Does Option Trading Have a Pervasive Impact on Underlying Stock Prices? *

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1 Does Opion Trading Have a Pervasive Impac on Underlying Sock Prices? * Neil D. Pearson Universiy of Illinois a Urbana-Champaign Allen M. Poeshman Universiy of Illinois a Urbana-Champaign Joshua Whie Universiy of Illinois a Urbana-Champaign February 23, 2007 * Deparmen of Finance, College of Business, Universiy of Illinois a Urbana-Champaign, 340 Wohlers Hall, 1206 Souh Sixh Sree, Champaign, IL Phone: 217) , phone: 217) , and phone: 217) , We hank Joe Levin, Eileen Smih, and Dick Thaler for assisance wih he daa used in his paper. We hank Michael Barclay, Sewar Mayhew, and seminar paricipans a Case-Wesern, Florida, Illinois, Oklahoma, Rocheser, and he Uah Winer Finance Conference for heir helpful commens. Qian Deng provided excellen research assisance. We bear full responsibiliy for any remaining errors.

2 Does Opion Trading Have a Pervasive Impac on Underlying Sock Prices? ABSTRACT The quesion of wheher and o wha exen opion rading impacs underlying sock prices has been a focus of inense ineres since opions began exchange-based rading in Despie considerable effor, no convincing evidence for a pervasive impac has been produced. A recen srand of heoreical lieraure predics ha rebalancing by raders who hedge heir opion posiions increases decreases) underlying sock reurn volailiy when hese raders have ne wrien purchased) opion posiions. This paper ess his predicion and finds a saisically and economically significan negaive relaionship beween sock reurn volailiy and ne purchased opion posiions of invesors who are likely o hedge. Hence, we provide he firs evidence for a subsanial and pervasive influence of opion rading on sock prices.

3 1. Inroducion Ever since individual equiy opions began rading in 1973, invesors, exchange officials, and regulaors have been concerned ha underlying sock prices would be impaced. 1 Despie a subsanial effor o idenify such impac and he exisence of a srand of heoreical lieraure modeling he effecs of opion hedge rebalancing on underlying sock prices, lile evidence has been produced ha opion rading influences he prices of underlying socks. 2 Indeed, he only convincing evidence ha opion aciviy alers underlying socks involves sock price deviaions righ a opion expiraion. The presen paper invesigaes wheher opion marke aciviy has a subsanially more pervasive influence on underlying sock prices. A firs srand of research on he impac of opion rading on underlying socks examines wheher opion inroducion generaes a one-ime change in sock price level. Earlier papers by Conrad 1989) and Deemple and Jorion 1990) indicae ha opion inroducion produces an increase in he level of underlying sock prices. These findings, however, do no appear o be robus. Sorescu 2002) and Ho and Liu 1997) show ha in a laer ime period sock prices decrease upon opion inroducion, and Mayhew and Mihov 2004) find ha he price level effecs disappear when benchmarked agains he price changes of mached firms ha do no have opions inroduced. Mos recenly, Lundsrum and Walker 2006) provide evidence ha he inroducions of LEAPS are associaed wih small declines in he prices of underlying socks. A second srand of research invesigaes wheher opion aciviy causes sysemaic changes in he prices of he underlying socks a opion expiraion daes. An early CBOE 1976) 1 See Whaley 2003) for an accoun of he early period of exchange raded opions. 2 A separae srand of lieraure examines he impac of fuures rading on he volailiy of underlying sock indices e.g., Bessembinder and Seguin 1992) and Gulen and Mayhew 2000)), especially a index expiraion e.g., Barclay, Hendersho, and Jones 2006) and he references herein). Researchers have also considered he possible impac of morgage hedging and OTC derivaives dealers hedging on ineres raes Perli and Sack 2003), Chang, McManus, and Ramagopal 2005), Kambhu 1998), and Kambhu and Mosser 2004)). 1

4 repor does no find evidence of abnormal underlying sock price behavior leading up o opion expiraion. Using small samples, Klemkosky 1978) documens negaive reurns on underlying socks in he week leading up o expiraion and posiive reurns in he week afer expiraion while Cinar and Vu 1987) find ha he average reurn and volailiy of opioned socks on he Thursday o Friday of expiraion week are largely he same as from he Thursday o Friday of non-expiraion weeks. Ni, Pearson, and Poeshman 2005), on he oher hand, provide srong evidence ha he prices of opioned socks cluser a srike prices and herefore are alered on opion expiraion daes. A final srand of research on he impac of individual equiy opions examines wheher opions produce pervasive changes in underlying sock price movemens changes no limied o he imes ha opions are inroduced or expire. Bansal, Prui, and Wei 1989), Conrad 1989), and Skinner 1989) all find ha being opioned yields a decrease in he volailiy of underlying sock prices. However, Lamoureux and Panikkah 1994), Freund, McCann, and Webb 1994), and Bollen 1998) demonsrae ha he apparen decrease in volailiy is probably rooed in he fac ha exchanges end o inroduce opions afer increases in volailiy. In paricular, hey show ha he decrease in volailiy ha occurs afer opion inroducion is also observed in samples of mached conrol firms ha lack opion inroducion. All in all, he lieraure conains lile evidence ha opion rading has a significan impac on underlying sock prices. The only compelling evidence ha sock prices are alered is limied o expiraion daes. Specifically, Ni, Pearson, and Poeshman 2005) documen ha he prices of opionable socks i.e., socks wih exchange-raded opions) cluser a opion srike prices on opion expiraion daes, and show ha sock rading underaken by opion marke paricipans in order o keep heir porfolios hedged as he delas of heir expiring opion posiions change 2

