Momentum Trading, Individual Stock Return. Distributions, and Option Implied Volatility Smiles

Transcription

1 Momentum Trading, Individual Stock Return Distributions, and Option Implied Volatility Smiles Abhishek Mistry This Draft: June 15, 2007 Abstract I investigate the sources of variation in observed individual stock return distributions and options implied volatility smiles. Highly volatile stocks tend to have fatter tailed distributions and steeper option smiles. Stocks with higher systematic risk tend to exhibit more smirkiness. In addition to these fundamental factors, I analyze the effect of momentum trading on return distributions and smiles. Using a measure of momentum trading constructed from mutual fund holdings, I find that stocks which experience a high level of momentum trading have approximately 1% higher implied volatilities for deep OTM options, even after controlling for the effect of ATM volatility. Lastly, I analyze the changes in return distribution over the course of a price run. During a price run, option implied volatilities adjust to place higher weight on future movements in the same direction as the run. Following a positive price run, ATM and OTM put option volatilities decrease while deep OTM call options become more expensive. My findings point to the development of a volatility model that incorporates short term price momentum. Stern School of Business, New York University, amistry@stern.nyu.edu. I thank my dissertation committee Robert Whitelaw, Robert Engle, and Stephen Figlewski for helpful discussions and comments. 1

2 The Black and Scholes (1973) model of option pricing serves as a benchmark for the equity options literature. Much research has focused on the failings of the model, in particular, the phenomenon known as the implied volatility smile. Researchers have examined this puzzle in several ways. A vast literature has attempted to model the data generating process in a continuous time jump-diffusion setting to extend the Black and Scholes framework to one that matches the data 1. The volatility literature has focused on time series models that match the stochastic nature of volatility which holds implications for options pricing 2. Papers such as Bakshi, Kapadia, and Madan (2003) and Dennis and Mayhew (2002) have characterized skewness in equity returns and option-implied distributions. Lastly, demand-based stories such as Bollen and Whaley (2004) and Gârleanu, Pedersen, and Poteshman (2007) have linked demand effects arising from crashophobia or other sources to option prices. Many of these papers have focused on index options and return distributions. In this paper I study distributions of individual stock returns and option implied volatility smiles in an attempt to infer economic determinants of the non-normality present in individual stock returns. I consider sources of non-normality described in previous literature at the index level and find many of the same effects manifest in individual stocks and options. Furthermore, I analyze a novel source of return distribution non-normality: the presence of momentum trading. Mistry (2007) argues that momentum traders can temporarily push prices away from long run equilibrium values in such a way that generates fat tailed returns. In his simulation model, prices move in response to a fundamental shock. Momentum traders react to this movement, causing buying or selling pressure. Ordinary fundamental traders will provide liquidity, but due to random arrival of agents, a sequence of momentum traders may enter the market consecutively and temporarily push prices away from long run equilbrium. The 1 Recent papers include Broadie, Chernov, and Johannes (2007), Eraker (2004), Pan(2002), and Duffie, Pan, and Singleton (2000) 2 See Bae, Kim, and Nelson (2007), Engle and Mistry (2007), Bekaert and Wu (2000), and Campbell and Hentschel (1992) 2

3 jump is later reversed as fundamental traders return to the market; however, short term stock return kurtosis is altered. An implication of this theory is that individual stocks that experience heavy momentum trading should exhibit increased tail fatness and increased value for OTM options relative to a Black-Scholes benchmark. I find evidence consistent with this concept. After constructing a measure of momentum trading using mutual fund data, I find that stocks which experience heavier momentum trading tend to exhibit fatter tailed returns and steeper (more curved) implied volatility smiles, even after controlling for the effect of volatility and other factors. More surprisingly, I analyze price runs to see how the effect of momentum trading affects the evolution of the distribution of returns over time. I find that following a sequence of positive returns, the distribution of returns becomes considerably more right-skewed. In fact, equivalently out-of-the-money put options decline in value after a positive price run, a finding that cannot be reconciled with common time series models of equity returns and asymmetric volatility. The goal of this paper is threefold. First, I characterize the implied volatility smile in individual stock options and link the smile to the same factors described in the index option literature. Using a broad universe of options, I find that volatility and level of systematic risk affect the shape of the implied volatility smile. Second, I attempt to identify stocks that experience heavy momentum trading activity and link this activity to excess kurtosis in returns and a premium on OTM options, yielding evidence that momentum trading causes fatter tailed return distributions. Lastly, if momentum trading causes price runs, then I attempt to uncover the evolution of implied volatilities during the course of a price run to see how investors subjective return distributions change during a period of sustained momentum trading. I find that investors assign more weight to positive return tails following a positive price run and negative tails following a negative price run. My findings point to a potential extension of time series models that incorporates short-run momentum into the 3

