Characteristics and Expected Returns in Individual Equity Options

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 factors level, slope, and value in 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, Ľuboš Pástor and Harald Uhlig for their invaluable 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 Saltoğlu, Shri Santosh, Michael Baltasi and seminar participants at the University of Chicago Booth School of Business, Central Bank of Turkey. University of Chicago Department of Economics; Email: metekarakaya@uchicago.edu; phone: 312-213-8575; Web: home.uchicago.edu/metekarakaya

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 equity options 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 option s 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 find that 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 literature 1 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) threefactor (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 1 Frazzini and Pedersen (2012); Goyal and Saretto (2009) 2 This 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. 1

expected returns for long maturity options relative to short maturity ones. I identify three new return-based factors level, slope, and value which 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 and selling 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 the Fama and French (1993) threefactor 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[R e i] = β L i E[Level]+β S i E[Slope]+β V i E[Value] (1) R e i,t = α i +β L i Level t +β S i Slope t +β V i Value t +ε i,t (2) If the option three-factor model (OPT3: level, slope and value) in equation 1 and 2 holds, with α 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 direction, 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 result explain 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 = 1 N α ) against alternative models using thirty portfolios formed on moneyness and maturity. The mean absolute average excess return ( 1 N R e ) 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 MAE greater than 37 bps. On the other hand, OPT3 achieves MAE of only 15 bps. The results are qualitatively similar if we compare MAE s on decile portfolios formed on value. BS MAE is 38 bps and the CAPM, FF4 and FF5 produce a MAE of greater than 30 bps. In contrast OPT3 achieves MAE of less than 10 bps. 3 Black-Scholes model predicts zero expected delta-hedged excess return if hedging is done continuously. In practice I rely on daily delta-hedging. According to my simulation results, daily delta-hedging introduce a bias in the opposite direction. Bakshi and Kapadia (2003) get similar results. Therefore we should interpret 44 bps as lower bound to the MAE of BS. 2

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 the economic 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 Lehman s 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 puzzle by 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 ( 2 OptionPrice StockPrice 2 ). 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- 3

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 alleviate 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 of Garleanu, Pedersen, and Poteshman (2009) is related 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 option. 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 marketwide 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. Schürhoff 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 Schürhoff 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 Ihypothesizethatifmarketmakersareaversetohavingtoomuchgammaintheirportfolios(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. 4 I also find that low value option portfolios experience substantially higher growth in their total option market capital then high value option portfolios in the portfolio formation month. This might be a sign for higher demand pressure. If this is the case, then Garleanu, Pedersen, and Poteshman (2009) model can potentially explain the premium on the value factor. I define total option market capital as the sum of open interest times mid-price of all options on the underlying stock. 5 Following Frazzini and Pedersen (2010), I proxy funding liquidity by TED spread. 4

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 Sharpe ratio 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. Schürhoff 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. In Section II, I describe the data and explain the methodology. In Section III, I study patterns of expected option returns related to moneyness, maturity and value. In Section IV, I propose an empirical pricing model and conduct asset pricing tests. In Section V, 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 5

on them in Section VI. In Section VII, 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) and Goyal 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 are Bates (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 by Elkamhi and Ornthanalai (2010) and Christoffersen, Fournier, and Jacobs (2013), is to incorporate marketwide 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 shortterm dividend strips on the S&P 500 index have higher returns and Sharpe ratios than long-term dividend 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 6 See also Heston (1993); Bates (2000); Carr and Wu (2004); Liu, Pan, and Wang (2005). 7 See also Van Binsbergen, Hueskes, Koijen, and Vrugt (2011); Lettau and Wachter (2007); Hansen, Heaton, and Li (2008). 6

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 daily and leverage-adjusted monthly. A. Data Sources My primary data source is the OptionMetrics Ivy DB database, the industry standard for historical 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 following variables: 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 on Cox, Ross, and Rubinstein (1979) binomial 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, Iusefiltersinthesamemannerasinpreviousstudies, suchasgoyalandsaretto(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. 7

Following Frazzini 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 their return 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 P P / S S. 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 trading volume across 5 moneyness and 6 maturity groups. Panel A shows the results in market value, which is volume times mid-price of an option, while Panel B shows the results in notional, which is estimated as volume 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: Volume (in trillion dollars) 2 1.5 1 0.5 All Call Put Option Trading Volume (Market Value) Volume (in trillion dollars) 70 60 50 40 30 20 10 All Call Put Option Trading Volume (Notional) 0 1996 1998 2000 2002 2004 2006 2008 2010 2012 0 1996 1998 2000 2002 2004 2006 2008 2010 2012 DITM 0.8 < >12M DITM 0.8 < >12M Moneyness ITM ATM OTM DOTM 0.6 < 0.8 0.4 < 0.6 0.2 < 0.4 0.2 Maturity 7:12M 4:6M 3M 2M 1M Moneyness ITM ATM OTM DOTM 0.6 < 0.8 0.4 < 0.6 0.2 < 0.4 0.2 Maturity 7:12M 4:6M 3M 2M 1M 0 10 20 30 40 Percentage of total volume 0 10 20 30 40 Percentage of total volume 0 10 20 30 40 Percentage of total volume 0 20 40 60 Percentage of total volume 8

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, r S t the stock return and r f t the risk free rate. The starting value of my portfolio V 0 is $1 and I buy x = 1 F 0 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(f t F t 1 ), delta hedging x t 1 r S t S and lastly from the margin account r f t (V t 1 xf t 1 +x t 1 S t 1 ). I iterate the value of my portfolio using this algorithm from the beginning of holding period 0 to end of holding period T. V t = V t 1 + x(f t F t 1 ) } {{ } Option s price appreciation x t 1 r S t S t 1 } {{ } P&L from delta hedge + r f t (V t 1 xf t 1 + x t 1 S t 1 ) } {{ } gain or loss from margin account The delta-hedged return for option i is defined as the change in the value of portfolio during the holding period. r i,[0,t] = V T V 0 = V T 1 I subtract the monthly risk free rate in order to work with excess returns. 9

r e [0,T] = r i,[0,t] ( T t=0 ) ( ) 1+r f t 1 Lastly, I adjust for leverage, which is defined as the elasticity of the option price with respect to the stock price. F F Ω = = S F S F S = S F S I scale delta-hedged excess returns with leverage in order to derive leverage-adjusted excess returns. R e i,[0,t] = 1 Ω re i,[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. Let R p,[0,t be leverage-adjusted delta-hedged excess return of portfolio p. R p,[0,t] = N v i Ri,[0,T] e i=1 In all of these calculations, I use a closing bid-ask midpoint for the option price; hence, I implicitly 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: ( Rm r f ) = SMB [0,T] = T ( ) T 1+Rm d t t=1 T t=1 t=1 ( 1+SMB d t +r f t ( ) 1+r f t ) T t=1 ( ) 1+r f t 10

HML [0,T] = WML [0,T] = T t=1 T t=1 ( ) 1+HML d t +r f t ( ) 1+WML d t +r f t T t=1 T t=1 ( ) 1+r f t ( ) 1+r f t whererm d t isdailymarketreturn, SMB d t dailysmbreturn, HML d t dailyhmlreturn, WML d t 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 illiquidity, 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 < 0.40 3) At the money (ATM) 0.40 < 0.60 4) In the money (ITM) 0.60 < 0.80 5) Deep in the money (DITM) 0.80 < 1 Maturity: I define maturity as the number of holding periods (approximately 1 month) to expiration. I form 6 maturity groups; 1, 2, 3, 4 to 6, 7 to 12 and greater than 12 months (holding periods) to expiration. Option Value (Historical- Implied Volatility): I define value as the difference between historical volatility and Black-Scholes implied volatility. I estimate historical volatility as the standard deviation of daily realized volatility over the last 1 year. Goyal and Saretto (2009) show that this variable is related to the ATM one month to maturity straddle returns, but they don t call it value. I argue that this is a value-style investment strategy for options, because option value is mimicking valuation ratios such as dividend-price, earning-price, book-to-market ratio for stocks. We should interpret Black-Scholes implied volatility as a measure of market price and historical realized volatility as some measure of fundamental value. In fact, realized volatility is like a dividend for delta-hedged option traders, since they will be making money as they update their hedge ratio. Option traders, who are long (short) options, will be making more (less) money with 11

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: Following Koijen, 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. Let F t (S t,k,t,σ T ) be the price of an option contract (either call or put) at time t, with maturity T, strike K and implied volatility σ T. Koijen, Moskowitz, Pedersen, and Vrugt (2012) show that we can approximate the carry using the option s first time derivative theta (θ) and first volatility derivative vega υ. Let C t (S t,k,t,σ T ) denote the options carry. C t (S t,k,t,σ T ) = θ t(s t,k,t,σ T ) υ t (S t,k,t,σ T )(σ T σ T 1 ) F t (S t,k,t,σ T ) F t, θ, υ and σ T are available from Optionmetrics, I interpolate σ T 1 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 Ω C t(s t,k,t,σ T ) Variance Risk Premia (VRP): Following Bollerslev, 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 (IV i,t ) over the [t,t+1] time period and the ex post realized return variation (RV i,t ) over the [t 1,t] time period. VRP i,t = IV i,t RV i,t My estimation procedure of IV i,t is as in Han and Zhou (2012). IV i,t = 2 0 Ct i (t+t,k)/b(t,t) max(0,st/b(t,t) K) i dk K 2 where S i t represents stock price of stock i at time t, T = 1/12. C i t (t+t,k) stands for price of call option on stock i, with strike K and time to maturity T. B(t,T) denotes price of zero coupon 12

bond that pays one dollar at time t+t. I numerically estimate IV i,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 estimate RV i,t, I get TAQ intraday equity trading data spaced by 15 minute intervals. Let p i j,t denote log price of stock i at the end of j th 15-minute interval in holding period t and let N t denote number of trading days and n t number of 15-minute intervals in holding period t. RV i,t = 252 N t n t j=1 [ p i j,t p i j 1,t] 2 Volatility Reversals: Let σ 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 volatility surface file, I interpolate the implied volatility with the same delta and time to maturity σ t 1 (,T). Timing: if I am taking a position at the 4th Monday of month t, σ t (,T) is the implied volatility of an option from the 3rd Thursday of month t, σ t 1 (,T) is interpolated from the3rdthursdayofmontht-1. Idefinevolatility reversalsasσ t (,T) σ t 1 (,T). Thismeasure gives us the change in implied volatility at a specific point on the volatility surface. Systematic and Idiosyncratic Volatility: Following Cao and Han (2012), idiosyncratic volatility IVOL i,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 VOL 2 i,t IVOL2 i,t, where VOL i,t is the standard deviation of daily returns of stock i in period t. Stock Short-Term Reversal: Jegadeesh (1990) define short term reversal as the stock return over the previous month. Since I don t work with regular monthly data, I calculate the stock return over the previous holding period, which is about a month. Stock Size: Thefirmsizeisanaturallogarithmofthemarketvalueofequity, whichisestimated as the stock price times the number of shares outstanding. Stock Illiquidity: Following Amihud (2002), I define a stock s illiquidity as R i,t V i,t, where R i,t is the month t return of stock i, and V i,t the total dollar volume of stock i in month t. Option Illiquidity: I measure option illiquidity by the relative quoted spread. Previously Christoffersen, 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 13

the Black-Scholes model. I denote embedded leverage with Ω. Ω = F F S S = 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 market makers 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 moneynessmaturity 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 5 moneyness categories based on the range of : Deep out of the money 0 < 0.20, out of the money 0.20 < 0.40, at the money 0.40 < 0.60, in the money 0.60 < 0.80, deep in the money 0.80 < 1. I measure maturity with holding periods up to expiration. Holding periods are roughly one month in duration, so I will begin referring to holding periods as months. I assign options into 6 maturity categories: 1, 2, 3, 4 to 6, 7 to 12, and greater than 12 months to expiration. To set the stage, the Figure 2 presents expected excess returns as well 14

as two standard deviation confidence intervals across moneyness and maturity categories. You can find more detailed descriptive statistics across moneyness and maturity in the Table 1. 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 moneyness and 6 maturity groups. Red lines indicate two standard deviation confidence intervals for average excess returns. 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. Excess return (basis points per month) 300 250 200 150 100 50 0.2 209 53-6 0.2 < 0.4 175 46 4 0.4 < 0.6 132 28 17 0.6 < 0.8 94 25 10 46 1 month 2 months 3 months 4:6 months 7:12 months >12 months 0.8 < 11 5 0 Maturity Maturity Maturity Maturity Maturity DOTM OTM ATM ITM DITM Moneyness Maturity The most puzzling pattern is related to maturity: expected returns decrease and several measures of risk rise by maturity. If we go back to Markowitz (1959), risk is defined as the standard deviation of returns. If we consider the CAPM of Sharpe (1964), risk is market beta. If we consider more modern option pricing theories, we need to think about volatility and jump risk. Figure 3 presents several risk measures across the thirty moneyness-maturity portfolios. The first three panels report volatility of returns in basis points, CAPM β and negative of VIX β. To estimate CAPM β s, I regress excess returns on the thirty moneyness-maturity portfolios on CRSP value-weighted market return over the full sample. Similarly, I regress excess returns on the thirty moneyness-maturity portfolios on the change in VIX index to estimate VIX β s. I use holding-period excess returns and the change in VIX index in the regressions. Holding-periods are about a month. You can observe a clear rise in volatility, CAPM β and negative of VIX β. Note that VIX has a negative market price; hence, a rise in the negative of VIX β implies rise 15

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 with one 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 contemporaneous 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 in Figure 3 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 leverageunadjusted 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 long time, 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. Table 1 also 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 Table 1 also 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 16

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 displays average monthly volatility, capm β, negative of VIX β and negative of excess returns during price-jump episodes across moneyness-maturity portfolios. β s are estimated using holding period returns over the full sample. Blue lines indicate two standard deviation confidence intervals for β s. The full sample covers 204 months from January 1996 to January 2013. Price jump (nobs=14) 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%. See Table 1 for the details of moneyness and maturity groups. 700 600 0.2 Panel (A) Volatility 1 month 2 months 3 months 4:6 months 7:12 months >12 months 0.9 0.8 0.7 Panel (B) CAPM Beta 1 month 2 months 3 months 4:6 months 7:12 months >12 months 500 0.2 < 0.4 σ in basis points 400 300 200 0.4 < 0.6 0.6 < 0.8 0.8 < CAPM β 0.6 0.5 0.4 0.3 0.2 100 0.1 0 DOTM OTM ATM ITM DITM Moneyness Maturity 0 DOTM OTM ATM ITM DITM Moneyness Maturity -VIX β 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Panel (C) - VIX Beta 1 month 2 months 3 months 4:6 months 7:12 months >12 months - Excess return (basis points per month) 700 600 500 400 300 200 100 Panel (D) - Average Excess Returns During Price-Jumps 1 month 2 months 3 months 4:6 months 7:12 months >12 months 0.1 0 0 DOTM OTM ATM ITM DITM Moneyness Maturity 0.2 0.2 < 0.4 0.4 < 0.6 0.6 < 0.8 0.8 < DOTM OTM ATM ITM DITM Moneyness Maturity 17

