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dissertation entitled

PROPERTIES OF THE PRICING KERNEL AND
IMPLICATIONS

presented by
Ranadeb Chaudhuri

has been accepted towards fulfillment
of the requirements for the

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PROPERTIES OF THE PRICING KERNEL AND IMPLICATIONS
By
Ranadeb Chaudhuri

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment. of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Finance

2009

ABSTRACT
PROPERTIES OF THE PRICING KERNEL AND IMPLICATIONS
By
Ranadeb Chaudhuri

A fundamental property of many well known asset pricing models is that the
pricing kernel is a decreasing function of gross returns on a security. Recent empirical
evidence on the shape of the empirical pricing kernel suggests that for a range of
security returns, the pricing kernel is increasing. Such a violation in rnonotonicity of
the pricing kernel is inconsistent with traditional asset pricing theory as this would
suggest that utility functions are not concave and that the representative agent is
locally risk seeking. In the two chapters comprising this dissertation, I provide robust
tests of the monotonicity of the pricing kernel that do not depend on a particular
specification of consumer preferences or asset price stochastics.

My first chapter identifies a very simple framework to test the monotonicity of
the pricing kernel in the terminal stock price. I characterize the entire class of option
trading strategies whose expected returns are increasing in the strike price under a
monotonic pricing kernel. The class of strategies is shown to be uniquely identified
by the property that the logarithm of the payout function is concave over the interval
where the payout is positive. I then examine average returns of some common Option
trading strategies that are special cases of this class. Call and put Options are shown
to be special cases, but. the set also includes butterfly spreads, bullish call spreads,
and binary options.

A Violation of monotonicity of expected returns in the strike price for any of the
strategies implies a. violation of the nionotonicity of the pricing kernel, but the con—
verse is not true. Hence I also provide a set of weaker conditions for some of the option

tradinfT strategies which implies nmnotonicity of expected returns in the strike. Using

-‘fl

data on options on individual stocks, I find that the average returns of the option
strategies are increasing in strike price which is consistent with a monotonically de-
creasing pricing kernel in the terminal stock price. The monotonicity tests outlined in
this paper are particularly useful in the case of individual stocks where it is difficult to
explicitly estimate the shape of the projected pricing kernel. The framework outlined
in this chapter can also be used to test classes of asset pricing models like the CAPM,
representative agent models with expected utility and the Black-Scholes model.

The second chapter explicitly estimates the shape of the pricing kernel from data
on option prices. Traditional asset pricing theory suggests that the pricing kernel
should be a decreasing function of the security returns. Recent empirical work on
options on the S&P 500 index suggests that for a range of ending index levels, the
pricing kernel is increasing instead of decreasing. I estimate the shape of the kernel
as the solution to a constrained quadratic optimization problem. Unlike previous
approaches, the model appropriately accounts for bid-ask spreads, which are typically
a large proportion of the price of an option especially if the option is out of the money.
One advantage of my method is that it. can be applied to returns over any holding
period and therefore do not have to rely upon holding till maturity returns, thus
allowing the use of a much larger sample. The empirical evidence using options on
the S&P 500 index suggests that, contrary to previous empirical research, the pricing
kernel is a decreasing function of the index return. My approach is general enough to
allow the pricing kernel to depend on additional state variables. Using gross returns
on the VIX index as an additirmal state w-ii‘ialilc, I find evidence suggesting a 1:)ositive.

volatility risk premium in the market.

 

 

 

T o my family, who offered me unconditional love and support

throughout this journey.

iv

ACKNOWLEDGMENTS

I am deeply indebted to my dissertation committee- Dr. Mark Schroder (Co—Chair),
Dr. G. Geoffrey Booth (Co-Chair), Dr. Zoran Ivkovich, and Dr. Andrei Simonov for
their insightful comments, helpful guidance, and unwavering support. In particular,
I am indebted, to Dr. Mark Schroder, who, first as a teacher and later as a co-
author, taught me much of what I know about asset pricing theory. \Nithout his
constant guidance and support this dissertation would not have been possible. I owe a
huge debt of gratitude to Dr. G. Geoffrey Booth for his affection, encouragement, and
unflinching support through all phases of the doctoral program. I especially appreciate
his guidance in helping me to keep things in perspective, whether in regards to my
dissertation, or my career aspirations. I thank Dr. Zoran Ivkovich, and Dr. Andrei
Simonov who have also been instrumental in my development. as an academic.

I would like to thank Dr. Charles Hadlock for supporting me in every way possible
through the entire term of my doctoral program. I thank the other professors of the
finance department for their ei'icouragement and advice in the process of completing
my dissertation. Special thanks go to my colleague William Gerken for challenging me
at every stage of my dissertation which has gone a long way in improving my under-
standing of the subject. Will has also been a close and dependable friend throughout
this journey. I thank Iordanis Karagiannidis, Neslihan Yilmaz, Noolee Kim, my other
fellow graduate students, and all my friends for all their help and support. I thank
Celeste. Emily, and Jane for the for the responsibilities they undertake everyday to
make the finance (‘lepartment a better place to work.

On behalf of my wife, my daughter. and myself, I would like to express our sincere
gratitude for Elizabeth Boot-h for treating us like family and for her help and support
throughout our stay in East Lansing. I would like to express my sincere appreciation

to my cousin. Rini Banerji. and my brother—in—law, Dr. Iiunal Banerji for their un-

 

wavering faith and confidence in my abilities. Their encouragement, support and care
helped me overcome setbacks and stay focused on my graduate studies. Finally, none
of this would have been possible without the love and patience of my family back in
India, who have been a. constant source of love, concern, support and strength over
all these years. My deepest gratitude goes to my wife Malika and my daughter Eslia
for their understanding during all the time spent away from them. W ithout their

sacrifices. this dissertation would not have been possible.

vi

.‘:-"‘.l .

‘w .

 

TABLE OF CONTENTS

List of Tables .................................................... viii
List of Figures ................................................... ix

1 Monotonicity of the pricing kernel and expected option returns . . .
1.1 Introduction ................................. 1
1.2 Previous Research .............................. 8
1.3 Theoretical Results ............................. 12
1.3.1 I\»Ionotonicity of the SDF ...................... 12
1.3.2 Theoretical Expected Returns ................... 14
1.4 Data ..................................... 19
1.5 Empirical Results .............................. 21
1.5.1 Average returns ........................... 21
1.5.2 Page test. for ordered alternatives ................. 28
1.6 Robustness Tests .............................. 34
1.7 Conclusion .................................. 41
A1 Chapter 1 Appendix ............................................. 43
A1.1 Monotonicity of the SDF .......................... 44
A12 Monotonicity in expected returns ..................... 47
A13 Skewness-adjusted t statistics ....................... 50
A14 Proof of the Page test ........................... 52
2 Estimating the pricing kernel from option prices .................. 63
2.1 Introduction ................................. 63
2.2 Data ..................................... 71
2.3 Empirical Estimation ............................ 74
2.3.1 The Case with No State Variables ................. 74
2.3.2 Generalization to Bid-Ask Spreads ................ 80
2.3.3 The case with additional state variables .............. 85
2.3.4 Average call returns ........................ 94
2.4 Conclusion .................................. 96
A2 Chapter 2 Appendix ............................................. 98
A21 Risk premia and kernel projection with volatility as a state variziilne . . 99
A2.1.1 Discrete—t inie/instantanemis—tiu1e formula ............ 101
A212 Deterministic-coefficieut case. .................... 103
Bibliography ...................................................... 1 1 3

 

1.1

1.2

1.6

1.7

1.8

2.1

LIST OF TABLES

Average Return Differences ........................ 55
Page Test for Ordered Alternatives .................... 56
Appropriate measure of leverage ...................... 57
Average Deltas and Elasticities by strike group .............. 58
Average call returns sorted by elasticity .................. 59
Average put returns sorted by elasticity .................. 60
Regression Analysis ............................. 61
Robustness Tests .............................. 62
Average Returns Of call Options on the VIX index ............ 112

viii

 

to
p—A

2.2

2.3

2.4

2.6

2.7

LIST OF FIGURES

Empirical pricing kernel vs gross return .................. 105
Empirical pricing kernel vs gross return (allowing for bid-ask spread) . 106
2D scatter plot of V IX index return versus S&P 500 index return . . . 107

Delaunay Triangulation Of VIX index returns versus S&P 500 index
returns .................................... 108

3D scatter plot Of the estimated pricing kernel versus S&P 500 index
and VIX index returns ........................... 109

Interpolated surface Of the estimated pricing kernel as a function Of
return on the S&P 500 index and return on the VIX index. . . . . . . . 110

Interpolated surface Of the estimated pricing kernel as a function of
return on the S&P 500 index and return on the VIX index ....... 111

 

Chapter 1

Monotonicity Of the pricing kernel

and expected Option returns

1 . 1 Introduction

This cl'iapter identifies a simple framework to test the monotonicity Of the (projected)
stochastic discount factor (SDF) in the terminal stock price. I characterize the entire
class Of option trading strategies whose expected returns are increasing in the strike
price under a monotonic SDF. The class Of strategies is shown to be uniquely identified
by the property that the logarithm of the payout function is concave over the interval
where the payout is positive. I then examine average returns of some common Option
trading strategies that are special cases Of this class. Using data on Options on individ-
ual stocks, I find that the average returns of these option strategies are increasing in
strike price which is consistent with a monotonically decreasing SDF in the. terminal
stock price.

The relation between the stochastic discount factor and security payoffs is key to

many fuiulamental results in asset pricing theory. Tecl'micz-ilitics aside, the existence of

a SDF is equivalent to the law of one price. while the absence Of arbitrage corresponds
to the existence of a strictly positive SDF. In the absence of arbitrage, all asset prices
can be expressed as the expected value of the product of the SDF and the security
payoff. In a single period setting, the SDF, or pricing kernel is a random variable

~rm+1 which satisfies the equation

Et Imt+14lit+ll = Pt

for every security with payoff n+1 at time (t + 1) and price pt at time t, where
Et denotes the time-t conditional expectation Operator. In some sense, the SDF
can be viewed as a generalization Of the notion Of a discount factor to the world of
uncertainty. More precisely, the SDF is the ratio of a state contingent claim price and
the probability of occurrence of that particular state. Given a probability model for
the states, the SDF gives a complete description Of asset prices, expected returns, and

risk premia for any security in the economy.

Almost all asset pricing models can be expressed as special cases Of the pricing
equation (See. Cochrane (2001). Moreover, these models can be uniquely identified by
their s1')ecification of the SDF which are known as SDF representations of asset pricing
models. For exainple. in the Black-Scholes model, the SDF is affine in the return on
the underlying security. The SDF is decreasing if and only if the risk premium Of the
underlying security is positive. This equivalence between the form Of an SDF and an
asset pricing model also holds for non—redundant asset pricing where we require some
equilibrium model for the SDF. For example. a principal implication Of the CAPM is
that the SDF is linear in a single factor, the return on the portfolio Of aggregate wealth.
In a. rejn‘esei'itative agent model with time—additive utility the SDF is the ii‘itertemporal

margimil rate of substitution (IMRS) Of the consumer. The pricing equation, in this

A- inn"!!!

 

case, can be represented as the consumers intertemporal Euler equation which is
a necessary condition for an individual consumers optimization problem. In such a
setting, the SDF can be shown to decreasing in returns for all concave utility functions

but the exact form Of the SDF would depend on the specification Of the utility function.

The equivalence between the form of the SDF and the specification of an asset
pricing model suggests that any framework designed to test properties of the SDF can
also be used as a. test. for violations Of asset pricing models. For example, if we were
to empirically Observe that the SDF is not decreasing in terminal security prices (see
Brown and Jackwerth (2004)), then this would indicate a clear violation of some very
well known asset pricing models. As discussed above, well known examples would
include the CAPM, Black-Scholes model, and the representative agent model with
concave utility. The primary purpose of this chapter is therefore to identify a simple
framework to test the monotonicity Of the SDF in the terminal security price.

A straightforward implication Of the pricing equation is that a negative correlation
between returns and the SDF is equivalent to a positive risk premium on the security.
In the case where the SDF is the aggregate IMRS, a security will earn a positive
risk premium if it is positively correlated with aggregate consumption, and therefore
negatively correlated with marginal utility Of aggregate consumption. The intuition
being that, such a security is risky as it delivers less wealth precisely when it is more
valuable to the investor. The investor therefore demands a large risk premium to hold
the security. Although the relation between the risk premium and the SDF is easy to
understand from the intertemporal choice problem, it can be derived simply from the
absence of arbitrage, without assuming that investors maximize well-behaved utility
functions.

A stronger condition than a negative correlation is monoto1'1icity: the SDF pro—

jected on the terminal stock price is a (.Iecreasing function. A positive risk premium

3

on a security does not guarantee monotonicity Of the SDF. I show that. the mono-
tonicity of the SDF is equivalent to a negative correlation between security returns
and the SDF for all possible return distribution Of the security. Alternatively, strict
monotOi‘iicity Of the SDF is essentially equivalent to the positivity Of all “conditional"
risk premium, defined as the difference in returns between the security and unit pay-
outs, the payouts being made conditional on the terminal security price falling within
some specified interval. A simple example is the Black-Scholes model where the SDF
is always monotonic, and is decreasing if and only if the risk premium is positive.

h’lonotonicity also hold in a. lar ‘er class Of 0. tion ricinO‘ models.
t b

Although testing SDF monotonicity is equivalent to testing that all conditional risk
premia are positive, such a test. is difficult to conduct as we would need a continuum
Of binary cash-or—nothing and binary asset-or—nothing contracts. Instead, I provide a
much cleaner and practical way to test the monotonicity of the SDF using commonly
traded Option strategies. I characterize the entire class Of Option strategies whose
expected returns are increasing in the strike price under a monotonic SDF. The class
is uniquely identified by the property that the log-payout functions Of the strategies
are concave over the set. of ending security prices where the payouts are positive. Call
and put Options are special cases. as shown in Coval and Shumway (2001), but the
class also includes butterfly spreads, bullish call spreads, and binary Options. One
easy way to understand this characterization is that an increase in the strike price for
these strategies shifts the. payoff to lower utility states, requiring a higher return to
compensate. More precist-‘Iy, an increase in the strike shifts the probability weighted
DayOffs to lower values of the SDF. resulting in lower prices and higher expected

returns.

A violation Of monotonicity (in the strike) Of expected returns of any of the strate-

gies implies a violation of the nionotonicity of the SDF, but the com-'erse is not. true.

For example. the conditional covariance conditions in Coval and Shumway (2001) guar-
antees that the expected returns for call and put options are increasing in strike. but
they do not imply that the SDF is a decreasing function of the terminal security price.
Hence I provide a. set of weaker SDF monotonicity conditions for some of the option
trading strategies which implies monotonicity of expected returns in the strike. Based
on my empirical results on options on individual stocks, I find that expected returns
are increasing in the strike price for all the option trading strategies considered in
this paper, which is consistent with monotonicity of the SD-F in the terminal security

price.

This chapter cmrtributes to the literature in the following ways. I extend the work
in Coval and Shumway (2001) by providing necessary and sufficient clraracterization
of the entire class of option strategy payoffs whose expected returns are increasing in
strike when the SDF is monotonic in the terminal stock price. Tests of monotonicity
of the SDF also provide a test of a family of option pricing models. The Black-Scholes
model is a special case. as are models with independent stochastic volatility. My
characterization of monotonicity of the SDF can be applied to options on a proxy for
the market portfolio (say the 885.1) 500) to test classes of models such as the CAPM
(where the SDF is linear) or representative agent models with expected utility (where
the SDF is strictly monotonic under concave utility). This chapter also contributes to
the existing literature on viable price processes. Bick (1990) outlines conditions under
which the state price density is a decreasing function of the terminal security price,
under the assumption that the state price density is a function of only the terminal
security price. This assumption is very restrictive and is not gcmrrally true even
for many simple asset pricing models (The Black-Scholes model with deterministic
but time varying parameters being a. simple example). My characterization of SDF

mormtonicity is much more general than what is presented in Bick (1090).

5

 

Several researchers have attempted to estimate the shape of the SDF from index
prices and aggregate consumption data. However. such a. study is difficult to conduct
for individual stocks as we would have to estimate the shape of the projected SDF
on each individual stock. Moreover, the usefulness of options data is limited as the
number of available strikes is much fewer for individual stocks compared to a market
index like the S&P 500. This chapter contributes to the literature by providing a simple
framework to test properties of the projected SDF without explicitly estimating its
shape.

This chapter also contributes to the empirical literature on option returns. This is
one of the few papers ( Ni (2007), being one of the exceptions) that analyzes expected
returns of option strategies on individual stocks. The strategies that I Consider in this
chapter include traded strategies like calls, puts and butterfly spreads. as well as non-
traded strategies like binary callsland bullish call spreads. To test the hypothesis that
average strategy returns are increasing in the strike price, I also use a nonparametric
test that I borrow from the behavioral sciences. The Page test for ordered alternatives
tests the null hypothesis that groups (or measures) are same versus the alternate
hypothesis that the groups (or measures) are ordered in a specific sequence. An
advantage of the Page test is that it allows us to test the monotonicity of expected
returns by calculating a single test statistic. I also extend the original Page test
statistic to allow for the fact that different underlying stocks have options with varying
number of strikes. The procedure outlined for in'iplementing the Page test is quite
general and can be applied to any finance problem where a variable of interest is
expected to be increasing across multiple, but related, samples.

Finally, this chapter uses a. very simple method for calculating skewness-adjusted t-

statistics and standard errors. The distribution of option returns are extremely skewed

 

'1 CBOE has introduced binary options on SPX and VIX indices starting from July 2008

as out of the money exmrations generate a return of 400%. Central limit approxi-
i'nations are problematic because of the irregular nature of option return distributions
and usual standard errors and t-statistics cannot be relied upon for assessing the sig-
nificance of option returns. The skewness—adjusted t-statistic was developed in Chen
(1995) and has shown to be quite accurate for small sample sizes and for distributions
as asymmetric as the exponential distribution.

The remainder of this chapter is organized as. follows. In Section 1.2, I summarize
the prior relevant literature. In Section 1.3, I derive theoretical results regarding the
SDF and expected returns. In Section 1.4, I discuss my dataset, In Section 1.5, I test
the implications developed in Section 1.3. In Section 1.6, I provide some additional

robustness tests. Section 1.7 concludes this chapter.

 

 

1 .2 Previous Research

Academic research in options and other forms of derivative assets has grown expo—
nentially in the last few decades. A part of this interest in derivative research can be
attributed to the gradual growth in the derivatives market. A significant part, how-
ever, can be attributed to the seminal work on options by Black-Scholes and Merton.
Hence the Black—Scholes model would be an appropriate starting point when relating

this work to the existing literature.

