Are Options on Index Futures Profitable for Risk Averse Investors? Empirical Evidence

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1 Are Opions on Index Fuures Profiable for Risk Averse Invesors? Empirical Evidence George M. Consaninides Universiy of Chicago and NBER Jens Carsen Jackwerh Universiy of Konsanz Michal Czerwonko Concordia Universiy Sylianos Perrakis Concordia Universiy Absrac American call and pu opions on he S&P 500 index fuures ha violae he sochasic dominance bounds of Consaninides and Perrakis (007) over are idenified as poenially profiable invesmen opporuniies. Call bid prices more frequenly violae heir upper bound han pu bid prices do, while evidence of underpriced calls and pus over his period is scan. In ou-of-sample ess, he inclusion of shor posiions in such overpriced calls, pus, and, paricularly, sraddles in he marke porfolio is shown o increase he expeced uiliy of any risk averse invesor and also increase he Sharpe raio, ne of ransacion coss and bid-ask spreads. The resuls are srongly supporive of mispricing. (JEL G11, G13, G14) Curren draf: March 16, 008 Keywords: opion mispricing; fuures opions; derivaives pricing; sochasic dominance; ransacion coss; marke efficiency We hank Russell Davidson for insighful commens and consrucive criicism. We remain responsible for errors and omissions. Consaninides acknowledges financial suppor from he Cener for Research in Securiy Prices of he Universiy of Chicago; Czerwonko and Perrakis acknowledge financial suppor from he Social Sciences and Humaniies Research Council of Canada. addresses: gmc@chicagogsb.edu, m_czerwo@jmsb.concordia.ca, Jens.Jackwerh@uni-konsanz.de, SPerrakis@jmsb.concordia.ca

2 A large body of finance lieraure addresses he mispricing of opions. Rubinsein (1994), Jackwerh and Rubinsein (1996), and Jackwerh (000), among ohers, observed a seep index smile in he implied volailiy of S&P 500 index opions ha suggess ha ou-of-hemoney (M) pus are oo expensive. Indeed, a common hedge-fund policy is o sell M pus. Coval and Shumway (001) found ha buying zero-bea a-he-money (ATM) sraddles loses money. Consaninides, Jackwerh, and Perrakis (008) provided empirical evidence ha boh M pus and calls on he S&P 500 index are mispriced by showing ha hey violae sochasic dominance bounds pu forh by Consaninides and Perrakis (00). In his paper, we provide ou-of-sample ess of opion mispricing, ne of ransacion coss and bid-ask spreads. Specifically, we idenify American call and pu opions on he S&P 500 index fuures ha violae he sochasic dominance bounds of Consaninides and Perrakis (007) as poenially profiable invesmen opporuniies. In ou-of-sample ess over , we show ha rading policies ha exploi hese violaions provide higher Sharpe raios han policies wihou opion rading. We also show ha he expeced uiliy of any risk averse invesor, ne of ransacion coss and bid-ask spreads, increases when exploiing such opion rading. Below, we highligh novel feaures of our approach. Firs, we use he Chicago Mercanile Exchange (CME) daa base on S&P 500 fuures opions, , which is clean and spans a long period. Much of he earlier empirical work on he mispricing of index opions is based on daa on he S&P 500 index opions ha comes from wo principal sources: he Berkeley Opions Daabase ( ) ha provides relaively clean ransacion prices, bu misses imporan evens over he pas 1 years, such as he 1998 liquidiy crisis, he do-com bubble, and is 000 burs; and he OpionMerics ( ) daa base which, however, is of uneven qualiy and conains only end-of-day quoes. Second, we idenify mispriced opions wih a screening mechanism ha uses minimal assumpions abou marke equilibrium. This mechanism is based on he sochasic dominance bounds of Consaninides and Perrakis (007). These bounds idenify reservaion purchase and reservaion wrie prices such ha any risk averse invesor may increase her expeced uiliy by including he opion ha violaes hese bounds in her porfolio. The bounds are valid for any disribuion of he underlying asse and 1

3 accommodae jumps. They also recognize he possibiliy of early exercise of American opions. The only necessary assumpion abou he marke for he validiy of hese bounds is ha here exiss a class of raders holding porfolios conaining only he S&P 500 index and he riskless asse. 1 Ample evidence exiss ha his assumpion holds for US markes. Numerous surveys have shown ha a large number of US invesors follow indexing policies in heir invesmens. Bogle (005) repors ha in 004 index funds accouned for abou one hird of equiy fund cash inflows since 000 and represened abou one sevenh of equiy fund asses. The S&P 500 index is no only he mos widely quoed marke index, bu has also been available o invesors hrough exchange raded funds for several years. We find ha any such invesor would improve her uiliy by including in her porfolio an opion idenified as mispriced by he sochasic dominance bounds. As a hird novel feaure, we assess he profiabiliy of our rading policy by employing he powerful saisical ess of sochasic dominance by Davidson and Duclos (000 and 006) which can deal wih opion reurns even in a seing where we do no make assumpions abou he preferences of he invesors. These ess compare he profiabiliy of he opimal rading policies of a generic S&P 500 index invesor wih and wihou he opion in a seing ha recognizes he possibiliy of early exercise of he fuures opion. These profiabiliy comparisons are valid from he perspecive of any risk averse invesor. By conras, he ubiquious Sharpe raio measure of porfolio performance is valid only from he perspecive of a mean-variance invesor and suffers from well known problems when used o assess non-normal reurns such as hose encounered in porfolios ha include opions. Finally, boh he bounds employed in deecing mispriced opions and he porfolio reurns explicily ake ino consideraion bid-ask spreads and rading coss. Once a rading opporuniy is deeced, we execue he rade by buying a he nex ask price or selling a he nex bid price. We use hisorical daa on he underlying S&P 500 index reurns in order o esimae he bounds. We use several empirical esimaes of he underlying reurn disribuion, all of hem observable a he ime he rading policy is implemened. For each one of hese 1 The mean-variance porfolio heory ha gives rise o he Sharpe raio measure of porfolio performance is based on he sronger assumpion ha every invesor holds he marke porfolio and he risk free asse.

