Earnings Announcements in Options Expiration Weeks

by

Huakun Ding

AN ESSAY SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

IN MATHEMATICS

DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF BRITISH COLUMBIA

March 2009

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TABLE OF CONTENTS

Abstract …………………………………………………………………………….....2

Chapter 1. Introduction ……………………………………………………….…….3

Chapter 2. Evolution of Implied Volatility Around Earnings Announcements ……..7

2.1 The Theoretical Framework ……………………………………….7

2.2 Empirical Tests ……………………………………………..…….11

Chapter 3. Option Portfolio Strategy ………………………………………...…….14

3.1 Implied Volatility Mispricing …………………………….………14

3.2 Option Portfolio Strategy ……………………………………...…16

3.3 Sample Selection …………………………………………………18

Chapter 4. Portfolio Returns …………………………..………………...…………21

Chapter 5. Conclusions …………………..…………………………………..……26

Acknowledgement ……………………………………………………...……………27

References ……………………………………………………………...……………28

Appendix ……………………………………………………………………….……30

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Abstract

We study earnings announcements occurring in options expiration weeks. We focus on the dynamics of option implied volatility around the earnings announcements dates. Based on the behavior of the implied volatility at these special times and the term structure of implied volatility, we construct an option portfolio consisting of three groups of near-the-money straddles with three different maturities. The option trading strategy can produce an economically and statistically significant return. The return of the portfolio is lower for earnings announced on options expiration day (Friday). This can be explained by three potential reasons: earnings announcements on Friday have lower immediate stock return response; earnings announcements on Friday have higher delayed implied volatility response; earnings announcements on options expiration days have lower stock return volatility.

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Chapter 1

Introduction

Since Ball and Brown (1968) the impact of information disclosures on stock price returns has been the focus of many studies. An issue which has received far less attention in previous research is the impact of firm-specific news on the uncertainty regarding the evolution of future stock prices. The uncertainty surrounding a firm’s fundamentals effect the stock price closely, driving expected returns, volatility, and valuation ratios. Quantifying this uncertainty is important for empirical applications, but is difficult as the uncertainty is unobserved. Prior papers proxy uncertainty by variables such as firm age, return volatility, firm size, analyst coverage, or the dispersion in analyst earnings forecasts. In this paper, we consider option prices, a natural source of information about uncertainty. Specifically, we consider implied volatility, the only forward-looking measure of market uncertainty that can be computed from options prices. We study the behavior of implied volatility around scheduled earnings announcements occur in options expiration weeks. Different methods are available to estimate a volatility process that incorporates large movements in the asset prices. The most well-known models describing changing volatilities over time are the autoregressive conditional heteroskedasticity (ARCH) models, originally introduced by Engle (1982) and extended to Generalized ARCH (GARCH) models by Bollerslev (1986) and Exponential GARCH (EGARCH) models by Nelson (1990). Although these models are useful in valuing derivative instruments, they are not very suitable to describe some one shot increases in volatility due to scheduled news announcements. Donders and Vorst (1996) show that GARCH models with an additional structural break in volatility might better describe these situations. Some other people suggest use a jump diffusions process to value options around earnings announcements (see e.g. Merton (1976), Piazzesi (2004) and Johannes (2004)). However, although earnings announcements dates are known in advance, the exact time of a jump in the stock price is unknown. This causes difficulty to this method. Merton (1973) shows that the implied volatility represents the average instantaneous volatility until the expiration of the option if the instantaneous volatility is a deterministic function of time. Heynen, Kemna and Vorst (1994) show that a similar interpretation holds if the instantaneous volatility moves stochastically through time or follows a GARCH process. Based on the work of Merton (1973), Patell and Wolfson (1979, 1981) provide early descriptive work both theoretically and empirically on the time series behavior of implied volatility around earnings

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announcements. They assume that instantaneous volatility is constant except on the disclosure date, when it rises because of the uncertainty linked to the content of the announcement. They get the assumption that implied volatility increases before earnings announcements, reaches to a maximum immediately before the earnings

Figure 1: Implied volatility for the nearest maturity at-the-money call option for Intel Corporation from January 1996 to December 2002 (Dubinsky and Johannes (2005)). disclosure and collapses to its long-term level on the disclosure date. They also assume that the rise and subsequent decline of the implied volatility of the longer dated option will be less extreme than the implied volatility of the shorter dated option. They investigate empirically the evolution of the implied volatility around earnings announcements on the US equity market. Their results are consistent with their assumption. Donders and Vorst (1996), Donders, Kouwenberg and Vorst (2000), and Isakov and Perignon (2001) investigate the same issue on different equity markets. The conclusions are similar to those obtained in Patell and Wolfson (1979, 1981) except that it takes the implied volatility from minutes to days to decrease to normal levels (e.g. see fig. 1). Donders and Vorst (1996) also show that only on the event day the index adjusted instantaneous volatility differs significantly from the instantaneous volatility during other periods. Thus they conclude that Patell and Wolfson’s assumption that instantaneous volatility is constant except on the earnings disclosure date holds. Whether or not implied volatility provides accurate forecasts of future realized volatility is another interesting topic. In practical option market, “volatility trading” is one of the most common options trading strategies, many option traders think the implied volatility is somehow mispriced and does not correctly forecast the subsequent realized volatility. Several papers have investigated this issue. Their

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results show that implied volatility is often mispriced, in particular, it is often overpriced. Jackwerth and Rubinstein (1996) and Ni (2006) find that at-the-money implied volatilities of index options are systematically and consistently higher than historical volatilities. Stein (1989) studies the term structure of the implied volatility of index options and finds that investors overreact to the current information. They ignore the long-run mean reversion in implied volatility and instead overweight the current short-term implied volatility in their estimates of long-term implied volatility. It is not easy to quantify implied volatility accurately in regular trading days when no significant information is released, and it is more difficult to estimate implied volatility correctly before earnings announcements. Implied volatility is often distorted before earnings announcements. For stocks that have a history of large earnings-associated price spikes, the market tends to overprice options by setting implied volatility too high. McMillan (2002) studies what the profitability would be if one bought straddles on the day before earnings were announced. He finds that the straddle buyer lost money statistically, and the market expectation about volatility tends to be higher than the earnings announcement might cause. Hence he suggests that buying straddles in advance of earning is not recommended. The behavior of the implied volatility around earnings announcements and the implied volatility mispricing create many opportunities for option traders. Our paper provides an option trading strategy based on the behavior of implied volatility around earnings announced in options expiration weeks and the term structure of implied volatility. In the option expiration week of each month A, one day before the earnings announcements, we short 2 units of potentially overpriced near-the-money straddles which expire in month A+1. To hedge our positions, we simultaneously long 1 unit of near-the-money straddle which expire very soon on Friday of the option expiration week and long 1 unit of near-the-money straddle which expire after month A+1. We liquidate our positions next day. We find that this trading strategy generates a weekly average return of 26%. We conduct several tests to understand the nature of these profits. In one test, we compare the portfolio returns for earnings announced on different days in the options expiration week. Our result indicates that the return is lower for earnings announcements on Friday compared with earnings announced on other weekdays. We find three potential explanations for our low return on Friday: 1. Weekends distract investors and lower the quality of decision-making, the immediate response to Friday earnings surprises is less pronounced (Bagnoli et al. (2005) and Vigna and Pollet (2007)). This causes the stock price spikes to earnings on Friday lower than the spikes to earnings on other weekdays; 2. Earnings announcements on Friday have much higher delayed stock return response. The delay of the stock return response seems effect the implied volatility response, and it causes the drop of the implied volatility for earnings on Friday slower and smaller; 3. Option trading alters the distribution of underlying stock prices and returns for earnings announced on options expiration days.

