6The vega of an option measures the rate of change of the option price to a small increase in volatility.
75
premium. For a given option f jt , the daily dollar gains and losses from delta-hedging it
from day t� 1 to t is
�Hedgejt � f jt ��f jt�1 +�
jt�1St + (f
jt�1 ��
jt�1St�1) � r�t+�
jt�1St�1 � q�t
�; (2.4)
where �jt is option j delta, St denotes the value of the S&P 500, q is the dividend yield, and
the time-step �t = 1=365. On each week, we average the daily �Hedgejt for all options
for which 0:98 6 St=Kj 6 1:02 to obtain the weekly average gain and loss. Interestingly,
the large �uctuations in V RP (t; 30) during the crisis period are also apparent from the
time-series of delta-hedged gains and losses.
Inventory Risk and Market-Maker Wealth
About 273 SPX calls and puts with distinct moneyness and maturities are quoted on
each day. To assess market-makers� inventory across options, Table 2 reports the daily
average of implied volatility, inventory, and delta-hedged gains and losses for di¤erent
moneyness and maturity categories. Note how market-makers�positions are consistently
negative across moneyness and maturities. Overall, SPX intermediaries are short about
one hundred thousand contracts on a daily basis.
Inventory risk measures the aggregate exposure of CBOE market-makers�inventory to
S&P 500 volatility. On each day, we calculate inventory risk as
InvRiskt �Pj�MM;jt V egajt ; (2.5)
where �MM;jt denotes market-makers�inventory for option j, and V egajt � @f jt =@
pVt is the
option vega, which we calculate using Black-Scholes.7 At any point in time, 1%� InvRisk
indicates the way inventory would respond in dollar terms to a 1% increase in market
7Black-Scholes�s vega has also been used as a proxy for option vega in a stochastic volatility frameworkin Carr and Wu (2007), Trolle and Schwartz (2009), and Christo¤ersen, Heston, and Jacobs (2009) amongothers.
76
volatility.
Figure 2 plots the CBOE VIX index in the top panel, and the dynamic of inventory risk
in the bottom panel. Interestingly, no clear patterns are apparent between the VIX and
inventory risk time-series. Consistent with Table 2, we see that market-makers�exposure
to S&P 500 volatility is negative most of the time excepted during the �nancial crisis, and
in the beginning of 2011. Through their inventory, CBOE market-makers carry billion
dollars in risk exposure to market variance.
In order to measure the changes in market-makers�wealth overtime, we �rst compute
dealers�delta-hedged inventory pro�ts and losses as
P&Lt �Pj�MM;jt�1 �Hedgejt : (2.6)
Moreover, we estimate the bid-ask spread revenue of CBOE market-makers by calculating
for each day
BAt �Pjmin(BOjt ; SO
jt ) ��BidAskjt � 0:36
�; (2.7)
where BOjt denotes end-users� buy orders for option j, SOjt is end-users� sell orders,
BidAskjt denotes the option bid-ask spread, and $0:36 is the transaction fee charged to
dealers per contract traded.8 In our empirical analysis, we de�ne changes in wealth as the
sum of (2.6) and (2.7).
Figure 3 plots the daily pro�ts and losses from market-makers�delta-hedged inventory
in the top panel, the cumulative daily pro�ts and losses in the middle panel, and the
bid-ask spread revenue in the bottom panel. Market-makers face substantial risks. Their
daily pro�ts and losses �uctuate between +265 and �393 million dollars. Consistent with
the idea that large positions in options involve risks, the distribution of the daily delta-
hedged gains and losses is highly leptokurtic with an excess kurtosis of 73. Moreover, it is
asymmetric with a skewness coe¢ cient of �2:55. Market-makers have a 5% risk of losing
8This fee includes $0:33 charged by the CBOE and $0:03 charged by the Option Clearing Corporationfor clearing cost.
77
at least 21 million dollars on any given day.
The fact that market-makers earn a pro�t on average from delta-hedging their inventory
generates the positive trend observed in the middle panel of Figure 3. In aggregate, market-
makers earn 9 million dollars monthly from their delta-hedged positions.
Finally, comparing the top panel with the bottom panel reveals the substantial impact
inventory risk has on dealers�wealth. Unlike equity stock market-makers, a large portion of
changes in option market-makers�wealth is driven by the �uctuations in their delta-hedged
inventory. In our sample, the absolute value of P&Lt is on average 1:5 times bigger than
BAt.
So far, we have established that the representative SPX market-maker faces substantial
variance risks from carrying large inventories over time. In option markets, dealers often
trade among themselves in order to manage part of the risks they face. However, whenever
end-users have large net exposure to market variance, so do SPX market-makers in the
aggregate. Because of market-makers large exposure to market variance, part of their
required compensation should be embedded in the variance risk premium.
2.2.3 Methodology and Predictions
Our empirical analysis is based on two di¤erent speci�cations for the variance risk premium.
Our �rst battery of tests is conducted on Carr and Wu (2009)�s log-variance risk premium,
LogV RP (t; T ) � ln (RV (t; T )=RNV (t; T )).9 We then investigate the robustness of our
results using V RP (t; T ) as de�ned by (2.1).
The methodology we follow is adapted from Bollen and Whaley (2004). More precisely,
we regress the changes in the variance risk premium against the explanatory variables and
9Because the distributions of the two measures of variance are positively skewed, the log-speci�cationalleviates the e¤ect of extreme values.
78
lagged changes. When working with the log-speci�cation, we run
�LogV RP (t; T ) = Intercept+ �Inv1 InvRiskt�1 + �Inv2 (P&Lt +BAt) � InvRiskt�1
+�cControlt + �V RP�LogV RP (t� 1; T ) + "t;
(2.8)
where�LogV RP (t; T ) � LogV RP (t; T )�LogV RP (t�1; T ), and similarly for�V RP (t; T ).
If the variance risk premium captures part of dealers�required compensation, we should
expect a positive relationship between lagged inventory risk and changes in the variance
risk premium. The more negative (positive) inventory risk is the more negative (positive)
the variance risk premium should be as such a relationship implies positive returns for
dealers. Therefore, lagged inventory risk should positively impact changes in variance risk
premium. Our prediction is thus �Inv1 > 0.
As their wealth decreases, dealers should command a higher compensation. After con-
trolling for their lagged exposure, interacting cotemporaneous changes in wealth with
lagged inventory risk should capture the way dealers dynamically update their required
compensation as their wealth �uctuates. When market-makers are experiencing losses and
are negatively (positively) exposed to market variance, a higher compensation implies a
decrease (increase) in the variance risk premium. Therefore, our prediction is �Inv2 < 0.
Motivated by previous studies, we include a series of cotemporaneous control variables.
We discuss them in the next section.
2.2.4 Control Variables
Carr and Wu (2009) show that part of the variation in the variance risk premium are
related to market index returns. To control for the variation in variance risk premium
induced by contemporaneous changes in market conditions, we calculate the log-returns of
the S&P 500 index and denote this variable S&P500LogRett.
Eglo¤, Leippold, and Wu (2010), Todorov (2010), and Aït-Sahalia, Karaman, and
Mancini (2014) among others study the impact of jumps on the variance risk premium. We
follow Cremers, Halling, and Weinbaum (2014) and construct an aggregate jump factor.
