metekarakaya_optionpremia_jmp

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Characteristics and Expected Returns in

Individual Equity Options

Mete Karakaya

January 11, 2014

Abstract

I study excess returns from selling individual equity options that are leverage-adjusted

and delta-hedged. I find that options with longer maturities have higher risk yet lower

average returns. I identify three new factorslevel, slope, and valuein option returns,

which together explain the cross-section of expected returns on option portfolios formed on

moneyness, maturity, and value. This three-factor model also helps explain expected returns

on option portfolios formed on twelve other characteristics. While the level premium appears

to compensate investors for market-wide volatility and jump shocks, market frictions help us

understand the slope and value premiums.

Keywords: expected option returns, option characteristics, option strategies, factor pricing

JEL codes: G11, G12, G13

I thank my advisors John Cochrane, Bryan Kelly, Ralph Koijen, Lubos Pastor and Harald Uhlig for theirinvaluable guidance. I am grateful for comments, suggestions and support from Tobias Moskowitz, Pietro Veronesi,George Constantinides, Andrea Frazzini, Lasse Pedersen, Dacheng Xiu, Emre Kocatulum, Can Bayir, Rui Mano,Diogo Palhares, Burak Saltoglu, Shri Santosh, Michael Baltasi and seminar participants at the University of ChicagoBooth School of Business, Central Bank of Turkey.

University of Chicago Department of Economics; Email: [email protected]; phone: 312-213-8575;Web: home.uchicago.edu/metekarakaya
mailto:[email protected]:[email protected]://home.uchicago.edu/metekarakayahttp://home.uchicago.edu/metekarakayamailto:[email protected]


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I. Introduction

The market for individual equity options is huge but relatively understudied. I examine how

expected returns vary across individual equity options with a particular focus on moneyness,

maturity and option value. Specifically, I study excess returns from selling individual equityoptions that are leverage-adjusted monthly and delta-hedged daily.

Both delta-hedging and leverage adjustment are essential to my analysis. Delta-hedging strips

out most of the expected return variation that comes through stock returns. This allows me to

focus on the expected return variation that is unique to option markets. In the Black-Scholes

world, delta-hedged excess returns are zero on average, since agents can perfectly replicate options

by continuously trading the underlying stock and a risk-free bond. However, in the real world,

perfect replication fails due to discrete trading, transaction costs, and the presence of untradable

state variables such as stochastic volatility and jumps. As a result, delta-hedged excess returns

are not zero on average and they potentially compensate investors for risks, behavioral biases and

market frictions. Leverage adjustment is important because options with different moneyness and

maturity have vastly different degrees of leverage (e.g.,Frazzini and Pedersen(2012)). Differences

in leverage dominate differences in option behavior. To account for leverage effect, I use the

sensitivity of an options return to the return of the underlying stock. This definition of leverage

measures the options return magnification relative to the return of the underlying stock. By

leverage-adjusting, I am able to detect new risk and return patterns that were previously obscured

by leverage.

I uncover a puzzling connection between option maturity, risk and expected return. I findthat options with longer maturities have higher risk yet lower average returns. For example,

option portfolios with a long maturity have more volatile returns, higher market beta and lower

VIX beta compared to option portfolios with a short maturity. Moreover, they perform worse

during market-wide price and volatility jump episodes. According to popular belief, options earn

a premium as a compensation for market-wide volatility and jump shocks. Here, I argue that long

maturity options have higher volatility and jump risk yet lower average return. It is crucial to

adjust returns for leverage to see this maturity-risk pattern.

Earlier literature1 finds that the expected return on selling options decreases by moneyness,

maturity and option value (the spread between historical volatility and the Black-Scholes implied

volatility).2 Standard risk adjustment models, such as the CAPM,Fama and French(1993) three-

factor (FF3), and Carhart (1997) four-factor (FF4) or five-factor (FF4 plus a volatility factor)

models, all fail to explain these patterns. Even more strikingly, these models all predict higher

1Frazzini and Pedersen(2012);Goyal and Saretto(2009)2This spread is akin to the book-to-market ratio of a stock. For options, Black-Scholes implied volatility is a

measure of market value and realized volatility is a measure of fundamental value.

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expected returns for long maturity options relative to short maturity ones.

I identify three new return-based factorslevel, slope, and valuewhich together explain the

cross-sectional variation in expected returns on option portfolios formed on moneyness, maturity

and option value. The level factor is the average return on selling at the money (ATM) option

portfolios. The slope factor is the average return on buying long maturity option portfolios andselling short maturity ones. Lastly, the value factor is the average return on buying high value

option portfolios and selling low value ones. In the spirit of theFama and French(1993) three-

factor model, I argue that we can describe the expected excess return of option portfolios based

on their sensitivity to these three factors. In particular, the expected excess return of portfolio i

can be described as,

E[Rei ] =Li E[Level] +

Si E[Slope] +

Vi E[V alue] (1)

Rei,t = i+ Li Levelt+

Si Slopet+

Vi V aluet+ i,t (2)

If the option three-factor model (OPT3: level, slope and value) in equation1and2holds, with

i= 0 i, then the model is consistent with the idea that investors require a higher premium for

certain options because of their comovements with systematic factors.

The variation in the level betas capture the expected return variation along moneyness di-

rection, by gradually rising from in the money to out of the money option portfolios. Similarly,

the slope betas gradually decrease from short to long maturity option portfolios and as a resultexplain the variation along maturity. Lastly, the value betas rise smoothly from high to low value

portfolios and explain the expected return variation among value portfolios.

To show the economic success of the option three-factor model, I compare its mean absolute

pricing errors (MAE = 1N

||) against alternative models using thirty portfolios formed on

moneyness and maturity. The mean absolute average excess return ( 1N

|Re|) of the thirty

portfolios is 44 basis points (bps). This 44 bps can also be viewed as the MAE according to

Black-Scholes model (BS), since BS predicts that delta-hedged excess returns are on average equal

to zero.3 CAPM, FF4 and FF5 produce MAEgreater than 37 bps. On the other hand, OPT3

achieves MAE of only 15 bps. The results are qualitatively similar if we compare MAEs on

decile portfolios formed on value. BS MAE is 38 bps and the CAPM, FF4 and FF5 produce a

MAEof greater than 30 bps. In contrast OPT3 achieves M AEof less than 10 bps.

3Black-Scholes model predicts zero expected delta-hedged excess return if hedging is done continuously. Inpractice I rely on daily delta-hedging. According to my simulation results, daily delta-hedging introduce a bias inthe opposite direction. Bakshi and Kapadia (2003) get similar results. Therefore we should interpret 44 bps aslower bound to the M AEof BS.

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The option three-factor model (OPT3) is equally successful in sub-samples and it performs well

on several other sets of portfolios formed on twelve other characteristics. OPT3 is also economically

significant. Annualized Sharpe ratios of the level, slope and value factors are 0.86, 3.22 and

2.61, respectively. Moreover, the tangency portfolio of the factors is magnifying the economic

significance of OPT3, because the tangency portfolio has a Sharpe ratio of 5.15. Given theeconomic significance and empirical success of OPT3, it is economically interesting to understand

the level, slope and value factors.

I argue that the level factor is a risk factor, because it tends to crash during financial and

liquidity crises such as Lehmans bankruptcy, the European sovereign crisis, the Asian financial

crisis, the Russian default and the bankruptcy of WorldCom. Moreover, I find strong evidence

that the premium on the level factor represents a compensation for market-wide volatility and

jump shocks. First of all, it is highly correlated with the innovations in VIX (-0.7), which can be

considered as a proxy for volatility shocks. The unconditional expected return on the level is 46

bps, while expected returns conditional on market-wide price-jumps and volatility-jumps are -200

and -82 bps, respectively. This evidence suggests that the level factor has a high exposure to jump

shocks. During severe bear markets, expected returns on the level factor rise dramatically to 126

bps. This is in line with the central intuition of macro asset pricing models with time-varying

expected returns. Expected returns are higher during recessions, since marginal value of wealth is

high at those times.

In contrast to the level factor, the empirical properties of the slope factor do not appear to

coincide with standard risk measures. It is highly sensitive to market-wide volatility and jump

shocks; however, the sign of sensitivity is positive. Volatility and jump shocks amplify the puzzleby implying a negative premium for the slope factor. In order to explain the high premium on

the slope, we need to explain why short maturity options have higher expected returns than long

maturity options.

There are two key differences between short and long maturity options, which affect the supply

and demand for options. The first one is gamma ( 2OptionPriceStockPrice2 ). It tells us the sensitivity of hedge

ratio (delta) to the movements in the underlying stock. It is more costly to hedge options with a

higher gamma and the hedge is less effective. When there is a large movement in the underlying

stock, selling options with higher gamma result in larger losses. Short maturity options have

substantially higher gamma than long maturity options. Even in the Black-Scholes world, the

average hedging cost of one-day to maturity options can be more than 200 times the hedging cost

of one-year to maturity options. In fact, what market makers fear most is having too much gamma

in their portfolio. With high negative total gamma, they can lose an arbitrarily large amount of

money, while the premium is limited. As a result, market makers are less willing to supply short

maturity options.

The second difference is embedded leverage (the amount of market exposure per unit of com-

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mitted capital). Short term options enable investors to take on higher leverage than long term

options. Frazzini and Pedersen(2012) argue that securities with high embedded leverage allevi-

ate investors leverage constraints. Therefore, investors are willing to pay more for assets that

enable them to increase their leverage. Because of high embedded leverage in short term options,

constrained investors demand short-term options more, driving up their prices.The demand-based option pricing model ofGarleanu, Pedersen, and Poteshman(2009) is re-

lated to both gamma and embedded leverage. They argue that risk-averse financial intermediaries

require a higher premium on options with higher demand pressure, since they cannot hedge their

positions perfectly. Their theoretical model implies that demand pressure in one option contract

increases its price by an amount proportional to the variance of the unhedgeable part of the op-

tion. Because of gamma, the unhedgeable part is bigger for short term than long term options,

and because of embedded leverage, the demand pressure is bigger for short-term than long-term

options.