5 rapidly as he remaining ime o expiraion shrinks o zero is a major driver of his sock price clusering. 3 Avellaneda and Lipkin 2003) model his mechanism, focusing on he role of he ime derivaives of opion delas. These are large for opions ha are near he money and close o expiraion, and have signs depending upon wheher he opions are purchased or wrien. Due o hese ime derivaives, as ime passes dela-hedgers who have ne purchased wrien) opion posiions will sell buy) sock when he sock price is above he opion srike price and buy sell) sock when he price is below he srike price, ending o drive he sock price oward he opion srike price. As documened by Ni, Pearson, and Poeshman 2005), his causes clusering declusering) a opion expiraion when dela-hedging opion marke paricipans have ne purchased wrien) posiions in opions on an underlying sock. The finding ha re-hedging of opion posiions jus before expiraion produces measurable deviaions in sock price pahs leads naurally o he quesion of wheher re-hedging away from expiraion also changes sock price movemens. In he heoreical lieraure, Jarrow 1994), Frey and Sremme 1997), Frey 1998), Plaen and Schweizer 1998), Sircar and Papanicolaou 1998), Frey 2000), and Schönbucher and Wilmo 2000) model he effec of he dela-hedging of derivaive posiions on underlying asses ha are no perfecly liquid. The key resul in his lieraure is ha dynamic rading sraegies ha replicae purchased opion posiions i.e., posiions ha have convex payoffs) involve buying he underlying asse afer is price has increased and selling i afer is price has decreased. This paern of buying and selling causes he underlying asse o be more volaile han i oherwise would have been and may even exacerbae large movemens in he price of he underlying asse. The models also imply ha dynamic rading sraegies ha replicae wrien opion posiions i.e., posiions ha have 3 The dela of an equiy opion is he change in is value per uni increase in he value of he underlying sock. 3

6 concave payoffs) will cause volailiy o be lower han i oherwise would have been. The gamma of an opion is is change in dela per uni increase in he underlying asse, and he gamma of purchased wrien) opions is posiive negaive). The specific predicion of he heoreical models is formulaed in erms of opion gamma. In paricular, he models predic ha when he gamma of he ne opion posiion on an underlying sock of dela-hedging invesors is posiive negaive), hedge re-balancing will reduce increase) he volailiy of he sock. This predicion has no ye been empirically esed. 4 We invesigae wheher he ne gamma of dela-hedging invesors is indeed negaively relaed o he volailiy of he underlying sock by using a daase ha allows us o compue on a daily basis for each underlying sock he gamma of he ne opion posiion of likely dela hedgers. We indeed find a highly significan negaive relaionship beween he gamma of he ne opion posiion of likely dela-hedgers and he absolue reurn of he underlying sock. The finding is robus o conrolling for persisence in sock volailiy and also for he possibiliy ha he opion posiions of likely dela-hedgers are changed as he resul of invesors rading opions o profi from informaion abou he fuure volailiy of underlying socks. In addiion, he finding is presen for large and small underlying socks, in he firs and second half of our sample period, when we define likely dela hedgers o include firm proprieary raders as well as marke makers, and when we exclude he week of opion expiraion from our analysis. Hence, we provide evidence ha opion marke aciviy has a pervasive impac on he price pahs of underlying socks. In paricular, he impac is no limied o imes very close o opion expiraion. 4 Cein, Jarrow, Proer, and Warachka 2006) carry ou empirical work examining he effecs of sock illiquidiy on opion prices for five differen socks, bu do no address he impac of opion hedging on sock prices. 4