79
On each day, we calculate the returns on two zero-beta at-the-money SPX straddles of
maturities T1 and T2 with T1 < T2. We denote by rS1t and rS2t the returns on the two
straddles.10 These daily returns are then combined such that
JumpFactort � rS1t ��V egaS1tV egaS2t
�� rS2t ; (2.9)
where V egaS1t and V egaS2t denote the vega of each straddle. By construction JumpFactort
has zero delta, zero vega, and positive gamma, and thus captures the large �uctuations in
the S&P500 index.11
Bollen and Whaley (2004) document the e¤ect of net buying pressures on options
implied-volatility. Through their impact on implied-volatility, net buying pressures po-
tentially in�uence the variance risk premium. To disentangle the e¤ect of inventory risk
relative to net buying pressures on the variance risk premium, we calculate on each day
Bollen and Whaley�s net buying pressure variable
NetByingPressuret �Pj
�BOjt � SO
jt
��
abs��jt
�SPXV olumet
; (2.10)
where SPXV olumet is the aggregate volume of SPX options, and abs(:) denotes the ab-
solute value.
Buraschi, Trojani, and Vedolin (2014) establish that investors�disagreement in�uence
the variance risk premium. To construct a daily measure of investors�disagreement, we rely
on S&P500 index volume data to compute unexpected changes in index trading volume.
More precisely, for every day we calculate the di¤erence between S&P 500 index volume
on that day, and the average volume of the last 90 trading days. We denote this variable
Disagreementt.
10When constructing the �rst straddle, we choose T1 to be in the month following the current month,and at least �fteen days to maturity. Similarly, T2 is chosen to be in the month following T1.11The gamma of an option is the second derivative of the option price with respect to the underlying
asset price.
80
2.2.5 Results
Table 3 presents the results obtained from regressing the changes in the one-month log-
variance risk premium against the previous day�s inventory risk, the interaction of lagged
inventory risk with changes in market-makers�wealth, and the control variables. We report
the results for the full sample in regressions (1) and (2). To investigate the extent to
which the �nancial crisis drives the results obtained, we divide the sample in two sub-
samples of similar length, 1997-2004 and 2005-2011. To facilitate the interpretation of
the results, regressors are standardized to have unit variance. For each parameter, the
p-value is computed using Newey-West p-value with 8 lags to capture autocorrelation in
the residuals.
Regressions (2), (4), and (6) in Table 3 establish the importance of inventory risk
and market-maker wealth for explaining the variance risk premium. Both variables are
statistically signi�cant with their expected sign. For the two subsamples, their inclusion
results in a 9% increase in adjusted R-Squared relative to regressions (3) and (5).
Interestingly, it is for the sample period including the �nancial crisis that the impact of
inventory risk is the greatest. A one standard deviation decrease in inventory risk causes
a 1:2% decrease in the variance risk premium. When inventory risk equals its sample
average, a one standard deviation decrease in market-maker wealth is associated with a
4:47% decrease in the variance risk premium. The e¤ect of inventory is magni�ed when
market-makers experience dramatic losses. Conditioning on a market-makers� loss in its
90th percentile, a one standard deviation decrease in inventory risk results in a 9% decrease
in next day�s log-variance risk premium.12
The relation between S&P 500 log-returns and the variance risk premium is strongly
signi�cant. The variance risk premium tends to decrease when index returns decrease. In
regressions (1), (3), and (5) in Table 3, S&P 500 log-returns and lagged changes in variance
12The results documented in this section are robust and quantitatively similar when RV (t; T ) is set tothe one-step-ahead forecast of a time-series model in the spirit of Corsi (2009).
81
risk premium jointly account for 80% of the R-Squared documented.13
Note the way aggregate jumps are negatively related to changes in variance risk pre-
mium. By construction, the returns of the jump factor are high whenever the S&P 500
abruptly decreases. Thus, the negative loading found for the jump factor is consistent with
Todorov (2010), who shows that negative jumps result in a more negative variance risk
premium.
Bollen and Whaley (2004) demonstrate that net buying pressures increase implied
volatility. Through their e¤ect on implied volatility, high buying pressures should result in
a lower variance risk premium. From Table 3, we see that net buying pressure is negatively
related to the variance risk premium. However, the variable is only signi�cant in regression
(5).
Consistent with Buraschi, Trojani, and Vedolin (2014), the loadings estimated for dis-
agreement are consistently negative. When investors�disagreement increases, the di¤erence
between realized and implied volatility tends to become more negative.
In Table 4, we presents the full sample results obtained when the variance risk premium
is constructed for various horizons. This exercise allows us to quantify the impact each
variable has on the term structure of variance risk premia. In addition to the one-month
horizon, we focus on four additional horizons of 60, 90, 180, and 270 days respectively.
Table 4:A reports the result obtained for the log-variance risk premium. Several inter-
esting �ndings emerge. Note that the e¤ect of inventory risk and �uctuations in market-
makers wealth are robust across horizons. Interestingly, there is a term structure of the
coe¢ cients estimated. For most variables, the magnitude of their in�uence on the variance
risk premium decreases as the horizon increases. Accordingly, the e¤ect of inventory risk
and market-makers wealth is most prominent for short-term variance risk premia. For hori-
zons of at least sixty days, the two variables appear to have the second greatest economic
impact after the index returns, and before aggregate jumps.
13Bivariate regressions of changes in log-variance risk premium on index log-returns and lagged changeshave an adjusted R-Squared of 30% on average for the full sample and subsample tests.
82
In Table 4:B, we present the result obtained when the variance risk premium is mea-
sured as RV (t; T ) � RNV (t; T ). We see that the e¤ect of inventory risk and �uctuations
in market-makers wealth remain robust. Relative to inventory risk and market-makers�
wealth, the impact of jumps vanishes for the nine-month horizon. This result relates to
Aït-Sahalia, Karaman, and Mancini (2014), who conclude that investors�fear of a market
crash is captured in short-term variance risk premia.
2.2.6 The Financial Crisis
Typically, the index option market functions similarly to an insurance market. A clientele
of institutional investors predominantly buys SPX options, which causes market-makers�
inventory risk to be negative on average. On November 20, 2008, the VIX Index reached a
high of 80:86%. Around the same time, end-users are heavily shorting SPX options. These
large net sell orders resulted in an increase in inventory risk of more than 1:8 billion dollars
during the month of November. By November 20, CBOE market-makers carried more than
2:5 billion dollars positive exposure to market volatility. This risk exposure represents 8%
of the capitalization of SPX options at that time.
To understand the risk and reward incurred from being positively exposed to market
volatility during the �nancial crisis, Table 5 presents the return statistics of delta-hedged
near-the-money options. Because these options are close to the money, they are highly
sensitive to market variance. For comparison purposes, we report the statistics for the full
sample and the �nancial crisis separately.
While losing most of the time, the delta-hedged positions earn 6:26%monthly during the
crisis period. Thus, around the same time market-makers�exposure to market variance is
positive, long positions in near-the-money options are pro�table on average. Note however
the tremendous risk exposure from these options. During the �nancial crisis, the volatility
of daily returns peaks at 95%, and its excess kurtosis is about 11. This result further
stresses the critical role of SPX market-makers in undertaking large risk to allow end-users
to hedge against and speculate on market volatility.
83
Overall, the evidences presented so far are consistent with the idea that part of the
variance risk premium captures CBOE market-makers� compensation for their exposure
to market variance. Motivated by these �ndings, we now develop a theoretical model in
which the physical market variance is dynamic and a risk-averse representative market-
maker endogenously quotes index options, and thus in�uences the variance risk premium.
2.3 The Model
The economy is cast in a continuous time framework in which the underlying source of
uncertainty is driven by two independent Brownian motions ZS and ZV .14 Market par-
ticipants have a �nite investment horizon T , and can invest in the market index St, which
evolves according to
dStSt
= �dt+pVt
�q1� �2V dZ
St + �V dZ
Vt
�; S0 known, (2.11)
where � is the market premium and Vt is the market variance which follows a CEV dynamic
dVt = �(� � Vt)dt+ �V �t dZVt ; V0 known, (2.12)
where � denotes the unconditional variance, � is the speed of mean reversion, � is the
volatility of volatility, and � determines the variance elasticity.15 In (2.11), �V captures
the correlation between market return and market variance innovations. In addition to the
market index, market participants can invest in a risk-free bond
dBtBt
= rdt; B0 = 1; (2.13)
14The information available to agents consists of the trajectories generated by the two Brownian motions(Brownian �ltration F). The underlying probability space is (;F ; P ), P being the physical probabilitymeasure.15Equation (2.12) nests many stochastic volatility models. For instance, Heston (1993) is obtained when
� = 1=2, and Jones (2003) discusses the model for � > 1.