The value premium is difficult to explain as well. At the beginning of the financial crises, the

value factor tends to lose, but once the turmoil begins, volatility spreads widen and the expected

returns on value rise dramatically. It is not possible to explain the value premium with market-

wide volatility or jump shocks. It has almost no correlation with contemporaneous innovations in

VIX and it tends to perform better during jump episodes. Average variance risk premia (VRP) is

considerably larger for low relative to high value portfolios. Schurhoff and Ziegler(2011) develop

a model consistent with theories of financial intermediation under capital constraints, in which

both systematic VRP and idiosyncratic VRP is priced. The interpretation of the value premium

as compensation for risk-averse financial intermediaries is consistent with Schurhoff and Ziegler(2011) model.4

If market frictions drive the slope and value premiums, then we should expect that these

premiums are related to funding liquidity conditions.5 I find that the premium on the slope and

value factors are higher when funding liquidity conditions are tight. The finding is consistent with

the idea that capital is required to exploit the slope and value premiums, and capital becomes

more costly when funding liquidity conditions are tight

I hypothesize that if market makers are averse to having too much gamma in their portfolios (as

I argue to explain maturity premium), then they will require a higher premium for selling options

on stocks in which they have high total gamma. I base this on the fact that their portfolios with a

higher total gamma will lose more when there is a large movement in the underlying stock price.

4I also find that low value option portfolios experience substantially higher growth in their total option marketcapital then high value option portfolios in the portfolio formation month. This might be a sign for higher demandpressure. If this is the case, then Garleanu, Pedersen, and Poteshman(2009) model can potentially explain thepremium on the value factor. I define total option market capital as the sum of open interest times mid-price ofall options on the underlying stock.

5 FollowingFrazzini and Pedersen(2010), I proxy funding liquidity by TED spread.

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To test this hypothesis, I construct a new characteristic open interest gamma, which is defined

as the sum of open interest times gamma of all options for a given underlying stock divided by

the market capital of that underlying stock. I sort options into decile portfolios by open interest

gamma and I find that my hypothesis is correct. Excess returns and Sharpe ratios rise from low to

high in open interest gamma portfolios. The high minus low portfolio has a return with a Sharperatio of 1.7 and t-statistics of 7.1.

Lastly, I explore two new option investment strategies: option carry and volatility reversals.

Carry is traditionally applied only to currencies; but Koijen, Moskowitz, Pedersen, and Vrugt

(2012) generalize this concept to other asset classes such as global equities, bonds, commodities

and index options. I find that the carry trade is extremely successful in individual equity options,

with the high minus low carry portfolio achieving an annualized Sharpe ratio of 2.5, which is

greater than the premium on carry strategies in other asset classes. I define volatility reversals as

the change in implied volatility over the past month for a given delta and maturity. A high minus

low volatility reversals portfolio generates considerable return with an annualized Sharpe ratio of

1.4. I show that the option three-factor model (OPT3) performs well at explaining the returns of

option carry and volatility reversals strategies.

I contribute to the large literature on empirical option pricing by showing that the option

three-factor model (OPT3) explains a substantial portion of average returns on option portfolios

formed on embedded leverage, stock size, stock and option illiquidity, stock short-term reversal,

volatility term structure (only for short maturity options), idiosyncratic volatility and variance risk

premia. However, OPT3 fails to explain returns on portfolios formed on volatility term structure in

long maturity options. All of these patterns are established by previous researchers. For instance,Christoffersen, Goyenko, Jacobs, and Karoui(2011) find that illiquid options have higher expected

returns and options on illiquid stocks have lower expected returns for buyers. Vasquez(2012) finds

that option portfolios with a high slope of implied volatility have higher returns for buyers than the

ones with a lower slope of implied volatility term structure. Ang, Bali, and Cakici(2010) show that

call options on stocks with high returns over the past month (stock short-term reversal), display an

increasing implied volatility. Cao and Han(2012) find that the return on buying options decreases

with an increase in idiosyncratic volatility of the underlying stock. Schurhoff and Ziegler(2011)

show that variance risk is related to the cross-section of expected synthetic variance swap returns.

Di Pietro and Vainberg(2006) find that firm characteristics such as size and book-to-market ratio

are related to expected returns on synthetic variance swaps.

The paper consists of seven sections. InSectionII, I describe the data and explain the method-

ology. InSectionIII,I study patterns of expected option returns related to moneyness, maturity

and value. InSectionIV,I propose an empirical pricing model and conduct asset pricing tests. In

SectionV, I discuss economic interpretations for the pricing factors and option premia. I study

patterns of expected option returns related to other characteristics and conduct asset pricing tests

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on them inSectionVI. InSectionVII, I provide concluding remarks.

A. Literature Review

This paper is directly related to the recently growing literature on expected option returns. We

can broadly classify these papers into three categories. The first argues that option prices are

potentially affected by risk preferences. Coval and Shumway(2001) show that zero beta at the

money straddle returns are negative on average, which may suggest the presence of additional

priced factors such as systematic stochastic volatility. Bongaerts, De Jong, and Driessen(2011)

show that correlation risk is priced. Bakshi and Kapadia(2003) study delta-hedged option gains,

and argue that volatility risk is priced. The second category is about mispricings due to distorted

expectations (behavioral biases). Stein(1989),Poteshman(2001) andGoyal and Saretto (2009)

are examples. The final category ties option prices to market frictions. Two examples of market

frictions are liquidity and embedded leverage (Christoffersen, Goyenko, Jacobs, and Karoui(2011),Frazzini and Pedersen(2012)). I contribute to this literature in several ways. First, I document

that carry and volatility reversals are related to expected option returns. While most of the

existing studies are done in a small sample with only at the money one-month maturity options,

I extend their results to different moneyness and maturity groups.

Option pricing theories are also relevant for this paper. Based on option pricing theories,

excess option returns should compensate investors for a variety of different risk premia, such as

stochastic volatility and price-jump and volatility-jump risk premium. Some notable examples in

this area areBates(1996);Pan(2002);Broadie, Chernov, and Johannes(2007).6

Most existing option pricing theories are designed for index options. A promising area of

research is to extend these models for individual equity options, which requires the daunting task

of identifying priced factors affecting option premia. A good starting point, followed byElkamhi

and Ornthanalai(2010) andChristoffersen, Fournier, and Jacobs(2013), is to incorporate market-

wide jump and volatility as priced factors. My results show that we need more than market-wide

volatility and jump factors to explain option premia.

Finally, this paper is also related to the growing literature studying term structure of risk and

risk premia across asset classes. Van Binsbergen, Brandt, and Koijen (2010) show that short-

term dividend strips on the S&P 500 index have higher returns and Sharpe ratios than long-termdividend strips. Palhares(2012) find that average returns decrease by maturity in CDS markets.

Duffee(2011) show that Sharpe ratios decrease by maturity for nominal government bonds. 7

6See alsoHeston(1993);Bates(2000);Carr and Wu(2004);Liu, Pan, and Wang(2005).7See alsoVan Binsbergen, Hueskes, Koijen, and Vrugt (2011);Lettau and Wachter(2007);Hansen, Heaton, and

Li(2008).

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II. Methodology and Data

In this section, I first describe my data sources and then provide a detailed explaination of the

filters that I use. After that I present the summary information on my final sample, and from

there go on to explain holding period option excess return calculations that are delta-hedged dailyand leverage-adjusted monthly.

A. Data Sources

My primary data source is the OptionMetrics Ivy DB database, the industry standard for his-

torical option price information. The database was first launched in 2002 and has since been

compiling the dealers end-of-day quotes directly from the U.S. exchanges. This data includes all

of the U.S. exchange-listed individual equity options from January 1996 to January 2013. All of

the options are American style. From the OptionMetrics option price files, I use the followingvariables: the daily closing bid-ask quotes, open interest, trading volume, implied volatility and

the option Greeks (delta, gamma, vega, theta). Implied volatility and the Greeks are calculated

by using Optionmetrics proprietary algorithms based onCox, Ross, and Rubinstein(1979) bino-

mial tree model, which accounts for discrete dividend payments and the early exercise possibility

of American options. As an input to their algorithms, they use the term structure of interest rates

derived from both the LIBOR rates and the settlement prices of the Chicago Mercantile Exchange

for Eurodollar futures. From the security price files, I use the following variables: close price,

return, volume, and shares outstanding. Lastly, I get the interpolated implied volatility, implied

strike price, delta, and days to maturity from the volatility surface files.

In addition, I use accounting data from Compustat to calculate the book value of companies.

I also get daily returns, volume, closing price and shares outstanding information from CRSP.

Because the Optionmetrics security price file starts in 1996, I use data from CRSP whenever

Optionmetrics data are not available. Lastly, I use high frequency stock trading data from TAQ

to estimate realized variances.

B. Filters and Final Sample

Generally, I use filters in the same manner as in previous studies, such asGoyal and Saretto(2009);Frazzini and Pedersen (2012). I drop all observations where the bid is greater than the ask or

where the bid is equal to zero. Minimum tick size for options trading under $3 is equal to $0.05

and $0.10 for all others. I eliminate all observations where the bid-ask spread is smaller than

the minimum tick size and all observations that violate arbitrage bounds. I keep options with

standard settlements and expiration dates. I require positive open interest, non-missing delta,

implied volatility and spot price to keep the observations in the sample.