In the constant-parameter setting of Black-Scholes option pricing framework, where
asset prices follow a Geometric Brownian Motion (GBM), the SDF (projected on the
terminal stock price) is a strictly decreasing function of the stock price if and only if
the risk premium of the stock is positive. In such a setting, the instantaneous excess
expected rate of return (over the risk free rate) of an option is equal to the product
of the options elasticity with respect to the underlying price times the instantaneous
excess expected return of the underlying asset. In this framework, it is straightforward
to see that expected call and put returns should be increasing in the strike price
provided the risk premium of the stock is positive. l\~‘loreover, expected call returns
should never be lower than the return on the underlying stock, and expected put
returns should never be higher than the risk free rate. The above results can easily be
extended to any general one factor model. In particular the results would hold if the

volatility of the underlying stock is a function of the price of the underlying stock.

Coval and Shumway (2001) show that the above result hold under much weaker
conditions than implied by the Black—Scholes model. They show that. under no arbi-
trage crmditions, as long as the covariance of the SDF and the terminal stock price,
conditional on the option being in the money, is negative for all strikes. expected

option returns would be increasing in strike. Empirically. they show that the above

8

restrictions are satisfied by options on the 88:13 500 and S&P 100 indices. On the
other hand, Ni (2007) studies option returns on individual stocks and shows that
deep out-of-the-money (OTM) calls earn significantly lower average returns than deep
in-the—money (ITM) calls. She sorts call option contracts on each buying date into
portfolios based on the options moneyness and shows that the portfolio with OTM
options earns significantly lower returns than the portfolio with ITM option contracts.
She concludes that, unlike index options, individual stock options do not conform to
the monotonicity restrictions on call returns. She argues that investors are sometimes
risk—seeking, and a preference for idiosyncratic skewness leads to a premium for deep

OTM options and is a. possible explanation for the puzzling call returns.

The assumption that the SDF is negatively correlated with ending security prices
is fundamental in any asset pricing theory, the intuition being that, the SDF should
be high in bad states of the world and low in good states of the world. A sufficient
condition for this covariance to be negative is that the projection of the SDF on the
ending level of the security price is a monotonically decreasing function of the security
price. Several researchers have attempted to recover the stochastic discount factor
from security prices (eg. Hansen and Singleton (1982), Hansen and Singleton (1983)
and Chapman (1997)). Jackwerth and Rubinstein (1996), Jackwerth (1997), Brown
and Jackwerth (2004) and Ait-Sahalia and Lo (2000) provide empirical procedures for
estimating a stochastic discount factor(SDF) from option prices when a finite number
of options exist instead of a dense set. Option data is particularly helpful in this
context because it is possible to develop a model-free option based estimator that
does not depend on a particular specification of consumer preferences or asset. price

stochastics.

Brown and Jackwerth (2004) use the theory outlined in Breeden and Litzenberger

(1978) and Cox and Ross (1985) to estimate the pricing kernel as a ratio of the state

9

price density and the subjective probabilities. They recover the risk neutral probability
distribution from option prices on the S&P 500 index and the subjective distribution
from actual index returns. They estimate the SDF as a. function of the ending level
of the 885.1) 500 index and show that overall, the SDF is a. decreasing function of
the ending index level. However. for index levels approximately ranging from 0.97
to 1.03 (a 3% deviation from the current level), the SDF is increasing. They argue
that such a shape of the SDF function is inconsistent with traditional asset pricing
framework as this would suggest that the representative agent is locally risk seeking
and the utility function is non-concave or locally convex. They attempt to resolve
this puzzle by introducing additional state variables. Rosenberg and Engle (2002)
analyze the empirical cllaracteristic of investor risk aversion over equity return states
by estimating a time-varying SDF. They argue that the state prices and probabilities
in Jackwerth (1997) are averz-rged over time, so their estimates can be intepreted as a
measure of the average SDF over the sample period. Thus assets are correctly priced

only when risk aversion and state probabilities are at their average level.

Bick (1990) provides necessary and sufficient conditions for viability of diffusion
price processes. The paper identifies conditions under which the associated utility
function to any diffusion price process is consistent with an equilibrium. In particular,
assuming that the state price density (SPD) is only a function of the terminal stock
price. the paper identifies conditions under which the SPD is monotonic in the terminal
stock price. As discussed in the. previous section, this assumption turns out to be quite
restrictive for many simple asset pricing models (For exai‘nple. Black-Scholes model

with deterministic but time varying pz-u'z-ii'neters).

My work is related to the strands of literature discussed above. My work extends
the results in Coval and Shumway (2001) by characterizing the entire class of option

trading strategies whose expected returns are increasing in the strike price under a

10

monotonic. SDF. On the other hand, my work is related to the huge literature that
attempts to estimate the shape of the SDF from option prices. I do not try to es-
timate the. shape of the SDF directly. but instead use expected returns of a set of
option strategies to test for violations of the monotonicity of the SDF in the terminal
stock price. This chapter also extends the work in Bick (1990) as it provides char-
acterizations of SDF monotonicity under much general conditions. The next section
characterizes the class of option trading strategies that I use to test the monotonicity
of the SDF and also derives i'iecessary and sufficient conditions for the monotonicity

of the SDF.

ll

1 .3 Theoretical Results

This section provides the theoretical results that motivate our empirical tests of the
monotonicity of the SDF. Section 1.3.1 provide characterizations of the monotonicity of
SDF that is much general than what is provided in the previous literature. I show that
SDF monotonicity is essentially equivalent to positive “conditional” risk premia of the
security. I will also argue that to design an empirical framework based on testing the
positivity of all conditional risk premia is difficult to achieve using traded securities.
Section 1.3.2 therefore provides a cleaner and easier way to test the monotonicity of

the SDF using commonly traded option strategies.

1.3.1 Monotonicity of the SDF

One of the straightforward implications of the pricing equation is that a positive risk
premium on a security is equivalent to a negative correlation between the SDF and
the return on the security. A monotonically decreasing SDF is a stronger condition
than a negative correlation, and a positive risk premium is not enough to guarantee
monotonicity. The following lemma derives conditions under which the projected
SDF is a decreasing function of the terminal security price. Unlike a positive risk
premium, which guarantees a negative correlation between the SDF and the terminal
stock price, I show that the monotonicity of the SDF requires a positive “conditional”
risk premium on the security Consider the following two period setup.

Let us assume that the price of a stock S is P; today, and that the stock would
make a payout P75: at time T, if P; lies in an arbitrary interval [and]. Also as—
sume. there exists a. security, with current price [7,1, that makes a unit payout at T if
P; lies in [(1, ,d]. Let H5 and R1 represent the returns corresponding to these pay-

outs. The risk premium of the security. conditional on Pig. lying in [(1,113] is given by

1?.

E (RS _ R1

P; E [(1. 13]). Then, the following lemma shows that, a necessary and

 

sufficient condition for the monotonicity of the SDF is that the risk premium, condi-
tional on P15. lying in [(i. d]. is positive for all possible intervals [0,. ,3]. Alternatively.
I also show that a. strictly decreasing SDF would imply that the risk premium of the
stock is positive for all possible return distributions. The following lemma formally

states these results.

Lemma 1.3.1. Let 0 (ST) denote the information set generated by the strictly positive
terminal stock price ST. For any A E 0 (ST) satisfying P(A) > 0. let P}; and PE

denote the prices of payouts 1 and ST, respectively. in the sci A:
P}; = E(7121A), Pi = E(mST1A);

and let R1 = l/P}1 and R5 = SCI/P2w denote the returns on set .4. Furthermore,
lcf m denote the stochastic discount factor, and 9(51‘) = E(m[Sr1~) its projection on
ST.

(1) If g is continuous and strictly decreasing then E(RS — R1[ .4) > 0 for all
A E 0 (ST) satisfying P(A) > 0.

b) Ifg is continuously diffcrcnfuible. and the distribution function ofF is strictly in-
creusin.__q 0n (0,00), then 9 is strictly decreasing if and only if E ( RS — Rl[ ST E [(1, ,3])

> 0 for all 0 S (r < (‘3.
Proof. See appendix [:1

Lemma 1.3.1 provides char-acterizations of the SDF that is much general than what
is provided in the previous literature. In particular. it does not retpiire that the SDF is
solely a function of the termimd stock price (Bick (1990)). The above results suggest

that a possible way to test the monotonicity of the projected SDF would be to test

13

 

the positivity of all conditional risk premia of the stock. Unfortunately, such a test. is
difficult to conduct with available traded securities as we would need a. continuum of
binary cash-or—nothing and binary asset-or—nothing contracts.

I instead test. the n'ionotonicity of the SDF in a much cleaner way by characterizing
a class of option strategies whose. expected returns are increasing in the strike price
under a monotonic SDF. The class of strategies is uniquely identified by the property
that the log-payout function is concave over the continuous interval where the payout
is positive. A violation in the monotonicity of expected returns in the strike price
for any of the strategies implies a violation of the monotonicity of the SDF, but the
converse is not. true. Hence I also provide a. set. of weaker conditions for some of the
option strategies which implies monotonicity of expected returns in the strike. The

theoretical results are presented in the next subsection.

1.3.2 Theoretical Expected Returns

Let ST denote the stock price at expiration, and interpret the constant (shift parame—
ter) K 2 0 as the strike price. We characterize the class of functions G () such that the
expected return corresponding the payout G (ST — K) is increasing in the strike for all
stock-price distribution functions and all monotonically decreasing stochastic discount
factors (that is. for all stochastic discount factors whose projection on ST are mono-
tonically decreasing functions of ST) I then consider some common option trading
strategies that. are special cases. which I use to test weaker forms of monotonicity.
“'ithout loss of generality. let us assume tln‘oughout the following regularity con-

dition of G:

Condition 1.3.2. Assume G(.r) > 0 for .r E (11,12). where ~30 3 .r1 < 1‘2 3 00.

and G (I) = 0 for .r E [.11. 1'3]. Furthermore. G (1) is continuous for 1' E (11.;7‘2) and

14

 

G’ (.1) is piecewise continxuous for .r E (.1‘1..r2).

Let in denote a. strictly positive stochastic discount factor for pricing time-T pay-
outs. and let in (s) = E (nil ST 2 s). s E [0. 0c) denote the projection of in. on the

terminal stock price.

Proposition 1.3.3. The expected return

, _ E{G(ST-K)l
R (A) _ E [in (ST) C(ST — Kl}

 

is increasing in If for all distribution functions F (satisfying E {G (ST — IQ} # 0)

and any inonotimically decreasing m () if and only if

C" .-
—(:—(—I—) is decreasing in 1' for all .1.‘ E (.11'1. 1'2) . (1.3.1)
.1 'J'
Proof. See Appendix E]

Proposition 1.3.3 says that if the projected stochastic discount factor is monoton-
ically decreasing in the terminal stock price. then concavity of the logarithm of the
payoff function 2 is equivalent to monotonicity of the expected return in the strike for
all return distribution functions. The concavity of the log-payoff condition implies
that increases in the strike shifts (in the first—order dominance sense) the probability-
weighted payoffs to higher stock prices; the. lower values of the SDF correspomling to
higher stock prices implies a lower price (per unit probal'nlity-weighted payoff) and
therefore a higher expected return.

The following three examples present exchaiige-traded strategies with payoffs that

satisfy the condition in equation (1.3.1). Monotonicity of in () therefore implies

 

3 (:‘ij 1‘)
(111‘)

 

is decreasing in .r is etuiivalent to the concavity of the log-payout function

 

monotonicity of expected returns in the strike price. Conversely, monotonicityr of

expected returns in the strike price in each case implies a. weaker form of monotonicity.

Example 1.3.4 (Call options). The payofjr function is C(r) = .‘L'+. Coval and
Shumway {2001) show that the derivative of the expected return with respect to the
strike is preportional to the conditional covariance between the stock price and the

SDF on the set relieve the option empires in the money:

2

P S 1'
f T > ‘) Coe(ST,m [Sf > K)-

E {771- (ST — K)+}

 

R, (K) = —

Therefore monotonicity of erpected returns in the strike implies negative instantaneo-us

conditional covariances.

Example 1.3.5 (Put options). The payoff function G (J) = (——.r) .+ Analogous to
the call case. we get from, Coral and Shumway (2001)

R’ 1' =—- C s" .  sn< K.
(X) E{m-(K — ST)+} ()i( I m l 1 )

 

Example 1.3.6 (Butterfly spread). A strategy of long 1 call each at strike prices
K — AK and K + AK. and short two calls at strike K has the payofi’ function G (.r) 2
(AK — |:r|)+. The erpccted return is

) .. _ E{(AK— lzrl)+}
MM —' E{m(5’1‘)(AK " “ll—i}

 

The int-'crse of the (arpcctcd return is a weightt—d average of the stochastic discount
factor, “urith the weights proportional to the product of the density and payoff functions

and the result follows directly from. Proposition. 1.3.3

16

The following examples also satisfy the concavity condition in equation (1.3.1), but.

are not exchange-tradecl.

Example 1.3.7 (Binary cash—or—nothing option). The payofl function for the call

version is G (.1) = 1 {‘00} The erpected return is

P (5T 2 K)
Ei"’1{5r21\'}}

and is therefore increasing in K if and only if E (m. [ST > K) is decreasing in K ,

R(K)= ={E(In |ST2K)}—1,

 

which is equivalent to
m (K) > E(m IST 2 K) for all K 2 0.

The payofi‘ functm'n of the put version is G (I) = 1{;,7<()}, and the expected return is

increasing in K if and only 2f

m (K) < E (m I ST 3 K) for all K 2 0.

If the expected returns of vanilla call options, and the expected returns of binary

cash-or—nothing call options are, both increasing in the strike, then

E(m- (ST — K) lST 2 K)

i . all I'.
E(S-1~—K STZK) ‘ ‘

 

m (K) > E(m IST 2 K) >

 

The call-optii)n-return test puts much more weight on the right tail of the distribution
than the binary test. For example. if m (s) is not monotonic, but declines sharply
as .5 becomes large. then call option returns may be. increasing in the strike, but the

binary calls may not.

Example 1.3.8 (i\rlodified bullish call spread). Consider a portfolio that is long a
call with strike K. short a call with strike K + AK, where AK > 0. and short a
cash—or-nothing binary call "with payofl'AK- 1 {51> K+ AK } ( all on the same stock and
same eapiration T ) The portfolio payoff is (ST —— K) 1 { 5.116%.le AKll’ the errpected

return of the portfolio is

E {(ST — K) 1{5Te[1\11<+mr]}}

R(K) = i
[a {In (ST — 1‘) 1{‘5'Te[1\'.K+AKl}}

 

and the derivative [with respect to K is

P (ST E [K. K + AM)

R’ (K) = — .
E {m- (ST — 1‘) 1{STE[K.K+AI\'l}}

 

Coo (ST, in.

 

ST e [K K + AK])

As the interval shrinks to zero, 11” (K) becomes proportional to the slope of in ().

Cou(ST. ml ST E (K. K + AK])
1111 , . .
Akin Var(ST| ST e (K, Ii“ + AM)

 

I '
= 7n (Ix) .

The final example presents a payoff function which results in an expected return
invariant to K for all distribution functions F.

Example 1.3.9. (Strihe-Invariant Erpectcd Return.) Suppose C(17) = exp(—k.r),
some k 63 IR. Then. the (:trpected return (when it artists) is invariant to K for any

distribution fzmction F.

18

1.4 Data

The data on options are from the OptionMetrics Ivy DB database. The dataset
contains information on the entire US equity option market from 1996 to 2008 and
includes daily volume, open interest, best daily closings bid and ask quotes, option
Greeks and implied volatilities. The implied volatilities and Greeks are calculated
using a binomial tree model developed by Cox, Ross, and Rubinstein (1979). The data
set also includes information on daily prices, returns and distribution of all exchange

traded stocks.

I select option contracts that satisfy the folloWing criteria. Since the monotonicity
restrictions on options are applicable to European style options, I restrict my analysis
to only those option contracts for which underlying stocks do not have an ex-dividend
date during the remaining life of the contract.3 It is well known that it is never
optimal to exercise non—dividend paying American style call options early. American
put options however might be optimally exercised early irrespective of whether the

underlying stock pays dividend or not.

In accordance with the standard practice in empirical option studies, I choose
only those call and put option contracts for which the bid price is greater than or
equal to $0.125. I also eliminate contracts for which the recorded ask price is lower
than the bid price.4 The arbitrage bound filter that I use requires that option prices,
estimated as the bid—ask midpoint, should be greater than S — Ke‘TT for calls and
Kt?” — S for puts,where S is the price of the underlying asset, K is the Options
strike price, 7' is the risk free rate and T is the time to expiration. In an unreported

rolmstness check. my results hold when I restrict my sample to option contracts that.

 

3It. is not difficult to adjust the returns for the case when we have an ex-dividend date prior to
maturity.
4A significant. violation is observed in the data.

19

 

satisfy the put-call parity bounds. That is, the bid and ask prices should satisfy
Cb,” -— PM). 3 S —- Ke‘r” g Cask — PM]. where C and P are the call and put prices
respectively. On each expiration Friday from January. 1996 to June, 2005, I first
identify option contracts that expire on the next expiration Fridz-iy. I then use prices
observed on Tuesdays to calculate weekly returns for the. option contracts identified
in the first step. My initial sample consists of 2,123,004 call returns and the sample
reduces to 1,643,925 after all the restrictions have been applied. The mean (median)
number of unique stocks on each buying date is 1,380 (1,421) and the mean (median)
number of strikes for each stock on each buying date is 3.43 (3.00). Since deep in-the—
money (ITM) and deep out—of-the—money (OTM) options are thinly traded, I note the
possibility of prices being estimated from stale quotes. As a robustness test I repeat
my analysis with only those option contracts which have a positive volume on the
buying date. The number of calls in our sample that have a positive volume on the
buying date is 598,826.

The initial sample, for put options. consists of 2.224293 return observations.
1,982.501 observations satisfy the sample restrictions and 450,931 returns have a posi-
tive volume on the buying date. The mean (median) number of unique stocks on each
buying date. is 1,423 (1,461) and the mean (median) number of strikes for each stock

is 3.72 (3.00).

20

1.5 Empirical Results

The theoretical results in section 1.3 suggests an empirical framework that. we can use
to test for violations in the monotonicity of the SDF in the terminal stock price. In
particular, I show that. for any option strategy for which the log-payout function is
concave, expected returns would be increasing in the strike price under a monotonic
SDF. A violation in monotonicity in expected returns for any representative strategy
would indicate a clear violation in the monotonicity of the SDF. Section 1.3 provides
examples of some common option strategies for which the log-payout function is con-
cave. In this section I test empirically whether average returns for these strategies are
increasing in strike using a. set of different empirical methods. Section 1.5.1 assigns
strategies into strike groups and tests if average return differences are positive for all
strike groups. Return differences are defined as the difference in strategy returns for
a particular strike group and the immediate lower strike group. Section 1.5.2 compli—
ments the results in Section 1.5.1 by conducting a joint test using the. Page test for
ordered alternatives. The Page test has the added advantage that it provides us with

a single statistic to test monotonicity in expected returns.