84
with constant interest rate r. The economy is endowed with a stochastic discount factor
(SDF) which is the pricing tool used to discount state contingent cash �ows. This SDF
followsd�t�t= �rdt� �St dZSt � �Vt dZVt ; �0 = 1; (2.14)
where �St and �Vt are the market prices of risks. As in the portfolio literature (see Detemple,
Garcia, and Rinsdisbacher, 2003 and 2005, Detemple and Rinsdisbacher, 2010, and Elkamhi
and Stefanova, 2011), the form of the SDF is exogenously given in the model. However,
both �St and �Vt are to be determined.
The investment opportunity set is further composed of European index calls and puts
denoted f jt where j identi�es a particular option. In the option market, two types of agents
interact. End-users have an exogenous need to get exposure to index options. We denote by
�EU;jt end-users�net demand for option j. To meet this demand, a representative market-
maker provides liquidity for index options. The next proposition presents the pricing rule
used by the market-maker to quote each option.
Proposition 5 Given (2.11), (2.12), and (2.14) applying Ito�s lemma to f jt and imposing
the no-arbitrage condition �tfjt = EPt [�T f
jT ] implies the following dynamic for the price of
option j under the P-measure
df jt = d�Repjt + #jt � dF Vt (2.15)
= d�Repjt + #jt ��V RPtdt+ �V
�t dZ
Vt
�; (2.16)
where �Repjt corresponds to the delta replication of fjt , #
jt � V egajt=
�2pVt�is the sensi-
tivity of option j to the market variance risk factor F Vt , V RPt � EPt [Vt+dt]�EQt [Vt+dt] is
the (instantaneous) variance risk premium, and �V �t dZVt is the aggregate variance risk.
Proof. See Appendix A.
Under the Black and Scholes (1973) assumptions, index options can be perfectly repli-
cated by holding the appropriate amount of the market index and the risk-free bond. When
85
the market variance is stochastic, perfect replication is no longer achievable through the
trading of St and Bt only. As a result, the price dynamic of index options can be decom-
posed into two components. As in Black and Scholes (1973), the �rst component, denoted
d�Repjt , corresponds to the delta replication of fjt . In addition to d�Rep
jt , the entire
cross-section of index options is a¤ected by the market variance risk factor.
When �MM;jt denotes the market-maker�s inventory of option j, the market clearing
condition for index options is
�MM;jt +�EU;jt = 0 for all j: (2.17)
Equation (2.17) has an interesting implication. Whenever end-users�exposure to market
volatility,Pj �
EU;jt V egajt , does not cancel out across index options, then neither does
market-maker�s inventory risk (i.e.Pj �
MM;jt V egajt ). Consequently, the representative
market-maker will be non-trivially exposed to the market variance risk factor. In this
context, the variance risk premium can be interpreted as the market-maker�s compensation
for absorbing unhegeable variance risk.
As apparent in Figure 2, the time-series of inventory risk shares features with volatility
(i.e. clustering, autocorrelation, and reversal). To capture the �uctuations in the market-
maker�s variance risk exposure, we model inventory risk exogenously and adopt a mean-
reverting process in the spirit of stock volatility models
dInvRiskt = �(��InvRiskt)dt+ Vtdt+�InvRiskt�q
1� �2InvdZSt + �InvdZ
Vt
�; (2.18)
where InvRiskt =Pj �
MM;jt V egajt as in (2.5), � is the speed of mean reversion, � in�u-
ences the level of inventory risk, de�nes the loading on market variance, and � is the
volatility parameter. In previous equation, �Inv is the correlation of inventory risk with
the market variance risk factor.
We determine the variance risk premium through the maximization of the market-
maker�s expected utility of terminal wealth. For a given admissible investment strategy
86
~�t ���Bt ; �
St ; f�
f;jt g�, the market-maker�s self-�nanced wealth dynamic is
dWt
Wt= �Bt �
dBtBt
+ �St �dStSt
+Pj�f;jt � df
jt
f jt; with W0 = w, (2.19)
where each � is expressed in percentage of wealth, and w denotes his initial endowment.
The problem faced by the market-maker can be written as
max~�
EP [U(WT )] subject to � (2.17) and (2.18)
� (2.19) with Wt > 0: t 2 [0; T ];(2.20)
where U(:) is the market-maker�s utility function. In the model, the representative market-
maker determines his trading strategy given prices. Our goal is to invert this mapping and
infer the variance risk premium in (2.16) that induces the market-maker to clear the index
option market.
2.4 Model Implications
2.4.1 The Structure of the Variance Risk Premium
To understand the way the variance risk premium depends on inventory risk and the
market-maker�s wealth, the following proposition is needed.
Proposition 6 At time t 2 [0; T ], when the market-maker is myopic with U(Wt) = ln(Wt),
the variance risk premium is given by
V RPt = �V �V�t (Sharpet) + 0:5(1� �2V )�2V
2��0:5t
�InvRiskt
Wt
�; (2.21)
where Sharpet � ��rpVtis the market Sharpe ratio, InvRiskt = �
Pj �
EU;jt V egajt captures
the sensitivity of the market-maker�s inventory to the variance risk factor F Vt de�ned in
Proposition 1, and Wt is market-maker wealth.
Proof. See Appendix B.
87
This proposition provides several insights. The decomposition in (2.21) separates the
variance risk premium into two components. The �rst term is a function of the market
Sharpe ratio. Innovations in market variance are correlated with index returns. Conse-
quently, the variance risk premium inherits of the market Sharpe ratio. The greater the
abs(�V ) the higher the dependence of the variance risk premium on Sharpet is. Index
returns are in�uenced by the realisations of the market Sharpe ratio. Therefore, equa-
tion (2.21) helps understand the strong statistical signi�cance found when regressing the
changes in variance risk premium measures on S&P 500 log-returns.
Whenever abs(�V ) < 1, index options cannot be perfectly hedged through the trad-
ing of St and Bt. Consequently, the market is incomplete from the market-maker�s per-
spective. In this case, the market-maker requires an additional compensation relative to
�V �V�t (Sharpet). This additional premium is proportional to the ratio of inventory risk to
wealth. Since (1 � �2V )�2V 2��0:5t > 0 and Wt > 0, the variance risk premium is positively
in�uenced by inventory risk. The more negative is the exposure of market-maker to market
variance, the lower the variance risk premium will be. Consequently, the model captures
the positive sign of the coe¢ cient obtained when regressing the changes in variance risk
premium against market-makers�exposure to market variance.
Several studies have shown that the premium for market variance risk is negative on
average (see, among others, Bakshi and Kapadia, 2003, Vilkov, 2008, and Carr and Wu,
2009). Given (2.21), the variance risk premium is negative whenever
InvRisktWt
<��V (Sharpet)
0:5(1� �2V )�V��0:5t
: (2.22)
For the empirically relevant case �V < 0 and Sharpet > 0, a su¢ cient condition for
the variance risk premium to be negative is a negative inventory risk. Since SPX option
market-makers typically have a negative exposure to market variance, the model o¤ers an
interesting channel for explaining the negative variance risk premium found in previous
studies.
88
As apparent in the middle and bottom panels of Figure 1, the variance risk premium is
sporadically positive. When inequality (2.22) is not satis�ed, and �V < 0 and Sharpet > 0,
a positive inventory risk exposure results in a positive variance risk premium. Hence, the
model can also explain a positive variance risk premium at times when option market-
makers are positively exposed to market variance.