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In particular, for earnings announced on expiration days, stocks with options listed are more likely to experience returns that are small in absolute value and less likely to experience returns that are large in absolute value. The rest of the paper is organized as follows. The next section studies the evolution of implied volatility around earnings announcements. Section 3 presents the data and studies option portfolio strategy. Section 4 provides the results of the empirical analysis. We conclude in Section 5.

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Chapter 2

Evolution of Implied Volatility Around Earnings Announcements 2.1 The Theoretical Framework There are a number of papers that address issues related to earnings announcements and asset prices. We first briefly review some of the major contributions. Among the extensive literature related to earnings announcements, a significant portion of studies concentrate on the phenomenon called post-earnings-announcement drift (PEAD) which remains a puzzle for researchers. Early studies try to interpret PEAD in the context of efficient market theories but fail to fully account for the observed price drift. Recent research works (e.g. Bernard and Thomas (1989), Barberis, Shleifer and Vishny (1998)) have leaned toward behavior finance and attribute the cause of PEAD to investors’ under-reaction to earnings news. In this paper, we are primarily interested in the impact of earnings announcements on implied volatility and option prices. Many previous papers use time series data to analyze how scheduled announcements affect the level and volatility of asset prices. Ball and Brown (1968), Foster (1977), Morse (1981), Kim and Verrecchia (1991), Patell and Wolfson (1984), Penman (1984) and Ball and Kothari (1991) analyze the response of equity prices to earnings or dividend announcements, focusing on the speed and efficiency with which new information is incorporated into prices. Most of these studies are ex post analyses, in the sense that they demonstrate what security prices actually did on the date of an earnings disclosure. Patell and Wolfson (1979, 1981) focus on investors’ assessment of signal generating processes rather than on their reaction to particular signal realizations. They capture ex ante the anticipated information content of an earnings announcement by observing the behavior of option prices on dates leading up to and passing through the disclosure date. Options provide a particularly appropriate instrument for this type of research because of the relationship between their value as a contingent claim and investor beliefs about the future stochastic behavior of the underlying stock price over the remaining life of the option contract. The Black-Scholes (1973) option pricing model is one of the most widely used tools for valuing options. The Black-Scholes model defines the vale of an option as a function of five variables, four of which are directly observable. Specifically, the Black-Scholes model may be written as follows:

( , , , , )P P S X T r σ= ,

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Where P = option price, S = stock price, X = exercise price of the option T = time to expiration of the option r = continuous risk-free rate of interest σ = standard deviation of continuous returns on the stock per unit time. Using the Black-Scholes option pricing model, one can infer from option prices the expected stock price volatility (implied volatility) at dates surrounding the earnings announcement. In the model Black-Scholes assumed that the volatility of the underlying stock is constant over time. While the Black-Scholes valuation formula does not strictly hold if the volatility is stochastic, the results of Latane-Rendleman (1976) and Schmalensee-Trippi (1978) suggest that the model still performs well in the face of volatility changes. In particular, Merton (1973) showed that the Black-Scholes valuation formula is virtually unchanged if volatility is a deterministic function of time. The only difference is in the definition of 2σ . Whereas in the Black-Scholes formulation 2σ is the constant instantaneous volatility, 2σ can be somewhat more generally defined as the average volatility from the valuation date to the option expiration date,

2 1 2

0( ) ( )

TT T t dtσ σ−= ∫

where 2 ( )tσ is the instantaneous variance at time t .

Heynen, Kemna and Vorst (1994) show that a similar interpretation holds, under simplifying assumptions, when the instantaneous volatility moves stochastically through time, as in the model of Hull and White (1987), and also when the instantaneous volatility follows a GARCH process, as in the model of Duan (1995). The implied volatility is therefore a forward-looking measure of uncertainty. It provides the market's assessment of the average volatility that will affect stock prices until the expiration of the option. Based on the work of Merton, Patell and Wolfson (1979) provide early descriptive work on the time series behavior of implied volatility around earnings announcements. They assume that the instantaneous volatility is constant except on the disclosure date and then get the following hypothesis: In the case of scheduled earnings announcement, investors know that some information will be released on a precise date, prior to the maturity of the option, and expect a higher instantaneous volatility on that day as there is uncertainty with respect to the informational content of the announcement. The expected average volatility (i.e. implied volatility) to expiration rises to a maximum immediately before the disclosure due to the steady decrease in the time to expiration. Once the disclosure is made, assuming that no other event happens in this time period, one would expect that the implied volatility should drop

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to its normal level. Donders and Vorst (1996) rewrite the idea in a more formal way. They define implied volatility as the average volatility until the maturity of the option x :

2 21 1normal high

xIVx x

σ σ−= +

Where x is the number of days until the expiration date of the option, 2normalσ is the

volatility on a day when there is no news announcement and 2highσ is the volatility on a

day when scheduled earnings is released. With this set of assumptions the implied volatility, as a function of the time until and after the expected earnings announcement, can be depicted in Fig. 2. The bars represent the instantaneous volatility, which is constant except on the announcement date, and the line depicts the evolution of the implied volatility. According to this model, the implied volatility should increase gradually and reach a peak on the day before the announcement, as investors expect the instantaneous volatility to be higher on the earnings disclosure date. It should then drop to its long-term level on the announcement date, as the uncertainty linked to the content of the announcement is resolved and assuming that no more instantaneous volatility shocks are to be expected before the maturity of the option.