1.5.1 Average returns
A Call options

To test the monotonicity restriction on call returns, I first divide the calls into strike
groups based on the difference between their strike prices and the strike price of the
call which is closest to being at the money. In particular, for any tuiderlying stock
on any buying date. I identify the call option cont ‘act that is closest to being at the
money and assign a. strike group value of 3. The next two higher strikes are. assigned

strike group ralues of 4 and 5 respectively. Similarly, the immediate two lower strikes

21

are assigned strike group values of 2 and 1 respectively. Note that the option contracts
are first divided into strike groups before any filters are applied to the data. Excess
returns for any particular strike group are calculated by taking the return ("lifferences
between the return of an option in a particular strike group and the return in the
immediate lower strike group. Returns for strike group 1 are represented in excess of
the return on the underlying stock for the same holding period. Since a non-dividend
paying stock is essentially equivalent to a zero-strike call option, excess returns for
strike group 1 can be thought of as being calculated in excess of the return on a call

with the lowest possible strike price.

The return differences for each strike group are averaged across different stocks to
form a weekly time series of average return differences. The average of the weekly time
series and the corresponding t—statistics are reported in Table 1.1. Results indicate that
the average return differences for call options are positive and statistically significant

suggesting that average call returns are monotonically increasing in the strike price.

Table 1.1 also provides a skmvness-adjusted t-statistic for the average return dif—
ferences. The distribution of call option returns are extremely skewed as OTM ex—
pirations generate a return of 400%. Central limit approximations are problematic
because of the short sample size and irregular nature of option return distributions.
Thus usual standard errors and t statistics cannot be relied upon for assessing the
significance of option returns. I use a very simple method for calculating skewness-
adjusted standard errors and t—statistics. This statistic for testing the mean of pos-
itively skewed distributions was drweloped by Chen (1995) and is an extension of
Johnson (1978) and Sutton (1993). It is derived using Hall’s t-type inversion of the
Edgeworth expansion and has been shown to be quite accurate for sample sizes as
small as 13 and for distributions as asymmetric as the exponential distribution. Fur-

ther details on this test statistic are provided in the appendix. The skewness-adjusted

22

t-statistic would provide estimates closer to the usual t—statistics if the distribution is
close to normal. As one would expect, the adjusted t-statistics estimate for out of the
money calls differ significantly from the usual t statistic. This is because the return
distribution for OTM calls is much more positively skewed compared to the return

distribution for ATM or ITM calls.

B Put options

The possibility of early exercise complicates analysis of put option returns. For Amer—
ican put options early exercise might be optimal even if the underlying stock pays no
dividends. Optimal stopping theory implies that it is optimal to exercise a put option
at a given time if and only if the market price and the intrinsic price are identical.
Under complete markets and additional regularity conditions, it is suboptimal to not
exercise at the first instance when the market price coincides with the intrinsic value
of the put options [See Duffie, Lin, and Poteshman (2005)]. In practice, American put
option should always be exercised early if it is sufficiently in the. money.

Several researches have attempted to estimate this early exercise premium for puts
using different. valuation approaches. Engstrom and Norden (2000) uses Swedish equity
options data. and finds a substantial early exercise premium for American put options.
The premium increases with moneyness and time left to expiration. while the effect
of volatility and interest rates is dependent on the moneyness of the option 5. In my
analysis of average put option returns I believe that early exercise premium should be
small because of the short maturity of the option contracts and low interest rates for
my sample period.

Although. the monotonicity test result from Table 1.1 for put options is somewhat.

weak, it. is still consistent with the theoretical predictions. In-the-money puts earn

 

”This is surprising since the early exercise premium should be increasing in the "interest rate.

23

 

significantly higher average returns than at—the-money and out-of-the-money puts.
I\'Ioreover, the Page test and the elasticity test, to be discussed later, also provide
strong evidence in support of the monotonicity hypothesis.

To adjust the expected returns to account for an early exercise premium, we would
first have to estimate the critical stock price boundary below which early exercise
would be optimal. Using the idea behind the representation in Theorem 1 in Carr,
Jarrow, and Myneni (1992), the gross return on an American put option over the
interval (t, t + At), including any interest. earned on the strike over any subinterval in

which early exercise is optimal is

P(St+/Afn f. + Af) + TIX’ Ltd-At (‘flrUTIAt-u)1{SU_<BU}(IU
”51:13)

 

where 7‘ is the short rate, and Bt is the critical time—t stock price below which early ex—
ercise is optimal at t (we assume that the stock price is the only source of uncertainty).

I defer this adjustment of put option returns to future research.

C Butterfly spreads

A butterfly spread with call options is a strategy that is created by taking a long
position in a call with strike price (K — AK), a long position in a call with strike price
(K + AK), and a short position in two calls with strike K. To calculate the return
differences for the butterfly spreads. I impose the following restrictions: 1) There
should exist at least four valid strikes for any underlying stock on any buying date to
estimate at least one return difference. 2) The estimated price of the spread must be
greater than $0.01 to reduce the possibility of unrealistic returns 6 and 3) The value

of AK should be the same across all l‘mtterfiy spreads on the same underlying stock

 

"Although convexity in option prices should ensure that the price of a butterfly spread be always
positive. a significant violation is observed in the data

‘24

and the same buying date. I allow the spreads to have overlapping strikes, that is, the
middle and right strike of a. spread could be the left and middle strike respectively of
the next higher spread. Results are stronger if I restrict the butterfly spreads to have
non-overlapping strikes and / or the value of AK is increased but this comes at a cost
of losing a lot of observatimis. The total number of valid lmttcrfly returns after the
restrictions on the calls and the spreads are imposed is 192,823.

Results from Table 1.1 indicate that average butterfly returns are increasing in
the strike price. Recall that for call and put options, monotonicity in expected re-
turns require only a negative conditional covariance between the SDF and the terminal
stock price. On the other hand. monotonicity in expected returns for butterfly spreads
require that the projected SDF is strictly decreasing in the terminal stock price. Re-
sults on average butterfly returns reported in Table 1.1 therefore provide very strong

evidence in support of SDF monotoi‘iicity in the terminal stock price.

D Binary call options

Estimating average return differences for non-exchange traded securities is slightly
more involved. Recall that the cash-or-nothing binary option pays a fixed amount of
cash if the option expires in—the—money and nothing otherwise. The binary options
are not. exchange traded 7 and so I estimate their prices by using a smooth implied
volatility curve estimated using calls on all available strikes on the same underlying
stock and same trading day. Let B (K) denote the price of the binary option with

strike K that pays a dollar if it expires in the monev.

u

[3(K) = E{m1{ST>K}}.

In general.

 

7CBOE has introduced binary options on SPX and V IX indiccs from July 2008

25

(,1 (1

Wear) 2 (IKE {m- (ST — 1\)1{ST>K}} = ——B(I\).

 

If the short rate is deterministic, B (K) is the product of the risk—neutral probability

of the call being in the money, Pm (ST > K), and the discount. factor.

Because strikes are not dense enough, a better way to compute 7(11TC (K) (rather
than differencing premiums at different strikes and scaling) is to estimate a smooth
Black-Scholes implied volatility curve a (K) and then estimate B (K) from the BS
model in the following way. Assuming a deterministic short rate, and letting r denote
the annualized continuously compounded zero yield from 0 to T (on a discount bond),

then (assuming a smooth B-S implied volatility curve 0 0):

d

. l K
B (K) = --fi:CBS (S. K,0 (10) = —C}§~"(S.K.a(1<)) -— 507290.535 (3, K10 (10)
(1 \ ( \
Substituting, we get
B (K) = ("T {\ ((12) — Ix'N’(c12)x/Ta’(1{)}. (1.5.1)

The no-arbitrage bound on the binary call Options are calculated as follows: If

there is no arbitrage (and a continuum of strikes), the European calls must satisfy
(1C _ . .

_, E (_. I710).

(1A

ii‘nplying that the prices of the binary calls should lie in the interval (0, (771T). The

second derivative satisfies

26

 

(’26 —r;r { .I .. .n r , . { 1 a’ (10}
,. 6 ,\ '1: I\ A (l- '2 a I\ } _______ + d. .___'_

+ e—I‘TKN” ((12) fig” (K)

Substituting N” (I) = ——:rN’ (.I) we get.

(126' _,.T A" ((12) , 1 ,
——: =3 ——-—.— l 11' T K } — 1 K
(”(2 c 0(0) { +(2\\/—0'( ) {Kfi+(la( )}

+ e""TI\'i\-” (d2) fig” (K) (1.5.2)

and the lie-arbitrage condition (assuming a continuum) is

(.120
d K 2

 

20.

To estimate a smooth B-S implied volatility curve. I first compute the implied
volatilities of all the call options in our sample. The implied volatilities corresponding
to calls on the same stock and same buying date are then used to fit a smooth curve
using cubic spline interpolation. I also compute the first. and second derivatives of the
estimated implied volatility curve at the traded points. The first. derivative is used
to compute the price of the l‘Iinary call option using equatitni (1.5.1) and the second
derivative is used to estimate. equation (1.5.2) to check if the binary call satisfies the

lie—arbitrage condition. To ensure better estimation of the implied volatility curve.

27

 

I restrict my sample to only those calls whose underlying stocks have at least four
strikes on any trading day. 8 Finally, to reduce the possibility of unrealistic returns,
I restrict my sample to only those binary calls for which the estimated price is at
least $0.01 giving me a total of 162,964 valid binary call returns that satisfies all
the restrictions. Results indicate that the average returns of the binary calls are
monotonically increasing in the strike price. This is not surprising, since monotonicity
in expected returns for binary calls require a slightly weaker condition than what is

required for butterfly spreads.

E Modified bullish call spreads

Once we have the prices of the cash-or-nothing binary call option, computing the
prices of the modified bullish call spread is straightforward. Recall that we defined
a modified bullish call spread as a portfolio that is long a. call with strike K, short
a call with strike K + AK, where AK > 0, and short a cash-or-nothing binary call
with payoff AK - 1 {ST> K+AK}'(aH on the same stock and same expiration T). I
do not impose any other restriction on the bullish call spreads other than setting the
minimum bound on the prices to 0.01 giving us a total of 80,988 bullish call return
observations. Consistent with the rest of the results in Table 1.1, Average returns are

seen to be increasing in K.

1.5.2 Page test for ordered alternatives

In summary. the results in Table 1.1 are consistent with monotonicity of expected
returns in the strike 1‘)riee. In 1:)articular, I show that. average return differences between
a strike grmip and the. innnediate lower strike group are positive for all strategies.

However, testing the mmiotonicity restriction convincingly requires that we design a.

 

8I get similar results when I require at least three strikes instead of four.

28

test that allows us to compare more than two strike groups simultaneously. W hen
k(> 2) samples are to be compared, it is necessary to use a. statistical test which will
indicate whether there is an overall difference among the A: samples before choosing
any pair of samples to test the significance of the difference between them. In other
words, we need a single statistic for monotonicity of expected returns in the strike
price.

Consider the setting in Table 1.1 where I use a two—sample statistical test to test for
differences among five consecutive strike groups. I conduct five, two—sample statistical
tests in order to test if the average returns are increasing when we move from one
strike group to the next higher strike group. Such a procedure could lead to erroneous
conclusions because it capitalizes on chance. we are giving ourselves five chances
rather than one chance to reject the null hypothesis. Now, with an or of 0.05, we are
taking the risk of rejecting the null liypt‘Ithesis erroneously five percent of the time.
But if we make five statistical tests of the same hypothesis, we are increasing the
probability to 0.23 that. a. two sample statistical test will find one or more significant
positive differences. The “ actual significance ” level in such a procedure becomes a =
0.23.9 It is only when an overall k—sample test allows us to reject the null hypothesis
that. we are justified in employing a. procedure for testing differences between any two
of the k. samples [See Siegel and Castellan Jr. (1988) for a review of available k-sample
tests]. In other words, an appropriate k-sample test would compliment the results in
Table 1.1 and together would be a convincing and rigorous test. of the monotonicity
restrictions.

The k—sample test. that I use is the ” Page test. for ordered alternatives”. It tests
the null hypothesis that the groups (or measures) are the same versus the alternative

that the groups (or measures) are ordered in a specific sequence. In particular let. 6’]-

 

9“, Z 1 _ (opsf' : 0.23

be the population mean for the jth group. Then we may write the null hypothesis

that the means are the same as

IIO:()1=()2=---=()A.

and the alternative hypothesis may be written as

H.4191392S~-Sf)k

If the null hypothesis is rejected then at least one of the differences is a strict inequality
(<). To apply the Page test I first assign the option strategies to strike groups as in
Table 1.1. I then estimate expected returns for the strategy by computing the time
series average of the strategy return for each underlying stock and each strike group.
Assuming k,- strikes for the 1th underlying stock. I rank the average returns of the
strike groups for the ith stock on a scale of 1 to 13,-. Assuming we had k strikes for
each of the In. underlying stocks (I will relax this assumption momentarily), the. ranks
are cast into a two-way table having m. rows and 1: columns. The null hypothesis is
that the average rank in each of the columns are the same against the alternative that

the average rank increases across groups 1 to k. The Page statistic L,” is defined as

"I. In.

I; k
LIII :— Z Y} Z JYIj : Z Z 3):]le
j=1 ~1

i=1 i=1 j—

where i indexes the III observations (unique underlying stocks in our case), j indexes
the I; strike groups. Xij is the rank of the average return of the 21th stock and the ‘jth
strike group and Y]- is the a priori ordering of the. group which in our case is equal to

j. Then under the null hypothesis

k ‘)
k(k+1)“
EZYJ-Xij = —.

, 4
1:1
and
k 1
r r . .‘2 ,. ‘2 .
var Zippy- : mi, (1 +1) (A — 1).
i=1
1;
Assuming the weighted ranks Z: YJ-Xl-j are independent across different
1:1

i=1,....m

stocks. the CLT implies

I:
LII). - DIE 211/1ij
J:

 

k _1/2 —->N(0.1).
'7'72.lr"'(1.'l‘ . Yj )(i J

J=1
From the results above, this is equivalent to
12L,,, - 3m}.- (k +1)2

z: ‘ —+.'\7((),1).
(Ink?(k+1)(k2 —— 1))1/2

 

The expression above is the. original Page test statistic“) (see Page (1963) and Pirie
(1985)) and it is only valid for samples which have the same 1: treatments for each
observation. To account for the fact that underlying stocks have different number
of strikes. I modify the original Page test statistic to allow for varying number of
strike groups (treatments) for (.lifferent underlying stocks. Let k) be the number of
treatments for the 1th observation (eg. the number of strikes for the ith underlying

stock). \Ve then have

 

l”See appendix for a detailed proof of the original Page statistic

31

, 1
E§:bloi=;fidh+1f

 

j=l
i 1 )
, , , ,. _ g; , 2 ,
Vm zypw, —-fifiMM+U(M-U
Then, defining
"I
in X1.
i=1
the CLT implies
"I
1;)L171_3Zki(ki2+1)
JW“: ’4 -stJ) dam

(fr2t+)U—nyfl

The statisticzlI "1h essentially gives less weight to observations (is) with fewer treat-
ments(strike groups). Alternatively. we can standardize the weighted rank for each

observation, and define

12zyxaafl+n

7 1?.

L m :

 

1/2

- .2 . 2 .

2=1 (i,- <A.-+1> (A.- — 1))

Here we essentially scale up the statistics for observations with fewer strikes. Then

LN?

III,

Jou? _

 

-sxmd) use

 

Table 1.2 reports the number of unique underlying stocks and the values of the
modified Page test statistics for all the option strategies. Recall that the statistic

zh’gh refers to the case where we give less weight to stocks that have fmver munber

of strikes and 31m" refers to the rase where we scale up the statistics for stocks that
have fewer number of strikes. To be consistent with the analysis in Table. 1.1, the
maximum number of strike groups that I consider is five. Panel A reports the results
for the whole sample and panel B reports the results for subsamples which have to
satisfy an additional restriction. To ensure better estimation of the expected returns,
I require that the option strategies on each underlying stock and for each strike group
have a minimum number of weekly observations. For calls and puts I require at least
150 weekly observations. For the remaining strategies, I require at least 50 weekly
observations 11 to be included in the Page test. From Table 1.2, we see that the Page
statistics for all the option strategies are positive and very highly significant suggesting
that the average returns of the option strategies are ordered in an increasing sequence
across strike groups. As an added robustness test, I repeat the above analysis with
only those observations which have at. least five strike groups so that the assumptions
of the original Page test are satisfied. Results are similar to those reported in Panel

A.

\

llRaising the minimum number of required observations to 100 or 1:30 reduces the number of unique
& tocks significantly.

33

1 .6 Robustness Tests

The previous section provides strong evidence that. average returns of option strategies
whose log-payout functions are concave are increasing in strike price. This is consistent
with the hypothesis that. the (projected) SDF is a decreasing function of the terminal
stock price. This section provides several robustness test on call returns. I focus
primarily on calls because of the recent. empirical evidence suggesting that average
call returns violate nmnotonicity of expected returns in the strike price. hl'loreover, all
the option strategies considered in this chapter are combinations of call options and
it is imperative that the results on calls are robust to different testing procedures 12.

A traditional approach to undm'stand why expected call returns should be increas—
ing in strike is the following: A call option is equivalent to a levered long position in
the stock. In terms of rate of return, a levered position is more risky than an unlevered
one, and leverage amplifies the risk premium of the option. Since higher strike calls
are more levered positions than lower strike calls, expected returns would be larger for
higher strike prices. As moneyness increases with strike price of the option, expected
option returns should be increasing in the moneyness of the option.

A seemingly intuitive way to test the above hypothesis would be to sort option
contracts into portfolios based one their moneyness. Ni (2007) sorts call option con-
tracts on each buying date into five portfolios based on moneyness and shows that the
portfolio with OTM option contracts earns significantly lower returns than the portfo—

1 io with ITl\l option contracts. She suggests that investors are. sometimes risk—seeking

étnd a preference for idiosyncratic skewness leads to a premium for deep OTM op—

‘11— ions and is a possible explanation for the puzzling call returns. This section provides

(“:3 widence that such a. testing procedure could lead to erroneous conclusions primarily

\

13 Note that. the empirical tests conducted in this section are. also valid for other strategies and

1‘ (gsults could be provided if necessary

34

 

because of the heterogeneity in the underlying assets of the call options. Consider the
example provided in Table 1.3. The example looks at actual data on option contracts
on two underlying stocks on the same buying date having approximately the same
moneyness range. However, the Black-Scholes elasticity of the lowest strike call on
”AIG” is almost equal to the elasticity of the highest strike call on ”USRX”. Thus
call options on ”AIG” on this buying date, in general, are more levered positions than
call options on ”USRX” and should be expected to earn higher returns.