Finally, note that a positive inventory risk will not result in a positive variance risk
premium as long as (2.22) is satis�ed.
2.4.2 Market-Maker Optimal Wealth
In Proposition 2, inventory risk is normalized by the market-maker�s wealth. Consequently,
inventory risk will matter the most when the market-maker is poor (i.e. for low values of
Wt). In contrast, the e¤ect of inventory risk on the variance risk premium vanishes when
the market-maker wealth goes to in�nity. This result justi�es the strong signi�cance found
in all the empirical tests for the interaction of inventory risk with changes in market-makers�
wealth.
In the model, market-maker�s wealth is endogenous and satis�es
Wt = 1= ( �t) ; (2.23)
when U(Wt) = ln(Wt). Since the market-maker�s marginal utility is strictly increasing,
is uniquely de�ned by the static budget constraintW0 = EP [�TWT ] = 1= )Wt =W0=�t.
Consequently, at any point of time, the market-maker�s wealth is proportionally related to
the ratio of his initial endowment W0 to the SDF. Given (2.14), we can apply Ito�s lemma
to Wt to obtain the dynamic of the market-maker�s optimal wealth
dWt =�r +
��St�2+��Vt�2�
Wtdt+ �StWtdZ
St + �
Vt WtdZ
Vt . (2.24)
89
2.4.3 Risk-Neutral Dynamics
The model has implications for pricing. To discount the option payo¤ at maturity using
the risk-free rate, all underlying processes must be risk-neutralized.
Proposition 7 When the SDF follows (2.14), the risk-neutral processes driving the dy-
namic of the economy are
dSt = rStdt+pVtSt
�q1� �2V d ~Z
St + �V d ~Z
Vt
�(2.25)
dVt = �(� � Vt)dt� V RPtdt+ �V �t d ~ZVt (2.26)
dInvRiskt = �(�� InvRiskt)dt+ Vtdt+ �InvRiskt�q
1� �2Invd ~ZSt + �Invd
~ZVt
���InvRiskt
�q1� �2Inv�
St + �Inv�
Vt
�dt (2.27)
dWt = rWtdt+ �StWtd ~Z
St + �
Vt Wtd ~Z
Vt : (2.28)
where ~ZSt and ~ZVt are risk-neutral, �St = Sharpet=
q1� �2V � �V �
Vt =q1� �2V is obtained
by imposing the index no-arbitrage condition, �Vt = V RPt= (�V�t ), and V RPt satis�es
Proposition 2.
Proof. The Girsanov theorem implies that dZSt = d ~ZSt � �St dt and dZVt = d ~ZVt � �Vt dt.
Using this result in (2.11), (2.12), (2.18), and (2.24) delivers the risk-neutral processes.
The result in Proposition 3 can be used for pricing. When end-users�demand for SPX
options increases, it decreases inventory risk. A decrease in inventory risk implies a more
negative variance risk premium on average. From (2.26), we see that changes in the risk-
neutral market variance are negatively in�uenced by the variance risk premium. Thus,
an increase in end-users�demand will result in a higher risk-neutral variance. This model
prediction can be related to Bollen and Whaley (2004) who document that changes in the
implied volatility of SPX OTM puts are positively in�uenced by end-users� net buying
pressure. However, our model suggests that only the variance exposure of market-maker�s
total risk exposure should impact changes in risk-neutral volatility.
90
Overall, the predictions delivered by the model are in line with the empirical �ndings
documented. In the next section, we estimate the model and assess the role of inventory
risk and market-maker wealth for the valuation of SPX options.
2.5 Estimation and Fit
In this section, we �rst describe our estimation methodology. Subsequently, we report
on parameter estimates and model �t. Finally we relate the estimated parameters to the
economic impact of inventory risk and market-maker�s wealth on index option prices.
2.5.1 Estimation Methodology
Several approaches have been proposed in the literature for estimating stochastic volatil-
ity models. Aït-Sahalia and Kimmel (2007) and Jones (2003) estimate volatility model
parameters using bivariate time-series of returns and at-the-money implied volatility. Pan
(2002) uses GMM to estimate the objective and risk-neutral from returns and option prices.
Christo¤ersen, Jacobs, and Mimouni (2010) adopt a particle �ltering approach to estimate
various alternatives to Heston (1993) based on returns and a large panel of option prices
combined.
For estimating the model, we adopt a two steps procedure. In a �rst step, we �lter the
physical parameters describing (2.12), along with the spot variances Vt. Taking the �ltered
spot variances and the objective variance parameters as given, we estimate the dynamic
of inventory risk and market-maker�s wealth using a large panel of SPX put prices. Both
InvRiskt, and Wt are latent variables in the model. However, to avoid over�tting the data
we constrain InvRiskt to be equal to its empirical value (2.5). In addition, we �x the
market-maker�s wealth to W0 = w at the beginning of every day. Based on these initial
values and on the �ltered Vt, we exploit the results in Proposition 2 and 3 to simulate the
economy forward and infer the dynamics (2.27) and (2.28) consistent with observed index
option prices.
For estimation purposes, we set the time-step �t to 1=365 and the expected return �
91
to its sample average 1T�1
TPt=2(St � St�1) =St�1 � 365, where T denotes the last day in the
sample.
We now describe each step in more details.
Step 1: Filtering the Variance Dynamic Using S&P 500 Returns
We need to estimate the structural parameters �V � f�; �; �; �V ; �g in (2.12) along with the
vector of spot variances fVtgt=1;2;:::;T . To this end, we adopt the particle �ltering algorithm
(PF henceforth). The PF o¤ers a convenient way to estimate stochastic volatility models.
It has recently been applied in Johannes, Polson, and Stroud (2009), Christo¤ersen, Jacobs,
and Mimouni (2010), and Malik and Pitt (2011) among others.
LetnV jt
oNj=1
denotes the smooth resampled particles where N de�nes the number of
particles which is set to 10; 000. Using the algorithm described in Appendix C, we estimate
on each day the likelihood of observing St+1 given Vjt and St, and denote it ~P
jt
�V jt ;�
V�.
Based on the likelihood of each particle calculated on each day, the MLIS criterion applied
to estimate �V is
�V = argmax
TXt=1
Lt; (2.29)
where Lt � ln
1N
NPj=1
~P jt
�V jt ;�
V�!
is the model daily log-likelihood. On day t, the
�ltered spot variance is obtained by averaging the smooth particles
Vt =1
N
NXj=1
V jt : (2.30)
Next, we discuss the estimation of inventory risk and market-maker�s wealth based on SPX
options.
Step 2: Estimating Inventory Risk and Market-Maker�s Wealth
The model does not allow for closed-form solution for option prices. Consequently, we
rely on Monte-Carlo methods for estimating the inventory risk and market-maker�s wealth
92
dynamic embedded in SPX options. Taking �V andnVt; InvRiskt
ot=1;2;:::;T
as given where
InvRiskt corresponds to (2.5), we estimate �Inv � f�; �; ; �; �Invg and w by minimizing
the sum of implied volatility squared errors (SIVSE)
n�Inv; w
o= argmin
NJXj;t
�IVj;t � IVMj;t
��Inv; w; �V ; Vt; InvRiskt
��2; (2.31)
where NJ is the total number of observations, IVj;t is the option j implied volatility on
day t, and IVMj;t is the model implied volatility. We use Black-Scholes to calculate implied
volatilities for both market and model prices. When calculating model prices, we follow
the algorithm described in Appendix D using 10; 000 Monte-Carlo paths.
2.5.2 Parameter Estimates
In Panel A of Table 6, we present descriptive statistics on the daily S&P 500 returns.