7 Furhermore, he effec is economically significan. The average daily absolue reurn of he socks in our sample is 310 basis poins and a one sandard deviaion shock o he gamma of he ne opion posiion variable is associaed wih a 37 basis poin change in absolue reurn. Consequenly, we esimae ha on he order of 12 percen =37/310) of he daily absolue reurn of opioned socks can be accouned for by opion marke paricipans re-balancing he hedges of heir opion posiions. We also show ha he effec is no resriced o small or medium size absolue reurns. Examinaion of large absolue daily reurns reveals ha a one sandard deviaion shock o he gamma of he ne opion posiions changes he probabiliy of a daily absolue reurn greaer han ) basis poins by ) percen. Our resuls shed ligh on he lieraure ha invesigaes wheher opion inroducion i.e., he exisence of opion rading) leads o an overall increase or decrease in he variabiliy of underlying socks. As noed above, his lieraure finds ha wih proper benchmarking no overall increase or decrease in volailiy is deecable. We show, by conras, ha volailiy increases or decreases depending upon he sign of he ne gamma of dela-hedging invesors. Consequenly, even hough opion rading changes he variabiliy of underlying sock reurns, i is no surprising ha here is no evidence of an uncondiional increase or decrease of volailiy associaed wih opion rading. The remainder of he paper is organized as follows. Secion 2 develops our empirical predicions. The hird secion describes he daa. Secion 4 presens he resuls, and Secion 5 briefly concludes. 5

8 2. Empirical Predicions Dynamic rading sraegies ha involve replicaing or dela-hedging opions require buying or selling he underlying asse as he dela of he opion or opions porfolio changes. Unless he underlying asse is raded in a perfecly liquid marke, such rading will lead o changes in he price of he underlying asse. Boh inuiive argumens and a number of heoreical models imply ha his rading due o hedge rebalancing will eiher increase or decrease he volailiy of he underlying asse, depending upon he naure negaive or posiive gamma) of he opion posiions ha are being hedged. This secion develops he main esable predicion abou he relaion beween he ne posiions of dela-hedging opion invesors and he volailiies of underlying socks. Leing V, S) denoe he value of an opion or opions porfolio, recall ha he dela is Δ, S)= V, S)/ S and he gamma is Γ, S) = Δ, S)/ S = 2 V, S)/ S 2. Consider an opion marke maker who has wrien opions and wans o mainain a dela-neural posiion, ha is he or she wans he dela of he combined posiion of opions and he underlying sock o be zero. Because he opion posiion consiss of wrien conracs, is gamma is negaive, and o mainain dela-neuraliy he marke maker mus buy he underlying sock when is price increases and sell i when is price decreases. Similarly, he rading sraegy o dela-hedge a posiive-gamma opion posiion purchased opions) requires selling he underlying asse afer is price has increased and buying i afer is price has decreased. Inuiion suggess ha if he gamma of he aggregae posiion of marke makers and oher dela-hedgers is negaive, hen he rading due o hedge rebalancing buying if he sock price increases, and selling if i decreases) will have he effec of increasing he volailiy of he underlying sock. Conversely, if he gamma of he aggregae posiion of marke makers and oher dela-hedgers is posiive, hen he rading due o 6

9 hedge rebalancing selling if he sock price increases, and buying if i decreases) will have he effec of reducing he volailiy of he underlying sock. This reasoning predics ha he volailiy of he underlying sock will be negaively relaed o he gamma of he aggregae opion posiion of he opion marke makers and any oher dela hedgers. As briefly menioned in he inroducion, he possible effecs of he sock rading semming from hedge rebalancing have been he focus of a srand of he heoreical lieraure. Consisen wih he inuiion above, a number of models have he implicaion ha unless he marke for he underlying asse is perfecly liquid he associaed rading will cause he volailiy of he underlying asse o be greaer han or less han i would have been in he absence of such rading, depending on wheher he gamma of he aggregae opion posiion of he dela-hedgers is less han or greaer han zero. Below we briefly summarize he resuls of several models ha provide explici formulas showing he effec of hedge rebalancing on volailiy. As expeced, in hese models he gamma of he posiion being dela-hedged plays he key role. In addiion o formalizing he inuiion described above, he formulas also provide guidance for he empirical work abou how o normalize he gammas of he opion posiions so ha hey are comparable across firms. These models are buil so ha in he special cases of no dela hedgers he price dynamics of he underlying asse specialize o he usual geomeric Brownian moion wih consan insananeous volailiy σ ha underlies he Black-Scholes-Meron analysis. When here are dela hedgers, he insananeous volailiy is of he form volailiy = v )σ, where σ is a consan and he argumens of he scaling funcion v include he gamma of he dela hedgers aggregae opion posiion. 7