Table 1.4 reports the Black-Scholes deltas and elasticities of calls sorted into strike
groups based on the option’s moneyness. The sample and moneyness cutoffs 13 used
are from Ni (2007). We see that although the average deltas are monotonically increas-
ing across strike groups, the same cannot be said about the elasticities. The mean and
median elasticities of deep out of the money calls are smaller than that of at the money
calls. Also note that the standard deviation of the elasticities are enormous suggesting
huge variations within strike groups. In the Black-Scholes continuous time framework
the elasticity (or leverage) of an option would depend on the moneyness of the option
and the volatility of the underlying stock. For options on the same underlying stock,
elasticities and hence expected returns are monotonically increasing in the moneyness
of the option. When the underlying asset is not constant, a higher moneyness would
not necessarily imply a higher elasticity [This is also easy to confirm in a simulation
setting]. Note that monotonicity of the. elasticity in strike price would also hold for all
the option strategies considered in this paper. Of course, unlike call and put options,
the elasticities would not. necessarily be of the same sign. For example, the elasticity

()f a butterfly spread is negative for in the money and positive for out of the money

contracts.

 

1‘3 The moneyness cutoffs used are as follows: [If/S S 085,085 < K/S S 0.05.005 < K/S g
1.05.1.05 < K/S g 1.15. K/S > 1.15]

.IY ‘- .“ AV.

 

Another problem with testing the mmmtonicity hypothesis by sorting by money-
ness is illustrated through the following example: Consider a call option with a strike
price of 35 when the ui‘iderlying stock is worth $30. The moneyness of this option is
1.17 and would be assigned a strike group value of five. using the cutoffs used in Ni
(2007). In fact. all out of the money calls in this case would be assigned a strike group
value of five 14. Alternatively, when the stock price is high ($175 for example), most
of the out of the money calls will be assigned a strike group value of three. In other
words, when current stock prices are low or high, the sorting of option contracts into
bins depend more on the level of the current stock price than on its leverage or its
distance from being at the money. In summary, this procedure neither test whether
expected returns are increasing when moving from one strike to the next higher strike,
nor does it test whether expected returns are increasing in the leverage of the option.

As a robustness test, Table 1.5 reports the average of the monthly and weekly time
series of call returns and return differences sorted by leterage. I sort option contracts
into leverage groups by sorting option contracts on each buying date into quintiles
based on their elasticities with respect to the underlying stock. Panel A reports the

. . . 7
one month holding-till-maturity returns 1"

and Panel B reports the weekly buy-and-
hold returns. Since the underlying stock that does not pay dividend has the same
return structure as a call option with zero strike, I calculate all option returns in
excess of that of the return on the underlying stocks. 16 Options on individual
stocks earn higher returns than the underlying stock and returns are monotonically
increasing with leverage. The portfolio with the highest leverage call option contracts
earn significantly higher returns than the portfolio with lowest leverage call options.

My results are also robust to the cases when (1) I remove the top and bottom 1% of

 

HFor a. underlying stock price of 30. strikes would generally increase in intervals of 5
1"One month holding till maturity returns are calculated following Ni (2007).
1 ’Itemdts are stronger 1f returns are not. (alt ulatcd in excess of the return on the underly ing stock

36

 

the elasticities from our sample, and (‘2) when I consider only those option contracts

that hate a positive volume on the buying date.

I estimate expected put returns in the same way as call options and the results are
reported in Table 1.6. Since a put option with infinite strike price has an expected
return equal to the risk-free rate, expected put returns must always be lower than the
risk-free rate (if the risk premium of the underlying stock is positive). Expected stock
returns are typically higher than the risk—free rate; so expected put returns should
always be lower than the return on the underlying stock. I find that average put
returns are lower than the average return on the underlying stock and the returns are
increasing with leverage. I also confirm that the average put returns are never higher

than the average risk free rate.

In Table 1.7, I use regressions to test the monotonicity restriction in call option
returns. Note that. option payoffs are non-linear functions of the underlying stock
and any discrete time linear model that attempts to explain option returns would
be inherently wrong. In fact, my regression results confirm the presence of significant.
non-linearities in call returns. However, I use regression models as an added robustness

test to check if the results are consistent with the theoretical predictions.

In our regressions, the dependent variable is the excess call option return over the
risk free rate. Averages of the weekly cross-sectional regression estimates as well as
Fama—h‘lacBeth t statistics adjusted for third order serial dependency are reported.
Specificatimi (1) of Table 1.7 regresses the excess option return on dummy variables
for the strike groups where strike group 1 is considered as the base group for all our
regressions. The. strike groups are defined in the same way as before. The estimates
in specification (l) are consistent with the results in Table 1.1. The coefficients of the.
strike group dummy varial')l<_rs are all positive, significant and increasing with strike

group. In specification (‘2) I add the independent. variable carcass stock return which

37

 

is the excess return of the underlying stock over the risk free rate. Controlling for
the risk premium of the underlying stock, we still find that coefficients on the dummy
variables are positive and increasing except for the coefficient on strikeS which is the
dummy variable for strike group 5. Although, the coefficient on strikef) is slightly
lower than that on s/r'zlke4, the difference is not statistically significant. This can be

confirmed by repeating the above regression with strike group 4 or 5 as the base group.

The effect of leverage in option returns can be seen from the results in specifications
(3), (4) and (5). The basic idea is to test whether the increase in expected call option
returns for increases in the excess return on the underlying stock is the same for all
strike groups. I include interaction terms of the strike group dummy variables with
excess stock return to allow for a difference in slopes for different strike groups. In
specification (3) we see that the coefficients on the interaction terms are all positive,
significant and increasing except again for the coefficient 011 interaction term of Stink-€55
and ezrcess stock return, which although, is slightly lower than the interaction term
of strike group 4 but is not. significantly different. We also see that the intercepts of the
different strike groups are all negative. This is not surprising because the intercepts
indicate the return differentials when the excess return of the stock is zero. When
the excess return on the underlying stock is very low or zero, the expected call option
returns should be negative. Specification (3) captures the effect of leverage in call
Option returns. The results show that for an increase in the excess return of the
underlying stock. the increase in the excess return of the call option would be higher
for a higher strike price option.

As discussed earlier, option payoffs are nonlinear functions of the underlying stock
return and any discrete time linear model trying to explain option returns should be
incorrect. I confirm this by including («arccss stock return =1: clast)2 as a nonlinear

explamrtory variable in sliiecification (4). This variable is defined as the square of

38

 

the elasticity weighted excess stock return. The coefficient on this variable is very
significant suggesting significant nonlinearities in call option returns. The significance
of the nonlinear explanatory variable is robust to different choices of the explanatory
variables. The other variables that I have used are the square of the excess return of
the underlying stock, and the excess return of the stock multiplied by an indicator
function for a positive excess return. Note that I do not try to interpret the coefficients
on these nonlinear explanatory variables. These are just used as proxies to test for
nonlinearities in the call returns. Thus we can conclude that, although the results
from the regression analysis are consistent with our hypotheses, a linear model for call

option returns can be rejected.

In specification (5) I introduce other explanatory variables that have. been sug-

gested in the literature to have an effect on option returns. Coval and Shumway
(2001) and Buraschi and Jackwerth (2001) find evidence that volatility is priced in
Option markets. I use the change in implied volatility define! as an explanatory vari-
able in call option returns. In an efficient market, any deviation from no arbitrage
values should trigger an immediate response from investors leading to the rapid disap-
pearance of the arbitrage opportunity. However, the empirical asset pricing literature
suggests the existence of market frictions that impede the arbitrage process. Liquidity
has often been argued as an important limit to arbitrage because it makes arbitrage
more risky and costly and might lead to persistence in mispricing in assets. I use the
call trading volume on the buying date as a. proxy for liquidity in my regressions. From
results for specification (5) we see that the coefficients on delivel turn out to be sta-
tistically significant. Htm'ever, the value of the coefficient on vol um e is very small and
not statistically significant and does not. appear to have. a significant effect on option
returns. The sign of the coefficient on volume hmvever turn out to be consistent. with

theoretical predictions. The nonlinear term remains significant in this specification.

39

 

Finally, in specification (6) I regress excess option returns on elasticity weighted
excess return on the stock, vega—weighted change in implied volatility, trading volume
on the buying date and a nonlinear term. If discrete time option returns were just
explained by the option-CAPM. the coefficient on the el-(iisticity weighted excess return
would be one or at least Very close to one. A coefficient of 0.87 and significantly
different from 1 suggests that. discrete time option returns are not fully explained
by the CAPM. The other explanatory variables along with the nonlinear term are
also significant but the value of the coefficient on volume remains very small. Note
however, that elasticity weighted excess return of the stock, by itself, explains about
90% of the variation in call returns. Overall, the results from the regression analyis
provides evidence that is consistent. with monotonicity in expected returns for call
options.

Table 1.8 presents some more robustness tests on average call returns. To elimi-
nate call prices estimated from stale quotes, Panel A repeats the test for call options
reported in Table 1.1 with the added restrictions that the calls should have a positive
volume on the buying date. Consistent. with the results in Table 1.1, average call
returns are increasing in strike. Panel B tests the monotonicity restriction where the
average returns are calculated in the same way as for the Page test. I first estimate
expected returns for the calls by computing the time series average of the call returns
for each underlying stock and each strike group. Return differences are calculated for
each stock and then averaged across different stocks. Average return (.lifferences are all
positive and significant, consistent with the results in Table 1.1. The results in Panel
B are. robust to the case when I restrict. my analysis to the sub—sample where calls on
each umlerlying stock and each strike group have at lease 150 weekly observations.

The results are. reported in Panel C.

40

“11M

 

1 .7 Conclusion

This chapter examines the expected returns of a family of option trading strategies
on individual stocks to test whether the (projected) stochastic discount factor(SDF)
is monotonic in the terminal stock price. I show that strict monotonicity of the SDF
is essentially equivalent to the positivity of all “conditional” risk premia, defined as
the difference in returns between the stock and unit payouts, the payouts being made
conditional on the terminal stock price falling within some specified interval. Alter-
natively, I show that a strictly monotonic SDF in the terminal stock price guarantees
a positive risk premium for all possible return distributions of the underlying stock

return.

To test the monotonicity of the SDF, I characterize a class of strategies whose
expected returns are increasing in the strike price under a monotonic SDF. The class of
strategies include all payout functions for which the logarithm of the payout function
is concave. The concavity of the log-payout function implies that increases in the
strike shifts the pi'tgibal.)ility weighted payoffs to lower values of the SDF and therefore
a higher expected return. Call and put options have been shown to be special cases of
this class of strategies, but the class also include butterfly spreads, bullish call spreads

and binary calls.

A violation in monotonicity of expected returns for any of the strategies implies a
violation in monotonicty of the SDF, but the converse is not true. Hence, I also provide
weaker conditions for monotoncity of expected returns in the strike price for some of
the option trading strategies. Using data on option contracts on individual stocks,
I find that the average weekly returns of the option trading strategies are increasing
in the strike price of the strategy which is consistent with a monotonic SDF in the

terminal stock price. My theoretical results characterizes the entire class of option

41

 

trading stra.tegies whose expected returns are. increasing under a monotonic SDF in
the terminal stock price. My characterization of monotonicity of the SDF can be
applied, in future work. to test. classes of models such as the CAPM or representative

agent models with expected utility.

5'4 w I «In. J

 

 

 

Chapter 1 Appendix

43

 

A1.1 Monotonicity of the SDF

Lemma A1.1.1. (Lemma 1.3.1) Let 0 (ST) denote the in ormation set generated by
the strictly positive terminal stock price ST. For any A E 0 (ST) satisfying P (A) > 0,

let P}, and Pi denote the prices of payouts 1 and ST, respectively, in. the set A:

1:11: E (HtlA), P3 = E(IH.ST1A);

and let R1 = l/P}4 and R5 = ST/Pg denote the returns on set .4. Furthermore,
let m denote the stochastic discount factor, and 9 (ST) 2 E (ml ST) its projection on
ST.

a) If g is continuous and strictly decreasing then E( R5 — PM A) > 0 for all
A E 0 (ST) satisfying P (A) > 0.

b) Ifg is continuously differentiable. and the distribution function ofF is strictly in—
creasing on. (0, 00). then g is strictly decreasing if and only if E (R8 — Rh] ST E [0” #0

> 0 for all 0 g o: < ,1‘)’.

Proof. From the definitions of P}; and PE we get

ewes-Rae}

and therefore

Cov (771, RS‘ A)
E (nil A)

 

E(RS—R}4|A) :-

If A E o (5]) (that is. 1.4 is a. function of ST) then, letting 9 (ST) =E( ml ST),

44

~ C." s s ,4
E(15’5_.}:.>‘14 (4):“ ("(M1) [I )

PASE (nil A)

 

for all A E 0 (ST).

The result follows from Lemma Al.1.‘2 and the strict positivity of m, and therefore

E(m| A) and Pi. Cl

Lemma A1.1.2. Suppose Y is a random variable with distribution function F , and

let 0 (Y) denote the infomiation set generated by Y. a) Ifg is continuous and strictly

decreasing then.

C011,!(Y. g (3)] A) < 0 for all A E a (Y) and P(A) > O.

b) Ifg is contint/Misti; (ll/fore?)liable, lhc support on is la,bl, where —:>c S a <
b S ac, and F is strictly increasing on. la, b] { no zero probability internals) then. g is

strictly decreasing if and only if

Cou(Y,g(Y)l Y E la,;’ll) < 0 for all a g (1' <13 _<_ b. (A1.1.1)

Proof. Part a.) Define

(WW)
1’ (A) ‘

 

h (.1: A) = / {13(3’

 

A) — y} lit/6:1}

which satisfies h (a) 2 h (.t) = (7) and h. (:r) > 0 for all .r such that P (Y E A and Y g at)

E (0, 1). Using integration by parts:

 

dFW)
P (A)

 

Caveman/1) = / a—Eo'l A))1{,,€A}g<y)
= -/9@hmwwfl

= /h (y;A)(19(!1)

If g is strictly decreasing then the left hand side must be strictly negative.

Part b) Suppose (A1.1.1) holds, but g is not strictly decreasing. Then there must
exist points c < e such that g (c) g g (e), and therefore a point (1 E (c, e) such that
g’ (d) > 0 (we can rule out g’ (.1) = 0 for all 1' E (c.c), because this contradicts
Cov(Y,g(Y) 1{3’€l<'~€l}) < 0). Continuity of g’ then implies g' (1:) > 0 for .r in a.
neighborhood of d. This implies that Gov (Y,g(Y)1{Y€[d‘5‘d+€]}) > 0 for some

3 > 0, contradicting (Al.1.1). E]

 

A1.2 Monotonicity in expected returns

Proposition A1.2.1. ( Proposition 1.3.3) The earpected return

 

_ , _ EG(ST— K)
R (A) _ E {m (ST) G (sT — m}

is increasing in K for all distribution. functions F {satisfying E G (ST — K) % 0) and

any monotonically decreasing m () if and only if

G" (1:)
G (:r.)

 

is decreasing in .r for all :r E (11,12) . (A1.2.1)

Proof. We assume, for simplicity, that G is right continuous at 1:1 and left continuous
at 1'2 (that is, G (1:1) 2 limgll.1 G (s) and G (1'2) : linisll.2 G (5)).

a) Sufficiency of (Al.2.1): Let F be some absolutely continuous distribution function
F and let. (x = fox G (a — K) dF (a) (which represents the expected payout under F).

Define the distribution function HI":

ll' (5‘; K) = g /08 G (u — K) dF (a) s E [0. 00). (A122)

The inverse of the expected return is

 

x.
, = m (s) dH' (5; K . A123
R (A) f > < >

We show that R (K) is strictly increasing in I\' when in (s) is strictly decreasing in
.s by showing that H" (s: K) is decrmsing in K for all s E (0. 00) and. K 2 0. That
is. ”7(1163) stochastically dominates ll' (-; K1) in the first. order sense for any 0 S

K1 < [(2.17 If s — K < .1'1, then W (s: A?) = 0 for A: in some, neighbm'homl of K and

 

ITSee Huang and Litzenberger (1988. Ch 2).

4T

 

therefore HleV (s; K) = 0. We therefore consider only s 2 K + .r1. Differentiating

(.4122), we have 21—(lTW (s; K) S 0 is equivalent to H (s) S 0 where

3 CC 3
H (s) 2 —oz/ F/ (a) (1G (a — K) +/ F, (u) ([0 (u — K)/ C(u — K) (1F (11).
0

0 0

Substituting I’V’ (s; K) : (PIG (s -— K) F ’ (s) then iii—W (s; K) g 0 if and only if

()0 S 7/ r
,,rI _ , , “(C(11— K) H" (u; K) dG (u — Ii)
/0 111—K6m‘f21” (MK) C(u—K) S 0 1{“—I"El“'1’1’2l} W (s;K) C(u—Kl

 

which follows for all s because of the monotonicity condition in (A121).
b) Necessity of (A121): Consider first the case of a jump in G’ () /G (), and suppose

contrary to the hypothesis that

 

. . 3 A
hm < hm ——'— some s E (11.12) .

as 0(3) in 0(5)

and define the midpoint

 

 

1 . 0(3) . G’f‘“)
"’“iflil’lc(s)+1lf§e(s))'

Using piecewise continuity, choose 5 > 0 sufficiently small that G’ (1) /G (1) < m
for .r E [s -— 5,8) and G’(.r) /C (.r) > in. for .r E (S's + 5), and concentrate F on

lé -— e. s + El (let F (s — 3) z 0, and F (.9; + s.) :— 1). Define

 

\ S 7/(ll) '
(1’s 2 (ill 1.: .
(l l 0(a) ("0)

.
s—E

Then 11(5) 3 0 is equivalent to (I) (s + e) g (I) (s)/Iz1«"(s;0) But. (I) (s) /ll*' (s: O) < m.

48

and

 

 

<1) (1) a 5 _ (I, g.
<1><é+s> = H«'<.e:0>(,,.(l,;l,))+{1_ur<.a»;o>}( (133ml ))

> W" (s; 0) (”(11:51”) + {1 —- H" (S; 0)} Hi

 

Together with

<1)(.§+e) —<I>(s) (13(8)
. . > m > —.
1— ll’ (s;0) EV (s;0)

 

we get <I>(s + 5) > (P(s) /W(.§;0), a contradiction. Therefore G’() /G' () cannot
increase at a jump.