In-sample, the average market return is 5:86%. This low average return estimate is partly
in�uenced by the abrupt drop in the S&P 500 index during the crisis period (see Figure 1).
During the 1997-2011 period, the sample variance is 4:58% annually, which corresponds to
a 21% average volatility.
In Panel B, we report the estimated parameters for the CEV dynamic (2.12). The
sample MLIS is 11; 746. The estimated � is close to the sample variance in Panel A.
Moreover, �V is large and negative. In the model, a large and negative �V is important to
generate enough negative skewness in the S&P 500 return distribution. The estimate for
the CEV elasticity parameter is 90%. A high volatility of volatility generates additional
variance in the return process. Thus, the estimated � arguably captures the fat tails of
S&P 500 returns distribution. Finally, note how the index data require a slow variance
mean-reversion speed. The coe¢ cient of 2:91 corresponds to a daily variance persistence
of 1� �=365 = 0:9920.
For comparison purposes, we also report in Panel C the parameters obtained for the
Heston model for which � = 1=2. The model MLIS is lower than the likelihood obtained
93
by the CEV dynamic. The two models require di¤erent structural parameters to explain
the data. For instance, note the di¤erences of mean reversion speed and unconditional
variance between the two models. The Heston dynamic requires a higher speed reversion
but has a lower long-term variance. Moreover, the two models also require substantially
di¤erent volatility of volatility parameters. However, it should be noted that the di¤erence
in � between the two models in�uences the magnitude of the other parameter estimates.
In Figure 4, we plot the time-series of �ltered spot volatilitiespVt annualized and
expressed in percentage for both models. As expected, the time-series of objective spot
volatilities share common features with the VIX Index in Figure 2.
When calibrating inventory risk and market-maker wealth, we rely on OptionMetrics�
volatility surface data. In order to speed up estimation, we restrict attention to puts
observed on the �rst Wednesday of each month with moneyness between 0:9 and 1:1,
and with 2, 3, and 6 months to maturity. The �nal option sample consists of 6; 292 put
observations during the 1997-2011 sample period.
In Panel A of Table 7, we report the estimated coe¢ cients for �Inv obtained from
minimizing the sum of squared errors (2.5). The daily inventory persistence is 1��=365 =
0:9706, which is slightly less than the persistence of market variance. A high mean reversion
speed is necessary to explain the abrupt reversal of inventory risk observed during the
crisis period in the bottom panel of Figure 2. Moreover, we saw that inventory risk is
negative most of the time. This stylized fact is captured by � estimated to be large
and negative. Note however, that the unconditional expectation of inventory risk is also
in�uenced by . Interestingly, the loading of inventory risk on lagged spot variance is
positive. Thus, inventory risk tends to increase with market uncertainty. In-sample, the
volatility parameter for inventory risk is 16:55%. Note that the correlation of inventory
risk with market variance is close to zero. While the changes in inventory risk depend
positively on the level of market variance through , inventory risk is nearly independent
of contemporaneous innovations in variance.
Interestingly, the estimated wealth level is about 440 million dollars. Since w captures
94
the dealer�s initial wealth, the parameter estimated is comparable in magnitude to the
daily delta-hedged pro�ts and losses documented in Figure 3. Quantitatively, it represents
approximately 4 years of cumulative daily pro�t and losses. Finally, the model�s sum of
squared errors is about 6:40 at the optimum.
To benchmark the model performance, we �t Heston (1993) on option prices using a
similar methodology but for which we set the instantaneous V RPt = h � Vt, where h is a
constant to be estimated. This speci�cation for the variance risk premium is consistent
with Heston (1993), Bates (2000), and Pan (2002) among others. From Panel B, we see
that the variance risk premium parameter is �1:08. Note however that the �t obtained by
the Heston model is not as good as the �t of the inventory risk model. During our sample,
Heston�s sum of squared implied volatility errors is about 20% greater than the inventory
model.
2.5.3 Index Option Fit
To measure the performance of the model, we compute the percentage implied volatility
RMSE de�ned as
IV RMSE �
vuut 1
NJ
NJXj;t
�IVj;t � IVMj;t
�2� 100; (2.32)
where IVMj;t is the model implied volatility. We compute the performance criterion for
both models using all puts options with moneyness between 0:9 and 1:1, and with 2, 3,
and 6 months to maturity. The sample used to evaluate each model performance consists
of 131; 838 observations.
Table 8 presents the �t obtained for the inventory risk model and Heston. We report
the IVRMSE for all contracts for each year in Panel A, as well as separate results for in-,
at-, and out-of-the-money puts in Panel B. Several interesting �ndings emerge.
First, the average yearly IVRMSE for the inventory risk model is 3:14%, which is good
given that the sample includes the �nancial crisis. The �t of each model are close; however,
95
the inventory model dominates the Heston model particularly in the second part of the
sample period. Relative to the inventory model, Heston�s best pricing performance is
obtained in 1999 with a IVRMSE di¤erence between the two model of 0:48%. In contrast,
the inventory risk model achieves its best �t relative to the Heston model in 2009 with an
IVRMSE lower by 1:95%. Since 2003, the IVRMSE of the inventory model is lower than
Heston by more than 0:90%.
Second, the inventory model achieves a better �t for ATM puts relative to the Heston
model. ATM options are the most sensitive to change in variance risk premium. Thus, this
result potentially uncovers the inability of the Heston model combined with the speci�cation
V RPt = h�Vt to adequatly captures the variations in the variance risk premium. To further
assess this, we regress the daily implied volatility root mean squared errors against the
empirical proxy of one-month variance risk premium, and the daily log-likelihood Lt of the
physical returns. For the inventory risk model, we obtain
IV RMSEt = 3:142(0:000)
+ 0:021(0:901)
� V RP (t; 30)� 0:266(0:006)
� Lt + "t; (2.33)
where the regressors are standardized, and the Newey-West p-values reported under each
parameters are calculated with 8 lags. The adjusted R-Squared of the regression is 1:77%.
Arguably, the mispricing of the inventory risk model is not related to the realization of
the variance risk premium. We see however that part of its IVRMSE is correlated with
its log-likelihood. When the likelihood decreases the pricing error of the model tends to
increase. For Heston (1993), we have
IV RMSEt = 3:562(0:000)
� 0:813(0:000)
� V RP (t; 30)� 0:022(0:752)
� Lt + "t; (2.34)
with an adjusted R-Squared of 17:93%. The result is striking. In contrast to the inventory
risk model, a large part of the pricing errors of the Heston model can be attributed by its
inability to fully capture the �uctuations in the variance risk premium.
Third, from Panel B in Table 8 we see that the inventory risk model �ts OTM puts
96
better than Heston. Stochastic volatility models often have more di¢ culties in capturing
the implied volatility patterns of OTM option. Thus, the ability of the inventory model
to better price OTM options is particularly interesting. This result is partly due to the
non-linearities in the joint dynamic of risk-neutral variance, inventory risk, and market-
maker wealth. When volatility is high, the ratio of inventory risk to wealth tends to be
large and negative. This causes the variance risk premium to be large and negative, which
in turn implies an increase in risk-neutral volatility and a more negative ratio of inventory
risk to wealth through its feedback e¤ects on the risk-neutral dynamics (2.27) and (2.28).
A higher risk-neutral volatility during bad times increases the negative skewness in the
risk-neutral index returns distribution, which substantially improves the pricing of OTM
puts.
In Figure 5, we plot the daily one-month variance risk premium and IVRMSE for each
model. From Panel A, we see that the inventory risk model can deliver a wide range
of variance risk premium. In comparison, the one-month variance risk premium implied
by the Heston model is always negative. The ability of the inventory model to generate
substantial variation in the variance risk premium drives its performance.
The results suggest that accounting for inventory risk and market-maker wealth is
important to capture variation in the variance risk premium and thus explain index option
prices.