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FollowingFrazzini and Pedersen(2012), I apply filters to eliminate options with very little time

value F-V, where F is an the price of option and V is intrinsic value ( payoff you will receive if you

exercise, max(Spot-Strike,0) for call options and max(Strike-Spot,0) for put options). Specifically,

I eliminate options where the time value expressed as a percentage of option price (F-V)/F is

less than 0.05. These options might get exercised early, which can affect the accuracy of theirreturn calculations. To control for outliers, I drop observations with embedded leverage at the top

and bottom 1% of the distribution for both calls and puts separately. The embedded leverage is

defined as the elasticity of option price with respect to the underlying stock price PP /SS .

Unlike previous studies, I require that observations have positive volume to keep them in the

sample. Closing price data might not be reliable, if the option contract has not been traded for a

while.

Lastly, I find recording errors for 9 optionids (11928459, 33108873, 33108872, 44963448, 25242345,

45209832, 10758314, 46164533, 11932329) when searching for extreme observations. In each case

the option price jumped more than couple of thousand percent and reversed back a few days

later, when there was no significant move in the underlying equity. I reported each case to the

Optionmetrics support desk.

For my final sample, I use 11,008,246 option-months in total (6,066,370 call option-months and

4,941,876 put option-months). These observations come from 3,480,644 unique option contracts

(1,858,744 call and 1,621,900 put) and 7535 underlying stocks. On average there are 53,000 options

and 2500 underlying stocks per month.

Figure 1: Trading Volume

This figure displays aggregate trading volume in trillion dollars through years and allocation of tradingvolume across 5 moneyness and 6 maturity groups. Panel A shows the results in market value, which isvolume times mid-price of an option, while Panel B shows the results in notional, which is estimated asvolume times closing price of underlying stock. Moneyness groups are DOTM (deep out of the money,0 ||< 0.20), OTM (out of the money, 0.20 ||< 0.40), ATM (at the money, 0.40 ||< 0.60 ) ITM(in the money, 0.60 || < 0.80 ), DITM (deep in the money, 0.80 || < 1). Maturity groups are 1, 2,3, 4 to 6, 7 to 12, greater than 12 months.

Panel A: Panel B:

1996 1998 2000 2002 2004 2006 2008 2010 20120

0.5

1

1.5

2Option Trading Volume (Market Value)

Vo

lume(intrilliondollars)

0 10 20 30 40

DOTM

OTM

ATM

ITM

DITM

Percentage of total volume

Moneyness

|| 0.2

0.2 < || 0.4

0.4 < || 0.6

0.6 < || 0.8

0.8 < ||

0 10 20 30 40

1M

2M

3M

4:6M

7:12M

>12M

Maturity


AllCallPut

1996 1998 2000 2002 2004 2006 2008 2010 20120

10

20

30

40

50

60

70Option Trading Volume (Notional)

Vo

lume(intrilliondollars)

AllCallPut

0 10 20 30 40

DOTM

OTM

ATM

ITM

DITM


Moneyness

|| 0.2

0.2 < || 0.4

0.4 < || 0.6

0.6 < || 0.8

0.8 < ||

0 20 40 60

1M

2M

3M

4:6M

7:12M

>12M

Maturity


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C. Institutional Details on Individual Equity Option Market

At the end of 1990s, the individual equity options market was relatively small, with aggregate

trading volume less than 400 billion dollars of market value and 10 trillion dollars of notional

value. In 2000, aggregate trading volume spiked up to above 500 billion dollars market value,

but after the dot-com bubble burst, trading volume shrank back to lower levels. Between 2004

and 2012, volume increased at a rapid pace. As a result, the individual equity option market has

turned into a giant market with trading volume of about 60 trillion dollars in notional and 1.5

trillion dollars in market value.

There is a considerable variation of trading volume across maturity and moneyness groups.

Short maturity options are more liquid than long maturity options, since more than 30% of

market value and 50% of notional value is in options with a one month maturity. In terms of

market value, at-the-money options have the biggest share, which is more than 35%. On the other

side, in notional amount, deep-out-of-the-money options have largest share.

D. Return Calculations (leverage-adjusted monthly, delta-hedged daily)

I work with holding period returns for approximately one month. The expiration date of individual

equity options is every month following the third Friday of a given month. I open the position

following the expiration date, usually a Monday, and hold it until the expiration date of the next

month while delta-hedging for the underlying stock daily. My delta-hedged return calculations are

exactly the same as in Frazzini and Pedersen(2012). The exact algorithm for the delta-hedged

return calculations is as follows: let V be the value of my portfolio, F the option price, S the

stock price, x the number of option contract holdings, the Black-Scholes delta, rSt the stockreturn and rft the risk free rate. The starting value of my portfolio V0 is $1 and I buy x =

1F0

unit of option contracts, which I hold till the next expiration date. Every day, I gain or lose some

money from the change in option price x (FtFt1) , delta hedging xt1rStS and lastly from

the margin account rft (Vt1xFt1+xt1St1). I iterate the value of my portfolio using this

algorithm from the beginning of holding period 0 to end of holding period T.

Vt = Vt1 + x (FtFt1)

Options price appreciation xt1r

StSt1

P&L from delta hedge+ rft (Vt1xFt1 + xt1St1)

gain or loss from margin accountThe delta-hedged return for option i is defined as the change in the value of portfolio during

the holding period.

ri,[0,T] = VTV0 = VT1

I subtract the monthly risk free rate in order to work with excess returns.

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re[0,T]= ri,[0,T]

Tt=0

1 +rft

1

Lastly, I adjust for leverage, which is defined as the elasticity of the option price with respect

to the stock price.

=|FFSS

|= |S

F

F

S|=

S

F||

I scale delta-hedged excess returns with leverage in order to derive leverage-adjusted excess

returns.

Rei,[0,T]= 1

rei,[0,T]

In my analysis, I specifically work with option market cap weighted portfolio returns, where

option market cap is defined as mid-price times open interest of the option contract. LetRp,[0,T

be leverage-adjusted delta-hedged excess return of portfolio p.

Rp,[0,T] =Ni=1

viRei,[0,T]

In all of these calculations, I use a closing bid-ask midpoint for the option price; hence, Iimplicitly assume no transaction costs. In my margin account calculations, I assume there is no

spread between lending and borrowing rates and that the initial margin requirement is zero.

E. Fama-French-Carhart Four-Factor Model Calculations

Rather than working with the standard monthly data from the beginning of a given month to the

next, my holding period is from the fourth week of each month to the fourth week of following

month. I calculate four factors for my holding period. I start with daily excess returns and add

back risk-free rates, then I compound over the holding period and subtract monthly risk-free rates.My holding period return calculations for each factor are as follows:

Rmrf

=Tt=1

1 +Rmdt

Tt=1

1 +rft

SM B[0,T] =Tt=1

1 +SM Bdt +r

ft

Tt=1

1 +rft

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HML[0,T]=Tt=1

1 +HMLdt + r

ft

Tt=1

1 +rft

W ML[0,T]=T

t=1 1 +W MLdt +rft T

t=1 1 +rft whereRmdt is daily market return,SMB

dt daily SMB return,HML

dt daily HML return,W ML

dt

daily WML return from Kenneth French data library.

F. Characteristic Calculations

In this subsection, I give detailed explanations for the fifteen option-stock characteristics I used.

These characteristics are moneyness, maturity, value, option carry, variance risk premia, volatility

reversals, systematic volatility, idiosyncratic volatility, stock risk reversal, stock size, stock illiquid-

ity, option illiquidity, embedded leverage, slope of volatility term structure, open interest gamma.

As a precaution against recording errors and measurement errors, I keep a 1 day lag between the

date I estimate characteristics and the date I take position on an option. Moneyness and maturity

are the only two exceptions to this rule.

Moneyness: I define the five moneyness groups in terms of absolute value of delta || .

1) Deep out of the money (DOTM) 0 ||< 0.20

2) Out of the money (OTM) 0.20 ||


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the increase in realized volatility. This result is a direct consequence of positive (negative) gamma

of long (short) option positions and independent of the directions of stock price movements or the

final outcome of the stock price. We should understand that the option has a low value when the

price (implied volatility) is high relative to historical volatility and high value if the price (implied

volatility) is low. For stocks, we compare market price to some intrinsic assessment of stock(book value) to build value style investment strategies. For options, the same style of investment

corresponds to making an investment by comparing Black-Scholes implied volatility and historical

volatility. The essential philosophy of a value-style investment is just buy cheap assets and sell

expensive assets according to some measure.

Option Carry: FollowingKoijen, Moskowitz, Pedersen, and Vrugt(2012), I define carry as the

return of an option contract if the underlying stock price and implied volatility term structure do

not change. LetFt(St, K , T , T) be the price of an option contract (either call or put) at time

t, with maturity T, strike K and implied volatility T. Koijen, Moskowitz, Pedersen, and Vrugt

(2012) show that we can approximate the carry using the options first time derivative theta ( )

and first volatility derivative vega . Let Ct(St, K , T , T) denote the options carry.

Ct(St, K , T , T) =t(St, K , T , T)t(St, K , T , T) (TT1)

Ft(St, K , T , T)

Ft , , and T are available from Optionmetrics, I interpolate T1 from volatility surface

files. Lastly I adjust carry for leverage, since all my returns are leverage-adjusted. From now on,

I will always mean a leverage-adjusted carry when I talk about carry.

1

Ct(St, K , T , T)

Variance Risk Premia (VRP): FollowingBollerslev, Tauchen, and Zhou(2009), I define VRP

for each underlying stock as the difference between model-free implied volatility or, in other words,

ex-ante risk-neutral expectation of the future return variation (IVi,t) over the [t, t+ 1] time period

and the ex post realized return variation (RVi,t) over the [t1, t] time period.

