Clearly Irrational Financial Market Behavior: Evidence from the Early Exercise of Exchange Traded Stock Options

Transcription

1 Clearly Irraional Financial Marke Behavior: Evidence from he Early Exercise of Exchange Traded Sock Opions Allen M. Poeshman and Vialy Serbin Mark R. Manfredo Allen M. Poeshman Deparmen of Finance Universiy of Illinois a Urbana-Champaign and Vialy Serbin Deparmen of Finance Universiy of Illinois a Urbana-Champaign OFOR Paper Number Augus 2001

2 Clearly Irraional Financial Marke Behavior: Evidence from he Early Exercise of Exchange Traded Sock Opions Allen M. Poeshman Vialy Serbin * Augus 2001 * Assisan Professor and Ph.D. Candidae, respecively, Deparmen of Finance, College of Commerce and Business Adminisraion, Universiy of Illinois a Urbana-Champaign. Please send correspondence o: Allen Poeshman, 1407 W. Gregory Drive, Urbana, IL 61801, (217) , poeshma@uiuc.edu. We hank Joe Levin and Eileen Smih for generous assisance wih he daa used in his paper. We are also graeful for helpful commens from Louis Chan, Jenny France, Josef Lakonishok, Inmoo Lee, George Pennacchi, and Mike Weisbach. We bear full responsibiliy for all remaining errors.

3 Clearly Irraional Financial Marke Behavior: Evidence from he Early Exercise of Exchange Traded Sock Opions ABSTRACT This paper analyzes he early exercise of Chicago Board Opions Exchange lised calls by differen classes of invesors over he period. We presen wo main findings. Firs, here are a large number of early exercises ha can be idenified as clearly irraional wihou invoking any model of marke equilibrium, and hese exercises are no uniformly disribued across he invesor classes. Cusomers of discoun brokers and cusomers of full service brokers boh engage in a significan number of irraional exercises while raders a large invesmen houses exhibi no irraional early exercise behavior. Second, irraional exercise is riggered boh by he underlying sock price aaining is highes level over he pas year and by he underlying sock having high pas reurns. Our findings provide evidence ha prospec heory is operaive in he opions marke and ha i applies differenially across various classes of invesors.

4 I is well-known ha in he absence of marke fricions i is irraional o exercise American call opions early excep in some circumsances jus before he underlying sock goes ex-dividend. Even in he presence of marke fricions, i is possible o idenify wihou he imposiion of any model of marke equilibrium call exercises which are clearly irraional. In his paper, we employ a previously unavailable daa se o analyze he raionaliy of early exercises of Chicago Board Opion Exchange (CBOE) calls over he period by cusomers of discoun brokers (discoun cusomers), cusomers of full service brokers (full service cusomers), and raders a large invesmen houses rading for heir firms own accouns (firm proprieary raders.) We presen wo main findings. Firs, here are a large number of early exercises ha can be definiively idenified as irraional, and irraional exercise aciviy is no evenly disribued across he invesor classes. Discoun cusomers and full service cusomers boh execue a significan number of irraional exercises. Firm proprieary raders, by conras, carry ou no early exercises which can be shown o be irraional. Second, irraional exercise is riggered for boh discoun and full service cusomers by wo evens. The firs even is he underlying sock aaining is highes price level over he pas 52 weeks, and he second even is high reurns on he underlying sock over any of a number of pas ime periods. Each of hese evens riggers irraional exercise by discoun and full service cusomers even afer conrolling for he oher even. In addiion o providing new informaion on he behavior of differen ypes of opion marke invesors, our resuls conribue o an imporan debae over a large body of evidence ha has emerged over recen decades which suggess ha invesors behave irraionally. The 1

5 inerpreaion of his evidence has been conroversial, because in almos all cases i can be explained eiher by invesor irraionaliy or by misspecificaion of he model of marke equilibrium agains which he raionaliy of invesor behavior is judged. 1 Our evidence is a noeworhy addiion o he exising pool, because he inerpreaion of i as a manifesaion of invesor irraionaliy is no condiional on any model of marke equilibrium. Insead, i requires only he very weak assumpion ha invesors prefer more money o less. Our findings also conribue o a lieraure (Shefrin and Saman (1985) and Odean (1998)) which mainains ha financial marke paricipans behave in accordance wih he prospec heory of Kahneman and Tversky (1979). As we will see below, prospec heory predics ha invesors are mos likely o engage in irraional early exercise behavior afer exceeding a reference poin or afer a large gain. Consequenly, our findings on he riggers of irraional early exercise provide evidence ha some discoun and full service cusomers exhibi behavior ha conforms o prospec heory bu fail o provide evidence ha any firm proprieary raders behave in accordance wih prospec heory. We also examine he riggers of raional early exercise by he invesor classes and find ha discoun and full services cusomers ac in accordance wih he predicions of prospec heory while he firm proprieary raders do no. Insofar as we are aware, our evidence is he firs ha prospec heory is operaive in he exchange raded opions marke and is also he firs ha i funcions differenially across various classes of invesors. Even hough we do no es direcly for he impac of irraional early call exercise on he prices of securiies, our findings have implicaions for heoreical models which invesigae how 1 See Shleifer (2000) for a review of he evidence and he conroversy over is inerpreaion. 2