Figure 2: Implied volatility vs instantaneous volatility

denotes instantaneous volatility, denotes implied volatility. This figure is obtained assuming the following values: the maturity of the option is 20 days after the event date, instantaneous volatility=20% except on the announcement date when it is equal to 40%(Isakov and Perignon (2000)).

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Patell and Wolfson (1979) also compare the differences of the implied volatility around earnings announcements for options with different maturities. They get the following conclusions (see fig. 3): (1) The simultaneous implied volatility of the shorter dated option will exceed the simultaneous implied volatility of the longer dated option on dates preceding the earnings disclosure date. (2) The rise and subsequent decline of the implied volatility of the longer dated option will be less extreme than the implied volatility of the shorter dated option.

Figure 3: Implied volatility behavior around earning announcement for two options, one expiring at t = +2 and one expiring at t = +3 (Patell and Wolfson (1979)). Dubinsky and Johannes (2005) study the same issue using another method. They take seriously the timing of earnings announcements. They model equity prices as a process with randomly sized jumps at the time of the earnings release. To price the options in a no-arbitrage setting, they construct an equivalent martingale measure. They extend Black-Scholes model by incorporating deterministic jumps. In addition to the usual Brownian motion component, they assume there is a single jump at time

jτ , the time of an earnings release. The size of the jump, jZ , is log-normally

distributed with a volatility of jσ where is an equivalent martingale measure. If

there is an option maturing at time jT τ> , then the moment before an earnings

announcement, the Black-Scholes implied volatility is given by 2

2 2 ( )j

j

jTτ

σσ σ

τ− = +−

,

where σ is the diffusive volatility. After the announcement, the implied volatility

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falls to 2 2jτ

σ σ= . This implies that implied volatility drastically increases leading into

an announcement: it increases at rate 1( )T t −− as the maturity decrease. Second, it

implies that there is a discontinuous decrease in implied volatility immediately following the earnings release. Third, there are also term structure implications. Holding the number of jumps constant, implied volatility decreases as the maturity of the option increases. Thus longer dated options will have lower implied volatility than shorter dated options. These effects are consistent with Patell and Wolfson’s conclusions. 2.2 Empirical Tests Patell and Wolfson (1981) investigate empirically the evolution of the implied volatility around earnings announcement dates on a sample of 333 events (96 firms quoted on the Chicago Board Options Exchange over the period August 1976 - October 1977). The implied volatility is computed from call options prices by inverting the Black-Scholes formula. The results display a pattern which is relatively similar to their assumptions. They report a significant increase in the implied volatility on the 20 days before the announcement date and a significant drop two days after. Applying the Patell and Wolfson’s approach, Donders and Vorst (1996), Donders, Kouwenberg and Vorst (2000), Sahlstrom (2000), Isakov and Perignon (2000) got similar conclusions by investigating empirically in different equity markets. Donders and Vorst (1996) is of particular interest. Under the assumption that the volatility during non-announcement days can be described by a GARCH model, they use the following extended GARCH model to exhibit a structural break in volatility in the earnings announcement date:

211 1 1 1

1ln( )2

tt t t t

t

X rX

λσ σ σ ε++ + + += + − +

* *2 2 2 2 2

1 0 1 2 1 1 1[ ( ) ][1 ]t t t t u dt t t t

σ α α σ α σ ε κ χ β χ β+ + = + = += + + − + −

where tX is the underlying stock price, r is the riskless interest rate, λ is the market

price of risk, tσ is the volatility, 1tε + , conditional on the time t information, is a

standard normal random variable, *t is the event date and χ is an indicator

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function. 0α , 1α , 2α , uβ and dβ are constants. The parameters uβ and dβ

determine the magnitude of the increase and decrease in volatility around the event day. Donders and Vorst (1996) illustrate the properties of this extended GARCH model and compare it with the simple average volatility model used by Patell and Wolfson (1979). They find that although there are some differences, both models basically reveal the same pattern in implied volatilities around event days (see fig. 4).

Figure 4: (Donders and Vorst (1996)) Average volatility and GARCH volatility around announcement days (- average volatility; ...GARCH volatility). Donders and Vorst (1996) test the pattern in implied volatilities around earnings announcements on the Dutch market. They use a sample of 96 interim and annual earnings announcements of 23 firms over the period 1991-1992. Their results display a pattern which is similar to what they assume and is also similar to Patell and Wolfson’s results. In addition to the pattern in implied volatilities Donders and Vorst (1996) also test the assumption that the underlying asset has a significantly higher instantaneous volatility only on the scheduled news announcement day and not in the pre- and post-event periods. They compare cross-sectional instantaneous volatilities during the periods around earnings announcements to the instantaneous volatilities during normal periods. Specifically, they divide the earnings period into three sub-periods: the

pre-event period 1I , the event-day 2I and the post-event period 3I , then they use a

GMM regression with Newey-West standard errors to estimate excess instantaneous

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volatilities tEV on day t :

1 1 2 2 3 3t t t t tEV c x x xα α α ε= + + + +

Where 1tx , 2tx and 3tx are dummy-variables, 1α , 2α and 3α are t-values of the

coefficients indicating whether the volatilities in the three sub-periods are statistically different from those in the control period. Considering volatilities of individual stocks might not only result from firm specific circumstances, but can also be caused by general market trends, they apply regression to both raw stock instantaneous volatilities and the European Option Exchange (EOE) index adjusted instantaneous volatilities. Their results (see fig. 5) show that only on the event day the EOE-index adjusted instantaneous volatility differs significantly from the volatility during other periods. Thus they conclude that Patell and Wolfson’s assumption that instantaneous volatility is constant except on the earnings disclosure date holds.

Figure 5: Stock instantaneous volatilities around event days (Donders and Vorst (1996)) - raw instantaneous volatilities, 1 2 30.00015 0.00108 0.01256 0.00058t t t t tEV x x x ε= + + + + ;

… EOE-index adjusted instantaneous volatilities, 1 2 30.00012 0.00046 0.01312 0.00071t t t t tEV x x x ε= + + + + ).

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Chapter 3

Option Portfolio Strategy 3.1 Implied Volatility Mispricing Volatility is critical to the pricing of options as there is a one-to-one correspondence between the price of an option and the volatility of the underlying asset. However, characterizing and quantifying implied volatility is difficult as it is not directly observed. All option pricing models require at least an estimate of the parameters that characterize the probability distribution of future volatility. So implied volatility mispricing is the most obvious source of options mispricing. The following is an example of the implied volatility mispricing. From Figure 6, we can see that during the period of April 2007 – March 2008 the implied volatility of SPX options was substantially and consistently higher than the historical volatility under almost all kinds of market situations.