In the next section, we quantify the economic magnitude of the response of SPX option
prices to changes in inventory risk and market-maker�s wealth.
2.5.4 Economic Magnitude
Little is known about the way market-makers adjust their quotes when they absorb large
buy orders, which cause their exposure to market variance to become more negative (i.e.
decrease their inventory risk). Similarly, the magnitude of the impact of market-makers�
losses and gains on index option prices is also an open question.
To understand the way inventory risk and market-maker�s wealth in�uence market-
97
makers�quotation behaviour, in Figure 6 we plot the dollar sensitivity of SPX put options
to a decrease in the state variables implied by the model. To calculate the model sensitivies
@Pt=@InvRiskt and @Pt=@Wt, we use the estimated parameters �V , �Inv, and w, and set
r = 4%, St = 1183 , and Vt = �: Moreover, inventory risk is set to its unconditional
mean InvRiskt = �9:03E + 09, and market-maker wealth is initialized to Wt = w. Based
on the sensitivities calculated, we then compute the dollar response of each option as
�InvRiskt � @Pt=@InvRiskt and �Wt � @Pt=@Wt, and plot the result across moneyness.
In both panels, each line corresponds to di¤erent level of decrease in inventory risk and
wealth. The circles correspond to the dollar response to an average decrease in the state
variables. The diamonds identify the response when the decrease of each latent variable is
within its 90th percentile decrease.
Figure 6 provides several insights. First, we see that a decrease in inventory risk results
in an increase in index option prices. This is consistent with the theoretical predictions in
Proposition 2. When market-makers�risk exposure decreases, they require a more negative
variance risk premium which translates into an increase in index option prices. Similarly,
market-makers� losses result in an increase in the price of index options. Interestingly,
market-maker�s wealth has the biggest impact on option prices.
Note also that the e¤ect of inventory risk and market-maker�s wealth on SPX options
across strike prices is non-linear, and is most prominent for at-the-money options. In-
terestingly, it is these options that make up most of market-makers�inventory (see Table
2).
When all variables are set to their mean values, the average decrease in inventory risk
and the average wealth loss cause between a 10 and 50 dollars increase in price. Given an
average option price of 6; 500 dollars, these e¤ects correspond to a 0:15% and 0:77% daily
increase in price. This result echoes the �ndings of Murayev (2013) who shows that the
average trade for equity option has inventory risk impact of 0:4%. Note however that this
study is the �rst to document and quantify the impact of option market-makers�wealth
on index option prices.
98
During turbulent times, when market-makers� aggregate loss is within its 90th per-
centile decrease, it causes a 150 dollars increase in price, which corresponds to a 2:31%
daily increase. These results further highlight the non-trivial role of market-makers in the
determination of index option prices through their e¤ect on the variance risk premium.
2.6 Summary and Conclusions
Using aggregate market-makers�positions at the CBOE, we examine how inventory risk and
market-makers�wealth jointly determine the value of index options through their e¤ects
on the variance risk premium. The analysis demonstrates that part of the variance risk
premium corresponds to option market-makers�compensation for their exposure to market
variance. We �nd that inventory exposure to market variance and changes in market-
makers�wealth explain the variance risk premium. Using daily observations, we �nd that
a one standard deviation decrease in inventory risk causes a 1:2% decrease in the variance
risk premium next period. In addition, the interaction of lagged inventory risk with changes
in market-maker wealth is also strongly signi�cant.
Based on these �ndings, we then develop a model in which market variance is stochastic
and a representative market-maker accumulates inventory over time by absorbing end-
users�net demand for index options. Starting from the market-maker�s optimal trading
strategy, we derive an explicit formula linking the variance risk premium to inventory
risk and market-maker wealth. The model predictions provides several insights that help
understand empirical stylized facts.
Finally, we estimate the model on S&P 500 returns and options. Overall, the model
performs well. Our results suggest that accounting for inventory risk and market-maker�s
wealth in pricing models can help for index option valuation. Based on the estimated
parameters, we �nd that the e¤ect of inventory risk and market-maker�s wealth on SPX
options across strike prices is non-linear, and is most prominent for at-the-money options.
Several issues are left for future research. First, it would be interesting to develop
and test the implications of various inventory risk dynamics for option pricing. Second,
99
it would be interesting to extend the model by allowing for jumps in the market price.
Finally, investigating the pricing implications inventory risk and market-makers�wealth
have for other derivative markets would also be of signi�cant interest.
Appendix
This appendix collects the proofs of the propositions and presents the algorithms used for
estimating the model.
A. Proof of Proposition 1
For ease of notation, we de�ne at � EPt [dVt]=dt, bt � �V �t , �jt � @f jt =@St (i.e. the
delta of option j), �jt � @�jt=@St (i.e. the gamma of option j), and #jt � @f jt =@Vt (i.e.
the sensitivity of option j to market variance). Applying Ito�s lemma to f j implies the
following dynamic under the P -measure
df jt =�@fjt@t +�
jtSt�+ #
jtat +
@�jt@Vt
�VpVtStbt +
12
��jtVtS
2t +
@#jt@Vt
b2t
��dt
+�jtStpVt
�q1� �2V dZSt + �V dZVt
�+ #jtbtdZ
Vt :
(2.35)
Given (2.14) and the no-arbitrage condition �tfjt = EPt [�T f
jT ], we have
rf jt ��jtStr � #
jtaQt =
@f jt@t
+@�jt@Vt
�VpVtStbt +
1
2
�jtVtS
2t +
@#jt@Vt
b2t
!; (2.36)
where aQt � EQt [dVt]=dt. Combining (2.35) with (2.36), we obtain
df jt =�rf jt +�
jtSt(�� r) + #
jt
�at � aQt
��dt+�jtSt
pVt
�q1� �2V dZ
St + �V dZ
Vt
�(2.37)
+#jtbtdZVt :
Since the delta replication of f j satis�es �Repjt = �Bt Bt+�St St where �
B and �S denote the
units of bond and market index to hold. For replication strategy with rebalancing horizon
100
dt, we have �St St = �jtSt and �
Bt Bt = f jt ��
jtSt. Consequently,
d�Repjt =�rf jt +�
jtSt(�� r)
�dt+�jtSt
pVt
�q1� �2V dZ
St + �V dZ
Vt
�, (2.38)
with �Repjt = f jt . Combining (2.37) and (2.38), and the de�nition of b, we get
df jt = d�Repjt + #jt ��V RPtdt+ �V
�t dZ
Vt
�(2.39)
= d�Repjt + #jt � dF Vt ; (2.40)
where V RPt � at � aQt = EPt [Vt+dt] � EQt [Vt+dt] is the (instantaneous) variance risk pre-
mium, and dF Vt = V RPtdt + �V �t dZVt is the market variance risk factor. We can now
re-express #jt in terms of sensitivity to volatility
#jt =@f jt@Vt
=@f jt@pVt
@pVt
@Vt=V egajt2pVt; (2.41)
which completes the proof.
B. Proof of Proposition 2
The strategy we adopt is as follows. First, we solve the market-maker�s (unconstrained)
portfolio allocation when the market clearing condition is not imposed. Based on the
investment strategy obtained, we then require it to satisfy the market clearing condition
to infer the structure of the variance risk premium.