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 first trading 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 option positions till 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 portfolio excess returns by taking option market capital weighted average return of each option in a given portfolio, where option market capital is open 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 volatilities are calculated by OptionMetrics based on Cox, Ross, and Rubinstein (1979) model. Lastly table presents percentage of total trading volume in market value and in notional. The sample covers 204 months from January 1996 to January 2013. Moneyness Maturity 1 2 3 4:6 7:12 >12 1 2 3 4:6 7:12 >12 Mean Standard Deviation DOTM 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 11.32 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 0.03 ITM 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 18

% 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 0.31 DITM 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 between one-year historical realized volatility (calculated using daily stock returns) and Black-Scholes implied volatility. For each month, the Saturday following the 3rd Friday of the month is standard expiration date. Each month, at the first trading day following the expiration date, I assign options into decile portfolios based on 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 portfolio excess returns by taking value weighted average return of each option in a given portfolio, where values are 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 average implied volatilities, delta, gamma, vega, theta, days to maturity of options in a given portfolio. Option Greeks 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 market capital of options on a given portfolios. I define total option market capital as the sum of open interest times mid-price of all options on the underlying stock. The sample covers 204 months from January 1996 to January 2013. High Low 1 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 1.85 Standard 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 11.78 Sharpe Ratio -0.16 0.11 0.11 0.26 0.38 0.56 0.65 0.81 1.15 2.08 2.86 Skewness -1.18-1.41-1.25-1.14-1.06-1.48-1.31-0.93-0.85-0.35 1.07 Kurtosis 6.81 8.95 8.47 7.86 10.00 11.86 11.53 9.20 9.59 6.68 6.48 Value 0.25 0.11 0.07 0.04 0.02-0.00-0.02-0.05-0.08-0.23-0.47 Implied Vol 0.56 0.46 0.42 0.40 0.38 0.38 0.39 0.42 0.47 0.71 0.15 abs(delta) 0.59 0.59 0.59 0.59 0.60 0.61 0.62 0.62 0.63 0.59-0.00 Gamma 0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.03 0.03 0.04-0.00 Vega 13.24 16.17 16.74 16.70 17.85 18.77 19.20 19.65 20.31 10.91-2.33 Theta -8.00-7.70-7.05-6.96-7.12-7.53-8.22-9.35-10.51-11.37-3.36 Days To Maturity 213 212 207 205 203 200 195 187 173 143-70 VRP -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. 19

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. Table 2 displays 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 of underlying 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 as the 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 subsamples 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 20

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 the bootstrap procedure. Table 3 presents the α s, slope coefficients, and t-statistics, adjusted r-squares from multiple time-series regressions, as well as the GRS statistics and it s p-value. The FF4 model is strongly rejected by the data. The GRS statistic is 16.43 and the null hypothesis that α s (pricing errors) are jointly 0, is rejected at a 0.01 significance level. More importantly, the average portfolio returns and α s 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 their average. I test FF5 (FF4 plus a volatility factor) on the thirty moneyness-maturity portfolios. Adding the volatility factor does not help. Actually MAE of FF5 is four basis points greater than MAE of FF4, because the volatility factor predicts a higher premium for long maturity options. You can see excess returns, FF5 predicted returns and α s in Figure 4. Adding the liquidity factor as in Pastor and Stambaugh (2003) does not help to price these portfolio returns either. 21

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-Carhart four-factor model on excess returns (delta-hedged, leverage-adjusted) of the thirty portfolios formed on moneyness-maturity. There are 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 report α 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 the null hypothesis of zero pricing errors to estimate t-statistics and p-values. β 1, β 2, β 3, β 4 are coefficients of excess market return, SMB (small minus big), HML (high minus low), WML (momentum); respectively. To save space, I don t report coefficient of WML. The data on pricing factors are available from Kenneth French data library. I convert daily returns to holding period returns (details are explained in methodology section). The sample covers 204 months from January 1996 to January 2013. R e i,j,t = α i,j +β 1 i,j (Rm t Rf t )+β 2 i,j SMB t +β 3 i,j HML t +β 4 i,j WML t +ε i,j,t i {DOTM, OTM, ATM, ITM, DITM} j {1M, 2M, 3M, 4 : 6M, 7 : 12M, > 12M} Moneyness Maturity 1 2 3 4:6 7:12 >12 1 2 3 4:6 7:12 >12 α DOTM 1.90 0.66 0.15-0.24-0.32-0.63 8.72 2.39 0.58-0.90-1.13-1.86 OTM 1.57 0.77 0.22-0.00-0.05-0.28 10.63 4.63 1.35-0.01-0.32-1.58 ATM 1.21 0.59 0.12 0.00-0.03-0.04 10.64 5.13 1.06 0.02-0.24-0.33 ITM 0.89 0.42 0.17 0.05 0.00-0.02 12.31 4.72 2.48 0.70 0.02-0.26 DITM 0.47 0.18 0.06-0.02-0.05-0.00 6.81 2.63 1.05-0.35-0.68-0.07 β 1 t(β 1 ) DOTM 0.24 0.39 0.42 0.51 0.48 0.68 5.27 6.93 7.76 9.23 8.05 9.75 OTM 0.19 0.22 0.29 0.30 0.31 0.37 6.05 6.50 8.44 8.97 8.92 10.02 ATM 0.09 0.14 0.19 0.19 0.20 0.24 3.95 6.01 8.40 7.71 8.02 9.88 ITM 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 3.95 β 2 t(β 2 ) DOTM 0.02 0.01 0.13 0.08 0.15 0.29 0.28 0.09 1.72 0.99 1.82 2.90 OTM 0.11 0.06 0.08 0.10 0.15 0.22 2.46 1.33 1.69 2.12 3.06 4.17 ATM 0.07 0.07 0.09 0.10 0.10 0.17 1.96 2.02 2.90 2.83 2.75 4.96 ITM 0.04 0.05 0.03 0.06 0.11 0.13 1.85 1.71 1.44 2.79 3.79 5.31 DITM 0.03 0.05 0.07 0.05 0.12 0.11 1.36 2.31 3.61 2.79 5.56 5.01 β 3 t(β 3 ) DOTM 0.26 0.40 0.36 0.37 0.32 0.41 3.66 4.49 4.18 4.25 3.46 3.68 OTM 0.16 0.21 0.22 0.22 0.22 0.26 3.23 3.84 4.06 4.11 3.96 4.43 ATM 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 2.53 DITM 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 0.47 ATM 0.14 0.26 0.36 0.35 0.37 0.47 ITM 0.13 0.22 0.34 0.34 0.26 0.35 DITM 0.02 0.07 0.20 0.13 0.20 0.21 t(α) 22

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. The bottom panel presents alphas as red bars and their two standard deviation confidence intervals as blue lines across moneyness and maturity groups. See Table 3 for details of moneyness-maturity groups. Excess return (basis points per month) 250 200 150 100 50 0 0.2 0.2 < 0.4 0.4 < 0.6 0.6 < 0.8 0.8 < 1M 3M 12M 1M 3M 12M 1M 3M 12M 1M 3M 12M 1M 3M 12M DOTM OTM ATM ITM DITM 200 Alpha 100 0 100 DOTM OTM ATM ITM DITM Moneyness Maturity B. Empirical Option Pricing Model with Three Factors In the spirit of the Fama-French three-factor model, I propose applying an empirical pricing model, which I name the option three-factor model. In the first subsection, I explain how I construct factors. In the second subsection, I test this model on the thirty moneyness-maturity portfolios and the decile value portfolios. 1. Pricing Factors I construct pricing factors as linear combinations of the excess returns of the moneyness-maturity and value portfolios. Note that the thirty moneyness-maturity portfolios are formed on five moneyness categories based on the range of : Deep out of the money (DOTM) 0 < 0.20, out of the money (OTM) 0.20 < 0.40, at the money (ATM) 0.40 < 0.60, in the money (ITM) 0.60 < 0.80, deep in the money (DITM) 0.80 < 1 and, 6 maturity categories: 1, 2, 3, 4 to 6, 7 to 12, and greater than 12 months to expiration. The decile value portfolios are formed on value, which is defined as the difference between historical volatility and Black- Scholes implied volatility. Portfolio (expensive) 1 consists of options with high value (cheap), while portfolio 10 consists of low value options. 23

Level Factor : I define level as the average return from selling at-the-money (ATM) option portfolios. Ri,j,t e denotes excess return of option portfolio with moneyness category i and maturity category j at time t. Level t = 1 6 i j R e i,j,t i {ATM}, 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. Slope t = 1 12 i 1 j 1 R e i,j,t 1 6 i 2 j 2 R e i,j,t i 1 {DOTM, OTM, ATM, ITM, DITM}, j 1 {1M, 2M} i 2 {DOTM, OTM, ATM, ITM, DITM}, j 2 {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. R e i,t denotes excess return on the i th decile value portfolio at time t, where 10 th decile portfolio consists of options with low value. VAL t = 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 equation 1 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 1 30 i j α i,j under 7 different pricing models. These models are BS (Black-Scholes), FF4 (Fama-French-Carhart fourfactor 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 Figure 5 shows the success of the option factor models. The MAE for 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

successful either. The option factor models, however, reduce the M AE to 13-15 basis points.table 4 reports the results of the following time series regressions. R e i,j,t = α i,j +β L i,jlevel t +β s i,jslope t +β v i,jvalue t +ε i,j,t i {DOTM, OTM, ATM, ITM, DITM} j {1M, 2M, 3M, 4 : 6M, 7 : 12M, > 12M} Figure 5: Average α : Moneyness-Maturity Portfolios The figure presents average α 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 principal component of the thirty moneyness-maturity portfolios. OPT2 refers to the option two-factor model (level and slope), OPT3 the option three-factor model (level, slope and value). 45 40 35 MAE Basis Points 30 25 20 15 10 5 0 BS CAPM FF4 FF5 PCA2 PCA5 OPT2 OPT3 I estimate coefficients using OLS. I calculate the GRS statistics as they do in Gibbons, 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 threefactor model describes the expected excess returns of moneyness-maturity portfolios, then the α s should be close to zero. The model seems to do well on average, with mean absolute α 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 M AE of 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 in market 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 α 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%. 25

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 threefactor model (OPT3) on excess returns (delta-hedged, leverage-adjusted) on the thirty portfolios formed on moneyness-maturity. There are 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 report α 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 the null hypothesis of zero pricing errors to estimate t-statistics and p-values. β L, β S, β V are factor sensitivities of the level, slope, value factors respectively. The sample covers 204 months from January 1996 to January 2013. R e i,j,t = α i,j +β L i,j Level t +β s i,j Slope t +β v i,j Value t +ε i,j,t Moneyness i {DOTM, OTM, ATM, ITM, DITM} j {1M, 2M, 3M, 4 : 6M, 7 : 12M, > 12M} Maturity 1 2 3 4:6 7:12 >12 1 2 3 4:6 7:12 >12 α DOTM -0.13-0.80-0.65-0.27 0.01-0.31-0.52-2.27-2.16-1.08 0.04-0.97 OTM 0.14 0.03-0.27-0.01 0.14 0.02 1.03 0.18-1.88-0.07 1.12 0.15 ATM 0.06 0.03-0.18-0.04-0.06 0.19 0.62 0.33-2.36-0.87-0.77 2.15 ITM 0.19-0.17-0.06-0.02 0.04 0.13 2.31-1.47-0.79-0.34 0.38 1.83 DITM 0.06-0.18-0.07-0.05 0.00 0.12 0.58-1.77-0.78-0.71 0.03 1.08 β L t(β L ) DOTM 1.83 2.21 2.13 2.19 2.03 2.54 18.51 16.52 18.43 22.90 18.24 21.01 OTM 1.40 1.47 1.49 1.39 1.35 1.46 27.21 27.31 27.34 39.72 27.99 30.23 ATM 1.04 1.01 1.00 0.97 1.02 0.96 27.44 29.09 34.54 53.20 34.51 28.73 ITM 0.59 0.65 0.59 0.62 0.67 0.61 19.04 15.15 21.99 29.64 17.31 22.00 DITM 0.32 0.40 0.37 0.29 0.33 0.30 7.89 10.31 11.12 10.17 7.67 6.88 β S t(β S ) DOTM 1.50 0.73 0.11-0.78-0.95-1.65 7.54 2.68 0.46-4.03-4.25-6.74 OTM 1.15 0.40 0.03-0.50-0.67-0.98 11.04 3.67 0.26-7.00-6.80-10.11 ATM 0.99 0.29-0.02-0.29-0.32-0.65 12.89 4.02-0.30-7.83-5.32-9.69 ITM 0.48 0.23 0.05-0.18-0.30-0.35 7.74 2.64 0.93-4.17-3.84-6.23 DITM 0.28 0.28 0.04-0.09-0.18-0.26 3.36 3.53 0.58-1.54-2.02-2.98 β V t(β V ) DOTM 0.12 0.14 0.12 0.09-0.05 0.54 0.96 0.84 0.86 0.74-0.40 3.64 OTM -0.01-0.09 0.03 0.05 0.02 0.21-0.18-1.39 0.41 1.14 0.33 3.56 ATM -0.07-0.03 0.02-0.00-0.01 0.11-1.58-0.80 0.47-0.12-0.40 2.65 ITM 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 0.38 Adj 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 0.92 ATM 0.79 0.84 0.90 0.96 0.92 0.91 ITM 0.65 0.60 0.78 0.89 0.75 0.84 DITM 0.23 0.36 0.47 0.49 0.38 0.38 t(α) 26

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. The bottom panel presents alphas as red bars and their two standard deviation confidence intervals as blue lines across moneyness and maturity groups. See Table 4 for details of moneyness-maturity groups. Excess return (basis points per month) 250 200 150 100 50 0 50 0.2 0.2 < 0.4 0.4 < 0.6 0.6 < 0.8 0.8 < 1M 3M 12M 1M 3M 12M 1M 3M 12M 1M 3M 12M 1M 3M 12M DOTM OTM ATM ITM DITM 0 Alpha 50 100 150 DOTM OTM ATM ITM DITM Moneyness Maturity Despite the fact that the option three-factors α s are substantially less than the absolute excess returns, the F-test of Gibbons, Ross, and Shanken (1989) still rejects the null hypothesis that all α s are jointly zero. Furthermore, I added a couple of new factors that do not solve the problem. I interpret this with the existence of small non-systematic priced factors. In the next subsection, I will diagnose the rejection by applying a principal component analysis. I test linear factor models consisting of principal components of moneyness-maturity portfolios. I find that we need first 15 principal components to not to reject a model in the data. One interpretation of these small priced factors: they might be reflecting the difference of expected return between portfolios due to liquidity. Another interpretation is that these factors are unreal and sample specific. The option three-factor model (OPT3) is actually not rejected in recent sample (January 2005 to January 2013). In the following sections I will show that OPT3 performs even better for the other sets of portfolios I considered. Another interesting thing about these regressions concerns the pattern of factor β s. The coefficients of level β L does not change much across different maturities.β M is about 2 for DOTM 27