6 non-sandard invesor behavior influences he prices of financial asses. Barberis and Huang (2001) show ha he high mean of sock reurns, heir excessive volailiy, and he large crosssecional value premium can be explained by a model in which invesors become less cauious abou an individual securiy afer ha securiy experiences a gain or surpasses a dynamically deermined reference poin. Since our analysis shows ha real world invesors (as opposed o laboraory subjecs) behave in his way, our resuls lend plausibiliy o he explanaion provided in Barberis and Huang (2001) for a number of imporan sock marke anomalies. A he same ime, he variaion in behavior across invesor classes ha we documen suggess ha i would be ineresing o exend he represenaive invesor framework of Barberis and Huang (2001) o one ha includes heerogeneous agens. Our findings are also relaed o hose of Finucane (1997) and Heah, Huddar and Lang (1999). Finucane (1997) provides evidence ha here are a large number of irraional early exercises of CBOE calls over he period. We exend his resuls by invesigaing he disribuion of irraional exercise across invesor classes and he evens ha rigger irraional exercise. 2 Heah, Huddar and Lang (1999) find ha he early exercise of execuive sock opions is also riggered by he underlying sock price reaching a yearly high and by posiive reurns on he underlying sock. Alhough he Heah, Huddar and Lang (1999) resuls are ineresing, heir inerpreaion is no sraighforward, because an exension of he Black-Scholes model is used as he benchmark for assessing execuive exercise decisions. The Black-Scholes benchmark is problemaic, because he Black-Scholes model is known o perform poorly even when pricing liquid, European exchange raded opions (Bakshi, Cao and Chen (1997).) This poor 2 We also improve upon Finucane s paper by avoiding wo errors in his mehodology which will boh be seen below o inroduce a bias oward misclassifying early exercises as irraional. 3

7 performance is exacerbaed in he case of execuive sock opions by long mauriies and he fac ha execuives canno hedge (or sell) heir opions. The inabiliy o hedge implies ha he individual risk aversion of each execuive impacs he value of holding raher han exercising opions which is no aken ino accoun by Heah, Huddar and Lang (1999). The remainder of he paper is organized as follows. The firs secion discusses he relaionship beween prospec heory and irraional early exercise behavior. Secion wo describes he daa. The hird secion develops he procedure ha is used o classify early exercises as irraional. Secion four presens resuls on he incidence of irraional exercise behavior and analyzes is disribuion across invesor classes. The fifh secion provides resuls on sock price paerns ha rigger irraional exercise. Secion six concludes. I. Prospec Theory and Irraional Early Exercise Prospec heory is an experimenally based descripive model of decision making under uncerainy pu forward by Kahneman and Tversky (1979). Prospec heory has hree main feaures. Firs, agens derive value from gains and losses in wealh from a reference poin raher han from absolue levels as in radiional uiliy heory. Second, people are more sensiive o losses han o gains which is known as loss aversion. Finally, boh gains and losses from he reference poin exhibi diminishing sensiiviy. Panel A of Figure 1 depics a sandard prospec heory value funcion. The sudden decrease in he slope of he value funcion a he reference poin (i.e., he poin where here is no gain or loss) is a reflecion of loss aversion. The convexiy in he region of losses and he concaviy in he region of gains occurs because people are less sensiive o boh gains and losses furher from he reference poin. 4

8 Below we will define an early exercise as irraional if an invesor would have obained more money (for cerain) by selling he opion. Since he sandard prospec heory value funcion has a posiive slope across he range of losses and gains, invesors who behave in sric accordance wih he sandard prospec heory value funcion will never choose a smaller over a larger amoun of money. Hence, such invesors will never irraionally exercise opions early. We can, noneheless, derive predicions from he sandard prospec heory value funcion abou he siuaions under which invesors are mos likely o make careless decisions ha resul in a loss of value. The firs siuaion occurs when he reference poin is crossed from below. When he reference poin is exceeded, here is a marked decrease in he slope of he value funcion which enails ha an invesor would suddenly care significanly less abou hrowing away a fixed amoun of money by execuing an irraional early exercise. The second siuaion occurs when here has been a large gain. Afer a large gain, he value funcion has a small posiive slope which resuls in an irraional early exercise causing only a small loss of value for he invesor. The upsho is ha he sandard prospec heory value funcion predics ha he circumsances under which i is mos likely ha invesors will carelessly execue irraional early exercises are eiher afer a reference poin has been exceeded or afer a large gain. Under prospec heory as under radiional uiliy heory he value funcions of differen invesors will vary. We consider wo alernaives o he sandard prospec heory value funcion which would resul in irraional early exercise (wihou any carelessness) if invesors adhered o hem sricly. The slope of he sandard prospec heory value funcion decreases suddenly a he reference poin. Panel B of Figure 1 depics a value funcion whose slope decreases so markedly a he reference poin ha i becomes negaive in he region of small gains. An invesor whose preferences are described by his value funcion would exercise irraionally 5

9 when he reference poin is crossed from below. The slope of he sandard prospec heory value funcion reaches is minimum afer a large gain. 3 Panel C of Figure 1 depics a value funcion ha is so concave in he region of gains, ha i becomes negaively sloped for large gains. An invesor whose preferences are described by his value funcion would exercise irraionally afer a large gain. I should be borne in mind ha he value funcions depiced in Panels B and C of Figure 1 are boh irraional in he sense ha hey correspond o invesors who prefer less o more in he regions where he value funcions are negaively sloped. These value funcions are, noneheless, worh considering because hey have he poenial o provide insigh ino any irraional exercise behavior ha is observed in he marke. II. Daa The main daa for his paper were obained from he CBOE. The daa consiss of a daily record of exercise and volume aciviy broken down by differen ypes of invesors for all CBOE lised opions from he beginning of January 1996 hrough he end of December When a CBOE lised opion is also lised on anoher exchange, he daa covers exercises and volume for he opion from all he exchanges a which i rades. The differen ypes of invesors are discoun cusomers, full service cusomers, and firm proprieary raders. Brokerage houses are assigned o he discoun or full service caegory by an analys a he CBOE. E-Trade is an example of a discoun brokerage house, and Merrill Lynch is an example of a full service brokerage house. 3 The value funcion of he sandard prospec heory invesor may also have a posiive slope close o zero afer a large loss. We do no expec o observe irraional early exercises in his region, however, because call opions are likely o be ou-of-he money afer a large loss on eiher he opion or he underlying securiy. 6