4 return generating process. This paper is organized as follows. Section 1 analyzes the time series and aggregate risk premium determinants of the implied volatility smile. In section 2, I link cross-sectional variation in return distributions and implied volatility smiles to a measure of momentum trading in individual stocks. Section 3 presents how investors subjective probability distributions evolve during the course of a price run and how the data refutes certain aspects of current asymmetric volatility models. Lastly, in section 4 I conclude with a summary of the main results. 1 Time series and aggregate effects I begin by considering the effect of asymmetric time-varying volatility. Consider the following GARCH model. r t = µ + ε t, ε t = h t η t, η t N(0, 1) (1) h t = ω + αε 2 t 1 + γε 2 t 1I t 1 + βh t 1 This system captures stochastic mean-reverting volatility where negative returns are associated with larger shocks to volatility than positive returns. It has been documented that such a model can produce negative skewness and excess kurtosis in returns 3. These higher moments should also manifest in an implied volatility smile for options on a stock with this type of volatility process. To examine the implied volatility smile produced by the asymmetric GARCH process, I simulate one million returns at one month horizons with parameters set to match a 5% yearly risk-free rate and volatility of 15%. From these returns, which are simulated under 3 See for example, insert citation 4

5 the risk-neutral measure by setting the drift equal to the risk-free rate, I then calculate call option prices for a range of strikes as C = E Q[ (S 0 e r T K) +] (2) Using the prices along with risk-free rate and other option information, I then calculate Black-Scholes implied volatilities. Figure 1 presents implied volatility smiles plotted against option moneyness, defined in standard deviation units as ln(k/s) rt σ. In Panel (a), implied T volatilities are plotted for options with 15%, 20%, and 40% return volatilities. All three smiles exhibit a smirk shape, where the left tail (OTM puts) is priced higher than the right tail (OTM calls). The smile is steeper, meaning a greater difference between ATM and OTM implied volatilities, for the more volatile stocks. [Figure 1 about here.] One can control for the effect of differences in the level volatility when analyzing across stocks by dividing the implied volatilities by at-the-money volatility. I will refer to these quantities as scaled implied volatilities. Figure 1b shows the same three smiles when scaled by the level of ATM volatility. The smiles look very similar, with the major difference being that more volatile stocks have a 10% higher left tail and roughly 5% lower right tail. In addition to the effect of time-varying volatility, I also consider several other factors. Engle and Mistry (2007) documented the relation between individual stock skewness and aggregate market skewness. In a CAPM world, they argue the two are related by ( ) 3 βi σ m SKEW i = SKEW m + σ i ( σεi σ i ) 3 SKEW εi (3) where SKEW i, SKEW m, and SKEW εi are individual stock, market, and idiosyncratic ( ) 3 skewness respectively. They define Ri 3 as the quantity, which can easily be measured 5 β i σ m σ i

6 by regressing returns on market returns and computing (R 2 ) 3/2 with the regression R 2. Since the market is negatively skew, the authors argue that stocks with higher R 3 will be more negatively skewed. Hence, in analyzing implied volatilities, higher R 3 should correspond to higher OTM put option volatility and lower OTM call option volatility. Other notable explanatory variables include past year s return and firm size. A stock s past year return was found to be significantly positively correlated with return skewness, with a Pearson correlation of Hence, we would expect that stocks which have performed well would have lower OTM put option volatility and higher OTM call option volatility. Firm size is related to its systematic risk. Larger firms have a greater component of systematic risk and should therefore by more negatively skewed, resulting in higher OTM put volatility and lower OTM call volatility. I next evaluate these predictions using a universe of option and underlying stock data. 1.1 Data I use underlying stock data from the Center for Research in Security Prices. For each stock and date, I calculate the logarithm of the current market capitalization, past one year historical volatility of daily returns, past one year return, and next year kurtosis of returns. In addition, following Engle and Mistry I use the past year of returns and CRSP value-weighted market returns to estimate the equation r it = α + β 1 r m,t 1 + β 2 r mt + β 3 r m,t+1 + ε it (4). The stock R 3 is then calculated as (R 2 ) 3/2, where R 2 is the regression R 2 from ( 4). I calculate implied volatility smiles using data from the Optionmetrics IvyDB database spanning using a method similar to Engle and Mistry. I discard options with zero open interest, zero volume, zero bid price, non- standard size, and options with maturity 6

7 greater than three months. I then calculate each option s moneyness, as before defined in standard deviation units by m = ln(k/s) rt σ. I discard options more than two standard T deviations out of the money. For each stock and date, I check to see that there is at least one option with negative moneyness and one with positive moneyness. After averaging call and put implied volatilities reported in Optionmetrics for equivalent option terms, I regress implied volatilities on moneyness for each stock and day to fit the following quadratic spline function. IV = α 0 + α 1 m + α 2 m 2 + α 3 m 2 1 m>0 (5) Lastly, using the fitted spline, I calculate option implied volatilities at five moneyness points: -2, -1, 0, 1, and 2. These are labeled IV 2, IV 1, IV 0, IV 1, and IV 2 respectively. I also calculate the OTM volatilities scaled by the ATM volatility, denoted ScaledIV 2, ScaledIV 1, ScaledIV 1, and ScaledIV 24. After merging the option and stock data, I discard observations with zero volatility, R 3 of one, and any implied volatility less than 10%. I remove outliers by capping kurtosis to lie between 0 and 10. Table 1 lists descriptive statistics for the data. There are X observations and Y firms in the sample. A smirk pattern similar to the GARCH simulation is present in the data. Average implied volatility for put options two standard deviations OTM was 14% higher than average ATM volatility, whereas average implied volatility for equivalently OTM call options was only 6% higher than ATM volatility. The average stock also exhibited tail fatness in stock returns. Average kurtosis was 6.15, although there was considerable cross-sectional variation. [Table 1 about here.] Figure 2 shows the average implied volatility smile plotted against moneyness, where the 4 For example, ScaledIV 2 = IV 2 IV 0 7