Now consider the possibility of an increase in G’ () /G' () over any interval. Suppose

 

 

(A124)

for some s1 < s2, where sl, s2 E (£1.12). Let 5* E arg inf {G’ (s) /G (s) ; s E lsl, 32]}.
Concentrating F on ls*, 32] (let F (s*) = 0. and F (82) = 1). then, for any 3 E (s*, 82),

H (s) S 0 is ecplivalent to

 

 

.92 I , 8 1v! , .. ,' . ;
f G (u)dw, (11:0) S / C (a) (ll/1' (u 0)1
3* C(11) 3* C(11) H (3:0)

Letting s l s*,

 

 

"2 G" (u) G" (s)
.. III" :0 <1' -, .
l, C'm‘ (“ l-sl‘li as)

T

which contradicts (A 1.2.4 ). Cl

49

 

A1.3 Skewness-adjusted t statistics

The preposed statistic for testing the mean of positively skewed distributions is taken

from Chen (1995) and is derived using the Edgeworth expansion as follows:

 

   

 

Wt)? - I!) 1 . 2 —1 2

P ——————- < :. 13- 1 + 2.77 2 (I) ;r + o n / . A1.3.1

{ S _ M. n > < > < >. < >
- , ._ . . 7 _,2 , _ Z: (.\"-—X)2 A

where (l)(;r) is the standard normal distribution, .8 is defined as W’ and ,31

. . 72—12(X--—K)3 . .

1s (l(‘f1ll(‘(l as 53’ (For more details on the Edgeworth expansion, see. Hall

(1983)). To test the hypothesis H0 : Ila: = #0 against H0 : at > #0 is to reject H0

when

 

 

. 1 - 2

   

where 20 satisfies the equation 1 — (P(za) = (i. Chen (1995) argues that the critical
point of the above hypothesis test depends on the skewness of each sample. thus to find

a more aecurat e and powerful test with a common critical point. for a given significance

 

 

1') ’ f‘ ‘\ I . '- fl ' 1 \ I.\ '-- . " fi(—-l‘l ) ,
lead a, w. 11M. tlu following appioarh. \Vc first. solu. —S——O— > J— 6\/,—Ii)'1(1+212)
for r Let (i = "3L and l —— —3—‘/'—’(X—“) lhus when n is large such that 1— 80(l+a) > 0
u o ’ . GVIII ' I L.’ L ' O L .' ." .I — "

we have from equation A131,

 

 

1— fl—‘lat—l—a 1+ 1—8(1(t+a) _ .
P \/ 4 f ( ) > z” ()1‘ \/ 4 < .2” = (l + ()(n 1/‘2)
a «a

 

 

Thus we have

 

 

 

4a

— _L l .
P{1 \/1 Sa( -l-a)>:fl}<0

Using Taylor series expansion we then have

 

1— \/1— 8a(t + (1)
4a 1

 

= t + (1+ 201‘? + 4(12(t+ 2t3) + 0(n_1/2) (A1.3.3)

Equation A1.3.3 is the skewness adjusted t statistic that we have used in this paper.
An advantage of this test statistic is that it can be used for sample sizes as small as
13 and for distributions as asymmetric as the exponential distribution. Expressing

equation A1.3.3 in terms of the original variables, we get the following expression:

 

 

 

 

 

 

A1.4 Proof of the Page test
The Page statistic L," is defined as

m m

m-ZY 2X1] 2 2:1Yj/Yij

z—1j1=
where Ii indexes the m, observatitms (or ”levels” or ”ranking”) and j indexes the k
treatments. Let p( ) denote the 2th position of a random permutation of {1 ..... 1.}

(each permutation is equally likely). Then

I: k k 2
. . .~ _k___+1 . k(k+1)
j=1 j=1
The variance is derived next.
Lemma A1.4.1.
* 1
Var 2}}ij = m}? (1.: + 1)2 (k — 1) (A142)

Proof. The variance of any index value is

 

‘2
1
lm( :1) =—5(k+1)(k—1). fori€{l,...,k},

il

i=1

Filly—a

where we have used

 

 

We next show (recall that 13 (j) and 13(2') are the values 1th and jth positions of a

given random permutation 13)

Comm 3(1)) = —;—1—1v <13 0)) ’2: aé j, m e {1, . . . . Ar}

This is done by defining the random variable

13(1) with probability (k — 1) M-

 

1? =
15 (j) with probability 1 / k
Note that .r is uniformly distributed on {1. . . . . k} and is independent of [3(1) There—
fore
~ . k - 1 - . ~ . 1 ~ .
0 = COM. (1)91) = k CW (1) (J) ,1) ('21)) + gVaMP (1))-
Finally,

k k k
Va?" ZYJ-Xl-j = ZZ'é-JCW (13 (.1)...’ ('0)
j=1 . .

 

Substituting gives the result.

 

 

C]
A.
Assuming the weighted ranks Z YJ-ng are independent, the CLT im-
JZI i=1 ..... m
plies
L:
L") —7)1,E 2 53'1ij
i=1 .
1.. _1/2 -—>1’\’ (0.1).
fill/(If Z Yinj
i=1
From the results above, this equivalent to
12L — 317172 A“ +1 2
’" l ) —> N (0. 1). (A143)

(77211.2 (k +1) (k2 — 1))1/2

*-

lm.m.l ”I-

 

Table 1.1
Average Return Differences

This table reports the weekly buy and hold return differences for some common option trad-
ing strategies that are special cases of the class of payout functions for which the growth rate
is monotonic. Strike groups are defined using the following algorithm: For each underlying
stock and buying date we find the option contract which has a strike price is closest to the
price of the underlying stock and assign that to strike group 3. The next two higher strikes
are assigned to groups 4 and 5 respectively. Similarly, the previous two lower strikes are
assigned to groups 1 and 2 respectively. The options have to satisfy the following conditions
to be included in the sample: (1) The bid price is strictly larger than 330.125, (2) the ask
price is greater than the bid price, (3) the underlying stock does not have an ex—dividend
date. prior to maturity and (4) the option prices satisfies a lie-arbitrage restriction.

 

Table 1.1: Average return differences by strike group

 

 

Strike Group 1 2’—1’ 3’-2’ 4’-3’ 5’—4’
call returns 0.010 0.006 0.014 0.102 0.070
t statistics 2.33 2.25 2.23 8.81 3.82
Skewness adjusted t statistics 2.30 2.25 2.28 15.07 6.27
put returns -0.009 0.002 -0.004 0.013 0.012
t statistics -0.16 0.07 —0.36 0.97 1.73
Skewness adjusted t statistics -0.11 0.17 -0.39 0.85 1.63
butterfly spread returns 0.108 -0.009 0.079 0.193 0.333
t statistics 6.48 —0.53 3.95 4.94 3.08
Skewness adjusted t statistics 7.76 -0.55 4.31 6.74 5.21
binary call returns 0.050 0.000 0.006 0.059 0.225
t statistics 8.32 0.04 0.54 3.30 3.04
Skewness adjusted t statistics 7.80 0.04 0.63 3.70 5.63
modified bullish call returns 0.174 0.062 0.092 0.334 0.351
t statistics 9.28 3.53 4.98 6.67 6.06
Skewness adjusted t statistics 15.63 3.42 5.35 36.82 9.62

 

Cfi
CJ‘I

Table 1.2
Page Test for Ordered Alternatives

Table 1.2, Panel A reports the results of the ” Page Test for Ordered Alternatives” for option
strategies on individual stocks. Option contracts are first divided into strike groups using
the algorithm described in Table 1.1. Expected strategy returns for each underlying stock
and strike group are estimated by taking the average of the available weekly returns for the
same underlying stock and strike group. Average strategy returns for each underlying stock
are ranked in an increasing sequence across strike groups giving us a set of rankings for each
underlying stock. These sets of ranking are then used to calculate the statistics for the Page
test. Option contracts satisfy the same restrictions outlined in Table 1.1. Panel B repeats
the analysis in Panel A with the added restriction that there should be minimum number
of weekly observations available to estimate the expected returns. We require at least 150
weekly observations for the call and put options and a minimum of 50 observations for the
remaining strategies

 

Table 1.2: Page Test.

 

 

 

 

 

 

Panel A
No. of stocks 3mg}, Zlnw
call options 3531 41.23 40.73
put options 3532 17.39 16.60
butterfly spreads 2600 26.30 22.54
binary call options 1683 28.24 21.23
modified bullish ('all spreads 1771 26.52 25.24
Panel B
No. of stocks :high 31,)".
call options 1222 35.84 3636
put options 927 9.17 8.69
butterfly spreads 360 8.97 5.09
binary call options 69 4.72 3.53
modified bullish call spreads 75 2.90 2.44

 

Table 1.3
Appropriate measure of leverage

This table illustrates that sorting by moneyness might not be equivalent to sorting by lever—
age by using an example from the actual data

 

 

 

 

 

Table 1.3
date ticker option id stock price strike moneyness elasticity
19-Jan-96 AIG 10415728 94.25 80.00 0.85 6.24
19—Jan-96 AIG 10394678 94.25 85.00 0.90 9.19
19-Jan-96 AIG 11008635 94.25 90.00 0.95 14.86
19-Jan-96 AIG 11132774 94.25 95.00 1.01 24.34
19-Jan-96 AIG 10105028 94.25 100.00 1.06 33.24
date ticker option id stock price strike moneyness elasticity
19—Jan-96 USRX 11566685 89.50 75.00 0.84 4.27
19-Jan-96 USRX 10755696 89.50 80.00 0.89 4.54
19-Jan—96 USRX 11569192 89.50 85.00 0.95 5.29
19—Jan-96 USRX 10055356 89.50 90.00 1.01 5.95
19—Jan—96 USRX 11499707 89.50 95.00 1.06 6.90

 

Table 1.4

Average Deltas and Elasticities by strike group

This table reports the mean, median and standard deviation of the elasticities and deltas
of call Options on each buying date sorted by moneyness.
using the Ni (2007) moneyness cutoffs. They are as follows: [K/S S 0.085,0.85 < K/S :3
0.95.0.95 < K/S S 105,105 < K/S S 1.15, K/S' > 1.15]. Call options are selected on each
option expiration date that mature on the next expiration date if the following conditions
are satisfied: (1) The bid price is strictly larger than $0.125, (2) the ask price is greater than
the bid price, (3) the underlying stock does not have an ex-dividend date prior to maturity
and (4) the call and put prices satisfies a lie-arbitrage restriction.

Strike Groups are estimating

 

 

 

 

 

 

Table 1.4
Panel A: Call deltas sorted by moneyness
Strike Group 1 2 3 4 5
mean 0.92 0.81 0.54 0.30 0.20
median 0.94 0.81 0.55 0.30 0.19
std. deviation 0.05 0.09 0.12 0.09 0.09
Panel B: Call elasticities sorted by moneyness
mean 3.82 7.20 12.87 13.60 9.51
median 3.79 6.83 11.13 12.37 8.80
std. deviation 1.05 2.18 7.52 6.49 3.59

 

Table 1.5
Average call returns sorted by elasticity

This Table reports average returns and return differences of call options sorted by elasticity.
Panel A reports the one—month holding till maturity (HTM) returns and Panel B reports
the weekly returns. HTM returns are calculated following Ni (2007). Usual restrictions on
call options apply. HTM returns are calculated following Ni (2007).

 

 

 

 

 

Table 1.5

Panel A : Average call option returns sorted by elasticity (holding till maturity)
Leverage Group 1 2 3 4 5 5’-1’ 4’-2’ 5’-3’
Avg. Return 0.004 0.012 0.035 0.059 0.083 0.080 0.047 0.049
t-stats. 0.16 0.39 0.97 1.41 1.48 2.04 2.70 1.52
t—skewness adjusted 0.16 0.40 0.98 1.43 1.53 2.16 2.79 1.61

Panel B : Average call option returns sorted by elasticity (weekly buy-and-hold)
Leverage Group 1 2 3 4 5 5’-1’ 4’-2’ 5’-3’
Avg. Return 0.010 0.028 0.060 0.103 0.123 0.113 0.075 0.063
t-stats. 0.90 1.79 3.03 4.43 4.78 5.98 6.72 4.49
t-skewness adjusted 0.95 1.92 3.43 5.44 5.91 8.28 12.79 5.31

 

Table 1.6
Average put returns sorted by elasticity

This Table reports average returns and return differences of put. options sorted by elasticity.
Panel A reports the one-month holding till maturity (HTM) returns and Panel B reports
the weekly returns. Usual restrictions on put options apply.

 

 

 

 

 

 

 

Table 1.6

Panel A : Average put option returns sorted by elasticity (holding till maturity)
Leverage Group 1 2 3 4 5 5’-1’ 4’-2’ 513’
Avg. Return -0.210 -0.170 -0.143 -0.103 -0.063 0.146 0.067 0.080
t stats. -2.20 -2.34 -2.03 -1.65 -1.27 1.97 2.14 2.38

Panel B : Average put option returns sorted by elasticity (weekly buy-and-hold)
Leverage Group 1 2 3 4 5 5’-1’ 4’-2’ 5’-3’
Avg. Return —0.618 —0.507 -0.442 -0.338 -0.170 0.448 0.169 0.272
t stats. -24.87 -21.63 -19.37 -15.14 -9.25 22.52 14.58 26.56

 

60

Table l .7
Regression Analysis

In this table we regress excess option returns on a number of factors. ”strikeU)” is a
dummy variable that takes values 1 if the option contract is assigned to strike group "1'”.
”excess stock Tet” is the excess return of the underlying stock over the risk free rate,
”'col'zmzc" is the trading volume on each buying date, ”delivol” is the change in implied
volatility of the option, ”vega” is the option Vega, and ”elast” is the elasticity of the Option.
Averages of the cross-sectional estimates as well as Fama—MacBeth t statistics adjusted for
third order serial dependency are reported.

 

Table 1.7
Dependent variable is excess call return

(1) (2) (3) (4) (5) (6)

 

 

 

constant 0.014 -0.051 0.000 -0.020 -0.140 -0.218
(2.14) (—8.23) (0.27) (-10.66) (~28.84) (-53.08)
strikc2 0.014 -0.051 0.000 -0.020 -0.140 -0.21754
(2.42) (3.32) (-1.5) (-13.93) (7.82)
strike3 0.020 0.024 -0.015 -0.07 -0.004
(2.65) (3.08) (—5.65) (-29.78) (-0.75)
st7‘ikc4 0.084 0.066 —0.078 -0.237 -0.178
(4.25) (3.68) (—11.22) (43.82) (—26.84)
strike-:5 0.107 0.056 -0.177 —0.425 -0.398
(3.4) (2.26) (—11.17) (—31.69) (-29.58)
excess stock 'ret 7.244 3.162 3.833 3.448
(54.3) (89.09) (74.82) (22.2)
excess stock ret * strike2 1.065 0.970 1.562
(48.97) (27.88) (11.49)
ercess stock 'ret * strikc3 4.088 3.528 4.924
(32.38) (29.51) (24.37)
arcess stock ret =1: strike4 9.590 7.026 7.695
(20.52) (23.99) (24.22)
erwss stock: ret * strikc5 8.581 5.991 5.885
(16.75) (18.63) (14.91)
to] (1 me 0.000 0. 000
(—0.63) (—11.48)
dclirol 0.612
(29.85)
ea'eess stock ref * elast 0.870
(201.42)
delirol * I'cqu 0.359
(22.4)
((".l.'(‘(.‘S.S' stock 'I'ct * (zlrfzst):2 0.278 0.343 0.287
(38.66) (40.83) (68.09)
Averz-rge R-Squared 0.027 0.525 0.629 0.814 0.846 0.91

 

61

Table 1 .8
Robustness Tests

This table presents some more robustness tests on call returns. Panel A repeats the analysis
for call options reported in Table 1.1 with the added restriction that calls should have a
positive volume on the buying date. Panel B reports average return differences calculated as
follows: Option contracts are first. divided into strike groups using the algorithm described
in Table 1.1. Expected strategy returns for each underlying stock and strike group are
estimated by taking the average of the available weekly returns for the same underlying
stock and strike group. Return differences are calculated between consequent strike groups
and then averaged across different stocks. Panel C repeats the analysis in Panel B with the
added restriction that there should be at least 150 weekly observations available to estimate
the expected returns.

 

 

 

 

 

 

 

 

 

 

Table 1.8
Panel A
Strike Groups 1 2’-1’ 3'-2" 4’—3’ 5’-4"
Average call returns 0.011 0.005 0.015 0.072 0.055
t statistics 1.60 1.34 2.13 5.07 3.25
Panel B
Strike Groups 1 2‘-1’ 3"-2" 4'-3’ 5’—4’
Avg. call returns 0.001 0.008 0.016 0.124 0.040
t statistics. 1.11 8.27 13.59 25.74 3.95
Panel C
Strike Groups 1 2’-1’ 312’ 413’ 514’
Avg. call returns 0.013 0.010 0.019 0.094 0.033
t statistics. 16.52 16.95 15.83 33.83 3.82

 

Chapter 2

Estimating the pricing kernel from

option prices

2. 1 Introduction

This chapter estimates the shape of the pricing kernel and finds evidence in support
of a monotonically decreasing pricing kernel as a function of gross returns on the S&P
500 index. Unlike previous research, I appropriately adjust for bid-ask spreads in my
kernel estimation. The estimation procedure is general enough to allow the pricing
kernel to depend on additional state variables. I use volatility as the additional state
variable as this allows me to draw some general inferences about the volatility risk
premium. The VIX index is a. measure of the market’s expectation of 30-day volatility
and I use the gross return on the VIX index as the proxy for volatility in my kernel
estimation. Contrary to some previous research, I find evidence in support of a positive
volatility risk premium. Independent confirmation of this result. is provided through
an analysis of average returns of call options on the VIX index.

The asset pricing kernel model is rapidly developing as the most convmiient and

63

general method to price assets. Technicalities aside, the existence of a. pricing kernel
is equivalent to the law of one price while the absence of arbitrage corresponds to the
existence of a strictly positive kernel. In the absence of arbitrage, all asset prices can
be expressed as the expected value of the product of the pricing kernel and security
payoffs. In a single period setting, the pricing kernel or the stochastic discount factor

(SDF) is a random variable mt+1 which satisfies the equation

Et [771t+11?t+1l = Pt

for every security with payoff It+1 at time (t + 1) and price pt at time t, where E
denotes the time—t conditional expectation operator. Intuitively, we could think of the
pricing kernel as just a generalization of the familiar concept of a discount factor to
a world of uncertainty. More precisely, it is the ratio of contingent claim prices and
the probability of the realization of a particular state. Since a price today of a future
dollar is different. in different states of nature, the pricing kernel provides a summary
of investor preferences over all states of nature. Along with a probability model for
different possible states, the pricing kernel provides a complete description of prices,
expected returns and risk premia for any security. In a representative agent model,
the pricing kernel is the intertemporal marginal rate of substitution (IMRS) and is
a measure of the aggregate measure of discomfort in the economy. It is high in bad
states of the world when investors are not consuming a lot and are not feeling very

good about it and low in good states of the world when investors are consuming a lot.

