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Equity Volatility Term Structures

and the Cross-Section of Option Returns

Aurelio Vasquez

ITAM School of Business

First Draft: October 14, 2011

This Draft: November 25, 2012

Abstract

The slope of the implied volatility term structure is positively related with future option

returns. We rank rms based on the slope of the volatility term structure and analyze the

returns for ve dierent option trading strategies. Option portfolios with high slopes of the

volatility term structure outperform option portfolios with low slopes by an economically and

statistically signicant amount. The results are robust to dierent empirical setups and are

not explained by well-known market, size, book-to-market, or momentum factors. Additional

higher-order option-related factors, volatility risk premiums, jump risk, and existing option

anomalies cannot explain the large option returns.

JEL Classication: C21, G13, G14.

Keywords: Equity Options; Volatility Term Structure; Implied Volatility; Predictability; Cross-

Section.

This work is part of my dissertation at McGill University. I am grateful for helpful comments from Diego Amaya,Redouane Elkamhi, Alex Horenstein, Lars Stentoft, Richard Roll, and especially from my PhD supervisors PeterChristoersen and Kris Jacobs. I also want to thank seminar participants at EFA Boston, ITAM, FMA New York,Laval University, McGill University, MFA New Orleans, NFA Vancouver, TEC of Monterrey, Texas A&M, and UQAMfor helpful comments. I thank IFM2 and Asociacin Mexicana de Cultura A.C for nancial support. Correspondenceto: Aurelio Vasquez, Faculty of Management, ITAM, Rio Hondo 1, Alvaro Obregon, Mexico DF; Tel: (52) 55 56284000 x.6518; E-mail: [email protected].

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1 Introduction

We investigate if the shape of the term structure of implied volatility is related to future one-month

option returns in the cross-section. Every month, we sort stocks by the slope of the volatility term

structure, group them in deciles, and examine the average future returns for ve option trading

strategies: naked call, naked put, straddle, delta-hedged call, and delta-hedged put.

We nd a strong positive relationship between the slope of the volatility term structure and

future option returns for all ve trading strategies. The straddle, a trading strategy that bets on

the direction of volatility, has an average monthly return of9:6% for the decile with the lowest

slope of the volatility term structure and a return of10:0% for the decile with the highest slope.

The long-short strategy generates a monthly return of19:6%with a t-statistic of9:17. The other

four option trading strategies also report large positive and signicant returns for the long-short

strategy. The long-short returns for the naked call and put strategies are24:1% and 19:5% with

signicant t-statistics. The delta-hedged call and put strategies report a long-short return of2:7%

and2:2%with t-statistics of7:97and 8:35.

These ndings hold across dierent time periods, moneyness levels, and weighting method-

ologies. Naked put long-short returns are not signicant around earnings announcement dates.

Fama-MacBeth (1973) regressions and double sorts on rm characteristics further conrm our nd-

ings. The coecients of the slope of the volatility term structure and the long-short premia are

positive and signicant for all trading strategies when we control by rm size, book-to-market, re-

alized volatility, and option Greeks. After transaction costs, the straddle strategy is still protable.

The long-short straddle return decreases from 19:6% to a statistically signicant 5:5% monthly

return.

We compute the alphas of the long-short option strategies using the Carhart model as well

as the coskewness and cokurtosis factor models proposed by Vanden (2006) that include higher

order moments of the market return and the market option return. The alphas from the Carhart,

coskewness and cokurtosis models are large and signicant, and are very close to the raw returns.

The large option returns potentially represent a compensation for some risk. We perform an

exhaustive analysis using Fama-MacBeth regressions and double sorts to nd potential explanations

for our results. First, we show that the volatility risk premium cannot explain the large option

returns. Following Bollerslev, Tauchen, and Zhou (2009), we compute the volatility risk premium

as the dierence between risk-neutral volatility squared and realized volatility squared. When

the volatility risk premium is included in the Fama-Macbeth regressions, the coecient of the

slope of the volatility term structure remains positive and signicant for all ve trading strategies.

Additionally, the long-short premia are positive and signicant in the two-way sort analysis for all

ve option trading strategies.

Second, we check if our results are a compensation for jump risk. Bakshi and Kapadia (2003)

show that risk-neutral skewness and risk-neutral kurtosis proxy for jump risk. Yan (2011) demon-

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strates that the slope of the volatility smilethe dierence between out-of-the-money and in-the-

money volatilitiesis a proxy for the jump magnitude. We compute the slope of the volatility

smile using the methodologies proposed by Xing, Zhang and Zhao (2010) and Yan (2011). After

controlling for jump risk, the Fama-MacBeth coecients of the slope of the volatility term structure

and the long-short premia in the double sorts remain positive and signicant for the ve optiontrading strategies.

Third, we control for investor overreaction (underreaction) to current volatility changes (see

Stein (1989) and Poteshman (2001)). When investors overreact (underreact) to current volatility

changes, option volatility increases (decreases) and options become expensive (cheap). To ensure

that this phenomenon does not explain our results, we include ratios of current implied volatility and

several measures of lagged volatility in the Fama-MacBeth regressions. We nd that the coecients

of the slope of the volatility term structure remain positive and signicant for all trading strategies

except for the naked put strategy. For the other four trading strategies, misreactions to volatility

changes do not explain the relationship between the slope of the volatility term structure and theirreturns. In the double-sorting exercise, the long-short premiums are positive and signicant for all

trading strategies.

Finally, we examine whether the large option returns documented in our paper are related to

existing option anomalies. Specically, we control for idiosyncratic volatility and for the dierence

between historical and implied volatility. Cao and Han (2012) nd that delta-hedged call returns

are negatively related to idiosyncratic volatility, and Goyal and Saretto (2009) nd that straddle

and delta-hedged call returns are positively related to the dierence between historical and implied

volatility. Fama-MacBeth regressions and double sorts show that our large returns are not related

to existing option anomalies.

This paper contributes to the nance literature in two ways. It is the rst study to show that the

slope of the term structure of implied volatilities has a positive relationship with subsequent option

returns in the cross-section. The shape of the volatility term structure has previously been used

to test the expectations hypothesis and the overreaction of long-term volatilities. Notable research

in this area includes papers by Stein (1989), Diz and Finucane (1993), Heynen, Kemna, and Vorst

(1994), Campa and Chang (1995), Poteshman (2001), Mixon (2007), and Bakshi, Panayotov, and

Skoulakis (2011). However, to the best of our knowledge no existing paper uses the shape of the

term structure to identify mispriced options.

A second contribution relates to the forecasting of future realized volatility. Since the slope of

the volatility term structure is positively related with option returns, we test if it forecasts future

realized volatility. Recent studies by Cao, Yu, and Zhong (2010) and Busch, Christensen, and

Nielsen (2011) show that short-term implied volatility, historical volatility and realized volatility

are all good predictors of future volatility.1 We document that the long-term implied volatility also

contributes to the prediction of future realized volatility.

1 See Granger and Poon (2003) for a review on volatility forecasting.

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Empirical option research has focused primarily on indexes.2 This paper explores the cross-

section of equity options. In the literature on the cross-section of individual options, Cao and

Han (2012) nd that delta-hedged option returns are negatively related to total and idiosyncratic

volatility. Jones and Shemesh (2010) document the options weekend eect, option returns are lower

on weekends than on weekdays, and Choy (2011) reports a negative relationship between optionreturns and retail trading proportions. Bali and Murray (2012) create skewness-assets using options

and the underlying stock and nd a negative relationship between the skewness-asset returns and

their risk-neutral skewness. The most closely related paper is Goyal and Saretto (2009), who show

that option returns are positively related to the dierence between individual historical realized

volatility and at-the-money (ATM) implied volatility.

The implied volatility term structure is used in the option pricing literature. Christoersen,

Jacobs, Ornthanalai, and Wang (2008) and Christoersen, Heston, and Jacobs (2009) show that an

option pricing model that properly ts the volatility term structure has a superior out-of-sample

performance compared to classical option pricing models such as the Heston model. This resultsuggests that the volatility term structure contains crucial information on future option prices. Our

paper documents a positive relation between the slope of the volatility term structure and future

option returns.

The remainder of this paper is organized as follows. Section two describes the option data.

Section three describes the option trading strategies and the portfolio characteristics. The returns

from these strategies using dierent setups are in section four, and section ve contains a series of

robustness checks. Section six examines the forecasting power of the slope of the volatility term

structure on future volatility. Section seven concludes the paper.

