Investor Beliefs and State Price Densities

in the Crude Oil Market∗

Xuhui (Nick) Pan†

Tulane University

July 25, 2012

Abstract

Standard asset pricing theory suggests that state price densities (SPDs) monoton-

ically decrease with returns. We find that the SPDs implicit in the crude oil market

display a time varying U-shape pattern. This implies that investors assign high state

prices to both negative and positive returns. We use data of the crude oil market,

where speculation and short sales are not regulated, to document how the SPDs are

dependent on investor beliefs. Investors assign higher state prices to negative returns

when there are more net short positions, higher dispersion of beliefs in the futures mar-

ket, and higher demand for out-of-the-money put options. The increase in speculation

after 2004 reinforces this effect.

JEL Classification: G12, G13

Keywords: State price density; Skewness; Investor belief; Crude oil; Speculation.

∗Correspondence to: Xuhui (Nick) Pan, A. B. Freeman School of Business, 7 McAlister Dr., New Orleans,LA 70118, USA. Tel: 1-504-314-7031. Email: xpan@tulane.edu.†I am grateful to Peter Christoffersen and Kris Jacobs for their invaluable advice and helpful comments. I

also thank Laurent Barras, Sebastien Betermier, Jan Ericsson, Aytek Malkhozov, Kenneth Singleton (CFTC2011 discussant), Feng Zhao (CICF 2010 discussant), and seminar participants at the Bank of Canada andTulane University for their comments, and I would like to thank the Fonds de Recherche sur la Société et laCulture (FQRSC) for financial support.

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1 Introduction

State price densities (SPDs) contain information about investor preferences and essentially

determine expected returns and risk premia. According to the no-arbitrage principle, we can

price any asset as long as we know the SPDs and the final payoff of this asset. Although

several studies have estimated the SPDs using equity market data,1 few papers investigate

other financial markets.2 Given that (a) commodities have emerged as a fast growing asset

class, (b) each market only contains part of the wealth of the aggregate economy and a subset

of information about the aggregate pricing kernel, and (c) participants in the commodity

market are often different from investors in other markets, a systematic study of the SPDs

implicit in this market is warranted. Using the crude oil market, which is the largest and

most liquid commodity derivatives market, this paper estimates the SPDs and backs out

investor preferences towards different states of nature.

We estimate SPDs for one-, three-, and six-month horizons, using crude oil futures and

options, and we document time variation in the SPDs and their dynamic structure. We find

that the SPDs in the crude oil market exhibit a time varying asymmetric U-shape pattern:

Investors assign high state prices to both negative and positive returns. The U-shape of the

SPDs becomes more dispersed over time as more extreme returns are realized. For instance,

investors assigned a lower value in 2006 than in 2002 for payoffs in states with a given level of

negative or positive returns. The returns on out-of-the-money (OTM) call and put options

are negative, consistent with the U-shaped SPDs.

Another strand of the literature has argued that SPDs depend on differences in investor

beliefs, and that this heterogeneity affects expected returns, the price of risk, and trading

volume of assets (e.g., Anderson, Ghysels, and Juergens, 2005; Buraschi and Jiltsov, 2006;

Beber, Buraschi, and Breedon, 2010). In particular, Bakshi, Madan, and Panayotov (2010)

show that the pattern of the SPDs implied by the index option is compatible with theory

when investors have heterogeneous beliefs and are able to short sell equities. However, there

is no direct empirical test in the literature showing how heterogeneous beliefs affect the SPDs

due to various reasons: Estimating the dynamic structure of the SPDs is econometrically

demanding; heterogeneity of investor beliefs are diffi cult to be precisely measured; and short

selling is highly regulated in most financial markets, while it is a critical element for the

theory to explain the empirical SPDs.

1An incomplete list includes Aït-Sahalia and Lo (2000), Jackwerth (2000), Rosenberg and Engle (2002),Chernov (2003), Chabi-Yo (2011), Christoffersen, Heston, and Jacobs (2011).

2Notable exceptions are Beber and Brandt (2006) and Li and Zhao (2009) who study empirical SPDs inthe fixed income market.

2

For the following reasons, the crude oil market provides an excellent laboratory to examine

the dependence of SPDs on investor beliefs. First, data on speculators’positions are available

in this market. The level of speculation can be interpreted as a measure of heterogeneous

beliefs because speculators usually bet on certain price movements, and disagreements among

investors induce speculative trading (e.g., Scheinkman and Xiong, 2003). Second, there are

no restrictions on short sales in this market. While investors can trade derivatives to hedge

risks according to their real demand and supply of crude oil, they also take any positions

simply based on their beliefs. Therefore we can test whether investor beliefs affect SPDs.

Third, the crude oil derivatives market is fast growing, and historical data are available for

more than twenty years. We have large cross-sections of derivative prices, which allow us to

accurately extract SPDs. Moreover, data on trading volumes and open interests enable us to

construct various measures of investor heterogeneous beliefs as suggested by literature such

as Kandel and Pearson (2005), and Buraschi and Jiltsov (2006). Admittedly, these measures

are indirect and imperfect.

We investigate whether investor beliefs embedded in trading activities in crude oil futures

and options impact on the slope of the SPDs, one of the fundamental characteristics of the

SPDs. The slope can be directly related to investors’ risk aversion (e.g., Rosenberg and

Engle, 2002). Since the physical densities are relatively symmetric, the slope of the SPDs is

primarily characterized by risk-neutral skewness. If risk-neutral skewness is more negative

(positive), the left (right) side of the SPDs has a steeper slope, and investors assign higher

state prices to negative (positive) returns. The slope of the SPDs, however, can be very noisy

around the distribution tails. We therefore focus on more reliable risk-neutral skewness. We

calculate risk-neutral skewness from two distinct approaches and regress it on measures of

beliefs. To highlight how the SPDs depend on market participation of both hedgers and

speculators, we run regressions in two sub-periods: from January 1990 to December 2003,

a period marked by relatively stable participation and prices, and from January 2004 to

December 2010, a period marked by extreme price movements and increased participation

by various types of investors. The selection of these sub-sample periods is based on the

evidence by Tang and Xiong (2011) and Singleton (2011), who document more speculative

activities in the commodity market after 2004.

Our empirical results indicate that the slope of the SPDs is affected by our measures

of the heterogeneity of investor beliefs. When there are more net short positions by large

traders, a higher level of belief dispersion in the futures market, and more OTM put option

demand, the risk-neutral distribution is more negatively skewed. As such, the left side of the

SPD has a steeper slope and investors assign higher state prices to negative returns. This

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finding is consistent and robust across time and maturities. Regressions for two sub-periods

reveal that more intensive speculation after 2004 reinforces this effect and is associated with

more negative risk-neutral skewness. This is consistent with Singleton (2011) who finds

that large fund flows may cause high crude oil futures prices, which implies more negative

expected returns.

We also investigate the impact of investor beliefs about the equity market on the SPDs,

which we use as a proxy for the aggregate economy. We find that it is market-specific beliefs

that affect state prices of returns in the crude oil market, and that trading activities in

crude oil futures, as well as in options, play a role in determining the risk-neutral skewness.

Although investors’expectations about the equity market significantly affect index option

prices and return distributions in the equity market (Han, 2008), bearish stock market beliefs

do not imply more negative risk-neutral skewness of crude oil futures returns, and do not have

significant impacts on the slope of the SPDs in the crude oil market. It is the participants

in the crude oil market, rather than the investors in other markets, who assign values to

certain states.

This paper is part of a growing list of recent studies that examine how the activities of

investors, both hedgers and speculators, in the commodity market affect futures prices and

returns. Büyüksahin et al. (2008) show that prices of short maturity and long maturity

crude oil futures have become cointegrated since 2004 due to increased market activities by

swap dealers, hedge funds, and other financial traders. Motivated by the coincident price rise

and increased financial participation in the crude oil market, Büyüksahin and Harris (2011)

analyze whether the crude oil price is driven by hedge funds and other speculators. Acharya,

Lochstoer, and Ramadorai (2011) find that producers’hedging demand, captured by their

default risk, predicts commodity returns. Hong and Yogo (2012) show that open interest

growth contains useful information regarding future commodity returns. They argue that

the high level of commodity market activity, measured by the high open-interest growth,

predicts high commodity returns. Etula (2010) finds that the supply of speculator capital,

captured by changes in broker-dealer balance sheets, predicts commodity returns, especially

in energy commodities. However, this paper examines how investor beliefs embedded in

trading activities affect the SPDs, which determine not only the commodity futures prices

and returns, but also the risk premia.

The rest of the paper proceeds as follows. We present the estimation of the risk-neutral

densities and SPDs, as well as risk-neutral moments in Section 2. Section 3 discusses how

the SPDs are affected by investor beliefs. Section 4 concludes.

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2 Risk-Neutral Densities and SPDs in the Crude Oil

Market

In this section, we provide a general discussion of the economic framework to obtain the SPDs

in the crude oil market based on the no-arbitrage principle. We then discuss the estimation

methodology of the risk-neutral densities and SPDs using futures and option prices. Next

we describe futures and option data and we present the estimated densities and SPDs, as

well as option implied moments.

2.1 Economic Framework

If we denote the SPD by ξ, as long as we know the final payoff pT of any asset, the price of

the asset at time t can be obtained by

pt = E[ξTpT |Ft], (1)

where Ft denotes the investors’ information set at time t. Suppose at time t we have aEuropean call option written on a futures contract Ft,T with the strike price K, which

matures at time T .3 Its price is the final payoff discounted to time t,

C(Ft,T , K, t, T ) = E[ξT (FT −K)+|Ft]

=

∫ ∞K

ξT (x)(FT (x)−K)P (FT (x)|Ft)dx, (2)

where we use P (FT (x)|Ft) to denote the conditional physical density at time t. However,note that the general SPD or pricing kernel depends on many state variables and is unknown

to investors.4

In order to price derivatives, we usually rely on the price dynamics of underlying assets

under the risk-neutral measure Q. Under this measure, options discounted at the riskless

3Crude oil options expire three business days prior to the expiration of the underlying futures contract.To simplify the notation, I do not explicitly distinguish between the futures maturity date T and the optionmaturity date T ′ in this paper.

4Since we estimate the SPDs using certain specific assets, which are only a subset of the aggregatewealth, we can only obtain the SPDs projected onto those assets. For example, the SPDs estimated fromindex options are the projection of ξ onto the index returns.

5

rate are martingales. At time t, we price the call option by

C(Ft,T , K, t, T ) = e−r(T−t)EQ[(FT −K)+|Ft]

= e−r(T−t)∫ ∞K

(FT (x)−K)PQ(FT (x)|Ft)dx, (3)

where PQ(FT (x)|Ft) is the conditional density of FT under the risk-neutral measure. Basedon this equation, we can price any option with a known final payoffonce we have PQ(FT (x)|Ft).According to Breeden and Litzenberger (1978), PQ(FT (x)|Ft) can be obtained by taking thesecond order derivative of call prices with respect to the strike price K,

PQ(FT |Ft) = er(T−t)∂2C(Ft,T , K, t, T )

∂K2|K=FT . (4)

Although it is not possible to obtain a SPD ξ that is defined over the aggregate economy,

we estimate the SPD in the crude oil market, ξ, and we focus on this specific but relatively

segmented financial market. This allows us to infer relevant information about investors’

preferences and expectations for the purpose of pricing crude oil derivatives; and how in-

vestors value certain economic states and their expectations about the probability of those

states in the crude oil market. Combining equations (2) and (3), we get the SPD in the

crude oil market,

ξ(FT |Ft) = e−r(T−t)PQ(FT |Ft)P (FT |Ft)

. (5)

Defined as the Arrow-Debreu price of per unit of probability, SPDs reflect how investors

evaluate possible states of nature and their expectations of the probability of those states

happening. While many studies estimate the SPDs using index options (i.e., ξ in the equity

market [e.g., Jackwerth, 2000]) and interest rate derivatives (i.e., ξ in the fixed income market

[e.g., Li and Zhao, 2009]), this paper investigates the SPDs in the crude oil market. SPDs

depend on two components: risk-neutral densities and physical densities. We first discuss

estimation of the risk-neutral density.

2.2 Estimation of the Risk-neutral Density

We conduct two types of empirical exercises to estimate the option implied risk-neutral den-

sity. The first exercise uses the semi-parametric approach introduced by Aït-Sahalia and Lo

(1998) and further developed by Christoffersen and Jacobs (2004). In this paper, we follow

the implementation of Christoffersen, Heston, and Jacobs (2011). The other is the nonpara-

metric approach proposed by Aït-Sahalia and Duarte (2003). Li and Zhao (2009) extend

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this methodology to a multivariate setting and show that it has satisfactory finite-sample

performances in estimating risk-neutral densities. Both methodologies give conditional es-

timates since the density is computed using option prices only on a particular day. Our

estimation spans a long period while keeping the parametric assumptions to a minimum,

compared with the existing literature in the commodity market (e.g., Melick and Thomas

[1997] estimate the risk-neutral distribution from crude oil options around the time of the

first Gulf war under restrictive lognormal assumptions).