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. β L varies with moneyness, because the volatility of DOTM options is much higher than DITM options. If we adjust the returns with volatility, β L wouldbemoreorlessconstant. Incontrast, coefficientsofslopefactor, monotonically varies over the maturity dimension, and B S is positive for short maturity options and negative for long 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 characteristicbased 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 explained by the first n principal components. There is a strong factor structure: the first principal component explains more than 83% of the variation of returns, with the first two component numbers 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 the Figure 7 show the p(grs)(p-value of GRS statistics) and M 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 components 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 8 See Nozawa (2012),Palhares (2012),Lustig, Roussanov, and Verdelhan (2011). 28

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 M AE 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 moneynessmaturity portfolios. X-axis refers to the number of principal components. The first figure displays the cumulativer 2 explainedbyfirstnprincipalcomponents. P-valueoftheGRSstatisticsisdenotedbyp(GRS). MAE refers to the mean absolute pricing errors. The second and third figure displays p(grs) and MAE from 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. 100 Cumulative R 2 0.35 p(grs) 0.3 95 0.25 90 0.2 0.15 85 0.1 0.05 80 0 5 10 15 0 0 5 10 15 40 MAE 4 E[PC] 30 3 20 OPT3 2 1 10 OPT2 0 0 5 10 15 0 1 0 5 10 15 4. Value In this part, I argue that equation 1 is a good specification for the decile value portfolio s 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. R e i,t = α i +β L i Level t +β S i Slope t +β V i Value t +ε i,t i {1, 2,..., 10} 29

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. Table 5 presents the details of the asset pricing tests. If equation 1 is a good description of the expected returns, then the regression intercepts should be close to zero. The F-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 α 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 α 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 model on excess returns(delta hedged, leverage adjusted) of the decile portfolios formed on value(one year historical realized volatility minus Black-Scholes implied volatility). I report α 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 the null hypothesis of zero pricing errors to estimate t-statistics and p-value. β L, β S, β V are slope coefficients of the level, slope, value factors respectively. Ri,t e is excess return on portfolio i at time t. The sample covers 204 months from January 1996 to January 2013. Ri,t e = α i +βi L Level t +βislope s t +βivalue v t +ε 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[R e ] -0.11 0.06 0.06 0.12 0.18 0.25 0.30 0.41 0.62 1.72 1.84 α 0.20 0.17-0.09-0.04-0.04-0.03-0.02-0.05 0.01 0.32 0.11 t(α) (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) β L 1.07 0.90 0.89 0.80 0.77 0.74 0.75 0.80 0.84 1.23 0.15 t(β L ) (24.32) (32.52) (32.11) (32.67) (33.13) (28.94) (29.44) (30.82) (34.82) (24.68) (3.44) β S -0.35-0.27-0.12-0.17-0.20-0.17-0.20-0.25-0.28-0.21 0.14 t(β S ) (-4.01) (-4.79) (-2.22) (-3.38) (-4.43) (-3.33) (-3.92) (-4.86) (-5.76) (-2.13) (1.57) β V -0.55-0.32-0.17-0.07 0.04 0.10 0.16 0.34 0.51 1.12 1.67 t(β V ) (-10.33) (-9.36) (-4.90) (-2.31) (1.42) (3.15) (5.20) (10.64) (17.36) (18.49) (30.63) R 2 0.87 0.92 0.90 0.91 0.91 0.88 0.89 0.90 0.93 0.87 0.83 30

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 the cross-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, Figure 8 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 Lehman s 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 pattern is most likely a consequence of the rise in volatility spreads during the crisis. Figure 8: Cumulative Factor Returns and Mean-Variance Frontier The left panel displays log cumulative return of the level, slope, value factors. The right panel displays tangency portfolio and mean-variance frontier of the level, slope and value factors. SR stands for annualized Sharpe ratio. 2 1.8 1.6 Level Slope Value 110 100 90 Tangency (SR=5.15) Value (SR=2.61) (log) Cumulative Returns 1.4 1.2 1 0.8 0.6 Asian financial crisis Russian default crisis WorldCom bankruptcy European sovereign crisis Lehman bankruptcy Excess Return (basis points per month) 80 70 60 50 40 30 Slope (SR=3.22) Level (SR=0.86) 0.4 20 0.2 0 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 10 0 0 20 40 60 80 100 120 140 160 180 200 220 Standard Deviation Table 6 presents the summary statistics of these pricing factors. The level factor has a mean 46 basis point monthly return, 184 basis point volatility and 0.86 annualized Sharpe ratio. The level factor is very similar to the first principal component of the thirty moneyness-maturity portfolios with a correlation coefficient of 0.96. The slope factor has a mean 86 basis point monthly return, 31

92 basis point volatility and 3.22 annualized Sharpe ratio. Lastly, the value factor has a mean 91 basis point monthly return, 121 basis point volatility and 2.61 annualized Sharpe ratio. The correlation structure between pricing factors are surprising. The level and slope factors have high negative (-0.61) correlation with each other and they are essentially uncorrelated with the value factor. As a result, there is enormous diversification benefits in combining these three strategies. Table 6 also presents summary statistics of tangency portfolio of the level, slope and value factors. The tangency portfolio has an annualized Sharpe ratio of 5.15. From Figure 8, we can see that the tangency portfolio does not suffer from crashes. The evidence strongly indicates that the level factor behaves as a risk premium and is a compensation for market-wide volatility and jump shocks, in particular the fact that the level factor is highly sensitive to the contemporaneous innovations in VIX. Their correlation coefficient is more than -0.7, which is not surprising since the level factor is basically the return on selling volatility. Note that VIX has a negative market risk price since high volatility corresponds to the bad states of nature. We can also see from Table 6 that the level factor has a high correlation, around 0.6, with excess market returns. To develop this argument, I evaluate the performance of the level factor under different states of nature. To do this, I consider several sub-samples, which include months in which there are market-wide price jumps, volatility jumps and months without any jumps. I use jump definitions similar to the ones used in Constantinides, Jackwerth, and Savov (2011). Price jump is defined as the daily change in S&P 500 index of less than -4%, and volatility jump is defined as the daily change in VIX index greater than 4%. If there is more than one day with a price (volatility) jump day in a given month, I call that month price (volatility) jump. I also consider periods with NBER recessions, market distress (months with market return less than -5% or -10%), severe bear markets (market return over the past twelve months less than -25%), and high-low volatility (months with top-bottom 30% of VIX distribution over the sample period, exact thresholds are around 25% and 16%). I find that the level factor is extremely sensitive to the episodes of price and volatility jumps. Average returns during price jumps (14 months) are -200 basis points with a -1.7 Sharpe ratio; on the other hand, average returns are 64 basis points with a 1.6 Sharpe ratio during months without price jumps (190 months). We see a similar effect of volatility jumps. Average returns during volatility jumps (39 months) are -82 basis points with -0.9 Sharpe ratio; average returns go up to 76 basis point with a 2.1 Sharpe ratio during periods without any volatility jumps (165 months). I find that average return of the level factor is not sensitive to NBER recessions and expansions, but the volatility is highly sensitive. Volatility of the level during recessions is more than twice its volatility during expansions. During the severe bear markets (market return over the past twelve months less than -25%), average returns rise dramatically to 126 basis points. This is in line with the central intuition of macro asset pricing models with time-varying expected returns. Expected 32

returns are higher during recessions, since marginal value of wealth is high at those times. The level factor is also very sensitive to the market distress. The people who invest in the level lose 197 basis points and 485 basis points during the months with monthly market return less than 5% (25 months) and 10% (7 months). Table 6: Summary Statistics of Factors This table reports summary statistics for ten factors. 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 and selling short maturity ones; lastly, the value factor is the average return on buying high value option portfolios and selling low value ones. Tang is the tangency portfolio of the level, slope and value factors. P1 and P2 are first and second principal components of the thirty moneyness-maturity portfolios. Rm-Rf, SMB, HML, WML are factors of the Fama-French-Carhart four-factor model (excess market return, small minus big, high minus low, momentum). Lastly dvix refers to monthly (holding period) change in VIX. The sample covers 204 months from January 1996 to January 2013. Level Slope Value Tang P1 P2 Rm-Rf SMB HML WML dvix Mean 0.46 0.86 0.91 0.77 1.99 2.15 0.54 0.31 0.18 0.56 0.08 St.Dev. 1.84 0.92 1.21 0.52 14.00 3.39 5.47 3.50 3.14 5.67 6.10 Sharpe Ratio 0.86 3.22 2.61 5.15 0.49 2.20 0.34 0.31 0.20 0.34 0.04 T-statistics 3.56 13.28 10.78 21.25 2.03 9.07 1.40 1.26 0.81 1.41 0.18 Skewness -1.21 0.93 1.11 0.07-1.56 0.29-0.98 0.34-0.29-0.70 2.08 Kurtosis 8.69 6.58 6.03 5.33 8.27 8.84 5.92 5.46 6.84 8.02 11.82 Correlation Coefficients Level 1.00-0.61-0.03 0.17 0.96-0.08 0.57 0.08 0.36-0.24-0.70 Slope 1.00 0.16 0.63-0.62 0.54-0.52-0.05-0.20 0.27 0.57 Value 1.00 0.51-0.00 0.05-0.01 0.00 0.03-0.00-0.05 Tang 1.00 0.13 0.54-0.09 0.02 0.10 0.09 0.01 P1 1.00 0.00 0.61 0.01 0.39-0.19-0.73 P2 1.00-0.14-0.28 0.10 0.23 0.17 Unlike the level factor, I find it extremely challenging to explain the premium on the slope factor, which is essentially buying long maturity options and selling short maturity options. Based on modern option pricing theories, the slope premium is potentially a compensation for marketwide volatility, price-jump and volatility-jump risk. Table 6 reports the correlation coefficient between the slope factor and innovations in VIX. Indeed, the slope factor is moderately sensitive to the innovations on VIX since they have correlation of a 0.57. However the correlation has the wrong sign. Correlation with VIX implies a negative premium for the slope factor, not a positive one. The slope factor also has a negative market beta, which implies a negative premium. Traditionally, investors view short maturity options as being more sensitive to the jumps than long maturity options, since short maturity options are more levered positions. It is tempting to think that jumps can explain the slope premium. There is one problem with this interpretation: since I adjust for the leverage when forming positions, short maturity options are not necessarily more sensitive to the jumps. According to Figure 8, we don t see any negative jump for the 33

slope factor. Moreover, we can observe mild positive jumps during Lehman s bankruptcy and the European sovereign crisis. Evaluating the performance of the slope factor under sub-samples shed further light on the slope-jump link. The average return on the slope factor is 163 basis points during a price-jump period (14 months) as opposed to 80 basis points during period without any price jump (190 months). The pattern is similar for volatility jumps, with 130 basis points versus 75 basis points. From Table 6, we also see that the slope factor performs better during NBER recessions (116 basis points) than expansions (80 basis points). The slope factor does extremely well during market distress. The people who invest in the slope factor earn 190 basis points and 295 basis points during the months with monthly market return of less than 5% (25 months) and 10% (7 months). These results are puzzling, since the slope factor behaves as an insurance, yet it has a positive premium. It is true that the slope factor is highly sensitive to jumps, but jump risk implies a negative premium for the slope. As a result, we need to think beyond the standard market-wide volatility jump risk interpretation in order to understand the slope premium. Indeed, the slope premium puzzle and the maturity-risk puzzle are two sides of the same coin. In order to understand slope, we first need to understand why short-maturity options have higher expected returns and lower risk than long-maturity options. The answer lies in the effects of gamma and embedded leverage on the supply and demand for options. Gamma ( 2 OptionPrice StockPrice 2 ) is the second derivative of option prices with respect to stock price, and it tells us how much we need to update the hedge ratio delta when the underlying asset moves. As gamma rises, it becomes more costly to hedge options. In fact, option market makers are known to be averse to holding gamma risk in their portfolios, because they can lose arbitrarily large amounts of money by having negative gamma exposure, while the premium is limited. An anecdotal example is presented in the April 16, 2010 edition of the Wall Street Journal. The article, Option Market Makers Grapple with Gamma Risk in Goldman, describes the real life case in which option market makers on Goldman Sachs shares had a painful time when hedging their risk in short maturity options. Goldman Sachs shares decrease from $184 to almost $155 when the U.S. Securities and Exchange Commission charged the bank with fraud. Market makers had to buy and sell a massive amount of Goldman shares in a very short amount of time. The hedging process is especially more difficult for short maturity options, because delta the hedge ratio moves much faster for short maturity options. To show the importance of gamma, I prepare a simulation-based example. I consider a market makers hedging cost of for selling $1 worth of straddles consisting of 0.5 call and -0.5 put option across 8 maturities (1, 3, 5 days and 1, 3, 6 months and 1, 2 years). I simulate 100,000 a day of stock price under Black-Scholes dynamics with σ = 0.4, µ = 0.15, rf = 0.04. I assume that the market maker delta-hedges his position 16 times a day and each day consists of 16 periods in the simulations. I define the hedging cost as 0.2% times total dollar volume of hedging demand. Figure 9 presents the results. Panel A reports hedging cost per day for 1$ worth option. There 34