2 Data

In this section, we describe the data and explain the lters that are applied.

We use the cross-section of options from the OptionMetrics Ivy database. The OptionMetrics

Ivy database is a comprehensive source of high quality historical price and volatility data for the

US equity and index options markets. We use data for all US equity options and their underlying

prices for the period starting on January 4, 1996 through June 30, 2007. Each observation contains

information on the closing bid and ask quotes for American options, open interest, daily trading

volume, implied volatilities, and Greeks. Implied volatilities and Greeks are computed using the

Cox, Ross, and Rubinstein (1979) binomial model.

OptionMetrics also provides stock prices, dividends, and risk-free rates. A complete history of

splits is also available for each security. The risk-free rates are linearly interpolated to match the

2 Coval and Shumway (2001) study index option returns and nd that zero cost at-the-money straddle positionson the S&P 500 produce average losses of approximately 3% per week. Other studies of index option returns areBakshi and Kapadia (2003), Jones (2006), Bondarenko (2003), Saretto and Santa-Clara (2009), Bollen and Whaley(2004), Shleifer and Vishny (1997), Jackwerth (2000), Buraschi and Jackwerth (2001), and Liu and Longsta (2004).

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maturity of the option. If the rst risk-free rate maturity is greater than the option maturity, no

extrapolation is performed and the rst available risk-free rate is used.

Next, we apply standard lters for individual options as in Cao and Han (2012) and Goyal and

Saretto (2009). We eliminate the prices that violate arbitrage bounds. 3 That is, we eliminate call

option prices that fall outside of the interval (S Ke

r

De

r

; S), and put option prices thatfall outside of the interval (S+ Ker + Der; S); where Sis the price of the underlying stock,

K is the strike of the option, r is the risk-free rate, D is the dollar dividend, and is the time to

expiration. An observation is eliminated if the ask is lower than the bid, the bid (ask) is equal to

zero, or the spread is lower than the minimum tick size. The minimum tick size is $0:05for options

trading below $3and$0:10 for other options. Whenever the bid and ask prices are both equal to

the previous days quotes, the observation is also eliminated. We lter one-month options with zero

volume or zero open-interest to ensure that the one-month option prices are valid. Options with

underlying stock prices lower than $5 are removed from the sample. Finally, the moneyness of the

options must be between 0:95and 1:05, and volatilities should lie between 3% and 200%.

4

Each month, we compute the slope of the volatility term structure for each stock. The slope of

the volatility term structure is dened as the dierence between the long-term and the short-term

volatility. The short-term volatility, IV1M; is dened as the average of the one-month ATM put

and call implied volatilities. The long-term volatility, IVLT; is the average volatility of the ATM

put and call options that have the longest time-to-maturity available and the same strike as that

of the short-term options. The longest time to expiration is between 50and360days. Hence, the

maturity of the long-term options is dierent across stocks and, for any given stock, can change

across months.5

3 Portfolio Formation and Trading Strategies

In this section, we explain how portfolios are constructed and provide a summary of dierent

characteristics across portfolios. Then, we describe the return computation for ve option trading

strategies: naked call, naked put, delta-hedged call, delta-hedged put, and straddle.

3.1 Portfolio Formation

Each month, we form ten portfolios based on the slope of the volatility term structure, I VLTIV1M.

Decile portfolios contain the one-month ATM options that are available on the second trading day(usually a Tuesday) after the expiration of the previous one-month options, which occurs on the

third Saturday of the month. We extract the ATM put and call options that are one-month away

3 Duarte and Jones (2007) point out that options that violate arbitrage bounds might still be valid options. Theinclusion of options that violate arbitrage bounds does not change the conclusions.

4 The conclusions hold when the volatility range is 3% to 100%.5 Note that option returns are computed only for short-term options. Long-term options are only used to extract

long-term volatility to compute the slope of the volatility term structure.

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from maturity. The one-month option maturity ranges from26 to 33 days. The strike price is as

close as possible to the closing price. For example, if the stock price is 17, and the two closest

strikes are 15 and20, we select the options with strike price 15. If the selected put or call options

do not pass the ltering process, we choose the two options with a strike price of20. If either of

those two options is ltered out, that particular stock is excluded for that month because it has novalid options.

On the option expiry date, we compute the option returns for all ve trading strategies: naked

call, naked put, delta-hedged call, delta-hedged put and straddle. Then, we form decile portfolios

based on the slope of the volatility term structure. Since decile portfolios are formed based on the

options availability, stocks drop in and out of the sample from month to month. On average, there

are386 stocks per month.

3.2 Characteristics of Portfolios Sorted by the Slope of the Volatility Term

Structure

Table 1 reports the time-series averages for dierent rm characteristics for the ten portfolios

ranked by the slope of the volatility term structure. The characteristics included are divided into

three groups: the variables related to the slope of the volatility term structure, rm and option

characteristics and higher moment measures.

[ Insert Table 1 here ]

The variables related to the slope of the volatility term structure are I V1M; IVLTand the slope

of the term structure IVLTIV1M. The rm and option characteristics are the options size (in $

thousands), dened as the open interest for calls and puts multiplied by their price, the average

maturity of IVLT, option Greeks, rm size, and book-to-market. Finally, the higher moment

measures are the risk-neutral volatility, skewness and kurtosis (RNVol, RNSkew and RNKurt)

extracted from one-month options using the methodology proposed by Bakshi, Kapadia, and Madan

(2003), the risk-neutral jump dened as the slope of the option smirk as proposed by Yan (2011),

idiosyncratic volatility computed from the one month daily returns using the Fama-French factors,

future volatility (F V , dened as the standard deviation of the underlying stock return over the life

of the option), F VI V1M, and the dierence between the current one-month implied volatility

and the average of the one-month implied volatility over the previous six months, IV1MIVavg1M .

As reported in Table 1, the slope of the volatility term structure increases from 13:7%to 7:6%

for portfolios1 to 10. Companies with the lowest and highest slope of the volatility term structure

are the most volatile. Portfolios1and10have the highestI V1M(with portfolio2);the highestI VLT,

the highest risk-neutral volatility and the highest idiosyncratic volatility. Companies in extreme

portfolios tend to be small, with a high risk-neutral jump, and have a lower risk-neutral skewness,

and vega exposure. No pattern is observed between the slope of the volatility term structure and

risk-neutral kurtosis, book to market, option size, and option delta.

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From Table 1, the average maturity of long-term options is approximately 220 days (7months)

for all decile portfolios. Thus, the slope of the volatility term structure is computed using volatili-

ties that are on average six months apart. Extreme portfolios have the lowest vegas of7:3 and 7:2

compared with the vegas for portfolios 5 to 8 that are all over 9:3. According to vega, extreme port-

folios are the least sensitive to volatility movements. Additionally, these portfolios (with portfolio2) hold the rms with the lowest value of $4:5 billion for portfolio 1 and $9:3 billion for portfolio

10. Firms in portfolios 5 and 6 have an average size of $15.1 and $18.7 billion, respectively.

While the slope of the volatility term structure increases from portfolio 1 to portfolio 10, the

dierence between IV1M and IVavg1M decreases. Portfolio 1 has an averageIV1MIV

avg1M of11:1%

that decreases to 10:2% for portfolio 10. However, F V I V1M increases from portfolio 1 to

portfolio 10. Portfolio 1 has an average F VIV1M of7:4% that increases to 1:6% for portfolio

10.

In summary, Table 1 shows that the slope of the volatility term structure appears to be related

to past and future volatility. We now attempt to establish a cross-sectional relationship betweenthe slope of the volatility term structure and future option returns.

3.3 Trading Strategies and Option Returns

The analysis of option returns is not as straightforward as that of stock returns. Option investors

have several degrees of freedom when buying an option. Calls and puts with dierent maturities and

strike prices are available. Hence, many dierent trading strategies can be implemented. Saretto

and Santa-Clara (2009) analyze 23 dierent option trading strategies for the S&P 500 index. Since

liquidity is a major constraint when studying individual stock options, we work with the most

liquid options: at-the-money options that are close to expiration. In particular, we study veoption trading strategies constructed with one-month ATM options: naked calls and puts, delta-

hedged calls and puts, and straddles.

The returns on these strategies are computed following Goyal and Saretto (2009). As reported in

Table 1, the average bid to mid option spread for portfolios 1 and 10 is 6:7%and8:3%;respectively.