Figure 6: The implied volatility and historical volatility of S&P 500 index options in the period of April 2007 – March 2008. The line above denotes implied volatility, and the line below denotes historical volatility. (Source: www.interactivebrolers.com)

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Many researchers investigate implied volatility mispricing and find that implied volatility is often overpriced. Jackwerth and Rubinstein (1996) and Ni (2007) find that at-the-money implied volatilities of index options are systematically and consistently higher than historical volatilities. Stein (1989) studies the term structure of the implied volatility of index options and finds that investors overreact to the current information. They ignore the long-run mean reversion in implied volatility and instead overweight the current short-term implied volatility in their estimates of long-term implied volatility. Poteshman (2001) also finds evidence of overreaction in the index options market. Previous CBOE’s report shows that eighty percent of options expire worthless. Most options decay to the point of no value at expiration, and option writers are often the winners. Goyal and Saretto (2007) study why options are mispriced. They find two potential reasons: One is that investors do not use all the available information in forming expectations about future stock volatilities. In particular, they might ignore the information contained in the cross-sectional distribution of implied volatilities and consider assets individually when forecasting volatility; the other potential reason is that investors over-react to the current information. It is even not easy to quantify implied volatility accurately in regular trading days when no significant information is released, it is more difficult to estimate implied volatility correctly before earnings announcements or some other important events that contain much unobservable uncertainty and are expected to produce a large stock price response. The literature on the forecasting of implied volatility is extensive. A number of papers use accounting variable-based asset pricing models to study equity prices and option prices. The interested reader is referred to Ohlson(1995), Britten-Jones and Neuberger (2000), Pastor and Veronesi (2003) and Dubinsky and Johannes (2005). However, up to now no models have been found that can forecast implied volatility perfectly. Implied volatility mispricing still exist extensively. In particular, implied volatility is often greatly distorted before earnings announcements. Because of the mispricing of implied volatility, quarterly earnings create tremendous opportunities for option traders. For stocks that have a history of large earnings-associated price spikes, the market tends to overprice options by setting implied volatility too high. The distortion is the largest for at-the-money strikes and decreases as we move away from the stock price. Another distortion occurs just after earnings are announced and volatility collapses back to an appropriate level. In practice, based on this phenomenon, hedge funds often short inflated straddles before earnings announcements. They may get busted one time, but in general short expensive options before earnings is one way they make money. McMillan (2002) studies what the profitability would be if one bought straddles on the day before earnings were announced. He finds that the straddle buyer lost money statistically, and option traders were overly optimistic about the volatility than the earnings announcement might cause. Hence he suggests that buying straddles in advance of

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earning is not recommended.

In this study we explore trading opportunities that arise from earnings-driven implied volatility prices distortions. We will focus on two specific categories: trades that benefit from increasing volatility during the days that precede an earnings release; trades that benefit from post-earnings volatility collapse.

3.2 Option Portfolio Strategy

Option prices depends on five variables. A successful option trading strategy must rely on a signal about at least one of these variables. The most common options trading strategies often involve the underlying volatility, because volatility is the only one variable that is not directly observables, and the market expectation about future volatility is often somehow not correct. Recently, there is a growing literature analyzing trading in options. Coval and Shumway (2001) and Bakshi and Kapadia (2003) study trading in index options. Donders and Vorst (1996), Ni, Pan, and Poteshman (2006) and Goyal and Saretto (2007) study volatility trading in individual equity options. Donders and Vorst (1996) is of particular interest. They study the impact of earnings announcements on option implied volatility. They simulate a trading rule based on the observed feature of implied volatility that implied volatility increases before earnings announcements: ten days before the event they buy a call option and short the underlying stock to obtain a delta-neutral portfolio, each trading day the stock position is adjusted to changes in the hedge ratio of the option. After test they find the strategy does not yield significant return.

In this study, we develop an option strategy based on the observed feature of implied volatility that implied volatility rises before earnings announcements, reaches a peak one day before the announcement and falls dramatically on the disclosure date. Our strategy differs previous research in that it also makes use of the feature of option prices in options expiration weeks to decrease portfolio risk and increase potential profits. Specifically, for each month A, we consider firms which release earnings announcements in the 3rd week (options expiration week). In particular, we focus on earnings announcements that occur on Tuesday, Wednesday, Thursday and before the market open on Friday in the week. On the last trading day before the earnings announcements, we select firms whose implied volatility is higher than their historical volatility. For each firm, we long one unit of near-the-money straddles (S1) which expire in month A; and we short two units of near-the-money straddles (S2) which expire in month A+1; also, we long one unit of near-the-money straddles (S3) which

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expire in month A+x, where x=2, 3 or 4, the smaller the better, depending on which option is available. In brief, we construct the following positions: S1-2*S2+S3. Next day we liquidate all the positions, i.e. sell S1 and S2 and buy back 2*S2. Note that since the early 1990s, almost all of the earnings announcements are released after the close of the equity and options market, i.e., in the evening after 4pm EST or in the morning before 9:30am EST. So for earnings announcements released in the evening, we open our positions before the market is closed on the earnings announcements day, and we liquidate our positions next day; for earnings announcements released in the morning, we open our positions one day before the earnings announcements day, and we liquidate our positions on the earnings announcements day. We do not consider earnings announcements released in the evening of Friday, after market is closed.

We now discuss the theories behind our strategy. We study the short position on S2 first since it is one important source of the profits for our strategy. From section 2, we know that implied volatility rises before earnings announcements and reaches a peak one day before the announcements, in addition, a lot of implied volatility tends to be overpriced before earnings announcements. Based on these observations, we select firms whose implied volatility is much greater than their historical volatility, we assume these firms’ options are potentially overpriced and then short 2 units of their near-the-money straddles (S2). We expect that after earnings announcements, the implied volatility of S2 collapses dramatically, which will make our short position of S2 profitable. We now study our long positions on S1 and S3. The main role of our long positions is to hedge our short position. S3 will expire at least two months later, from Patell and Wolfson (1979), the rise and subsequent decline of the implied volatility of S3 will be less extreme than the implied volatility of S1 and S2. The change of the price of S3 is relatively smaller after earnings announcements. Thus S3 can serve to hedge our short positions. The role of S1 is a little different. On one hand, S1 also serves to hedge our short position. On the other hand, S1 can produce potential profits. S1 will expire in less than five days, its time value is very low, so a major change in stock prices can lead to huge profits on S1. In brief, we use the combination of the long positions on S1 and S3 to hedge our short positions on S2. We expect the risk of our portfolio is low regardless of the magnitude of stock price spike after earnings announcements: If the spike is in a preferable range, we expect our short position on S2 profitable, and our long position on S1 and S3 may be also profitable, this is the best case; if the spike is too small, we expect the profit from our short position on S2 can offset some loss from the long positions, mainly from S1; if the spike is too large, we expect the profit from our long positions on S1 and S3 can offset some loss from the short position on S2.