10 The daily opening, high, low, and closing ransacion price for each of he opions in he CBOE daa se were obained from Prophe Financial Sysems. The CBOE daa conains he icker symbol for he sock ha underlies each opion. This icker symbol is used o exrac informaion on he underlying sock for each opion from he Cener for Research in Securiies Prices (CRSP) files. When a given opion observaion on a paricular rade dae canno be mached wih a CRSP sock, i is dropped from he analysis. For each opion on each rade dae, he informaion exraced from CRSP on he underlying sock is (1) wheher he rade dae is an ex-dividend dae, (2) he high price for he rade dae, (3) he daily closing prices for he previous year (adjused for splis and sock dividends), and (4) he daily reurns for he previous six monhs. Alogeher here are 74,523 disinc call opions in he CBOE daabase where disinc calls are defined by an underlying sock, srike price, and expiraion dae. These calls were wrien on 708 differen underlying socks. Table I conains basic descripive informaion on he call opions in he sample, he underlying sock price, and he exercise of he calls. Discoun cusomers exercised 4851 disinc call opions on disinc rade daes prior o mauriy on which he high price of he underlying sock was greaer han he srike price. Full service cusomers had 6458 such exercises, and firm proprieary raders III. Classificaion of Exercises as Irraional We now urn o he ask of idenifying irraional opion exercises under he weak assumpion of non-saiaion ha invesors prefer more money o less. Alhough we would like o enumerae crieria ha single ou exercises as irraional if and only if hey violae his 7

11 assumpion, uncerainy abou marke imperfecions makes his impossible. Consequenly, we develop crieria ha are saisfied by as many exercises as possible subjec o he consrain ha every idenified exercise almos surely violaes non-saiaion. Sandard opion pricing heory demonsraes ha in he absence of marke fricions i is always irraional o exercise an American call opion early excep possibly a imes immediaely prior o he underlying sock going ex-dividend (Hull (2000)). Accordingly, our iniial se of candidaes for irraional exercises are call opion exercises wih a leas one day lef o mauriy on days ha are no he rade dae before an ex-dividend dae for he underlying sock. In he presence of marke fricions, i is no necessarily irraional for an invesor o exercise a call opion early on a rade dae ha does no precede an ex-dividend dae. An invesor who exercises a call opion may (1) keep he share in his porfolio a leas for a shor period of ime, (2) sell he share immediaely o ge cash, or (3) use he share o close ou a shor posiion in he underlying sock. Since our daa do no indicae wha invesors do wih shares obained from exercise, we will only classify an exercise as irraional if i would be irraional under all hree of hese alernaives. In order o evaluae he hree possibiliies, i is necessary o consider he commissions and axes associaed wih he exercise and sale of calls. The commission for exercising a call is equal o he commission for buying a he srike price he number of shares called. No axes are paid upon he exercise of a call. When he invesor disposes of he shares obained from exercising, however, capial gains axes are due. If he invesor sells he shares, a commission is paid on he sale and capial gains axes are paid on he difference beween he sale price of he shares and he sum of he exercise price and he purchase price of he call. If he invesor uses he shares o cover a shor posiion, hen capial gains axes are paid on he difference beween he price a 8

12 which he shares were shored and he sum of he exercise price and he purchase price of he call. When he shares obained from call exercise are used o cover a shor posiion here is no commission beyond ha paid for exercising he call. The clock for deermining wheher he capial gain is shor erm or long erm sars when he call is exercised regardless of wheher he shares obained are sold or are used o cover a shor posiion. The commission paid when an invesor sells a call is, in general, differen han he commission paid upon exercise. When a call is sold, capial gains ax is paid on he difference beween he sale and he purchase price of he opion, and he ime ha has elapsed beween he purchase and he sale is used o deermine wheher he capial gain is shor erm or long erm. All commissions discussed in his and he previous paragraph are ax deducible. Using hese facs abou commissions and axes, we can analyze he invesor s decision o exercise a call. Consider firs he possibiliy ha he invesor exercises he call early a a ime ha does no immediaely precede he underlying sock going ex-dividend and holds he share obained in his porfolio for a leas one rade dae. This sraegy is sricly dominaed by waiing o exercise on he nex rade dae. Eiher way he invesor pays he exercise premium and obains a share of sock. By waiing, however, he invesor can earn an exra day of risk free reurn on his exercise premium and commission. Under boh scenarios, no ax is paid a he ime of exercise, and he basis for paying ax when he share is evenually sold is he exercise price plus he purchase price of he opion. Consider nex he case where he invesor exercises he call and hen immediaely sells he sock o obain cash. The invesor could have chosen insead simply o sell he call. If afer aking accoun of bid-ask spreads, commissions, and axes he invesor would have received 9

13 more cash from selling he call, hen he exercise is irraional. Hence, he exercise decision is irraional if he following inequaliy is saisfied ( ) τ ( 1 τ ) Bid Bid Purch Exercise Call Sell Sock > S K ( S K C ) τ Comm ( 1 τ) Comm ( 1 τ) C C C Comm Bid Bid Purch Sell Call (1) where Bid C is he price ha he call can be sold for a ime, Purch C is he price for which he call was purchased a some ime before, τ is he shor erm capial gains ax rae a ime, Bid S is he price ha a share of sock can be sold for a ime, K is he exercise price of he call, and X Comm is he commission for underaking acion X a ime. The commissions and oher quaniies are all on a per share basis. The capial gain for exercising he call and hen selling he sock is shor erm, because he invesor exercises he call and hen immediaely sells he sock. The capial gain associaed wih selling he call will almos always be shor erm as well, because almos all calls are purchased when heir imes o expiraion are shorer han he period required for a capial gain o be long erm. If for a paricular exercise he capial gain ha would have resuled from selling he call would have been long erm, hen here is a furher disincenive for exercising he call. This disincenive arises, because exercising he call convers a long erm capial gain ino a shor erm capial gain. Hence, if assuming ha he capial gains ax rae in inequaliy (1) is he same for boh selling or exercising he call inroduces any disorion ino he assessmen of he raionaliy of exercises, he disorion will be a bias agains classifying exercises as irraional. We make his assumpion in order o be conservaive when deciding wheher exercises are irraional. Using he fac discussed above ha he commission for exercising a call is he same as he commission for selling a sock (because he commission on sock rades does no generally vary 10