8 average was calculated across all stocks and dates in the sample. Again, the characteristic smirky smile is evident, both using implied volatilities as in Panel (a) or scaled volatilities as in Panel (b). Panel (c) plots average implied volatility smiles for three groups of historical volatility. Highly volatile stocks exhibit a slightly steeper smile than less volatile stocks, as predicted in the simulations. However, Panel (d) presents a discrepancy. More volatile stocks have a lower left tail relative to ATM volatility. This is counter to the prediction of the GARCH simulations. It seems that in the data higher volatility is associated with a higher left tail of the smile, but not quite as high as the GARCH model with market parameters would predict. [Figure 2 about here.] Having established the basic pattern of the smile, I next turn to a multivariate regression of hypothesized determinants of the smile. 1.2 Regression Results Table 2 presents several regressions of return kurtosis and smile components on predictor variables. In analyzing the non-scaled implied volatilities, I regressed the OTM volatilities on the common predictor variables along with ATM volatility. Thus, the coefficients on the predictor variables for columns 2, 3, 5, and 6 represent the effect of the variable on implied volatility beyond the effect on ATM volatility. For example, in the regression with dependent variable IV 2, the coefficient on R 3 represents the additional effect of R 3 on IV 2 beyond any effect it may have on ATM volatility. The positive value means that greater R 3 corresponds to a steepening of the smile. [Table 2 about here.] [Table 3 about here.] 8

9 The coefficients are estimated using ordinary least squares with Rogers (1993) clustered standard errors, where the clustering is done by firm. Petersen (2006) shows that in the presence of serial correlation within firm, clustered standard errors are able to approximate the error structure well. Year dummies were also included in all regressions. Results using the methodology of Fama and MacBeth (1973) are provided in Table 3. The estimated coefficients are very similar across both methods; however, the standard errors are much larger with the Fama-MacBeth procedure. Examining Table 2, volatility appears to increase kurtosis and implied volatilties as predicted. A 10% increase in historical volatility is associated with a increase in kurtosis of future returns and a increase in ATM volatility. Furthermore, higher volatility is associated with a steepening of the smile. The coefficients on historical volatility in the OTM implied volatility regressions are all positive and strongly significant. The same volatility prediction does not entirely appear to hold in analyzing scaled implied volatilties. In the simulations, higher volatility was associated with higher left tails and lower right tails of the scaled smile. The regression results show a lower right tail but also a lower left tail. Empirically, it appears that OTM put volatility is less sensitive to differences in the level of stock volatility than the asymmetric model would predict. This may be an artifact of the particular parameters chosen for the asymmetric term in the simulation. The parameters were chosen from estimation on market data; the coefficient of asymmetry may vary for individual stocks. The R 3 variable, on the other hand, matches the above predictions perfectly. It is significant across almost all regressions, and higher R 3 corresponds to an increased left tail and decreased right tail of the smile. It is interesting to note also that high R 3 is associated with decreased future kurtosis of returns. The evidence for other variables is more mixed. The past year return predicts right skewness with non-scaled implied volatilties; however, after scaling, a large positive past 9

10 year return is associated with a flattening of the smile, albeit more flattening on the left tail than on the right tail. According to the previous discussion, larger firm size should be associated with negative skewness. Indeed, one standard deviation OTM put volatility increases and OTM call volatility decreases with size. However, deep OTM put volatility also appears to increase with size. This puzzling effect is significant with either clustered standard errors or the Fama-MacBeth methodology. Now that I have examined the effect of volatility and other firm characteristics, I proceed to examine the impact of momentum trading on return distributions and implied volatility smiles. 2 Momentum Trading and the Distribution of Returns According to the simulation model of Mistry (2007), although momentum trading may not affect long run prices, heavy momentum trading creates overreaction to fundamental news shocks, which can generate temporary jumps in prices. These jumps create tail fatness, which manifests in kurtosis of returns as well as an option implied volatility smile. The goal of this section is to construct a measure of momentum trading in individual stocks and use this measure to identify a group of stocks with substantial momentum trading activity and examine whether such stocks have fatter tails and steeper implied volatility smiles than stocks which experience little or no momentum trading. 2.1 Mutual Fund-Based Measure Grinblatt, Titman, and Wermers (1995) studied mutual fund trading behavior and found that as many as 70% of mutual funds pursue positive-feedback trading strategies. Motivated by this finding, I propose a measure of momentum trading in individual stocks based on fund holdings data as reported in the Thomson Financial database. At the end of each 10