To avoid paying high transaction costs more than once, options are held until maturity. By holding

the options until maturity, the large transaction costs for the option are only paid when opening

the position and are avoided at expiration. If the option expires in-the-money, only the stock incurs

transaction costs (i.e., bid-ask spread).

3.3.1 Naked Option

A naked option consists of buying either a call or a put option with no underlying security protec-

tion. Naked options are very risky. If the underlying asset moves in the expected direction, large

returns are made. However, huge losses of up to 100% plus accrued interest are recorded if the

underlying stock moves in the opposite direction. A zero-cost naked option traded at time t has a

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return equal to

rcallt;T = max(STK; 0)

ctrft;T; and (1)

rputt;T = max(KST; 0)

pt

rft;T (2)

wherect andpt are the average of the bid and ask prices of a call and a put option, on trading day

t, rft;T is the future value of one dollar from time t to T, Kis the strike price, and ST is the stock

price at maturity T :

3.3.2 Straddle

A straddle is an investment strategy that involves the simultaneous purchase (or sale) of one call

option and one put option. A long straddle return is dened as

rstraddlet;T = jSTKj

pt+ctrft;T (3)

wherect andpt are the average of the bid and ask prices of a call and a put option, on trading day

t, rft;T is the future value of one dollar from time t to T; K is the strike price, and ST is the stock

price at maturity T :

3.3.3 Delta Hedged Option

A more sophisticated trading scheme involves the delta of the option. Delta is the rate of change

of the option with respect to the stock price. A delta neutral or delta hedged position consists of

selling the option and buying delta units of the stock so that small stock movements do not aect

the prot and loss function of the investor. For simplicity, no rebalancing is performed and the

number of stocks is kept constant until the option expiration date. The return of a delta hedged

optiona combination of going long delta number of shares and short on the optionis

rHedgedCallt;T = (ST

callt max(STK; 0))

Stcallt ct+rft;T (4)

rHedgedPutt;T = (ST

putt max(KST; 0))

Stputt pt

rft;T (5)

where ct and pt are the average of the bid and the ask prices of a call and a put option, K is the

strike price,St is the stock price, t is the delta of the option at time t and rft;T is the future value

of one dollar from time t to T.

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4 Slope of the Volatility Term Structure and the Cross-Section of

Option Returns

In this section, we rst analyze the relationship between the slope of the volatility term structure

and the one-month option returns for the ve trading strategies: naked call, naked put, delta-hedged

call, delta-hedged put, and straddle. Second, we use the Fama and MacBeth (1973) cross-sectional

regressions and double sorts to determine the signicance of the slope of the volatility term structure

when controlling by volatility risk premium, jump risk, investor misreaction to volatility changes,

option anomalies, and rm characteristics. Then, we report the coskewness and cokurtosis risk

adjusted alphas for the long-short portfolio for all ve option trading strategies. Finally, we assess

the impact that transaction costs have on the returns of the ve trading strategies.

4.1 Sorting Option Returns by the Slope of the Volatility Term Structure


Each month, we rank stocks by the slope of their volatility term structure and form ten option

portfolios. Table 2 reports equally weighted portfolio returns for straddles, delta-hedged calls and

puts, and naked calls and puts. The option returns for all trading strategies increase from portfolio 1

to portfolio 10. In particular, the straddle returns are negative when the slope of the volatility term

structure is negative, and are positive when the slope of the volatility term structure is positive.

The long-short straddle strategy (portfolio 10 minus portfolio 1) yields a 19:6% monthly average

return with a t-statistic of9:17. Both portfolios contribute equally to the long-short portfolio returnsince the straddle returns are 9:6%and 10:0% for portfolios 1 and 10, respectively.

The other four option trading strategies also report signicant returns on the long-short trading

strategy. The naked call strategy yields an average long-short return of24:1% with a t-statistic

of5:6. The long-short portfolio for the delta-hedged call strategy has a return of2:7% with a t-

statistic of7:97. For both trading strategies, the return of portfolio 1 is negative and the return of

portfolio 10 is positive. The results for put options display the same trend. The long-short return

for the naked put strategy is 19:5% with a t-statistic of4:79 while that for the delta-hedged put

strategy is 2:2%with a t-statistic of8:35.

[ Insert Figure 1 here ]

Figure 1 displays the time series of the long-short portfolio returns. For all ve option trading

strategies, more than50% of the returns are positive: 81%of the returns are positive for straddles,

83% for delta-hedged calls and puts, and 67% for naked calls and naked puts. The long-short

straddle positive returns decrease through the sample period; while in 1996 and 1997; long-short

returns of up to150%are observed, after 2005 they are below 50%. For delta-hedged calls and puts,

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the maximum positive long-short returns occur in 2001. Finally, the naked call and put strategies

do not show any pattern.

[ Insert Figure 2 here ]

Figure 2 displays the qq-plots of the long-short portfolio returns for all option trading strategies.All ve option trading strategies have a right fat tail compared to the normal distribution. A fat

left tail is present in the long-short return of delta-hedged puts, delta-hedged calls and naked puts.

For naked calls, the left tail is thinner than that of the normal distribution. Fat tails in all option

return distributions are conrmed by the excess kurtosis, reported in Table 2, that ranges from 0:2

to3:7:Finally, the long-short returns of all ve option trading strategies report a positive skewness.

In conclusion, we nd a clear positive relationship between the slope of the volatility term

structure and the cross section of option returns. Additionally, the option returns for all ve

trading strategies are not normally distributed; they have positive skewness and positive excess

kurtosis.

4.2 Controlling for Volatility Risk Premium and Jump Risk

To conrm that the slope of the volatility term structure is positively related with option returns

in the cross section, we run the two-stage Fama and MacBeth (1973) regressions. An advantage

of the Fama and MacBeth (1973) regressions is that they do not impose breakpoints for portfolio

formation but allow for an evaluation of the interaction among variables and the slope of the

volatility term structure. In the rst stage, for each month t; a regression is run with the option

return on the left hand side and the slope of the term structure along with other variables on the

right hand side. From stage one, we obtain a time series oft coecients that are averaged in the

second stage to obtain an estimator for each coecient. We evaluate the coecients signicance

using the Newey-West t-statistic with 3 lags.

Using Fama and MacBeth (1973) regressions, we examine whether the option returns are ex-

plained by the volatility risk premium or jump risk. Following Bollerslev, Tauchen, and Zhou

(2009), the individual volatility risk premium, V RP, is computed as the dierence between risk-

neutral volatility squared and realized volatility squared. Risk-neutral volatility is extracted from

the model free measure proposed by Bakshi, Kapadia, and Madan (2003) and realized volatility is

computed with intraday 5-minute returns over the previous month. To account for volatility risk,

we include risk-neutral volatility, RN V ol, in the regressions.

Four proxy variables account for jump risk. Bakshi and Kapadia (2003) show that risk-neutral

skewness and risk-neutral kurtosis proxy for jump risk. Using the model free methodology by

Bakshi, Kapadia, and Madan (2003), we compute risk-neutral skewness and kurtosis, RNSkew

and RNKurt. Another proxy for jump risk is the slope of the volatility smile, the dierence

between out-of-the-money and at-the-money volatilities. Two measures are used for the slope of

the volatility smile: OptionSkew by Xing, Zhang and Zhao (2010) and RNJump by Yan (2011).

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Table 3 reports the regressions of option returns on the slope of the volatility term structure,

volatility risk premium, and jump risk measures. The coecient of the slope of the volatility term

structure is positive and signicant for all ve option trading strategies. Comparing the univariate

and the multivariate regressions, we conclude that the strong positive relation between the slope of

the volatility term structure and option returns is not explained by the volatility risk premium or

by jump risk. The univariate regression for straddle returns shows that the coecient for the slope

of the volatility term structure is 0:840with a t-statistic of8:08. After including the volatility risk

premium and jump risk proxies, the coecient of the slope of the volatility term structure increases

to 0:952 and the t-statistic is now 10:44. Therefore, including volatility risk premium and jump

risk exacerbates the eect of the slope of the term structure instead of reducing it.

The results for the other four option trading strategies are very similar. For all four strategies,

the coecient of the slope of the term structure is positive and signicant. While for the delta-

hedged call and the naked call the signicance of the coecient increases, for the put strategies

the coecient signicance has a small decrease but remains signicant. Two control variables are

signicant for most of the ve regressions: OptionSkew and RNJump, the measures of the slope

of the volatility smirk. While OptionSkew has a positive and signicant relation with the four

option trading strategies aside from naked call, RNJump has a negative relation with three option

strategies and a positive relationship with the remaining two.