14 wih he sock price or depend upon wheher he rade is a buy or a sell), sraighforward algebra shows ha inequaliy (1) simplifies o ( ) 2. C S K > Comm Comm (2) Bid Bid Sell Call Sell Sock If inequaliy (2) is saisfied, hen exercising a call and selling he sock immediaely o ge cash is irraional. Noe ha inequaliy (2) differs from he analogous inequaliy (3) in Finucane (1997). This is because Finucane incorrecly assumes ha he commission for selling an opion is he same as he commission for exercising an opion. As a resul, he righ hand side of his Sell Sock inequaliy is ( 1 ) Comm τ which can be smaller han our expression and herefore inroduce a bias oward classifying exercises as irraional. 4 Consider finally he case where an invesor exercises a call in order o use he sock obained o cover a shor posiion in he sock. In his case, he exercise is irraional if he invesor would have ended up wih more money by selling he call and buying he sock o cover he shor posiion. Hence, he exercise decision is irraional if he following inequaliy is saisfied: ( ) τ ( ) τ ( )( 1 τ ) Shor Purch Exercise Call K ( S K C ) τ Comm ( 1 τ) C S C C S S Comm + Comm Bid Ask Bid Purch Shor Ask Sell Call Buy Sock > (3) where Ask S is he price a which a share of sock can be bough a ime, Shor S is he price a which he share was shored a some ime before, and all oher variables are as defined above. Once again making use of he fac ha he commission for exercising a call is he same as ha for buying a sock, sraighforward algebra shows ha inequaliy (3) is equivalen o: 4 Noe ha Finucane s expression is always negaive while ypical commission levels can resul in he righ hand side of inequaliy (2) being posiive. 11

15 ( ). C S K > Comm (4) Bid Ask Sell Call There is one difference on he lef hand side and one difference on he righ hand side of inequaliies (2) and (4). On he lef hand side, inequaliy (4) has Bid S insead of S. Since he Ask ask price is higher han he bid price, his difference will make i more difficul o saisfy inequaliy (4) han inequaliy (2). On he righ hand side, inequaliy (4) is missing 2 Comm Sell Sock. This will also make inequaliy (4) more difficul o saisfy. Consequenly, inequaliy (4) implies inequaliy (2) bu no vice-versa. As a resul, in order o classify an exercise as irraional, we require inequaliy (4) o be saisfied. 5 In order o use inequaliy (4) o classify he raionaliy of early exercises, we mus Sell Call deermine an upper bound on he quaniy Comm. We did his by surveying commission charges from 37 brokerage houses over our daa period and found ha he greaes commission charged for selling opions was a fla fee of $40 plus $2 per conrac. This commission implies a maximum commission charge on a per share basis when jus one call conrac is sold. The corresponding maximum commission is $0.42 (since each call conrac is for 100 shares.) Hence, in order o be conservaive we will assume an upper bound on Sell Call Comm of $ Anoher issue ha arises when empirically implemening inequaliy (4) is ha we do no know a wha poin during he day exercise orders are issued. The Opions Clearing Corporaion 5 Finucane (1997) does no consider he possibiliy ha an invesor exercises a call o use he sock obained o cover a shor posiion. This omission also biases Finucane s procedure oward misclassifying early call exercises as irraional, because i is he siuaion under which i is leas likely ha he invesor is behaving irraionally (i.e., inequaliy (4) implies inequaliy (2) bu no vice-versa), 6 As a robusness check, we also ran he ess below assuming an upper bound increased by 50 cens o $0.92. This change did no aler any of he main feaures of our resuls. I migh also make sense o use a lesser upper bound for discoun cusomers and firm proprieary raders. We did no do his, because (1) we are no cerain how much o reduce he upper bound for hese cusomers and (2) he resuls repored in Table II below sugges ha lesser upper bounds for hese invesors would no make much of a difference. 12

16 assigns exercised calls o call wriers a he end of he day, and he assigned call wriers hen have a fixed number of business days o deliver he underlying shares. Consequenly, if he call owners could no shor he underlying sock, hen i would always be raional for hem o exercise as lae in he day as possible in order o reain he opion of foregoing exercise if he underlying sock price moves adversely (e.g., o below he srike price) during he remainder of he day. However, since he invesors can, in fac, shor he underlying asse, i can poenially be raional o simulaneously place an exercise order and shor a corresponding number of share a some ime oher han he end of he day. This pair of acions would lock in a cash flow equal o he difference beween he sock price a he ime of exercise and he srike price regardless of how he sock price moves subsequenly. Ideally, ime-samped daa on Bid C and Ask S would be available. If his were he case, hen we would selec he minimum value aained by C Bid S a any poin during he day o Ask use in inequaliy (4). For mos of our daa period, however, ime-samped daa is no available on CBOE opions. Accordingly, in order o ensure ha we do no misclassify any possibly raional exercise as irraional, we se Ask S and Bid C in inequaliy (4) o, respecively, he sock s High high ransacion price during he day, S, and he calls low ransacion price during he day C. The value of Low High S almos cerainly reflecs a ransacion a he ask price, and he value of Low C almos cerainly reflecs a ransacion a he bid price. Since he call price will end o be a is highes value of he day when he sock price is a is highes value, i is likely ha he compued value C Low Bid Ask S is smaller han he minimum value during he day of C S. High Consequenly, i is improbable ha he use of C Low S as a proxy for he minimum value ha High 13