11 month a group of funds reports their positive holdings of all stocks 5. For each fund j and stock i, construct an indicator m i jt which is one if the fund executed a momentum trade for stock i in the quarter ending on month t. I classify fund trading activity in a stock as momentum trading if the fund increased its holdings of that stock over the previous quarter and the return over the same quarter was positive or if the fund decreased its holdings and the return was negative. Then let x i jt be the quantity of stock i held be fund j on in quarter t. For each stock on each month, aggregate across funds to calculate the proportion of reported trading volume that was classified as momentum trading. M i t = x i jt p i t x i j,t 1p i t 1 m i jt x i jt p i t x i j,t 1 pi t 1 (6) This quantity measures the percentage of trading by mutual funds that appears to be behavior akin to positive-feedback trading. Values of this measure close to one indicate a high degree of momentum trading by mutual funds in the stock. Values close to zero indicate contrarian activity. Values close to 0.5 indicate an equal degree of buying and selling by funds. Since stock returns and fund flows are positive on average, the mean of this measure should be greater than 0.5. Indeed, as Table 1 indicates, the mean of this measure is , with considerable cross-sectional variation. It is important to note that this measure is not only sensitive to explicit momentum trading strategies but will also detect implicit momentum strategies. For example, window dressing is often considered a common practice in the mutual fund industry as Lakonishok, Shleifer, Thaler, and Vishny (1991) document. Funds will buy winners and sell losers at the end of a quarter in order to report holding winner stocks. Although the goal of this behavior may not be explicitly to chase momentum, the net trading activity is the same and 5 Note that although fund holdings are reported quarterly, different funds report on different months, which makes it possible to construct a monthly time series using the holdings data. 11

12 hence should have the same implications on return distributions according to the simulation model. A similar effect is found if investors chase mutual fund performance. After funds report high performance, cash inflows to the fund increase 6. Supposing the fund simply scales the same strategy it is currently employing, mechanically it will purchase stocks that have performed well. Hence, investors chasing fund performance will cause funds to behave as if they are chasing momentum. The important distinction in both of these scenarios is that the fund s trading activity is indistinguishable from that of a positive-feedback trader. 2.2 Regression Results Using the Thomson Financial fund holdings database and return data from CRSP, I construct the momentum trading measure for each stock in each month from 1996 to I clean the fund holdings and then merge the end of month calculated momentum measure with all option and stock return characteristics within that month as calculated in the previous section 7. Next, I sort each stock and month into three groups based on the level of momentum trading activity and construct indicator variables for the high momentum measure group and the low momentum measure group. I expect that the high group, which contains stock with heavier momentum trading activity, will exhibit fatter tailed returns and steeper implied volatility smiles than the base group (the group with a medium level of the momentum trading measure). The low group contains stocks with potentially contrarian trading activity. I have no predictions on the return distributions of these stocks. Table 4 reports the panel regression results for regressing kurtosis and implied volatili- 6 See Chevalier and Ellison (1997) and Sirri and Tufano (1998) 7 The fundno identifiers in the holdings data is reused. To remedy this, I create a new fund identifier that classifies two funds as separate if they have different fundnos or investment objective codes, or if a fundno has a gap between reports greater than one year, in which case I identify the pre-gap fund as separate from the post-gap fund. I also remove observations where the report date is after the file date and where the report date is not at the end of the month. 12

13 ties on the momentum and contrarian group indicators along with the predictor variables discussed earlier. The coefficients on the high momentum indicator are all positive and significant; hence, heavy momentum trading is associated with fatter tailed returns and higher implied volatilities. [Table 4 about here.] [Table 5 about here.] The economic significance of the results is also strong. Kurtosis for heavy momentum stocks was an average of higher than for base stocks. By contrast, a 10% higher volatility corresponded to an increase in kurtosis of Momentum trading stocks had 2.6% higher ATM implied volatility. More interestingly, put options two standard deviations OTM on stocks with heavy momentum had nearly 2% higher implied volatilities, even after accounting for the effect of ATM volatility. In other words, the smile steepened on the left tail in addition to shifting upward. The same is true for OTM calls. Call options two standard deviations OTM on momentum stocks had 1.3% higher implied volatilities after accounting for ATM volatility. The effect remains if the implied volatilities are scaled by ATM volatility. On a scaled basis, two standard deviation OTM puts were 2.4% of ATM volatility more expensive for heavy momentum stocks. Equivalently OTM calls were 0.8% of ATM volatility more expensive. These results point to a strong cross-sectional effect where stocks which experience stronger momentum trading have fatter return distribution tails, higher implied volatilities, and steeper implied volatility smile curvature. Next I will consider the evolution of volatility smiles over the course of price runs to ascertain the impact of momentum trading on changes in the distribution of returns over time. 13