To further test that the abnormal option returns are not driven by the volatility risk premium

or the jump risk, we perform the double sorting methodology between the slope of the volatility

term structure and each measure. In the rst stage, we rank the stocks by the rm characteristic

and form ve portfolios. Portfolio 1 (5) has stocks with low (high) values of the characteristic.

In the second stage, we sort the stocks into ve portfolios using the slope of the volatility term

structure within each rm characteristic portfolio. Then, we compute the average option return for

each level of the slope of the volatility term structure and report the long-short option return.


Table 4 reports the long-short option returns and the t-statistics using two-way sorts for the

volatility risk premium and the jump risk measures for the ve trading strategies: straddle, delta-

hedged put, delta-hedged call, naked put, and naked call. The long-short option returns are positive

and signicant for the ve trading strategies across all measures. The long-short straddle returns

are between 12:4% and 14:3%; while the t-statistics are between 7:11 and 9:42. The long-short

delta-hedged put returns are signicant and range from 1:5%to 1:6%. Similar results are obtained

for delta-hedged call, naked put, and naked call returns: long-short returns are positive, signicant

and within a small range for all control variables.

The results for all trading strategies across dierent characteristics are comparable to those of

Table 2. The main dierence between the two tables is the number of portfolios used; in Table 2,

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stocks are sorted into ten portfolios, while in Table 4 they are sorted into ve portfolios. When

the number of portfolios is reduced from ten to ve, the long-short returns are lower. However, the

reduction in the number of portfolios ensures that they are well populated.

To summarize, we have shown that the positive relation between option returns and the slope

of the volatility term structure is not explained by the volatility risk premium or jump risk. Fama-Macbeth regressions and two-way sorts conrm that there is a positive relation between the returns

of ve option trading strategies and the slope of the volatility term structure. Moreover, controlling

for jump risk and volatility risk premium in the Fama-MacBeth regressions increases the eect of

the slope of the volatility term structure for all option trading strategies.

4.3 Controlling for Option Anomalies

Table 5 reports the results for the Fama and MacBeth (1973) regressions of option returns on

variables related to option anomalies and investor misreaction to volatility changes. The rst

variable is the dierence between historical and implied volatility. As previously stated, Goyal andSaretto (2009) nd that straddle and delta-hedged call returns have a positive relation with the

dierence between historical and implied volatility,H VIV1M. The second variable is idiosyncratic

volatility,IdioV ol. Cao and Han (2012) report that delta-hedged call returns decrease with the level

of idiosyncratic volatility. Finally, we include a set of variables that control for investor misreaction

to volatility changes. Poteshman (2001) and Stein (1989) document that investors can underreact

or overreact to changes in volatility. Hence, investors might be buying (selling) options that are

overpriced (underpriced) and that will generate negative (positive) future returns. To account for

high volatility periods and investor misreactions, we include the ratios of current volatility against

measures of previous implied volatilities. The measures of previous volatility are the one-month(IVt1

1M), 3-month (IVt31M), and 6-month (IV

t61M) lagged implied volatility as well as the maximum

(IVmax1M ) and the average implied volatility (IV

avg1M) over the previous 6-months.


Table 5 reports the coecients and t-statistics for the Fama-MacBeth regressions of the ve

option trading strategies. In the rst regression we include the slope of the volatility term structure,

idiosyncratic volatility and the dierence between historical and implied volatility. The second

regression includes all the ratios of short-term volatility and measures of lagged volatility. In the

rst regression, the coecient of the slope of the volatility term structure is positive and signicant

for all trading strategies. In the second regression, the coecient is also positive and signicant for

all trading strategies except for the naked put strategy. Additionally, compared with the results

of the univariate regression, the magnitude of the t-statistics for the slope of the volatility term

structure decreases for all regressions but remains signicant. In the rst regression, the decrease

is explained by the inclusion ofH V IV1Mthat shows a positive and signicant coecient in four

regressions: straddle, delta-hedge put, delta-hedge call, and naked put. These results conrm the

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ndings by Goyal and Saretto (2009), who show that HVI V1Mhas a positive and signicant

coecient for straddles and delta-hedged calls. In the second regression, the ratio between current

volatility and three-month lagged volatility has a negative and signicant relation with option

return. This relation explains the decrease in the coecient of the slope of the volatility term

structure.Idiosyncratic volatility is signicant for naked put and naked call returns only. Conrming

the results by Cao and Han (2012), delta hedged call returns and idiosyncratic volatility show a

negative relation. However, the coecient of idiosyncratic volatility is not signicant for the delta-

hedged call long-short return. This lack of signicance might be caused by the rebalancing of the

delta-hedge strategy. While Cao and Han (2012) rebalance the delta-hedge several times before the

call expires, we set the delta-hedge one month before expiration without rebalancing.

Finally, we show that the slope of the volatility term structure is not a proxy for changes

in implied volatility. The coecient of the slope of the volatility term structure is positive and

signicant in all ve regressions. The coecient of most volatility ratios is negative and in somecases signicant, conrming the negative relation observed in Table 1 between IV1MIV

avg1M and

the slope of the volatility term structure:Two volatility ratios are consistently negative, and at least

one of them is signicant in the ve regressions: ln(IVt1M=IV

t11M ) and ln(IV

t1M=IV

t31M ): The one-

month lagged volatility ratio is signicant for delta-hedged put, delta-hedged call and naked put,

and the three-month lagged volatility ratio is negative and signicant for all ve trading strategies.


To ensure that the slope of the volatility term structure is not a proxy for an existing option

anomaly, we use the double sorting methodology. Table 6 reports the long-short option returns andthe t-statistics for the option anomaly measures for the ve trading strategies. Once again, all the

long-short option returns are positive and signicant for the ve trading strategies. The long-short

straddle returns are between8:2%and14:3%;while the t-statistics are between4:97and8:56. Note

that the lowest long-short return occurs when double sorting by the dierence between historical

volatility and implied volatility, H V IV1M. This result is not surprising since Goyal and Saretto

(2009) show that HVI V1M is positively related to the straddle and delta-hedged call returns.

What is important is that the slope of the volatility term structure predicts option returns over

and above HVI V1M. As for the other four trading strategies, the long-short returns are once

again positive and signicant.Using Fama-MacBeth regressions and double sorts, we conclude that the slope of the volatil-

ity term structure detects option mispricing even in the presence of variables that are related to

option anomalies such as idiosyncratic volatility, historical minus implied volatility, and measures

of investor misreaction to volatility changes. Only the naked put strategy is explained by investor

misreaction.

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4.4 Controlling for Stock Characteristics

Table 7 reports the results of the Fama and MacBeth (1973) regressions of option returns on rm

characteristics to further control that the slope of the volatility term structure predicts option

returns. In addition to the slope of the term structure, the regression includes the size of the rm,

book-to-market, historical volatility, skewness and kurtosis, realized volatility computed with veminute returns, and the option Greeks: delta, gamma and vega.


Table 7 presents the results of two regressions for each option trading strategy. The coecient

of the slope of the volatility term structure is positive and signicant in all regressions for the ve

trading strategies. The rst regression includes the slope of the volatility term structure and the

option Greeks. For the long-short straddle returns, the coecient of the slope of the volatility term

structure is0:687with a Newey-West t-statistic of5:6. For delta-hedged put and delta-hedged call,

the coecients are 0:078 and 0:089 with a Newey-West t-statistic of 5:03 and 4:98. Finally, for

naked put and naked call, the coecients are 0:861 and 0:605 with signicant t-statistics of4:20

and 2:69. The relationship between the slope of the volatility term structure is not explained by

the option Greeks.

In the second regression, we add the six rm characteristics to the Fama-MacBeth regressions:

size, book-to-market, historical volatility, skewness and kurtosis, and realized volatility. For strad-

dles, the coecient of the slope of the volatility term structure slightly increases to 0:738with a

Newey-West t-statistic of5:90. For the other four strategies, the magnitude of the coecient of

the slope of the volatility term structure increases in all of the strategies and the t-statistics are

above 4:03. Realized volatility reports a negative and signicant relation for all trading strategies

except for the naked put strategy. The higher the volatility from intraday returns is, the lower

the option return. The same negative relation applies for size: small rms report a higher return

than large rms. Finally, historical volatility displays a positive relation with option returns. The

other variables (book-to-market, skewness, kurtosis, gamma, and vega) show no strong relation

with options returns.