4 esimaes we evaluae he corresponding bounds over he period , and hen idenify he observed S&P 500 fuures opions prices ha violae hem. For each violaion, we idenify he opimal rading policy of a generic invesor wih and wihou he mispriced opion, using he observed pah of he underlying asse ill opion expiraion and recognizing realisic rading condiions such as possible early exercise and ransacion coss. We idenify he profiabiliy of he pair of policies for each observed violaion, and hen conduc sochasic dominance comparison ess over he enire sample of violaions. We find a subsanial number of violaions of he upper bounds, bu relaively few violaions of he lower bounds. Since he frequency of violaions of he lower bounds is oo low for saisical inference, we focus on violaions of he upper bounds. The resuls are srongly supporive of mispricing. The paper is organized as follows. In Secion 1, we presen he resricions on fuures opion prices imposed by sochasic dominance and discuss he underlying assumpions. In Secion, we describe he daa and he empirical design. In Secion 3, we presen he empirical resuls and discuss heir robusness. We conclude in Secion 4. 1 Resricions on Fuures Opion Prices Imposed by Sochasic Dominance We assume ha marke agens are heerogeneous and invesigae he resricions on opion prices imposed by one paricular class of agens ha we simply refer o as raders. We allow for oher agens o paricipae in he marke bu his allowance does no invalidae he resricions on opion prices imposed on raders. We consider a marke wih several ypes of financial asses. Firs, we assume ha raders inves only in wo of hem, a bond and a sock wih naural inerpreaion as a marke index. Subsequenly, we assume ha raders can inves in a hird asse as well, an American call or pu opion on he index fuures. The bond is risk free and has oal reurn Essenially, we model buy-and-hold invesors who rade infrequenly and incur low ransacion coss. A leas for large invesors who earn a fair reurn on heir margin, ransacion coss are even lower in he index fuures marke han he sock marke. In pracice, however, buy-and-hold invesors inves in he sock and bond markes because of he inconvenience and cos of he frequen rolling over of shor-erm fuures conracs and he illiquidiy of long-erm fuures and forward conracs. 3

5 R. The sock has ex dividend sock price S a ime and pays cash dividend γ S, where he dividend yield γ is deerminisic. The oal reurn on he sock, (1 ) ( S / S ) assumed i.i.d. wih mean expiraion dae T. F F mauriy T, T T +, is γ + 1 R S. The call or pu opion on he index fuures has srike K and The underlying fuures conrac is cash-seled and has. We assume ha he fuures price F is linked o he sock price by he approximae cos-of-carry relaion ( ) ( F ) T F T F F = 1 + γ R S + ε, T, ε ε, where he basis risk variables { ε } are disribued independenly of each oher and of he sock price series{ S }. Transfers o and from he cash accoun (bond rades) do no incur ransacion coss. Sock rades decrease he bond accoun by ransacion coss equal o he absolue value of he dollar ransacion, imes he proporional ransacion coss rae, k,0 k < 1. Opion rades incur ransacion coss, exchange fees, and price impac which are incorporaed in wha we refer o as heir bid and ask prices. We assume ha raders maximize generally heerogeneous, sae-independen, increasing, and concave uiliy funcions. We furher assume ha each rader s wealh a he end of each period is monoone increasing in he sock reurn over he period. For example, a rader who holds 100 shares of sock and a ne shor posiion in 00 call opions violaes he monooniciy condiion, while a rader who holds 00 shares of sock and a ne shor posiion in 00 call opions saisfies he condiion. Essenially, we assume ha he raders have a sufficienly large invesmen in he sock, relaive o heir ne shor posiion in call opions (or, ne long posiions in pu opions), such ha he monooniciy condiion is saisfied. We do no make he resricive assumpion ha all marke agens belong o he class of uiliy-maximizing raders. Thus, our resuls are robus and unaffeced by he presence in he marke of agens wih beliefs, endowmens, preferences, rading resricions, and ransacion coss schedules ha differ from hose of he uiliy-maximizing raders modeled in his paper. A rader eners he marke a ime zero wih x 0 dollars in bonds and y 0 dollars in ex dividend shares of sock. We consider wo scenarios. In he firs scenario, he rader may 4

6 rade he bond and sock bu no he opions. The rader makes sequenial invesmen F decisions a discree rading daes ( = 0, 1..., T' ), where T', T' T T, is he finie erminal dae. The rader s objecive is o maximize expeced uiliy, Eu [ T' ( W T' )], where W T' is he rader s ne worh a dae T'. 3 Uiliy is assumed o be concave and increasing and defined for boh posiive and negaive erminal worh, bu is oherwise lef unspecified. We refer o his rader as he index (and bond) rader,, and denoe her maximized expeced uiliy by (, ) V x y In he second scenario, as in he firs scenario, he rader eners he marke a ime zero wih x 0 dollars in bonds and y 0 dollars in ex dividend shares of sock, bu (in addiion o he firs scenario) immediaely wries an American fuures call opion wih mauriy T, T T F, where C are he ne cash proceeds from wriing he call. 4 We assume ha he rader may no rade he call opion hereafer. 5 A each rading dae ( = 0, 1..., T ) he rader is informed wheher or no she has been assigned (ha is, assigned o ac as he counerpary of he holder of a call who exercises he call a ha ime). If he rader has been assigned, he call posiion is closed ou, he rader pays F cash accoun decreases from x o x ( F K) K in cash, and he value of he. The rader makes sequenial invesmen decisions wih he objecive o maximize expeced uiliy, Eu [ T' ( W T' )]. We refer o his rader as he opion (plus index and bond) rader,, and denoe her maximized expeced uiliy by V ( x C, y ) Alernaively, he objecive may be he maximizaion of he discouned sum of he uiliy of consumpion u( c) a each rading dae, including he erminal dae. In his case, he erminal dae may be finie or infinie. Alhough he Consaninides and Perrakis (007) bounds are derived under he erminal wealh objecive, hey remain valid wihou any reformulaion under he alernaive objecive. 4 The reservaion wrie price of a call is derived from he perspecive of a rader who is marginal in he index, he bond, and only one ype of call or pu opion a a ime. Therefore, hese bounds allow for he possibiliy ha he opions marke is segmened. 5 The reservaion wrie price of a call is derived under his consrained policy. Under his policy, he invesor increases her expeced uiliy by wriing a call a price C and refraining from rading he call hereafer. If he consrain on rading he call is relaxed, he policy which he invesor follows under he consrain policy remains feasible and increases her expeced uiliy by wriing a call a price C. Therefore, C remains an upper bound on he reservaion wrie price of a call. Whereas he upper bound may be ighened when he consrain on rading he call is relaxed, here is no known igher bound ha is preference free. For furher discussion on his poin, see Consaninides and Perrakis (007). 5