17 C Bid S aains during he day will lead o he misclassificaion of any possibly raional Ask exercises as irraional. Combining he foregoing consideraions on commissions and inraday exercise, we will classify early call exercises on non-ex-dividend rade daes as irraional when hey saisfy Low High C ( S K) > $0.42. (5) In order o ensure ha sale prices are no used o conclude ha an exercise is irraional, we do no classify an exercise as irraional if he closing ransacion price for he call or he high price for he underlying sock are no available on he rade dae of he exercise. For he same reason, we require ha a leas 10 conracs of he call were ransaced on he rade dae of he exercise. Finally, we exclude from he irraional caegory call exercises ha occur on rade daes for which he high sock price is less han he srike price, because hese exercises may correspond o daa errors. The discussion below will be faciliaed by explicily saing he condiions ha an opion exercise on a paricular rade dae mus mee in order for i o be classified as irraional and by defining classes of exercises as irraional or raional in erms of hese condiions. An observed opion exercise will be classified as irraional if and only if (C1) The opion is a call. (C2) The call is no a is expiraion dae. (C3) I is no he day before an ex-dividend dae for he underlying sock. (C4) Daa are available on he daily low ransacion price for he call and he daily high ransacion price for he underlying sock. (C5) The daily rading volume for he call is a leas 10 conracs. 7 (C6) The day s high price for he underlying sock exceeds he call s srike price. 7 Requiring ha a leas 50 conracs of he call were ransaced on he rade dae of he exercise does no change any of he main feaures of our resuls. 14

18 Low High (C7) The following inequaliy is saisfied: C ( S K) > $0.42. Opion exercises ha conform o (C1)-(C6) will be referred o as poenially raional or irraional exercises. Poenially raional or irraional exercises ha also saisfy (C7) will be referred o as irraional exercises. Poenially raional or irraional exercises ha do no saisfy (C7) will be referred o as raional exercises. According o his axonomy, many exercises will be neiher raional nor irraional since many exercises do no saisfy (C1)-(C6). Some of hese unclassified exercises may be irraional. I is also possible ha because of he conservaive way crierion (C7) was developed, exercises which we caegorize as raional are, in fac, irraional. IV. Irraional Exercise Behavior and is Disribuion across Invesor Classes This secion of he paper analyzes he quaniy of irraional exercise behavior in our sample and is disribuion across he invesor classes. We begin by deermining he number of irraional exercises by each class of invesors. We hen evaluae hese numbers by comparing hem o he number of raional exercises by each invesor class and o he number of opporuniies ha each invesor class had o exercise irraionally. Our daa provide us wih he number of conracs exercised by each ype of invesor on each rade dae for each ype of call, where a ype of call is defined by an underlying sock, a srike price, and an expiraion dae. When one of he invesor classes exercises more han one conrac of some ype of call on a given rade dae, we do no know wheher his corresponds o exercise orders from one or more invesors. As a resul, he uni of analysis for his secion of he paper will be a ype of call on a paricular rade dae which we will refer o as a call-rade dae. 15

19 Whenever we observe an invesor class exercising a sricly posiive number of conracs of some ype of call on a rade dae, we coun ha as one exercise for ha class of invesors on ha callrade dae. To illusrae, suppose ha on June 23, 1998, discoun cusomers exercised a oal of 25 calls on IBM wih a srike price of 40 ha expire in July 1998, full service cusomers exercised none of hese calls, and firm proprieary raders exercised a oal of 10 such calls. Furhermore, suppose ha hese exercises conform o (C1)-(C7). We would coun his exercise daa as one irraional exercise for discoun cusomers, zero irraional exercises for full service cusomers, and one irraional exercise for firm proprieary raders of he July expiraion, 40 srike IBM call on June 23, Even hough we know ha here mus have been a leas wo disinc exercise orders (and here may have been as many as 65), we will coun his daa as one exercise for he invesors aggregaed ogeher in order o rea consisenly he call-rade dae as he uni of analysis. Nex we re-wrie inequaliy (5) which mus be saisfied in order for a call exercise o be classified irraional as and define he quaniy E by Low High C ( S K) $0.42 > 0 (6) Low High E C ( S K) $0.42. (7) Exercises are irraional hen, if hey conform o (C1)-(C6) and E is sricly posiive. Table II conains he disribuion of E for exercises ha saisfy (C1)-(C6) for all invesors (Panel A), discoun cusomers (Panel B), full service cusomers (Panel C), and firm proprieary raders (Panel D). Each panel also conains he disribuion of E condiional on E > 0 which 16

20 corresponds o irraional exercises and condiional on E 0 which corresponds o raional exercises. Panel A of Table II shows ha irraional exercise akes place on 191 call-rade daes in our sample. Hence, i is a regular occurrence. This finding is consisen wih he resuls repored in Finucane (1997) for CBOE exercises over he period. Our resuls, however, add o wha is already known from Finucane (1997), because (as noed in he previous secion) his mehodology is biased oward misclassifying exercises as irraional. Our resuls also exend hose in Finucane (1997) by breaking down he irraional exercise behavior by classes of invesors. Panels B hrough D of Table II indicae ha here are 85 irraional exercises by discoun cusomers, 110 irraional exercises by full service cusomers, and zero irraional exercises by firm proprieary raders. The raw numbers of irraional exercises show ha discoun cusomers and full service cusomers habiually engage in irraional exercise behavior while firm proprieary raders do no. Figures 2 and 3 display bar chars of, respecively, he number of raional and irraional exercises as a funcion of he number of calendar days o expiraion for each invesor ype. A comparison of he figures reveals ha raional early exercises are more concenraed near opion expiraion han irraional early exercises. The more uniform disribuion of he irraional exercises wih respec o opion expiraion suggess ha some exogenous facors may be riggering irraional exercise. This possibiliy will be explored in he nex secion. Figure 3 does no provide a char for firm proprieary raders, because his class of invesors had no exercises ha saisfied crieria (C1)-(C7). Omied analogous chars for all early call exercises (i.e., hose where condiions (C3)-(C7) are no imposed) look very similar o hose in Figure 2 for he raional exercises. 17