To give a better understanding of the strategy, we give the following specific examples. We select several companies who reported their earnings announcement on Oct 18, 2007 ( The third Thursday of October 2007, that is, one day before the options expiration day in that month). The results below show that our strategy can generate profits in different circumstance.

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Before Earning Announcement After Earning Announcement

Long 2*Short Long Sell 2*Buy Sell

Current

Month

Next

Month Far Month

Current

Month

Next

Month Far Month Earning

Date Stock Call Put Call Put Call Put Invest Call Put Call Put Call Put Result Profit

HON 1.3 0.6 2.43 1.6 3.3 2.33 -0.53 0 1.63 0.98 2.55 1.85 3.3 -0.28 0.25

MMR 0.65 0.13 1.2 0.63 1.8 1.18 0.1 0.1 0.6 0.58 1.05 1.18 1.55 0.17 0.07

ACI 1.28 0.38 2.18 1.2 3.4 2.15 0.45 0.7 0.1 1.88 1.15 3.2 2.23 0.17 -0.28

SLB 2.63 1.2 5.85 4 9.55 7.1 0.78 0 9.45 1.2 10.3 4.2 12.7 3.35 2.57

SXT 0.48 0.73 0.95 1.3 1.65 1.83 0.19 0.8 0.1 1.4 0.68 2.15 1.25 0.14 -0.05

18-Oct,2007 MMM 1.08 1.4 2.43 2.43 4.2 4 0.96 0 6.4 0.43 6.6 1.68 7.7 1.72 0.76

We compare our strategy with some previous methods and find it has the following advantages: First, its investment period is very short. We open our positions before the earnings announcements and close them next day. It takes only two days to complete the trading. Considering borrowing cost and margin requirements, this trading strategy’s high efficiency makes it quite attractive. Second, its risk is low and it is suitable for small investors as well as for big institutions. Many trading strategies in previous papers win statistically, based on trading on quite a lot of samples. However, for a given trading, the trading method is often very risky and is not suitable for small investors with only limited capital. For example, Goyal and Saretto (2007) develop a trading rule that longs straddles that are possibly undervalued, and shorts straddles that are possibly overvalued. They do not construct positions to hedge their portfolio. Their method works well statistically. However, for a given trading, their method, especially the shorting straddles method, can lead to huge loss in the case of a blow-up earnings announcement or a disaster like the one in 1987. This kind of strategy is not suitable for small investors with limited capital, because one big loss can wipe out their stock accounts. On the other hand, our positions are hedged for every trade we make. This efficiently limits our loss for each trade. So our method is also quite attractive to small investors.

3.3 Sample Selection

We investigate earnings announcements occur in options expiration weeks for the period from October 2007 to September 2008. We first collect the timing of the earnings announcements from the Wall Street Journal. The accuracy of earnings

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announcements dates is very important for such research related to earnings announcements. Vigna and Pollet (2007) showed that before 1995, a high number of earnings announcements were recorded with an error of at least one trading day; however, during the recent years, the accuracy of the earning date has increased substantially, and is almost perfect after December 1994. Doyle and Magilke (2007) tested the accuracy of earnings announcements dates of the Wall Street Journal from 2000 to 2005, they found that the accuracy rate was approximately 95.6%. In our study, we first directly use the announcement dates recorded in the Wall Street Journal, later we devise a rule to filter out some possible recording errors. We will discuss the details in the next section.

To analyze the effect of earnings announcements on options, we obtain options data from the Interactive Broker (IB) option database for the period from October 2007 to September 2008. The dataset contains information on the entire US equity option market and includes daily closing bid and ask quotes, volume, open interest and weighted average implied volatility. Stock options are traded at the American Stock Exchange, the Boston Options Exchange, the Chicago Board Options Exchange, the International Securities Exchange, the Pacific Stock Exchange, and the Philadelphia Stock Exchange. In our study, we consider near-the-money options: contracts which are at-the-money (ATM) or very near to ATM. Generally, option strike prices are spaced every $2.5 apart when the strike price is between $5 and $25, $5 apart when the strike price is between $25 and $200, and $10 apart when the strike price is over $200. It is not always possible to select option with the desired moneyness. Options with moneyness lower than 0.9 or higher than 1.1 are therefore eliminated from the sample. During our calculation, we use the average of the closing bid and ask quotes as options’ price. However, when we liquidate our positions, if the bid quote is 0 we treat this option worthless. We should note that the transaction costs, measured by the bid-ask spread, impact an options trading strategy greatly. Generally, the bid-ask spread is quite wide for options, it is much wider than the spread of stocks. "De Fontnouvelle, Fisher, and Harris (2003) and Mayhew (2002) document that the effective spreads for equity options are large in absolute terms but small relative to the quoted spreads. Typically the ratio of effective to quoted spread is less than 0.5" (Goyal and Sarette (2007)). Computing by the method of buying options at ask price and selling options at bid price, most option portfolio returns will decrease substantially. Goyal and Sarette (2007) test that when they use the mid-point price of the quoted spread, the returns of their two portfolios are 11.88 and 7.38, respectively; when using the effective spread equal to 100% of the quoted spread, their portfolio returns decrease to 3.12% and 0.77%, respectively. The bid-ask spread also impact our strategy greatly, because any unit of our position consists of six different legs of options. Especially, when we construct high volume of positions, it might not be possible to trade at the mid-point

20

price in every circumstance. Due to this reason, we expect that individual investors can get higher returns using our strategy than institutional investors do. Out of the universe sample, in each month A, we only consider stocks which have near-the-money options expire in three different months: in month A, in month A+1 and in a month between A+2 and A+4. To filter out some options not liquidate, we require that for each stock, the open interest of every near-the-money option in the three months is greater than 200 contracts. Also we eliminate from the sample all the observations for which both the bid and the ask are equal to the previous day prices. In addition, we exclude stocks priced below $10. After screening, the main sample includes 723 earnings announcements occur during the six option expiration weeks over the period October 2007 – September 2008.