2.2.1 Semi-parametric Approach

The semi-parametric approach is designed to utilize all available information implicit in the

entire cross-section of option prices, while keeping the parametric assumptions to a minimum.

For a given day, we first fit Black (1976) implied volatilities of the cross-sectional option data

as a second order polynomial function of strike price and maturity. Then we construct a

grid of strike prices and obtain at-the-money Black (1976) implied volatilities from the fitted

polynomial function for each maturity T − t. With these implied volatilities, we back outcall prices C (Ft, K, t, T, σ(K,T )) on the desired grid of strike prices, and then calculate the

risk-neutral density (4) for the spot price at the maturity date T. It is equal to the second

order derivative of the semi-parametric option price with respect to the strike price

PQ(FT |Ft) = er(T−t) · ∂2C (Ft,T , K, t, T, σ(K,T ))

∂K2|K=FT . (6)

Let the return of longing a futures contract maturing at T be Rt,T = log(FT/Ft,T ). It is

the same as log(ST/Ft,T ), since the futures price eventually converges to the spot price. We

obtain the density of futures return over the period of T − t as

PQ(Rt,T |Ft) =∂

∂uPr (log(FT/Ft,T ) ≤ u|Ft) =

∂

∂uPr (FT ≤ Ft,T exp (u) |Ft)

= PQ(St exp (u) |Ft) · St exp (u) , (7)

where Pr(.) denotes the cumulative distribution function.

2.2.2 Nonparametric Approach

Alternatively, we obtain the risk-neutral density of futures returns using the nonparametric

approach, which is the locally linear estimator of the second order derivative of call prices.

Following existing studies (Aït-Sahalia and Duarte, 2003, Li and Zhao, 2009), we assume

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the option price is homogeneous of degree one in the current futures price:

C(Ft,T , K, t, T ) = Ft,TC(Mt, t, T ), (8)

where Mt = K/Ft,T is the moneyness of call options written on futures contracts. Then we

estimate the risk-neutral density of returns Rt,T = log(FT/Ft,T ) ≈ FT/Ft,T as

PQ(Rt,T |Ft) = er(T−t)∂2C(Mt, t, T )

∂M2|R=FT /Ft,T . (9)

Normalizing by Ft,T has another advantage. Since futures prices and strike prices vary every

day, results based on moneyness are more comparable across time than results based on

prices.

Estimation of the second order derivative (9) using the locally linear method proceeds as

follows. First, with the arbitrage-free call option prices across all moneyness {Ci,Mi}, i =1, ...n, where C and M denote observed call price and moneyness, we estimate β0(M) and

β1(M) using a kernel regression:

minβ0,β1

n∑i=1

{Ci − β0(M)− β1(M)× (M −Mi)}2Kh(Mi −M), (10)

where Kh(Mi −M) = K ((Mi −M) /h) and K(.) is the Gaussian kernel function and h isthe bandwidth. Second, the partial derivative of the call price C with respect to moneyness

M is given by ∂β1(M)/∂M , where

β1(M) =

∑n−1i=1

∑nj=i+1(Mi −Mj)(Ci − Cj)kikj∑n−1

i=1

∑nj=i+1(Mi −Mj)2kikj

(11)

and ki = Kh(Mi −M). In the empirical estimation, the impact of the bandwidth is two-

fold: First, the size of h determines how much data are selected into the cell on which the

calculation is performed; second, h enters into the kernel function and affects the shape and

smoothness of the estimators. We choose the optimal bandwidth h∗ via numerical simulation.

The details are provided in the appendix. All results reported in the paper are based on the

optimal bandwidth h∗.

2.3 Estimation of the Physical Density and the SPD

Once we have obtained the risk-neutral density PQ(Rt,T |Ft) from option prices, the other

component needed to compute the SPD is the physical density P (Rt,T |Ft). We estimate the

8

physical density from the historical daily futures return, Rt,T = log(ST/Ft,T ). It is equivalent

to the continuously compounded returns of holding the futures contract to maturity and

realizing returns by closing out the position at the maturity date T . Estimation of the

historical distribution needs to take two factors into account. First, one needs to use a time

series of data as long as possible to increase precision. The longer the sample period is,

the more effi cient the estimator. Second, the estimation methodology needs to account for

the potential time-varying nature of physical density, especially to allow for the presence of

stochastic volatility in the crude oil market, as suggested by Trolle and Schwartz (2009). We

follow Christoffersen, Heston, and Jacobs (2011) in the estimation of the physical density.

We first calculate the time series of daily futures returns {Rt,T}Nt=1 from 1990 to 2010

using futures prices and spot prices. At each time t, we normalize the time series of returns

with its sample mean R and conditional volatility σt, which is filtered from a daily non-

linear GARCH model. This gives a time series of return innovation {zt,T}Nt=1 , defined as({Rt,T}Nt=1 − R)/σt. Then, similar to Jackwerth (2000), we estimate the density with a

Gaussian kernel using the return innovation at t. The physical density of returns is then

obtained by P (Rt,T |Ft) = P (R + σt · zt,T ).

Finally, we interpolate the physical density of returns onto the same spacing as the risk-

neutral density so that the SPD in the crude oil market is obtained by

ξ(Rt,T |Ft) = e−r(T−t)PQ(Rt,T |Ft)P (Rt,T |Ft)

. (12)

2.4 Futures and Option Data

Daily crude oil futures and option data from January 2, 1990 to December 30, 2010 are

obtained from the Chicago Mercantile Exchange (CME Group, formerly NYMEX). Crude

oil traded on the CME is the world’s largest and most liquid commodity. The range of

maturities of futures, the range of maturities and strike prices of options are also larger than

other commodity derivatives.5 An advantage of the data is that crude oil futures and options

have been traded on this exchange for more than 20 years, which allows us to study a long

time series spanning recessions and many geopolitical events such as the gulf wars, the 9/11

terrorist attacks and especially the recent financial crisis. This dataset also provides open

5Futures contracts expire on the third business day prior to the 25th calendar day (or the business dayright before it if the 25th is not a business day) of the month that precedes the delivery month. Optionwritten on futures expires three business days prior to the expiration date of futures.

9

interest and trading volume of both options and the underlying futures contracts.6 Relative

demand and trading volume in futures and options reveal investor beliefs and expectations

(e.g., Buraschi and Jiltsov, 2006), and therefore are informative when studying SPDs.

The calculation of the risk-neutral density is based on European options, but the crude

oil option data are American type, and the early exercise premium could be rather large

for options with a long maturity. We convert American option prices into European option

prices following Trolle and Schwartz (2009) who use the methodology of Barone-Adesi and

Whaley (1987). After obtaining European option prices, we exclude those observations with

Black (1976) implied volatility less than 1% or greater than 200%; we exclude those options

with prices less than $0.01 and contracts violating standard no-arbitrage constraints. The

empirical analysis is at the weekly frequency and uses OTM calls and puts. Using OTM

options is due to two motivations: First it will minimize the effect of possible approximation

errors in the early exercise premium; and second, OTM options are usually more liquid than

in-the-money (ITM) options. Each week, we use Wednesday data since it is the day least

likely to be a holiday during a week. In addition, it is also less likely to be affected by

day-of-the-week effects. This selection of data has been widely employed in the literature

(e.g., Bates, 1996, 2000; Heston and Nandi, 2000; Christoffersen, Heston, and Jacobs, 2011).

Since calculation of risk-neutral density is based on call prices, we transform OTM puts into

ITM calls. Together with observed OTM calls, call option data effectively span the entire

moneyness to apply formula (4).7

Panel A of Table 1 provides descriptive statistics for the futures data by maturity. Al-

though the number of contracts and average prices are relatively constant across maturities,

average open interest and trading volume decrease sharply beyond the two-month maturity.

Open interest of six-month futures contracts is less than 20% of one-month contracts, while

the trading volume of six-month contracts is only about 4% of one-month contracts. This

shows that long maturity futures often lack liquidity, which is also true for options as re-

ported in Panel B. Trading volume of all option contracts beyond six months is only about

5% of one-month contracts. Based on these observations, in this study we use futures data

with maturity of one, three and six months, and options written on these futures contracts.

Panel C of Table 1 reports option data by moneyness. We observe that although deep

OTM (ITM) options have large amount of contracts, at-the-money options are most heavily

traded. OTM puts have the highest open interest. Across moneyness, the average Black6Futures and option trading volume data are missing from December 15, 2006 to May 21, 2007 due to

technical reasons when the CME group converted data from the NYMEX database.7The discount factor rt,T is obtained by fitting a Nelson and Siegel (1987) curve using the 1-, 3-, 6-, 9-

and 12-month LIBOR rates and the 2-year swap rate downloaded from Bloomberg.

10

implied volatility displays a smile pattern with deep OTM options having higher implied

volatility than ITM options. Across maturities, short maturity options on average have a

higher implied volatility.

2.5 Empirical SPDs in the Crude Oil Market

2.5.1 Risk-neutral and Physical Densities

We estimate conditional risk-neutral densities on each Wednesday using both the semi-

parametric and nonparametric approach as discussed in Section 2.2. Estimation of the

conditional physical density uses historical futures returns. Figure 1 plots the weekly risk-

neutral density (black line) estimated from the semi-parametric method and physical density

(grey line) for three-month futures contracts for the firstWednesday of the years 1999 to 2010.

Estimates of one-month and six-month densities are not reported due to space constraints.

The horizontal axis denotes returns, and the sample year is indicated in the title of each

graph. The densities are scaled so that they integrate to one. Two key points are evident.

First, there are substantial differences between the two densities. While the physical density

is always relatively symmetric, the risk-neutral density can be either negatively or positively

skewed. The risk-neutral density always has fatter tails than the physical density. The

differences between the two densities are most pronounced in 2003 and 2009, when the risk-

neutral density displays excess negative skewness and excess kurtosis compared with the

physical density. Second, we can observe that volatility has clear time variations. Stochastic

volatility plays a significant role in determining the distribution, especially for the periods

of 2002-2003 and 2008-2009, with the later period exhibiting extremely high volatilities.

The time variation in the risk-neutral density is also documented in Figure 2, in which

we report the estimated risk-neutral density (black line) using the nonparametric approach

and the physical density (grey line). The dashed black lines in each graph represent the

95% confidence bounds of estimation. The main conclusions are compatible with Figure 1.

First, the risk-neutral density significantly deviates from the lognormal distribution. It can

be negatively or positively skewed and always has fat tails. Second, the shape of the SPD,

which is the ratio of the risk-neutral and physical densities, is mainly driven by the risk-

neutral density since the physical density is relatively symmetric. Lastly, one has to model

stochastic volatility to correctly price commodity derivatives. Both the risk-neutral density

and the physical density show extremely high volatility during the recent crisis. Although

here we only provide a snapshot of the conditional densities for the three-month contract

on the first Wednesday of 2009, we observe that the return distribution becomes much

11

more dispersed (meaning higher volatility) across one-, three- and six-month maturities in

2009 compared with other historical periods. Next, we check the implication of distribution

properties on the SPD.

2.5.2 SPDs

The SPD is obtained as the discounted ratio of estimated density under the two measures

as in (12). Figure 3 shows the SPDs in the crude oil market for one-, three- and six-month

maturities for the period from 1990 to 2010. Because the semi-parametric estimate of the

risk-neutral density is smoother than the nonparametric estimation, we report the empirical

SPDs estimated from the semi-parametric approach. We trim 1.5% extreme returns in the

left and right tails and return intervals of [−0.22, 0.22], [−0.5, 0.5] and [−1.0, 0.5] are reportedfor one-, three- and six-month maturities, respectively. This is because the majority of the

probability mass is concentrated in this return range.8

There are two remarkable patterns across the three time horizons. First, SPDs are

nonlinear and in general display an asymmetric U-shape as a function of returns. Although

risk-neutral densities estimated from index options have a different pattern from the crude

oil market, the U-shaped SPDs are also obtained from index options (e.g., Bakshi, Madan,

and Panayotov, 2010; Christoffersen, Heston, and Jacobs, 2011), as well as from interest

rate derivatives (Li and Zhao, 2009). At the aggregate level, investors in the crude oil

market regard the states with extremely low returns or extremely high returns as bad states

and assign a high value for a one-dollar payoff in those states. This might be due to the

heterogeneous nature of investors in the crude oil derivative market: Investors (such as long

speculators), who have net long futures positions, will bear losses in the case of futures price

decreasing if their positions are not protected. They regard negative returns as bad states

and value a one-dollar payoff in these states; investors (such as short speculators), who hold

net short futures positions, will suffer from increasing futures price and consider those states

with positive returns as bad states. They assign a higher value for a one-dollar payoff in

states with large positive returns.