10 Frey and Sremme 1997), Sircar and Papanicolaou 1998), and Schönbucher and Wilmo 2000) analyze essenially he same model, wih differen focuses and emphases. In his model here are reference raders whose demands are driven by an underlying Brownian moion and are decreasing in he price of he underlying asse, and also program raders who follow a pre-specified dynamic rading sraegy ha can be inerpreed as he sraegy o delahedge an opion posiion. When he demand funcions and oher assumpions are chosen so ha he model reduces o geomeric Brownian moion and he Black-Scholes-Meron model in he special case of no program raders, he form of he scaling funcion v is Δ, S) / M S / M ) Γ, S) v, S) = = 1 1 +, 1) 1 + Δ, S) / M + S / M ) Γ, S) 1 + Δ, S) / M where M is he number of shares of sock ousanding, S is he price per share, Δ = V, S)/ S and Γ = 2 V, S)/ S 2 are he dela and gamma of he dela-hedgers aggregae opion posiion, and V, S) is he value of he opion posiion of he dela-hedgers. Plaen and Schweizer 1998) describe a similar model in which he scaling funcion is 6 1 v, S) =, 2) 1 + S / γ ) Γ, S) where γ is a parameer ha appears in he demand funcion. In his model i seems naural o assume ha he demand parameer is proporional o he number of shares ousanding, i.e. ha γ = M/α, where α is consan. 7 Making his assumpion, he scaling funcion in 2) becomes 5 See equaion 24) on p. 55 of Sircar and Papanicolaou 1998), he definiion of ρ in erms of ζ on page 51, and he meaning of ζ on p. 50. The signs on Δ and Γ differ from hose ha appear in Sircar and Papanicolaou 1998) because here he symbols Δ and Γ represen he parial derivaives of he dela hedgers aggregae opion posiion, while he resuls in Sircar and Papanicolaou are expressed in erms of he rading sraegy in shares. The hedging sraegy involves a posiion of Δ shares.) 6 This is based on equaion 2.7) of Plaen and Schweizer 1998), where we have used he fac ha ξ/ log s) = s ξ/ s) and also adjused he equaion o reflec he fac ha equaion 2.7) of Plaen and Schweizer 1998) provides he volailiy raher han he scaling funcion v. 8

11 1 v, S) =. 3) 1 + α S / M ) Γ, S) Finally, Frey 2000) presens a simple model in which he scaling funcion is 1 v, S) =, 4) 1+ ρsγ, S) where he parameer ρ measures he sensiiviy of he sock price o he rades semming from hedge rebalancing. In his case, i seems reasonable o assume ha ρ is inversely proporional o he shares ousanding, i.e., ha i can be wrien as ρ = λ/m. Under his assumpion, he scaling funcion in 4) becomes 1 v, S) =. 5) 1+ λ S / M ) Γ, S) Recalling ha he insananeous volailiy is given by he produc v,s)σ, he main esable predicion ha comes from hese analyses is ha hedge rebalancing will impac he variabiliy of he reurns of he underlying socks. In paricular, here will be a negaive relaionship beween he ne gamma of dela-hedging invesors opion posiions on an underlying sock and he variabiliy of he sock s reurn. Noably, in all models Γ, S) is eiher he key or excep for he parameers) he only deerminan of he scaling funcion v. Furher, scaling by S/M is eiher par of he model i.e., equaion 1)), or a consequence of auxiliary assumpions ha seem naural equaions 3) and 5)). 8 For hese reasons, our empirical analysis below focuses on he relaion beween gamma and sock reurn volailiy using he normalized gamma S/M) Γ, S). In he empirical work we use he Black-Scholes model o compue he ne gamma of he hedge rebalancer s opion posiion on an underlying sock. As a 7 The demand funcion is equaion 2.3) of Plaen and Schweizer 1998). 8 Dimensional analysis also suggess scaling Γ, S) by he raio S/M. The unis of Δ, Γ, S, and M are shares, shares) 2 /$, $/share, and shares, respecively, implying ha he raio S/M) Γ, S) is dimensionless. 9

12 robusness check, we also re-esimae he empirical models using opion gammas from he OpionMerics Ivy DB daabase. 3. Daa The primary daa for his paper were obained from he Chicago Board Opions Exchange CBOE). These daa include several caegories of daily open ineres for every equiy opion series ha rades a he CBOE from he beginning of 1990 hrough he end of When equiy opions on an underlying sock rade boh a he CBOE and also a oher exchanges, he open ineres daa cover he opion series on he underlying sock from all exchanges. If equiy opions on an underlying sock are no raded a he CBOE, hen hey are no included in he daa. The daa se conains four caegories of open ineres for each opion series a he close of every rade day: purchased and wrien open ineres by public cusomers and purchased and wrien open ineres by firm proprieary raders. The caegorizaion of invesors as public cusomers or firm proprieary raders follows he Opion Clearing Corporaion OCC) classificaion. Since he OCC assigns an origin code of public cusomer, firm proprieary rader, or marke maker o each side of every ransacion, he CBOE daa encompass all non-marke maker open ineres. Invesors rading hrough Merrill Lynch or E*rade are examples of public cusomers while an opion rader a Goldman Sachs who rades for he bank s own accoun is an example of a firm proprieary rader. Daily reurns, closing prices, and number of shares ousanding are obained for he underlying socks for which we have opion daa from he Cener for Research in Securiies Prices CRSP). For some analyses we use opion gammas aken from he Ivy DB daabase produced by OpionMerics LLC. 10