It has been known for a while now that almost any asset pricing model tan be
expressed as a. special case of the asset. pricing equation described above (See Cochrane
(2001)). Moreover, these models can distinguished by their specification of the pricing

kernel and is known as the pricing kernel representation or SDF representation of an

64

asset pricing model. For example, in the CAPM, the pricing kernel is a decreasing
linear function of the return on the market portfolio. In a representative agent model,
the pricing kernel is a decreasing function of the terminal security prices if utility
functions are strictly concave. In a Black-Scholes model. the pricing kernel is affine
in terminal security prices and decreasing if and only if the risk premium of the
underlying security is positive. Hence, any study that allows us to draw inferences
about the shape and properties of the pricing kernel can also be used as a test for
classes of asset pricing models. For example, if the empirically observed shape of the
pricing kernel is not decreasing in ending security prices, it would indicate a clear
violation of many well known asset pricing models. For tests on the market portfolio
for example, it would also mean that representative agent utility functions are not
strictly concave for intervals where the pricing kernel is not decreasing (Jackwerth
(2000), Brown and Jackwerth (2004)). Tests of monotonicity of the pricing kernel
also provide tests for a class of option pricing models, the Black-Scholes model, as
discussed above, is an obvious example. as are. models with independent stochastic

volatility.

Several researchers have attempted to estimate the shape of the pricing kernel from
consumption data and security prices (e. g, Hansen and Singleton (1982), Hansen and
Singleton (1983), and Jagannathan and Wang (2002)). The use of option data has
been particularly useful in estimating the pricing kernel since option data. provides a
model free estimation that does not depend on a particular specification of consumer
preferences or asset pricing models. Jackwerth and Rubinstein (1996), Jackwerth
(2000), Brown and Jackwerth (2004) and Ait-Sahalia and Lo (2000) provide empirical
procedures for estimating a. pricing kernel from option prices when a. finite number
of options exist instead of a dense set. Brown and Jackwerth (2004) use the theory

outlined in Breeden and Litzenberger (1978) and Cox and Ross (1985) to estimate

65

the pricing kernel as a ratio of the state price density and the subjective probabilities.
State price densities are recovered from option prices on the S&P 500 index where as
the subjective distribution is obtained from 88513 500 index returns. Their evidence
shows that although overall, the kernel is a decreasing function of gross returns, for
index levels that are a 3% deviation from the current level, the pricing kernel is
actually increasing. They argue that such a shape of the pricing kernel is inconsistent
with traditional asset pricing theory as this would imply that utility functions are

non-concave and that the representative agent is locally risk-seeking.

The above results however rely strongly on the existence of frictionless markets.
One of the building blocks of standard asset pricing theory is the assumption of fric-
tionless markets, which means that every security can always be traded at no cost. In
perfect markets, the absence of arbitrage is equivalent to the existence of state prices
or risk neutral probability measures under which the price process becomes a martin-
gale. Furthermore, it implies the existence of a positive pricing kernel. Finally, when
markets are complete, the pricing kernel can be interpreted as the density function
between the risk neutral and actual probabilities and implies the existence of a unique
pricing kernel that prices all securities in the economy. In this case, every security
has a unique price. When markets are incomplete, the absence of arbitrage no longer
guarantee a unique pricing kernel that prices all securities. The existence of bid—ask
spreads for example implies that we now have a set of admissible pricing kernels in—
stead of a. unique pricing kernel and the pricing equation is replaced by an inequality,
implying that the law of one price does not hold anymore. Securities with identical
cash flows might have. different prices without introducing arbitrage opportunities as
long as they lie between the bounds of transaction costs. More importantly, the pric-
ing kernel can no longer be interpreted simply as the ratio of the risk neutral and

subjective probabilities.

66

In this chapter, I provide a kernel estimation procedure using option returns which
is much more general than what is provided in the previous literature. Estimates of
the pricing kernel are obtained as the solution to a. constrained quadratic optimization
problem. I initially start by assuming that markets are perfect and that the pricing
kernel is a function of only the gross returns on the S&P 500 index returns. Option
prices in this case are calculated as the bid-ask midpoint. I define the optimization
function in this case as the sum of the squared numerical second derivative estimate of
the pricing kernel. The optimization problem is solved to find estimates of the pricing
kernel that minimizes the objective function under the constraint that the pricing
equation holds identically for the S&P 500 index returns and call options returns on
the 88:13 500 index. The basic idea is to find the smoothest pricing kernel function
that satisfies the pricing equation for the call and index returns. My results reflect

anomalous findings that are similar to those reported in Brown and J ackwerth (2004).

To account for imperfect markets, I then reestimate the pricing kernel, but replace
the equality constraint with an inequality constraint that appropriately accounts for
bid-ask spreads. The two bounds of the inequality are calculated by assuming a long-
side of a trade where we buy at the ask price and sell at the bid price in the next
period and the short side where we initially sell at the bid and buy back the security
in the next period at the ask price. Accounting for bid-ask spreads. I find evidence of

a pricing kernel function that is strictly decreasing in gross returns.

My estimation procedure is general enough to allow the pricing kernel to depend
011 additional state variables. The assumption that the pricing kernel is a function
of only the terminal security price is restrictive and not. gt—inerally true even for many
simple asset pricing models. The Black-Scholes model with (.le‘terministic but time

varying parameters being a simple exanmle.

67

I use. volatility as an additional state \v'ariable 1 as this allows us to draw inferences
about the volatility risk premium in the market. In particular, I choose the gross
return on the VIX index as my proxy for volatility in the kernel estimation. The
volatility measure from the VIX index is meant to be forward looking and is being
used extensively by both academic researchers and practitioners as a reliable measure
of market risk. The objective in the two-dimensional case is to minimize the rate
of change in the gradient of the surface which is now defined as a function of gross
returns on the S&P 500 and VIX indices. The objective function remains quadratic
and I impose the same constraints as in the one-dimensional case.

My evidence suggests that the projected pricing kernel is i'nonotonically decreasing
in the return on the S&P 500 index as well as the return on the VIX index which is
consistent. with a positive volatility risk premium. Asset pricing theory suggests that
a. projected pricing kernel is monotonically decreasing in the return on the VIX index
if and only if the volatility risk prei’nium is positive. The underlying theory is just a
generalization of the one-dimensional case where a negative covariance between the
pricing kernel and the terminal stock price is equivalent to a positive risk premium
on the underlying stock. A simple model illustrating the relation between the pricing

kernel and the volatility risk premium is provided in the appendix of this chapter.

The current empirical evidence on volatility risk premium is mixed although it
leans towards finding a negative volatility risk premium. Carr and “’11 (2007) finds
evidence of a negative premium suggesting that investors consider increases in market
volatility as negative shocks to their investment opportunity and are thus willing to
accept a negative. return in order to hedge away upward movements in the market

volatility. Eraker (2004) on the other hand finds evidence of a. positive volatility risk

 

1The structure of the optimization problem is general enough to include more than two state
varia-tbles

68

premium whereas Pan (2002a) shows that the volatility risk premium although positive
is insignificant compared to the jump risk premium in the market. As an independent
verification of my results on the. volatility risk premium, I analyze daily returns on
options on the VIX index. I find evidence that call options on the VIX index have
average returns increasing in the strike price of the option. In line with the theoretical
results in Chapter 1. a monotonically decreasing pricing kernel in the return on the
VIX index guarantees monotonicity in the strike price of average call returns on the

VIX index and is consistent with a positive risk premium on the VIX index.

My kernel estimation procedure has several advantages over methods suggested in
the previous literature. To my knowledge, this is the first empirical work to provide
a model-free estimation procedure that allows the pricing kernel to depend jointly
on the returns on the S&P 500 index and an additional state variable that proxies
for volatility. Unlike many previous papers that use data on options to estimate the
kernel, I do not have to rely on using holding—till-maturity options. The advantage of
using hr)lding-till—maturity returns is that option returns depend only on the terminal
index levels. However its usefulness is limited to the case where we assume that the
pricing kernel depends only on the the terminal index levels. The dependence on
additional state variables makes the problem almost equally restrictive irrespective of
whether option returns depend only on the terminal stock price or not. Hence using
only holding—till-mati1rity returns restricts us from using valuable information from
shorter holding periods while not improving the estimation procedure significantly.
My work also contrilmtes to the literature by appropriately using the bid—ask prices
separately for the kernel estimation instead of using the bid-ask midpoint. as the option
price. I show that the use of bid-ask midpoint as the option price induces error in
the kernel estirnatirm and the error increases with the widening of the bid-ask spread.

Finally. my method is general enough to allow the pricing kernel to depend on more

69

than two state varial_)les.

The rest of this chapter is organized as follows: Section 2.2 describes the data in
the empirical estimation. Section 2.3 provides the empirical methodology. Section
2.3.1 estimates the pricing kernel as a function of 88;? 500 index returns. Section
2.3.2 allows for bid-ask spreads. Section 2.3.3 estimates the pricing kernel jointly as a
function of S&P 500 index returns and VIX index returns. Section 2.3.4 provides inde-
pendent. verification of our results on the volatility risk premium by analyzing average

returns of call options on the V IX index. Section 2.4 provides a short conclusion.

2.2 Data

The data on options are from the OptionMetrics Ivy DB database. The dataset
contains information on the entire US equity and index option market from 1996
to 2007 and includes daily volume, open interest, best daily closings bid and ask
quotes, option Greeks and implied volatilities. The implied volatilities and Greeks
are calculated using a binomial tree model developed by Cox, Ross, and Rubinstein
(1979). The data set also includes information on daily prices, returns and distribution
of all exchange traded stocks. From this data set, I first filter out market prices of
options on the S&P 500 index in the following way. On each expiration Friday of a
month, I first identify those option contracts that expire on the next expiration Friday.
I then look at prices observed on Tuesdays to calculate weekly returns for the option

contracts identified in the first. step.

Option contracts in my initial sample are first. assigned to strike groups based on
the following procedure. On each buying date I first identify the option contract that
is closest to being at the money and assign it to strike group 7. The next 6 higher
strikes are assigned strike group values of 8—13 and the previous 6 lower strikes are
assigned strike group values of 1-6 giving us a total of 13 option contracts on each
buying date. The use of 13 option contracts is arbitrary and our results hold even
if we add more deep in-the-money or deep out-of-the—money contracts on any buying

date.

In accordance with the standard practice in en'ipirical option studies, I choose only
those call option contracts for which the bid price is greater than or equal to $0.125.
I also eliminate contracts for which the recorded ask price is lower than the bid price.
The arbitrage bound filter that I use requires that call option prices, estimated as
I'T

the bid-ask midpoint, should be greater than S — Kc- ,where S is the price of the

71

underlying asset, K is the options strike price, 7‘ is the risk free rate and T is the
time to expiration. Note that the main results in this paper still hold even if I do
not impose any of these filters. Call excess returns are calculated by subtracting the
risk—free rate from the call return over the same holding period. The total number of

call option contracts in my sample is 20525.

Unlike some of the previous research, my kernel estimation method is general
enough to allow for the possibility that the pricing kernel is a function of additional
state variables other than the terminal stock price. I choose the weekly gross return on
the VIX index as an additional state variable. The Chicago Board Options Exchange
(CBOE) Volatility Index is constructed as a measure of the market’s expectation of
30-day volatility. The CBOE first introduced the VIX index in 1993 and the index was
then constructed as the weighted I‘neasure of the implied volatility of eight S&P 100
at-the-money put and call options. Since the S8513 500 index provides a more accurate
measure of investors’ expectatimi of future market volatility, CBOE expanded the
VIX index in 2003 to use options on the S&P 500 index. The volatility measure from
the. VIX index is meant to be forward looking and is been used extensively by both
practitioners and academic researchers as a reliable measure of market risk. As a
rule of thumb, values of VIX index greater than 30 usually reflect a large amount of
uncertainty in the market where as VIX values less than 20 are usually associated
with a less stressful or stable market. Note that the VIX index quotes volatility in
percentages. My data on the weekly quotes of the V IX index are also obtained from
OptionhIetrics.

Allowing the pricing kernel to be a. function of volatility along with return allows me
to draw conclusions about the volatility risk premium in the market. As a independent
Verification of my results on the volatility risk premium. I also test whether call options

on the VIX index have expectml returns that are monotonic in the strike price of the

72

option. The theory behind this and the estimation procedure are from Coval and
Slnunway (2001) and chapter 1. Options on the VIX index were first introduced in
2003 and the data is also obtained through Oji)tion;\letrics. Consistent with Coval and
Shumway (2001), we only choose option contracts that. expire. in the next calendar
month. That is, the option contracts have time to expiration roughly ranging between
20 and 50 days. I use daily, instead of weekly, prices to calculate returns as this allows
me to use a reasonably big sample to estimate expected option returns. VIX call
option contracts have strike prices that usually increase in intervals of 2.5. I use 5
different. strike groups and a total of 2877 call option contracts for the monotonicity

test.

2.3 Empirical Estimation

This section provides the empirical framework and estin‘iation of the pricing kernel
as a function of the terminal security price. I start by assuming that the pricing
kernel is solely a function of ending level of security prices. Subsequently, I relax
this assumption and provide estimation procedures that allows the pricing kernel to
depend on additional state variables. Following previous studies, I also assume that
prices of a market index like the S&P500 index is a. reasonable proxy for aggregate
wealth in the economy.

Let each observation corresponding to time t for the S&P500 index be composed
of a set of state variables Xi ( including the time-t index price St), and a set. of returns

from I. to I + T: the index return, 3.9+, = SHT/St, the riskless return 1?;er (which is

in the time-t information set). and the returns 4+7, 2' = 1, . . . , kt, of options with kt
different strikes on the index. Letting 7r be a state—price density, we have

7ft+r j
1 = E —R‘
7t t+T

 

ft). ]E{f.0....,kf}

2.3.1 The Case with No State Variables

Assume

1) The option returns RAT, i E {1, . . . . k1}, conditional on time-t information
(including Sf) are functirms of 13%,; that is, [If-+7 E .7} V 0 (Raw)

2) There exists a function f such that E (7ft+7-/Tr'fj Raw, ft) = ( fl.) —1 f (139%,)

that is, RATE (71+T/7ffj Rik,” ff) is conditionally independent of .7} given R94”.

Define the excess return vector

.. _ )0 )f . 1 .. f kt f I
’f _ (htwLT — ‘t+r’ RH—T _ “t—Ht ' ' ' ‘ Rt+r _ Rt+r

74

(note that {mt = 1,. . . . T} are approximately independent). I estimate f using the

restricticms
Etf<a+.)=1. E. {fame-rt} = 0. 1:1,...1 (2.3.1)

Suppose that the number of strikes is the constant k — 1 for all t, and define the length-
k vectors at = (7; 1)I and d = (0, . . . ,0. 1). The restrictions in equation (2.3.1) can

then be written as

Et{f(Rt+T)at—L3} =0, t=1,...,T. (2.3.2)
Defining
1 _
.A=:p1“.anL FzziflftRHq)a.wf(RT+ay,

then the restrictions are

AF=3 was

Typically, T is large compared to k and the system in equation (2.3.3) is underde-
termined and hence cannot be solved to find a solution for F. However, by Farkas‘

Theorem exactly one of the following systems has a solution

System 1: AF = J and F 2 0 for some F 6 RT

System ‘2: .r'A 2 0 and 198 < 0 for some I 6 RI"
The secoml system is equivalent to the existence of some .1? 6 IR!“ satisfying

A.

23npaH—AJ>OMM=LHHT
i'—‘-‘(l

That. is. either there is an “empirical pricing kernel" F, or tlmre exists an ”empirical

75

arbitrage,” in the sense, that the fixed trading strategy .1' (where 1‘" represents the
proportion invested in the 1th option) results in returns exceeding the riskless rate
every sample period. As T gets large relative to k, System ‘2 should be less likely to

have a. solution. Consider the following two cases.

A Suppose a Solution to System 1 Exists

This should be the typical case. I will find the estimate of the pricing kernel in the
following way. I will choose the (smooth) solution that minimizes the squared sum
of the numerical second derivative estimates under the constraint that the pricing
equation holds identically for the index and all options on the index. Note that the
objective is to find numerical values of a function that satisfies the pricing equation.
The idea behind minimizing the sum of squared second derivative estimates is that I
choose the function that has minimum number of kinks. In other words, I choose the
function for which changes in the slope are minimum under the constraint that the
frmction satisfies the pricing equation.

Defining ARI- = 1?? — R94, the. estimate of the second derivative over the interval

[Rf—r R9+1l is

 

 

Fi+1—Fz‘ _ Fi‘Fi—l
3" Ri+1 3R2? 2 Fi+1ARi '— Fi (A37: + ARHI) + Ft—iéle'H
(R?+1_ R§’_1)/2 ARiARi+1(ARi+ ARi+1)/2

Note here that the index returns If? are first. sorted in an increasing sequence. Now

define.

(s,- = {AIL-AR,” (ARI-1L Alt-+1) /‘2}‘1

and the (T — 2) x T matrix {5}): '1' = l ..... T — 2zj = 1,. . . ,T} as follows:

5i+1ARi+2 if J=i

S —(i,-+1(AR,-+1 + ARHQ) if .i = 77 +1
ij = -
0i+1ARi+l if j=i+2
0 otherwise

The pricing kernel estimation problem can now be expressed as follows:
, 1
mm— F's/SF
2
subject to AF 2 ,8 (234)

and F > 0.

The Lagrangian is

1
L = iF’S’SF + 9’ (u — AF) — 7,’F

with FOC
S’SF — .4'6 2 7) .7). F 2 0. 77'F = 0

Note that multiplying the FCC by F and substituting F’A’ = [3' im lies F’S’SF =
P

9r)

B Suppose a Solution to System 2 Exists

If there is no solution to AF — ,d = 0, then for any 7 > 0, we can solve. the following

pioblem:

min (AF — tau/T1 (AF — .3) + ., F’S’SF

subject to F Z 0.

~ I
xi

The parameter 7 detern‘iines the weight given to smoothness in the estimation. There’s
no analytic solution, but a simple iterative technique is providtxl in Sha, Saul, and
Lee (2002).

An estimate of the optimal weighting matrix when we assume serially uncorrelated
{f (Rl+T) at} and stationarity, including across strike groups) is

- 1 T -
MT = T 2 {f (Rt+r) (It — 13} {f (Rf+T) Oct — t3},
t=l

An iterative scheme is to choose some initial estimate f, compute the corresponding
NIT, substitute into equation (2.3.5) to estimate f, then re—estimate All-T, and so on

which is similar to standard GMM estimation.