Second, the U-shaped SPDs display dramatic variations over time and across maturities.

The six-month SPDs have a more pronounced U-shape after 2000 and this U-shape becomes

more dispersed over time. As the crude oil derivative market develops, more extreme returns,

especially negative returns, are realized and the skewness of the return distribution changes.

8The [1.5%, 98.5%] percentile of returns of one-, three- and six-month futures contracts in the data sampleis [−0.21, 0.19], [−0.44, 0.36] and [−0.94, 0.45], respectively.

12

Investors update their information set and compare states over a more widely spread return

region. In other words, they become more tolerant than before to the same returns and

only assign high values to payoff in states which they regard as extreme at each time t. As

such, SPDs stretch out over returns and the slope of the U-shape flattens as time evolves.

Although the SPDs of the one-month maturity in Panel A are clustered, the time variation

pattern is obvious for three- and six-month maturity SPDs in Panels B and C. For example,

investors could have assigned a lower value for payoffs when the three-month return was

−25% in 2006 than they had in 2002. This is because a much worse scenario with return

far less than −25% could have occurred in 2006. Furthermore, for a given decrease in the

three-month return (e.g., from −25% to −30%), the change of the SPD was smaller in 2006than in 2002. Investors would have required a lower payoff for exposure of the same negative

return in 2006 than in 2002. This variation is mainly driven by the differences in skewness

under the risk-neutral and physical measures.

2.5.3 OTM Option Returns Support the U-shaped SPDs

Net long investors in the futures market face the possibility of huge losses when futures and

spot prices decrease, and they are willing to pay a premium and buy OTM puts to hedge

their potential loss. Similarly, net short investors in the futures market can seek protection

by buying OTM calls. Compared with producers and inventory operators who have physical

assets as the natural hedge of the futures market, speculators have more risk exposure to

the futures market and have stronger motivation to hedge with options. This argument is

supported by futures and option positions taken by hedgers and speculators, as in Figure 5

and 6 (we will illustrate them more in Section 4). Figure 6 shows that option positions held

by speculators increase faster than those held by hedgers. Especially after 2008, speculators

hold a higher level of option positions than hedgers in order to reduce their risk exposure in

the futures market, as shown in Figure 5.

If the above statement is correct, the premia paid by investors for insurance purposes

will increase OTM option prices and cause negative expected option returns. In particular,

returns of OTM calls will be negative and will decrease with strike prices since deep OTM

calls (with large strike prices) provide protection for extremely bad states for short investors;

returns of OTM puts are negative and will increase with strike prices since deep OTM

puts (with small strike price) provide protection for extremely bad states for long position

investors. Moreover, investors are willing to pay higher premia to buy insurance from the

option market after experiencing the dramatic price movement of 2008. This implies that

the pattern of returns would be expected to be more pronounced after 2008.

13

We compute option returns as follows. Since we work with weekly option data, for the

first Wednesday of each month, we take a long position in available calls and puts with

maturity as close to 30 calendar days as possible. Hold-to-maturity returns of call and put

are calculated as

rcallt,T = max(Ft,T eRt,T −K, 0)/Ct,T − 1,

rputt,T = max(K − Ft,T eRt,T , 0)/Pt,T − 1, (13)

where Ft,T is the price of underlying futures, Rt,T = log(ST/Ft,T ) is the return of underlying

futures contract over the period T − t, K is the strike price, Ct,T and Pt,T are prices of

European style call and put options. We do not create artificial option prices by interpolation

or extrapolation. For each Wednesday, we assign available options to various bins according

to their moneyness defined as K/Ft,T , and returns are averaged within each bin. This

provides a non-overlapping return time series with various moneyness. For each moneyness,

we show in Table 2 the average return, its standard error, the 10% and 90% percentile from

January 1990 to December 2010, as well as average returns for three sub-sample periods:

January 1990 to December 2003, January 2004 to December 2010, and January 2009 to

December 2010.

For the period of 1990 to 2010, as well as for other three sub-sample periods, OTM calls

(with moneyness> 1) always have negative returns and returns generally decrease with strike

prices. This pattern becomes more pronounced during the 2009 − 2010 period, suggestinginvestors are paying higher premia to buy OTM calls after 2008. Similar observations apply

to OTM puts (with moneyness < 1). OTM puts always have negative returns; and returns

are more negative for deep OTM puts. Again, investors are paying higher premia in OTM

puts after 2008, compared with other sub-sample periods. The findings lend support to the

U-shaped SPDs and the time variation in the U-shape, especially after 2008.

One may ask if the expensiveness of OTM calls and puts is due to illiquidity premia. We

do not find supporting evidence for this. Based on our investigation of trading volume and

ratio of trading volume over open interests, OTM calls and puts are usually more liquid than

ITM calls and puts. For instance, during the period from 1990 to 2010, trading volume of

OTM calls is 591.4% higher than ITM calls; and ratio of trading volume over open interests

of OTM calls is 4.5% higher than ITM calls. The numbers for OTM puts over ITM puts are

much higher.9 Therefore, we conclude that OTM calls and puts are actively traded, and the

9Being consistent with Table 2, we define OTM calls (ITM puts) when moneyess belongs to [1.04, 1.10],and ITM calls (OTM puts) when moneyess belongs to [0.90, 0.96].

14

negative returns cannot be imputed to illiquidty.

2.6 Option Implied Risk-neutral Moments

We are not only interested in the empirical shape of SPD in a particular market, but also

how SPDs are affected by investors’heterogeneous beliefs. This is related to the economic

question of how much more investors in the aggregate are willing to pay for securities in

one state over another. Since the historical distribution of futures returns is approximately

symmetric, the shape of the SPDs is mainly driven by the properties of the risk-neutral

distribution. Therefore, we calculate risk-neutral moments and link their time variation to

investor beliefs and trading activity. Risk-neutral moments are calculated from the estimated

risk-neutral densities:

V olt,T =

{EQt

[(Rt,T − EQ

t [Rt,T ])2]}1/2

(14)

Skewt,T =

EQt

[(Rt,T − EQ

t [Rt,T ])3]

{EQt

[(Rt,T − EQ

t [Rt,T ])2]}3/2 (15)

where EQt [x] =

∫∞−∞ xP

Q(x)dx is the expected value under the risk-neutral measure and

PQ(x) is the density estimated from option prices. Since risk-neutral moments are calculated

on each Wednesday using the option implied density and available returns, they are truly

model free and conditional. Because the trading of options with maturity beyond six months

is not active, we compute moments of returns only for the one-, three- and six- month horizon.

This choice of horizons ensures a continuous and long time series of moments while allowing

us to examine the term structure of the moments.

Figure 4 displays the time series of weekly risk-neutral variance and skewness from 1990 to

2010 for one, three and six-month maturities. From left to right, the first two vertical dotted

lines denote two significant days of the first Gulf War: the Iraq invasion of Kuwait on August

2, 1990, and the liberation of Kuwait on January 17, 1991. Other vertical dotted lines denote

the September 11, 2001 terrorist attacks; the second Gulf War on March 20, 2003; the week

the WTI crude oil spot price reached the highest level in history (July 23, 2008 [$145.31]); the

week the Lehman Brothers filed for Chapter 11 bankruptcy protection (September 15, 2008);

and the week the crude oil spot price reached its lowest level during the recent crisis periods

(December 23, 2008 [$30.28]). Variance for all three maturities rises sharply on those days.

15

In addition, long maturity contracts gradually move in sync with short maturity contracts,

suggesting investors’expectations become simultaneously incorporated in all contracts. This

is consistent with the findings of Büyüksahin et al. (2008) who argue that the short maturity

futures market and the long maturity futures market become cointegrated due to increased

market activity. However, there is no clear pattern for skewness. One may observe that

from 2005 to 2010 skewness is almost always negative except on a few occasions, especially

for long maturity contracts. Investors expect a high chance of negative returns, and holding

long maturity futures contracts is risky during this period.

There are many other ways to calculate risk-neutral moments. We also extract risk-

neutral moments from option prices using the model-free methodology developed by Bakshi,

Kapadia, and Madan (2003; BKM hereafter), which are widely used in the literature. We

focus on the same three horizons as above. The numerical implementation follows Duan

and Wei (2009), which we describe in the appendix. We use this alternative measure of

risk-neutral moments in the regression analysis in section 3 as a robustness check.

In summary, the empirical results show that the risk-neutral distribution can be nega-

tively or positively skewed and always has fat tails compared with the physical distribution.

The SPDs in the crude oil market display an asymmetric U-shape as a function of returns,

and exhibit remarkable variations along time series and across maturities. Returns for OTM

options are consistent with this pattern.

3 Investor Beliefs and SPDs

The literature has shown that the SPDs depend on investor disagreements. Heterogeneity is

represented not only in asset prices and returns, but also in the relative positions and trading

volumes in equilibrium. Dependence of the SPDs on heterogeneous beliefs is present in the

equity market (e.g., Anderson, Ghysels, and Juergens, 2005; Buraschi and Jiltsov, 2006),

as well as in the foreign currency market (Beber, Buraschi, and Breedon, 2010). When

agents have different beliefs about future market movements, they engage in trading either

for speculation or hedging.

How do investor beliefs affect SPDs in the crude oil market? In this section, we investigate

how investor beliefs embedded in crude oil derivatives trading affect the slope of the SPDs,

because it is one of the fundamental characteristics of the SPDs and can be directly related

to investors’risk aversion. We further substitute the slope with risk-neutral skewness due

to the following rationale. The asymmetric U-shape pattern of the SPDs arises from the

16

differences between risk-neutral skewness and physical skewness. Because the skewness of

the risk-neutral density has more pronounced time variation compared with the physical

density, the slope of the U-shaped SPDs mostly depends on the skewness of the risk-neutral

density. Therefore, we regress the risk-neutral skewness on measures of investor beliefs. We

first describe how investors’ trading positions have evolved over the past twenty years in

crude oil futures and options. We then discuss the measures of investor beliefs in both the

crude oil futures and option markets, as well as in the equity market. Subsequently, we

discuss the regression model and results.

3.1 Market Participation and Measure of Beliefs

Generally there are two major types of players in the crude oil derivative market: commercial

traders or hedgers (who use derivatives to hedge their real supply or demand for crude

oil in business activities) and non-commercial traders or speculators (who use derivatives

mainly for financial profits without real positions in the physical assets).10 Each week the

CFTC publishes trading positions of hedgers and speculators. Figure 5 shows long and short

positions taken by hedgers and speculators in the futures market as well as their ratios, which

we obtain from the CFTC futures-only Commitments of Traders (COT) report. Although

participation in the futures market by hedgers and speculators has experienced steady growth

from 1990 onwards, the increase of positions has been faster since 2004. While both positions

of hedgers and speculators increase over time, speculators take relatively more long than short

positions. In particular, the relative long positions taken by speculators jumped to a high

level around 2004 and account for one-third of the entire market after 2009.

On the other hand, the crude oil option market has experienced even more dramatic

growth, as shown in Figure 6. We retrieve option positions taken by hedgers and speculators

from the futures-and-options combined COT reports and futures-only COT reports. Weekly

futures-and-options combined reports are available only after March 21, 1995. Hedgers’

long (short) option positions are obtained by subtracting futures positions from combined

futures-and-option positions. Speculators’long option positions are estimated as the sum of

combined option-futures long position and combined option-futures spread minus the sum of

futures long positions and futures spread. Speculators’short option positions are obtained

in a similar way. Therefore speculators’long option positions include naked long positions,

option-option spread, and futures-option spread. We observe from Figure 6 that both hedgers10Starting in 2006, the CFTC began to report positions of traders in a finer category: commercials,

managed money, commodity dealers and others. One may argue that the classification of hedgers andspeculators does not cover all investors in the crude oil market. But the fundamental distinction amongtraders as to whether they have physical attachment in the crude oil market or not still holds.

17

and speculators actively trade in the option market. Interestingly, we see that around 2008,

when the crude oil market experienced historically high volatility, speculators had larger

option positions than hedgers. This may be due to the fact that hedgers have physical assets

as the natural hedge, while speculators have more exposure to risks in the futures market

and need to protect themselves from losses using options. Overall, speculators take net long

positions, meaning they buy options most of the time. Given the active participation in

the futures and option market, we consider measures of investor beliefs in both markets and

investigate how they affect the SPDs.

Investors could take positions based on their beliefs about possible market movements.

The first measure of investor beliefs in the futures market is the net short position of total

reported large traders, which is calculated as

Net Short Positiont =(Num. of short positions)t − (Num. of long positions)t

(Num. of total positions)t. (16)

We use the column of Total Reported Positions Long (Short) in the futures-only COT reports.