is an enormous variation in the hedging cost across maturities. If a market maker sells 1$ worth straddle with 1 day to maturity, he needs to spend on average 0.25 cents in transaction costs per day. This magnitude is more than 200 times the cost of 1$ worth straddle with 1 year to maturity. Panel B plots gamma of the same straddle positions (per dollar) across maturities. Specifically, I calculate a gamma of a straddle as the sum of call and put option gamma divided by sum of call and put option price. Variation in gamma is very similar to the variation in hedging-cost across maturities. Gamma of one day to maturity straddle is more than 200 times the gamma of one year to maturity straddle. The results are not sensitive to the choice of µ and rf. Hedging cost per dollar rises with σ. Reducing the frequency of hedging reduces the cost, but then market maker can face a large stock price movement with a large delta. Figure 9: Why Gamma Matters? Panel A reports the average hedging cost per day as a percentage of option price across maturities from 10,000 simulated samples. I simulate a day of stock price under Black-Scholes dynamics with σ = 0.4, µ = 0.15, rf = 0.04. I assume that there are 16 time periods a day. I consider the hedging cost of a market maker for selling $1 worth of straddles consisting of 0.5 call and -0.5 put option across 8 maturities (1, 3, 5 days and 1, 3, 6 months and 1, 2 years). I assume that market maker delta-hedges his position 16 times a day. I define the hedging cost as 0.2% times total dollar volume of hedging demand. Red bars denote the average hedging cost and blue lines indicate two standard deviation confidence intervals. Panel B plots gamma of same straddle positions (per dollar) across maturities. Specifically, I calculate the gamma of a straddle as the sum of call and put option gamma divided by the sum of call and put option price. 45 25 40 Panel (A) 35 20 Panel (B) 100 Gamma Hedging Cost 30 25 20 Hedging Cost Option P rice Per Day 100 Gamma 15 10 15 10 5 5 0 1d 3d 5d 1m 3m 6m 1y 2y Maturity 0 1d 3d 5d 1m 3m 6m 1y 2y Maturity I consider the same straddle positions under six hypothetical scenarios in which the stock price jumps, implied volatility jumps, and both stock price and implied volatility jump 5% and 10%. I compare the instantaneous loss to market makers from selling straddles across maturities. Figure 10 Panel A and F report results with regular straddle loss and leverage-adjusted straddle loss for 5% 35

and 10% stock price jump. In both cases, we observe that the loss on straddles sharply decreases by maturity. Panel D and E plots present the results for the third and fourth scenarios, where implied volatility jumps 5% and 10%. Leverage-unadjusted only mildly decreases, while leverageunadjusted loss sharply increases. As a result, we see that price jumps impact short maturity options and volatility jumps impact long maturity options for leverage-adjusted positions. In the fifth and sixth scenarios, I consider both price and volatility jumps. From Panel E and F, we observe that the leverage unadjusted loss decreases only for very short maturity options (1, 3, 5 days) and rises after one month. Table 7: The Level, Slope, Value during Price-Volatility Jumps, Recessions, Severe Bear Markets, Market Distress and Funding Liquidity Conditions This table reports summary statistics of the level, slope and value factors under a series of market conditions. For each factor mean, standard deviation, t-statistics and annualized Sharpe ratios are displayed. 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 of less than -4%. Non price jump denotes rest of the months. Volatility jump indicates months with at least one day with a jump in VIX index, which is defined as daily change in VIX greater than 4%. Non volatility jump free denotes rest of the months. Recession implies NBER recessions. Expansion denotes all the other months. Severe bear market represents months with past 12 month S&P 500 index returns of less than -25%. Rising market stands for rest of the months. Market return <-5% and <%-10 represents months with contemporaneous monthly S&P 500 returns of less than -5% and -10%. High VIX (Low VIX) corresponds to the months at the top (bottom) 25% the distribution of VIX in the sample. Thresholds are approximately 16% and 25%. Rest of the months are denoted with Medium VIX. High (Low) funding liquidity corresponds to the months at the bottom (top) 25% of the distribution of Ted spread in the full sample. Thresholds are approximately 23 and 55 basis points. Rest of the months are denoted with medium funding liquidity. High (Low) market liquidity corresponds to the months at the top (bottom) 25% of the distribution of Pastor and Stambaugh (2003) aggregate liquidity measure. Rest of the months are denoted by medium market liquidity. The sample covers 204 months from January 1996 to January 2013. Level (bps) Slope (bps) Value (bps) Nobs Mean Std T-stat SR Mean Std T-stat SR Mean Std T-stat SR Full Sample 204 46 184 3.56 0.9 86 92 13.28 3.2 91 121 10.78 2.6 Price Jump Period 14-200 416-1.80-1.7 163 164 3.73 3.5 156 151 3.86 3.6 Non Price Jump Period 190 64 141 6.27 1.6 80 82 13.37 3.4 87 118 10.15 2.6 Volatility Jump Period 39-82 304-1.68-0.9 130 133 6.12 3.4 108 158 4.25 2.4 Non Volatility Jump Period 165 76 126 7.77 2.1 75 77 12.61 3.4 88 111 10.15 2.7 Recession 32 45 324 0.78 0.5 116 137 4.81 2.9 119 169 4.00 2.5 Expansion 172 46 146 4.14 1.1 80 81 13.02 3.4 86 110 10.29 2.7 Severe Bear Market 13 126 370 1.23 1.2 76 173 1.57 1.5 135 201 2.41 2.3 Rising Market 191 41 165 3.39 0.9 86 85 14.10 3.5 88 114 10.73 2.7 Market Return <-5% 25-197 279-3.54-2.5 190 124 7.67 5.3 101 138 3.65 2.5 Market Return <-10% 7-485 270-4.75-6.2 295 156 5.01 6.6 123 175 1.86 2.4 Low VIX 51 31 115 1.89 0.9 80 79 7.25 3.5 61 67 6.46 3.1 Medium VIX 101 37 157 2.40 0.8 85 81 10.58 3.6 77 112 6.90 2.4 High VIX 52 78 269 2.08 1.0 93 122 5.48 2.6 149 157 6.88 3.3 Low Funding Liquidity 51 12 265 0.33 0.2 104 118 6.31 3.1 139 141 7.05 3.4 Medium Funding Liquidity 102 50 147 3.46 1.2 81 79 10.41 3.6 91 108 8.43 2.9 High Funding Liquidity 51 71 151 3.38 1.6 76 87 6.23 3.0 45 107 3.03 1.5 Low Market Liquidity 51 20 261 0.55 0.3 109 113 6.86 3.3 135 155 6.22 3.0 Medium Market Liquidity 102 41 154 2.72 0.9 86 90 9.61 3.3 84 107 7.94 2.7 High Market Liquidity 51 81 142 4.08 2.0 62 64 6.91 3.4 63 99 4.56 2.2 36

From this simple exercise, we can see why option market makers are averse to having high gamma in their positions. They cannot replicate options using stocks and bonds for short maturity options. Even when they do, they will face large jumps. On the other hand, in my empirical analysis, I show that short maturity options (with high gamma) do not suffer from crashes as much as long maturity options. There are two possible explanations. One, most of the price jumps which market makers fear can be idiosyncratic and they don t impact the results from my well-diversified portfolios. Second, market-wide price jumps which would affect my short maturity option portfolio returns tend to occur with volatility jumps. In the simulations, I show that for options greater than one month to maturity, the impact of volatility jumps tends to dominate for leverage-adjusted positions. The second important difference between long and short maturity options is embedded leverage (elasticity of option price with respect to stock price). Frazzini and Pedersen (2012) argue that securities with high embedded leverage alleviate investors leverage constraints. Therefore, investors are willing to pay more for assets that enable them to increase their leverage. Options with one month to maturity have about six to seven times more embedded leverage than options with more than twelve months to maturity. The demand-based option pricing model of Garleanu, Pedersen, and Poteshman (2009) is related to both gamma and embedded leverage. Garleanu, Pedersen, and Poteshman (2009) argue that risk-averse financial intermediaries require a higher premium on options with higher demand pressure. I don t have data on demand-pressure, but the data on trading volume shows that more than 55% of the trading volume in notional is in the options with less than one month to maturity during 199601-201301 sample. If most of these transactions are demand-driven, then demand-pressure can potentially explain the maturity puzzle. Garleanu, Pedersen, and Poteshman (2009) also show that demand pressure in one option contract increases its price by an amount proportional to the variance of the unhedgeable part of the option. I argue that because of gamma, the unhedgeable part of an option is larger for short maturity options. The premium on the value factor is difficult to explain as well. It has very low correlation with the level, the slope, dvix and Fama-French-Carhart factors. Market-wide volatility premium can not explain the value premium, since contemporaneous correlation between the value factor and the innovations in VIX is very low, just -0.05. Market-wide jump premium can not be an explanation either. The value factor tends to perform better in jump periods then non-jump periods. Average return on the value factor is 156 basis points during price jump period (14 months) as opposed to 87 basis points during period without any price jump (190 months). The pattern is similar but less strong during volatility jump periods, 108 basis points versus 88 basis points. The value factor also performs slightly better during NBER recessions (119 basis points) than expansions (86 basis points). Based on this evidence, it is hard to reconcile the premium on value and market-wide volatility and jump premium. 37

Figure 10: Instantaneous Jump Loss on Selling ATM Straddles (Example) Figure reports instantaneous jump loss on selling $1 worth of straddles consisting of 0.5 call and -0.5 put option across 8 maturities (1, 3, 5 days and 1, 3, 6 months and 1, 2 years). I assume that annual Black-Scholes implied volatility is equal to 0.4. I consider 6 hypothetical scenarios. Panel A and B report the result for the first scenario in which stock price jumps 5% and 10%. Panel C and D report the result for the second scenario in which Black-Scholes implied volatility jumps 5% and 10%. Panel E and F report the result for the third scenario in which both stock price and Black-Scholes implied volatility jumps 5% and 10%. Panel A, C and E report results for regular return on selling straddle. Panel B, D and F report results for leverage-adjusted return on selling straddle. 450 400 Panel (A) %10 Price Jump %5 Price Jump 10 9 Panel (B) %10 Price Jump %5 Price Jump Jump Loss 350 300 250 200 150 100 Instantaneous Price-Jump Loss Leverage-Adjusted Jump Loss 8 7 6 5 4 3 2 Instantaneous Price-Jump Loss (Leverage-Adjusted) 50 1 0 1d 3d 5d 1m 3m 6m 1y 2y Maturity 0 1d 3d 5d 1m 3m 6m 1y 2y Maturity 40 35 Panel (C) %10 Vol Jump %5 Vol Jump 14 12 %10 Vol Jump %5 Vol Jump Panel (D) Jump Loss 30 25 20 15 10 Instantaneous Vol-Jump Loss Leverage-Adjusted Jump Loss 10 8 6 4 Instantaneous Vol-Jump Loss (Leverage-Adjusted) 5 2 0 1d 3d 5d 1m 3m 6m 1y 2y Maturity 0 1d 3d 5d 1m 3m 6m 1y 2y Maturity 400 350 Panel (E) %10 Price & Vol Jump %5 Price & Vol Jump 15 %10 Price & Vol Jump %5 Price & Vol Jump Jump Loss 300 250 200 150 100 Instantaneous Price-Vol-Jump Loss Leverage-Adjusted Jump Loss 10 5 Panel (F) Instantaneous Price-Vol-Jump Loss (Leverage-Adjusted) 50 0 1d 3d 5d 1m 3m 6m 1y 2y Maturity 0 1d 3d 5d 1m 3m 6m 1y 2y Maturity Previously, Goyal and Saretto (2009) propose an alternative explanation. They conjecture that overreaction to current stock returns leads to misestimation of future volatility, which cause 38

the value premium. Their main evidence is the underlying stock of the options in the extreme portfolios (decile 1 and 10), which show extreme returns at the portfolio formation date. I find that this is true for call options. Stocks on low value (expensive) call options perform worse at the portfolio formation month. There is still a value premium for put options, but no pattern in the underlying stock at the portfolio formation date. Stocks on low value (expensive) put options do not perform worse at the portfolio formation month then high value (cheap) put options. This evidence contradicts the overreaction story. Schürhoff and Ziegler (2011) propose an explanation for option portfolios sorted on variance risk premia. They develop a model consistent with theories of financial intermediation under capital constraints, in which both systematic VRP and idiosyncratic VRP are priced. I calculate average variance risk premia (VRP) for value portfolios. VRP of low value (expensive) options are about 0.06, compared to -0.01 VRP of high value (cheap) option portfolios, is substantially higher. The interpretation of the value premium as compensation for risk-averse financial intermediaries can be considered as consistent with Schürhoff and Ziegler (2011) model. I find another interesting pattern related to the value portfolios. The monthly growth in the option market capital (open interest times mid-price of option) is substantially higher for low value (expensive) option portfolios than high value (cheap) option portfolios. For instance, monthly growth is about 20% in decile 10 (low value) option portfolio as opposed to 8% in decile 1 option portfolio at the portfolio formation month. The pattern of growth in option market capital persists for both call and put options. We can consider the unusual growth of option market capital as a sign of higher demand pressure for low value (expensive) option portfolios. If so then demand-based model of Garleanu, Pedersen, and Poteshman (2009) appear to be a valid explanation for the value premium. If market frictions are affecting the slope and value premiums, then we should expect they are related to funding liquidity conditions. Following Frazzini and Pedersen (2010), I proxy funding liquidity conditions by TED spread (the difference between 3-month LIBOR rate and 3-month US Treasury yield) and funding liquidity risk as volatility of TED spread. Based on TED spread full sample breakpoints (bottom and top 25%), I form three sub-samples: low, medium and high funding liquidity samples. From Table 7, we see that the premium on the slope factor and value factor is higher when funding liquidity conditions are tight (high TED spread). Average excess return on the value factor rises from 45 to 139 basis points from high to low funding liquidity sample. Similarly average excess return on the slope factor rises from 76 to 104 basis points. This evidence shows that it is harder to exploit the slope and value premiums when funding liquidity conditions are tight. I get similar results in sub-samples based on market liquidity. I use Pastor and Stambaugh (2003) aggregate liquidity measure as a proxy for market liquidity. 39

Figure 11: Funding Liquidity and Excess Returns The figure displays average monthly excess returns on selling option portfolios across moneyness-maturity groups during different funding liquidity conditions. I measure funding liquidity by the TED spread at the beginning of the holding period. I consider three sub-samples (low, medium, high funding liquidity) based on full sample breakpoints (top and bottom 25% of TED spread). High TED spread corresponds to the time periods with low funding liquidity. The full sample covers 204 months from January 1996 to January 2013. See Table 4 for details of moneyness-maturity groups. 250 0.2 0.2 < 0.4 Excess Return (basis points per month) 200 150 100 50 0 50 0.4 < 0.6 0.6 < 0.8 0.8 < 100 High Medium Low DOTM OTM ATM ITM DITM Moneyness Maturity 200 150 Excess Return (basis points per month) 100 50 0 50 1 2 3 4 5 6 7 8 9 10 Value Low (Expensive) High (Cheap) 40

Figure 12: Market Liquidity and Excess Returns The figure displays average monthly excess returns on selling option portfolios across moneyness-maturity groups during different market liquidity conditions. I measure market liquidity by the lag of Pastor and Stambaugh (2003) aggregate liquidity measure. I consider three sub-samples (low, medium, high market liquidity) based on full sample breakpoints (top and bottom 25% of aggregate liquidity). The full sample covers 204 months from January 1996 to January 2013. See Table 4 for details of moneyness-maturity groups. 250 0.2 0.2 < 0.4 Excess Return (basis points per month) 200 150 100 50 0 50 0.4 < 0.6 0.6 < 0.8 0.8 < 100 High Medium Low DOTM OTM ATM ITM DITM Moneyness Maturity 200 Excess Return (basis points per month) 150 100 50 0 50 1 2 3 4 5 6 7 8 9 10 Low (Expensive) Value High (Cheap) 41