V RPi,t = I Vi,tRVi,t

My estimation procedure ofIVi,t is as inHan and Zhou(2012).

IVi,t = 2

0

Cit(t+ T, K) /B (t, T)max (0, Sit/B(t, T)K)

K2 dK

whereSit represents stock price of stockiat timet,T = 1/12. Cit(t+ T, K) stands for price of

call option on stocki, with strike K and time to maturity T. B(t,T) denotes price of zero coupon

12


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bond that pays one dollar at time t+T. I numerically estimate IVi,t. At the end of the previous

holding period, I extract implied volatilities of 30 day call options from standardized Volatility

Surface files provided by OptionMetrics. I then transform these implied volatilities into option

prices using the Black-Scholes model.

To estimateRVi,t , I get TAQ intraday equity trading data spaced by 15 minute intervals. Letpij,t denote log price of stock iat the end ofjth 15-minute interval in holding period tand let Nt

denote number of trading days and nt number of 15-minute intervals in holding period t.

RVi,t =252

Nt

ntj=1

pij,tp

ij1,t

2Volatility Reversals: Let t(, T) denote Black-Scholes implied volatility at time t, which

belongs to an option with delta and time to maturity T. From the last holding period volatil-

ity surface file, I interpolate the implied volatility with the same delta and time to maturityt1(, T). Timing: if I am taking a position at the 4th Monday of month t, t(, T) is the

implied volatility of an option from the 3rd Thursday of month t, t1(, T) is interpolated from

the 3rd Thursday of month t-1. I define volatility reversals as t(, T)t1(, T). This measure

gives us the change in implied volatility at a specific point on the volatility surface.

Systematic and Idiosyncratic Volatility: FollowingCao and Han(2012), idiosyncratic volatility

IV OLi,t is defined as the standard deviation of residuals from the Fama French three-factor model,

which is estimated using daily data over the previous holding period. Systematic volatility is

defined as the V OL2i,tIV OL2i,t, where V OLi,t is the standard deviation of daily returns ofstock i in period t.Stock Short-Term Reversal: Jegadeesh(1990) define short term reversal as the stock return

over the previous month. Since I dont work with regular monthly data, I calculate the stock

return over the previous holding period, which is about a month.

Stock Size: The firm size is a natural logarithm of the market value of equity, which is estimated

as the stock price times the number of shares outstanding.

Stock Illiquidity: FollowingAmihud(2002), I define a stocks illiquidity as |Ri,t|

Vi,t, where Ri,t is

the month t return of stock i, and Vi,t the total dollar volume of stock i in month t.

Option Illiquidity: I measure option illiquidity by the relative quoted spread. PreviouslyChristoffersen, Goyenko, Jacobs, and Karoui(2011) use this measure.

bid ask

(bid+ask) /2

Embedded Leverage: Following Frazzini and Pedersen (2012), I define embedded leverage as

the elasticity of the option price with respect to the stock price. In practice I use the delta from

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the Black-Scholes model. I denote embedded leverage with .

=

FFSS

=

S

F

F

S

= S

F ||

Slope of Volatility Term Structure: I define slope of volatility structure as the difference between

implied volatility of at-the-money (|| = 0.5) options with 365 days to maturity and 30 days

to maturity. Since it is not possible to have 365 and 30 days to maturity, implied volatilities

interpolated numbers from Optionmetrics volatility surface files. This characteristics is defined

separately for both call and put options.

Open Interest Gamma: I define open interest gamma as the sum of open interest times gamma

of all options for a given underlying stock divided by market capital of that underlying stock.

This characteristic measures the total gamma of all investors who short options on a given stock.

My implicit assumption is that this measure is correlated with total gamma of option marketmakers on a given stock. Because of their risk aversion, option market makers will charge a higher

premium when they have high total gamma. I base this on the fact that their portfolios with

a higher total gamma will lose more when there is a large movement in the underlying stock

price. I scale by the market capital of the underlying stock, because I expect that there are more

option market makers with larger capital for options on large stocks. You should consider this

characteristic as a noisy measure of total gamma of option market makers. Note that options on

the same stock share the exact same number for this characteristic.

III. Moneyness, Maturity and Value Patterns

In this section, I summarize the descriptive statistics for the thirty portfolios formed on moneyness-

maturity and the decile portfolios formed on value. For every month, the standard expiration date

is the Saturday immediately following the 3rd Friday of the month. For the first trading day

following the expiration date each month, I assign all available options into thirty groups based

on their moneyness and maturity. Note that each group includes both call and put options. The

moneyness and maturity groups as well as general definitions all follow Frazzini and Pedersen

(2012). I measure moneyness with the absolute value of delta || and assign options into 5moneyness categories based on the range of ||: Deep out of the money 0 ||


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as two standard deviation confidence intervals across moneyness and maturity categories. You

can find more detailed descriptive statistics across moneyness and maturity in the Table1. I will

briefly summarize the most important patterns.

Figure 2: Average Monthly Excess Returns Across Moneyness-Maturity

This figure displays average excess returns (basis points per month) from selling options across 5 moneynessand 6 maturity groups. Red lines indicate two standard deviation confidence intervals for average excessreturns. Moneyness groups are DOTM (deep out of the money, 0 || < 0.20), OTM (out of the money,0.20 ||< 0.40 ), ATM (at the money, 0.40 ||< 0.60) ITM (in the money, 0.60 ||< 0.80), DITM(deep in the money, 0.80 ||< 1). Maturity groups are 1, 2, 3, 4 to 6, 7 to 12, greater than 12 months.

DOTM OTM ATM ITM DITM

0

50

100

150

200

250

300

MoneynessMaturity

Excessreturn(basispointspermon

th)

|| 0.2

0.2


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in risk. The rise in these risk measures with maturity is more pronounced in DOTM, OTM and

ATM option portfolios. Panel D report average excess returns across maturities during price jump

episodes. Price jump indicates months with at least one day with a jump in S&P 500 index, which

is defined as daily change in index less than -4%. You can see that long maturity options perform

worse during episodes of price jump. For example, in the full sample ATM option portfolio withone month to maturity and three months to maturity have 132 bps and 28 bps of average excess

return per month, while during episodes of price jump, average return of option portfolios are

-62 and -252 bps, respectively. This evidence suggests that long maturity options are more prone

to crash risk. I also consider episodes of volatility jump and market distress. Volatility jump

indicates months with at least one day with a jump in the VIX index, which is defined as the daily

change in VIX greater than 4%. This definition of price jump and volatility jump closely follow

Constantinides, Jackwerth, and Savov(2011). I define market distress as the months with con-

temporaneous monthly S&P 500 returns of less than -5%. In total there are 14 months with price

jumps, 39 months with volatility jumps and 25 months with contemporaneous market returns less

than -5%. The average excess return pattern on the thirty moneyness-maturity portfolios during

volatility jump episodes and market distress are similar to the average excess return pattern during

price jumps, which is reported inFigure3 panel D.

Previous researchers did not notice a maturity-risk pattern, because they focus on Index options

and study leverage-unadjusted option returns. If we look at the same risk measures for leverage-

unadjusted returns of S&P 500 index options, patterns are completely reversed. Short maturity

options appear to be more risky than long maturity options. For leverage-unadjusted returns of

individual equity options, patterns are reversed or not monotonic. This suggests that for a longtime, we were blinded by the leverage of short maturity options.

The second pattern concerns moneyness, where the average return and volatility from selling

OTM options is much higher than the ITM options for short term options (up to 3 months).

For long term options, the volatility persists; OTM options are more volatile, though there is

no pattern in average returns. Table1also presents the t-statistics and Sharpe ratios of excess

returns from selling options. The Sharpe ratios and t-statistics decrease by maturity. In fact, only

short term options (up to 3 months) have statistically significant excess returns. The OTM, ATM,

and ITM option portfolios have higher Sharpe ratios and t-statistics than the DOTM and DITM

options. Usually skewness and kurtosis are concerns for option returns and so Table1also reports

those statistics. Most of the skewness estimates are greater than -1.2 and kurtosis estimates are

less than 10. Though there are a few exceptions, the DOTM options have a skewness of up to

-2.3 and DITM long maturity options have a kurtosis of 20. Broadie, Chernov, and Johannes

(2009) report skewness and kurtosis of OTM standard put option returns all the way up to 5.5

and 34; Constantinides, Jackwerth, and Savov (2011) report even more extreme estimates for

standard option returns ( skewness greater than 10 and kurtosis greater than 100). Compared to

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standard option returns, leverage-adjusted and delta-hedged returns have much lower skewness

and kurtosis. The return distribution is obviously not normal, but they are not significantly worse

than stock portfolios. For instance,Frazzini and Pedersen(2012) report the skewness and kurtosis

of a momentum portfolio -3.04 and 26.67; for Fama French HML 1.83 and 15.54 for the 1926-2010

sample.Figure 3: Risk Across Moneyness-Maturity

The figure displaysaverage monthly volatility, capm , negative of VIX and negative of excess returnsduring price-jump episodesacross moneyness-maturity portfolios. s are estimated using holding periodreturns over the full sample. Blue lines indicate two standard deviation confidence intervals for s. Thefull sample covers 204 months from January 1996 to January 2013. Price jump (nobs=14) indicates monthswith at least one day with a jump in S&P 500 index, which is defined as daily change in index less than -4%.SeeTable1 for the details of moneyness and maturity groups.