Since the position data are not separated by maturities, this measure reflects investors’beliefs

about the general movement of futures prices. A positive net short position number implies

that, overall, investors take more short positions and exert higher selling pressure. Panel A

of Figure 7 displays the time series of this variable from 1990 to 2010. While the net short

position fluctuates around zero before 2004, it is almost always negative after 2004 until

2009, suggesting that, overall, investors take net long positions and are generally optimistic

during this period.

Second, we use the speculation index, which is the level of speculation activity, as a

measure of investor beliefs. Several studies, such as Scheinkman and Xiong (2003) and

Xiong and Yan (2010), argue that agents with heterogeneous beliefs engage in speculative

trading among each other. The speculation index is to gauge intensity of speculation relative

to hedging (Working, 1960; Büyüksahin and Harris, 2011). If we denote SS (SL) as the

speculative short (long) position, and HS (HL) as the hedger short (long) position, we define

Speculation Indext =1 + SSt

HLt+HStif HSt ≥ HLt,

1 + SLtHLt+HSt

if HSt < HLt.(17)

Since speculators take positions in crude oil futures by anticipating certain price movements,

this speculation index contains information on belief differences among investors.11 The11Singleton (2011) discusses how disagreements among investors induce speculative activities, price drift,

and high volatility in the crude oil market.

18

speculation index has to be interpreted together with hedging activity in the futures market.

When long and short hedging positions do not match, speculation absorbs the residual

demand. The speculation index measures the extent by which speculative positions exceed

the necessary level to offset hedging positions. For instance, a 1.15 speculation index implies

that there is 15% more speculative positions than what is needed to offset the hedging

demand. Panel B of Figure 7 plots the speculation index from 1990 to 2010. We observe

that there has been a high level of speculative activities in recent years. Before 2002, the

speculative index was around 5%; however, it has risen steadily over time to 1.20 in 2010.

It suggests that speculation in excess of hedging needs increases to 20% in the crude oil

market.12 As such, the premium that hedgers are required to pay for insurance against

futures price risk is highly affected by the active participation of speculators.

Besides positions taken by investors, actual trading volume of futures may reflect the

degree of heterogeneity and how investors speculate and share risks among each other. Lit-

erature, e.g., Kandel and Pearson (1995), has documented the positive relationship between

investor heterogeneous beliefs and volume of trade.13 Buraschi and Jiltsov (2006) show that,

the trading volume of stocks and options is positively correlated with the dispersion in be-

liefs. Therefore, we use the trading volume of futures with maturity τ as another measure

of investor belief dispersion in the underlying futures market. Panel C plots the actual fu-

tures trading volume, which was relatively stable before 2004 and significantly magnified

afterwards.

Next we discuss two measures of heterogeneous beliefs in the option market suggested

by Han (2008). One is the open interest ratio of OTM puts to calls for maturity τ , which

measures the relative demand for insurance against downside risks and reflects the hedging

needs of heterogeneous agents; the other is the trading volume of options with maturity τ ,

which is a proxy for the level of disagreement among investors. Open interests and volume do

not capture the same information since open interests are the outstanding positions investors

take, while trading volume can be due to opening or closing a contract. A high open interest

ratio of OTM puts to calls suggests more investors expect downward price movement, and

they need more puts to hedge this risk. Panel D shows that the open interest ratio of OTM

puts to calls. It tends to be high around 2004 and 2008 when the futures market experiences

dramatic price changes. Panel E shows that, the actual option trading volume starts to rise

12We do not have a disaggregated speculative index for different maturities. However, as documented byBüyüksahin and Harris (2011), from 2004 to 2008, the speculation index in general displayed a higher valuein nearby contracts relative to all contracts.13One may also argue that volume has other interpretations such as a proxy for liquidity since it measures

the ease of constructing replicating portfolios.

19

steadily from 2003 and becomes much more volatile afterwards. An inspection of Panels C

and E reveals that large belief dispersion takes place during the market decline post-2008.

In the subsequent regression analysis, we adjust the time series of futures trading volume

and option trading volume with a deterministic time trend following the literature.

To what extent do investor beliefs about the general economy affect the slope of the SPDs

in the crude oil market? We consider two proxies of beliefs to address this question: investor

beliefs about the market level and investors’expectations about the market volatility. The

first measure is the bull-bear spread based on the survey of Investors Intelligence which

has been used by Brow and Cliff (2004, 2005) and Han (2008), among others. Every week,

Investors Intelligence sends out 150 surveys to institutional investment advisors and collects

their expectations of future market movements as bullish, bearish, or neutral. The forecast

horizon is from one to three months. The bull-bear spread is then calculated as the percentage

of bullish investors minus the percentage of bearish investors, and it is often used as a proxy

for beliefs of institutional investors. The second proxy we use to measure investor beliefs

about the stock market is the CBOE VIX. The VIX has become a benchmark for measuring

investors’expectations of market volatility and is widely interpreted as a fear indicator. As

plotted in Panels G and H, the VIX is in general negatively correlated with the bull-bear

spread. The VIX tends to be higher when more investors believe the market will go down;

and the VIX tends to be lower when more investors are bullish. The VIX is also positively

correlated with implied volatility of crude oil futures, which we back out from option prices

as shown in Panel F.

3.2 Other Control Variables

We consider two sets of control variables in the regression analysis. One set includes several

variables specific to the crude oil market that determine futures returns and therefore may

affect risk-neutral skewness. We first follow Hong and Yogo (2012) and calculate the basis

for a particular maturity contract as

Basist,T = (Ft,TSt)

1T−t − 1, (18)

where Ft,T is the price of futures with maturity of T − t at time t, St is the spot price attime t. Basis is the spread between futures prices and spot prices, and it can be interpreted

as the implied net convenience yield.

Next we include option implied volatility due to two considerations. First, Trolle and

20

Schwartz (2009) present strong evidence of stochastic volatility in the crude oil market; and

stochastic volatility has to be incorporated into the model to price commodity derivatives.

Therefore, volatility could affect futures prices and returns. Second, contrary to the tra-

ditional theory of storage, which claims volatility tends to be high when the futures price

is in backwardation, Carlson, Khokher, and Titman (2007) and Kogan, Livdan, and Yaron

(2008) document that the relationship between the futures term structure slope (similar to

the basis) and volatility is non-monotone. While the sign of the slope of the futures term

structure implies investors’demand for the convenience yields, volatility captures comple-

mentary information, indicating investors’exposure to risk.

We also consider the storage level of crude oil and historical returns of futures, both

of which affect futures returns as documented in the literature (e.g., Bessembinder, 1992;

Bessembinder and Chan, 1992; de Roon, Nijman, and Veld, 2000; Erb and Harvey, 2006). We

use the storage level defined as U.S. total stocks of crude oil, excluding strategic petroleum

reserves, based on the report from Energy Information Administration (EIA). Following

Acharya, Lochstoer, and Ramadorai (2011), a Hodrick-Prescott filter is applied to remove

the trend, where the smoothing parameter is set to a number appropriate for weekly data.

The historical return is calculated as the return of holding a specific maturity futures contract

over the past month. It can be interpreted as the momentum of returns.

The other set of control variables includes the one-month treasury bill rate, the yield

spread between the 10-year treasury bond and three-month treasury bill, the yield differ-

ence between Baa and Aaa corporate bonds, and the log growth of aggregate industrial

production over the last 12 months. These variables are used by Han (2008). As proxies for

economic conditions and investors’expectations of the aggregate economy, they may influ-

ence investors’trading in the crude oil market. We obtain the one-month treasury bill rate

from the online data library of Kenneth French. The other macro variables are from the

Federal Reserve Bank of St. Louis.

Table 3 reports the summary statistics of risk-neutral skewness, and regressors and control

variables in the crude oil market. Necessary filters are applied to variables to remove time

trends, and the unit-root tests are rejected for all time series at the significant level of

1%. Note that overall skewness is negative for all three maturities; large traders on average

take a similar amount of long positions as short positions during the whole sample period;

investors always demand more OTM puts than OTM calls; and historical annualized monthly

return of holding a three-month futures contract is 6% with a sharpe ratio of 0.68, which is

slightly higher than the calculation of Gorton and Rouwenhorst (2006). Table 4 displays the

21

correlation matrix of regressors, as well as control variables, in the crude oil market for the

three-month horizon regression. The correlation matrix for the one- and six-month horizons

has a similar pattern. We observe that, except for volatility having a correlation of 0.65 with

VIX, the storage level having a correlation of 0.45 with basis, futures trading volume and

option trading volume having a correlation of 0.42, all other correlations are below 0.40 with

the majority below 0.20.

3.3 Regression Model

As discussed above, we consider three levels of measures to examine how investor beliefs

affect the risk-neutral skewness (which characterizes the slope of the SPD) in the crude oil

market: 1) belief dispersion in the crude oil futures market; 2) belief dispersion in the crude

oil option market; and 3) investor beliefs in the equity market, which we use as a proxy for the

general economy. We also consider other control variables which may affect crude oil futures

returns and reflect macroeconomic conditions. For each maturity τ , the full regression model

is given by

Risk-neutral skewnesst,τ = ατ + βτ · Beliefs in the crude oil futures markett,τ+ γτ · Beliefs in the crude oil option markett,τ+ δτ · Beliefs in the equity markett+ ητ · Risk-neutral skewnesst−1,τ+ Control variablest. (19)

3.4 Regression Results

To assess the relative importance of the investor beliefs for risk-neutral skewness, we run

the regression (19) for one-, three- and six-month maturities. Dependent variables are risk-

neutral skewness obtained from (15). We include the lagged dependent variable to control

for its positive autocorrelation. We ensure the stationarity of all variables at the 1% sig-

nificance level by detrending certain regressors. Moreover, standard errors are corrected for

heteroskedasticity and autocorrelation using Newey and West (1987), where the lag length

is chosen as the maximal lag with significant autocorrelation in the regression residuals. To

directly compare regression results with risk-neutral moments obtained from the alternative

method (BKM, 2003), we only include those days with at least two OTM calls and two OTM

puts in the regression sample. We start with the basic regression results.

22

3.4.1 Basic Results

Table 5 contains the basic regression results for the entire sample period 1990—2010 and

for two sub-sample periods 1990—2003 and 2004-2010. Results for one-, three- and six-

month contracts are displayed in Panels A, B and C, respectively. The first four columns

of each panel show results for 1990 to 2010. The results suggest that risk-neutral skewness

is negatively associated with net short positions of large traders, the dispersion of beliefs

in the futures market measured by the futures trading volume, and the OTM puts to calls

open interest ratio. One possible explanation for negative skewness when belief dispersion

is large, is that a large dispersion in beliefs usually takes place when the market declines

with investors demanding more OTM puts. Comparing the pattern of coeffi cients for the

sub-sample periods of 1990 to 2003 and 2004 to 2010, the most striking point is that the

speculation index becomes a significant factor for risk-neutral skewness in the later period.

More speculative trading implies a more negatively skewed distribution, especially for three-

and six-month horizons. The effect of speculations also magnifies after 2004 as the absolute

value of coeffi cients strongly increases. It suggests that the increasing speculative trading

after 2004 does have a significant impact on the futures market, lending support to the

argument of Singleton (2011), that more speculation after 2004 may cause the high crude

oil price, meaning higher chances of negative returns. Furthermore, for all three maturities,

skewness becomes less persistent after 2004. One possible explanation is that investors start

to update the information set and change their expectations more often than before. Lastly,

regression results show that investor beliefs about the equity market, no matter the level or

volatility, do not have a significant impact on the return distribution of crude oil futures. It

is the crude oil market specific beliefs that affect the SPDs in this market.

Two comparisons with the existing literature can be made here. First, the empirical

evidence that the OTM puts to calls ratio affecting the risk-neutral skewness aligns with

findings in the stock and index option literature (Dennis and Mayhew, 2002; Buraschi and

Jiltsov, 2006; Han, 2008). Since heterogeneous investors have different demands for OTM

options due to their various expectations of market fundamentals, pessimists can share risks

with others by buying insurance from optimists. The larger the difference in beliefs, the

higher the demand for OTM puts which drives up the prices of options with low strike

prices, and the distribution is therefore more negatively skewed. Second, Han (2008) only

uses option trading volume as a proxy for dispersion of beliefs in the regression analysis,

but we document that trading volume of both underlying futures and options are negatively

associated with risk-neutral skewness. One possibility is that investors in the crude oil market

actively engage in trading in both the futures and options market to share risk or speculate

23

among each other. This possibility is also supported by the positions of traders in Figures 5

and 6, which show that positions in both futures and options have grown tremendously over

the past few years.