7 For a given pair( x, y ), we define he reservaion wrie price of a call as he value 0 0 of C such ha V ( x C, y ) V ( x, y ) + =. The inerpreaion of C is he wrie price of he call a which he rader wih iniial endowmen (, ) x y is indifferen beween wriing he call or no. Consaninides and Perrakis (007) sae a igh upper bound on he reservaion wrie price of a European fuures call opion ha is independen of he rader s uiliy funcion and iniial endowmen and independen of he early exercise policy on he calls: C( F, S, ) = k max N( S, ), F K 1 k, T. (1) The funcion NS (, ) is defined as follows: ( ) ( F 1 γ ) F 1 T T 1 (, ) = ( S) [max{ ε, ( + 1, + 1)} = ], 1 NS R E R S KNS S S T = 0, = T. () The economic inerpreaion of he call upper bound is as follows. If we observe a call bid price above he reservaion wrie price, C, hen any rader (as defined in his paper) can increase her expeced uiliy by wriing he call. Transacion coss on he index have only a small effec on he upper bound. Specifically, wihou ransacion coss on he index, he upper bound is ( ) max N S,, F K ; wih ransacion coss on he index, he upper bound merely increases by he muliplicaive facor ( 1 k ) /( 1 k ) +. The explanaion is ha his 1 paricular bound is based on a comparison of he uiliy of an index rader and he uiliy of an opion rader. Boh raders follow he rading policy which is opimal for he index rader bu is generally subopimal for he opion rader. This policy incurs very low ransacion coss because he rader rades infrequenly, as shown in Consaninides (1986). 6

8 If we furher assume ha he rader can buy a call a price (,, ) rade he fuures and do so coslessly, we obain he following pu upper bound: 6 C F S or less and ( ) ( ) ( T ) P F, S, = C F, S, R F + K, T. (3) The inerpreaion of he pu upper bound is as follows. If we observe a pu bid price above he reservaion wrie price P, hen any rader can increase her expeced uiliy by wriing he pu. Consaninides and Perrakis (007) also saed a igh lower bound on he reservaion purchase price of an American fuures pu opion. The cash payoff of he pu exercised a ime is K F, T. As in he case of a call opion, we define he reservaion purchase price of a pu as he value of P such ha he rader wih iniial endowmen (, ) x y is indifferen beween purchasing he pu or no. The following is a igh lower bound on he reservaion purchase price of an American fuures pu opion ha is independen of he rader s uiliy funcion and iniial endowmen: k PF (, S, ) max K- F, MS (, ), T 1+ k. (4) The funcion M ( S, ) is defined as follows: ( ) ( ) ( ) ( F 1 T 1 ) F T 1 M S, = RS E max K 1 γ R S 1 ε, M ( S 1, 1 ) S S =, T 1 = 0, = T. (5) 6 We prove equaion (3) by noing ha an invesor achieves an arbirage profi by buying a call ( T ) a C( F, S, ), wriing a pu a P, P> P( F, S, ), selling one fuures, and lending K R F. In he proof, we ignore he daily marking-o-marke on he fuures unil he exercise of he pu or he opions mauriy, whichever comes firs. 7

9 If we observe a pu ask price below he reservaion purchase price P, hen any rader can increase her expeced uiliy by buying he pu. As in he case of he upper call bound, ransacion coss on he index have only a small effec on he lower pu bound. If we furher assume ha he rader can wrie a pu a price PF (, S, ) or more, and rade he fuures and do so coslessly, hen we obain he following call lower bound, wih corresponding inerpreaion: 7 ( ) ( ) ( T ) C F, S, = P F, S, + R F K, T. (6) If we observe a call ask price below he reservaion purchase price C, hen any rader can increase her expeced uiliy by buying he call. Empirical Design We describe our empirical design, saring wih a descripion of he daa, he calibraion of a ree of he daily index reurn, and he consrucion of he porfolio of he index rader (who does no rade in he opion) and of he opion rader. This allows us o inroduce he wellknown Sharpe raio es and we discuss he problems associaed wih using his es. To address problems wih he Sharpe raio es, we inroduce ess based on second order sochasic dominance..1 Daa and esimaion We obain he ime-samped quoes of he 30-calendar-day S&P 500 fuures opions and he underlying 1-monh fuures for he period February 1983-July 006 from he Chicago Mercanile Exchange (CME) apes. This resuls in 47 sampling daes. We obain he ineres rae as he hree-monh T-bill rae from he Federal Reserve Saisical Release. The daa sources are described in furher deail in Appendix A. 7 We prove equaion (6) by noing ha an invesor achieves an arbirage profi by wriing a pu a P( F, S, ), buying a call a CC, CFS (,, ) <, selling one fuures, and lending ( T ) K R F. 8