DOTM OTM ATM ITM DITM0

100

200

300

400

500

600

700

MoneynessMaturity

inbasispoints

Panel (A) Volatility

|| 0.2

0.2 < || 0.4

0.4 < || 0.6

0.6 < || 0.8

0.8 < ||

1 month

2 months3 months

4:6 months7:12 months

>12 months


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

MoneynessMaturity

CAPM

Panel (B) CAPM Beta

1 month

2 months

3 months

4:6 months

7:12 months>12 months


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

MoneynessMaturity

-VIX

Panel (C) - VIX Beta

1 month

2 months

3 months

4:6 months

7:12 months

>12 months


0

100

200

300

400

500

600

700

MoneynessMaturity

-Excessreturn(bas

ispointspermonth)

Panel (D) - Average Excess Returns During Price-Jumps

|| 0.2 0.2 < || 0.4 0.4 < || 0.6 0.6 < || 0.8 0.8 < |||| 0.2 0.2 < || 0.4 0.4 < || 0.6 0.6 < || 0.8 0.8 < ||

1 month

2 months

3 months

4:6 months

7:12 months

>12 months

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Table 1: Summary Statistics of Portfolios formed on Moneyness and Maturity

This table reports summary statistics of portfolios formed on moneyness and maturity. For each month,the Saturday following the 3rd Friday of the month is standard expiration date. Each month, at the firsttrading day following the expiration date, I assign options into thirty portfolios based on five moneyness(absolute value of delta) and six maturity (months to expiration) groups. Moneyness groups are DOTM(deep out of the money, 0 ||< 0.20), OTM (out of the money, 0.20 ||< 0.40), ATM (at the money,

0.40 || < 0.60), ITM (in the money, 0.60 || < 0.80), DITM (deep in the money, 0.80 || < 1).Maturity groups are 1, 2, 3, 4 to 6, 7 to 12, greater than 12 months (holding periods). I keep the optionpositions till the next expiration date. For each option in the portfolio, I calculate excess returns thatare delta-hedged daily and leverage-adjusted monthly. I calculate portfolio excess returns by taking optionmarket capital weighted average return of each option in a given portfolio, where option market capital isopen interest times option price. I report means, standard deviations, t-statistics, annualized Sharpe ratios,skewness and kurtosis of monthly (percentage) portfolio excess returns (delta-hedged, leverage adjusted).Table also presents delta, gamma, vega of options in a given portfolio. Option Greeks and implied volatilitiesare calculated by OptionMetrics based on Cox, Ross, and Rubinstein (1979) model. Lastly table presentspercentage of total trading volume in market value and in notional. The sample covers 204 months fromJanuary 1996 to January 2013.

Moneyness Maturity1 2 3 4:6 7:12 >12 1 2 3 4:6 7:12 >12

Mean Standard DeviationDOTM 2.10 0.97 0.53 0.15 0.08 -0.06 3.44 4.64 4.54 4.92 4.91 6.27

OTM 1.76 0.96 0.47 0.25 0.20 0.04 2.37 2.74 2.96 2.95 3.08 3.47

ATM 1.32 0.71 0.28 0.15 0.12 0.17 1.73 1.87 1.94 2.01 2.16 2.29

ITM 0.94 0.51 0.26 0.14 0.10 0.10 1.10 1.42 1.19 1.31 1.62 1.48

DITM 0.46 0.24 0.12 0.01 0.03 0.05 0.98 1.02 0.96 0.84 1.15 1.16

T-statistics Sharpe Ratio

DOTM 8.71 3.00 1.68 0.43 0.22 -0.14 2.11 0.73 0.41 0.11 0.05 -0.03

OTM 10.61 5.01 2.25 1.22 0.95 0.17 2.57 1.22 0.55 0.30 0.23 0.04

ATM 10.92 5.44 2.06 1.09 0.78 1.07 2.65 1.32 0.50 0.26 0.19 0.26

ITM 12.27 5.13 3.07 1.48 0.85 0.99 2.98 1.24 0.74 0.36 0.21 0.24

DITM 6.80 3.31 1.79 0.21 0.38 0.65 1.65 0.80 0.43 0.05 0.09 0.16

Skewness Kurtosis

DOTM -1.51 -2.28 -2.29 -1.97 -1.42 -1.46 7.80 11.68 11.63 9.33 6.62 7.75

OTM -1.35 -1.00 -1.50 -1.36 -0.90 -1.46 7.05 5.71 8.68 7.76 7.90 9.74

ATM -0.83 -0.88 -1.03 -1.24 -0.89 -1.60 5.52 7.52 7.18 8.53 7.98 11.46

ITM 0.42 1.30 -1.01 -0.76 -0.12 -1.57 5.55 12.44 6.25 7.10 14.15 10.90

DITM 0.97 0.58 -0.60 -1.70 -1.57 -0.46 8.46 1 1.32 8.04 9.66 14.35 20.26

abs(Delta) Gamma

DOTM 0.12 0.11 0.12 0.12 0.12 0.12 0.04 0.03 0.03 0.03 0.02 0.01

OTM 0.30 0.30 0.30 0.30 0.30 0.30 0.10 0.08 0.06 0.05 0.04 0.03

ATM 0.50 0.50 0.50 0.50 0.50 0.50 0.12 0.09 0.08 0.06 0.05 0.03ITM 0.70 0.70 0.70 0.70 0.69 0.69 0.10 0.08 0.07 0.06 0.04 0.02

DITM 0.85 0.86 0.86 0.86 0.86 0.87 0.06 0.05 0.04 0.03 0.02 0.01

Vega Embedded Leverage

DOTM 2.62 3.51 4.36 5.96 8.90 14.00 14.61 11.48 9.69 7.75 6.24 3.88

OTM 4.07 5.67 6.88 8.87 13.14 22.60 12.65 9.55 7.95 6.29 5.20 3.36

ATM 4.62 6.47 7.85 10.07 14.46 25.38 10.57 7.79 6.47 5.19 4.40 3.03

ITM 4.05 5.65 6.93 8.96 13.24 23.27 8.33 6.12 5.12 4.20 3.59 2.61

DITM 2.84 3.90 4.76 6.06 8.92 15.06 6.84 5.04 4.25 3.52 2.94 2.22

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% of Trading Volume (Market Value) % of Trading Volume (Notional)

DOTM 2.25 1.00 0.55 0.93 0.70 0.61 21.89 5.68 2.36 2.74 1.69 0.92

OTM 7.15 3.39 2.04 3.75 2.87 2.28 14.37 5.00 2.35 3.26 1.99 0.96

ATM 12.87 7.04 4.80 5.62 4.22 3.36 13.39 5.74 3.22 2.55 1.58 0.84

ITM 7.17 2.62 1.55 2.74 1.94 2.17 3.94 1.08 0.48 0.63 0.39 0.31DITM 6.32 2.62 1.72 2.47 1.94 1.29 1.47 0.39 0.21 0.28 0.17 0.10

Table 2: Summary Statistics of Value Portfolios

This table reports summary statistics for portfolios sorted on value. I define value as the difference betweenone-year historical realized volatility (calculated using daily stock returns) and Black-Scholes implied volatil-ity. For each month, the Saturday following the 3rd Friday of the month is standard expiration date. Eachmonth, at the first trading day following the expiration date, I assign options into decile portfolios basedon their value. I keep the option positions until the next expiration date. For each option in the portfolio,I calculate excess returns that are delta-hedged daily and leverage-adjusted monthly. I calculate portfolioexcess returns by taking value weighted average return of each option in a given portfolio, where valuesare option market value (open interest times mid price). I report means, standard deviations, t-statistics,annualized Sharpe ratios, skewness and kurtosis of portfolio excess returns. Table also presents averageimplied volatilities, delta, gamma, vega, theta, days to maturity of options in a given portfolio. OptionGreeks and implied volatilities are calculated by OptionMetrics based on Cox, Ross, and Rubinstein(1979)model. Lastly, the table presents average VRP (variance risk premia) and lag growth in total option marketcapital of options on a given portfolios. I define total option market capital as the sum of open interesttimes mid-price of all options on the underlying stock. The sample covers 204 months from January 1996 toJanuary 2013.

High Low1 2 3 4 5 6 7 8 9 10 10-1

Mean -0.11 0.06 0.06 0.12 0.18 0.25 0.30 0.41 0.62 1.74 1.85Standard Deviation 2.50 1.97 1.83 1.65 1.61 1.57 1.61 1.74 1.87 2.89 2.24

T-statistics -0.66 0.46 0.46 1.08 1.56 2.30 2.69 3.35 4.72 8.57 11.78Sharpe Ratio -0.16 0.11 0.11 0.26 0.38 0.56 0.65 0.81 1.15 2.08 2.86Skewness -1.18 -1.41 -1.25 -1.14 -1.06 -1.48 -1.31 -0.93 -0.85 -0.35 1.07Kurtosis 6.81 8.95 8.47 7.86 10.00 11.86 11.53 9.20 9.59 6.68 6.48Value 0.25 0.11 0.07 0.04 0.02 -0.00 -0.02 -0.05 -0.08 -0.23 -0.47Implied Vol 0.56 0.46 0.42 0.40 0.38 0.38 0.39 0.42 0.47 0.71 0.15abs(Delta) 0.59 0.59 0.59 0.59 0.60 0.61 0.62 0.62 0.63 0.59 -0.00Gamma 0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.03 0.03 0.04 -0.00Vega 13.24 16.17 16.74 16.70 17.85 18.77 19.20 19.65 20.31 10.91 -2.33Theta -8.00 -7.70 -7.05 -6.96 -7.12 -7.53 -8.22 -9.35 -10.51 -11.37 -3.36Days To Maturity 213 212 207 205 203 200 195 187 173 143 -70VRP -0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.06 0.06

Option Market-Capital 0.08 0.06 0.05 0.05 0.05 0.05 0.06 0.07 0.10 0.20 0.12

Table 1 also reports the trading volume in market value and notional of each portfolio as a

percentage of total trading volume. Market value is calculated as open interest times option price

and notional value is calculated as open interest times closing price of underlying stock. More

than 97% of the trading volume in notional derives from DOTM, OTM, ATM, and ITM options;

therefore, we might consider giving less weight to the results from the DITM option portfolios.