Figure 11 presents average excess returns of the thirty moneyness-maturity portfolios and decile value portfolios under low-medium-high TED spread conditions. From the figure, we see a clean pattern. Average excess returns decrease from low to high TED spread conditions, but long maturity options decrease more than short maturity options. As a result, the maturity premium rises when funding liquidity conditions are tight. I find that the average return on low (expensive) value portfolio rises and on high (cheap) value portfolio decrease. Thus, the value premium is higher when TED spread is high. In fact, almost all option portfolio average returns are lower when the TED spread is high. This result is counter-intuitive, since selling options requires capital. Yet it is consistent with the findings of Frazzini and Pedersen (2010) for BAB portfolios across asset classes. Frazzini and Pedersen (2010) argue that a high TED spread can also mean a worsening of funding conditions. According to this interpretation, the decrease in the average option return makes sense. However, selling $1 worth of option with a short maturity require higher capital than selling $1 worth of long maturity option. As a result, we should expect short maturity options to be more sensitive to a high TED spread, but this is not what we see in the data. In any case, low average returns during high TED spread episodes appear to be robust findings for options and BAB portfolios. To further investigate this finding, I regress the level, slope, and value factors on the contemporaneous change in the TED spread, the lag TED spread, and VIX. I find that the lag TED spread predicts the level with a negative coefficient and the slope and value factors with positive coefficients. All three coefficients are economically and statistically significant. Detailed results on regressions are in the appendix. VI. Other Characteristics To further investigate the determinants of expected option returns, I build thirteen decile portfolios formed on eleven characteristics. These characteristics are option carry, variance risk premia, volatility reversals, systematic volatility, idiosyncratic volatility, stock short-term reversal, stock size, stock illiquidity, option illiquidity, embedded leverage and the slope of volatility term structure. I show that expected option returns are related to these characteristics. I also consider several other stock and option characteristics, such as book-to-market ratio, market beta, stock momentum, slope of volatility surface in moneyness direction. I do not report them, since they do not show any significant patterns, but the results are available upon request. Then I test seven alternative models on decile portfolios and compare the mean absolute pricing (M AE) implied by each model. These models are the Black-Scholes (BS), CAPM, Fama-French- Carhart four factor model (FF4), empirical pricing model with the first two and first five principal components of the thirty moneyness-maturity portfolios, the option-two-factor model(opt2: level and slope), the option three-factor model (OPT3: level, slope and value). Results are on Table 8. 42

I also report details of multiple time-series regression results for the option three-factor model. T- statistics and p-values are based on bootstrap procedure. I simulate 10,000 samples from the fitted regression residuals. I calculate standard errors of coefficients are sample standard deviations and the p-values of GRS and M AE statistics are estimated using the empirical distribution of these statistics. Table 8: Model Comparison This table presents average α (pricing errors) under different models. BS refers to Black-Scholes, FF4 refers to the Fama-French-Carhart four-factor model. FF5 is five factor model with an volatility factor in addition to FF4 factors. PCA2 and PCA5 refers to pricing model with first two and five principal components of the thirty moneyness-maturity portfolios. OPT2 refers to the option two-factor model (level and slope), OPT3 the option three-factor model (level, slope and value). The sample covers 204 months from January 1996 to January 2013. Dependent Portfolios Alternative Models M AE BS CAPM FF4 FF5 PCA2 PCA5 OPT2 OPT3 Moneyness-Maturity 44 38 37 41 31 21 13 15 Value 38 32 31 30 34 26 21 10 Carry 66 53 47 39 43 19 13 10 Variance Risk Premia 29 19 15 12 24 13 14 9 Volatility reversals 31 21 17 16 25 10 9 6 Systematic Volatility 33 23 18 5 26 8 11 7 Idiosyncratic Volatility 34 24 19 9 25 7 10 8 Stock Risk Reversal 32 27 22 11 24 14 14 14 Size 54 44 36 25 41 22 23 17 Stock Illiquidity 50 39 33 23 41 21 20 13 Option Illiquidity 48 35 28 9 30 18 13 10 Embedded Leverage (1) 240 196 162 110 146 83 62 61 Embedded Leverage (2) 34 24 19 13 27 14 14 12 Slope of Volatility Term Structure (S) 64 57 54 39 40 11 7 8 Slope of Volatility Term Structure (L) 18 15 15 25 26 22 22 22 Open Interest Gamma 32 22 20 20 26 19 16 12 A. Carry Expected return of an asset can be broken down into two components, the carry and the expected price change components. A carry trade involves going long in high carry assets and short in low carry assets. Traditionally, carry is applied only to currencies, but Koijen, Moskowitz, Pedersen, and Vrugt (2012) generalize carry to other asset classes such as global equities, bonds, currencies, and commodities, as well as within US Treasuries, credit, and equity index options. I complement 43

their work by applying carry to individual equity options. According to my knowledge, this is the first study of carry trade in individual equity option market. Particularly for equity options, carry is defined as expected return if the underlying stock price and implied volatility term structure do not change. Option carry has time-decay and roll-down components. I calculate the time-decay component using Black-Scholes theta and the roll-down component using Black-Scholes vega and implied volatility surface. See the Methodology section for the details of carry definitions. I sort options into decile portfolios based on their carry. Table 9 Panel A reports the summary statistics as well as the results from multiple time-series regressions. I find that carry strategies perform extremely well in individual equity options. Both annualized Sharpe ratios and monthly average returns rise smoothly from decile one (9 basis point, 0.26) to decile ten (257 basis point, 2.18). The spread between the excess return on decile ten and one portfolio has an enormous Sharpe ratio of 2.5, which is larger than the Sharpe ratio of carry strategies considered in Koijen, Moskowitz, Pedersen, and Vrugt (2012) in any asset class. Table 8 compares the M AE s from seven different models. Mean absolute average return or MAE of BS is 66 basis points. CAPM and FF4 can reduce the pricing errors to around 50 basis points. I find that the option three-factor model performs really well in explaining the returns of these portfolios. MAE is less than 10 basis points. Except for the decile 8 portfolio, all the t-statistics of alpha s are insignificant. P-value of M AE and GRS statistics are 0.38 and 0.04. Cross-sectional variation of expected returns on carry portfolios are captured by both the level and the slope coefficients. The level coefficients are low for low carry portfolios just like long maturity in-the-money options. Coefficients of slope factor are negative for low carry options and positive for high carry option portfolios. This is not surprising since carry varies systematically by moneyness and maturity characteristics. The option three-factor model also explains the realized return variation, average R 2 is close to 90%. Therefore we can interpret the results on carry portfolios as an application of Arbitrage Pricing Theory of Ross (1976). B. Variance Risk Premia (VRP) VRP is defined as the difference between the expected future stock return variation in the riskneutral measure and physical measure. For expected return variation in risk-neutral measure, I estimate model-free implied volatility as in Han and Zhou (2012) using a cross-section of option prices. For the expected stock return variation in the physical measure, I estimate ex-post return variation in the past month using high frequency data. There is a large literature on VRP. Previously Bollerslev, Tauchen, and Zhou (2009) show that VRP on the S&P 500 index forecast future index returns. Han and Zhou (2012) show that a stock s expected return increases with its VRP. Schürhoff and Ziegler (2011) constructed synthetic individual equity variance swaps using a cross-section of individual equity option prices. 44

Then, they show that both systematic and idiosyncratic VRP are related to the average returns on variance swaps. In this section, I examine the relation between VRP and individual option returns, instead of synthetic variance swaps. In order to build VRP portfolios, I initially sort options into decile portfolios by size (shares outstanding time stock price), then within each size group I form decile portfolios formed on VRP. As a result, I have 100 portfolios formed by sequentially sorting options first by size then by VRP. This way, I have ten separate decile one VRP portfolio across ten different size groups. Then I aggregate decile one VRP portfolios into one portfolio by estimating option market capital weighted average. I do same thing from decile two to ten. Let Ri,j,t e be excess return of portfolio that belongs to size group i and VRP group j at time t. Note that there is one VRP group j within each size group. Let v i,j,t be option market capital of portfolio that belongs to size group i and VRP group j at time t. I define option market capital of a portfolio as the sum of open interest times mid-price of all options that make up the portfolio. R e j,t = 10 i=1 v i,j,tr e i,j,t 10 i=1 v i,j,t I call Rj,t e as the excess return on decile j VRP portfolio at time t. Table 8 Panel B reports average returns across decile portfolios. Average return on decile 10-1 is 49 basis points (5.3 t- statistics and 1.29 Sharpe ratio). The option three-factor explains the excess returns on these portfolios. GRS statistics do not reject the model (p-value 0.18). Generally, the coefficient of value factor captures the average return differential between low VRP and high VRP portfolios. A fair question is why do I bother with controlling for size, when I form VRP portfolios? The VRP characteristic tends to be extreme in small companies, which have high expected option returns. If I don t control for size, decile 1 and 10 portfolios will consists of options on small companies. As a result, we observe a U-shaped expected return. I don t report results for those portfolios, but they are available upon request. C. Volatility Reversals Reversal effect is the relation between the expected return of a security and its recent performance. Options do not have a long maturity and they change character very quickly; therefore, it is not good idea to define reversal in terms of an option s past return. The primary determinant of a expected option return is moneyness and maturity; hence I decided to estimate reversal of an option using Black-Scholes implied volatility for a given moneyness ( ) and maturity. Suppose we have an option on stock i with delta, time-to-maturity T and Black-Scholes implied volatility σ i,t (,T)attimet. Idefinereversalforthisoptionasσ i,t (,T) σ i,t 1 (,T)themonthlychange in the implied volatility for a given moneyness and maturity. The implicit assumption here is that 45

the name of asset does not matter, characteristics matter. We can describe an option by its moneyness, maturity and underlying stock rather than optionid. Note that I interpret Black- Scholes implied volatility as a measure of price, and therefore volatility reversals is basically price appreciation. In order to build volatility reversals portfolios, I initially sort options into thirty portfolios by five moneyness and six maturity groups, then within each moneyness-maturity group, I build decile portfolios formed on volatility reversals. As a result, I have 300 portfolios formed by sequentially sorting options first by moneyness-maturity then by volatility reversals. This way, I have thirty separate decile one volatility reversals portfolio across thirty different moneyness-maturity groups. Then I aggregate decile one volatility reversals portfolios into one portfolio by estimating option market capital weighted average. I do same thing from decile two to ten. Let Ri,j,k,t e be excess return of portfolio that belongs to moneyness group i, maturity group j and volatility reversals group k at time t. Note that there is one volatility reversals group k within each moneyness-maturity group. Let v i,j,k,t be option market capital of portfolio that belongs to the same moneyness, maturity and volatility reversals group. I define option market capital of a portfolio as the sum of open interest times mid-price of all options that make up the portfolio. R e k,t = 5 i=1 5 i=1 6 j=1 v i,j,k,tr e i,j,k,t 6 j=1 v i,j,k,t I call Rk,t e as the excess return on decile k volatility reversals portfolio at time t. According to Table 9 Panel C, both average returns and Sharpe ratios rise from low to high volatility reversals portfolios. The difference between high and low volatility portfolios have a considerable Sharpe ratio of 1.4. GRS statistics (joint test of all pricing errors equal to zero) fail to reject the option three-factor model. M AE of the model is about six basis points, which is considerable progress compared to mean absolute average return of 31 basis points. Generally, coefficients of the slope factor and the value factor capture the expected return variation between high-low portfolios. D. Systematic and Idiosyncratic Volatility Cao and Han (2012) define idiosyncratic volatility IVOL i,t as the volatility of residuals from the Fama-French three-factor model, which is estimated using daily data over the previous month. They define systematic volatility as the VOL 2 i,t IVOL2 i,t, where VOL i,t is the volatility of daily returns of stock i in month t. Cao and Han (2012) find that option returns are related to idiosyncratic volatility of the underlying asset. In their analysis, they use at-the-money short maturity (shortest maturity greater than one month) options. I extend their analysis to the all moneyness and maturity groups. Their final sample has about 400,000 observations, while I conduct this study with more than 11 million observations. 46

My results are consistent with Cao and Han (2012). Expected option returns on high idiosyncratic volatility portfolios are higher than low ones. This pattern can not be explained by usual risk adjustment models. I find that the option three-factor model successfully explains this pattern. GRS statistics do not reject the model with a p-value of 0.16. None of the t-statistics of alphas are significant. Alpha of high minus low volatility portfolio is just 3 basis point, while the average return of high minus low portfolio is 58 basis points. Generally, variation on the coefficient of the level factor captures the expected return variation. I did not find any robust pattern related to systematic volatility. E. Stock Short-Term Reversal Ang, Bali, and Cakici (2010) define short-term reversal as the stock return over the previous month. They find that the past stock return predicts future call implied volatilities on the same stock, but they did not find any pattern for put option implied volatilities. Their findings are based on standardized at-the-money options with thirty days to maturity. In order to investigate the economic significance of this finding, I build portfolios formed on past stock return using the actual option prices for all moneyness and maturity groups. I form separate portfolios for call and put options. My results are consistent with findings of the Ang, Bali, and Cakici (2010). I find an economically significant pattern for call options and no pattern for put options. In Table 9 Panel F, I report results for only call option portfolios. Average returns rise smoothly from 11 basis points to 88 basis points. Sharpe ratio of decile 10-1 portfolios is more than one. The option three-factor model does a decent job at pricing these portfolios. MAE is only 14 basis points and p-value of GRS statistics is 0.03. Except for decile 9 portfolios, t-statistics of all alphas are insignificant. F. Stock Size The firm size is a natural logarithm of the market value of an equity, which is estimated as the stock price times the number of shares outstanding. Although size premium attracts enormous amount of attention in the stock market, not much attention is paid to it in the empirical option pricing literature. Previously, Di Pietro and Vainberg (2006) document that size of the underlying stock is related to the returns of synthetic variance swaps. I find a considerable amount of size premium in the option market. Expected returns on options on small firms are substantially larger than large ones. Average returns rise from 24 basis points to 138 basis points. Small minus big portfolio has a Sharpe ratio of almost 2.3. GRS statistics decisively reject the option three-factor model, but the model still shows some success. Mean absolute average return is 54 basis points on size portfolios. The option three-factor model leaves M AE of only 17 basis points. Compared 47