10 For he daily index reurn disribuion, we use he hisorical sample of log reurns from January 198 o January However, when looking forward for each of our 47 opion sampling daes, we adjus he firs four momens of he index reurn disribuion in various ways which we now describe in deail. We se he mean index reurn a 4% plus he observed 3-monh T-bill rae insead of esimaing he mean index reurn from he daa in order o miigae saisical problems in esimaing he mean. 8 We implemen his by adding a consan o he observed logarihmic index reurns so ha heir sample mean equals he above arge. We esimae he 3 rd and 4 h momens of he index reurn as heir sample counerpars over he preceding 90 days. Finally, we esimae boh he uncondiional and condiional volailiy of he index reurn as follows. We esimae he uncondiional volailiy as he sample sandard deviaion over he period January 198 o January We esimae he condiional volailiy in hree differen ways: (1) he sample sandard deviaion over he preceding 90 rading days; 10 () he a-he money (ATM) implied volailiy (IV) on he preceding day, adjused by he mean predicion error for all daes preceding he given dae (ypically some 3%), where we drop from he preceding days all 1 pre-crash observaions; and (3) he GARCH volailiy using GARCH coefficiens esimaed for S&P 500 daily reurns over January 198 o January 1983 applied o residuals observed over he 90 days preceding each sample dae o form projecions of he volailiy realized ill he opion expiry dae. In Table 1, we repor saisics of he predicion error of he above volailiy esimaes. The bes overall predicors are he adjused ATM IV and he 90-day hisorical volailiy.. Calibraion of he index reurn ree and calculaion of he opion bounds We model he pah of he daily index reurn ill he opion expiraion on a T-sep ree, where T is he number of rading days in ha paricular monh. 11 The ree is recombining wih m 8 Shor-horizon forecass of he condiional mean equiy premium are nooriously unreliable. Fama and French (00), Consaninides (00), and Dimson, Marsh and Saunon (006) esimaed he adjused uncondiional mean equiy premium o be 4-6% per year. For our main resuls, we se he mean reurn a 4% plus he observed 3-monh T-bill rae. We also repor resuls when we se he mean reurn a 6% plus he observed 3-monh T-bill rae. 9 We have also esimaed he uncondiional volailiy over he 4 years prior o January The resuls remain essenially unchanged and are no repored in he paper. 10 We have also esimaed he condiional volailiy over he preceding 360 days. The resuls remain essenially unchanged and are no repored in he paper. 11 For example, if he 3 rd Friday of July is on July 7, we record he price of he July opion on June 7, 9

11 branches emanaing from each node. Every monh we calibrae he ree by choosing he number of branches and he reurn a each node o mach he firs four momens of he daily index reurn disribuion, as described in Appendix B. The upper and lower bounds on he call and pu prices are given in equaions (1)-(6). We numerically calculae he bounds by ieraing backwards on he calibraed ree..3 Porfolio consrucion and rading For each monhly sock reurn pah, we employ he following rading policies. For he index rader (who manages a porfolio of he index and he risk free asse in he presence of ransacion coss), we employ he opimal rading policy, as derived in Consaninides (1986) and exended in Perrakis and Czerwonko (007) o allow for dividend yield on he sock. Essenially, his policy consiss of rading only o confine he raio of he index value o he bond value, y / x, wihin a no-ransacions region, defined by lower and upper boundaries. We derive hese boundaries for he following parameer values: one-way ransacion cos rae on he index of 0.5%; annual reurn volailiy of he index of , he sample volailiy over ; ineres rae equal o he observed 3-monh T-bill dae; risk premium 4%; and consan relaive risk aversion coefficien of. 1 For his se of parameers, he lower and upper boundaries are y 0 / x 0 = 1.06 and 1.559, respecively. A he beginning of each monh and before he rader rades in opions, we se x 0 = 73,300 and y 0 = 100,000, which corresponds o he midpoin of he no-ransacions region, y0/ x 0 = For he opion rader (who manages a porfolio of he opion, index, and he risk free asse in he presence of ransacion coss), we employ he rading policy which is opimal for he index rader bu is generally subopimal for he opion rader. Recall ha he goal is o demonsrae ha here exis profiable invesmen opporuniies for he opion rader. Given his goal, i suffices o show ha here exis profiable invesmen opporuniies for he which is 30 calendar days earlier. (If June 7 is a holiday, we record he price on June 6.) If here are 1 rading days beween June 7 and July 7, we model he pah of he daily index reurn ill he opion expiraion on a 1-sep ree. 1 We clarify ha he upper and lower sochasic dominance bounds on opion prices apply o any risk averse rader, independen of her paricular degree of risk aversion. In our empirical work, we make an assumpion abou he relaive risk aversion coefficien in order o calculae he boundaries of he no- 10

12 opion rader even if he opion rader follows a generally subopimal policy. We se x 0 and y 0 o he same values as for he index rader. However, his porfolio composiion changes, depending on he assumed posiion in fuures opions, as explained in Appendix C. We focus on he cases where he basis risk bound, ε, is 0.5% of he index price. Over he years , 95% of all observaions have basis risk less han 0.5% of he index price. For reference purposes, we also consider he case ε = 0. As o be expeced, when we suppress he basis risk, he bounds are igher and here appear o be more violaions..4 Descripion of he empirical ess For each one of our mehods of esimaing he bounds, we obain 47 monhly porfolio reurns for he index rader and he opion rader, respecively. Our goal is o es wheher he porfolio profiabiliy of he index and opion raders are saisically differen in he monhs in which we observe violaions of he bounds. In our firs se of ess, we compare he Sharpe raios of he wo porfolios. Despie he well-known limiaions of he Sharpe raio, we repor hese resuls because he Sharpe raio is one of he mos popular measures of porfolio performance. 13 We use he approach of Jobson and Korkie (1981) wih he Memmel (003) correcion ha accouns for differen variances of he wo porfolios. Deails of he es are described in Appendix D. In our second se of ess, we compare he reurns of he wo porfolios in erms of he crierion of sochasic dominance, which saes ha he dominaing porfolio is preferred by any risk-averse rader, independen of disribuional assumpions such as normaliy and preference assumpions such as quadraic uiliy. Specifically, we es he null hypohesis H / 0 :, which saes he opion rader s porfolio reurn does no sochasically dominae he index rader s porfolio reurn, agains he alernaive hypohesis H :, which saes he opion rader s porfolio reurn sochasically dominaes he A ransacions region for a specific rader. We presen resuls for relaive risk aversion and The Sharpe raio ignores momens of he reurn disribuion beyond he mean and variance and his is heoreically jusified only he special cases where eiher invesors have quadraic uiliy or he porfolio reurns are normally disribued. The laer assumpion is obviously violaed in porfolios ha include opions. 11