14 3 Price Runs and the Evolution of Return Distributions In the previous section I directly examined trading behavior to identify momentum trading. Next I look for momentum trading in the form of impact on prices. One possibility is that momentum trading activity can generate short term trends in prices, for instance a price run. Again, my hypothesis is that momentum trading affects the distribution of future returns. To see this effect, I study how option implied volatilities change over the course of a price run. As an example of a price run, consider Qualcomm (ticker: QCOM) stock during On November 3, Qualcomm announced earnings 3 cents per share greater than the expected 88 cents per share. Furthermore, the company announced the stock would split four-for-one in December. As Figure 3 shows, this sent investors scrambling to buy. From October 26 to November 12, the stock rose sharply from $ to $ More strikingly, by the end the stock had risen for 11 out of 13 days. [Figure 3 about here.] During the course of this run, option implied volatilites rose in a particular way. Panels (b) and (c) of Figure 3 present non-scaled and scaled implied volatilities for Qualcomm as calculated previously. ATM volatility rose from 60% to 68%. Volatilities for put options two standard deviations OTM rose more than proportionally, from 69% to 82%. Call option volatilities, however, skyrocketed. Implied volatilities for equivalently OTM call options rose from 71% to a high of 164% on November 10 and ended at 126%. Investors placed a large premium on high strike call options. 14

15 3.1 Run Sorts Motivated by the Qualcomm anecdote, I next consider the evolution of implied volatilities during a price run using my broad sample of firms. Here, the length of a run is defined as the number of consecutive days for which the price moved in the same direction. A single day movement is considered a one day run. Runs are nested so that a five day run includes a four, three, two, and one day run. My goal is to measure the change in implied volatility smile from the first day to each day of the run. In doing so, there are two theories as to how implied volatilities move as prices change absent any change in underlying return volatility: sticky moneyness and sticky strike 8. In the sticky moneyness framework, as prices move implied volatilities stay constant for options with the same moneyness. For example, suppose a put option one standard deviation OTM has an implied volatility of 60%. If the stock price rises by some amount, the put option that is now one standard deviation OTM (which is a different option) will have an implied volatility of 60%. The smile stays constant when viewed in moneyness space. An alternative theory for implied volatility changes is called sticky strike. In this view, options retain a constant implied volatility independent of price movements. The Black- Scholes model is correct when looking at each option separately. In this framework, ATM volatility may vary as prices fluctuate Sticky moneyness With sticky moneyness as a benchmark, I analyze changes in the implied volatility smile during a price run as follows. For each stock and date where the run length is greater than one, I use the implied volatilities at five moneyness points (-2, -1, 0, 1, and 2) calculated previously and compute the difference between implied volatility on the current day and the first day of the current run. I then average these differences for a given length of run. 8 See Derman (1999) for a discussion 15

16 The first five columns of Table 6 present the average change in implied volatilities since the first day of a run for each subsequent day. For example, the difference in ATM volatility between day one and day five of a negative price run with length at least five days was 2.83%. Thus, on average from day one to day five of a run, ATM implied volatilities rise 2.83%. [Table 6 about here.] The negative run portion of the table paints a picture potentially consistent with the asymmetric volatility model. Negative returns are associated with increased volatility and more negative return skewness. This is evident in higher volatilities across all strikes and in particular a larger increase in implied volatilities for OTM puts than for OTM calls. After a 10 day negative run, for example, put options two standard deviations OTM had a 12.9% higher implied volatility than on the first day, whereas equivalently OTM call options experience an increase of only 6.0%. Positive runs, however, directly contradict the asymmetric volatility model. ATM volatilities actually decline following a positive price run. After a 10 day positive run, ATM volatilities on average declined by 3.1%. Furthermore, both tails of the smile do not decline with ATM volatilities. The right tail actually increases; that is, deep OTM call options become more expensive following a positive price run. The asymmetry between the OTM puts and calls during a positive run is not supported by the asymmetric volatility framework. The next four columns of Table 6 presents an even stronger refutation. After scaling implied volatilities by ATM volatility, it is strikingly clear that following a negative run, OTM put option volatilties increase relative to ATM volatility and OTM call option volatilities decrease relative to ATM volatility. The opposite is also true for positive runs. OTM call volatilities increase and OTM put volatilities decrease relative to ATM volatilities. The result here implies that the investors subjective return distribution implied by option prices changes during the course of a price run. Following a positive run, the distribution 16

17 shifts to put more weight on a future positive price movement. Similarly, following a negative run, the distribution changes to weight negative future movements more greatly. In other words, investors appear to expect future price momentum after witnessing a string of returns in the same direction. This pattern could manifest through two channels. First, option market participants may set option prices to account for momentum in the underlying stock. During the course of a price run, the market puts a greater weight on underlying price movements in the same direction. Alternatively, momentum traders may trade options directly. Following a positive price run, momentum traders may increase demand for OTM call options, causing price pressure on these options. Similarly, following a negative price run, momentum traders may buy OTM put options, causing implied volatilities on these options to rise Sticky strike Of course, the previous results may simply be an artifact of the sticky moneyness assumption, since I compared implied volatilities on different options over the course of a price run. To account for this possibility, I recalculate implied volatility changes over the course of a price run, holding fixed the option terms from the first date of the run. For example, if IV 2 corresponds to an option with strike 120, regardless of where the stock price moves over a run, I calculate the difference in implied volatility with an option with strike 120 on each day of the run using the fitted spline for that day s smile. Holding option terms fixed from the first day of the run, Table 7 presents the changes in implied volatilities. The results here appear to contradict the previous table. Following a negative run, OTM put option implied volatilities decrease slightly whereas OTM call options become more expensive. Analogously, OTM put (call) options become more (less) expensive following a positive price run. 17