Example 2.3.1. Consider a binomial model with gross returns u and d with proba-
bilities q and. 1 — q. Let the weighting matrix M;- be the identity matrla: for each t.
Suppose we observe only the stock and bond returns (no options), and in the exact
frequencies as their probabilities. Suppose also that the short rate is the constant r

(and therefore 8,; (r) = 63—” for all t). Then

(6”.21)’ with prob q

(6”, at)I with prob l — q
and the F 0C is

"7‘7” [‘7'

v; qf* (n) J +(1 — q) f* ((l) — l = 0 all s

Qt'vl‘ng the eatbcrcted solution (setting {} Z 0)

u — e" er” — (l

l—(' T '(l =(::“"T——————i ( * u. :c-T"T——.
( [>1 () 1f() 0H,,

11. — (l 7

78

This is equivalent to solving

(IF (U)
(l-mefl

= (X’X)_1X’1, where

C Results

Figure 2.1 shows the estimation of the pricing kernel by solving the optimization
problem in equation (2.3.4). The plot shows the pricing kernel as a function of the
gross return on the Sid) 500 index. The vertical axis represents the estimated value
of the pricing kernel and the horizontal axis represents the gross return on the S&P
500 index. Call option returns are calculated by setting option prices as the bid-ask
midpoint. That is, we assume that there exists an equilibrium price for the option
contracts at which the contracts are both bought and sold. The elements F (R?) of
the vector F which is the solution to the optimization problem (2.3.4) is multiplied by
T/R{ to get the estimate of the pricing kernel. Option contracts are first assigned to
strike groups based on the following procedure. On each buying date we first identify
the option contract that is closest to being at the money and assign it to strike group
7. The next 6 higher strikes are assigned strike group values of 8-13 and the previous
6 lower strikes are assigned strike group values of 1—7 giving us a total of 13 option
contracts on each buying date.

The shape of the pricing kernel in Figure 2.1 is not consistent. with the notion that.
the pricing kernel is a (flecreasing function of ending level of security prices. Traditional

asset pricing theory assumes that a representative agent exists. A market index like

79

the 88:13 500 have often been assumed to represent the aggregate wealth held by the
representative agent. The estimate of the pricing kernel in Figure 2.1 would then
suggest that over a range of ending values of the market index, utility functions are
not concave. Brown and Jackwerth (2004) finds similar anomalous results in their
paper. Although. the estimation method and the smrmthing algorithm employed are
quite different, the anomalous results occur for approximately the same range of ending
values of the S&P .500.

As discussed before, the estimation procedure in this subsection and in Brown and
Jackwerth (2004) do not explicitly account for bid-ask spreads. For the remainder of
this paper, We will estimate the pricing kernel using procedures where we correctly

account for bid—ask spreads.

2.3.2 Generalization to Bid-Ask Spreads
A Overview

Market imperfections or market frictions are receiving increasing attention from re-
searchers since they are an important property of all financial markets. The most
intensely studied market friction is illiquidity which leads to the existence of bid—ask
spreads, that is, the ('lifference between the prices at which one can buy or sell a par-
ticular asset. Previous research outlines numerous explanations for liquidity which
include exogenous transaction costs, inventory risk, private information, counterparty
search frictions. and short. sale constraints. If markets are complete, then there exists
one equilibrium price and there exists a pricing kernel that prices all assets in the
economy such that equation (2.3.1) is satisfied. In the presence of bid—ask spreads,
there exists a band of admissible pricing kernels such that the expectation of the prod-

uct of a pricing kernel and the future payoff lies liietween the bid and ask prices. In

80

particular, let 1'0 and .1'1 denote the initial and possibly random end-of-period incre—
mental cash flows. respectively. from some marketed trading strategy. It can be show
that (ignoring teclmicalities) the absence of arbitrage is equivalent to the existence of

a. strictly positive stochastic discount factor In. satisfying

—:r0 2 E [our 1] for all 111arketed (;1%(),.r1).

In our setting, if at time-t we buy at the ask price Pf“, and sell at t + T at the bid

price HIT then the incremental cash flows are (—P{wk, PM

t+r)v resulting in

 

PIES/ii 2 Et

 

WELT bid.
1’ .
7ft t+T

The iin-remental cash flow (Film, —Pt”:f;) (sell at t at the bid price and buy at t + T

at the ask price) implies

 

(>1 7Tt+r .flg
7ft
Together (using the ask buying price at t + T) these imply

7ft Pbid 7ft _ ask
Et ”LT—til 3131;}, +7 ”T . (2.3.6)
7ft. Pt” s k 7ft Ptbl (l

 

 

 

A stronger set. of restrictions is based 011 trades from a different initial position

t+ 7', the incremental cash flows are. (Pth"1,—Ptlfl’rdT). An agent initially short could

offset by buying at t instead of t + T. with incremental cash flow (—Pt”“,", Pfié).

Together these strategies imply

"7f Pas/c Ti [)bid
I _ 7- I -— '7

7"! Ptusk 7f Pg)!”

 

 

 

 

|/\
...;
l/\
lv
w
Rl
V

\

Obviously the restrictions (2.3.7) imply (2.3.6). Alternatively, we could consider an
agent initially short an option who offsets by going long at t + T instead of t.
Note that defining Pt 2 % (Ptb’d + 11"“) for all t, and averaging the restrictions

in (2.3.7) and (2.3.6) results in

7Tt+T Pt+T 1
7ft Ptusk

 

 

 

 

, P
E, 7t“ ”T . (2.3.8)

E .
t 7ft Ptbzd

l/\
|/\

Furthermore, Jensen‘s inequality implies (assuming Ptb’d < Ptf‘Sk)

 

 

 

E, 7W+T Pt+r : Er 7Tt+r Pt+r
7ft Pt 7ft % (Ptbid + Ptask)
< lEt 7ft+r Pt+T l Wt+r Pt+r
2 ' 7ft Pthid 2 t 7ft Pics/a:

The difference between the left and right side is increasing in the bid—ask spread.
Therefore even if the conditional expectations in (2.3.8) are symmetric around one, we
will find that the expected discounted price ratio is strictly less than one when prices
are midpoints of bids and asks.

The error from using the bid-ask midpoint price in the denominator can be approx—
imated as follows. Fixing the prices Plf’id and 1193;", we take a Taylor series expansion

around Pt 2 % (Find + 13:93,"):

, . 2 2
l b I 1 s};
11 +1 1 ~1+§(Pr”—Pt) +20”? T”)
2 lib”! ‘2 Ptask W Pt [)3

 

at 1 d tl‘ierefore

77t+T Pt+T 1 77t+T Pt+T

7ft Ptbid é t 7ft ptask

 

 

 

 

 

Et 7Tt+T Pf+T

l

1 3'! 1. 2 —dt
’T ,lq'
It 2(Ptm +Ptu )

it
where the difference (It is approximately the average squared difference between the

bid and ask prices and the bid—ask midpoint:

. 2 , 2
d ... 1 P?” — P, + 1 Pg“ — Pt
t N 2 Pt 2 Ft

The above results show that the use of bid-ask midpoint as the price of a security
leads to erroneous estimation of the pricing kernel and the error increases with the

widening of bid-ask spreads.

B Results

I modify the original optimization problem in (2.3.4) to account for bid-ask spreads
in the following way. First consider tests of the restrictions (2.3.6). (It is easy to

modify the definitions to test (2.3.7) instead.) Define Rfiirt’o = ngi/Slf’id, and

Rs/zorti

t +7 2 Pl’aSk/Ptz‘bld for the returns from a long position (over [23, t + T]) in the

t+T

stock and options 2'. 2 1, . . . , kt. Now define

,short _ short.0 f shortl f shortkt f I
7t _(Rt+T —Rt+Tth+T _t+T....,Rt+T —Rt+T

Hg

10 ... . .. . .
and "'t analogously. With assumptions 1 and 2 above, we estimate f using the

restrictions

Etfth-+r) = 1, Et {fer+r)7‘iong} S 0 S E {ffRHrl’fhm}, i=1s---,T-

83

Suppose the number of strikes is the constant Ir —- 1 for all t, and define the length-k

7 , I ’ I ’
vectors along 2 [(riong) ,1] , (1";"0’" : [Of/’0’") ,1] and i3 = (0,. . . ,0. 1). Then

the restrictions are

13(Heasafiw—s}gogsfijuaapsMN—s) t=LHHT.
Defining.
Alonf] = [(IIIOHQ, . . . “iii-my] ,
.43th = [afhm‘fi . oilm'f] ,
1 I
and F = ? lf(R1+T)a---af(RT+T)l a

then the restrictions are

xWWF—J<O<AWmF—&

~ _ ., . I C” . . . .
Define A = [(Along)’, (—.4”""’t)'] and d = [,3’, —,:’)”]I. Then the pricmg kernel esti-

mation problem can now be expressed as follows:

1
min 2 F’S’SF

subject to +lF g 3 (2-3-9)

and F > 0.

Figure 2.2 provides the plot of the empirical pricing kernel as a function of the gross

84

return on the S&P 500 index allowing for bid-ask spreads. The solution F(R,-) to the
optimization problem (2.3.9) is multiplied by T/RZ-f to get the estimate for the pricing
kernel. I use the MATLAB quadratic optimization function quadprog to solve for F.
All the inetpiality constraints are satisfied and the value of the objective function is
4.3617. Figure 2.2 shows that the pricing kernel is a monotonically decreasing function
of the gross return.

The relationship between the pricing kernel and gross returns in Figure 2.2 contra-
dict the. results in Brown and Jackwerth (2004). In particular, Brown and Jackwerth
(2004) finds evidence that the pricing kernel is not monotonically decreasing in gross
returns. Over the range of returns that. are a 3% deviation from current levels, the
pricing kernel is actually increasing. As discussed before, the estimation procedure
outlined in Brown and Jackwerth (2004) do not appropriately account for bid-ask
spreads. Although. they include a penalty term in their optimization routine to ac-
count. for the possibility that equality constraints might be violated, but penalizing for
violations from bid—ask midpoint might not be appropriate. Any solution that allows
for prices to lie anywhere in the bid-ask spread is a valid solution, and a penalty for de-
viating from the bid-ask midpoint is over-restrictive. h-‘Ioreover, the theory underlying
the estimation of the pricing kernel as a ratio of risk neutral and subjective proba—
bilites is valid only under frictionless market conditions. In the presence of bid-ask
spreads, a band of pricing kernels exist instead of an unique one and the estimation

procedure outlined in Brown and J ackwerth (2004) would be no longer appropriate.

2.3.3 The case with additional state variables

In this subsection I extend my kernel estimation method to allow the. pricing kernel
to be a function of the gross return on the S&P 500 index and an additional state

variable. I use the gross return on the VIX index for the same holding period as an

additional state variable since this also allows us to draw some general conclusions
about the volatility risk premium in the market 2.

Although the basic structure of the optimization problem remains same as the
one-dimensional case, we need to choose a different smoothing criterion. Recall that
in the one-dimensional case the objective function or the smoothing function was the
sum of the squared second derivative estimates of the pricing kernel. Minimizing the
objective function was equivalent to finding the smoothest function of S&P 500 gross
returns that satisfied the pricing equation. In the two—dimensional case. our objective
is to minimize the rate of change in the gradient of the surface F (R? +T, a”), where
R?+T and a, are gross returns for the holding period from t to t+ T on the S&P 500
Index and the V IX index respectively . With a objective function defined as above
the problem is still quadratic and the constraints remain the same as in the one-
dimensional case. Note that the basic structure is general enough to accommodate
additional state variables but it would be difficult to provide a geometric interpretation

of a pricing kernel with three state variables.

A Overview of the Two-Dimensional Smoothing Criterion

The problem of fitting a surface of the form F (R? +T. hr) where 1?? +17 f+T and

F are known to us is not a very difficult problem with uniform or grid data. The
problem is considerably harder when we are dealing with scattered data. Fortunately,
the last few decades have witnessed considerable research in developing algorithms
that approximate a surface with a high degree of complexity. Mathematical packages
like MATLAB provide functions that do reasonably well in interpolating a surface
from scattered data. Our problem l'iowever adds an additional lay-fer of complexity to

the already difficult task of constructing a. smooth surface F since we do not have, the

 

2] get similar results if I use changes in the VIX index instead of gross returns

86

numerical values of F to begin with. To implement a. reasonable smoothing criterion
to find optimal values of F it is necessz-u'y to create a. partition of the scattered sample
v

points in the plane defined by ((R§)+T, 1+T). One of the most applicable and popular

partitions in this case is triangulation.

A triangulation of a finite set of n. sample points p,- = ((R?, R;")),i = 1. ...,n in
a plane is defined as a collection of triangles satisfying (1) each vertex of a triangle
corresponds to a sample point pi, (2) the union of all the triangles must. form a
connected set, and (3) the intersection of any two triangles is either a common edge
or a common vertex. There are clearly many distinct triangulations for a sufficiently
large finite set of vertices but a Delaunay triangulation of the sample points seems to

be the most appropriate because of some very unique ad vintages.

A Delaunay triangulation for a. set of points in a plane is a triz'mgulation such that
no point is in the circumcircle of any triangle formed. There are several different exist-
ing algorithms to accomplish this but the advantage of using a Delaunay triangulation
is that it maximizes the minimum angle of all the angles of the triangles. Thus the
method avoids forming triangles that are very long and thin. Moreover, Delaunay
trizu‘igulation is particularly non-restrictive and is ideal for interpolation algorithms.
In summary, the Delaunay condition in the two—dimensional case states that a set of
triangles is a Delaunay triangulation if all the circumcircles of all the triangles in the
set are empty. As mentioned earlier, this method is robust enough to accommodate
additional state variables. It is possible to extend the triangulation method to higher
dimensions. For example we can use Delaunay Triangulation in three—dimensimial
spaces by using a circumscribed sphere in place of the circumcircle. Figure 2.3 shows
the 2D scatter plot of the gross return on the 88:13 500 index (R0)and the gross return
on the VIX index (R‘) and Figure 2.4 shows the corresprmding Delaunay triangulation

of the scattered points.

87

Now that we have a triangulation of the sample with vertices at the data points,
where the data points are pairs of returns on the 88:13 500 and VIX indices correspond—
ing to each observation. the basic theory behind defining the objective or smoothing
function is as follows:

Let (R9. RE", 17,-), i E {0,1.2} denote the three data points of one triangle, and
let 1:,- = R? — R8. g/l- = R? — 18, and z,- = F,- — F0, for , i E {0,1,2}, denote the
coordinates of the points when the triangle is shifted so the first point is at the origin.

A necessary and sufficient condition for the (1'. y) coordinates to from a triangle is:

II 91 , _
lS nonsmgular. (2.3.10)
5‘2 92
This rules out all three points, including the origin, lying on one line. Under assump-

tion (2.3.10), the equation of the. plane passing through the points is z = or. + ,Jy

where the coefficients (1' and [J satisfy

 

 

a = Zw'z — 223/1, ’3 = 1'132 - 1231.
$1y2—'J2U1 1192“$2y1
Consider two contiguous triangles, labeled a. and b. To ensure continuity of our
surface, these triangles must share two vertices. Without loss of generality, we can
translate the triangle so that the sl‘iared vertices are (0, 0,0) and (3:1,y1, :1). Let
(13,112. 2:2) be the third vertex of triangle (1, and (.r3 y3, :3) the third vertex of tri-
angle l). The difference between the center points of each of the two triangles is
({zr3 — .1'2}/3. {,1/3 — ,1/2} /3 {.23 -— .32} /3).

A necessary and sufficient condition for triangles a. and b to form a. quadrilateral is

.[1 ”I I1 yl _
(let - det < 0.
12 yZ ‘I3 y3

88

Note that, if the rotation from the vector (11.111) to the vector (12.112) is clockwise,

then the rotation from (1:1, yl) to ($3,113) must be counterclockwise, and vice versa.
The objective is to penalize the rate of change in the gradient. One measure of

the rate of change of the vector (Oz/01', 03/03)) is simply the element-by—eleniei1t rate

of change (from center to center of the triangles), which we will denote V:

 

 

v = [W W]
where
: I — :- l 3 z — : z 3
V1,- : ( 1J3 3.11 _ 1.12 2.11) ’ and
.1'1y3 — 1'3'3/1 Ill/‘2 — 1723/1 1'3 " 172

 

 

_ 11:3 — 1‘3 :1 171:2 — 313321 3
vy _ . . _ . .
-’ 1313 ‘— 1311/1 111/2 — 1211/1 .3/3 - .7/2

Alternatively, we can measure the rate of change using the Euclidean distance

between the two center points. Defining

 

_ 1 ‘
o = 3 \/(l‘3 — 1172)2 + (.113 ‘12)2

then

V _ 6—1 [( 31.113 — 23,111 _ 3:1}[2 — 32.111) (.1133 - I321 _ 1:122 - 1231)]
11.113 — ~1'3U1 J'w‘z — 1291 ' 113/3 — fat/1 III/2 — $2.111

With a pmialty proportional to V’V the estimation problem is still quadratic. For

 

 

 

 

each triangle, there will be up to three V terms (interior tria-mgles will have three
cmitiguous triangles. and triangles on the bmnidzuy may have one or two). These

penalties are added over all the triangles to get our final objective function.

89

Sketch of the Algorithm

0 Let n denote the number of vertices (the number of stock return ol‘)servations),

and N the number of triplets (triemgles).

T

0 Suppose the triplets of vertices are labeled 1, . . . , A . Now fix a particular triplet

i.
0 Search the remaining triplets for any sharing two vertices with triplet i.
0 Suppose. triple j shares two vertices with triplet i (the triangles share an edge).

0 Let a, b. c, (l E {1, . . . ,n} denote the vertices of the two triplets, with a. and b the

two shared ones.

0 Check that the vertices form a true quadih—iteral by confirming

0 0 .v p!
R, — 1?. R; — R;

det - det < 0.
R2 — H2 11);: — R5 R3 — H5} 3 — R};

0 Define
= (R2 - R2) ( 25- Hz) — (HS — Hf?) (Hi; — R22)
0 = (R2 — RS) (R: — R32.) — (1e— RE!) (R5 — R5)
From the above formula. the rate of change in the gradient from triplet i to j is

Vi-J Z (“(zFu + (1be + ”(Fe + adpd: dupe + ”’be + 3ch + .x’ildpdl’

90

where

 

- 1 2 ’1 '9 2
0 = 3; (R91 “ 39.) + (3.31— RE)
and
R}; v Re_ Re
Ga. : O_1(-—b—i—_ (l_ ____)
CD
—1 ( RX. 3'3 R___“)
ab = — ..
0,7

RE— R33
five—t >
Re _ R1:
ad = -5_1(—b———£)
q,

and (multiplying by —1 after interchanging the roles of y and R)

R0 — R0 R0 — R9
#30 z (5.]. (_ I) . d + b ‘ (,

u e
- R0 — I?" 120. _ n0
53b : 0—1 <_ d x a + C I (7.