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Chapter 4

Portfolio Returns

Table 1 reports the return on option portfolios. The consistent high weekly return shows that we successfully make use of the dynamics of implied volatility around both earnings announcements and options expiration weeks.

We now conduct several tests to understand the nature of these profits. For our portfolio we long S1 and S3 and short S2, thus the possible profits come from S2’s depreciation and S1 and S3’s appreciation. Around the earnings announcements that occur in options expiration weeks, the price of S1 and S2 often change dramatically, while the change of the price of S3 is relatively smaller. Since the prices of S1 and S2 are more critical for our strategy than the price of S3, we focus on S1 and S2 from now on. To find out the relationship between the prices of S1 and S2 and the portfolio returns, we sort our sample into deciles based on the difference between the value of price(S1)/price(S2) before earnings announcements. Decile ten consists of stocks with the highest ratio of price(S1)/mrice(S2) while decile one consists of stocks with the lowest ratio of price(S1)/price(S2). Here high ratio of P(S1)/P(S2) also means high ratio of ImpliedVolatility(S1)/ImpliedVolatility(S2). This perspective does not require one to take a stand on the correct option pricing model. The reader should view our portfolio sorts as sorts based on the relative over-(under-)pricing between S1 and S2.

Table 2 presents the returns for the ten deciles. From table 2, we can see that for most samples, the value of price(S1)/price(S2) is between 0.3 and 0.65, and for these samples the mean portfolio return is high. For a small percentage of the samples, the value of price(S1)/price(S2) is greater than 0.65, and the portfolio return is relatively low. This is to be expected. Since we long S1 and short S2, when the implied volatility of S1 is low and the implied volatility of S2 is high we get more profits. If price(S1)/price(S2) is to high (greater than 0.65 in this case), it is very possible that S1 is overpriced compared with S2. For the other samples, the value of price(S1)/price(S2) is smaller than 0.3, and the portfolio return is also low. This is somewhat unexpected for us at first view, because in this case S1 is relatively cheap, and this is just what we want. To find out what happened, we random selected some samples whose ratio of price(S1)/price(S2) is low than 0.3 and studied their option price changes around earnings announcements. We find that for most of theses samples, the changes of their stock prices are relatively small after earnings announcements, this makes S1 expired worthless or only worth a little, while the price of S2 did not change much. One explanation for this phenomenon is that the earnings dates are not accurate and thus should be eliminate. The other explanation is that the

22

current earnings announcements are not important enough for these firms. For example, earnings announcements are not important for some drug manufacturers which are in the early stage of developing new drugs and do not have some successful drugs on sale. For these companies, the critical factors for their stock prices are their trial results and FDA’s attitude to their trial other than their earnings announcements. In addition, if there will be some more important things (e.g. buyout, lawsuit, pending for FDA approvals…) happen after the expiration of S1 and before the expiration of S2. In these cases, S1 also seem underpriced with respect to S2, although in fact they may not be underpriced. These earnings announcements can be expected in advance not substantial enough to produce a large stock price response, they make our strategy does not work and therefore should be eliminated from our sample. If the sample of earnings announcements is reduced to eliminate those whose price(S1)/price(S2) is greater than 0.65 and smaller than 0.3, our portfolio return will increase substantially.

Next, we compare portfolio returns on different weekdays in table 3. From table 3 we can see that the portfolio returns for earnings announcements occur on Friday are substantially lower than earnings announcements occur on other days. This result is unexpected for us. For earnings announcements occur on Friday, S1 is bought on Thursday, it will expire one day later, and its time value is the lowest among all S1, the portfolio returns on Friday should not be lower, if not higher than the other days. There must be some special things that make the returns for earnings announcements on Friday low. Theoretically, there are two possible factors that can reduce the returns of our portfolio: 1. The depreciation of S1 is too big. This may happen if the earnings announcements do not produce a large enough stock price response; 2. The depreciation of S2 is too small. This may happen if the earnings announcements do not produce a large enough drop of the implied volatility of S2. To find out the differences between earnings announcements on Friday and on other days, we first compare the aggregate market conditions on different days. In particular, we compare different states of aggregate volatility (proxied by the implied volatility of the S&P 500 index, VIX) on different days, since an option’s implied volatility is closely related to aggregate volatility. Table 4 reports daily VIX in the 12 option expiration weeks. The average daily VIX from Monday to Friday are 25.04, 23.69, 24.05, 23.77 and 23.38 respectively. From table 4 we can see that in our sample the average daily VIX from Monday to Friday is very closed, and the VIX on Friday does not make much difference. So it seems the low return on Fridays is not caused by aggregate volatility. It also implies that the low return is not caused by the market performance on Fridays. Since the low return on Friday is not caused by general market trends, we now concentrate on potential reasons of firm specific circumstances. Friday is always a special day for earnings announcements. Prior research provides extensive evidence that managers strategically time the release of financial information. Using a seasonal random walk model of earnings expectations and/or the market’s reaction to earnings

23

announcements, Patell and Wolfson (1982), Penman (1987), Damodaran (1989) and others have shown that announcements made after the close of trading and on Friday tend to contain more bad news than announcements made at other times. Bagnoli et al. (2005) perform a more recent study and continue to find that the conventional wisdom that announcements made on Friday are more negative still holds. They find that the conventional wisdom is not only the result of managers’ desire to take advantage of limited media coverage but also the result of managers’ desire to take advantage of investors’ muted response to earnings announcements made on Friday. In particular, they use the following model to exam differential market responses to earnings news announced Monday, midweek and Friday:

0 1 2 3 4 5 6

7 8 9 10

11

( * ) ( * )

( * ) ( * ) ( * ) ( * * )

( * * )

iq iq iq iq iq iq iq iq iq

iq iq iq iq iq iq iq iq iq

iq iq iq iq

RET UE NegUE Mon Fri NegUE UE Mon UE

Fri UE NegUE Mon NegUE Fri NegUE Mon UE

NegUE Fri UE

β β β β β β β

β β β β

β ε

= + + + + + +

+ + + +

+ +

where iqRET is the announcement return for firm i in quarter q, adjusted for the

return on the S&P 500 index on the same day; iqUE is the earnings surprise for firm i

in quarter q, scaled by stock price at the end of quarter q; iqNegUE is a dummy

variable that equals one if the earnings surprise for firm i in quarter q is negative;

iqMon and iqFri are dummy variables that equal 1 if firm i announced on Monday or