13 index rader s porfolio reurn. We repor he resuls of ess proposed by Davidson and Duclos (006), using he algorihm developed by Davidson (007). An earlier es, proposed by Davidson and Duclos (000), ess he null hypohesis H agains he alernaive, which is ha eiher or ha neiher one of 0 : he wo disribuions dominaes he oher. Hence, rejecion of he null hypohesis fails o rank he wo disribuions in he absence of informaion on he power of he es, which is generally no available. We repor resuls of his es as well because i has cerain saisical advanages over he Davidson and Duclos (006) es. Appendix D provides deails for boh ess. 3 Empirical Resuls In Secion 3.1, we describe he empirical resuls. We compare he porfolio reurn of an opion rader who wries overpriced calls, pus, or sraddles a heir bid price wih he porfolio reurn of an index rader who does no rade in he opions over he period In ou-of-sample ess, we find ha he reurn of an opion wrier sochasically dominaes he index rader s reurn, ne of ransacion coss and he bid-ask spread. We also find ha he Sharpe raio of he opion rader s reurn is higher han he Sharpe raio of he index rader s reurn and he difference is ofen saisically significan. In Secion 3., we esablish ha he empirical resuls are robus. In Secion 3.3, we demonsrae ha rading policies riggered by violaions of he sochasic dominance bounds consisenly ouperform naïve filer rules of buying low and selling high. 3.1 Resuls In Figure 1, we plo he four bounds for one-monh opions, expressed in erms of he implied volailiy, as a funcion of he moneyness, K/ F 0. We se σ = 0% and ε = 0. The figure also displays he 95% confidence inerval, derived by boosrapping he 90-day disribuion. Regarding he upper bounds, we observe ha he call upper bound is igher han he pu upper bound. Also, he call and pu upper bounds are igher when he (K/F) raio is high, ha is when he calls are M or he pus are M. Regarding he lower 1

14 bounds, we observe ha he pu lower bound is igher han he call lower bound. Also, he call and pu lower bounds are igher when he (K/F) raio is low, ha is when he calls are M or he pus are M. In Figure, we display he ime paern of acual violaions of he call upper bound. For all differen ways of esimaing volailiy, we observe violaions afer significan down moves in he index, i.e., when we expec he implied volailiy of raded opions o be high. We do no presen ime paerns for he remaining hree bounds since we do no observe many violaions of hese bounds. In Table, we presen he cases of call and pu bid prices violaing heir upper bound, when we se he basis risk bound a 0.5% of he index price. We do no presen he cases of call and pu ask prices violaing heir lower bound because we do no have a sufficien number of such violaions o be able o draw saisical inference, as we observed in Figure. We find a higher frequency of violaions of he upper call bound han of he upper pu bound since he upper call bound is igher han he upper pu bound, as we observed in Figure 1. The Sharpe raio of he call rader s reurn is uniformly higher han he Sharpe raio of he index rader s reurn, irrespecive of he mode of predicing he volailiy as an inpu o he call upper bound. When he call upper bound is calculaed using he adjused IV or he GARCH volailiy, he difference in Sharpe raios exceeds 9% annually and is saisically significan a he 10% level. There are far fewer violaions of he pu upper bound and, herefore, he resuls are saisically weaker. Neverheless, when using he uncondiional predicion of volailiy as an inpu o he pu upper bound, we find 3 violaions of he pu upper bound and he pu rader s porfolio has a Sharpe raio ha exceeds he index rader s porfolio by 1.8%, saisically significan a he 10% level. These Sharpe raio preliminary resuls moivae and reinforce our main resuls on sochasic dominance which are discussed nex. The DD (000) es does no rejec he hypohesis H0:, which saes ha he opion rader s reurn dominaes he index rader s reurn; and rejecs he hypohesis H :, which saes ha he index rader s reurn dominaes he opion rader s reurn. Thus, we unforunaely canno decide beween dominance of he sraegy or a ie where we canno esablish dominance one way or anoher. Luckily, he DD (006) es 13

15 allows us o make sronger claims of dominance. The DD (006) es srongly rejecs he null hypohesis H0: /, which saes ha eiher he index rader s reurn dominaes he opion rader s reurn or ha neiher disribuion dominaes he oher. The p-values of he hypohesis H0: / are equal o one and are no repored here. Here, we can uniquely esablish dominance of he sraegy. Nex, we explore he performance of he policy of wriing overpriced calls hrough he policy of wriing sraddles. Sraddles are popular rading policies and have been previously invesigaed in he lieraure. For example, Coval and Shumway (001) show ha a long ATM sraddle on he S&P 500 index or he S&P 100 index produces subsanial negaive reurns. 14 Each monh, we look for call bid prices ha lie above he upper call bound. If we find a leas one call bid prices ha lie above he upper call bound and if we find a leas one pu bid price (irrespecive of wheher he pu bid price violaes he pu upper bound or no) we proceed as follows. We shor equal fracions of he calls ha violae he call upper bound, such ha he fracions add up o one; we shor equal fracions of he pus for which we have bid prices, such ha he fracions add up o one; and we sell one fuures on he index. The resuls are repored in Table 3, panel A. The annualized Sharpe raio differenials are large and significan a he 5% or 1% level. These resuls are consisen wih he resuls of Coval and Shumway (001). The DD (000) es does no rejec he hypohesis H0:. I ofen rejecs he hypohesis H 0 :, bu no consisenly so. Finally, he DD (006) es srongly rejecs he hypohesis H0: /. We conclude ha he resuls in Table 3, panel A, are consisen wih hose in Table. We noe ha he numbers of cross-secions for which shor sraddles are raded in Table 3, panel A, is significanly lower han he corresponding numbers for calls is Table. Since, in our approach, he sraddle sales are solely deermined by he violaions of he call upper bound, we increase he number of cross-secions by relaxing he requiremen ha he pu sale has o occur a he same srike price as ha of he call ha riggers he violaion. Insead, we require ha he moneyness of he pu remains wihin imes he moneyness of he riggering call. The firs pu quoe wihin his bound following he call 14 The paper of Coval and Shumway (001)) focuses on he relaion beween he CAPM bea and he reurn of sraddles and, as such, differs from our goal of measuring he performance of sraddles, ne of bid-ask spreads and hrough he broader crierion of sochasic dominance. 14