21 We assess wheher he propensiy o exercise irraionally varies across he invesor classes in wo ways. The firs approach examines wheher he exercises of one invesor class are more likely o be irraional han hose of he oher classes. To deermine wheher his is he case, we compue he percenage of poenially raional or irraional exercises ha acually are irraional. More specifically, we deermine he percenage of exercises ha conform o (C1)-(C6) which also saisfy (C7). Panel A of Table III repors for all invesors and for each invesor ype he percenage of poenially raional or irraional exercises ha are acually irraional. For all invesors, 2.14% of poenially raional or irraional exercises are acually irraional. This percenage is 2.57% for discoun cusomers, 2.32% for full service cusomers, and zero percen for firm proprieary raders. Permuaion ess were conduced o deermine wheher he observed differences in he percenages beween pairs of invesor classes are likely o occur by chance if here is no difference in he rue underlying disribuion of percenages across pairs of invesors. For example, o evaluae he saisical significance of he observed difference in he percenages exercised irraionally by discoun cusomers and full service cusomers, we begin by pooling ogeher he 3303 call-rade daes on which discoun cusomers exercised and he 4745 call-rade daes on which full service cusomers exercised. When boh discoun and full service cusomers exercised on he same call-rade dae, i is included in he pool wice. We hen (1) Randomly choose 3303 call-rade daes (wihou replacemen) from he pool and rea hem as he discoun cusomer observaions. We rea he oher 4745 call-rade daes as he full service cusomer observaions. (2) Compue he difference of he percenage of observaions assigned o discoun cusomers ha are irraional and he percenage of observaions ha are assigned o full service cusomers ha are irraional. 18

22 Seps (1) and (2) are repeaed 1000 imes, and we coun he number of imes ha he difference compued in sep (2) exceeds he acual in-sample difference of 0.25%. In he es for he discoun cusomers and he full service cusomers, he difference was greaer han 0.25% 156 imes which yields he p-value of repored in Panel B of Table III. The ess for he oher pairs of invesors were conduced similarly, and he p-values from hese ess are also repored in Panel B of Table III. The permuaion ess indicae ha he differences in he percenages beween discoun and full service cusomers is no saisically significan a convenional levels. I is surprising ha he exercises of full service cusomers who receive professional advice are no less likely o be irraional han hose of discoun cusomers who do no receive such advice. In fac, i is puzzling ha full service cusomers are no alogeher prevened from making obviously irraional financial decisions by heir advisors. The p-values in he final column of Panel B of Table III indicae ha i is more likely ha discoun cusomer and full service cusomer exercises are irraional han hose of firm proprieary raders. We nex es for he likelihood ha various invesor classes will ac on opporuniies o exercise eiher irraionally or raionally. In order o conduc hese ess, we define an irraional exercise opporuniy for an invesor class as a call-rade dae for which he invesor class has sricly posiive open ineres and for which exercising would saisfy crieria (C1)-(C7). Similarly, a raional exercise opporuniy for an invesor class is defined as a call-rade dae for which he invesor class has sricly posiive open ineres and for which exercising would saisfy crieria (C1)-(C6) bu would violae crierion (C7). Panel A of Table IV repors by invesor class he number of irraional and raional exercise opporuniies, he number of irraional and raional exercises, and he percenage of 19

23 irraional and raional exercise opporuniies in which exercises were carried ou. Panel B of Table IV repors he resuls of permuaion ess for he significance of differences in hese percenages across invesor classes. Discoun cusomers exercise on 0.035% of heir irraional exercise opporuniies and full service cusomers exercise on 0.041% of such opporuniies. Firm proprieary raders exercise on none of heir 46,261 irraional exercise opporuniies. The p- values in Panel B indicae ha he difference in he percenage beween discoun and full service cusomers is no significan while he difference in he percenage beween eiher discoun cusomers or full service cusomers and firm proprieary raders is saisically significan. Hence, given he opporuniy o exercise irraionally, discoun and full service cusomers are significanly more likely o ac on i han firm proprieary raders. Panel A of Table IV indicaes ha discoun cusomers ac on 1.31% of heir opporuniies o exercise raionally while full service cusomers ac on 1.66% of such opporuniies and firm proprieary raders ac on 2.69% of raional exercise opporuniies. The difference in percenages beween each pair of invesor ypes is saisically significan a convenional levels. Accordingly, firm proprieary raders who have he lowes propensiy for irraional exercise have he highes propensiy for raional exercise. V. Reference Poins, Pas Reurns, and Irraional Exercise This secion of he paper invesigaes wheher irraional exercise for he various invesor classes is riggered by he underlying asse crossing dynamically defined reference poins or experiencing a period of posiive or negaive reurns. We address hese issues by performing logi regressions where he dependen variable indicaes wheher an irraional exercise has 20