18 [Table 7 about here.] An explanation for these seemingly contradictory results follows. Suppose sticky moneyness is the true baseline specification. In other words, disregarding any momentum effects, options with the same moneyness tend to have the same implied volatilities from day to day. Fix a put option with a deep OTM strike. During a negative price run, the moneyness of this option will increase (ie become less negative). Due to sticky moneyness, the implied volatility of the option will decrease. Similarly, an OTM call option will become even further out-of-the-money, and hence its volatility will increase. The pattern evident in Table 7 can be explained simply with sticky moneyness. But what if sticky strike were the true baseline specification? Compare the implied volatilities of OTM put options with -2 moneyness on the first and last day of a negative price run. The option at the end of the run has a much lower strike price than on the first day and hence has a higher implied volatility. This is consistent with the results seen in Table 6. However, the OTM call option analogue is not consistent with the table. An OTM call option with 2 moneyness on the last day of a negative price run will have lower strike than the option with equivalent moneyness on the first day of the run. This option is closer to ATM on the first day and hence should have a lower implied volatility. In contrast, the volatility of this option is higher, as seen in the sticky moneyness table. The results here point to sticky moneyness as the more correct specification of implied volatility dynamics, at least during a price run. Indeed, Derman (1999) claims that sticky moneyness is the appropriate choice when markets are trending, whereas sticky strike is more appropriate when prices are rangebound Robustness to return size The sorts above do not distinguish the effect of the size of the price movement during a run from the run itself. For example, a large one day return may affect the volatility 18

19 smile similarly to the equivalently large five day run. To determine the effect of a run separately from the size of return, I estimate the following specification on the change in implied volatility from the first day of the run to each subsequent day, as calculated before. δiv it = α 0 + α 1 Cumret it + α 2 Cumret it Run it + ε it (7) Here, Cumret is the cumulative return during the run so far and Run is the run length in days, both calculated for each day of each run. I calculate this regression by OLS for the five moneyness points and the four scaled implied volatilities. For changes in implied volatility at moneyness points -2, -1, 1, and 2, I also include the change in ATM volatility as a regressor to capture the additional effect on the curvature of the smile. The results assuming sticky moneyness are given in Table 8. Again, clustered standard errors are reported, where the clustering is done by firm. [Table 8 about here.] A large magnitude return is associated with steeper smiles for both negative and positive runs, as seen by the positive and highly significant coefficients on Cumret for the implied volatility changes at -2, -1, 1, and 2 moneyness. These coefficients were positive even after scaling, albeit less significant. ATM volatility increases following a large negative return and decreases following a large positive return. The coefficient on Cumret Run represents the additional effect of each day of a price run on the change in implied volatility. For example, each day of a negative run is associated with increased volatility of 2.67% of Cumret on options with moneyness of - 2. The estimated coefficients on this interaction term point to OTM put options growing expensive and OTM call options growing relatively less expensive following a negative price run. The effect on OTM call options is robust to scaling by ATM volatility. The coefficients 19

20 on the interaction term are negative and significant for the right tail of the smile. The analogous effect appears for positive runs. The coefficient on the interaction term is negative and significant on the left tail of the smile. Following a positive run, OTM put options become cheaper relative to OTM call options. The results indicate a similar momentum effect as seen in the sorts. Following a run, investors subjective probability distributions appear to shift to place more weight on movements in the same direction as the run. These results can also be seen using the sticky strike baseline. Table 9 reports the regression results. Here, the Cumret variable controls for the mechanical relationship I documented previously where misspecification may cause the smile to shift as a result of price movements. Indeed, the coefficients and significance of the Cumret variable seem to indicate this pattern. Following a negative return, OTM call option volatilities increase substantially whereas OTM put option volatilities decrease. [Table 9 about here.] After controlling for this effect, however, the results portray the same relationship between run length and implied volatility as seen in the sticky moneyness regressions. Negative runs are associated with higher OTM put implied volatility and lower OTM call implied volatility. Positive runs are associated with lower OTM put implied volatility than OTM call implied volatility. 4 Conclusion I study the effect of various factors on return distributions and option implied volatility smiles. High volatility is associated with fatter tailed distributions and steeper smile. High systematic risk is associated with more expensive OTM put options and cheaper OTM call 20

21 options. I then construct a measure of momentum trading from mutual fund holdings and find that stocks that experience greater momentum trading have fatter tailed distributions and steeper implied volatility smiles. Lastly, I study the evolution of return distributions during a price run. During a negative run, ATM and OTM volatilities increase. However, during a positive run, ATM and OTM put option volatilities decrease while OTM call option volatilities increase, a find that contradicts the asymmetric GARCH model. Future research could explore ways in which bursts of momentum could be modeled into a GARCH or continous time options framework. Further analysis could also look at other idiosyncratic determinants of higher moments of return distributions and their impact on options prices. For example, measures of corporate governance, liquidity, or credit may have influence on return behavior. Additional research could study demand patterns to determine whether the momentum effect is due to price pressure in options or through the effect of momentum trading on the underlying. In other words, do momentum traders trade directly in the options market? This would have implications on pricing and hedging of options as well as market efficiency and segmentation. 21