13 4. Resuls In order o address he quesions of wheher rebalancing of dela hedges impacs sock price pahs we need daily measures of he ne gamma of he opion posiions of likely dela hedgers. This secion of he paper begins by defining hese measures and hen goes on o invesigae he impac of dela hedger gamma on sock reurn volailiy Ne gamma of likely dela-hedgers The number of purchased and wrien posiions in each opion series is necessarily idenical. Consequenly, a any poin in ime for any underlying sock, he ne gamma of he opion posiions in each opion series and, hence, in he opions on any underlying sock) of all invesors is zero. Some invesors, however, are more likely han ohers o dela-hedge heir opion posiions. Cox and Rubinsein 1985) mainain ha marke makers are he opion marke acors who are mos likely o dela-hedge heir ne opion posiions on underlying socks. They wrie: many Marke Makers aemp o adhere quie sricly o a dela-neural sraegy. However, a dela-neural sraegy usually requires relaively frequen rading. As a resul, i is no advisable as a consisen pracice for invesors wih significan ransacion coss. While public invesors fall ino his caegory, Marke Makers do no. p. 308) Hull 2003, pp. 299, 309) similarly mainains ha marke makers and firm proprieary raders bu no public cusomers are likely o dela-hedge heir ne opion posiions. He explains ha delahedging is relaively more aracive o invesors who hold large porfolios of opions on an underlying sock. These invesors can dela-hedge heir enire porfolios wih a single ransacion in he underlying sock and hereby offse he hedging cos wih he profis from many opion rades. Dela-hedging by invesors who hold only a small number of opions on an underlying 11

14 asse, on he oher hand, is exremely expensive. McDonald 2006) devoes an enire chaper of his exbook o Marke Making and Dela-Hedging. Based on he widely held view ha nonpublic invesors are he predominan dela-hedgers in he opion marke, our ess assume ha dela-hedging is concenraed eiher in marke makers or in marke makers plus firm proprieary raders. We denoe by negamma he ne gamma of he likely dela-hedgers opion posiions on an underlying sock a he close of rade dae. The likely dela hedgers are eiher marke makers or marke makers plus firm proprieary raders, who ogeher consiue all non-public raders. Alhough we do no have daa on marke maker open ineres, we do have daa on he purchased and wrien open ineres of public cusomers and firm proprieary raders. We use he fac ha he sum of he marke maker, public cusomer, and firm proprieary rader open ineres on any opion series a any poin of ime mus be zero o consruc he negamma variable, as follows. For each underlying sock, we measure he likely dela hedgers ne open ineres a he close of rade dae in he jh opion series as he negaive of he ne open ineres of he oher invesor classes. Specifically, he dela hedger ne open ineres in opion series j a he close of rade dae is neopenineres = 1 OpenIneres OpenIneres Purchased, FirmProp Wrien, FirmProp j, { MM} j, j, Purchased, Public Wrien, Public + OpenIneres j, OpenIneres j,, ) 6) where neopenineres j, is he ne open ineres in unis of opion conracs) of he likely dela hedgers in opion series j and OpenIneres x, y j, is he open ineres of ype x i.e., purchased or wrien) by invesor class y i.e., Firm Proprieary or Public) in opion series j a he close of rade dae. The indicaor funcion 1{ } akes he value one if he se of likely dela hedgers is 12

15 assumed o consis only of marke makers MM), and zero if he se of likely dela hedgers is assumed o include boh marke makers and firm proprieary raders. The dela hedger ne gamma due o opion series j is jus he produc of he dela hedger ne open ineres in ha series and he gamma of series j for ime and sock price S, denoed Γ j, S ), muliplied by 100 o accoun for he fac ha opion gammas convenionally are expressed on a per-share basis and each opion is for 100 shares of he underlying sock. Summing over he differen opion series, he normalized dela hedger ne gamma on an underlying sock a he close of rade dae is S negamma = 100 neopenineres Γ, S ) N j, j, 7) M j= 1 where N, S, and M are respecively he number of opion series available for rading, he underlying sock price, and he number of shares ousanding, all as of he close of rading on dae. As discussed in Secion 2, he normalizaion by S /M is eiher required or suggesed by he heoreical models. When compuing Γ j, S ) all quaniies oher han he sock price i.e., he ime o expiraion of he jh opion, he risk free rae, and he volailiy and dividend raes of he underlying sock) are a heir ime values. In he empirical work below, we use Black-Scholes gammas as proxies for Γ j, S ). When compuing he Black-Scholes gammas, he risk-free rae is se o day s coninuously compounded, annualized 30 day LIBOR rae, he volailiy of he underlying asse is se o he annualized sample volailiy esimaed from daily log reurns over he 60 rading days leading up o, and he dividend rae is se equal o he coninuously compounded, annualized rae ha produces a presen value of dividends over he inerval from o he opion expiraion equal o he presen value of he acual dividends paid over he inerval. The assumpions of he Black- 13