Friday, respectively, in quarter q. Their regression results show that the negative surprise for bad news is significantly smaller when it is announced on Friday instead of midweek. At last, they find while investor respond strongly to bad new announced during the midweek period, investor reaction to Friday bad new is muted and it is at least partly anticipated by investors. Vigna and Pollet (2006) investigate the same issue and get similar conclusions. They find limited attention among investors indeed affect stock price, Friday earnings announcements have 20 percent lower immediate response and the abnormal volume is lower for Friday announcements than for non-Friday announcements. The results of Bagnoli et al. (2005) and Vigna and Pollet (2006) are particularly useful for our study, since they provide a reasonable explanation for our low portfolio return on Friday. From their theories, weekends distract investors and lower the quality of decision-making, the immediate response to Friday earnings surprises is less pronounced. For a negative earnings surprise on Friday, its stock price does not drop as much as the stock price of the same negative earnings surprise does on other weekdays. Similarly, for a positive earnings surprise on Friday, the spike of its stock prices is smaller than the spike of the same positive earnings surprise on other weekdays. Thus we can infer that the stock return volatility to earnings on Friday should be lower than the volatility to earnings on other weekdays. From our previous portfolio return analysis, a small stock price response to earnings announcements will

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decrease the profit of S1. If earnings announcements on Friday do not produce a large enough stock price response, the value of S1 will depreciate quickly. The worst case is that S1 expire worthless after the market is closed on Friday (in our study Friday is also options expiration day). So the lower immediate response to earnings announcements on Friday provides an explanation for our low portfolio return on Friday. Vigna and Pollet (2006) also find that earnings announcements on Friday have 60 percent higher delayed stock return response. They find that while investors are more likely to underreact initially to Friday announcements, eventually, investor become aware of the information they neglected and trade accordingly. From their observation, we can infer that, if the market is efficient, the decrease of the implied volatility for earnings announcements on Friday should be slower and smaller than the decrease on other days. Because both option writers and buyers think the stock movement on Friday does not fully reveal the information of the earnings announcements, and they anticipate larger post-announcement price movements for Friday earnings compared to earnings on other days. The delayed stock return response will also delay the implied volatility response. This will causes the drop of the implied volatility for earnings on Friday slower and smaller. In our portfolio, for earnings announcements on Friday, if the implied volatility of S2 drops less compared to earnings on other days, then the returns on Friday should be lower than the returns on other days. There is also a third explanation for the low portfolio return on Friday. In our study, we only consider earnings announced in options expiration weeks, thus the Fridays we discussed here are all options expiration days. From authors’ trading experience, some unusual things often happen on options expiration days. One of the unusual phenomenons is that on expiration dates the closing prices of stocks with listed options are often near option strike prices. Fig. 7 is an amazing example. From fig. 7 we can see that on almost every option expiration day the closed price of GOOG is closed to a number 5*n (n is an integer) which is an option strike price. Furthermore, we observe that, in company with this phenomenon, on each expiration day, the absolute value of the returns of stocks with options listed is much smaller. Ni, Pearson and Poteshman (2004) study this issue. Their reports confirm our observations. They find striking evidence that option trading alters the distribution of underlying stock prices and returns. In particular, they show that on expiration dates the closing prices of optionable stocks cluster at option strike prices. Also, they show that on expiration Fridays optionable stocks are more likely to experience returns that are small in absolute value and less likely to experience returns that are large in absolute value. Their reports indicate that hedge re-balancing by option market-makers and stock price manipulation by option writers contribute to the clustering. The discussion above naturally raises the following question: For earnings announced on options expiration days, whether the distribution of their stock prices and returns is also altered because of option trading. Theoretically, decreasing the volatility of stock

25

prices for earnings announced on options expiration day is beneficial to option writers. It is not difficult to image that for earnings announced on Friday, if the earnings results are not too much surprising option writers will intentionally manipulate the underlying stock price. They will manage to decrease the stock price response and delay the response to next Monday so that options finish at-the-money or just out-of-the-money and consequently are not exercised. If this assumption holds, we can expect that for earnings announced on options expiration days, the absolute value of the returns of optionable stocks is smaller and this can explain our low portfolio returns on Friday. As a test of our assumption, we use our previous data sample to compare the absolute value of optionable stock price returns for earnings announced on different weekdays. We find that the absolute value of optionable stock price returns to earnings announced on options expiration day is 1.62% lower compared to earnings announced on other days. This result is consistent with our assumption, and it is also consistent with the previous discussion that earnings announcements on Friday have lower immediate stock return response.

Figure 7: The closed stock prices for Google Corporation on option expiration days during the period November 2004 - March 2007.

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Chapter 5

Conclusions Implied volatility rises before earnings announcements, it reaches a maximum one day before the announcements and drop sharply after the disclosure. Based on implied volatility’s this behavior and its term structure we develop an option trading strategy. We test the strategy and find that it can produces an economically and statistically significant average weekly return. Our strategy also documents the existence of mispricing in the U.S. equity options. We conduct several tests to understand the nature of our strategy. We find the return of the portfolio is lower for earnings announced on options expiration day (Friday). This can be explained by three potential reasons: earnings announcements on Friday have lower immediate stock return response; earnings announcements on Friday have higher delayed implied volatility response; earnings announcements on options expiration day have smaller stock return volatility. The first and the third reasons can be supported by the result that the absolute value of optionable stock price returns is 1.62% lower for earnings announced on options expiration day. The second reason may cause the drop of the implied volatility for earnings on Friday slower and smaller.

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Acknowledgement

I would like to thank Professor Ekeland, my supervisor, for his useful guidance and suggestions. I am also grateful to the people that make my stay at UBC enjoyable. This paper is submitted in partial fulfillment of the requirements of the degree of Master of Science, the University of British Columbia, Vancouver, B.C.