to MAE of CAPM 44 and FF4 of 36 basis points, 17 basis points is considerable progress. The model fails to price the smallest two portfolios. G. Stock Illiquidity and Option Illiquidity I follow Amihud (2002) for the definition of stock illiquidity and Christoffersen, Goyenko, Jacobs, and Karoui (2011) for the definition of option illiquidity. Stock illiquidity is defined as the ratio of the absolute value of stock return in the past month to the total dollar volume on that stock R i,t V i,t, where R i,t is the month t return of stock i, and V i,t the total dollar volume of stock i in month t. Option illiquidity is defined as the ratio of the bid-ask spread to mid price bid ask (bid+ask)/2. My results on stock illiquidity are consistent with the findings of Christoffersen, Goyenko, Jacobs, and Karoui (2011). Options on illiquid stocks are more expensive, since the cost of replicating options on illiquid stocks are higher for market makers. Consequently, the expected return on selling options is considerably higher for options on illiquid stocks than liquid stocks. Results of Christoffersen, Goyenko, Jacobs, and Karoui (2011) is based on the options on stocks that are exclusively S&P 500 index constituents. They also limit their sample to firms that have option trading throughout the entire sample period, which reduces their sample to 341 firms. They consider options with delta range 0.125 0.875 and time-to-maturity range 20 to 180 days. I confirm their results with more than 7,500 stocks and all moneyness-maturity categories. Table 9 Panel H reports the results on decile portfolios formed on stock illiquidity. There is a rising pattern of average returns and Sharpe ratios between option portfolios on liquid stocks and illiquid stocks. Decile 10-1 (illiquid-liquid) has an average return of 103 basis points and Sharpe ratio of 1.64. Table 8 report the M AE s across alternative models. The option three-factor model successfully explains the returns on these portfolios, while the usual risk-adjustment models all fail. GRS statistics fail to reject the option three-factor model with p-value of 0.20. Moreover, MAE of the model is just 13 basis points, while mean( E ( R e i,t) )=50. My results on option illiquidity contradicts the results of Christoffersen, Goyenko, Jacobs, and Karoui (2011), which imply that buyers of options earn an illiquidity premium. My results imply an illiquidity premium for the sellers. In other words, Christoffersen, Goyenko, Jacobs, and Karoui (2011) find that illiquid options are cheaper and I find that illiquid options are more expensive. Theoretical models on zero net supply derivatives such as Bongaerts, De Jong, and Driessen(2011), tells that the sign of the illiquidity premium depends on the risk aversion of the buyers and sellers. My results are consistent with more risk averse buyers and Christoffersen, Goyenko, Jacobs, and Karoui (2011) s results are consistent with more risk averse sellers. Given that options are a type of insurance, I argue that buyers should be more risk averse. Average returns rise on selling options rise from 26 basis points to 80 basis points, as we go from liquid to illiquid option portfolios. Although GRS statistics rejects the option three-factor 48

model, the model does a decent job at explaining the expected returns MAE of the model is just 10 basis points, while mean( E ( R e i,t) ) = 48. H. Embedded Leverage Frazzini and Pedersen (2012) show that investors require lower returns on assets with higher embedded leverage. The definition of embedded leverage is the elasticity of the option price with respect to the stock price. In practice I use the delta from the Black-Scholes model. I denote embedded leverage with Ω. Ω = F F S S = S F F S = S F I build two sets of decile portfolios formed on embedded leverage. In the first set, I directly sort options into portfolios by their embedded leverage, but I use leverage-unadjusted returns of options. When I use leverage-adjusted returns, I can not find a clear pattern. In the second set of decile portfolios, I use standard leverage-adjusted returns but control for moneyness-maturity when building portfolios. My portfolio construction methodology is exactly same as the way I construct volatility reversals portfolios, so see the volatility reversals part for details. Table 9 Panel J reports the results for the first set of embedded leverage portfolios with leverageunadjusted returns. The results are consistent with the findings of Frazzini and Pedersen (2012). Both average returns and Sharpe ratios rise from low to high embedded leverage portfolios from 0.7% to 9.4% monthly average returns and 0.64 to 1.82 annualized Sharpe ratio. Mean( E ( R e i,t) ) or MAE according to Black-Scholes model is 240 basis points. CAPM and FF4 reduce MAE to only 196 and 192 basis points, while the option three-factor reduce it to 61 basis points. The explanatory power of the option three-factor model comes from the variation in the level and slope betas. The beta of the level rises smoothly from 1.5 to 8.5 from low to high embedded leverage portfolios. Similarly, the beta of the slope rises from -0.7 to 2.6. Table 9 Panel K reports the results for the second set of embedded leverage portfolios in which I control for moneyness-maturity when forming portfolios. This time, the expected return embedded leverage pattern is completely reversed. This result seems counter intuitive at first, but it has a reasonable explanation. When I control for moneyness and maturity, the only variation in embedded leverage (Ω = S ) will come from the variation of the ratio of stock price to option F price S/F. High Ω becomes high S/F, which means option price is low relative to stock price. As a result, bet on embedded leverage turning into a value style investment. High embedded leverage corresponds to cheap options. As expected, expected return on selling cheap options (high embedded leverage) is lower than selling expensive options (low embedded leverage). Note that here cheap means the option price is cheap relative to stock price. In the second set of portfolios, average returns decrease slowly from 113 basis points to 10 49

basis points. Although the option three-factor model explains about two thirds of the expected returns (mean( E ( R e i,t) ) = 32, MAE = 12), GRS test rejects the model. Specifically, the option three-factor model has difficulty at pricing low embedded leverage (expensive) option portfolios. I. 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. This characteristic is defined separately for both call and put options. Vasquez (2012) show that there is a negative relation between the slope of volatility term structure and expected return on selling options. Their analysis is based on at-the-money options with one month to maturity. I interpret the findings of Vasquez (2012) as another application of value-style investment. Most of the variation in slope of volatility term structure comes from the variation in the implied volatility of short term options, since the implied volatility of long term options is more stable. Low slope of volatility term structure means high implied volatility for short-term options. In other words price of short-term options is high relative to the price of long-term options. I build two sets of decile portfolios formed on the slope of volatility term structure. The first decile portfolios consists of short-term options with less than or equal to three months to maturity. The second decile consists of long-term options with greater than 3 months to maturity. I do this separation, because I observe completely opposite patterns related to expected returns between short-term and long-term decile portfolios. Actually opposite pattern is consistent with my interpretation of value style investment for these portfolios. In both set of decile portfolios, expected returns are higher for expensive options. My results on short-term option decile portfolios are consistent with the findings of Vasquez (2012). Average returns rise from 57 basis points to 187 basis points from low to high portfolios. High minus low portfolio has an Sharpe ratio of 1.84. Usual risk-adjustment models don t explain these patterns, but the option three-factor model does. GRS statistics fail to reject the option three-factor model with p-value of 0.86. MAE of the model is only 8 basis points. while the mean( E ( Ri,t) e ) is 64 basis points. The average return pattern is less strong and reversed in the second set of decile portfolios formed on slope of volatility term structure. Average returns decrease from 57 basis points to 20 basis points, but the pattern is not smooth. Yet, the high minus low portfolios have significant average return with t-statistics -2.78. The option three-factor model fail at pricing return on these option portfolios in every aspect. GRS test rejects the model. Moreover, MAE of the model is 22 basis points, which is even greater than mean( E ( Ri,t) e ) of 18 basis points. 50

J. Open Interest Gamma I hypothesize that market makers are averse to having too much total gamma on options on a given stock. Based on this motivation, 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. Note that options on the same underlying stock share the exact same open interest gamma. If my hypothesis is correct, options with high open interest gamma should be expensive, hence selling them should generate a higher expected return. In order to test this hypothesis, I sort options into decile portfolios by their open interest gamma. Table 9 Panel N reports the results, which confirm my hypothesis. Expected returns rise from 9 basis points to 95 basis points, Sharpe ratios rises from 0.18 to 1.35 from low the high open interest gamma portfolios. 10-1 decile portfolio excess return has a Sharpe ratio of 1.72 and t-statistic of 7.11. I test OPT3 on decile portfolios formed on open interest gamma. GRS test rejects OPT3 at 0.05 significance level, but not at 0.01 significance level. Overall, OPT3 does a decent job at pricing these portfolio returns. Only α of decile 8 portfolio has a significant t-statistics. M AE of the model is only 12 basis points. while the mean( E ( R e i,t ) ) is 32 basis points. Table 9: Characteristic Based Option Portfolio Summary Statistics and Asset Pricing Tests This table reports summary statistics, results of multiple time-series regressions and asset pricing tests. Summary statistics are on excess returns (delta-hedged, leverage-adjusted) of decile portfolios based on several characteristics. Mean, standard deviation, t-statistics and annualized Sharpe ratio of monthly (holding period) percentage excess returns are reported. Then I test the option three-factor model on decile portfolios. I report α 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 the null hypothesis of zero pricing errors to estimate t-statistics and p-values. β L, β S, β V are slope coefficients of the level, slope, value factors respectively. Ri,t e is excess return on portfolio i at time t. The sample covers 204 months from January 1996 to January 2013. Panel A reports the results for decile portfolios formed on carry. Panel B reports the results for decile portfolios formed on variance risk premia. I control for the size of underlying stock, when forming variance risk premia portfolios. Panel C reports the results for decile portfolios formed on volatility reversals. Note that I control for moneyness and maturity when forming volatility reversals portfolios. Panel D and E reports results for systematic and idiosyncratic volatility. Panel F report the results for decile portfolios formed on stock risk reversal. Note that I only keep call options when forming stock short-term reversal portfolios. Panel G, H and I report the results for decile portfolios formed on stock size, stock illiquidity and option illiquidity. Panel J and K report the results for decile portfolios formed on embedded leverage. In Panel J, excess returns are leverage-unadjusted returns. In Panel K, excess returns are standard leverage-adjusted returns, but I control for moneyness and maturity when forming portfolios. Panel L and M report the results for decile portfolios formed on slope of volatility term structure. In Panel L, I use options with less than or equal to three months when forming portfolios. In Panel K, I use long maturity options (greater than three months to maturity) when forming portfolios. Panel N report results for open interest gamma. See Section II for the characteristic definitions. R e i,t = α i +β L i Level t +β s i Slope t +β v i Value t +ε i,t i {1, 2,..., 10,10 1} 51

Panel A: Option Carry Portfolios Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) Carry Low High Mean 0.09 0.13 0.15 0.19 0.24 0.40 0.58 0.92 1.32 2.57 2.48 Std. Dev. 1.21 1.51 1.70 1.90 2.06 2.25 2.56 2.69 3.06 4.08 3.44 t(mean) 1.07 1.25 1.25 1.41 1.67 2.55 3.23 4.88 6.18 9.00 10.30 Sharpe R 0.26 0.30 0.30 0.34 0.41 0.62 0.78 1.18 1.50 2.18 2.50 α 0.09 0.03-0.02-0.04-0.04-0.06-0.06 0.24 0.18 0.21 0.11 t(α) 1.38 0.47-0.36-0.61-0.52-0.84-0.67 2.36 1.59 0.81 0.40 1.94 0.04 β L 0.50 0.68 0.80 0.90 0.98 1.12 1.29 1.36 1.66 2.20 1.70 β S -0.28-0.29-0.27-0.27-0.28-0.18-0.16-0.10 0.28 1.17 1.45 β V 0.01 0.04 0.04 0.05 0.08 0.11 0.19 0.16 0.15 0.38 0.37 R 2 0.81 0.91 0.93 0.91 0.92 0.93 0.93 0.91 0.91 0.75 0.58 Panel B: Variance Risk Premia Portfolios Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(gdrs) VRP Low High Mean 0.16 0.13 0.17 0.23 0.23 0.25 0.35 0.33 0.37 0.65 0.49 Std. Dev. 2.31 1.91 1.64 1.75 1.59 1.69 1.80 1.79 1.98 2.27 1.32 t(mean) 1.00 0.94 1.49 1.90 2.02 2.13 2.79 2.67 2.65 4.09 5.30 Sharpe R 0.24 0.23 0.36 0.46 0.49 0.52 0.68 0.65 0.64 0.99 1.29 α -0.04-0.16-0.11-0.03-0.01-0.02-0.01 0.15 0.24 0.17 0.21 t(α) -0.32-2.01-1.50-0.36-0.12-0.19-0.13 1.51 1.88 1.43 1.28 1.40 0.18 β L 1.06 0.92 0.79 0.79 0.71 0.75 0.81 0.77 0.79 1.07 0.01 β S -0.24-0.16-0.13-0.22-0.24-0.24-0.18-0.30-0.36-0.17 0.07 β V -0.09 0.00 0.03 0.09 0.12 0.14 0.16 0.10 0.08 0.15 0.24 R 2 0.82 0.88 0.87 0.83 0.84 0.82 0.80 0.80 0.73 0.84 0.05 Panel C: Volatility reversals Portfolios Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) Vol. Mom. Low High Mean 0.23 0.14 0.18 0.21 0.21 0.22 0.22 0.30 0.40 1.02 0.79 Std. Dev. 2.70 2.12 1.98 1.86 1.77 1.63 1.55 1.49 1.78 2.27 1.90 t(mean) 1.24 0.93 1.28 1.59 1.69 1.89 2.01 2.83 3.19 6.42 5.91 Sharpe R 0.30 0.23 0.31 0.39 0.41 0.46 0.49 0.69 0.77 1.56 1.43 α 0.16 0.01 0.06 0.06-0.01-0.05-0.11-0.00-0.02 0.09-0.07 t(α) 1.03 0.12 0.63 0.71-0.16-0.67-1.39-0.03-0.17 0.72-0.31 0.92 0.52 β L 1.11 0.95 0.88 0.83 0.82 0.76 0.73 0.69 0.84 1.10-0.01 β S -0.55-0.28-0.34-0.32-0.25-0.19-0.12-0.12-0.11 0.08 0.63 β V 0.04-0.08 0.00 0.04 0.06 0.09 0.10 0.09 0.13 0.39 0.35 R 2 0.79 0.83 0.86 0.86 0.88 0.86 0.83 0.82 0.82 0.80 0.17 52