16 Scholes model are violaed in a number of ways e.g., he volailiies of he underlying socks are no consan, here may well be jumps in he underlying sock reurn process, and he opions are American raher han European.) We believe he Black-Scholes model provides adequae approximaions o gamma for our purposes. Any noise in our esimaes of gamma should bias agains finding significan resuls. Noneheless, as a robusness check we also presen some resuls using opion gammas aken from he Ivy DB daabase from OpionMerics LLC in order o verify ha our findings are no affeced in any imporan way by our use of he Black-Scholes model. The OpionMerics gammas are compued using he binomial model o incorporae he possibiliy of early exercise, and ake accoun of he volailiy skew by using for each opion series he implied volailiy of ha series Relaion beween marke maker ne gamma and sock reurn volailiy Figure 1 is a bar char ha depics average absolue sock reurn on day +1 as a funcion of marke maker ne opion gamma on he underlying sock a he close of day. We consruc Figure 1 in he following way. Firs, for each underlying sock for which here are daa available for a leas 200 rade days, we use equaions 6) and 7) o obain a he end of each rade dae he marke maker ne opion gamma. Recall ha his marke maker ne gamma is normalized by muliplying by he rade day s closing sock price and dividing by he number of shares ousanding. Nex, we sor he sock s daily normalized marke maker ne gamma ino en equally sized bins and compue for each bin he sock s average nex day absolue reurn. The heigh of each bin in he figure is he average of his quaniy across underlying socks. Figure 1 makes i clear ha here is a negaive relaionship beween marke maker ne opion gamma and he variabiliy of sock reurns. Indeed, he negaive relaionship is monoonic and economically meaningful: he average daily absolue reurn of he low ne marke maker 14

17 gamma group is 100 basis poins greaer han he average absolue reurn for he high ne marke maker gamma group. 9 In addiion, he relaionship is very srong saisically. We do no, however, repor he resuls of saisical ess, because we recognize ha here is a possible alernaive explanaion for he negaive relaionship. In paricular, if invesors rade on privae volailiy informaion in he opion marke, hen we would expec hem o buy sell) opions when hey have informaion ha he variabiliy of underlying socks is going o increase decrease). As a resul, opion marke makers will sell buy) opions, and, herefore, decrease increase) he ne gamma of heir posiions before volailiy increases decreases). Our concern abou rading based on privae informaion abou volailiy is miigaed by Lakonishok, Lee, Pearson, and Poeshman s 2006) finding ha explici volailiy rading hrough sraddles, srangles, and buerflies consiues a small fracion of opion marke aciviy. Noneheless, he evidence in Ni, Pan, and Poeshman 2006) ha volailiy informaion rading is deecable from oal opion marke demand for volailiy leads us o develop a specificaion ha recognizes he possibiliy of informed volailiy rading in he opion marke. Our specificaion also addresses he possibiliy ha invesors migh rade opions based on pas public informaion ha is correlaed wih fuure sock reurn volailiy Impac of opions on underlying sock price pahs The key o developing a specificaion ha recognizes he possibiliy of rading in he opion marke based on privae informaion abou fuure sock reurn volailiy is he idenificaion of changes in he ne opion gamma of likely dela hedgers ha do no resul from invesors buying or selling opions on he basis of such informaion. We isolae such changes by recognizing ha par of he change in he dela hedgers ne gamma from ime τ o ime 9 The figure is similar if he marke maker ne gamma is no normalized or if marke maker plus firm proprieary ne gamma is used in place of marke maker ne gamma. 15

18 comes from changes in he gammas of he old opion posiions held by he dela hedgers a τ, and argue below ha his componen of he change in he ne gamma is likely o be uncorrelaed wih rading based on privae informaion abou volailiy. While mos of he discussion is cas in erms of privae informaion, he idenificaion sraegy also applies o public informaion ha is no accouned for by he conrol variables included in he regression model ha we will develop. In he conex of he regression model, public informaion ha is no capured by he conrol variables plays he same role as privae informaion. We begin by decomposing he dela hedgers ne gamma a ime ino he par ha is due o posiions ha exised τ daes earlier a ime τ and he par ha is due o new posiions ha were esablished beween τ and. To decompose he ne gamma, firs define o be he number of differen conracs on an underlying sock ha were available for rading a ime τ and expire afer, and hen define N τ N τ S negamma τ, S) = 100 neopenineres j, τ Γ j, S ) 8) M j = 1 o be he ne gamma a ime of he likely dela hedger opion posiions ha exised a dae τ. This definiion uses he ne open ineres from ime τ, neopenin eres j, τ, bu sums only over he opion series ha expire afer ime and uses for each series he gamma as of ime. In oher words, his is he ne gamma, a ime, of he old posiions ha exised τ days earlier and ha expire afer ). In erms of his more general noaion, he ne gamma variable defined earlier in equaion 7) is negamma = negamma, S ). Given he definiion in equaion 8), we can decompose he ne gamma a ime ino he par due o he old posiions ha exised a dae τ and he par due o he new posiions esablished beween τ and, ha is 16