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References Amin, K., 1993, Jump diffusion option valuation in discrete time, Journal of Finance 48, 1833-1863. Ball, Ray and Phillip Brown, 1968, An empirical evaluation of accounting income numbers. Journal of Accounting Research 6, 159-178. Bagnoli, M., M. Clement, and S. Watts, 2005. Around-the-Clock Media Coverage and the Timing of Earnings Announcements. Working paper, McCombs School of Business at the University of Texas at Austin (December). Bakshi, Gurdip, and Nikunj Kapadia, 2003, Delta-hedged gains and the negative market volatility risk premium, Review of Financial Studies 16, 527–566. Black F. and M. Scholes, 1973. The pricing of options and corporate liabilities, Journal of Political Economy 81, 637-654. Bollerslev, T., 1986, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, 307-327. Bollerslev T., R. Chou and K. Kroner, 1992. ARCH modeling in finance: A review of the theory and empirical evidence, Journal of Econometrics 52, 5-59. Chava, Sudheer, and Heather Tookes, 2006, Where (and how) does news impact trading, Working paper. Coval, Joshua D., and Tyler Shumway, 2001, Expected option returns, Journal of Finance 56, 983–1009. Damodaran, A., 1989. The Weekend Effect in Information Releases: A Study of Earnings and Dividend Announcements. The Review of Financial Studies, Vol. 2, No. 4, 607-623. De Fontnouvelle, Fisher, and Harris, 2003. The behavior of bid-ask spreads and volume in options markets during the competition for listings in 1999. Journal of Finance 58, 2437-2463 Donders M. and T. Vorst, 1996. The impact of firm specific news on implied volatilities, Journal of Banking and Finance 20, 1447-1461. Donders M., R. Kouwenberg and T. Vorst, 2000. Options and earnings

29

announcements: An empirical study of volatility, trading volume, open interest and liquidity, forthcoming in European Financial Management 6 (2). Doyle, J. and M. Magilke, 2007, The Timing of Earnings Announcements: An Examination of the Strategic-Disclosure Hypothesis, working paper. Duan J.-C., 1995. The GARCH option pricing model, Mathematical Finance 5, 13-32 Dubinsky A. and Johannes M. 2005, Earnings announcements and equity options, working paper. Engle, R., 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of U.K inflation, Econometrica 50, 987-1008. Foster, George, 1977, Quarterly accounting data: time series properties and predictive ability results, The Accounting Review, 1—21. Goyal A. and Saretto A. 2007, Option Returns and Volatility Mispricing, Working Paper. Heynen, R., A. Kemna and T. Vorst, 1994, Analysis of the term structure of implied volatilities, Journal of Financial and Quantitative Analysis 29, 31 56. Hull J., 2000. Options, futures, and other derivatives (Prentice Hall International, New York). Hull, J. and A. White, 1987, The pricing of options on assets with stochastic volatilities, Journal of Finance 42, 281-300. Isakov, D. and Perignon, Christophe, 2001, Evolution of market uncertainty around earnings announcements, Journal of Banking and Finance 25, 1769-1788. Jackwerth, J. and M. Rubinstein (1996). Recovering probability distributions from option prices. Journal of Finance 51, 1611–1631. Johannes, Michael, 2004, The statistical and economic role of jumps in interest rates, Journal of Finance 59, 227-260. Kim, Oliver and Robert Verrecchia, 1991, Market reaction to anticipated announcements. Journal of Financial Economics 30, 273-309. Latané H. and R. Rendleman, 1976. Standard deviations of stock price ratios implied in options prices, Journal of Finance 31, 369-381.

30

Mayhew, Stewart, 2002, Competition, market structure, and bid-ask spreads in stock option markets, Journal of Finance 57, 931-958 McMillan L.G, 2002, Options As a Strategic Investment, Prentice Hall Pr. Merton R., 1973. Theory of rational option pricing, Bell Journal of Economics and Management Science 4, 141-183 Morse, D., 1981, Price and trading volume reaction surrounding earnings announcements: A closer examination. Journal of Accounting Research 19, 374-383. Nelson. D., 1990, Conditional heteroskedasticity in asset returns: A new approach, Econometrica 59, 347-370. Ni, Sophie Xiaoyan, 2006, Stock option return: A puzzle, Working Paper. Ni S., Pearson N. and Poteshman A., 2004, Stock Price Clustering on Option Expiration Dates, Journal of Financial Economics, vol. 78, issue 1, pages 49-87 Ni S., Jun Pan, and Allen M. Poteshman, 2006, Volatility Information Trading in the Option Market, Working Paper. Patell J. and M. Wolfson, 1979. Anticipated information release reflected in call option prices, Journal of Accounting and Economics 1, 107-140. Patell J. and M. Wolfson, 1981. The ex ante and ex post price effects of quarterly earnings announcements reflected in option and stock prices, Journal of Accounting Research 19 (2), 434-458. Patell, James and Mark Wolfson, 1984, The intraday speed of adjustment of stock prices to earnings and dividend announcements, Journal of Financial Economics 13, 223-252. Penman, Stephen, 1984, Abnormal returns to investment strategies based on the timing of earnings reports, Journal of Accounting and Economics 1984, 165-84. Piazzesi, Monika, 2005, Bond yields and the Federal Reserve, Journal of Political Economy 113, 311-344. Poteshman, Allen M., 2001, Underreaction, overreaction, and increasing misreaction to information in the options market, Journal of Finance 56, 851–876. Stein, Jeremy, 1989, Overreactions in the options market, Journal of Finance 44, 1011–1023.

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Vigna S. and Pollet J., 2007, Investor Inattention, Firm Reaction, and Friday Earnings Announcements, working paper.

Appendix

Table 1: Summary Statistics

We selected 723 companies who reported earnings on Tuesday, Wednesday, Thursday or before the market open on Friday in the 12 options expiration weeks during the period Oct. 2007 - Sep. 2008.

Table 2: Portfolio Returns for Deciles Sorted on the value of Price(S1)/Price(S2)

High value of Price(S1)/Price(S2) means high value of ImpliedVolatility(S1)/ImpliedVolatility(S2). The value of Price(S1)/Price(S2) can be used to compare the relatively over-(under-)pricing between S1 and S2. We sort stocks independently into deciles based on the value of Price(S1)/Price(S2). Decile ten consists of stocks with the highest value of Price(S1)/Price(S2) while decile one consists of stocks with the lowest value of Price(S1)/Price(S2)

Decile 1 2 3 4 5 6 7 8 9 10

P(S1/S2) 0-0.2 0.2-0.3 0.3-0.35 0.35-0.4 0.4-0.45 0.45-0.55 0.55-0.6 0.6-0.65 0.65-0.75 0.75-1

% of stocks

2% 5% 7% 11% 15% 23% 14% 10% 9% 4%

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Table 3: Portfolio Returns on Different Weekdays

Earning Day Tue Wed Thu Fri

Return 22% 35% 39% 10%

Table 4: VIX -CBOE Volatility Index (S&P500) on different weekdays in options expiration weeks during Oct. 2007 – Sep. 2008

std 0.275 0.368 0.254 0.237 0.201 0.198 0.192 0.216 0.263 0.203

Mean

Return -20% 15% 30% 36% 41% 33% 31% 27% 13% -16%

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