The static maximization
max~�
EP [U(WT )] subject to � W0 > EP [�TWT ]
� Wt > 0;(2.42)
is the dual problem of the unconstrained utility maximization (2.20) (see, among others,
Karatzas, Lehoczky, and Shreve, 1987, and Cox, and Huang, 1989). To solve this, we form
101
the lagrangian
L( ) = EP [U(WT )] + (w � EP [�TWT ])
= EP [U(WT )� �TWT ] + w; (2.43)
where is the lagrangian coe¢ cient associated to the static budget constraint W0 >
EP [�TWT ]. A point wise maximization of (2.43) implies the following optimality condition
Wt = Iu( �t); (2.44)
where Iu(:) represents the inverse function of market-maker�s marginal utility. When
@U(:) > 0, is uniquely de�ned by W0 = EP [�TWT ] = EP [�T Iu( �T )]: Using the re-
sult in Proposition 1, the fact that �f;jt Wt = �MM;jt f jt , and the de�nition of inventory risk,
we can re-write the wealth process (2.19) as
dWt
Wt= ��Bt �
dBtBt
+ ��St �dStSt
+InvRiskt
2pVtWt
� dF Vt ; (2.45)
where ��Bt = �Bt �Pj �
MM;jt
�f jt ��
jtSt
�=Wt and ��St = �St �
Pj �
MM;jt �jtSt=Wt. In this
economy, the discounted wealth process is
d (�tWt)
�tWt=dWt
Wt+d�t�t+d h�;W it�tWt
; (2.46)
where d h:; :it is the covariance operator. Applying the previous equation on the SDF
dynamic (2.14) and the wealth process (2.45), we obtain
d (�tWt)
�tWt=
���StpVt
q1� �2V � �
St
�dZSt +
���StpVt�V + 0:5�V
��0:5t
InvRisktWt
� �Vt�dZVt ;
(2.47)
for which we have imposed the no-arbitrage conditions
�tSt = EPt [�TST ], Sharpet =q1� �2V �
St + �V �
Vt (2.48)
102
�tfjt = EPt [�T f
jT ], �f;jt � r = gjt�
St + h
jt�Vt ; (2.49)
where Sharpet ����rpVt
�is the market Sharpe ratio, and
�f;jt � EPt [dfjt =f
jt ]
gjt � �jt
�St=f
jt
�pVt
q1� �2V
hjt � �jt
�St=f
jt
�pVt�V + #
jt
��V �t =f
jt
�:
By application of the Clark-Ocone formula, for a FT�measurable random variable F , we
have
F = EP [F ] +TR0
EPt [DSt (F )]dZ
St +
TR0
EPt [DVt (F )]dZ
Vt ; (2.50)
where Dit (F ) is the time t Malliavin derivative of F with respect to Z
i.16 This representa-
tion of F is unique and can be exploited to obtain explicit formulas for ��S . Using (2.47),
we can express �TWT in its integral form
�TWT = w +TR0
�tWt
���StpVt
q1� �2V � �
St dZ
St
�(2.51)
+TR0
�tWt
���StpVt�V + 0:5�V
��0:5t
InvRisktWt
� �Vt dZVt�;
Setting F = �TWT in (2.50) combined with w = EP [�TWT ], the uniqueness of (2.50)
requires the integrands in (2.50) and (2.51) to be equal. De�ningAit � EPt [Dit(�TWT )]=�tWt
for i = S; V implies
��StpVt
q1� �2V = ASt + �
St (2.52)
��StpVt�V + 0:5�V
��0:5t
InvRisktWt
= AVt + �Vt ; (2.53)
which jointly de�ne the market-maker�s investment strategy. In (2.52) and (2.53), the mar-
ket clearing condition imposes �MM;jt = ��EU;jt for all j and InvRiskt = �
Pj �
EU;jt V egajt
16The variable F must be FT�measurable and must belong to the stochastic Sobolev space (i.e. F 2D1;2). We refer to Di Nunno, Økskendal, and Proske (2009) for a formal description of the D1;2 space.
103
in aggregate. Solving for ��S in (2.52) and using the result in (2.53), we get
�AVt + �
Vt
��m
�ASt + �
St
�= 0:5�V ��0:5t
�InvRiskt
Wt
�; (2.54)
where m � �V =q1� �2V .17 By the properties of Malliavin derivatives, we have
Dit(�TWT ) = Di
t(Iu( �T ))�T +Dit(�T )WT = (@Iu( �T ) �T +WT )D
it(�T ):
The facts that @U(Iu(y)) = y and @U(WT ) = �T imply
@Iu( �T ) =1
@2U(WT )) @Iu( �T ) �T =
@U(WT )
@2U(WT )= ��T ; (2.55)
where � is the Arrow-Pratt measure of market-maker�s absolute risk tolerance. Therefore,
Dit(�TWT ) = (WT � �T )Di
t(�T ): (2.56)
Combining (2.56) and the de�nition of Ai, we get
Ait = �EPt [�t;T(WT � �T )
WtDit(�T )];
where �t;T � �T =�t. When U(x) = ln(x), we have
@U(WT ) =1
WTand @2U(WT ) = �
1
W 2T
;
which implies
�T =WT ) ASt = AVt = 0:
Using this into (2.54), we obtain
�Vt �m�St = 0:5�V��0:5t
�InvRiskt
Wt
�: (2.57)
We also know from (2.48) that the market price of risks �S and �V are linked through the
17Note that m is well-de�ned whenever abs(�) 6= 1.
104
market index no-arbitrage condition which can be re-written as
�St = Sharpet=q1� �2V �m�
Vt : (2.58)
Combining (2.57) and (2.58), we can express the market price of variance risk as
�Vt = �V (Sharpet) + 0:5(1� �2V )�V��0:5t
�InvRiskt
Wt
�: (2.59)
Now, we can make use of the de�nition of the variance risk premium when the physical
volatility process is described by (2.12) and the SDF takes the form (2.14), which is
V RPt = (�V�t )�
Vt : (2.60)
Combining (2.59) with (2.60), we �nally get
V RPt = �V �V�t (Sharpet) + 0:5(1� �2V )�2V
2��0:5t
�InvRiskt
Wt
�; (2.61)
where Sharpet =���rpVt
�, and InvRiskt = �
Pj �
EU;jt V egajt .
C. Particle Filter Estimation
The following algorithm describes the way we evaluate the likelihood ~P jt
�V jt ;�
V�of ob-
serving St+1 given the smooth resampled particles Vjt , and the structural parameters �
V .
For estimation purposes, we set the number of particles denoted N to 10; 000.
Using the Euler discretization for dln(St) and (2.12), one can simulate the state of the
N raw particlesn~V jt
oNj=1
forward givennV jt�1
oNj=1
according to
ZV;jt =
0B@ ln�
StSt�1
����� V jt
2
��tq
V jt
�q1� �2V Z
S;jt
1CA =�V (2.62)
~V jt = V jt�1 + �(� � Vjt�1)�t+ �
�V jt�1
��ZV;jt ; (2.63)
where ZS;jt is N(0;p�t), and � is �xed to the sample average. Using the set of raw
105
particles, the likelihood of observing St+1 given ~Vjt and St is
~P jt
�~V jt ;�
V�=
1q2� ~V jt
exp
0B@��ln�St+1St
�����
~V jt2
��t�2
2 ~V jt
1CA : (2.64)
Based on the set of normalized weights
�P jt
�~V jt ;�
V�=
~P jt
�~V jt ;�
V�
Pj
~P jt
�~V jt ;�
V� ; (2.65)
and the raw ~V jt , the method of Pitt (2002) can be applied to resample the smoothed
particlesnV jt
oNj=1
and evaluate their corresponding weights ~P jt�V jt ;�
V�.18
D. Risk-Neutral Pricing
Suppose that we want to price an index put option on day t with T days to maturity and
strike price K based on N = 10; 000 simulations. For each simulation n, we initiate the
state variables St, Vt, and InvRiskt to their respective values on the day of the pricing.