16 violaions is included in he ensuing sraddle posiion. The resuls for his approach are presened in Table 3, panel B. Compared o Table 3, panel A, we observe an improvemen in he sochasic dominance ess resuls and a sysemaic increase in he Sharpe raios. Since he resuls for sraddles imply ha he call upper bound is an efficien selecor of overpriced pus, we apply his selecion crierion o all available pu quoes. Specifically, for every pu bid we derive he call upper bound. Then we sell he pu if is implied volailiy exceeds he corresponding quaniy for he call upper bound. The resuls are repored in Table, panel C. The reurns of he pu selling policy sochasically dominae he reurns of he index rader s porfolio, irrespecive of he way in which volailiy is esimaed. The pu selling policies produce Sharpe raios ha are higher han he Sharpe raios of he index rader s porfolio reurns, irrespecive of he way in which he volailiy is esimaed; he Sharpe raio differences are saisically significan when he volailiy is esimaed as he adjused IV or as he GARCH volailiy. 3. Robusness ess In Tables 4-9, we demonsrae ha he resuls of Tables and 3 are robus. Table 4 differs from Table only in ha he basis risk is se a zero, ε = 0, insead of bounding he basis risk by ε = 0.5. There are now more opions across he board violaing he bounds because all he bounds become igher: he upper bounds are lowered and he lower bounds are raised. We presen he cases of call and pu bid prices violaing heir upper bound. We do no presen resuls for he cases when he call and pu ask prices violae heir lower bound because we sill do no have a sufficien number of such violaions o be able o make saisical inference. Since he upper call and pu bounds are lower, he opions rader is less selecive han before in wriing opions ha violae heir upper bounds and we find ha he differences of he Sharpe raios are smaller in Table 4 han in Table. However, since here are more observaions in Table 4, he differences of he Sharpe raios are saisically more significan han in Table. The DD (000) es does no rejec he hypohesis H0: and rejecs he hypohesis H 0 :. Finally, he DD (006) es 15

17 srongly rejecs he hypohesis H0: /. We conclude ha he resuls in Table 3 are consisen wih hose in Table. Table 5 differs from Table 3 on sraddles only in ha he basis risk is se a zero, ε = 0, insead of bounding he basis risk by ε = 0.5. Again, we conclude ha he resuls in Table 5 are consisen wih hose in Table 3. Table 6 differs from Table only in ha he relaive risk aversion coefficien is se a 10 insead of. Since he upper and lower sochasic dominance bounds on opion prices are independen of he rader s uiliy, we observe he same number of violaions in Table 6 as we do in Table. The change in he risk aversion coefficien does change he boundaries of he no-ransacions region and, herefore, he rading policy of he index rader and he opion rader. The Sharpe raio differences are subsanially higher in Table 6 bu hese differences are no saisically significan. (Recall ha he differences in Panels A and B of Table are only marginally significan). The sochasic dominance resuls in wriing calls are as srong in wriing calls and sronger in wriing pus. Table 7 differs from Table only in ha he expeced premium on he index is se a 6% insead of 4%. The differences in Sharpe raios are comparable o hose in Table bu hese differences are no saisically significan. The sochasic dominance resuls in wriing calls are as srong in wriing calls and sronger in wriing pus. We conclude ha he resuls in Table are robus o he assumpion ha he expeced premium on he index is 4%. Table 8 differs from Table only in ha we exclude from he sample he seven monh from Ocober 1987 o April 1988 in order o absrac from effecs associaed wih he crash. The difference in Sharpe raios beween he reurns of he opion rader and index rader are comparable o hose in Table, bu are no saisically significan. The sochasic dominance resuls in wriing calls and pus are he same as in Table. This is parly due o he fac ha sock prices recovered on he days following he crash and he Ocober reurn on he index is fla. The bounds ha are used in idenifying mispriced opions in our empirical work are calculaed wih parameer inpus which are poin esimaes and vary for each ime poin of our sample for all bu he hisorical mehod of esimaing he bounds. These varying parameers imply ha he screening rules for mispriced opions become condiional on he ime poin of our sample. Since he earlier ess do no recognize his condiionaliy, we 16

18 develop in Appendix E an alernaive se of ess ha explicily ake ino accoun he ime varying naure of our sample and conclude ha condiional and uncondiional ess lead o same conclusions. The resuls are repored in Table 9 and discussed in Appendix E. They are also consisen wih he main resuls of Table and supporive of he mispricing hypohesis, even hough hey are derived wih a differen mehod. 3.3 Comparison wih naïve rading The opion rading policies riggered by violaions of he sochasic dominance bounds superficially resemble a well-known naïve rading policy of buying (or selling) an opion when is IV is a he low (or, high) end of he IV disribuion of opions wihin a cerain range of moneyness. In his secion, we demonsrae ha a paricular form of his naïve rading policy consisenly underperforms he earlier rading policies riggered by violaions of he sochasic dominance bounds. We derive he 90 h, 97.5 h, 10 h, and.5 h perceniles of he IV disribuion for a given range of moneyness by applying he mehod of Yu and Jones (1998). 15 Table 10 presens he resuls for he naïve rading policy. In all rading policies, we mirrored he policies applied for he opion rader (). We observe ha he number of cross-secions for which we find quanile violaions is relaively low. This observaion is caused by he clusering of violaions in some cross-secions. We conclude ha he naïve rading policy deecs parallel shifs in he implied volailiy insead of singling ou unusual observaions in he majoriy of cross-secions. These shifs appear o be inefficien: he naïve bounds only capure some of hese parallel shifs in implied volailiy, namely violaions a he op. The naïve rading policy performs well on he sell side bu performs disasrously on he buy side, as shown by he sochasic dominance saisics and Sharpe raios. 15 The quanile regression is a kernel regression in wo dimensions, in our case in he dimensions of moneyness and IV. As is usual in a kernel regression, he criical par is in deermining he kernel bandwidh. To deermine his quaniy in he moneyness dimension, we use he Leave-One-Ou mehod, as described in Härdle (1990), for which we use he ransformaion given in Table 1 in Yu and Jones (1998). To deermine he bandwidh in he implied volailiy dimension, we use (1) in Yu and Jones (1998). As our sample o derive he criical quanile funcion, we use five pas observaions wih 30 days o mauriy. Using en pas observaions yields similar resuls, no repored here. We verified ha in-sample he min q, 1 q. likelihood of observaions ouside any criical quanile q is close o ( ) 17