Panel D: Systematic Volatility Portfolios Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) Sys.Vol. Low High Mean 0.30 0.32 0.33 0.26 0.23 0.17 0.27 0.30 0.23 0.55 0.24 Std. Dev. 1.14 1.23 1.31 1.36 1.53 1.49 1.65 1.76 2.16 2.45 2.13 t(mean) 3.80 3.68 3.58 2.76 2.19 1.59 2.38 2.44 1.53 3.19 1.64 Sharpe R 0.92 0.89 0.87 0.67 0.53 0.39 0.58 0.59 0.37 0.77 0.40 α 0.17 0.02 0.05 0.02 0.03-0.26-0.01-0.03-0.29 0.04-0.13 t(α) 1.83 0.24 0.47 0.22 0.26-2.31-0.09-0.20-1.79 0.21-0.54 1.64 0.11 β L 0.42 0.48 0.54 0.52 0.62 0.65 0.69 0.71 0.99 1.04 0.62 β S -0.16-0.10-0.06-0.08-0.09 0.05-0.07-0.01 0.19 0.11 0.28 β V 0.09 0.18 0.09 0.08-0.01 0.10 0.03 0.01-0.11-0.08-0.17 R 2 0.58 0.60 0.63 0.56 0.62 0.63 0.63 0.56 0.64 0.57 0.24 Panel E: Idiosyncratic Volatility Portfolios Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) Idio. Vol. Low High Mean 0.19 0.14 0.13 0.23 0.29 0.27 0.32 0.40 0.44 0.77 0.58 Std. Dev. 1.09 1.43 1.34 1.47 1.65 1.78 1.69 1.87 1.84 2.42 1.95 t(mean) 2.49 1.40 1.40 2.25 2.52 2.18 2.73 3.07 3.38 4.52 4.21 Sharpe R 0.60 0.34 0.34 0.54 0.61 0.53 0.66 0.75 0.82 1.10 1.02 α -0.08 0.24-0.06 0.04-0.09-0.12-0.14-0.08 0.17-0.07 0.01 t(α) -0.84 1.92-0.56 0.33-0.74-0.85-1.12-0.59 1.24-0.35 0.03 1.44 0.16 β L 0.44 0.45 0.51 0.56 0.74 0.75 0.80 0.84 0.79 1.08 0.64 β S -0.04-0.34-0.15-0.15 0.05-0.03 0.16 0.10-0.04 0.36 0.40 β V 0.11-0.01 0.09 0.07-0.01 0.07-0.05 0.01-0.07 0.03-0.08 R 2 0.59 0.53 0.59 0.59 0.67 0.62 0.67 0.64 0.64 0.56 0.27 Panel F: Stock Short-Term Reversal Portfolios (Only Call) Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) S. Rev. High Low Mean 0.11 0.16 0.16 0.21 0.26 0.31 0.28 0.38 0.47 0.88 0.78 Std. Dev. 1.76 1.51 1.28 1.50 1.39 1.52 1.55 1.74 1.93 2.69 2.38 t(mean) 0.87 1.50 1.84 1.97 2.63 2.89 2.59 3.12 3.45 4.69 4.65 Sharpe R 0.21 0.36 0.45 0.48 0.64 0.70 0.63 0.76 0.84 1.14 1.13 α -0.24-0.18-0.09 0.14-0.05-0.11-0.12 0.09 0.31 0.08 0.33 t(α) -1.52-1.46-0.88 1.13-0.43-0.95-1.07 0.73 2.06 0.38 1.18 2.03 0.03 β L 0.67 0.63 0.54 0.54 0.59 0.65 0.70 0.74 0.76 1.13 0.46 β S 0.09 0.01-0.02-0.21-0.02 0.03 0.02-0.14-0.19 0.15 0.06 β V -0.03 0.04 0.03-0.00 0.05 0.10 0.07 0.08-0.04 0.16 0.20 R 2 0.46 0.60 0.61 0.56 0.62 0.61 0.68 0.68 0.61 0.56 0.13 53

Panel G: Stock Size Portfolios Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) Size High Low Mean 0.24 0.22 0.23 0.30 0.38 0.36 0.60 0.66 1.06 1.38 1.14 Std. Dev. 1.64 1.88 1.98 1.93 2.00 2.04 2.36 2.33 2.06 2.48 1.75 t(mean) 2.07 1.69 1.67 2.25 2.71 2.50 3.61 4.03 7.37 7.95 9.34 Sharpe R 0.50 0.41 0.40 0.55 0.66 0.61 0.88 0.98 1.79 1.93 2.27 α 0.03-0.05-0.11 0.00 0.20-0.02 0.12-0.10 0.57 0.48 0.45 t(α) 0.50-0.62-1.24 0.02 2.03-0.18 0.72-0.63 4.32 2.58 2.15 3.49 0.00 β L 0.75 0.90 0.95 0.90 0.89 0.95 0.99 1.09 0.88 1.10 0.35 β S -0.27-0.14-0.19-0.22-0.33-0.17-0.14 0.05-0.21 0.12 0.39 β V 0.10-0.02 0.07 0.08 0.06 0.09 0.15 0.23 0.29 0.32 0.22 R 2 0.89 0.87 0.88 0.85 0.84 0.82 0.66 0.73 0.74 0.65 0.12 Panel H: Stock Illiquidity Portfolios Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) S.Illiq. Low High Mean 0.28 0.20 0.28 0.22 0.32 0.43 0.58 0.53 0.91 1.30 1.03 Std. Dev. 1.65 1.76 1.91 1.89 2.02 2.13 2.08 2.31 2.43 2.71 2.17 t(mean) 2.38 1.65 2.07 1.65 2.29 2.88 3.98 3.25 5.32 6.85 6.76 Sharpe R 0.58 0.40 0.50 0.40 0.55 0.70 0.96 0.79 1.29 1.66 1.64 α 0.09-0.04-0.12 0.06 0.01 0.02-0.01 0.21-0.05 0.67 0.58 t(α) 1.39-0.55-1.44 0.65 0.08 0.17-0.10 1.51-0.31 2.53 2.16 1.36 0.20 β L 0.78 0.84 0.91 0.83 0.92 0.98 0.94 0.97 1.11 0.89 0.11 β S -0.21-0.20-0.18-0.36-0.27-0.20-0.08-0.38 0.10-0.09 0.12 β V 0.00 0.04 0.14 0.10 0.14 0.15 0.25 0.22 0.39 0.33 0.32 R 2 0.89 0.89 0.88 0.86 0.84 0.82 0.74 0.77 0.70 0.40 0.04 Panel I: Option Illiquidity Portfolios Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) O.Illiq Low High Mean 0.26 0.24 0.26 0.34 0.41 0.49 0.62 0.66 0.78 0.80 0.54 Std. Dev. 1.44 1.76 1.89 2.03 2.22 2.36 2.54 2.74 3.09 3.65 2.72 t(mean) 2.55 1.92 1.95 2.40 2.67 2.94 3.48 3.44 3.63 3.11 2.84 Sharpe R 0.62 0.47 0.47 0.58 0.65 0.71 0.84 0.83 0.88 0.76 0.69 α 0.08 0.02-0.06-0.01 0.00 0.16 0.18 0.14-0.02-0.34-0.43 t(α) 1.28 0.36-1.05-0.12 0.01 1.94 1.54 1.16-0.14-1.49-1.71 3.29 0.00 β L 0.67 0.83 0.91 0.99 1.08 1.12 1.18 1.30 1.48 1.69 1.02 β S -0.17-0.28-0.27-0.27-0.23-0.33-0.30-0.28-0.15-0.04 0.13 β V 0.01 0.08 0.14 0.14 0.13 0.11 0.17 0.18 0.28 0.44 0.43 R 2 0.87 0.94 0.94 0.95 0.92 0.92 0.87 0.88 0.83 0.75 0.47 54

Panel J: Embedded Leverage Portfolios (Leverage-Unadjusted Returns) Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) E.Lev.(1) Low High Mean 0.66 0.63 0.63 0.75 0.88 1.31 1.92 2.84 5.06 9.37 8.72 Std. Dev. 3.55 4.18 5.18 6.19 7.20 8.60 9.85 11.58 13.71 17.81 15.50 t(mean) 2.63 2.16 1.73 1.72 1.75 2.18 2.78 3.50 5.27 7.51 8.03 Sharpe R 0.64 0.52 0.42 0.42 0.42 0.53 0.67 0.85 1.28 1.82 1.95 α 0.45 0.44 0.27 0.41 0.18-0.06-0.03-0.18 0.58 3.47 3.02 t(α) 2.26 1.99 1.07 1.39 0.59-0.15-0.06-0.28 0.75 2.69 2.33 2.36 0.01 β L 1.47 1.77 2.25 2.72 3.29 4.10 4.78 5.82 7.17 8.63 7.17 β S -0.77-0.88-1.02-1.16-1.14-0.76-0.48 0.53 1.81 2.60 3.36 β V 0.20 0.15 0.22 0.09 0.18 0.15 0.17-0.13-0.40-0.32-0.52 R 2 0.80 0.83 0.85 0.86 0.88 0.87 0.85 0.81 0.80 0.67 0.56 Panel K: Embedded Leverage Portfolios Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) E.Lev.(2) Low High Mean 1.13 0.48 0.38 0.30 0.26 0.20 0.19 0.19 0.16 0.10-1.03 Std. Dev. 3.20 2.37 2.11 1.94 1.80 1.70 1.54 1.43 1.28 1.14 2.49 t(mean) 5.05 2.89 2.59 2.23 2.09 1.70 1.77 1.86 1.80 1.21-5.94 Sharpe R 1.22 0.70 0.63 0.54 0.51 0.41 0.43 0.45 0.44 0.29-1.44 α 0.46 0.24 0.25-0.01-0.04 0.00-0.08 0.02-0.03-0.06-0.52 t(α) 2.47 2.41 3.10-0.15-0.50 0.02-1.31 0.21-0.40-0.87-2.47 3.30 0.00 β L 1.38 1.09 0.98 0.95 0.89 0.81 0.74 0.63 0.58 0.50-0.89 β S -0.43-0.36-0.31-0.15-0.11-0.18-0.15-0.22-0.15-0.15 0.27 β V 0.43 0.05-0.06 0.01-0.02-0.02 0.07 0.07 0.05 0.06-0.37 R 2 0.78 0.88 0.90 0.90 0.89 0.89 0.89 0.83 0.82 0.78 0.54 Panel L: Slope of Volatility Term Structure Portfolios (Short) Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) Slope(S) High Low Mean 0.57 0.41 0.37 0.43 0.48 0.43 0.48 0.62 0.75 1.85 1.28 Std. Dev. 1.56 1.21 1.40 1.30 1.37 1.48 1.65 1.74 2.01 2.77 2.40 t(mean) 5.22 4.87 3.82 4.76 5.00 4.18 4.16 5.12 5.34 9.53 7.58 Sharpe R 1.27 1.18 0.93 1.16 1.21 1.01 1.01 1.24 1.29 2.31 1.84 α 0.13 0.12 0.08 0.03 0.06-0.02-0.10-0.05 0.02-0.18-0.31 t(α) 1.12 1.45 0.74 0.42 0.68-0.24-0.98-0.40 0.18-0.83-1.17 0.53 0.86 β L 0.68 0.55 0.62 0.65 0.68 0.73 0.84 0.89 1.02 1.41 0.72 β S 0.03 0.03 0.06 0.15 0.12 0.14 0.19 0.32 0.29 1.16 1.14 β V 0.11 0.01-0.04-0.03 0.01 0.00 0.03-0.02 0.02 0.42 0.30 R 2 0.64 0.69 0.64 0.75 0.76 0.73 0.78 0.72 0.74 0.63 0.24 55

Panel M: Slope of Volatility Term Structure Portfolios (Long) Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) Slope(L) High Low Mean 0.57 0.37 0.20 0.16 0.14 0.04 0.02-0.06-0.03 0.20-0.37 Std. Dev. 2.32 1.79 1.80 1.79 1.82 2.12 2.13 2.29 2.55 3.01 1.90 t(mean) 3.54 2.95 1.55 1.29 1.07 0.29 0.10-0.40-0.16 0.96-2.78 Sharpe R 0.86 0.71 0.38 0.31 0.26 0.07 0.03-0.10-0.04 0.23-0.68 α 0.47 0.38 0.20 0.24 0.12 0.06-0.22-0.00-0.19-0.31-0.79 t(α) 3.51 3.78 2.03 2.64 1.44 0.57-2.11-0.03-1.53-1.58-3.46 3.94 0.00 β L 0.90 0.69 0.73 0.74 0.78 0.90 0.96 0.96 1.11 1.26 0.37 β S -0.61-0.52-0.44-0.42-0.39-0.49-0.33-0.48-0.48-0.38 0.24 β V 0.24 0.13 0.05-0.06-0.01-0.00 0.08-0.10 0.07 0.29 0.05 R 2 0.78 0.80 0.81 0.84 0.85 0.86 0.85 0.83 0.84 0.72 0.09 Panel N: Open Interest Gamma Deciles 1 2 3 4 5 6 7 8 9 10 10-1 GRS p(grs) OpenGamma Low High Mean 0.09 0.06 0.17 0.23 0.14 0.21 0.23 0.47 0.62 0.95 0.86 Std. Dev. 1.76 1.78 1.55 1.65 1.82 1.93 2.00 2.03 2.21 2.45 1.73 t(mean) 0.75 0.48 1.54 1.96 1.08 1.59 1.67 3.30 4.03 5.55 7.11 Sharpe R 0.18 0.12 0.37 0.48 0.26 0.39 0.41 0.80 0.98 1.35 1.72 α -0.22 0.04 0.07 0.00-0.15 0.09-0.00 0.30-0.00 0.28 0.50 t(α) -1.80 0.46 0.85 0.03-1.57 0.89-0.01 2.84-0.00 1.94 2.58 2.03 0.03 β L 0.77 0.76 0.67 0.77 0.85 0.85 0.90 0.88 1.04 1.08 0.32 β S -0.09-0.33-0.25-0.17-0.13-0.33-0.29-0.37-0.05-0.22-0.13 β V 0.04-0.06 0.00 0.02 0.01 0.02 0.08 0.09 0.21 0.40 0.36 R 2 0.70 0.82 0.81 0.86 0.83 0.84 0.84 0.83 0.78 0.77 0.20 VII. Conclusion I examine the discount-rate variation in individual equity options by studying the excess returns from selling option portfolios that are leverage-adjusted monthly and delta-hedged daily. I uncover a puzzling connection between option maturity, risk and expected returns. Different measures of risk return volatility, market beta, VIX beta, average returns during price-jump, volatility-jump episodes and market distress all indicate that long-maturity options are riskier relative to shortmaturity options, yet expected returns are lower for long maturity options. I identify three new return-based pricing factors level, slope, and value in option returns. Cross-sectional variation in expected returns on option portfolios formed on moneyness, maturity and value can be explained by comovements of their excess returns with these three systematic 56

factors. Sensitivities of option portfolios to the level, slope and value factors capture the expected return variation in the moneyness, maturity and value direction respectively. The premium on the level factor is compensation for market-wide volatility and jump shocks. Understanding the slope factor (or maturity premium) is a challenge, because market-wide volatility and jump risk indicates a negative rather than a positive premium. I argue that the gamma aversion of market makers and the leverage preference of investors are the main drivers of the premium on the slope factor. It is likewise difficult to explain the value factor, because it has almost no correlation with contemporaneous innovations in VIX and overall it tends to perform better during jump episodes. I argue that the premium on value factor is related to variance risk premia and demand pressure. Theories of risk averse financial intermediaries such as the demand-based option pricing model of Garleanu, Pedersen, and Poteshman (2009), help us to understand the slope and value premiums. Consistent with the friction-based interpretation, the premiums on the slope and value factors are higher when funding liquidity conditions are tight. I explore three new option investment strategies open interest gamma, option carry and volatility reversals which are successful at generating statistically significant returns with high Sharpe ratios. Previous researchers document that expected returns on options are related to variance risk premia, idiosyncratic volatility, stock short-term reversal, stock size, stock illiquidity, option illiquidity, embedded leverage and slope of volatility term structure. My three-factor model helps explain all of these patterns. 57