19 [ τ ] negamma = negamma τ, S) + negamma negamma, S ). 9) This decomposiion is useful because he opion posiions ha exised a τ canno have been esablished based on privae volailiy informaion acquired subsequen o he close of rading a day τ. If volailiy informaion were sufficienly shor-lived, and in paricular if volailiy informaion obained prior o τ were no useful in predicing volailiy afer, hen his decomposiion would be sufficien. Specifically, we could include he variables negamma τ, S ) and negamma negamma τ, S ) separaely in he regression specificaion and he coefficien on negamma τ, S ) would reflec only he effec of hedge rebalancing on volailiy. Volailiy informaion, however, may no be sufficienly shor lived, and we address his possibiliy by furher decomposing he ne gamma of he dela hedgers old posiions ha exised a τ as [ τ τ ] negamma τ, S ) = negamma, S ) negamma, S ) τ + negamma τ, S ), τ 10) where negamma τ, S ) is defined by subsiuing S τ for S in equaion 8). The second τ componen on he righ hand side of equaion 10), negamma τ, S τ ), is he gamma of he dela hedgers posiions held a τ, compued using he ime τ sock price, while he firs componen negamma τ, S ) negamma τ, S ) represens he change in he ne τ gamma of he dela hedgers posiions a τ ha is due o changes in he sock price from S τ o S. Variaion in his laer variable comes from he fac ha he gamma of an opion is greaes or smalles, for a wrien opion) when he sock price is close o he opion srike price, and close o zero when he sock price is disan from he srike. Because he likely dela hedgers ne 17

20 opion posiion will be differen a differen srikes, movemen of he sock price oward or away from a srike, or from he neighborhood of one srike o he neighborhood of anoher, leads o variaion in he variable negamma τ, S ) negamma τ, S ). τ We use his variaion o idenify he effec of hedge rebalancing on volailiy as follows. Opion posiions ha exised a τ canno have been esablished based on privae volailiy informaion acquired subsequen o he close of rading a dae τ. Hence, he change in he dela hedgers ne gamma due o he changes in he gammas of hese opions canno resul from volailiy informaion acquired by raders beween volailiy informaion prior o τ and. Furhermore, alhough privae τ may well be responsible for some of he opion posiions held a τ, such volailiy informaion is highly unlikely o induce a negaive correlaion beween he change in he gammas of hose opion posiions beween τ and, negamma τ, S ) negamma τ, S ), and he absolue reurn beween and +1, r +1. τ Indeed, in order for any par of he correlaion wheher posiive or negaive) beween he variable negamma τ, S ) negamma τ, S ) and he absolue reurn r +1 o be due o τ privae volailiy informaion abou r +1 acquired on or prior o τ, wo condiions mus boh be me. Firs, some par of he volailiy informaion mus be realized prior o dae and hus conribue o he sock price change from τ o and hereby he change in gamma negamma τ, S ) negamma τ, S τ ) ) and some par of he informaion mus be realized afer in he reurn r +1. Second, any dependence beween he sock price change from τ o and r +1 mus no be capured by he lagged absolue reurn conrol variables ha we will include in our regression specificaion. 18

21 Since hese condiions canno be enirely ruled ou a priori, i is worh noing ha even if hey are saisfied i is more likely ha he correlaion beween volailiy informaion and negamma τ, S ) negamma τ, S ) will be posiive han negaive. A posiive τ correlaion will increase he esimaed coefficien on he negamma τ, S ) negamma τ, S ) variable and hence bias agains any finding ha τ hedge rebalancing affecs sock reurn volailiy. In order o see why any correlaion is likely o be posiive, suppose ha jus prior o τ some public cusomer e.g., a hedge fund) obains privae informaion ha volailiy will increase and buys a large number of near-he-money opions in order o profi from he informaion. Marke makers will wrie hese opions, and he gamma of he corresponding marke maker posiion will be negaive. As he underlying sock price changes from S τ o S, he near-he-money opions will move away from he money which will cause he gammas of he opions o decrease and he marke maker gamma o increase since hey have wrien he opions). As a resul, he change in gamma negamma τ, S ) negamma τ, S ) will be posiively relaed o he cusomers privae τ informaion abou r +1. Conversely, if a cusomer obains privae informaion ha volailiy will decrease he or she will wrie opions, he corresponding marke maker gamma will be posiive, and he change in gamma due o a sock price change from S τ o S will likely be negaive and hus posiively correlaed wih he negaive) privae informaion abou r +1. In ligh of hese consideraions, he main variable in our specificaion is negamma τ, S ) negamma τ, S ), ha is he change in he ne gamma beween τ τ and of opion posiions held by he likely dela hedgers a ime τ ha resuls from he change in he underlying sock price from S τ o S. In addiion, we also include he oher wo 19