19 4 Concluding Remarks We search for mispriced American call and pu opions on he S&P 500 index fuures by employing sochasic dominance upper and lower bounds on he prices of opions. We idenify call and pu bid prices on index fuures ha violae he upper bounds and call and pu ask prices ha violae he lower bounds. We find a subsanial number of violaions of he upper bounds, bu relaively few violaions of he lower bounds. Since he frequency of violaions of he lower bounds is oo low for saisical inference, we focus on violaions of he upper bounds. We compare he porfolio reurn of an opion rader who wries overpriced calls or pus a heir bid price wih he porfolio reurn of an index rader who does no rade in he opions over he period In ou-of-sample ess, our main resul is ha he reurn of a call or pu wrier sochasically dominaes (in second order) he index rader s reurn, ne of ransacion coss and he bid-ask spread. The dominance holds under a variey of mehods in esimaing he underlying reurn disribuion. I also holds wih or wihou he assumpion ha he porfolio reurns are drawn from he same disribuion each period. We also find ha he Sharpe raio of he call rader s reurn is uniformly higher han he Sharpe raio of he index rader s reurn and is ofen saisically significan. The Sharpe raio of he pu rader s reurn is uniformly higher han he Sharpe raio of he index rader s reurn bu he resuls are less saisically significan. Finally, he policy of wriing sraddles produces reurns ha srongly sochasically dominae he index rader s reurn and have subsanially higher Sharpe raios. The resuls are supporive of he hypohesis ha he opions idenified by violaions of he CP (007) bounds are mispriced. 18

20 Appendix A: Daa We obain he ime-samped quoes of he one-monh S&P 500 fuures opions and he underlying one-monh fuures for he period February 1983-July 006 from he CME apes. From he fuures prices, we calculae he implied S&P 500 index prices by applying he cos-of-carry relaion ( ) ( F γ ) T F T F = 1+ R S + ε, assuming away basis risk, ε We obain he daily dividend record of he S&P 500 index over he period from he S&P 500 Informaion Bullein and conver i o a consan dividend yield for each 30-day period. Before April 198, dividends are esimaed from monhly dividend yields. We obain he ineres rae as he hree-monh T-bill rae from he Federal Reserve Saisical Release. We esimae he variance of he basis risk, var( ε ), from he observed fuures prices and he inraday ime-samped S&P 500 record obained from he CME. We rescale he index price S by he muliplicaive facor 100,000/ S 0 so ha he index price a he beginning of each 30-day period is 100,000. Accordingly, we rescale he fuures price, index fuures opion price, and srike by he same muliplicaive facor. We consider opions mauring in 30 calendar days, which resuls in 47 sampling daes. 17 Since he firs mauriy of serial opions was in Augus 1987, he firs 19 periods occur wih quarerly periodiciy. Overall, we record 36,91 raw call quoes and 4,881 raw pu quoes. Afer eliminaing obvious daa errors, we apply he following filers: minimum 15 cens for a bid quoe and 5 cens for an ask quoe; K/F raio wihin for calls and wihin for pus; and maching he underlying fuures quoe wihin 15 seconds. Par of he daa is los due o he CME rule of flagging quoes, i.e. bids (asks) are flagged only if a bid (ask) is higher (lower) han he preceding bid (ask); in addiion, no ransacion daa is flagged. We recover a large par of he daa by analyzing he sequence beween consecuive bid-ask flags; however, his recovery is no possible in all cases. As a resul of he applied filers, we obain 9,8 quoes for calls and 30,81 quoes for pus in our final 16 Recall ha our goal is o compare he invesmen policies, of he index rader and he opion rader. Since boh policies sipulae approximaely he same sock componen, he effec of his componen cancels each oher ou. Also, i is a common empirical approach o derive he index value from he index fuures; see, for example, Jackwerh and Rubinsein (1996). 17 The 30-day rule eliminaes he occurrence of he Ocober crash from our sample. Therefore, we use one 40-day period o have he crash (he 48 h observaion) and verified ha he inclusion of he crash does no aler our resuls. 19

21 sample. These quaniies ranslae ino roughly 60 daa poins for all srikes for eiher bid or ask prices for an average day. Appendix B: Calibraion of he index reurn ree We model he pahs of he daily index reurn on a recombining ree wih m branches emanaing from each node. Every monh we calibrae he ree, including m, o mach he nd, 3 rd and 4 h momens of he daily index reurn disribuion. For he 3 rd and 4 h momens, we always use he observed sample momens over he 90 preceding calendar days. In he firs sep of our algorihm, we pick a value for he number of branches m and group he sample of daily reurns in a hisogram wih m bins of equal lengh (on he log scale) such ha he exreme bins are cenered on he exreme observed reurns. The cener of each bin hen becomes a sae in he equally spaced laice, wih he ordered saes and he corresponding probabiliies denoed respecively as x i and p i, i = 1... m. In he second sep, we impose he desired firs hree momens by alering he laice from he sep one. We derive he adjusmens by solving he following se of hree non-linear equaions ha are simply hree momen condiions for he quaniies a, b, and c: m i= 1 p exp ax + b exp = 0 i ( ) ( μ ) i m i= 1 pi exp( axi + b) exp( μ) σ = 0 (B.1) m i= exp( ) ( ) ˆ i + exp μ μ3σ = 0 pi ax b where exp( μ ) and σ are he firs and second arge momens, respecively, ˆμ 3 is he sample skewness, and p i p 1 i + c ( i n ) 1( pi 0) p + 1 i c ( i n ) 1( pi 0) m i= 1 ( ), where 1 (.) is he indicaor funcion, n is he index o his x i which brackes from above he arge expeced log-reurn μ. The 0