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A.I. Additional Empirical Results and Robustness Tests In this section, I consider a series of robustness checks to show the empirical success of the option three-factor model. First of all, I test the option three-factor model on the thirty moneynessmaturity portfolios and the decile value portfolios in 6 additional sub-samples. Table A.1 reports mean absolute average returns, MAE and GRS statistics with their p-values and average R 2 from multiple time-series regressions. The results are promising. Excluding the 1996-1999 sample, the option three-factor model actually performs better than original results both for moneynessmaturity and value portfolios. While the GRS test rejects the option three-factor model on moneyness-maturity portfolios in the full sample, it fails to reject in sub-samples excluding the 1996-1999. For example in the 2006-2012 sample p-value of GRS is 0.27, in the 2000-2012 sample p-value is more than 0.07. MAE test also do not reject the model in any sub-sample except the 1996-1999. In the full sample, GRS or MAE fail to reject the option three-factor model on value portfolios. Both of the test statistics fail to reject the model in all the sub-samples that I considered except the 1996-1999. We should not give much attention to the 1996-1999 period, because trading volume in the option market was very low. In market value trading volume was lower than 200 billion dollars. Nowadays trading volume is about 1.5 trillion dollars. Moreover option price data before 2002 is less reliable, because the data vendor Optionmetrics was launch in 2002, since then they have been daily collecting dealers end-of-day quotes directly from the U.S. Exchanges. They collect the data on the earlier period from the market-makers. Table A.1: Robustness Analysis: Asset Pricing Tests in Sub-Samples This table reports the results of asset pricing tests of option three-factor model using two groups of dependent portfolios and different sub-samples. Dependent portfolios are the decile value portfolios and the thirty moneyness-maturity portfolios. Sub-sample periods are 1996-1999, 2000-2005, 2006-2009, 2010-2012, 2006-2012. I also report the results of full sample 1996-2012 for comparison. Ave Rx is the average of absolute excess returns of dependent portfolios. M AE is mean absolute pricing errors (average α ). GRS is the F-statistics of Gibbons, Ross, and Shanken (1989) testing the null hypothesis of all α s are jointly equal to zero. p(grs) and p(mae) are the p-values of GRS and MAE statistics. I run OLS with 10,000 bootstrap simulations under the null hypothesis of zero pricing errors to estimate t-statistics and p-value. β L, β S, β V are slope coefficients of the level, slope, value factors respectively. Ri,t e is excess return on portfolio i at time t. The sample covers 204 months from January 1996 to January 2013. R e i,t = α i +β L i Level t +β s i Slope t +β v i Value t +ε i,t i {1, 2,..., 10} Dependent Portfolios Period Ave R e (bps) MAE (bps) p(mae) GRS p(grs) Ave R 2 30 moneyness-maturity 1996-1999 42 36 0.0035 2.32 0.3554 0.66 30 moneyness-maturity 2000-2005 54 19 0.4479 1.86 0.0722 0.63 30 moneyness-maturity 2006-2009 51 28 0.1687 0.92 0.8202 0.82 30 moneyness-maturity 2010-2012 84 27 0.1233 0.56 0.1154 0.81 30 moneyness-maturity 2006-2012 64 20 0.1663 1.19 0.2790 0.81 30 moneyness-maturity 2000-2012 59 17 0.1309 1.54 0.0418 0.74 30 moneyness-maturity 1996-2012 44 15 0.1037 1.95 0.0032 0.73 10 Value 1996-1999 41 31 0.0001 2.22 0.0309 0.78 10 Value 2000-2005 47 7 0.8703 0.63 0.6575 0.85 10 Value 2006-2009 30 19 0.1565 1.58 0.0960 0.94 10 Value 2010-2012 71 11 0.5170 0.51 0.7535 0.94 10 Value 2006-2012 47 13 0.1679 2.18 0.0151 0.94 10 Value 2000-2012 46 8 0.2877 0.88 0.4326 0.91 10 Value 1996-2012 38 10 0.0777 1.15 0.2428 0.90 1

As an additional robustness check, I report leverage-unadjusted returns on the thirty moneynessmaturity portfolios in the appendix. I find that the option three-factor model does decent job at explaining leverage-unadjusted returns as well. The model explains more than half of the average return variation. I also build the thirty moneyness-maturity portfolios just for call and put options. The model does well in explaining the return on those portfolios too. To save space, I report the results in the web appendix. Table A.2: Alternative Return Calculations This table reports summary statistics of unlevered returns on selling option portfolios across moneynessmaturity groups. At each month, Saturday following the 3rd Friday of the month is standard expiration date. Each month, at the first trading 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 option positions till the next expiration date. Panel A, B and C considers alternative return calculations. Panel A reports summary statistics of excess option returns that are delta-hedged daily, but unadjusted for leverage. Panel B and C report excess option returns that are unhedged and unadjusted for leverage. I calculate portfolio excess returns by taking option market capital weighted average return of each option in a given portfolio, where option market capital is open interest times mid price. I report means, standard deviations, t-statistics, annualized Sharpe ratios, skewness and kurtosis of portfolio excess returns. The sample covers 204 months from January 1996 to January 2013. Moneyness Maturity 1 2 3 4:6 7:12 > 12 1 2 3 4:6 7:12 >12 Excess Returns (Unadjusted for Leverage) Mean Standard Deviation DOTM 25.85 8.80 3.89 0.55-0.05-0.93 35.73 38.43 30.81 27.26 21.81 15.54 OTM 19.02 7.61 3.18 0.95 0.27 0.11 21.10 19.73 16.53 13.93 11.98 8.86 ATM 11.72 4.58 1.14 0.30-0.01 0.43 13.52 10.75 9.37 8.05 7.46 5.58 ITM 6.14 1.97 0.94 0.23 0.02 0.17 6.62 5.89 4.73 4.27 4.37 3.26 DITM 2.52 0.86 0.34 0.02-0.03 0.03 5.20 3.90 3.35 2.45 2.97 2.22 Excess Returns (Unadjusted for Leverage,Unhedged,Call) Mean Standard Deviation DOTM -14.23-6.45-11.48-4.79-5.04-1.11 98.97 67.51 63.70 46.09 47.04 30.20 OTM -9.43-2.19-5.21-2.14-2.31-1.77 80.43 56.71 47.06 36.34 31.19 23.53 ATM -8.67-2.02-3.67-3.05-2.18-1.49 64.99 46.92 38.92 31.38 26.28 18.90 ITM -3.88-1.32-2.23-2.64-2.08-1.57 49.36 36.62 30.08 25.61 21.99 16.28 DITM -3.59-2.15-2.55-1.29-1.14-1.39 37.60 29.10 23.09 21.41 17.64 13.77 Excess Returns (Unadjusted for Leverage,Unhedged,Put) Mean Standard Deviation DOTM 7.91 1.78 4.70 0.95 0.18 0.30 107.41 94.58 70.32 58.29 45.92 28.41 OTM 10.20 6.14 5.32 3.31 2.26 1.43 89.43 67.69 53.82 41.76 32.41 20.74 ATM 6.96 4.65 3.53 2.96 1.78 1.17 68.71 49.50 41.00 31.71 26.16 16.82 ITM 4.34 2.65 3.21 2.08 2.02 1.52 49.89 37.91 29.55 23.31 19.45 13.60 DITM 3.98 4.43 3.79 3.35 3.00 0.25 40.02 31.66 28.76 21.89 19.36 11.97 2

Table A.3: Asset Pricing Test of OPT3 (Leverage-Unadjusted Returns) This table reports the results of multiple regressions and asset pricing tests. I test the option three-factor model on excess returns (delta hedged, leverage unadjusted) of the thirty portfolios formed on five moneyness and six maturity groups. I report α 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 the null hypothesis of zero pricing errors to estimate t-statistics and p-values. β L, β S, β V are factor sensitivities of the level, slope, value factors respectively. The sample covers 204 months from January 1996 to January 2013. R e i,j,t = α i,j +β L i,j Level t +β s i,j Slope t +β v i,j Value t +ε i,j,t Moneyness i {DOTM, OTM, ATM, ITM, DITM} j {1M, 2M, 3M, 4 : 6M, 7 : 12M, > 12M} Maturity 1 2 3 4:6 7:12 >12 1 2 3 4:6 7:12 >12 α DOTM 10.37-2.67-2.16-1.29 0.57-1.42 3.45-0.88-0.94-0.79 0.41-1.64 OTM 10.83 2.42 0.37 0.53 0.80 0.43 7.32 1.80 0.40 0.82 1.29 0.93 ATM 4.72 1.40-0.35-0.05-0.43 0.72 5.01 2.08-0.65-0.13-1.23 2.36 ITM 2.93-0.02-0.02-0.03 0.18 0.33 5.15-0.03-0.07-0.12 0.64 1.62 DITM 0.63-0.44-0.25-0.16-0.13 0.12 1.08-1.12-0.79-0.73-0.45 0.54 t(α) β M t(β M ) DOTM 16.95 17.18 13.21 11.60 8.10 6.29 14.91 14.70 15.08 18.63 15.14 19.17 OTM 10.95 9.56 7.74 6.06 4.68 3.42 19.18 18.69 21.56 24.28 19.68 19.58 ATM 7.36 5.45 4.35 3.59 3.29 2.14 20.07 21.31 21.19 25.61 24.63 18.70 ITM 3.18 2.68 2.10 1.83 1.65 1.24 14.70 14.96 15.58 19.80 15.74 16.03 DITM 1.60 1.46 1.24 0.84 0.89 0.58 7.10 9.79 10.12 9.92 8.18 7.14 β T t(β T ) DOTM 10.12 3.74-0.92-4.18-5.97-3.73 4.36 1.57-0.52-3.33-5.48-5.54 OTM 5.90 2.19-0.79-2.85-3.53-2.73 5.13 2.09-1.09-5.68-7.24-7.64 ATM 6.05 1.48-0.54-1.45-1.42-1.65 8.28 2.85-1.29-5.13-5.17-6.96 ITM 2.46 0.69 0.09-0.73-1.11-0.81 5.54 1.93 0.31-3.90-5.13-5.10 DITM 1.52 0.83 0.19-0.25-0.47-0.49 3.34 2.75 0.77-1.45-2.10-2.93 β V t(β V ) DOTM -1.07 0.40 0.84 0.10 0.85 0.87-0.76 0.28 0.79 0.13 1.30 2.13 OTM -2.08-1.18-0.08 0.08 0.37 0.48-3.00-1.91-0.17 0.28 1.25 2.25 ATM -1.72-0.65-0.05-0.07 0.14 0.15-3.83-2.10-0.21-0.38 0.84 1.06 ITM -0.39 0.18-0.07 0.04 0.03-0.04-1.47 0.84-0.46 0.36 0.23-0.41 DITM -0.15-0.10-0.15 0.01 0.09 0.08-0.56-0.51-0.99 0.06 0.69 0.77 Adj R-square GRS (p-val) DOTM 0.55 0.59 0.65 0.77 0.74 0.80 7.65 (0.0000) OTM 0.69 0.70 0.79 0.86 0.83 0.83 ATM 0.68 0.75 0.79 0.87 0.86 0.81 ITM 0.53 0.60 0.65 0.79 0.74 0.75 DITM 0.19 0.34 0.42 0.48 0.41 0.39 3

Table A.4: Correlation Matrix This table reports 100 times correlation coefficients for several characteristic-based trading strategies. Strategies are 10-1 decile portfolio returns except the level, slope and value factors. The sample covers 204 months from January 1996 to January 2013. 100 Correlation Level Slope Value Carry VRP Vol.Mom. Sys.Vol. Idio.Vol. S.Rev. Size S.Illiq O.Illiq. E.Lev.(1) E.Lev.(2) Vol.Slope(S) Vol.Slope(L) O.Gamma Level Slope -62 Value -4 16 Carry 67-15 16 VRP -2 7 22-8 Vol.Mom. -21 34 27-3 15 Sys.Vol. 53-31 -12 66-32 -25 Idio.Vol. 57-30 1 61-14 -17 62 S.Rev. 33-18 9 33-1 17 20 25 Size 23 0 17 42-10 -2 30 25 22 S.Illiq 5 2 18 20 7-7 17 25 3 44 O.Illiq. 65-35 17 60-13 -29 46 37 33 44 21 E.Lev.(1) 73-33 -4 43-1 -22 29 24 20 19 10 69 E.Lev.(2) 71-48 14 73-4 -26 61 72 40 25 17 50 30 Vol.Slope(S) 28 12 20 58 5 14 35 47 26 18 16 16 2 48 Vol.Slope(L) -29 9-4 -41-4 -6-36 -40-31 -7-10 -17-8 -46-47 O.Gamma 37-24 22 47 2-4 28 47 37 44 33 40 20 60 32-28 Table A.5: Regression Results This table reports the results from multiple time-series regressions. Independent variables are the level, slope and value factors at time t. The dependent variables are Ted Spread (change in Ted spread from time t-1 to t), lag Ted spread (at time t-1), lag Ted Spread Vol (standard deviation of daily Ted spreads in the previous holding period from t-2 to t-1) and Ted Spread Vol ( change in the holding period standard deviation of Ted spread from t-1 to t). I bootstrap 10,000 samples under the null of zero predictability to calculate t-statistics. The sample covers 204 months from January 1996 to January 2013. Level (bps) Slope (bps) Value (bps) (1) (2) (3) (4) (5) (6) Ted Spread -2.12 0.68-0.51 t( Ted Spread) (-2.66) (1.92) (-1.49) Lag Ted Spread -1.82 0.64 0.53 t(lag Ted Spread) (-2.68) (2.10) (2.07) Ted Spread Vol -9.73 2.32-0.66 t( Ted Spread Vol) (-4.84) (2.60) (-0.50) Lag Ted Spread Vol -9.12 3.94 2.47 t(lag Ted Spread Vol) (-2.57) (2.77) (1.93) VIX 5.01 4.32-0.44-0.70 2.24 2.58 t(vix) (2.27) (1.88) (-0.42) (-0.68) (1.89) (2.24) Adj R 2 0.18 0.20 0.07 0.10 0.12 0.11 4