22 References [1] J. Bae, C. Kim, and C. Nelson. Why are stock returns and volatility negatively correlated? Journal of Empirical Finance 14(1): 41-58, [2] G. Bakshi, N. Kapadia, and D. Madan. Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options. Review of Financial Studies 16(1): , [3] G. Bekaert and G. Wu. Asymmetric volatility and risk in equity markets. Review of Financial Studies 13(1): 1-42, [4] N. Bollen and R. Whaley. Does net buying pressure affect the shape of implied volatility functions? Journal of Finance 59(2): , [5] M. Broadie, M. Chernov, and M. Johannes. Model specification and risk premia: evidence from futures options. Journal of Finance 62(3): , [6] J. Campbell and L. Hentschel. No news is good news - an asymmetric model of changing volatility in stock returns. Journal of Financial Economics 31(3): , [7] J. Chevalier and G. Ellison. Risk taking by mutual funds as a response to incentives. Journal of Political Economy 105(6): , [8] P. Dennis and S. Mayhew. Risk-Neutral Skewness: Evidence from Stock Options. Journal of Financial and Quantitative Analysis 37(3): , [9] E. Derman. Regimes of volatility; some observations on the variation of S&P 500 implied volatilities. Goldman Sachs Quantitative Strategies Research Notes, Jan [10] J. Duan and J. Wei. Is systematic risk priced in options? Working paper, [11] D. Duffie, J. Pan, and K. Singleton. Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6): , [12] R. Engle and A. Mistry. Priced risk and asymmetric volatility in the cross-section of skewness. Working paper, [13] B. Eraker. Do equity prices and volatility jump? Reconciling evidence from spot and option prices. Journal of Finance 59(3): , [14] E. Fama and J. MacBeth. Risk, return, and equilibrium: empirical tests. Journal of Political Economy 81(3): , [15] N. Gârleanu, L. Pedersen, and A. Poteshman. Demand-based option pricing. Working paper,

23 [16] M. Grinblatt, S. Titman, and M. Wermers. Momentum investment strategies, porfolio performance, and herding: a study of mutual fund behavior. American Economic Review 85(5): , [17] J. Lakonishok, A. Shleifer, R. Thaler, and R. Vishny. Window dressing by pension fund managers. American Economic Review 81(2): , [18] A. Mistry. Momentum trading and large movements in asset prices. Working paper, [19] J. Pan. The jump-risk premia implicit in options: evidence from an integrated timeseries study. Journal of Financial Economics 63(1): 3-50, [20] M. Petersen. Estimating standard errors in finance panel data sets: comparing approaches. Working paper, [21] W. Rogers. Regression standard errors in clustered samples. Stata Technical Bulletin 13: 19-23, [22] E. Sirri and P. Tufano. Costly search and mutual fund flows. Journal of Finance 53(5): ,

24 % 20% 40% % 20% 40% (a) IV by Long-run Vol (b) Scaled IV by Long-run Vol Figure 1: Implied volatility smiles resulting from asymmetric GARCH simulations. One month returns were simulated one million times using equation (1) with parameters estimated from Fama-French market factor returns, adjusted to match a 5% annual growth rate and several different volatilities. From these returns, options prices and Black-Scholes implied volatilies are calculated. Panel (a) plots the resulting implied volatilities against long-run volatility. Panel (b) divides the implied volatilities by ATM volatility. 24

25 (a) Implied Volatility Smile (b) Scaled Implied Volatility Smile Low Medium High Low Medium High (c) Smile by Historical Volatility (d) Scaled Smile by Historical Volatility Figure 2: Implied volatility smiles of individual stock options. Panels (a) and (b) plot the average implied volatility and scaled IV of individual stocks against option moneyness. Panels (c) and (d) report average implied volatility and scaled IV sorted by three groups of volatility. 25

26 (a) QCOM Price (b) Implied Volatilities During Run (c) Scaled IV During Run Figure 3: The case of Qualcomm. Panel (a) plots Qualcomm prices during the period Oct. 26, 1999 to November 12, Panels (b) and (c) plot implied volatilities and scaled implied volatilities over the same period respectively. The heading IV(2) corresponds to the implied volatility for an option with moneyness equal to two. 26

27 Table 1: Descriptive statistics for implied volatilities, momentum trading variables, and other firm characteristics. The sample includes 1,171,903 firm-date observations spanning 4,459 firms in the Optionmetrics universe from Variable Mean Std Dev Min Median Max IV IV IV IV IV ScaledIV ScaledIV ScaledIV ScaledIV Fundmom Avgrunret Run Cumrunret Kurt Overnight Histvol Logsize Yearret R