3.4.2 Robustness Check

To better illustrate the dependence of risk-neutral skewness on investor beliefs, we perform

two types of robustness exercises. The first exercise is to include control variables in the

crude oil market which affect futures returns, as well as macroeconomic variables which

represent economic conditions and investment opportunities, in the regression model (19).

Table 6 shows results when adding control variables in the crude oil market, namely, the

basis, storage level, historical returns, and option implied volatility. For ease of comparison,

we only report coeffi cients of the same variables as in Table 5, i.e., βτ , γτ , δτ and ητ in (19).

The coeffi cients of the additional control variables are omitted for brevity. If the regression

results in Table 5 were biased due to omitted variables in the model specification, then the

coeffi cients in Table 6 would be very different, which is not the case. Comparing with Table

5, adding control variables does increase R2 for the one-month maturity regression. But the

main results still hold, regardless of the significance level or the magnitude of coeffi cients of

interest. Higher net short positions and higher demand for OTM puts are associated with

more negative skewness. More speculative positions after 2004 induce a higher probability

of negative returns. Therefore, this confirms that the conclusions drawn from Table 5 are

robust and consistent. However, the coeffi cients of VIX get larger and become significant

for one-month contracts when adding the control variables. One possible explanation is that

risk-neutral skewness increases with volatility14 and volatility in the one-month horizon is

highly correlated with VIX (correlation is 0.67). The coeffi cient of VIX adjusts to offset the

effects of volatility on skewness.

Table 7 displays regression results when adding the macro control variables. The main

conclusions remain the same and are consistent with Tables 5 and 6. Again, one observes

that the risk-neutral skewness is more negative when there are more net short positions by

large traders and higher OTM put open interest over OTM calls; a higher level of specu-

lative activities after 2004 causes a higher probability of negative returns. Adjusted R2 in

regressions, including macroeconomic variables, are generally lower than regressions includ-

ing control variables in the crude oil market, as shown in Table 6. This is natural since

14Han (2008) finds S&P 500 index risk-neutral skewness is less negative when volatility is higher. In manyasset pricing models, such as Heston (1993), risk-neutral skewness is an increasing function of volatility.

24

those crude oil market specific variables determine futures returns, and therefore may affect

risk-neutral skewness.

The other robustness check replaces the dependent variable by another model-free mea-

surement of risk-neutral skewness (BKM, 2003). A brief description of this methodology and

implementation is provided in the appendix. We use this measure of risk-neutral skewness

as the dependent variable in the regression analysis, and check whether the results obtained

from density-based skewness still hold. Since these two methodologies of calculating mo-

ments are very different by nature, one can not expect estimated moments to perfectly

accord with each other. For all three maturities, the two measures of risk-neutral variance

have a correlation of about 95%; while the two measures of skewness have a correlation of

around 50%.

Table 8 reports the regression results using the alternative measure of risk-neutral skew-

ness, as well as control variables in the crude oil market. All regressions have the same model

specification as the regressions in Table 6, except for a different measurement of the depen-

dent variable. The main conclusions still remain valid: Higher OTM put demand and larger

net short position imply more negatively skewed densities. A higher level of speculation after

2004 is associated with more negative skewness in the three- and six-month horizon. How-

ever, there are several differences. First, risk-neutral skewness from BKM (2003) is much

less persistent than density-based risk-neutral skewness for the period of 1990-2003. The

persistence level of the two measurements of skewness during the second period is similar.

This may be due to the fact that there are more cross-sectional option data available in the

second period, and that the risk-neutral skewness can be estimated more precisely. Second,

the coeffi cient magnitude on the OTM puts to calls ratio and the dispersion of beliefs in

the option market (i.e., option volume) are larger. In BKM (2003), moments are directly

estimated from prices of traded options, whose trading volume and open interest are used

to construct explanatory variables. Naturally, regressors can explain variations in the di-

rect measurement of skewness with a higher intensity. However, in the other case, we first

estimate densities from option prices, and skewness is calculated in the second step. The

two-step procedure may introduce noise. Third, the coeffi cient magnitude on the net short

position also rises. This may be due to the fact that risk-neutral moments estimated using

BKM (2003) are volatile. When a small position change is associated with a fluctuation

of the risk-neutral skewness, the coeffi cient will be large. We also include macroeconomic

control variables in the regression, results are similar to those in Table 7, and again the main

conclusions hold.

25

In summary, regression results indicate that investors assign higher state prices in the

crude oil market (i.e., the left side slope of the SPDs is steeper) when there are more net short

positions, a higher level of dispersion of investor beliefs, and higher OTM put demand. More

speculative positions after 2004 induce more negative risk-neutral skewness and reinforce

this effect. This implies that the relative state price change with respect to negative returns

becomes higher. In contrast, investors’bearish beliefs about the equity market do not imply

more negative risk-neutral skewness and do not affect the SPDs in the crude oil market. The

findings are consistent and robust, regardless of whether we add more control variables or

use another measurement of risk-neutral skewness.

4 Conclusion

Using more than twenty years of futures and option data, this paper backs out investor

preferences in the crude oil market. We first estimate return distributions under the physical

and risk-neutral measures; and we characterize the time variation in the SPDs for three

different maturities. Moreover, we investigate the dependence of the SPDs on investors’

heterogeneous beliefs by utilizing the informative features of the crude oil market. Comparing

with other financial markets, speculation data is available in this market and investors can

take any long or short positions based on their beliefs without being subject to regulations.

We investigate how our measures of investor beliefs affect the slope of the SPDs, which can

be directly related to investor risk aversion. We analyze the entire sample period, as well as

the period from 2004 to 2010, when the crude oil market experiences a historical boom and

bust.

We obtain four main results. First, we find that the risk-neutral densities exhibit excess

skewness and excess kurtosis compared with the physical densities; and risk-neutral densities

critically determine the marginal rates of substitution across states. Second, we find that

the SPDs in the crude oil market display a time varying U-shaped pattern. This implies

that investors with various hedging or speculative purposes regard extreme low and high

returns as bad states. We show that realized returns on OTM calls and puts are negative

and are consistent with the U-shaped SPDs. Third, we document the dependence of the

SPDs on investor beliefs. When there are more net short positions from large traders, higher

levels of belief dispersion, and higher demands for OTM puts, the left side slope of the

SPDs is steeper, and investors assign higher state prices to negative returns. This finding

is significant and robust across different maturities and under different model specifications.

Fourth, the increase in speculation after 2004 reinforces this effect and is associated with

26

more negative risk-neutral skewness.

The empirical findings in this paper suggest several extensions in the broader context of

commodity markets. First, to price commodity derivatives, a model needs to have the SPDs

that reconcile the deviation between physical and risk-neutral densities. Second, it would

be interesting to develop theoretical commodity pricing models with heterogeneous agents.

Prices are determined in an equilibrium where investors share risks and speculate based on

their beliefs about the market. Finally, our results highlight the importance of exploring the

implications of heterogeneous beliefs for the relative demand and trading volume of com-

modity futures and options. When agents have different beliefs of future market movements,

they engage in trading derivatives either for hedging or speculation.

27

Appendix

1. Selection of bandwidth in the nonparametric estimation.We choose the optimal bandwidth h∗ by minimizing the finite sample integrated MSE

of the locally linear estimation via simulation. Closely following Li and Zhao (2009), we

simulate option prices from a nonparametric call price function estimated using the actual

data. First, from the actual data we obtain the locally linear estimator of Ch0 from equation

(10), where the initial bandwidth h0 is chosen from the standard plug-in and cross validation

method. Then we resample the estimated residuals {εi = Ci − Ch0(Mi), i = 1, ...n}. Wegenerate a new set of residuals {ε∗i , i = 1, ..., n}, where the residuals ε∗i is drawn from a two-

point distribution, with probability (5 +√5)/10 to get ε∗i = εi(1 −

√5)/2 and probability

(5−√5)/10 to get ε∗i = εi(1+

√5)/2, such that Eε∗i = 0, Eε

∗2i = ε2i and Eε

∗3i = ε3i . This gives

us a simulated sample of call prices {C∗i = Ch0(Mi) + ε∗i , i = 1, ...n} to calculate the locallylinear estimator again. We simulate 100 sets of option prices to obtain the locally linear

estimators of Ch. Since our objective is to correctly estimate the second order derivative, we

choose the optimal h∗ to minimize the following integrated MSE:

minh

∫E∗[Ch(x)− Ch0(x)]2dP (x), (20)

where P (x) is the distribution of x and the expectation E∗ is calculated with the distribution

of the simulated samples. After getting the optimal h∗, the final result is obtained from the

locally linear estimation using h∗. The 95% confidence bounds in Figure 2 are based on the

results of simulated data sample {C∗i , i = 1, ...n}.

2. Calculating risk-neutral skewness using BKM (2003).As a robustness check, we also extract risk-neutral moments using the model free method-

ology developed by BKM (2003). This methodology has been widely used in literature. Based

on the theoretical results of BKM (2003), risk-neutral moments can be extracted from OTM

option prices. We define returns from holding a futures contract from t to maturing at T as

Rt,T = log(ST/Ft,T ), where ST is the spot price at T and Ft,T is the futures price. At date

t, risk-neutral skewness of futures return with maturity (T − t) is backed out as

Skewt,T =er(T−t)Wt,T − 3µt,T er(T−t)Vt,T + 2µ3t,T[

er(T−t)Vt,T − µ2t,T]3/2 ,

where

µt,T = er(T−t) − 1− er(T−t)

2Vt,T −

er(T−t)

6Wt,T −

er(T−t)

2Xt,T

28

and Vt,T ,Wt,T , and Xt,T are calculated from OTM calls an puts,

Vt,T =

∫ ∞Ft,T

2 (1− log(K/Ft,T ))K2

C(Ft,T , K, t, T )dK+

∫ Ft,T

0

2 (1 + log(K/Ft,T ))

K2P (Ft,T , K, t, T )dK

Wt,T =

∫ ∞Ft,T

6 log(K/Ft,T )− 3 (log(K/Ft,T ))2

K2C(Ft,T , K, t, T )dK

−∫ Ft,T

0

6 log(K/Ft,T ) + 3 (log(K/Ft,T ))2

K2P (Ft,T , K, t, T )dK

Xt,T =

∫ ∞Ft,T

12 (log(K/Ft,T ))2 − 4 (log(K/Ft,T ))3

K2C(Ft,T , K, t, T )dK

+

∫ Ft,T

0

12 (log(K/Ft,T ))2 + 4 (log(K/Ft,T ))

3

K2P (Ft,T , K, t, T )dK

C(Ft,T , K, t, T ) and P (Ft,T , K, t, T ) are time t price of call (put) written on contract Ft,Tand matures at T with strike price K. Following Duan and Wei (2009), we only consider

those days with at least two OTM calls and two OTM puts in our sample. We calculate the

integrals on each Wednesday from 1990 to 2010 for one, three and six-month maturity using

the trapezoidal approximation.

29

References

[1] Acharya, V., L. Lochstoer, and T. Ramadorai, 2011. Limits to Arbitrage and Hedging:

Evidence from Commodity Markets. Working Paper, New York University and NBER.

[2] Aït-Sahalia, Y., and A. Lo, 1998. Nonparametric Estimation of State-Price Densities

Implicit in Financial Asset Prices. Journal of Finance 53: 499-547.

[3] Aït-Sahalia, Y., and A. Lo, 2000. Nonparametric Risk Management and Implied Risk

Aversion. Journal of Econometrics 94: 9-51.

[4] Aït-Sahalia, Y., and J. Duarte, 2003. Nonparametric Option Pricing under Shape Re-

strictions. Journal of Econometrics 116: 9-47.

[5] Anderson, E., E. Ghysels, and J. Juergens, 2005. Do Heterogenous Beliefs Matter for

Asset Pricing? Review of Financial Studies 18: 875-924.

[6] Bakshi, G., N. Kapadia, and D., Madan, 2003. Stock Return Characteristics, Skew

Laws, and the Differential Pricing of Individual Equity Options. Review of Financial

Studies 16: 101-143.

[7] Bakshi, G., D. Madan, and G. Panayotov, 2010. Returns of Claims on the Upside and

the Viability of U-Shaped Pricing Kernels. Journal of Financial Economics 97: 130-154.

[8] Barone-Adesi, G., and R. Whaley, 1987. Effi cient Analytic Approximation of American

Option Values. Journal of Finance 42: 301-320.

[9] Bates, D. 1996. Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in

Deutsche Mark Options. Review of Financial Studies 9: 69-107.

[10] Bates, D. 2000. Post-’87 Crash Fears in the S&P 500 Futures Option Market, Journal

of Econometrics 94: 181-238.

[11] Beber, A., and M. Brandt, 2006. The effects of Macroeconomic News on Beliefs and

Preferences: Evidence from the Options Market. Journal of Monetary Economics 53:

1997-2039.