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Trading volume is also concentrated in short maturity options with less than three months to

maturity. In fact, more than 80% of trading volume in notional value and 60% of trading volume

in market value is concentrated in options with less than three months to maturity.

Table2displays on the summary statistics for the portfolios based on value. Following Goyal

and Saretto (2009), I define value as the difference between the 1 year historical volatility ofunderlying stock returns and the Black-Scholes implied volatility. The returns on selling low value

(expensive) options is much higher than high value (cheap) options. The average returns decrease

smoothly from 1.74% (t-statistics: 8.57) to -0.11% (t-statistics: -0.66), but there is no clear pattern

to the standard deviation of returns. In the lowest decile, the implied volatility is 23% higher than

the realized volatility; in the highest decile, it is 25% lower than realized volatility. However, there

is no economically and statistically significant return in buying high value portfolios.

Goyal and Saretto(2009) report that this characteristic is associated with the expected return

of ATM options with one month to maturity from the 1996 to 2006 sample. In my analysis, I

extend the sample from 1996 to 2013, and I use all moneyness and maturity categories. I find

this pattern very pervasive. In unreported results, I build portfolios within each moneyness and

maturity group; high minus low value portfolios generate a high expected return spread within

almost all groups.

I also consider alternative decile portfolio formations. Initially, I build decile portfolios within

each maturity moneyness group, and then I aggregate them by taking their weighted average,

where the weights are option market capital (mid price times open interest). For instance, I build

a lowest decile portfolio within each of the 30 maturity moneyness group, then I take the weighted

average of all the lowest decile portfolios within each group, and call the resulting portfolio asthe lowest decile of all options. This way I can keep the moneyness and maturity of the decile

portfolios relatively stable.

The expected return pattern is still very smooth and strong (low-minus-high Sharpe ratio: 2.3)

in the moneyness-maturity controlled decile portfolios. The pattern is very robust in various sub-

samples too. For example, I started the sub-sample from the end of Goyal and Saretto(2009)s

sample, 2007:2013 a low-minus-high portfolio generates a Sharpe ratio of greater than 2 with

reasonable skewness kurtosis estimates. Regarding stock return anomalies, patterns often vanish

once they are known; value strategy with regard to options seems to persist well after Goyal and

Saretto(2009)s paper. In fact nowadays this strategy is very famous in the industry, and some

financial intermediary firms report this for their clients (e.g., Fidelity).

IV. Pricing

In the first part of this section, I test the Fama-French-Carhart four-factor model on the thirty

portfolios formed on moneyness-maturity. In the second, I propose a new empirical pricing model

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and then test it on the moneyness-maturity portfolios and value portfolios.

A. Asset Pricing Test of The Fama-French-Carhart Four-Factor Model

In the previous section, we saw various patterns of expected returns. The question then becomes,

can these be explained with usual stock market risk factors? For this reason, I test the Fama-

French-Carhart four-factor (FF4) model on moneyness-maturity portfolios. I start with daily

factor returns and calculate holding period returns. Details are in the methodology section.

My testing methodology is as follows: I run time-series regressions of excess returns of option

portfolios on FF4 factors; I estimate coefficients with OLS. If FF4 describes the expected returns,

then the regression intercepts should be close to 0. To test this, I calculate the F-statistics

of Gibbons, Ross, and Shanken (1989) (GRS). Under the null hypothesis of 0 pricing errors, I

bootstrap 10,000 samples. I run the same regressions and estimate GRS statistics for each sample.

The t-statistics of coefficients and the p-value of GRS statistics are then calculated from thebootstrap procedure.

Table3presents the s, slope coefficients, and t-statistics, adjusted r-squares from multiple

time-series regressions, as well as the GRS statistics and its p-value. The FF4 model is strongly

rejected by the data. The GRS statistic is 16.43 and the null hypothesis that s (pricing errors)

are jointly 0, is rejected at a 0.01 significance level. More importantly, the average portfolio returns

ands are almost the same, and so the model has very little explanatory power for a cross section

of option returns. For instance, the average return of an ATM 1m option is 132 basis points; the

model could explain only 11 basis points of this return. The mean of the absolute excess returns

for the thirty portfolios is 44 basis points, the FF4 can reduce it only to 37 basis points, not very

much.

Market betas are significant and rise from ITM to OTM and from short maturity to long

maturity options. This implies that market betas predict a higher return for long maturity options

than short maturity options. R-squares are around 20 to 40%, hence the model fails to explain

the variance of returns as well as means.

I also consider extending FF4 with a volatility factor. To achieve this, I estimate the option

market capital weighted portfolio excess return (delta-hedged, leverage-adjusted) of options (ATM,

1-month to maturity) on SP 500. I do this separately for call and put options, then take theiraverage.

I test FF5 (FF4 plus a volatility factor) on the thirty moneyness-maturity portfolios. Adding

the volatility factor does not help. Actually M AEof FF5 is four basis points greater than M AE

of FF4, because the volatility factor predicts a higher premium for long maturity options. You

can see excess returns, FF5 predicted returns and s inFigure4. Adding the liquidity factor as

inPastor and Stambaugh(2003) does not help to price these portfolio returns either.

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Table 3: Asset Pricing Test of FF4 on Moneyness-Maturity Portfolios

This table reports the results of multiple regressions and asset pricing tests. I test the Fama-French-Carhartfour-factor model on excess returns (delta-hedged, leverage-adjusted) of the thirty portfolios formed onmoneyness-maturity. There are five moneyness (absolute value of delta) and six maturity (months to expi-ration) groups. Moneyness groups are DOTM (deep out of the money, 0 || < 0.20), OTM (out of the

money, 0.20 ||< 0.40), ATM (at the money, 0.40 ||< 0.60), ITM (in the money, 0.60 ||< 0.80),DITM (deep in the money, 0.80 || < 1). Maturity groups are 1, 2, 3, 4 to 6, 7 to 12, greater than 12months (holding periods). I report s, slope coefficients, t-statistics and R-squares. GRS is the joint testof all pricing errors. I run OLS with 10,000 bootstrap simulations under the null hypothesis of zero pricingerrors to estimate t-statistics and p-values. 1, 2, 3, 4 are coefficients of excess market return, SMB(small minus big), HML (high minus low), WML (momentum); respectively. To save space, I dont reportcoefficient of WML. The data on pricing factors are available from Kenneth French data library. I convertdaily returns to holding period returns (details are explained in methodology section). The sample covers204 months from January 1996 to January 2013.

Rei,j,t = i,j+ 1i,j(Rmt Rft) +

2i,jSM Bt+

3i,jHM Lt+

4i,jW MLt+ i,j,t

i {DOTM, OTM, ATM, IT M, DITM} j {1M, 2M, 3M, 4 : 6M, 7 : 12M, >12M}


t()

DOTM 1.90 0.66 0.15 -0.24 -0.32 -0.63 8.72 2.39 0.58 -0.90 -1.13 -1.86OTM 1.57 0.77 0.22 -0.00 -0.05 -0.28 10.63 4.63 1.35 -0.01 -0.32 -1.58ATM 1.21 0.59 0.12 0.00 -0.03 -0.04 10.64 5.13 1.06 0.02 -0.24 -0.33ITM 0.89 0.42 0.17 0.05 0.00 -0.02 12.31 4.72 2.48 0.70 0.02 -0.26DITM 0.47 0.18 0.06 -0.02 -0.05 -0.00 6.81 2.63 1.05 -0.35 -0.68 -0.07

1 t(1)

DOTM 0.24 0.39 0.42 0.51 0.48 0.68 5.27 6.93 7.76 9.23 8.05 9.75OTM 0.19 0.22 0.29 0.30 0.31 0.37 6.05 6.50 8.44 8.97 8.92 10.02ATM 0.09 0.14 0.19 0.19 0.20 0.24 3.95 6.01 8.40 7.71 8.02 9.88ITM 0.06 0.11 0.12 0.11 0.12 0.13 3.79 5.81 8.23 7.23 5.84 7.60

DITM -0.00 0.05 0.07 0.04 0.06 0.06 -0.14 3.12 5.28 3.26 4.24 3.952 t(2)

DOTM 0.02 0.01 0.13 0.08 0.15 0.29 0.28 0.09 1.72 0.99 1.82 2.90OTM 0.11 0.06 0.08 0.10 0.15 0.22 2.46 1.33 1.69 2.12 3.06 4.17ATM 0.07 0.07 0.09 0.10 0.10 0.17 1.96 2.02 2.90 2.83 2.75 4.96ITM 0.04 0.05 0.03 0.06 0.11 0.13 1.85 1.71 1.44 2.79 3.79 5.31DITM 0.03 0.05 0.07 0.05 0.12 0.11 1.36 2.31 3.61 2.79 5.56 5.01

3 t(3)

DOTM 0.26 0.40 0.36 0.37 0.32 0.41 3.66 4.49 4.18 4.25 3.46 3.68OTM 0.16 0.21 0.22 0.22 0.22 0.26 3.23 3.84 4.06 4.11 3.96 4.43ATM 0.13 0.14 0.12 0.13 0.12 0.16 3.40 3.73 3.39 3.39 3.10 4.00

ITM 0.06 0.08 0.07 0.07 0.06 0.07 2.36 2.76 2.93 2.86 1.80 2.53DITM 0.00 0.04 0.02 0.03 0.04 0.02 0.11 1.79 0.98 1.66 1.64 0.74

Adj R-square GRS (p-val)

DOTM 0.22 0.32 0.34 0.42 0.34 0.43 16.43 (0.0000)

OTM 0.23 0.28 0.37 0.40 0.41 0.47ATM 0.14 0.26 0.36 0.35 0.37 0.47ITM 0.13 0.22 0.34 0.34 0.26 0.35DITM 0.02 0.07 0.20 0.13 0.20 0.21

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Figure 4: Asset Pricing Test of FF5 on Moneyness-Maturity Portfolios

The top panel presents average excess returns and predicted returns by the FF5 (FF4 plus a volatility factor)across moneyness and maturity. Red bars are average excess returns, green bars are predicted values. Thebottom panel presents alphas as red bars and their two standard deviation confidence intervals as blue linesacross moneyness and maturity groups. SeeTable3 for details of moneyness-maturity groups.