Moreover, we initialize the market-maker wealth to w. For a given path n, the forward
state of the discretized processes in Proposition 3 given the information on day t is
ln(Snt+1) = ln(Snt ) + (r � V nt =2)�t+
pV nt
�q1� �2V ~Z
S;nt + �V ~Z
V;nt
�(2.66)
V nt+1 = V nt + �(� � V nt )�t� V RPnt �t+ � (V nt )� ~ZV;nt (2.67)
InvRisknt+1 = InvRisknt + �(�� InvRisknt )�t+ V nt �t
+�InvRisknt
�q1� �2Inv ~Z
S;nt + �Inv ~Z
V;nt
���InvRisknt
�q1� �2Inv�
S;nt + �Inv�
V;nt
��t (2.68)
18The method proposed in Pitt (2002) involves smoothing the �P jt to a continuous CDF from which theset of smooth particle V j
t can be resampled.
106
ln(Wnt+1) = ln(W
nt ) +
�r �
���S;nt
�2+��V;nt
�2�=2
��t+ �S;nt
~ZS;nt + �V;nt~ZV;nt ; (2.69)
where ~ZS;nt and ~ZV;nt are independent N(0;p�t). In the previous system, we set
V RPnt = �V � (Vnt )
� Sharpent + 0:5(1� �2V )�2 (V nt )2��0:5
�InvRisknt
Wnt
�; (2.70)
where Sharpent =���rpV nt
�. Moreover, the prices of risk are calculated according to
�S;nt = Sharpent =q1� �2V � �V �
V;nt =
q1� �2V and �V;nt = V RPnt = (� (V
nt )
�) : (2.71)
Simulating the system forward from day t to T , the price of the index put option on day t
is equal to
P��Inv; w;�V ; Vt; InvRiskt
�=
NXn=1
max (K � SnT ; 0) � exp(�r � (T � t) =365)N
; (2.72)
where �V and �Inv are the structural parameters of the market variance and inventory
risk processes respectively, and w is the market-maker�s wealth parameter.
Combining (2.59) with (2.60), we �nally get
V RPt = �V �V�t (Sharpet) + 0:5(1� �2V )�2V
2��0:5t
�InvRiskt
Wt
�; (2.73)
where Sharpet =���rpVt
�, and InvRiskt = �
Pj �
EU;jt V egajt .
107
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111
Figure 1: S&P 500 Index, Variance Risk Premium, and Delta-Hedged Gains and Losses
of Near-the-Money Options
1998 2000 2002 2004 2006 2008 2010 2012600
850
1100
1350
1600S&P 500 Index
1998 2000 2002 2004 2006 2008 2010 201234
17
0
17
34OneMonth Variance Risk Premium
1998 2000 2002 2004 2006 2008 2010 2012175
0
175
350
525Weekly DeltaHedged Gains and Losses of NeartheMoney Options
Notes to Figure: The top panel plots the time-series of the S&P 500 index. The middle
panel plots the one-month variance risk premium expressed in percentage and measured by
the di¤erence between the one-month ex-post realized variance and one-month expected
risk-neutral variance. In the bottom panel, we graph the weekly average of daily delta-
hedged gains and losses of all options with moneyness (S=K) between 0:98 and 1:02.
112
Figure 2: CBOE VIX Index and
Market-Makers�Inventory Risk
1998 2000 2002 2004 2006 2008 2010 20125
25
45
65
85CBOE VIX Index
1998 2000 2002 2004 2006 2008 2010 20127500
5000
2500
0
2500Inventory Risk ($ Millions)
Notes to Figure: The top panel plots the CBOE VIX Index which represents the implied
volatility of an at-the-money option with exactly 30 days to maturity expressed in percent-
age. The bottom panel plots CBOE market-makers�inventory risk dynamic measured as
the vega-weighted sum of inventories across all contracts expressed in $ Millions.
113
Figure 3: Market-Makers�Daily and Cumulative Pro�ts and Losses, and
Market-Makers�Bid-Ask Spread Revenue
1998 2000 2002 2004 2006 2008 2010 2012398
199
0
199
398Profits & Losses of DeltaHedged Inventory ($ Millions)
1998 2000 2002 2004 2006 2008 2010 2012500
0
500
1000
1500Cumulative Profits & Losses of DeltaHedged Inventory ($ Millions)
1998 2000 2002 2004 2006 2008 2010 20120
75
150
225
300BidAsk Spread Revenue ($ Millions)
Notes to Figure: The top panel plots the daily pro�ts and losses of market-makers�delta-
hedged inventory expressed in $ Millions. The middle panel graphs the cumulative daily
pro�ts and losses over the sample period of market-makers�delta-hedged inventory in $
Millions. In the bottom panel, we plot the SPX market-makers�bid-ask spread revenue
expressed in $ Millions.
114
Figure 4: Filtered Spot Volatilities Using Daily S&P 500 Returns
1998 2000 2002 2004 2006 2008 2010 20125
25
45
65
85CEV Spot Volatility
1998 2000 2002 2004 2006 2008 2010 20125
25
45
65
85Heston (1993) Spot Volatility
Notes to Figure: The �gure plots the daily spot volatilities �ltered from S&P 500 daily
returns using particle �ltering. In the top panel, we plot the spot volatilities estimated for
the CEV dynamic. In the bottom panel, we graph the �ltered spot volatilities obtained
for Heston (1993). For both panels, the daily volatilities are annualized and expressed in
percentage.
115
Figure 5: One-Month Variance Risk Premium, IVRMSE, and
Implied Volatility Smile by Model
2000 2004 2008 201234
17
0
17
34
VR
P
Panel A: VRP Inv. Risk Model
2000 2004 2008 20120
6
12
18
24
IVR
MSE
Panel C: IVRMSE Inv. Risk Model
0.90 0.95 1 1.05 1.1017
21
24
27
30
Impl
ied
Vol
.
Moneyness (S/K)
Panel E: Market and Inv. Risk Model IV
Model IVMarket IV
2000 2004 2008 201234
17
0
17
34Panel B: VRP Heston Model
2000 2004 2008 20120
6
12
18
24Panel D: IVRMSE Heston Model
0.90 0.95 1 1.05 1.1017
21
24
27
30
Moneyness (S/K)
Panel F: Market and Heston Model IV
Notes to Figure: For each model, we plot the daily one-month variance risk premium (VRP)
and IVRMSE expressed in percentage terms in Panel A to D. To obtain the models�one-
month VRP on each day, we simulate 10,000 paths, calculate the 30 days integrated VRP
for each path, and take the average. For the inventory risk model, the VRP is calculated
using estimated parameters and latent variables. For the Heston model, the instantaneous
VRP is set to h � Vt where h = �1:08. In Panel E and F, we graph market-implied (solid)
and model-implied (dashed) volatility smile.
116
Figure 6: The Dollar Response of Index Options to Inventory Risk
and Market-Maker�s Wealth
0.8 0.9 1 1.1 1.20
15
30
45
60 D
olla
r R
espo
nse
Response to Inventory Risk Decrease
50th Quantile Decrease10th Quantile Decrease
0.8 0.9 1 1.1 1.20
50
100
150
200
Moneyness (S/K)
Dol
lar
Res
pons
e
Response to Wealth Decrease
Notes to Figure: We plot the dollar response of SPX puts with 90 days to maturity to a
decrease in inventory risk (top panel), and to a decrease in market-maker wealth (bottom
panel). To calculate the model sensitivies @P@InvRisk and
@P@W , we use the estimated parame-
ters �V , �Inv, and w, and set r = 4%, St = 1183 , and Vt = �: Moreover, inventory risk
and market-maker wealth are set to InvRiskt = �9:03E+09 and Wt = w. Based on these
sensitivities, we then calculate the dollar response of each option as �InvRiskt � @P@InvRisk
and �Wt � @P@W , and plot the result across moneyness. In both panels, each line corresponds
to di¤erent level of decrease in inventory risk and wealth. The circles correspond to the
dollar response to an average decrease in the state variables. The diamonds identify the
response when the decrease of each latent variable is within its 90th percentile decrease.
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120
121
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123
124
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