28 Table 2: Baseline regressions with OLS and clustered standard errors. Each column heading correponds to the dependent variable in the regression. The data span unbalanced daily observations from There are 1,171,903 observations spanning 4,459 firms. ATM volatility is included as a regressor variable for the OTM volatility regressions to get the effect of various factors, controlling for any effect they may have on ATM volatility. R 3 is the measure of systematic risk, whereas the last row R 2 is the R 2 of the regression. Standard errors are computed with clustering by firm. Year dummies were included in all regressions. Kurt IV 2 IV 1 IV 0 IV 1 IV 2 ScaledIV 2 ScaledIV 1 ScaledIV 1 ScaledIV 2 Intercept t (21.90) (15.21) (0.34) (27.13) (35.81) (31.76) (48.33) (108.55) (115.60) (42.77) Atmvol (132.17) (410.48) (400.49) (109.21) Histvol (4.34) (23.21) (15.05) (73.90) (24.20) (27.26) (-22.38) (-14.85) (-25.48) (-26.18) Logsize (-8.54) (-10.55) (3.07) (-23.81) (-31.06) (-24.61) (-2.38) (9.61) (-16.68) (-7.14) Yearret (-6.35) (-16.81) (-13.80) (3.16) (3.10) (0.35) (-20.03) (-16.11) (-4.46) (-5.16) R (-14.94) (2.59) (6.77) (5.87) (-13.36) (-10.12) (0.52) (7.03) (-12.43) (-7.40) R

29 Table 3: Baseline Fama-MacBeth regressions. The same data is used as previously. Each regression is computed per year. The time series mean and t-statistic are then reported below. Year dummies were included in all regressions. Kurt IV 2 IV 1 IV 0 IV 1 IV 2 ScaledIV 2 ScaledIV 1 ScaledIV 1 ScaledIV 2 Intercept t (13.89) (6.00) (0.27) (9.67) (13.95) (11.38) (13.11) (28.45) (53.57) (12.72) Atmvol (23.13) (64.55) (78.00) (19.83) Histvol (4.51) (6.00) (4.44) (14.86) (7.54) (7.02) (-4.18) (-2.35) (-9.63) (-6.43) Logsize (-6.11) (-4.19) (0.69) (-8.88) (-14.91) (-10.13) (0.03) (3.67) (-5.30) (-1.41) Yearret (-1.54) (-3.58) (-2.51) (0.94) (0.25) (0.73) (-4.70) (-4.30) (-1.29) (-1.29) R (-5.89) (1.54) (2.64) (2.51) (-5.59) (-3.67) (-0.55) (2.88) (-8.51) (-6.42) 29

30 Table 4: Momentum trading cross sectional regressions. Two dummy variables are added to the baseline regressions. The f undmomhi dummy indicates stocks which are in the top third of stocks ranked by the fund-based momentum measure. The f undmomlo dummy indicates stocks which are in the bottom third of stocks ranked by the fund-based measure. All regressions are again computed with OLS and standard errors clustered by firm. Year dummies were included in all regressions. Kurt IV 2 IV 1 IV 0 IV 1 IV 2 ScaledIV 2 ScaledIV 1 ScaledIV 1 ScaledIV 2 Intercept t (21.73) (14.07) (-0.50) (26.41) (35.36) (31.04) (47.15) (106.92) (115.58) (42.60) Fundmomhi (2.68) (18.61) (15.90) (24.28) (10.70) (12.54) (9.77) (9.15) (2.36) (3.16) Fundmomlo (-0.06) (6.94) (6.71) (2.94) (-0.51) (0.78) (6.18) (6.68) (-0.03) (1.21) Atmvol (131.31) (410.52) (399.02) (108.59) Histvol (4.32) (23.54) (15.34) (74.00) (24.44) (27.55) (-22.59) (-15.10) (-25.41) (-26.14) Logsize (-8.44) (-9.72) (3.66) (-23.27) (-30.75) (-24.10) (-1.65) (10.20) (-16.56) (-6.95) Yearret (-6.27) (-16.27) (-13.39) (3.78) (3.45) (0.81) (-19.68) (-15.77) (-4.35) (-5.00) R (-14.96) (2.47) (6.63) (5.80) (-13.43) (-10.19) (0.42) (6.93) (-12.45) (-7.42) R

31 Table 5: Momentum trading Fama-MacBeth regressions. The same data is used as previously. Each regression is computed per year. The time series mean and t-statistic are then reported below. Year dummies were included in all regressions. Kurt IV 2 IV 1 IV 0 IV 1 IV 2 ScaledIV 2 ScaledIV 1 ScaledIV 1 ScaledIV 2 Intercept t (13.48) (5.59) (0.07) (9.26) (13.79) (10.96) (13.11) (28.78) (55.95) (13.06) Fundmomhi (1.57) (9.73) (8.45) (4.92) (7.94) (9.34) (4.93) (4.64) (1.31) (2.32) Fundmomlo (0.45) (5.88) (4.23) (2.02) (0.17) (0.52) (8.81) (5.59) (0.01) (0.71) Atmvol (23.42) (65.34) (77.58) (19.77) Histvol (4.46) (6.24) (4.61) (14.92) (7.66) (7.17) (-4.17) (-2.37) (-9.51) (-6.37) Logsize (-5.99) (-3.92) (0.82) (-8.58) (-14.78) (-9.85) (0.25) (3.91) (-5.31) (-1.38) Yearret (-1.49) (-3.45) (-2.36) (1.08) (0.35) (0.88) (-4.55) (-4.11) (-1.30) (-1.27) R (-5.92) (1.51) (2.62) (2.51) (-5.60) (-3.74) (-0.60) (2.83) (-8.53) (-6.44) 31