Table 8 reports the long-short option returns and the t-statistics for each rm characteristic for

the ve trading strategies using the two-way sort methodology. As in the previous double sorting

tables, all of the long-short option returns are positive and signicant for the ve trading strategies.

In this case, the long-short straddle returns range between 12:3%and14:4%;while the t-statistics

are between 7:51 and 9:89. Conrming the main nding of this paper, the long-short returns for

the other four trading strategies are also positive and signicant.

In conclusion, the slope of the volatility term structure is positively related to the option returns

according to the Fama-MacBeth regressions and the two-way sorting methodology. We show that

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the slope of the volatility term structure is not a proxy for rm characteristics such as realized

volatility, rm size, book-to-market, higher historical moments or option Greeks.

4.5 Alphas of Portfolios from Coskewness and Cokurtosis Pricing Models

Index options are nonredundant securities (Bakshi, Cao, and Chen (1997) and Buraschi and Jack-werth (2001)). Since we are studying option returns, it is crucial to compute the alphas using

factor models that include the market return and the market option return. The coskewness and

cokurtosis pricing models developed by Vanden (2006) comply with this requirement. The coskew-

ness model incorporates not only the market return and the square of the market return but also

the option return, the square of the option return, and the product of the market and the option

returns. Similarly, the cokurtosis model includes the cubes of the market return and the option

return, as well as the product between the market return and the option return squared, and the

product between the market return squared and the option return.

The general version of Vandens model is dened as

rP = P+1Rm+2SM B+3HML+4U M D+ (6)

5(RoRf) +6(R2mRf) +7(R

2oRf) +8(RoRmRf) (7)

9(R3oRf) +10(R

3mRf) +11(R

2oRmRf) +12(RoR

2mRf) +"; (8)

whereRm is the market return, Ro is the market option return,Rf is the risk-free rate, andS M B,

HM Land U M D are the Fama-French and momentum factors. This equation embeds three factor

models: the Fama-French-Carhart model (the rst line of the equation), the coskewness model

(the rst and second lines), and the cokurtosis model (the entire equation). For the market optionreturn, we use the delta-hedged call return of the S&P 500 since it is related to the volatility risk

premium as documented by Bakshi and Kapadia (2003).6


Table 9 contains the results of the coskewness and the cokurtosis model regressions for the

ve option trading strategies. The rst column presents the results of the regression of each option

trading strategy on the market return, the Fama-French and momentum factors, and the coskewness

factors. The alphas for all ve trading strategies are positive and statistically signicant at the

10% level. For example, the alpha of the long-short straddle portfolio is 26:2%with a signicant t-

statistic at the 1% level. This alpha is larger than the long-short portfolio return of19:6%reported

in Table 2. The alphas for the other four trading strategies are positive and signicant and are

larger than the raw returns of the long-short trading strategy reported in Table 2.

6 Using any of the other four option strategies for the S&P 500 option return, that is naked call, naked put,delta-hedged put or straddle, does not change the results.

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The second column presents the results for the cokurtosis model. The alphas for all ve trading

strategies are positive and signicant at the 5% level. In particular, the alpha for the straddle

strategy is 19:5% with a highly signicant t-statistic. The alphas for the delta-hedged put, the

delta-hedged call and the naked put are also positive and signicant at the 1% level, and the alpha

of the naked call strategy is positive and signicant at the 5% level.Three factors are signicant for most regressions: R2m Rf, R

3m Rf, and R

2mRo Rf. The

square of the market return is a proxy for market volatility; its coecient is negative and signicant

for 8 out of 10 regressions. The higher the market volatility is, the lower the option returns. The

coecient ofR3m Rfis also negative and signicant for 4 out 5 regressions, and the coecient of

R2mRoRfis positive and signicant for 4 out of 5 regressions.

We conclude that the coskewness and cokurtosis factor models that include the market return

and the market option return do not explain the long-short returns for straddle, delta-hedged put,

delta-hedged call, naked put, and naked call.

4.6 Transaction Costs

The results presented so far do not include transaction costs: the trading strategies are executed at

mid prices. As reported in Table 1, the average bid-to-mid percent spread for option prices is 6:7%

and8:3%for portfolios 1 and 10. Hence, the bid-ask spreads will reduce the large prots of the ve

long-short trading strategies. To mitigate the eect of the bid-ask spreads, options are held until

maturity. When expired, the payo for the option is based only on the stock price and the strike

price. If the option expires in-the-money, the stock incurs transaction costs.7

Financial research has reported that the eective option spreads are, in some cases, higher than

the quoted bid-ask spreads. Battalio, Hatch, and Jennings (2004) show that the ratio betweeneective spreads and quoted spreads is higher than 1 in June 2000; however, after June 2002, the

ratio signicantly decreases, varying between 0:927and 1:155; and is lower than 1 for two option

exchanges out of ve. In this study, we evaluate two ratios: 1:0and1:25. A ratio of1:0is equivalent

to paying the full spread when executing the option trading strategy. A ratio of1:25corresponds

to a spread that is 25% higher than the quoted spread.


Table 10 presents the long-short returns for the ve trading strategies for bid-ask ratios of1:0

and1:25. When the ratio is set to 1:0, the return of the long-short straddle portfolio is 5:5%with

a t-statistic of2:67: At mid prices, the long-short return is 19:6%. Hence, bid-ask spreads reduce

the average return by 14:1%. When the ration is increased to 1:25, the long short straddle strategy

is no longer protable. It is important to note that most of the return is produced by the short

position in portfolio 1.

7 For stocks, bid and ask quotes are obtained from the CRSP database.

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As explained by Saretto and Santa-Clara (2009), another source of transaction costs are margin

requirements. Saretto and Santa-Clara (2009) document that margin requirements are very high

when shorting options. To write an option, an investor needs to borrow about one and a half times

the cost of the written option to meet margin requirements. This borrowing cost further reduces

the returns and the capacity to implement the strategy. However, large rms can dedicate enoughcash to allow the execution of these strategies.

Table 10 shows that long-short returns for the other four trading strategies are not signicant

when transaction costs are introduced. We conclude that the straddle strategy is the only strategy

to report a long-short return that is protable and signicant for a bid-ask ratio of1:0.

5 Robustness Analysis

In this section, we check the robustness of the relationship between the slope of the volatility term

structure and option returns. First, we investigate the robustness for dierent levels of moneyness

and dierent subsamples. Second, we relax the data lters to include options that violate arbitrage

bounds since they could be valid options. Then, we ensure that the results hold on earnings

announcement periods. Finally, we analyze the impact of portfolio weightings on the long-short

portfolio return. Table 11 presents the robustness analysis the long-short option returns for all ve

trading strategies. To facilitate a comparison between the primary results and the robustness test

returns, the rst row of Table 11 has the long-short returns of the baseline portfolio from Table 2.

Below, we mainly focus on the long-short straddle returns since the conclusions are very similar for

the other four trading strategies.


5.1 Moneyness

In our study, the moneyness level for call and put options is between 0:95and 1:05. Table 11 shows

that when the moneyness bounds are changed to 0:975and 1:025;the long-short straddle return is

20:8%with a t-statistic of9:08. In this case, the number of stocks per month decreases from386

to 210. If the moneyness is not bounded, the long-short straddle return decreases to 15:3%with a

t-statistic of7:77, and the number of stocks per decile increases from 386to 650. Table 11 shows

similar results for the other four trading strategies. Therefore, the moneyness level does not aectthe conclusions. The magnitude of the returns and the t-statistics remains very similar to those

reported in the primary analysis.

5.2 Sub-samples

In August 1999, stock options began to be listed in more than one U.S. option exchange. As a

consequence, option trading volume increased and bid-ask spreads decreased (see Fontnouvelle,

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Fishe, and Harris (2003), Battalio, Hatch, and Jennings (2004), Hansch and Hatheway (2001)). To

assess the impact that multiple exchange listings had on option returns, we divide the sample into

two dierent sub-periods: 1996 to 2000 and 2001 to 2007. The long-short straddle returns decrease

from the rst period to the second. For the 1996-2000 period, the long-short straddle portfolio

has an average monthly return of22:3% with a t-statistic of7:41, and the 2001-2007 period hasa return of17:5% with a t-statistic of5:85. The decrease in option returns from the rst to the

second sub-period is compensated by a 32% decrease in trading costs as reported by Fontnouvelle,

Fishe, and Harris (2003).