[12] Beber, A., A. Buraschi, and F. Breedon, 2010. Differences in Beliefs and Currency Risk

premiums. Journal of Financial Economics 98: 415-438.

[13] Bessembinder, H. 1992. Systematic Risk, Hedging Pressure, and Risk Premia in Futures

Markets. Review of Financial Studies 5: 637-667.

30

[14] Bessembinder, H., and K. Chan, 1992. Time Varying Risk Premia and Forecastable

Returns in Futures Markets. Journal of Financial Economics 32: 169-193.

[15] Black, F. 1976. The Pricing of Commodity Contracts. Journal of Financial Economics

3: 167-79.

[16] Breeden, D., and R. Litzenberger, 1978. Prices of State-Contingent Claims Implicit in

Option Prices. Journal of Business 51: 621-651.

[17] Brown, G., and M. Cliff, 2004. Investor Sentiment and the Near-Term Stock Market.

Journal of Empirical Finance 11: 1-27.

[18] Brown, G., and M. Cliff, 2005. Investor Sentiment and Asset Valuation. Journal of

Business 78: 405-440.

[19] Buraschi, A., and A. Jiltsov, 2006. Model Uncertainty and Option Markets with Het-

erogeneous Agents. Journal of Finance 61: 2841-2897.

[20] Büyüksahin, B., M. Haigh, J. Harris, J. Overdahl, and M. Robe, 2008. Fundamentals,

Trader Activity and Derivative Pricing, Working Paper, Commodity Futures Trading

Commission (CFTC) and Securities and Exchange Commission (SEC).

[21] Büyüksahin, B., and J. Harris, 2011. Do Speculators Drive Crude Oil Futures Prices?

The Energy Journal 32: 167-202.

[22] Carlson, M., Z., Khoker, and S. Titman, 2006. Equilibrium Exhaustible Resource Price

Dynamics. Journal of Finance 42: 1663-1703.

[23] Chabi-Yo, F., 2011. Pricing Kernels with Stochastic Skewness and Volatility Risk.Man-

agement Science, forthcoming.

[24] Chernov, M., 2003. Empirical Reverse Engineering of the Pricing Kernel, Journal of

Econometrics 116: 329-364.

[25] Christoffersen, P., and K. Jacobs, 2004. Which GARCH Model for Option Valuation?

Management Science 50: 1204-1221.

[26] Christoffersen, P., S. Heston, and K. Jacobs, 2011. A GARCH Option Model with

Variance-Dependent Pricing Kernel. Working Paper, University of Toronto.

[27] Dennis, P., and S. Mayhew, 2002. Risk-Neutral Skewness: Evidence from Stock Options.

Journal of Financial and Quantitative Analysis 37: 471-493.

31

[28] De Roon, F., T. Nijman, and C. Veld, 2000. Hedging Pressure Effects in Futures Mar-

kets. Journal of Finance 55: 1437-1456.

[29] Duan, J.-C., and J. Wei, 2009. Systematic Ris and the Price Structure of Individual

Equity Options. Review of Financial Studies 22: 1981-2006.

[30] Etula, E. 2010. Broker-Dealer Risk Appetite and Commodity Returns. Working Paper,

Federal Reserve Bank of New York.

[31] Gorton, G., and K. Rouwenhorst, 2006. Facts and Fantasies about Commodities Futures.

Financial Analysts Journal 62: 47-68.

[32] Han, B. 2008. Investor Sentiment and Option Prices. Review of Financial Studies 21:

387-414.

[33] Heston, S. 1993. A Closed-Form Solution for Options with Stochastic Volatility with

Applications to Bond and Currency Options, Review of Financial Studies 6: 327-343.

[34] Hong, H., and M. Yogo, 2012. What Does Futures Market Interest Tell Us about the

Macroeconomy and Asset Prices? Journal of Financial Economics, forthcoming.

[35] Jackwerth, J. 2000. Recovering Risk Aversion from Option Prices and Realized Returns.

Review of Financial Studies 13: 433-451.

[36] Kandel, E., and N. Pearson, 1995. Differential Interpretation of Public Signals and Trade

in Speculative Markets, Journal of Political Economy 103: 831-872.

[37] Kogan, L., D. Livdan, and A. Yaron, 2008. Oil Futures Prices in a Production Economy

with Investment Constraints. Journal of Finance 64: 1345-1375.

[38] Li, H., and F. Zhao, 2009. Nonparametric Estimation of State-Price Densities Implicit

in Interest Rate Cap Prices. Review of Financial Studies 22: 4335-4376.

[39] Melick, W., and C. Thomas, 1997. Recovering an Asset’s Implied SPD from Option

Prices: an Application to Crude Oil During the Gulf Crisis. Journal of Financial and

Quantitative Analysis 32: 91-115.

[40] Nelson, C., and A. Siegel, 1987. Parsimonious Modeling of Yield Curves. Journal of

Business 60: 473-489.

[41] Newey, W., and K. West, 1987. A Simple Positive Semi-Definite, Heteroscedasticity and

Autocorrelation Consistent Covariance Matrix. Econometrica 55: 703-708.

32

[42] Rosenberg, J., and R. Engle, 2002. Empirical Pricing Kernels. Journal of Financial

Economics 64: 341-372.

[43] Scheinkman, J., and W. Xiong, 2003. Overconfidence and Speculative Bubbles. Journal

of Political Economy 111: 1183-1219

[44] Singleton, K. 2011. Investor Flows and the 2008 Boom/Bust in Oil Prices, Working

Paper, Stanford University.

[45] Tang, K., andW. Xiong, 2011. Index Investing and the Financialization of Commodities.

Working Paper, Princeton University.

[46] Trolle, A., and E. Schwartz, 2009. Unspanned Stochastic Volatility and the Pricing of

Commodity Derivatives. Review of Financial Studies 22: 4423-4461.

[47] Working, H. 1960. Speculation on Hedging Markets. Stanford University Food Research

Institute Studies 1: 185-220.

[48] Xiong, W., and H. Yan, 2010. Heterogeneous Expectations and Bond Markets. Review

of Financial Studies 23: 1433-1466.

33

34

Figure 1: Three-Month Risk-Neutral Semi-Parametric Densities and Physical Densities

Notes: This figure shows risk-neutral densities estimated using the semi-parametric method (Ait-Sahalia and Lo,

1998, 2000) and physical densities of the three-month futures returns on the first Wednesday of each year from

1999 to 2010. The horizontal axis is futures returns defined as log(ST/Ft,T). The risk-neutral density is obtained

using option prices. On each Wednesday, we construct a surface of option prices as a function of maturity and

moneyness, and the density is then estimated using Breeden and Litzenberger (1978). For physical densities, we

use realized return innovations on the three-month futures contracts from 1990 to 2010. On each Wednesday,

we scale the time series with the conditional volatility from a non-linear GARCH model and then estimate

physical densities.

35

Figure 2: Three-Month Risk-Neutral Non-Parametric Densities and Physical Densities

Notes: This figure shows risk-neutral densities estimated using the non-parametric method (Ait-Sahalia and

Duart, 2003, Li and Zhao, 2009) and physical densities of three-month futures returns on the first Wednesday

of each year from 1999 to 2010. The horizontal axis is futures returns defined as log(ST/Ft,T). The risk-neutral

density is obtained using option prices. The dotted lines represent 95% confidence bounds of risk-neutral

densities. For physical densities, we use realized return innovations on the three-month futures contracts from

1990 to 2010. On each Wednesday, we scale the time series with the conditional volatility from a non-linear

GARCH model and then estimate physical densities.

36

Figure 3: State Price Densities in the Crude Oil Market

Notes: This figure shows weekly estimates of the state price densities (SPDs) in the crude oil market at the one-, three- and six-month horizons.

SPDs are estimated on each Wednesday from 1990 to 2010. Return is defined as log(ST/Ft,T), and we trim 1.5% extreme returns in the left and

right tails. The [1.5% 98.5%] percentile of returns of one-, three- and six-month futures contracts are [-0.21 0.19], [-0.44 0.36] and [-0.94 0.45],

respectively.

37

Figure 4: Option Implied Variance and Skewness

Notes: This figure shows option-implied variance and skewness of one-, three- and six-month maturity

crude oil futures returns from January 1990 to December 2010. We first obtain risk-neutral densities on

each Wednesday using the semi-parametric method. Variance and skewness are then calculated based on

estimated densities. From left to right, the first two vertical dotted lines denote two significant weeks of

the first Gulf War: the Iraq invasion of Kuwait on August 2, 1990, and the liberation of Kuwait on

January 17, 1991. Other vertical dotted lines denote the September 11, 2001 terrorist attacks; the second

Gulf War on March 20, 2003; the week the WTI crude oil spot price reached the highest level in history

(July 23, 2008 [$145.31]); the week the Lehman Brothers filed for Chapter 11 bankruptcy protection

(September 15, 2008); and the week the WTI crude oil spot price reached its lowest level during the

recent crisis periods (December 23, 2008 [$30.28]), respectively.

38

Figure 5: Futures Positions Taken by Hedgers and Speculators

Notes: This figure shows futures positions taken by hedgers and speculators from 1990 to 2010.

Positions of hedgers and speculators are from the U.S. Commodity Futures Trading Commission

(CFTC) futures-only Commitments of Traders (COT) report. As defined in the report, hedgers

(commercials) are those investors who have direct exposure to the underlying crude oil commodity

and use futures and options for hedging purposes; and speculators (non-commercials) are those

investors who are not directly engaged in crude oil business activities and use derivatives markets for

the purpose of financial profits. Before September 30, 1992, the CFTC publishes reports twice every

month; then reports are available every week. Panel A (B) displays long (short) futures positions taken

by hedgers and speculators, as well as their ratios.

39

Figure 6: Option Positions Taken by Hedgers and Speculators

Notes: This figure shows option positions taken by hedgers and speculators from 1995 to 2010. Option

positions are constructed from the CFTC futures-only and futures-and-options combined reports.

Hedgers (speculators) are commercials (non-commercials) as defined in the CFTC report. Hedgers’ long

(short) option positions are obtained by subtracting futures positions from combined futures-and-option

positions. Speculators’ long option positions are estimated by (combined futures-and-option long +

combined futures-and-option spread) - (futures long positions + futures spread). Speculators’ short

option positions are obtained in a similar way. Note that weekly futures-and-options combined reports

are available only after March 21, 1995. Panel A (B) displays long (short) positions taken by hedgers and

speculators, as well as their ratios.

40

Figure 7: Belief Measures in the Crude Oil and Equity Markets

Notes: This figure shows measures of investor beliefs in the crude oil futures and option, and the equity

markets. Net short position is defined as short positions minus long positions divided by total positions of

large traders, including hedgers and speculators. Speculation index measures the extent by which speculative

positions exceed the necessary level to offset hedging positions. Futures volume is based on the three-month

futures contracts. OTM open interest ratio is open interest of OTM puts divided by open interest of OTM

calls. Volatility is calculated from the densities implied by option prices. OTM open interest ratio, option

volume and volatility are based on options written on the three-month futures contracts. From Dec 15, 2006

to May 21, 2007, futures and option trading volume data are missing because of technical reasons when the

CME group converted data from the NYMEX database. Bull spread is defined as the percentage of bullish

investors minus the percentage of bearish investors based on the survey of Investors Intelligence. VIX is the

CBOE’s Volatility Index.

41

Table 1: Crude Oil Futures and Options Data

Panel A: Futures Data by Maturity (months)

1 2 3 6 12 24 All

Number of Contracts 5254 5273 5275 5276 5276 4196 30550

Average Price 37.676 37.754 37.777 37.678 37.389 41.330 38.160

Average Open Interest (OI) 133304 111450 56653 26024 13030 5290 59415

Average Volume 97272 56798 19228 3691 964 269 30693

Average Volume/OI Ratio 0.73 0.51 0.34 0.14 0.07 0.05 0.52

Panel B: Option Data by Maturity (days)

(0,30] (30,60] (60,90] (90,120] (120,180] (180,∞) All

Number of Contracts 50467 60332 61114 55668 84049 308343 619973

Average IV 0.509 0.450 0.419 0.395 0.370 0.311 0.367

Average Price 5.275 5.189 5.351 5.782 6.024 10.571 8.055

Average Open Interest (OI) 3577 2997 2482 2216 1914 1542 2053

Average Volume 349 186 99 66 45 20 78

Average Volume/OI Ratio 0.10 0.06 0.04 0.03 0.02 0.01 0.04

Panel C: Option Data by Moneyness (K/F)

(0,0.94] (0.94,0.96] (0.96,0.98] (0.98,1.02] (1.02,1.04] (1.04,1.06] (1.06,∞) All

Number of Contracts 223154 25563 27412 54858 22969 20115 245902 619973

Average IV 0.365 0.317 0.313 0.312 0.318 0.325 0.399 0.367

Average Price 5.635 5.796 5.688 5.683 5.557 5.614 11.713 8.055

Average Open Interest (OI) 2337 2084 2040 1966 2002 1972 1824 2053

Average Volume 65 105 130 186 131 112 49 78

Average Volume/OI Ratio 0.03 0.05 0.06 0.09 0.07 0.06 0.03 0.04

Notes: This table reports descriptive statistics of crude oil futures and option data from January 2, 1990 to December 30, 2010. Panel A

presents daily futures contracts across maturities. Panel B and C display option data across maturities and moneyness on each Wednesday,

where moneyness is defined as the strike price K divided by the underlying futures price F.