24 occurred and he independen variables eiher signify wheher he price of he underlying sock has crossed a reference poin or conain informaion on he accumulaed pas reurns on he underlying asse. We focus on he price and reurn pahs of he underlying asses raher han ha of he calls. We do so, because mos opions on individual equiies are illiquid when expiraions are longer han a couple of monhs so ha for mos exercises i would be impossible o ge reliable pas ime series of prices or reurns on he exercised calls. In he presen conex he call-rade dae is no an appropriae uni of analysis, because on a given rade dae all calls wrien on he same sock would have exacly he same values for he dependen variables ha are defined in erms of he price pah of he underlying asse. The resuling mulicollineariy would make i difficul o inerpre any findings. For his reason, we choose sock-rade daes as he uni of analysis for he regressions where a sock-rade dae refers o all calls on a given rade dae ha are wrien on a paricular underlying sock. In he regressions below, we include sock-rade daes only if he invesor class under consideraion has sricly posiive open ineres on a leas one call wrien on he underlying sock ha conforms o crieria (C1)-(C7) (i.e., whose exercise would be classified as irraional.) The dependen variable is a dummy variable ha is equal o one when an invesor class has a leas one exercise of a call ha saisfies (C1)-(C7) which is wrien on a specific underlying sock on a paricular rade dae: Exercise Irraional ij 1 if invesor class i exercises a (C1)-(C7) call on sock j on rade dae = 0 if invesor class i has open ineres in bu no exercises of hese calls (8) The subscrip i denoes eiher all invesors, discoun cusomers, or full service cusomers. 8 8 We do no run he regressions for firm proprieary raders, because his class of invesors has no exercises which conform o (C1)-(C7). 21

25 The firs independen variable is a dummy variable ha akes he value one if he closing price of he underlying sock on rade dae is sricly higher han any of he daily closing prices of he underlying sock over he pas 52 weeks. RefPoin j 1 if closing price of sock j on rade dae is a 52 week high = 0 oherwise (9) Sock splis and sock dividends are aken ino accoun when compuing his variable. The nex se of independen variables are cumulaive reurns on he underlying asse over various ime periods prior o he exercise on rade dae. ReWeek1 = Cumulaive reurn on sock j for rade daes 5 hrough 1 (10) j ReWeek2 = Cumulaive reurn on sock j for rade daes 10 hrough 6 (11) j ReWeek3 = Cumulaive reurn on sock j for rade daes 15 hrough 11 (12) j ReWeek4 = Cumulaive reurn on sock j for rade daes 20 hrough 16 (13) j ReMonh2 = Cumulaive reurn on sock j for rade daes 42 hrough 21 (14) j ReMonh3To6 = Cumulaive reurn on sock j for rade daes 126 hrough 43 (15) j We also include a conrol variable on he righ hand side of he regressions. This variable provides a measure of he average cash flow ha would be los from exercising raher han selling each of he N opion held by invesor class i on rade dae ha are wrien on sock j and which saisfy (C1)-(C7): CashFlowLoss ij 1 N = Ekij (16) N S = Close ( j ) k 1 where Close S j is he closing price of sock j on rade dae and E kij is he value from equaion (7) on rade dae for he h k opion on sock j held by invesor class i ha saisfies (C1)-(C7). Scaling by he sock price sandardizes he cash flow loss across differen socks. 22

26 Now ha all of he necessary variables have been defined, we urn o he analysis of he following regression equaion: Exercise = β + β RefPoin + β ReWeek1 + β ReWeek2 + β ReWeek3 + β ReWeek4 Irraional ij 0 1 j 2 j 3 j 4 j 5 j + β ReMonh2 + β ReMonh3To6 + β CashFlowLoss + ε (17) 6 j 7 j 8 ij Descripive saisics on he variables in equaion (17) are provided in Table V. Table VI presens esimaes of his regression equaion when all of he invesors are aggregaed ino one class as well as for discoun cusomers and full service cusomers separaely. The esimae on he reference poin variable is significanly posiive in all hree regressions which provides evidence ha a 52 week high price for he underlying sock riggers irraional exercise by discoun cusomers and full service cusomers. This is rue even hough here are conrols for he reurn of he underlying sock over various periods leading up o he poenial exercise and for he cash flow difference beween exercising and selling calls. The discoun cusomer and full service cusomer regressions boh have posiive coefficien esimaes for all of he pas reurn variables. Four of hese six coefficiens from each regression are saisically significan a convenional levels and he oher coefficiens are all close o saisically significan. The posiive coefficien esimaes are consisen wih irraional opion exercise by discoun cusomers and full service cusomers being riggered by posiive sock reurns over he corresponding pas horizons. The riggering effec is presen over he various horizons, even afer conrolling for reurns over he oher horizons, for he underlying sock price being a is yearly high, and for he cash flow difference beween exercising and selling he opion. As explained in Secion I above, he sandard prospec heory value funcion predics ha invesors are mos likely o carelessly execue irraional early exercises eiher afer a reference 23

27 poin has been exceeded or afer a posiive gain. Our finding ha irraional early exercise is riggered for discoun and full service cusomers by he underlying asse exceeding is 52 week high or by a period of gains on he underlying asse indicaes ha a leas a subse of discoun and full service cusomers behave in accordance wih his predicion of prospec heory. An alernaive inerpreaion of hese findings is ha a leas some subse of discoun cusomers and full service cusomers have prospec heory value funcions like hose depiced in Panels B and C of Figure 1. According o his inerpreaion, invesors behave in accordance wih value funcions ha someimes reflec a preference for less over more. To he bes of our knowledge, our resuls provide he firs evidence ha prospec heory is operaive in he exchange raded opions marke and ha i applies differenially o various classes of invesors. Barberis and Huang (2001) incorporae recen experimenal evidence ha invesor cauiousness depends dynamically on prior gains and losses ino a heoreical asse pricing model. In paricular, hey make use of experimenal evidence ha a loss ha is preceded by gains causes a smaller loss in uiliy han one ha is preceded by losses. Barberis and Huang show ha combining his dynamic loss aversion wih narrow framing in which invesors evaluae he gains or losses on each sock in heir porfolios separaely produces equilibrium sock reurns ha have high mean, excessive volailiy, and a large cross-secional value premium. Hence, dynamic loss aversion and narrow framing can accoun for many of he imporan sock marke anomalies. Our findings ha irraional early exercise is riggered by he underlying sock price being a is yearly high and by a period of high reurns on he underlying asse provide evidence ha invesors in acual financial markes (as opposed o hose in arificial experimenal siuaions) make decisions abou individual asses less carefully afer a dynamically deermined reference poin for he asse has been surpassed or afer here has been a period of gains for he asse. 24