0

50

100

150

200

250

Excessreturn(basispointspermonth)

|| 0.20.2


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Level Factor : I define level as the average return from selling at-the-money (ATM) option

portfolios. Rei,j,t denotes excess return of option portfolio with moneyness category iand maturity

categoryjat time t.

Levelt =

1

6 i

j

Rei,j,t

i {AT M} , j {1M, 2M, 3M, 4 : 6M, 7 : 12M, >12M}

Slope Factor: I define slope as the average return from selling option portfolios with 1-2 months

to maturity minus average return on selling option portfolios with 4 to 6 months to maturity.

Slopet = 1

12 i1 j1Rei,j,t

1

6 i2 j2Rei,j,t

i1{DOTM, OTM, ATM, IT M, DIT M}, j1{1M, 2M}

i2{DOTM, OTM, ATM, IT M, DIT M}, j2 {4 : 6M}

Value Factor : I define VAL as the average return from selling the lowest three value decile

portfolios minus the highest three value decile portfolios. Rei,t denotes excess return on the ith

decile value portfolio at time t, where 10th decile portfolio consists of options with low value.

V ALt=

10

i=8 R

e

i,t

3

i=1 R

e

i,t

2. Asset Pricing Test of The Option Three-Factor Model on the Moneyness-Maturity

Portfolios

In this part, I test the option three-factor model on the excess returns on the thirty portfolios

formed on moneyness-maturity and argue that equation1 is a good description of those expected

returns. To show the progress made by option factor models, I begin by estimating the mean

absolute pricing errors (MAE) of the thirty moneynes-maturity portfolios 130 ij|i,j|under 7different pricing models. These models are BS (Black-Scholes), FF4 (Fama-French-Carhart four-

factor model), PCA2-PCA5 ( the models with the first 2 and 5 principal components of the thirty

moneyness-maturity portfolios), OPT2 ( the option two-factor model with level and slope), and

OPT3 ( the option three-factor model with level, slope and value). The BS is the benchmark

case, since BS implies that these returns should be on average equal to 0. The Figure5 shows the

success of the option factor models. TheMAEfor the benchmark BS is about 44 basis points.

The CAPM and FF4 models reduce it to only 36 basis points. The PCA based models are not

24


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successful either. The option factor models, however, reduce theM AEto 13-15 basis points.Table

4reports the results of the following time series regressions.

Rei,j,t = i,j+ Li,jLevelt+

si,jSlopet+

vi,jV aluet+i,j,t

i {DOTM, OT M, AT M, ITM, DITM} j {1M, 2M, 3M, 4 : 6M, 7 : 12M, >12M}

Figure 5: Average || : Moneyness-Maturity Portfolios

The figure presents average || under different models. BS refers to Black-Scholes, FF4 refers to the Fama-French-Carhart four-factor model. PCA2 and PCA5 refers to pricing model with first 2 and 5 principalcomponent of the thirty moneyness-maturity portfolios. OPT2 refers to the option two-factor model (leveland slope), OPT3 the option three-factor model (level, slope and value).

BS CAPM FF4 FF5 PCA2 PCA5 OPT2 OPT30

5

10

15

20

25

30

35

40

45

MAEB

asisPoints

I estimate coefficients using OLS. I calculate the GRS statistics as they do inGibbons, Ross,

and Shanken(1989) to test the joint significance of pricing errors. Under the null hypothesis of

zero pricing errors, I bootstrap 10,000 samples from the fitted regression residuals. At each sample

I run the same regressions and estimate GRS statistics. The t-statistics of the coefficients and

the p-value of the GRS statistics are all calculated from bootstrap procedure. If the option three-

factor model describes the expected excess returns of moneyness-maturity portfolios, then the s

should be close to zero. The model seems to do well on average, with mean absolute s (MAE)

of about 15 basis points, but the model still leaves large intercepts for some of the portfolios. In

particular, it tends to leave intercepts far from zero for DOTM options. For example, the DOTM

option portfolios with 2 and 3 months to maturity have intercepts of -80 and -65 basis points.

Excluding the DOTM option portfolios, the MAEof the volatility surface portfolios is about 9

basis points. I should also note that the DOTM and DITM option portfolios are relatively less

liquid than the OTM, ATM and ITM option porfolios which cover most of the trading volume inmarket value. The ITM option portfolio with 1 month to maturity and the ATM option portfolios

with 3 months to maturity and greater than 12 months to maturity leave about 18-19 basis point

absolute intercepts. This is not much, but statistically significant. However, this might be sample

specific: the statistical significance of the s do not seem to be robust across sub-samples. The

option three-factor explains a considerable portion of the realized volatility of excess returns. The

average R-squares for the ATM and OTM option portfolios are high, about 88%, it decreases to

75% for the ITM and DOTM options, it is lowest for the DITM options at 38%.

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Table 4: Asset Pricing Test of OPT3 on Moneyness-Maturity Portfolios

This table reports the results of multiple regressions and asset pricing tests. I test the option three-factor model (OPT3) on excess returns (delta-hedged, leverage-adjusted) on the thirty portfolios formedon moneyness-maturity. There are five moneyness (absolute value of delta) and six maturity (months toexpiration) groups. Moneyness groups are DOTM (deep out of the money, 0 ||< 0.20), OTM (out of the

money, 0.20 ||< 0.40), ATM (at the money, 0.40 ||< 0.60), ITM (in the money, 0.60 ||< 0.80),DITM (deep in the money, 0.80 || < 1). Maturity groups are 1, 2, 3, 4 to 6, 7 to 12, greater than12 months (holding periods). I report s, slope coefficients, t-statistics and R-squares. GRS is the jointtest of all pricing errors. I run OLS with 10,000 bootstrap simulations under the null hypothesis of zeropricing errors to estimate t-statistics and p-values. L, S, V are factor sensitivities of the level, slope,value factors respectively. The sample covers 204 months from January 1996 to January 2013.

Rei,j,t = i,j+ Li,jLevelt+

si,jSlopet+

vi,jV aluet+ i,j,t

i {DOTM, OTM, ATM, IT M, DITM} j {1M, 2M, 3M, 4 : 6M, 7 : 12M, >12M}


t()

DOTM -0.13 -0.80 -0.65 - 0.27 0.01 - 0.31 -0.52 -2.27 -2.16 -1.08 0.04 -0.97OTM 0.14 0.03 -0.27 -0.01 0.14 0.02 1.03 0.18 -1.88 -0.07 1.12 0.15ATM 0.06 0.03 -0.18 -0.04 -0.06 0.19 0.62 0.33 -2.36 -0.87 -0.77 2.15ITM 0.19 -0.17 -0.06 -0.02 0.04 0.13 2.31 -1.47 -0.79 -0.34 0.38 1.83DITM 0.06 -0.18 -0.07 -0.05 0.00 0.12 0.58 -1.77 -0.78 -0.71 0.03 1.08

L t(L)

DOTM 1.83 2.21 2.13 2.19 2.03 2.54 18.51 16.52 18.43 22.90 18.24 21.01OTM 1.40 1.47 1.49 1.39 1.35 1.46 27.21 27.31 27.34 39.72 27.99 30.23ATM 1.04 1.01 1.00 0.97 1.02 0.96 27.44 29.09 34.54 53.20 34.51 28.73ITM 0.59 0.65 0.59 0.62 0.67 0.61 19.04 15.15 21.99 29.64 17.31 22.00DITM 0.32 0.40 0.37 0.29 0.33 0.30 7.89 10.31 11.12 10.17 7.67 6.88

S t(S)DOTM 1.50 0.73 0.11 -0.78 -0.95 -1.65 7.54 2.68 0.46 -4.03 -4.25 -6.74OTM 1.15 0.40 0.03 -0.50 -0.67 -0.98 11.04 3.67 0.26 -7.00 -6.80 -10.11ATM 0.99 0.29 -0.02 -0.29 -0.32 -0.65 12.89 4.02 -0.30 -7.83 -5.32 -9.69ITM 0.48 0.23 0.05 -0.18 -0.30 -0.35 7.74 2.64 0.93 -4.17 -3.84 -6.23DITM 0.28 0.28 0.04 -0.09 -0.18 -0.26 3.36 3.53 0.58 -1.54 -2.02 -2.98

V t(V)

DOTM 0.12 0.14 0.12 0.09 -0.05 0.54 0.96 0.84 0.86 0.74 -0.40 3.64OTM -0.01 -0.09 0.03 0.05 0.02 0.21 -0.18 -1.39 0.41 1.14 0.33 3.56ATM -0.07 -0.03 0.02 -0.00 -0.01 0.11 -1.58 -0.80 0.47 -0.12 -0.40 2.65ITM 0.08 0.20 -0.00 0.02 0.01 -0.01 2.06 3.74 -0.05 0.92 0.21 -0.34

DITM 0.02 -0.01 -0.02 0.01 0.03 0.02 0.36 -0.18 -0.49 0.26 0.53 0.38Adj R-square GRS (p-val)

DOTM 0.64 0.63 0.72 0.84 0.78 0.83 1.95 (0.0027)

OTM 0.79 0.83 0.85 0.94 0.89 0.92ATM 0.79 0.84 0.90 0.96 0.92 0.91ITM 0.65 0.60 0.78 0.89 0.75 0.84DITM 0.23 0.36 0.47 0.49 0.38 0.38

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Figure 6: Asset Pricing Test of OPT3 on Moneyness-Maturity Portfolios

The top panel presents average excess returns and predicted returns by the option three-factor model (OPT3)across moneyness and maturity. Red bars are average excess returns, green bars are predicted values. Thebottom panel presents alphas as red bars and their two standard deviation confidence intervals as blue linesacross moneyness and maturity groups. SeeTable4 for details of moneyness-maturity groups.