42

Table 2: Average Returns of Crude Oil Options

Panel A: Call Returns (%)

Moneyness (K/F) 0.90 0.92 0.94 0.96 1.04 1.06 1.08 1.10

01/1990 to 12/2010 7.844 14.506 1.492 10.476 -2.813 -27.328 -19.245 -36.304

Std Err 77.724 88.242 95.974 114.300 180.222 190.391 271.436 258.402

10% Percentile -100 -100 -100 -100 -100 -100 -100 -100

90% Percentile 121.309 138.560 130.472 167.404 218.232 171.580 153.900 -25.119

Subsamples

01/1990 to 12/2003 18.123 18.488 -9.351 7.011 -0.111 -36.117 -12.705 -4.222

01/2004 to 12/2010 3.347 12.747 8.314 12.461 -4.014 -21.814 -22.796 -51.535

01/2009 to 12/2010 -1.415 2.326 -0.595 3.880 -20.808 -31.054 -35.109 -87.721

Panel B: Put Returns (%)

Moneyness (K/F) 0.90 0.92 0.94 0.96 1.04 1.06 1.08 1.10

01/1990 to 12/2010 -45.686 -52.721 -24.198 -17.994 -35.151 -24.960 -18.904 -19.566

Std Err 251.008 159.352 204.530 180.884 99.355 89.806 82.085 78.130

10% Percentile -100 -100 -100 -100 -100 -100 -100 -100

90% Percentile -83.385 46.909 122.464 168.954 99.593 87.654 86.673 67.362

Subsamples

01/1990 to 12/2003 -25.890 -62.975 -19.497 -14.355 -36.614 -29.773 -21.719 -31.998

01/2004 to 12/2010 -53.265 -47.930 -26.636 -21.483 -33.624 -17.973 -17.877 -11.955

01/2009 to 12/2010 -95.980 -82.378 -73.467 -59.382 -60.663 -39.331 -29.469 -42.707

Notes: This table reports the average returns of options, with the maturity as close to 30 calendar days as

possible. Moneyness is defined as the strike price K divided by the futures price F. OTM calls (puts) denote

those calls (puts) with moneyness > (<) 1. On the first Wednesday of each month, we calculate expected

hold-to-maturity returns of available options, with the maturity as close to 30 calendar days as possible.

Without interpolation or extrapolation, we assign available options to various bins according to their

moneyness, and returns are averaged within each bin. For each moneyness, we show the average return, its

standard error, the 10% and 90% percentile from January 1990 to December 2010, as well as average returns

for three subsample periods: January 1990 to December 2003, January 2004 to December 2010, and January

2009 to December 2010. Returns of call and put are calculated as

⁄ and

⁄ ,

where is the price of underlying futures, ⁄ is the return of the underlying futures contract

over the period , is the strike price, and are prices of European style call and put options.

43

Table 3: Summary Statistics

Variable Mean Std Dev Percentile Augmented Dickey-

Fuller (test-stat) 25% 50% 75%

Skewness (1Mon) -0.135 0.261 -0.285 -0.171 -0.037 -11.557

Skewness (3Mon) -0.078 0.463 -0.380 -0.198 0.116 -9.091

Skewness (6Mon) -0.085 0.523 -0.434 -0.215 0.102 -7.758

Total Net Short -0.002 0.017 -0.010 -0.001 0.008 -8.742

Speculation Index 1.062 0.044 1.027 1.043 1.097 -3.426

Speculation Index (Adjusted) 0.000 0.016 -0.011 0.000 0.011 -9.826

Futures Volume 21186 19552 8910 14803 26083 -12.049

Futures Volume (Adjusted) 0.002 0.055 -0.028 -0.008 0.018 -17.201

OMT Put/Call 1.557 1.486 0.670 1.141 1.859 -11.563

Option Volume 5852 6272 2103 4000 7306 -17.985

Option Volume (Adjusted) 0.001 0.003 -0.001 0.001 0.003 -21.603

Bull-Bear Spread -0.069 0.335 -0.274 -0.090 0.148 -5.244

VIX 0.135 0.155 0.038 0.152 0.247 -5.280

Basis -0.032 0.214 -0.145 -0.037 0.058 -10.136

Storage Level 322230 24627 303883 324203 339895 -2.724

Storage Level (Adjusted) 0.007 0.145 -0.080 -0.010 0.062 -4.018

Historical Return (1Mon) 0.005 0.103 -0.058 0.011 0.070 -20.433

Historical Return (3Mon) 0.005 0.088 -0.044 0.012 0.058 -18.726

Historical Return (6Mon) 0.008 0.074 -0.032 0.012 0.050 -30.822

Volatility 0.163 0.057 0.117 0.162 0.192 -3.719

Notes: This table reports the summary statistics of variables used in the regression analysis (19). Variables are

measured as closely as possible to each Wednesday from January 1990 to December 2010. Skewness of one-,

three- and six-month futures returns are calculated from the risk-neutral densities implied by option prices. Total

Net Short is the relative net short futures positions taken by all reported large traders (including commercials and

non-commercials). Speculation Index measures the extent by which speculative positions exceed the necessary

level to offset hedging positions, and a Hodrick-Prescott filter is applied to remove the trend. Both variables are

based on the CFTC futures-only COT report. OTM Put/Call is the ratio of total open interest for OTM puts over

OTM calls. Futures Volume and Option Volume are the total trading volume of crude oil futures and options

with a specific maturity, and they are adjusted for a deterministic time trend. Bull-bear Spread is the proportion

of bullish investors minus bearish investors based on the survey conducted by Investors Intelligence. The VIX is

the CBOE’s Volatility Index. Basis is the discounted spread between futures and spot prices. Storage Level is

U.S. total stocks of crude oil, excluding SPR, based on the report from Energy Information Administration (EIA),

and a Hodrick-Prescott filter is applied to remove the trend. Historical Return is the return of holding a long

position of a specific maturity futures contract over one month. Volatility is the volatility of futures returns

implied by option prices. Although Futures Volume, OTM Put/Call, Option Volume, Basis and Volatility are

calculated for each maturity, we only report those variables for the three-month maturity to save space.

44

Table 4: Correlation Matrix

Skew

Net

Short

Spec

Index

Fut

Vol

Put/

Call

Opt

Vol

Bull-

Bear VIX Basis

Stor

Level

Hist

Ret

Total Net Short -0.131 .

Speculation Index 0.039 -0.071 .

Futures Volume -0.113 0.043 0.030 .

OMT Put/Call -0.325 0.021 0.079 0.046 .

Option Volume -0.068 -0.019 0.001 0.415 0.035 .

Bull-Bear Spread 0.036 0.010 -0.120 -0.186 0.184 -0.138 .

VIX -0.035 0.097 0.122 0.078 -0.047 0.007 -0.253 .

Basis 0.126 -0.147 0.073 0.180 -0.193 0.074 -0.025 0.193 .

Storage Level 0.024 0.013 -0.167 0.026 -0.191 -0.045 -0.020 -0.058 0.453 .

Historical Return 0.069 0.037 -0.027 -0.062 -0.009 -0.027 -0.001 -0.015 0.097 0.049 .

Volatility 0.043 0.112 0.087 -0.015 0.073 -0.024 0.082 0.651 0.212 -0.114 0.075

Notes: This table reports the correlation matrix of variables used in the regression (19) for the three-month maturity.

Variables are measured as closely as possible to each Wednesday from January 1990 to December 2010. Skewness of

three-month futures returns is calculated from the risk-neutral densities implied by option prices. Total Net Short is

the relative net short futures positions taken by all reported large traders (including commercials and non-

commercials). Speculation Index measures the extent by which speculative positions exceed the necessary level to

offset hedging positions, and a Hodrick-Prescott filter is applied to remove the trend. Both variables are based on the

futures-only traders report published by the CFTC. OTM Put/Call is the ratio of total open interest of OTM puts over

OTM calls. Futures Volume and Option Volume are the total trading volume of crude oil futures and options and they

are adjusted for a deterministic time trend. Bull-bear Spread is the proportion of bullish investors minus bearish

investors based on the survey conducted by Investors Intelligence. VIX is the CBOE’s Volatility Index. Basis is the

discounted spread between futures and spot prices. Storage Level is U.S. total stocks of crude oil, excluding SPR,

based on the report from Energy Information Administration (EIA), and a Hodrick-Prescott filter is applied to remove

the trend. Historical Return is the return of holding a specific maturity futures contract over one month. Volatility is

the volatility of futures returns implied by option prices. To save space, we do not report correlation matrixes for the

one-month and six-month regressions, which have a similar pattern.

45

Table 5: Investor Beliefs and Risk-Neutral Skewness

Panel A: One-Month

1990-2010 1990-2003 2004-2010

Net Short Position -0.951** -0.408 -0.898** -0.469 -2.535 -2.474

(-2.28)

(-0.98) (-2.19)

(-1.03) (-1.60)

(-1.40)

Speculation Index -0.066

0.468 0.522

0.847 -1.191*

-0.085

(-0.16)

(0.94) (0.70)

(0.98) (-1.78)

(-0.15)

Future Volume -0.007

-0.005 -0.013

-0.005 -0.013

-0.013

(-0.70)

(-0.51) (-0.42)

(-0.15) (-1.20)

(-1.20)

OTM Put/Call

-0.054***

-0.056***

-0.041***

-0.040**

-0.092***

-0.102***

(-3.79)

(-3.44)

(-2.76)

(-2.51)

(-5.21)

(-5.04)

Option Volume

-0.011

0.00827

-0.069

-0.097

-0.036

0.032

(-0.17)

(0.13)

(-0.40)

(-0.49)

(-0.54)

(0.54)

Bull-Bear Spread

0.018 0.054

-0.052 -0.043

0.056 0.130

(0.41) (1.14)

(-0.81) (-0.68)

(0.78) (1.45)

VIX

-0.000 -0.001

0.000 -0.000

0.000 -0.000

(-0.37) (-1.08)

(0.17) (-0.12)

(0.05) (-0.44)

Lagged Skew 0.720*** 0.671*** 0.725**

* 0.660*** 0.784*** 0.732*** 0.796*** 0.733*** 0.495*** 0.507*** 0.534*** 0.479***

(17.08) (15.34) (16.84) (13.59) (14.93) (12.29) (14.95) (10.84) (5.78) (7.21) (6.32) (6.60)

Observations 840 840 840 840 530 530 530 530 310 310 310 310

Adjusted R2 53.2% 57.1% 52.9% 57.2% 62.6% 64.8% 62.3% 64.8% 29.7% 38.8% 28.4% 40.4%

46

Panel B: Three-Month

1990-2010 1990-2003 2004-2010

Net Short Position -0.883** -0.934*** -0.898*** -0.848** -2.543 -3.104

(-2.53)

(-2.66) (-2.64)

(-2.40) (-1.16)

(-1.30)

Speculation Index -0.093

0.040 0.570

0.757 -1.905*

-2.351**

(-0.23)

(0.10) (1.07)

(1.39) (-1.88)

(-2.02)

Future Volume -0.055

-0.017 -0.234**

-0.205* 0.028

0.014

(-1.54)

(-0.43) (-2.42)

(-1.68) (0.59)

(0.28)

OTM Put/Call

-0.017*

-0.017*

-0.033*

-0.034**

-0.003

-0.000

(-1.68)

(-1.66)

(-1.93)

(-1.98)

(-0.29)

(-0.02)

Option Volume

-0.237**

-0.222*

-0.454**

-0.239

0.026

-0.023

(-2.08)

(-1.70)

(-2.12)

(-0.99)

(0.20)

(-0.15)

Bull-Bear Spread

-0.014 0.008

-0.052 -0.080

-0.002 -0.071

(-0.33) (0.18)

(-0.90) (-1.37)

(-0.01) (-0.48)

VIX

0.000 0.000

0.001 0.001

0.001 0.001

(0.05) (0.25)

(0.98) (1.30)

(0.34) (0.61)

Lagged Skew 0.852*** 0.839*** 0.857*** 0.834*** 0.890*** 0.867*** 0.907*** 0.860*** 0.568*** 0.592*** 0.592*** 0.555***

(38.62) (33.52) (38.65) (33.32) (41.58) (30.44) (43.33) (31.07) (7.73) (8.12) (8.09) (7.04)

Observations 1058 1058 1058 1058 717 717 717 717 341 341 341 341

Adjusted R2 73.9% 74.1% 73.8% 74.1% 82.1% 82.3% 81.9% 82.5% 35.9% 35.0% 34.9% 35.6%

47

Notes: This table presents the results of the weekly regression that examines the dependence of risk-neutral skewness on investor beliefs in the crude oil futures, crude

oil options, and equity markets. Panel A contains the results for one-month maturity, while Panels B and C display the results for three- and six-month maturities. The

dependent variables are skewness based on the risk-neutral densities estimated from option prices. For each maturity, we report the regression results from the entire

data period: 1990 to 2010; and two sub-sample periods: 1990 to 2003, and 2004 to 2010. Standard errors are adjusted for heteroskedasticity and serial correlation

according to Newey and West (1987); and t-statistics are reported in parentheses below the coefficients. ***, ** and * denote the significant level of 1%, 5% and 10%,

respectively.