28 Consequenly, our findings lend plausibiliy o he explanaion provided in Barberis and Huang (2001) for a number of imporan sock marke anomalies. A he same ime, since we do no find ha firm proprieary raders become less careful afer surpassing a yearly high price or afer high pas reurns, i would be ineresing o exend he represenaive invesor framework of Barberis and Huang (2001) o one ha also includes agens who do no display dynamic loss aversion or narrow framing. 9 We nex invesigae wheher he evens ha rigger irraional opion exercise also rigger he opion exercises which we classify as raional. In order o explore his issue, we run he logi regressions for sock-rade daes on which an invesor class is long opions whose exercise would be raional (i.e., is long opions ha saisfy (C1)-(C6) and ha have an E value less han or equal o zero.) In hese regressions he Exercise ij variable is redefined as follows: Exercise Raional ij 1 if class i exercises a (C1)-(C6), E 0 call on sock j on dae = 0 if class i has open ineres in bu no exercises of hese calls (18) The subscrip i denoes eiher all invesors, discoun cusomers, full service cusomers, or firm proprieary raders. Table VII presens coefficien esimaes for he regressions when Raional Exercise ij is he dependen variable. As for he case of irraional exercises, he discoun cusomers and he full service cusomers have coefficiens on he reference poin variable and all of he pas reurns variables ha are posiive. Now all of hese coefficien esimaes are significan a he one percen level wih he excepion of one of he pas reurn variables for he full service cusomers which is close o saisically significan. Hence, for discoun cusomers 9 The Barberis and Huang (2001) model is in discree ime wih each ime sep corresponding o one year. Consequenly, heir dynamically evolving reference poin is se o an asse s value one year ago. The reference poin ha we invesigae, by conras, is he highes price of he underlying asse over he pas year. In ligh of our 25

29 and full service cusomers he opion exercises which we classify as raional are riggered by he same evens as he irraional opion exercises. These riggers of raional exercise by discoun cusomers and full service cusomers are also consisen wih prospec heory. To see ha his is he case, firs noe ha because of he conservaive way ha we developed he crieria for classifying exercises, a number of he exercises ha we classify as raional may, in fac, be irraional. Insofar as his is he case, invesors who behave in accordance wih prospec heory may become more careless afer exceeding a reference poin or afer a posiive gain, as explained above, and execue irraional exercises ha have been misclassified as raional. Prospec heory also predics ha exercises which are properly classified as raional hose for which an invesor receives more cash by exercising han selling he call will have more of a endency o occur afer a reference poin is exceeded or afer a period of gains. To undersand why his is he case, consider once again he sandard prospec heory value funcion in Panel A of Figure 1. This value funcion abruply swiches from convex o concave when he reference poin is exceeded, and i becomes uniformly more concave (i.e., he convexiy decreases or he concaviy increases) as one moves from lef o righ on he figure. As wih sandard uiliy funcions, invesors are risk-seeking when he value funcion is convex and riskaverse when he value funcion is concave. Consequenly, when invesors move from below o above he reference poin, ceerus paribus, hey suddenly swich from preferring risky gambles over cerain paymens o preferring cerain paymens over risky gambles. More generally, as he value funcion becomes less convex or more concave, invesors become more risk-averse. evidence, i would also be ineresing o incorporae a reference poin of he highes price over he pas year ino a coninuous ime version of he Barberis and Huang model. 26

30 Hence, as heir level of loss relaive o a reference poin decreases or heir level of gain relaive o a reference poin increases, heir preference for cerain paymens over risky gambles increases. As a resul, prospec heory predics ha invesors will have a endency o rade he risky gamble inheren in holding a call opion for he sure paymen ha is obained by selling he call or exercising he call and disposing of he called sock when hey exceed a reference poin or when here is a gain (i.e., a decrease in losses or an increase in gains.) If he exercises we classify as raional are indeed ones for which an invesor obains more cash by exercising he call and disposing of he called sock han by selling he call, hen prospec heory predics ha invesors will have a greaer propensiy o engage in raional exercise afer a reference poin has been exceeded or afer a period of gains. Hence, our findings on he riggers of raional exercise by discoun cusomers and full service cusomers are consisen wih hese invesors behaving in accordance wih he sandard prospec heory value funcion. Nex we consider he riggers of raional exercise for firm proprieary raders. The final column of Table VII conains he raional exercise regression esimaes for firm proprieary raders. The coefficien on he reference poin variable is close o zero and saisically insignifican. This coefficien esimae indicaes ha unlike for he oher invesors exercise is no riggered for he firm proprieary raders by he underlying asse aaining a yearly high price. The coefficiens on he pas reurn variables are all negaive and saisically significan. These coefficien esimaes indicae ha firm proprieary rader exercise is riggered by pas negaive reurns on he underlying asse. This sands in conras o he discoun cusomers and full service cusomers whose exercise is riggered by posiive reurns on he underlying asse. The resuls on he riggers of raional exercise by firm proprieary raders violae boh of he predicions of prospec heory. Hence, he resuls from he riggers of raional exercise, like hose from he 27