Next, we ensure that the triple witching Friday is not driving the results. The triple witching

Friday refers to the third Friday of every March, June, September, and December when three

dierent types of securities expire on the same day: stock index futures, stock index options and

stock options. Since the market is particularly active in these months, we divide the sample into

two groups: options that expire on the triple witching-Friday and options that expire in any other

month. The two groups obtain similar option returns. Table 11 shows that the triple-witchingFriday group has a long-short straddle return of 18:0% with a t-statistic of 4:23 and the other

group has a return of20:4%with a t-statistic of8:52.

We also control for the January eect that causes stock prices to increase during that month.

Option returns in the month of January are compared to those for the rest of the year. As Table

11 reports, the January group has an average long-short straddle return of25:0%with a t-statistic

of3:23while the non-January group has a return of19:1%with a t-statistic of8:58.

In conclusion, the relationship between the slope of the volatility term structure and the long-

short straddle returns holds for dierent sub-samples. This nding is also true for the other four

trading strategies.

5.3 Filters, Implied Volatility, and Arbitrage Bounds

Options that violate arbitrage bounds are excluded from the analysis. Duarte and Jones (2007)

note that options that violate arbitrage bounds might be valid options that, at some point in

time, have their prices below intrinsic value making it impossible to solve for an implied volatility.

To account for this bias and to include as many options as possible, we relax the lters. First,

all options with a positive volume are included even if they do not have an implied volatility.

Since some options do not have volatility, we now extract all of the implied volatilities from the

standardized OptionMetrics Volatility Surface database. IV1MandI VLTare dened as the averageimplied volatility of the call and put options with 30and 365days to expiration, and an absolute

delta of0:5. When no volatility is available, we look for a valid volatility on the 10 days before

the transaction date and select the volatility from the closest date to the transaction date. Third,

all options have exactly 30 days to expiration. When options with 30 days to maturity are not

available, we extract all of the at-the-money options with expirations between 20 and 40 days, and

choose the pair with the closest maturity to 30days. When two pairs are available, say 28and32

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days to expiration, we select the one with the highest total volume. All the other lters are applied:

positive bid-ask spread, volatility between 3% and 200%, moneyness between 0:95 and 1:05, and

underlying price above $5.

The results are robust to the inclusion of options that violate arbitrage bounds. As reported in

Table 11, the long-short straddle return is 17:7%with a t-statistic of10:40. The average numberof stocks per month increases from386 to 814. Options that violate arbitrage bounds only account

for 0:4% of the sample data. The increase in the number of stocks comes from the usage of the

OptionMetrics Volatility Surface to extract implied volatilities.

The delta-hedged put and delta-hedged call long-short returns are 2:2% and 2:7% with t-

statistics of12:29and13:50; respectively. Naked put and naked call returns are17:3%and22:4%

with signicant t-statistics of5:11 and 6:50. Note that the t-statistics for all strategies are larger

than those reported in Table 2.

In summary, option returns for all trading strategies are robust to the inclusion of options that

violate arbitrage bounds.

5.4 Earnings Announcements

Dubinsky and Johannes (2005) report that earnings announcements enhance the uncertainty of a

company, dened as the implied volatility. Volatility increases before earnings are announced and

decreases after the announcement. To conrm that the returns occur in periods other than the

earnings announcement periods, we exclude all rms that have an earnings announcement date

that falls between the transaction day and the expiration day.

When rms with earnings announcements are excluded, the magnitude of the long-short straddle

return increases to21:0%with a signicant t-statistic of8:56as reported in Table 11. When rmswith earnings announcements are included, the long-short straddle return is14:4%with a t-statistic

of2:86. The long-short returns are also robust for delta-hedged puts, delta-hedged calls, and naked

calls. However, on earnings announcement dates, the naked put premium is not signicant. Thus,

earnings announcements cannot explain the large option returns for four option trading strategies:

straddle, delta-hedged put and call, and naked call.

5.5 Controlling for Portfolio Weightings

In the primary analysis, the portfolios are equally weighted. We now explore the robustness of the

results for two dierent weighting schemes. First, we study value-weighted portfolios which are

based on the option dollar volume for each stock. Second, stocks are weighted on the option dollar

value of their open interest. For straddles, portfolios are weighted by the minimum dollar value of

the volume or the open interest between the put and the call. With the new portfolio weightings,

the long-short option returns are signicant and of the same magnitude as the original returns for

all trading strategies. As shown in Table 11, the long-short straddle returns for the value-weighted

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and the open-interest weighted portfolios are 16:1% and 18:6%, respectively, and the t-statistics

are above 3:40. Hence, the results are robust to the weighting methodology.

6 Relation between the Slope of the Volatility Term Structure

and Future Realized Volatility

Since the slope of the term structure of volatilities is related to future option returns, the next step

is to assess whether the slope of the volatility term structure is related to future realized volatility.

Future realized volatility is dened as the standard deviation of the underlying stock return over

the life of the option. Following the analysis of Cao, Yu, and Zhong (2010), we perform time-

series regressions of future realized volatility (F V) on implied volatility (IV1M), long-term implied

volatility (IVLT), and historical volatility (HV). We report the average coecients and their t-

statistics. Cao, Yu, and Zhong (2010) show that IV1M is superior to HV in forecasting future

volatility. However, does IVLT contribute to better forecast future realized volatility? We alsoinclude realized volatility from intraday returns since Busch, Christensen, and Nielsen (2011) use it

to forecast future realized volatility. To further control by rm specic volatility and higher moment

variables, we also include idiosyncratic volatility, risk-neutral skewness, risk-neutral kurtosis, the

slope of the option smirk (Option Skew), and risk-neutral jump. For each rm, we perform the

following regression:

F Vi;t = B0;t+B1;tIV1Mi;t+ B2;tIVLTi;t+ B3;tHV1;t+Control V ariables+"i;t:

We run the two stage Fama and MacBeth (1973) regressions. In the rst stage, we run the

regression for each rmi across time. In the second stage, we obtain the average for each regressor.

To account for autocorrelation and heteroscedasticity, the regressors signicance is evaluated using

the Newey-West t-statistic with 5 lags.


Table 12 summarizes the results for three univariate and four multivariate regressions. The

univariate regressions show that IV1M, IVLT, and HV contain information for future realized

volatility. The average adjusted R2 is 45%, 42%, and 30%, respectively. However, the bivariate

regressions suggest that IV1M and IVLThave a forecast ability for future realized volatility that

is superior to that ofH V. In bivariate regressions 5 and 6, the coecient ofHV is signicant for

only8%and3% of the rms, while the coecient ofIV1M is signicant for 86%of the companies

(regression 5) and the coecient ofIVLT is signicant for79%of the companies (regression 5). In

the bivariate setting,I V1MoutperformsI VLTgiven that42%of theI V1Mcoecients are signicant

to only 11%of those for IVLT.

In regression 7, all control variables are included. The variables that better forecast future

realized volatility areIV1M, realized volatility, andIVLTsince32%,13%, and8%of their coecients

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are signicant. In this regression, risk-neutral skewness, risk-neutral kurtosis, option skew, and

risk-neutral jump are signicant whereas historical volatility and idiosyncratic volatility are only

marginally signicant.

We conclude that IV1M is the single best predictor of future realized volatility. Given that

IV1Ms coecient is lower than one for all regressions, I V

1M is a biased estimator of future realized

volatility. As a predictor variable, historical volatility is outperformed by I VLT, realized volatility,

risk-neutral skewness, risk-neutral kurtosis, option skew, and risk-neutral jump.

7 Conclusions

This paper documents a positive relation between the slope of the implied volatility term structure

and ve option trading strategies in the cross section. The slope of the volatility term structure is

dened as the dierence between implied volatilities of long- and short-dated at-the-money options.

Every month, we rank stocks according to the slope of the volatility term structure and study

subsequent one month option returns. We nd that as the slope of the volatility term structure

increases, so do the one-month future returns for ve option trading strategies: straddle, naked

call, naked put, delta-hedged call, and delta-hedged put. For straddles, the portfolio of stocks with

a high slope of the volatility term structure outperforms the portfolio with a low slope by 19:6%

per month.

The results for the other four trading strategies are very similar to those of straddles. A portfolio

that buys stocks with a high slope of the volatility term structure and sells those with a low slope

generates a signicant monthly return of2:2% and 2:7% for delta-hedged puts and delta-hedged

calls, respectively. For naked puts and naked calls, this long-short strategy has a signicant return

of19:5%and 24:1%per month.