Panel C: Six-Month

1990-2010 1990-2003 2004-2010

Net Short Position -0.943** -1.066** -1.022** -1.076** -0.232 -1.898

(-2.07)

(-2.27) (-2.34)

(-2.39) (-0.11)

(-0.76)

Speculation Index -0.212

-0.178 0.361

0.303 -1.316

-1.905

(-0.51)

(-0.42) (0.64)

(0.52) (-1.28)

(-1.63)

Future Volume -0.002

-0.119 -0.147

-0.414 0.006

-0.010

(-0.02)

(-1.20) (-0.45)

(-1.13) (0.07)

(-1.00)

OTM Put/Call

-0.004*

-0.004*

-0.010

-0.012*

-0.002

-0.001

(-1.94)

(-1.95)

(-1.46)

(-1.67)

(-0.95)

(-0.45)

Option Volume

0.089

0.185

0.280

0.628

0.070

0.097

(0.55)

(0.97)

(0.35)

(0.72)

(0.44)

(0.54)

Bull-Bear Spread

-0.048 -0.032

-0.086 -0.071

0.053 -0.001

(-0.99) (-0.67)

(-1.30) (-1.04)

(0.45) (-0.01)

VIX

0.000 0.001

0.001 0.002

0.003 0.003*

(0.35) (0.74)

(0.89) (1.56)

(1.59) (1.71)

Lagged Skew 0.880*** 0.875*** 0.881*** 0.872*** 0.912*** 0.905*** 0.920*** 0.901*** 0.657*** 0.664*** 0.649*** 0.626***

(44.43) (42.52) (43.20) (43.30) (53.69) (46.39) (54.70) (46.22) (10.42) (10.83) (9.58) (8.61)

Observations 954 954 954 954 614 614 614 614 340 340 340 340

Adjusted R2 77.8% 77.9% 77.8% 77.9% 84.2% 84.3% 84.2% 84.4% 44.2% 44.0% 44.4% 44.5%

48

Table 6: Investor Beliefs and Risk-Neutral Skewness

(Robustness to Control Variables in the Crude Oil Market)

One-Month Three-Month Six-Month

1990-2010 1990-2003 2004-2010

1990-2010 1990-2003 2004-2010

1990-2010 1990-2003 2004-2010

Net Short Position -0.614 -0.492 -1.308

-0.789** -0.765* -3.269

-0.844* -0.634 -2.286

(-1.46) (-1.01) (-0.68)

(-2.04) (-1.92) (-1.24)

(-1.66) (-1.12) (-0.79)

Speculation Index 0.461 1.281 -0.422

-0.078 0.976 -2.728***

-0.326 0.452 -2.428**

(1.02) (1.44) (-0.79)

(-0.18) (1.64) (-2.70)

(-0.68) (0.69) (-2.58)

Future Volume -0.002 0.004 -0.028**

-0.011 -0.199* 0.014

-0.118 -0.198 -0.139

(-0.22) (0.13) (-2.49)

(-0.26) (-1.65) (0.27)

(-1.06) (-0.52) (-1.20)

OTM Put/Call -0.062*** -0.039** -0.119***

-0.016 -0.033* -0.006

-0.004* -0.0105 -0.00252

(-3.61) (-2.35) (-5.96)

(-1.37) (-1.67) (-0.35)

(-1.75) (-1.29) (-1.01)

Option Volume 0.046 -0.141 0.109*

-0.249* -0.231 -0.047

0.174 0.782 0.198

(0.81) (-0.81) (1.92)

(-1.93) (-0.96) (-0.29)

(0.82) (0.83) (0.94)

Bull-Bear Spread 0.024 -0.050 0.040

-0.016 -0.127** -0.113

-0.030 -0.156* -0.0401

(0.61) (-0.84) (0.54)

(-0.36) (-2.17) (-0.85)

(-0.57) (-1.94) (-0.32)

VIX -0.004*** -0.004* -0.003**

-0.001 -0.002 -0.001

0.000 -0.001 -0.000

(-3.14) (-1.93) (-1.97)

(-1.21) (-0.92) (-0.43)

(0.04) (-0.62) (-0.02)

Lagged Skew 0.644*** 0.704*** 0.460***

0.833*** 0.850*** 0.531***

0.870*** 0.877*** 0.589***

(10.42) (7.77) (6.09)

(32.83) (28.81) (6.83)

(43.30) (34.83) (8.14)

Observations 838 528 310 1055 714 341 953 613 340

Adjusted R2 58.7% 66.2% 45.0% 74.1% 82.7% 36.0% 77.8% 84.6% 45.4%

Notes: This table reports the results of the weekly regression that examines the robustness of the relation between investor beliefs and risk-neutral

skewness for one-, three- and six-month maturities. Control variables in the crude oil market are the basis, storage level, historical returns, and option

implied volatility. The dependent variables are skewness based on the risk-neutral densities estimated from option prices. For each maturity, we report the

regression results from the entire data period: 1990 to 2010; and two sub-sample periods: 1990 to 2003, and 2004 to 2010. Standard errors are adjusted

for heteroskedasticity and serial correlation according to Newey and West (1987); and t-statistics are reported in parentheses below the coefficients. ***,

** and * denote the significant level of 1%, 5% and 10%, respectively.

49

Table 7: Investor Beliefs and Risk-Neutral Skewness

(Robustness to Control Variables in the Macroeconomy)

One-Month Three-Month Six-Month

1990-2010 1990-2003 2004-2010

1990-2010 1990-2003 2004-2010

1990-2010 1990-2003 2004-2010

Net Short Position -0.515 -0.544 -4.860**

-1.055*** -0.929** -3.980

-1.229** -1.210** -3.953

(-1.02) (-1.13) (-2.43)

(-2.83) (-2.49) (-1.37)

(-2.45) (-2.57) (-1.18)

Speculation Index 0.267 0.723 -0.625

-0.114 0.759 -2.714**

-0.276 0.125 -2.312**

(0.50) (0.76) (-1.13)

(-0.26) (1.38) (-2.43)

(-0.60) (0.21) (-2.12)

Future Volume -0.003 0.017 -0.028*

0.005 -0.190 -0.009

-0.065 -0.302 -0.166

(-0.27) (0.48) (-1.82)

(0.12) (-1.48) (-0.16)

(-0.62) (-0.88) (-1.60)

OTM Put/Call -0.055*** -0.039** -0.110***

-0.018* -0.034** 0.000

-0.005** -0.013* -0.001

(-3.33) (-2.44) (-5.69)

(-1.76) (-2.00) (0.03)

(-2.09) (-1.82) (-0.56)

Option Volume -0.012 -0.123 0.014

-0.189 -0.263 0.034

0.172 1.102 0.197

(-0.19) (-0.62) (0.21)

(-1.53) (-1.03) (0.23)

(0.89) (1.35) (1.01)

Bull-Bear Spread 0.023 -0.043 0.027

-0.007 -0.071 -0.113

-0.052 -0.083 -0.049

(0.41) (-0.63) (0.28)

(-0.14) (-1.24) (-0.77)

(-0.89) (-1.15) (-0.36)

VIX -0.001 -0.000 -0.000

0.001 0.002 0.002

0.003** 0.002 0.002

(-0.62) (-0.27) (-0.10)

(1.42) (1.10) (0.72)

(2.18) (1.37) (0.63)

Lagged Skew 0.646*** 0.723*** 0.419***

0.826*** 0.855*** 0.543***

0.862*** 0.885*** 0.605***

(12.18) (10.24) (5.61)

(32.33) (30.82) (6.65)

(42.92) (43.41) (7.93)

Observations 840 530 310 1058 717 341 954 614 340

Adjusted R2 57.6% 64.8% 43.7% 74.2% 82.4% 35.7% 78.0% 84.4% 44.8%

Notes: This table reports the results of the weekly regression that examines the robustness of the relation between investor beliefs and risk-neutral

skewness for one-, three- and six-month maturities. Control variables are the 1-month treasury-bill rate, the spread between yields on the 10-year

treasury bond, and the 3-month treasury bill, the difference in yields on Baa and Aaa corporate bonds, and the log growth in aggregate industrial

production over the last 12 months. The dependent variables are skewness based on the risk-neutral densities estimated from option prices. For each

maturity, we report the regression results from the entire data period: 1990 to 2010; and two sub-sample periods: 1990 to 2003, and 2004 to 2010.

Standard errors are adjusted for heteroskedasticity and serial correlation according to Newey and West (1987); and t-statistics are reported in parentheses

below the coefficients. ***, ** and * denote the significant level of 1%, 5% and 10%, respectively.

50

Table 8: Investor Beliefs and Risk-Neutral Skewness

(Robustness to the Alternative Measure of Skewness)

One-Month Three-Month Six-Month

1990-2010 1990-2003 2004-2010

1990-2010 1990-2003 2004-2010

1990-2010 1990-2003 2004-2010

Net Short Position -0.279 -0.101 -10.470*

-1.415* -1.070 -12.940***

-1.754 -0.601 -5.212*

(-0.20) (-0.07) (-1.81)

(-1.77) (-1.22) (-3.13)

(-1.39) (-0.43) (-1.77)

Speculation Index -0.063 0.372 0.369

-0.481 0.429 -3.172**

0.128 1.836 -2.032**

(-0.05) (0.22) (0.24)

(-0.69) (0.41) (-2.51)

(0.17) (1.49) (-2.57)

Future Volume -0.105*** -0.272*** -0.109***

-0.135** -0.343 -0.045

-0.212 -0.654 -0.066

(-3.45) (-2.74) (-2.80)

(-2.00) (-1.45) (-0.56)

(-1.15) (-0.95) (-0.44)

OTM Put/Call -0.244*** -0.176*** -0.488***

-0.0251* -0.0843** -0.034*

-0.008* -0.037*** -0.009*

(-3.82) (-2.67) (-9.41)

(-1.68) (-2.39) (-1.74)

(-1.78) (-3.71) (-1.76)

Option Volume -0.094 -0.545 0.111

-0.059 -0.499 -0.121

-0.413 -1.258 -0.486**

(-0.51) (-1.19) (0.64)

(-0.30) (-0.91) (-0.49)

(-1.63) (-0.62) (-2.34)

Bull-Bear Spread 0.139 -0.262 -0.003

0.267*** 0.119 -0.025

0.311** 0.299* -0.022

(1.09) (-1.40) (-0.02)

(3.04) (1.05) (-0.14)

(2.58) (1.71) (-0.18)

VIX 0.003 0.003 -0.003

-0.003* 0.001 -0.005

-0.001 -0.000 -0.002

(1.19) (0.78) (-0.72)

(-1.83) (0.54) (-1.58)

(-0.61) (-0.02) (-1.07)

Lagged Skew 0.396*** 0.381*** 0.267***

0.690*** 0.664*** 0.447***

0.705*** 0.655*** 0.613***

(8.29) (5.95) (3.38)

(18.63) (12.89) (4.88)

(20.31) (14.89) (9.18)

Observations 838 528 310 1055 714 341 953 613 340

Adjusted R2 37.4% 40.8% 45.8% 66.2% 72.0% 38.4% 67.4% 68.5% 53.5%

Notes: This table reports the results of the weekly regression that examines the robustness of the dependence of risk-neutral skewness on investor

beliefs for one-, three- and six-month maturities by using another measure of risk-neutral skewness. The dependent variables, risk-neutral skewness, are

obtained by using the methodology of Bakshi, Kapadia, and Madan (2003). All regressions include control variables in the crude oil market, which are

the basis, storage level, historical returns, and option implied volatility. Standard errors are adjusted for heteroskedasticity and serial correlation

according to Newey and West (1987); and t-statistics are reported in parentheses below the coefficients. ***, ** and * denote the significant level of

1%, 5% and 10%, respectively.