0

50

100

150

200

250

Excessreturn(basispointspermonth)

|| 0.20.2


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option portfolios, gradually decreases to 1.5 and 1 for OTM and ATM portfolios and lastly it

decreases to 0.6 and 0.3 for ITM and DITM option portfolios. L varies with moneyness, because

the volatility of DOTM options is much higher than DITM options. If we adjust the returns with

volatility,L would be more or less constant. In contrast, coefficients of slope factor, monotonically

varies over the maturity dimension, and BSis positive for short maturity options and negative forlong maturity options. The magnitude of positiveness and negativeness is higher for DOTM

options since they are more volatile. The value factor does not have any explanatory power for

the moneyness-maturity portfolios. I include it only to show that it does not make the results

substantially worse. It does not explain these portfolios, because it is not correlated with them. We

will however find it crucial in explaining the decile value portfolios as well as other characteristic-

based option portfolios.

3. Diagnosing Rejection : Principal Component Analysis (PCA)

In this subsection, I will answer two important questions:

1. Why do we rely on ad-hoc factor construction rather than principal component analysis?

2. We rejected the option three-factor model, but how many factors do we need to accept a

linear factor pricing model?

To answer these questions, I extract the principal components of the thirty moneyness-maturity

portfolios. Figure 7 presents the results. The first figure displays the cumulative R-square ex-

plained by the first n principal components. There is a strong factor structure: the first principalcomponent explains more than 83% of the variation of returns, with the first two component num-

bers going up to 88%. I form empirical pricing models using the first n principal components from

n=1 to n=16, and then I test the principal component models on the thirty moneyness-maturity

portfolios. The second and third figure of theFigure7show the p(GRS) (p-value of GRS statistics)

andM AE(mean absolute pricing error) from multiple asset pricing tests.The last figure displays

the mean of principal components (PC) with two standard deviation confidence intervals. Mean

of PC is not statistically different than zero, if the confidence interval passes from zero.

The results are surprising. To explain the thirty portfolios, we need fifteen principal compo-

nents not to reject the model. It seems like there are many small priced factors. The PCA method

gives us the model of variances, but usually model variance is also a good model of mean. At

least this was the case in the previous applications of the PCA method on stocks, government

bonds, currencies, CDS and corporate bonds.8 Equity options seem to be an exception. One of

the main reasons for this result is the relationship of mean and volatility with maturity. Long

maturity option portfolio excess returns are more volatile, but have smaller means. The principal

8SeeNozawa(2012),Palhares(2012),Lustig, Roussanov, and Verdelhan(2011).

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components explain mainly unpriced variations in long maturity options. In fact, we can see from

Figure 7 that there are factors that are not statistically different than zero, yet they explain a

big part of return variation. This is why I rely on an ad-hoc method rather than a PCA one to

construct factors. The PCA models need 5-6 factors and 13-14 factors in order to achieve MAE

and GRS statistics similar to the option three-factor model (15 basis points, 1.95).

Figure 7: Principal Component Analysis

This figure display the results of principal component analysis and asset pricing tests on the thirty moneyness-maturity portfolios. X-axis refers to the number of principal components. The first figure displays thecumulative R2 explained by first n principal components. P-value of the GRS statistics is denoted by p(GRS).M AErefers to the mean absolute pricing errors. The second and third figure displays p(GRS) and M AEfrom testing pricing model with first n principal components on the thirty moneyness-maturity portfolios.The last figure displays mean of principal components (PC) with two standard deviation confidence intervals.The sample covers 204 months from January 1996 to January 2013.

0 5 10 1580

85

90

95

100

Cumulative R2

0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.30.35

p(GRS)

0 5 10 150

10

20

30

40MAE

0 5 10 151

0

1

2

3

4E[PC]

OPT3

OPT2

4. Value

In this part, I argue that equation1is a good specification for the decile value portfolios expected

excess returns. To show this, I test the option three-factor model on decile value portfolio excess

returns using time-series regressions. In particular, I estimate the following equation for the

January 1996 to January 2013 sample period.

Rei,t = i+Li Levelt+

Si Slopet+

Vi V aluet+i,t

i {1, 2,...,10}

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The methodology of the asset pricing test is the same as before. I bootstrap 10,000 samples

from the fitted regression residuals. At each sample, I run the same regressions and estimate GRS

statistics. The t-statistics of coefficients and the p-value of GRS statistics are calculated from the

bootstrap procedure. Table5presents the details of the asset pricing tests. If equation1is a good

description of the expected returns, then the regression intercepts should be close to zero. TheF-test of Gibbons, Ross, and Shanken (1989) fails to reject the option three-factor model with

1.15 GRS statistics and a corresponding 0.24 p-value. Hence, the intercepts from the time-series

regressions are not jointly statistically different than zero. Most of the s are small and their

t-statistics are not statistically significant as well. Portfolio 10, however, is statistically significant

but the model is still a clear improvement. This is because the average excess return of Portfolio

1 is 175 basis points, while the from the three-factor model is only 32 basis points. Most of the

explanatory power comes from the value factor, whose coefficient monotonically varies from 1.12

for low value portfolios to -0.55 for high value portfolios. The coefficients of level and slope do

not have any clear pattern, but if I exclude any one of them then the model is economically and

statistically rejected. The option three-factor seems to be a good model for the realized volatility

of value portfolios as well, the average R-square is about 90%.

Table 5: Asset Pricing Test of OPT3 on the Decile Value Portfolios

This table reports results of multiple regressions and asset pricing tests. I test the option three-factor modelon excess returns (delta hedged, leverage adjusted) of the decile portfolios formed on value (one year historicalrealized volatility minus Black-Scholes implied volatility). I report s, slope coefficients, t-statistics and R-squares. GRS is the joint test of all pricing errors. I run OLS with 10,000 bootstrap simulations under thenull hypothesis of zero pricing errors to estimate t-statistics and p-value. L, S, V are slope coefficients

of the level, slope, value factors respectively. Rei,tis excess return on portfolioiat timet. The sample covers

204 months from January 1996 to January 2013.

Rei,t = i+Li Levelt+

siSlopet+

viV aluet+i,t i {1, 2, ..., 10}

Deciles

1 2 3 4 5 6 7 8 9 10 10-1 GRS p(GRS)

Value High Low

E[Re] -0.11 0.06 0.06 0.12 0.18 0.25 0.30 0.41 0.62 1.72 1.84

0.20 0.17 -0.09 -0.04 -0.04 -0.03 -0.02 -0.05 0.01 0.32 0.11

t() (1.74) (2.32) (-1.28) (-0.57) (-0.63) (-0.48) (-0.34) (-0.71) (0.13) (2.43) (0.98) 1.15 (0.24)

L

1.07 0.90 0.89 0.80 0.77 0.74 0.75 0.80 0.84 1.23 0.15t(L) (24.32) (32.52) (32.11) (32.67) (33.13) (28.94) (29.44) (30.82) (34.82) (24.68) (3.44)

S -0.35 -0.27 -0.12 -0.17 -0.20 -0.17 -0.20 -0.25 -0.28 -0.21 0.14

t(S) (-4.01) (-4.79) (-2.22) (-3.38) (-4.43) (-3.33) (-3.92) (-4.86) (-5.76) (-2.13) (1.57)

V -0.55 -0.32 -0.17 -0.07 0.04 0.10 0.16 0.34 0.51 1.12 1.67

t(V) (-10.33) (-9.36) (-4.90) (-2.31) (1.42) (3.15) (5.20) (10.64) (17.36) (18.49) (30.63)

R2

0.87 0.92 0.90 0.91 0.91 0.88 0.89 0.90 0.93 0.87 0.83

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V. Understanding Factors and Option Premia

In the previous sections, I first show that expected option returns are related to moneyness,

maturity and option value (the spread between historical volatility and the Black-Scholes implied

volatility), then I show that three systematic return-based factors (level, slope, value) explain thecross-sectional variation on expected option returns related to moneyness, maturity and value. In

this section, I investigate the economics behind these factors. It is important to understand why

these factors have high risk prices and why they have information about the cross-section of option

premia.

To set the stage,Figure8 plots the time-series of the cumulative sum of excess log returns for

all three pricing factors. The figure give us the first clue about the economic meanings of pricing

factors. For example, there is a clear pattern related to the level factor in which it tends to crash

during financial and liquidity crises such as Lehmans bankruptcy, the European sovereign crisis,

the Asian financial crisis, the Russian default and the bankruptcy of WorldCom. The slope factor

is just puzzling, because it does not seem to suffer during crises. Meanwhile, the value factor shows

it own interesting pattern, in which it loses slightly at the beginning of a financial crisis, but once

the crisis starts, this factor tends to rise sharply. This pat

metekarakaya_optionpremia_jmp

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