22 firs indicaor funcion ensures ha he consan c is added only o he probabiliies in he righ ail of he disribuion; he second one ensures ha he consan c is added only o he posiive probabiliies. 18, 19 Noe ha he affine ransformaion of he log-saes x i preserves he equal disance beween he adjacen saes, which is necessary for he laice o recombine. To mach he fourh sample momen ˆμ 4, we resor o varying m, he number of nodes in he laice. Wih each new m he iniial disribuion derived from a hisogram changes providing some variabiliy in he fourh momen afer he adjusmens resuling from solving (B.1). Afer a search over a range of m s, we pick his disribuion which has he lowes absolue difference beween is kurosis and he sample kurosis ˆμ 4. I urns ou ha his search procedure ends up wih accepably small errors in maching ˆμ 4 for he daa ha we use. For he four volailiy predicion modes we apply in our work, he relaive error on he fourh momen had he following characerisics: median 0.003%, 99 h percenile 0.105%, maximum 1.659% across 973 observaions while we consrained he laice size m o be no larger han Insead of using a hisogram in he firs sep above, we could sar building our laice by discreizing a kernel-smoohed disribuion. However, since he kernel smoohing would involve more parameer choices and would resul in a significanly larger laice size o aain accuracy similar o he one of our mehod, we reain a preference for he hisogram approximaion By grouping he observaion in he hisogram as a rule we end up wih saes in our laice ha have zero probabiliies. We don invesigae here he precise recombinaion paern of a laice wih zero-probabiliy saes; we observe, however, ha he number of zero-probabiliy saes remains relaively consan as he number of convoluions of he laice wih iself increases, resuling in a decreasing proporion of such saes as he ime period increases. 19 Noe ha he presened adjusmen of he probabiliies in he righ ail may no yield an admissible soluion, i.e. we may end up wih some negaive probabiliies. If his is he case, we inroduce an analogous adjusmen in he lef ail of he disribuion. 0 This laice size appears unaracive o derive recursive condiional expecaions. However, he use of fas Fourier ransforms resuls in a fairly shor processing ime. See Cerny (004). 1 A criical parameer in he kernel densiy esimaion is he kernel bandwidh. In addiion, since he densiy esimae of he log-reurns covers he real line, he scope of he discreized disribuion would need o be chosen. 1

23 Appendix C: Trading policy We consider calls wih moneyness (K/S) wihin he range and pus wihin he range If we observe n call bid prices violaing he call upper bound, each wih differen srike price, he opion rader wries 1/n calls of each ype wih he underlying fuures corresponding o he index value of y 0. The rader ransfers he proceeds o he bond accoun: x = x0 + C / 1 i n and y = y0. n i= If we observe n pu ask prices violaing he pu lower bound, each wih differen srike price, he opion rader buys 1/n pus of each ype and finances he purchase ou of he bond accoun: x = x0 P/ 1 i n and y = y0. n i= However, when here is a violaion of he upper pu bound and he opion rader wries pus, he rader also sells one fuures conrac for each wrien pu. The inuiion for his policy may be gleaned from he observaion ha he combinaion of a wrien pu and a shor fuures amouns o a synheic shor call. In fac, he upper pu bound in equaion (3) is derived from he upper call bound in equaion () hrough he observaion ha if we can wrie a pu a a sufficienly high price we violae he upper call bound by wriing a synheic call. Finally, when here is a violaion of he lower call bound and he opion rader buys calls, he rader also sells one fuures conrac for each purchased call. The inuiion is he same as above. The early exercise policy of a call is based on he funcion N in equaion (). The early exercise policy of a pu is based on he funcion M in equaion (5). However, whenever he opion rader is shor an opion, each period we derive he funcions N and M based on he forward-looking disribuion of daily reurns, i.e. hese funcions are derived under he empirical disribuion of he daily index reurns beween he opion rade and he opion mauriy. Effecively, we endow he counerpary of he opion rader wih informaion on he nd, 3 rd, and 4 h momens of he forward disribuion, while imposing he In implemening he rading policy of eiher wriing pus or buying calls, he opion rader buys or sells a fuures conrac as well and his violaes he assumpion made in Secion 1 ha he opion rader does no rade in fuures. Even when we relax he assumpion on rading in fuures, in pracice, raders manage heir porfolio by rading in he index because of he inconvenience and cos of he frequen rolling over of shorerm fuures conracs and he illiquidiy of long-erm fuures and forward conracs.

24 firs momen. The early exercise policy of a call or pu is simplified by he observaion ha he decision is a funcion only of ime and he raio of he srike price o he index level. Appendix D: The Sharpe raio and he Davidson-Duclos (000, 006) ess For he Sharpe raio ess, we use he approach of Jobson and Korkie (1981) wih he Memmel (003) correcion ha accouns for differen variances of he wo porfolios. Specifically, given he sample of N realizaions of he index rader s () and opion rader s () porfolio oucomes wih ˆ μ, ˆ μ, ˆ σ, ˆ ˆ σ, σ, as heir esimaed excess means, variances, and covariances, we es he hypohesis H ˆ ˆ ˆ ˆ 0 : μ σ μ σ 0 wih he es saisic ẑ, which is asympoically sandard normal: where ˆ μ ˆ ˆ ˆ σ μσ zˆ = (D.1) ˆ θ ˆ ˆ μ ˆ μ θ = ˆ σ ˆ σ ˆ σ ˆ σ ˆ σ + ˆ μ ˆ σ + ˆ μ ˆ σ ˆ σ N,, ˆ σ ˆ σ. (D.) DD (000) provide a es of he null hypohesis H : 0 in erms of he maximal and minimal values of he exremal es saisic, T ( z ). The null is no rejeced, if he maximal value of he saisic is posiive and saisically significan and he minimal value of he saisic is eiher posiive or negaive and saisically no significan. The variable z denoes he logarihm of end-of-he-monh wealh of a rader, where he subscrips and disinguish beween he index rader and he opion rader. The saisic T ( z ) is defined as follows: Dˆ ( z) Dˆ ( z) = (D.3) V z ˆ( ) ˆ ( ) T z where 3