_ . R0 — 120
o
BO — R0
w

The jwnalty for the edge between triples ‘2'. and j is

91

 

VQJVU- = <(13+,i_)33) Fg-l- ... + ((13+133) Ff

+ 2 Z Z (Li/£011 + riff/1.1331) F“. (2.3.11)

ke{a,b.c,d} 17$];

0 These coefficients of Fa, . . . , Fd should be added to the appropriate terms of the
matrix Q in F ’ QF . For example, add (03 + 33) to Qua, (aaab + 5031,) to Qab

and Qba, and so on.

. i . I ,- - I
o In matrix terms. after dehmng a = (amalgawaed) and x3 = (kiln/ib--"/3(‘.9,x‘33([)a
we have a penalty term of F ’ (00" + (if) F. So each contribution to Q is a
matrix of rank one or two. The rank one if and only if ,3 is a scalar multiple

of a; equivalently. that is, if (R2, R8, R2, RS) is equal to a scalar multiple of

(R22, M2: 2;)-

0 Once we have the matrix Q, the structure of the quadratic optimization problem

is similar to the one dimensional case and can be'expressed as follows:
. 1 ,
nun —F QF
2
subject to AF 5 x3 (2-3-12)

and F > 0.

where F is the pricing kernel function and A and (3 are have the same definition

as in equation (2.3.9).

92

C Results

Figure 2.5 shows the the 3D scatter plot of the results from the optimization problem
in (2.3.12). The S&P 500 index return and the VIX index return are plotted on the
X and Y axes respectively and the estimated pricing kernel is plotted on the Z axis.
The solution F (R?) obtained from the optimization problem is multiplied by T/Rif to
get the estimate of empirical pricing kernel. All the inequality constraints are satisfied

and the sum of F over all holding periods equals 1.

I then use the 3D scattered data (R?,R§’,Fi) to interpolate a surface passing
through all the data points. I use the MAT LAB function griddata to interpolate
points on an uniform 3D grid from the 3D scattered data and then draw a surface
through the interpolated points. The interpolated surface is shown in Figure 2.6. From
the surface we see that the estimated pricing kernel is downward sloping in the gross
return on the S&P 500 index return even when we include volatility as an additional
state variable. As mentioned earlier, the inclusion of volatility as an additional state
variable allows us to draw inferences about the volatility risk premium in the market.
Surprisingly, the volatility risk premium appears to be positive which is contrary to

recent empirical evidence.

Figure 2.7 shows the rotation of the interpolated surface in Figure 2.6 to illustrate
that the pricing kernel is also (‘lecreasing in the gross return on the VIX index 3. The
pricing kernel projection is decreasing in returns on the VIX index if and only if the
\Iv'olatility risk premium is positive. The appendix in this paper shows a simple model

motivating the above result.

 

31 get the same result. if I changes in the V IX index instead of gross returns.

93

2.3.4 Average call returns

In this subsection I provide an independent verification of my results on the volatility
risk premium. I analyze average daily returns of call options on the V IX index. Since,
call options data on the VIX index are available for only a short time period, I use
daily returns instead of weekly returns that I use for kernel estimation. Coval and
Shumway (2001) show that as long as the covariance of the pricing kernel and gross
returns, conditional on the option being in the money, is negative, call option returns
would have expected returns that are increasing in the strike price of the option. A
violation in monotonicity of expected call returns however implies that the pricing
kernel is not a decreasing function of returns.

Call options on the VIX index were first. introduced in 2006 and hence we have
a relatively small sample to test the above hypothesis. My data on options on the
VIX index starts from February 2006 and ends in September 2008. I first assign the
call option contracts into five strike groups in the following way: Call options on the
VIX index have strike prices increasing in intervals of 2.5 and hence I define my strike
groups in intervals of 2.5. A call option contract is assigned a strike group value of 1
if the difference between the strike price and the current price of the underlying VIX
index lies between —7.5 and —5. Similarly, a call option contract is assigned a strike
group value of 5 if the difference between the strike price and the current. price of the
underlying VIX index lies between 2.5 and 5. Rows 2 and 3 of Table 2.1 show the value
of strike group C(in'responding to the range of values of K — St where K is the strike
price and St is the current price of the VIX index. After assigning the call contracts
to strike groups, the options are then checked for outliers in their bid ask quotes. In
particular, I remove those contracts from the sample for which the bid price is not
strictly greater than zero and/ or the differeiu-e between the ask price and the bid price

is less than the minimum tick. Note that. the minimum tick for the. series trading below

94

$3 is 0.05, and for series trading above $3, the minimum tick is 0.1. Results in Table
2.1 shows that call option contracts have average returns increasing in the strike price
of the option which is consistent with a monotonically decreasing projected pricing

kernel in gross return on the V IX index and hence a positive volatility risk premium.

2.4 Conclusion

This chapter estimates the shape of the pricing kernel from data on option prices.
Traditional asset pricing theory suggests that the pricing kernel should be a decreasing
function of security returns. Recent. empirical evidence using options on the S&P 500
index options suggests that the pricing kernel as a function of gross returns on the
S8513 500 index is increasing for a. range of ending index levels implying that utility
functions are non-concave and that the representative agent is locally risk—seeking. The
existence of additional state variables has been suggested as a possible explanation for

this puzzling behavior of the pricing kernel.

In this chapter, I estimate the shape of the pricing kernel as the solution to a
ccmstrained quadratic optimization problem. The objective is to minimize measures
of smoothness for the pricing kernel function under the constraint that the pricing
equation holds for all call option returns and the return on the underlying index.
Unlike previous approaches, my method appropriately accounts for bid-ask spreads
which are. typically a large proportion of the price of an option especially if the option
is out of the money. I show that the popular approach of using the bid-ask midpoint
as the price of the option induces error in the estimation of the kernel and the error
increases with a widening in the bid-ask spread. Accounting for Ind—ask spreads, I find
evidence of a pricing kernel that is monotonically decreasing in gross returns on the

SSJP 500 index.

My kernel (‘istximation method is general enough to allow the pricing kernel to
depend on additional state variables. Using gross returns on the V IX index as a. proxy
for the rate of change of volatility. I show that the projected pricing kernel is decreasing
in both. the gross return on the 8&1) 500 index and the gross return on the VIX index.

A monotonical1y decrmsing projected pricing kernel in the VIX returns is consistent

96

with a. positive volatility risk premium in the market. This is surprising considering
that. a significant bulk of previous research suggests that the v(_)latility risk premium
is negative. To provide an independent confirmation of my results 011 the volatility
risk premium, I also present results on the average daily returns of call options on the
VIX index. I show that average call returns are increasing in the strike price of the
option which is consistent with a the pricing kernel being a (.lecreasing function of the

return on the V IX index.

97

,A'J.’ _'

Chapter 2 Appendix

98

A2.1 Risk premia and kernel projection with volatil-

ity as a state variable

Let the stock price and stock—price variance processes satisfy

 

(IS

t = ptdt+fldBt1
St
(let ,
— = (rtclt-i-l-{tdBt
'Ut

where B is 2-dimensional standz-u'd Brownian motion under the original measure P.

(If)? = (1131‘, + mdt

Let a. state-price density satisfy

(17?
——3 = —rtdf —1}£dBt

fit

and define. as usual, the risk-neutral measure P through

E z 81.? ,-..ds:;
(1P 7m

Girsanov’s theorem implies that

(11?, : ([8; + 7m]!

is 2-dinwnsional standard Brownian motion under P. The risk-neutral dynamics are

99

[S r
(—-3- = I‘Hff-i-fldBtl

St
(let ~ ~
———’- 2 otdt + rtdet
’L‘t '
where
_, _ , I
lit — ’f - Vit'lt
_~ I
at — (If 2 Ift’lfi

Option price ('lynamics:

 

 

(lC(f.St,w) _ SCS dS vC—vdt' C; l . _ 2 , .2 . 2
C — T-S—d- C 7+ (.3 +2C ((55((15t) +Cmy ((111) +2C5v(dlzt) )

 

The option price PDE is (letting Z2 denote the drift operator under the risk-neutral

 

 

 

measure)
no (r. 5,. a) SOS 153 + ((7.131: + c.
z ]‘ : —— —__.___ __
C C S C e C
1 . 2 2 . 2
+ 9c ((55(([St) +0“, ((11.7) +2Cg,,((l~vt) )
Therefore
1C LS. ' IS 1.‘
3 ( (f ") —rc1t=§25 (‘7 — nu) + (2....(1 —a.(1r) (A2.1.1)
. ‘7‘ I,

100

where the elasticities of C with respect to S and ‘I’ are
(23 = ——-~— (2,. = —' (A2.1.2)

Recall the notation we used for the projection of the pricing kernel on stock returns

and changes in state variables:

 

E (”HT

7ft

I

 

ft,R9+,.AX,) : ( {+T)_1f (Egg/AM)

The state variable in this setting is X = e.

A2.1.1 Discrete-time/instantaneous—time formula

The stock and variance dynamics are

 

AS-
33 = Warm/5.33,?
t
A)
——” = atm+n§ABt
(It

From the dynamics of 7r, we, have

f ”HA1? N I

Defining the ”beta”

#3 : $2,;31S2831'

101

where

Var (as, /S,) Cov (ASt/St: Am/ w)
COV (ASf/Sf. Al’t/Ut) \Y'dI‘ (Al'f/Uf)

93,1? =
and where

Cov (As/St. Ava/m.)

Qsm :
Cov (Act/ct, Ant/7n)
which implies
_ ‘l‘t x/T'tb'lt .Ut. - 7‘
Id :
. _ _I ,I . ~
‘/ l'f/Ilt Ixtftt (If _ Of

the projection of the pricing kernel is

71' ,
E< t+At
7ft

 

H.138. Act) 2 1- E (l'lf-ABI‘I flABf‘ hiABt’)

= 1 — (flABtl, MAB.) .3

AS} AW
2 1— —— -— mAl. —— — (”At .3.
St ()1
Therefore the monotonicity of the projection depends on the signs of the two
elements of 3 \thn Cov(A.S}/St. Al’f/Uf) = 0, then the priciiig—kernel projection is
decreasing in 4351/5} if and only if [If > r; and is decreasing in Act/e; if and only if

(H > (1;.

102

 

A2.1.2 Deterministic-coefficient case

From

 

1+7 t+T
f fif‘i‘T _ ’ 1 , / _ . I
Rt+r 7ft — exp (—5 . t 081),..(15 — t 0511/33

we obtain

 

 

-f+T -t+T ~t+T t+T 't+’r

E (/ 77;.st \/i—,(IB}./ Hést) = ( \/v—,,.(IB;,/ I{;(1B3),13,

- f - f . t . t - t
where

.3 : Sign...
where
‘ SW) '( (“9’”) (”’“D
Q,“ 2 M11 (111 (T Coy ln —5t_ ,ln l-‘t

 

(he—>c—a) weer»

and where

 

 

 

V ‘53 3T "' T
f Cov (hi ( ’57?) ,111 (-€:—’)) FL ('11.. — 1‘...) (Is
25, T I , ll, = .
"I ‘ I' ‘7? ._,._ 7' ~
Cov (In ( {(7:7) .111 < '37.? )) f3+ (us - (1,5)(15‘
Defining

103

t+T ~t+T . t+T
Z = Var (/ 7):.st / Jeri-(133,] 5MB.)
3 f f

Va (111 (733+T)) — '3’ C033 (333 (SET) 211‘ (33%?»

 

 

 

7ft

 

 

 

ft.

 

 

E (”Her

7ft

 

SH—T L313+T
St 3 let

1 t+T t+T t+T
= exp {2 <2 —/ ngnsds) — ( fidBé./ sgng) 13}
t - t t
1 3
20x) —(7 — 1,1. .
I 2 t It. .

 

where
7ft
at = Z - Var ln +T ,
7ft
and

 

 

Sr t+T 13f t+T
m = ln 3T —— #413,111 +7 — / asds
5t t Ut t

\Vith regard to the monotoiiicity of the pricing-kernel projections on 111(ST/St) and

In (NT/ct), we obtain essentially the same result as in the discrete—time case.

104

Figure 2.1
Empirical pricing kernel vs gross return

 

Pricing
Kernel 5

 

 

 

 

0.9 0.95 1 1.05 1.1 1.15
Gross Return

This figure shows the plot of the pricing kernel estimated as a function of the gross return on
the S&P 500 index. The pricing kernel is estimated as a solution to a quadratic optimization
problem under the constraint that the pricing equation is satisfied for returns on the 88513 500
index and returns on call options written on the S&P 500 index. The objective function is
defined as the sum of the squared second derivative estimate of the pricing kernel. Estimates
of the pricing kernel are obtained by solving the optimization problem outlined in equation
2.3.4. The sample period is from 1996 to 2007. Call and S&P 500 index returns have a
weekly holding period.

Figure 2.2
Empirical pricing kernel vs gross return (allowing for bid-ask spread)

 

14 r r r r

1.3" -1

1.2- a

1.1 - -
Pricing
Kernel

0.9

r
l

l
l

0.8

0.7 r -

 

 

l l

0.9 0.95 1 1.05 1.1 1.15
Gross Return

 

This figure shows the plot of the pricing kernel estimated as a function of the gross return on
the S&P 500 index correctly accounting for bid-ask spreads. The pricing kernel is estimated
as a solution to a quadratic optimization problem under the constraint that the pricing
inequality is satisfied for returns on the S&P 500 index and returns on call options written
on the 88:13 500 index. The objective function is defined as the sum of the squared second
derivative estimate of the pricing kernel. Estimates of the pricing kernel are obtained by
solving the optimization problem outlined in equation 2.3.9. The sample period is from 1996
to 2007. Call and S&P 500 index returns have a weekly holding period.

106

VIX Index Return

Figure 2.3
2D scatter plot of VIX index return versus S&P 500 index return

1
0

1.6

1.5” °

1.4-

1.3- o

r
O
0

1.2

1.1

I

I

0.9

l

0.8

 

 

1 0 J

0'7 4 L l
0.9 0.95 1 1.05 1.1 1.15

S&P 500 Index Return
This figure shows the scatter plot plot of the gross return on the VIX index versus the gross

return on the S&P 500 index. The sample period is from 1996 to 2007. VIX and S&P 500
index returns have a weekly holding period.

107

VIX Index Return

Figure 2.4
Delaunay Triangulation of VIX index returns versus S&P 500 index
returns

 

 

0.9 0.95 1 1.05 1.1 1.15
S&P 500 Index Return

This figure shows the Delaunay triangulation of the gross return on the VIX index versus

gross return on the S&P 500 index. The sample period is from 1996 to 2007. VIX and S&P
500 index returns have a weekly holding period.

108

Figure 2.5
3D scatter plot of the estimated pricing kernel versus S&P 500 index and
VIX index returns

1.5—

 

Pricing Kernel
A
1

 

    

0.5 7 .. ‘.‘
2
O 7
0.9 9““

1.05
S&P 500 Index Return

This figure shows the 3D scatter plot of the estimated pricing kernel versus gross return on
the S&P 500 index and gross return on the VIX index. The pricing kernel is estimated as a
solution to a quadratic optimization problem under the constraint that the pricing inequality
is satisfied for returns on the S&P 500 index and returns on call options written on the S&P
500 index. The objective function is defined as the sum of the squared rate of change of
the gradient of the pricing kernel surface which is a function of S&P 500 index return and
VIX index return. Estimates of the pricing kernel are obtained by solving the optimization

problem outlined in equation 2.3.12. The sample period is from 1996 to 2007. Call and
index returns have a weekly holding period.

109

 

VF._.1-a.-.Ut‘m '5' u . 1'.

Figure 2.6
Interpolated surface of the estimated pricing kernel as a function of
return on the S&P 500 index and return on the VIX index.

Pricing Kernel

 

 

 

 

 

 

    

1.05 i. 0'5 VIX Return
S&P 500 Return

This figure shows the interpolated surface of the estimated pricing kernel. The pricing kernel
is estimated as a solution to a quadratic optimization problem under the constraint that. the
pricing inequality is satisfied for all S&P 500 index and call returns . The objective function
is defined as the sum of the squared rate of change of the gradient of the pricing kernel
surface which is a function of S&P 500 index return and VIX index return. Estimates of the
pricing kernel are obtained by solving the optimization problem outlined in equation 2.3.12.
The MATLAB function griddata is used to interpolate points on an uniform 3D grid from
the 3D scattered data and then draw a surface through the interpolated points.The sample
period is from 1996 to 2007. Call and index returns have a weekly holding period.

110

Figure 2.7
Interpolated surface of the estimated pricing kernel as a function of
return on the S&P 500 index and return on the VIX index

....................

..........

 

Pricing Kernel
l

 

 

 

    

1.2
VIX Return

This figure shows the interpolated surface of the estimated pricing kernel. The plot is just a
rotation of the plot in Figure 2.6. The pricing kernel is estimated as a solution to a quadratic
optimization problem under the constraint. that the pricing inequality is satisfied for all S&P
500 index and call returns . The objective function is defined as the sum of the squared rate
of change of the gradient of the pricing kernel surface which is a function of S&P 500 index
return and VIX index return. Estimates of the pricing kernel are obtained by solving the
optimization problem outlined in equation 2.3.12. The MATLAB function griddata is used
to interpolate points on an uniform 3D grid from the 3D scattered data and then draw a
surface through the interpolated points.The sample period is from 1996 to 2007. Call and
index returns have a weekly holding period.

111

Table 2.1
Average Returns of call options on the VIX index

 

This table reports the average daily buy and hold returns of call options on the VIX index.
The sample period is from February, 2006 to September, 2008. Option contracts have to
satisfy the following conditions: (1) The bid price is strictly positive and (2) The difference
between the ask and bid prices is greater than or equal to the minimum tick. K — St denotes
the difference between the options strike price and the price of the underlying VIX index
on the buying date. Option contracts are assigned to 5 strike. groups based on the value of
K — St. Strike groups are defined in intervals of 2.5.

 

Table 2.1: Call option returns on the VIX index

 

 

 

K — St -7.5 to —5 -5 to -2.5 -2.5 tot) 0 to 2.5 2.5 to 5

Group 1 2 3 4 5 3 -2 4 -2 5 -1

Mean 0.004 0.004 0.006 0.008 0.018 0.004 0.000 0.014
t-stat 0.573 0.667 0.884 0.714 1.161 1.523 0.061 1.342
median 40.002 -0.008 —0.012 -0.024 -0.024 -0.003 -0.014 -0.014
minimum -0.439 -0.500 -0.513 -0.588 -0.621 —0.181 —0.265 -0.337
maximum 0.541 0.782 1.323 3.958 6.250 0.477 0.718 1.264
Obs 325 457 586 558 510 440 409 257

 

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