Fama-MacBeth regressions and double sorts conrm the predictive power of the slope of the

volatility term structure. The abnormal returns for any of the ve option strategies are not ex-

plained by the volatility risk premium, jump risk, investor misreaction to volatility changes, option

anomalies or rm characteristics. Transaction costs, namely bid-ask spreads, reduce the straddle

monthly prots to a still attractive and statistically signicant return of5:5%per month. However,

the returns for the other four option trading strategies do not survive transaction costs. Finally,

the large abnormal returns hold for dierent time periods, portfolio weightings, and for options

that violate arbitrage bounds. The only strategy that does not show a strong relation to the slope

of the volatility term structure is naked put. Naked put long-short returns are not signicant on

earnings announcement periods, and appear to be explained by changes between current and lagged

volatility.

A potential explanation for the positive relation between the slope of the volatility term structure

and future option returns are demand pressures of the type studied by Garleanu, Pedersen, and

Poteshman (2009). On the one hand, whenIV1M is too high compared to IVLT, option investors

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demand more options to hedge away further volatility increments. This positive demand pressure

makes these options more expensive. On the other hand, whenI V1Mis too low compared toI VLT,

there is no demand pressure and options become cheap. Carr and Wu (2008) provide evidence of a

similar phenomenon for variance swaps, where variance swap buyers are willing to suer negative

returns to hedge away upward movements in the variance. Something similar is also shown byBlack and Scholes (1972), who nd that options of high variance stocks are overpriced and options

of low variance stocks are underpriced. Therefore, demand-pressure eects might be causing the

mispricing in current option prices that leads to large future returns.

The explanation underlying the large and signicant individual option returns is, however, not

clear. Option returns are not explained by any of the three factor models used: Fama-French-

Carhart, Coskewness or Cokurtosis. Even after controlling by higher-order factors of the market

model and option-related factors, the alphas are still large and signicant. These returns might be

explained by a yet unknown risk factor. A theoretical option pricing model that accounts for the

large option returns should be investigated in future research.

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Figure 1: Time Series of Option Returns

Portfolio returns are generated as in Table 2. The gures below display the option returns of the long-shortportfolios dened as the dierence between decile 10 (highest slope of volatility term structure) and decile 1(lowest slope of volatility term structure) portfolios. Panels A through E contain long-short returns for the

following strategies: straddle, delta-hedged put, delta-hedged call, naked put, and naked call. The sampleperiod is 1996 to 2007.

Panel A: Straddle Returns

1996 1998 2000 2002 2004 2006-50

0

50

100

150

200

Time

Raw-Retu

rns(%p

ermonth)

Panel B: Delta-Hedged Put Returns Panel C: Delta-Hedged Call Returns

1996 1998 2000 2002 2004 2006-10

0

10

20

30

Time

Raw-Returns(

%p

ermonth)

1996 1998 2000 2002 2004 2006-20

0

20

40

Time

Raw-Returns(%p

ermonth)

Panel D: Naked Put Returns Panel E: Naked Call Returns

1996 1998 2000 2002 2004 2006-200

-100

0

100

200

300

Time

Raw-Returns(%pe

rmonth)

1996 1998 2000 2002 2004 2006-200

0

200

400

Time

Raw-Returns(%p

er

month)

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Figure 2: QQ-plot of Option Returns

Portfolio returns are generated as in Table 2. The gures below display the qq-plots of the option returns ofthe long-short portfolios dened as the dierence between decile 10 (highest slope of volatility term structure)and decile 1 (lowest slope of volatility term structure) portfolios. Panels A through E contain qq-plots of

the long-short returns for the following strategies: straddle, delta-hedged put, delta-hedged call, naked put,and naked call. The sample period is 1996 to 2007.

Panel A: Straddle Returns

-2 0 2

0

0.5

1

1.5

Standard Normal Quantiles

Straddle

Quantiles

Panel B: Delta-Hedged Put Returns Panel C: Delta-Hedged Call Returns

-2 0 2-0.1

0

0.1

0.2


DeltaHedgedPut

Quantiles

-2 0 2

-0.1

0

0.1

0.2

0.3


DeltaHedgedCall

Quantiles

Panel D: Naked Put Returns Panel E: Naked Call Returns

-2 0 2-2

-1

0

1

2


NakedPut

Quantiles

-2 0 2

0

2

4


NakedCall

Quantiles

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Table 1Characteristics of Portfolios Sorted by

the Slope of the Volatility Term Structure

This table reports the characteristics of ten portfolios sorted by the slope of the volatility term structure,

(Slope V T Sdened as I VLT IV1M) for the period January 1996 to June 2007. Average characteristics ofthe portfolios are reported forI V1M(the one-month implied volatility dened as the average of the ATM calland ATM put implied volatilities),I VLT(the long-term implied volatility dened as the average of the ATMcall and ATM put implied volatilities of the options with the more distant time-to-maturity), $ Size Options(Open interest of the ATM call and put multiplied by their respective mid price, in $ thousands), DTM ofIVLT (average days to maturity of the long-term implied volatility), Delta Call, Delta Put, Gamma, Vega,Size (market capitalization in $ billions), BE/ME (book-to-market ratio), risk-neutral volatility (RNvol),skewness (RNskew)and kurtosis (RNkurt)as dened in Bakshi, Kapadia, and Madan (2003), risk-neutral

jump (RNjump) is the slope of the option smirk dened as in Yan (2011), idiosyncratic volatility ( idioV ol),future volatility (F V ) computed as the standard deviation of the underlying stock return over the life ofthe option), F VI V1M, and the dierence between IV1Mand the average ofIV1Mover the previous sixmonths, I Vavg

1M .

Deciles P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Slope of the Volatility Term Structure

Slop e VTS -0.137 -0.065 -0.041 -0.026 -0.015 -0.005 0.004 0.015 0.030 0.076

IV1M 0.720 0.578 0.504 0.463 0.427 0.403 0.389 0.391 0.414 0.495

IVLT 0.583 0.514 0.462 0.436 0.412 0.398 0.394 0.406 0.443 0.571

Option and Firm Characteristics

$ Size Options 613 560 588 591 558 612 615 708 767 599

DTM of IVLT 220 220 222 222 221 223 222 223 225 222

Put-Call Spread IV1M 0.013 0.008 0.008 0.008 0.009 0.008 0.009 0.010 0.011 0.018

Bid to Mid Spread IV1M 0.067 0.067 0.069 0.069 0.070 0.069 0.072 0.070 0.072 0.083

Delta call 0.556 0.555 0.551 0.551 0.546 0.539 0.534 0.527 0.527 0.525

Delta put -0.445 -0.448 -0.452 -0.453 -0.459 -0.467 -0.472 -0.480 -0.479 -0.479Gamma 0.202 0.207 0.212 0.214 0.221 0.226 0.234 0.240 0.253 0.304

Vega 7.3 8.3 8.8 9.3 9.6 9.8 9.9 9.9 9.1 7.2

Size 4.5 8.1 11.0 14.3 15.1 18.7 20.8 21.2 18.7 9.3

BE/ME 0.352 0.425 0.369 0.369 0.392 0.364 0.369 0.359 0.338 0.338

Higher Moments

RNVol 0.624 0.549 0.496 0.466 0.440 0.426 0.420 0.429 0.465 0.573

RNSkew -0.386 -0.427 -0.472 -0.484 -0.516 -0.539 -0.562 -0.539 -0.525 -0.453

RNKurt 3.966 4.106 4.264 4.300 4.441 4.510 4.610 4.520 4.471 4.311

RNJump 0.012 0.007 0.007 0.008 0.008 0.007 0.008 0.008 0.010 0.017

IdioVol 0.032 0.027 0.024 0.022 0.020 0.020 0.019 0.019 0.021 0.026

FV 0.646 0.548 0.478 0.446 0.410 0.392 0.387 0.389 0.418 0.510

FV-IV1M -0.074 -0.030 -0.026 -0.017 -0.017 -0.011 -0.003 -0.002 0.005 0.016

IV1MIVavg1M 0.111 0.037 0.017 0.002 -0.010 -0.019 -0.028 -0.041 -0.061 -0.102

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Table 2The Slope of the Volatility Term Structure and

the Cross-Section of Option Returns

Portfolios are constructed as in Table 1. This table reports the monthly equal-weighted returns of decileportfolios for ve option trading strategies: straddles, delta-hedged put, delta-hedged call, naked put, and

naked call. We report t-statistics (t-stat), standard deviation (StDev), skewness and kurtosis values of thereturns. The last column displays the dierence between decile portfolio 10 (highest slope of volatility termstructure) and decile 1 (lowest slope of volatility term structure). The sample period is 1996 to 2007.

Deciles P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P10-P1

Straddle Returns

Mean -0.096 -0.021 -0.046 -0.005 -0.031 -0.014 0.000 0.015 0.044 0.100 0.196

t-stat (-5.89) (-1.02) (-2.91) (-0.26) (-1.70) (-0.68) (-0.01) (0.68) (1.86) (4.22) (9.17)

StDev 0.189 0.242 0.185 0.235 0.213 0.239 0.237 0.261 0.276 0.276 0.248

Skewness 0.5 1.1 0.9 1.1 0.7 1.7 1.4 1.6 1.7 1.6 0.9

Kurtosis -0.1 2.5 1.7 1.8 0.9 5.4 4.9 5.7 6.2 7.1 2.5

Delta-Hedged Put Option Returns

Mean -0.016 -0.005 -0.009 -0.003 -0.006 -0.003 -0.003 -0.001 0.002 0.005 0.022

t-stat (-5.83) (-1.82) (-4.61) (-1.47) (-3.00) (-1.47) (-1.47) (-0.37) (1.13) (2.35) (8.35)

StDev 0.033 0.031 0.021 0.026 0.022 0.024 0.022 0.023 0.025 0.027 0.030

Skewness 0.0 1.1 1.1 1.4 1.0 1.7 2.0 1.7 1.5 1.2 0.3

Kurtosis 1.6 3.7 4.1 4.9 2.8 6.2 9.8 7.3 3.7 4.7 2.0

Delta-Hedged Call Option Returns

Mean -0.015 -0.001 -0.005 0.000 -0.002 0.000 0.002 0.003 0.007 0.012 0.027

t-stat (-4.25) (-0.47) (-2.64) (-0.08) (-0.94) (0.14) (1.01) (1.41) (2.74) (4.15) (7.97)

StDev 0.041 0.037 0.024 0.028 0.025 0.024 0.024 0.024 0.028 0.032 0.039

Skewness -0.4 1.0 1.3 1.3 1.4 1.6 2.2 1.3 1.7 1.3 0.7

Kurtosis 3.1 3.8 5.3 5.0 4.2 6.8 11.0 5.1 4.6 5.6 3.6

Naked Put Option Returns

Mean -0.165 -0.187 -0.205 -0.175 -0.209 -0.147 -0.157 -0.103 -0.038 0.030 0.195t-stat (-3.37) (-3.53) (-4.05) (-3.05) (-3.95) (-2.28) (-2.84) (-1.55) (-0.58) (0.43) (4.79)

StDev 0.570 0.618 0.587 0.669 0.614 0.749 0.643 0.770 0.753 0.818 0.473

Skewness 1.1 1.2 1.5 1.8 1.7 2.6 1.6 2.2 1.8 1.5 1.2

Kurtosis 0.4 1.3 3.0 4.6 3.7 9.2 3.2 7.7 5.6 3.5 3.7

Naked Call Option Returns

Mean -0.052 0.089 0.066 0.145 0.098 0.127 0.137 0.140 0.148 0.188 0.241

t-stat (-1.06) (1.48) (1.26) (2.53) (1.83) (2.20) (2.26) (2.34) (2.19) (2.72) (5.60)

StDev 0.574 0.703 0.605 0.663 0.622 0.674 0.704 0.696 0.784 0.804 0.500

Skewness 0.8 1.1 0.4 0.7 0.5 0.5 1.0 0.6 1.2 0.7 0.5

Kurtosis 0.5 1.6 -0.4 -0.1 0.0 -0.3 2.1 -0.1 1.7 -0.4 0.2

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Table 3

Controlling for Volatility Risk Premium and Jump Risk

Option trading strategies are constructed as in Table 2. This table reports the results from the Fama-MacBeth monthly cross-sectional regressions of option returns on the volatility risk premium and jump

risk variables. The variables included are the slope of the volatility term structure, variance risk premium(Bollerslev, Tauchen, and Zhou (2009)), risk-neutral volatility, skewness and kurtosis (Bakshi, Kapadia, andMadan (2003)), option skew (Xing, Zhang and Zhao (2010)) and risk-neutral jump (Yan (2011)). The rstrow gives the coecients of the regression and the second row gives the t-statistics (in parentheses). AdjustedR2 is reported at the bottom of the table. The sample period is 1996 to 2007.

Straddle Delta-Hedged Put Delta-Hedged Call Naked Put Naked Call

(1) (2) (1) (2) (1) (2) (1) (2) (1) (2)

Intercept -0.004 -0.041 -0.003 -0.007 0.001 -0.003 -0.149 -0.269 0.120 0.178

(-0.19) (-1.06) (-1.37) (-2.12) (0.62) (-0.93) (-2.62) (-3.44) (2.35) (2.61)

Slope VTS 0.840 0.952 0.094 0.103 0.107 0.112 1.013 1.056 1.009 1.246

(8.08) (10.44) (8.48) (9.52) (7.50) (8.40) (4.78) ( 4.46) (4.04) (4.90)

VRP -0.049 0.006 0.008 -0.063 0.011

(-0.78) (0.91) (0.94) (-0.54) (0.09)

RNVol 0.015 0.005 -0.001 0.244 -0.174

(0.23) (0.78) (-0.13) (1.98) (-1.12)

RNSkew 0.001 -0.001 0.000 -0.020 0.031

(0.08) (-0.31) (-0.13) (-0.62) (0.88)

RNKurt 0.001 0.000 0.000 -0.006 0.003

(0.17) (0.17) (0.45) (-0.67) (0.39)

OptionSkew 0.263 0.020 0.034 0.340 -0.024

(1.92) (1.32) (2.11) (1.45) (-0.11)

RNJump -0.314 -0.113 0.090 -0.579 0.543

(-1.71) (-5.93) (4.00) (-1.71) (1.87)

Adj. R2 0.006 0.019 0.012 0.034 0.014 0.040 0.004 0.027 0.005 0.028

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Table 4The Slope of the Volatility Term Structure,

Volatility Risk Premium and Jump Risk,and Option Returns

Each month, rms are rst sorted into quintiles based on the volatility risk premium and jump risk, and then,

within each quintile, rms are sorted by the slope of the volatility term structure, dened as IVLTIV1M. Theslope-of-the volatility term structure portfolios are averaged over each of the ve characteristic portfolios.The characteristic included are the variance risk premium (Bollerslev, Tauchen, and Zhou (2009)), risk-neutral volatility, skewness and kurtosis (Bakshi, Kapadia, and Madan (2003)), option skew (Xing, Zhangand Zhao (2010)) and risk-neutral jump (Yan (2011)). This table reports the average option return of thedierence between quintile 5 and quintile 1, and the t-statistics (in parentheses). The sample period is 1996to 2007.

Straddle DH-Put DH-Call Naked Put Naked Call

Control P5-P1 P5-P1 P5-P1 P5-P1 P5-P1

VRP 0.134 0.015 0.017 0.170 0.148

(8.87) (8.58) (8.39) (5.81) (4.34)

RNVol 0.142 0.016 0.018 0.166 0.184

(9.42) (10.47) (10.41) (5.52) (5.77)

RNSkew 0.132 0.015 0.018 0.166 0.160

(7.80) (8.62) (8.44) (5.04) (4.74)

RNKurt 0.143 0.016 0.018 0.167 0.170

(8.89) (9.49) (9.21) (5.28) (4.91)

OptionSkew 0.124 0.015 0.017 0.148 0.169

(7.11) (8.37) (7.84) (4.17) (4.74)

RNJump 0.132 0.015 0.017 0.158 0.166(7.41) (8.16) (7.77) (4.61) (4.60)

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Table 5

Controlling for Option Anomalies

Option trading strategies are constructed as in Table 2. This table reports the results from the Fama-MacBeth monthly cross-sectional regressions of option returns on option anomalies. Variables related to

option anomalies are the one-year historical volatility of daily return minus implied volatility (HV IV1M),idiosyncratic volatility (idioV ol), and the ratios between IVt

1Mand one-month (IVt11M

), 3-month (IVt31M

),6-month (IVt6

1M ) lagged implied volatility as well as the maximum (IVmax

1M ) and average (IVavg

1M) implied

volatilities over the previous 6-months. The rst row gives the coecients of the regression and the secondrow gives the t-statistics (in parentheses).

Documents

crosssec-optret