22 componens negamma τ, S ) and negamma negamma τ, S ) separaely as τ independen variables in our regression specificaions. These hree variables sum o he dela hedger ne gamma a ime, negamma. Our specificaion mus also conrol for oher variables ha predic volailiy and are correlaed wih he gamma measures. The leading candidaes for such conrol variables are funcions of pas reurns, as he exisence of volailiy clusering in reurns i.e., GARCH effecs) has been well documened. We conrol for volailiy clusering in a compuaionally racable fashion by including lagged absolue reurns in he regression specificaion. This approach o modeling volailiy clusering was proposed by Schwer and Seguin 1990) and recenly used by Barclay, Hendersho, and Jones 2006). Because we esimae he regressions firm-by-firm, we do no need o conrol explicily for firm characerisics ha affec volailiy, as hese will be subsumed in he consan erms. By aking accoun of boh firm characerisics and he effec of lagged reurns, our regression specificaion includes he mos imporan publicly available predicors of sock reurn volailiy. While he regression model almos cerainly does no capure all public informaion abou volailiy, he preceding discussion of privae informaion also applies o public informaion, because he par of public informaion ha is no capured by he conrol variables ends up in he regression residuals and plays he same role as privae informaion. As wih privae informaion, any public informaion ha is no capured by he conrols can induce a negaive correlaion beween he absolue reurn r +1 and he variable negamma τ, S ) negamma τ, S ) only if some par of i is realized in sock reurns τ on or prior o dae and hus conribues o he change in gamma negamma τ, S ) negamma τ, S ) ) and some par of i is realized in he reurn r +1, τ 20

23 and his dependence beween he reurns before and he absolue reurn r +1 is no capured by he conrols for volailiy clusering. Our specificaion has one ime-series equaion for each underlying sock, and he main variable ), ), τ τ τ S negamma S negamma is he firs one on he righ hand side of he following equaion: [ ] [ ] T r n r m r r k r j r i r h r g r f r e S negamma S negamma d S negamma c S negamma S negamma b a r 1,...,, ), ), ), ), ), = = + ε τ τ τ τ τ τ l 11) We will esimae model 11) for each underlying sock wih τ se equal o 3, 5, and 10 rade daes. Our primary predicion is ha he b coefficiens are negaive. For each underlying sock, he second independen variable ), τ τ S negamma measures he likely dela hedgers ne gammaτ rade daes in he pas. The dela-hedging effec also predics ha his variable s coefficien will be negaive. However, a negaive esimae for c will no provide unambiguous evidence ha dela-hedging impacs underlying sock variabiliy, because he volailiy informaion effec will also end o make his coefficien negaive. Of course, insofar as any increase or decrease in volailiy associaed wih volailiy informaion rading appears and disappears in fewer han τ days, a negaive c coefficien does in fac indicae ha dela-hedging effecs sock price variabiliy. We canno, however, be cerain of he horizon of volailiy changes prediced by volailiy informaion rading. The hird independen variable ), S negamma negamma τ measures he change in ne gamma from τ o ha resuls from he change in he dela hedgers opion posiion from τ o. Since boh he dela rehedging and volailiy informaion sories predic a negaive coefficien for his variable, a negaive coefficien esimae does no provide sraighforward evidence for eiher. These second 21

24 and hird independen variables also serve o conrol for volailiy rading based on privae informaion. The curren and nine pas daily lags of absolue reurns conrol for volailiy clusering in sock reurns. We esimae all 2,308 equaions one for each underlying sock) simulaneously in a sacked regression, allowing coefficiens in each equaion o be independenly deermined. We exclude socks for which here are fewer han 200 rade days for which observaions on all of he variables are available. Sandard errors for he coefficien averages are clusered by dae. Specifically, we firs form a covariance marix V of all coefficiens, clusered by dae. We hen derive he sandard error for he average direcly from his covariance marix as ΞVΞ', where Ξ is chosen o consruc he arihmeic average of individual equaion coefficiens from he sacked coefficien vecor. An advanage of his approach is ha sandard errors are robus o he crosssecional covariance srucure of he individual equaion regression errors, which is of unknown srucure. Table 1 conains descripive saisics on he absolue reurn variables r and he normalized ne posiion gamma negamma for he wo groups of likely dela hedgers, marke makers Marke Makers) and marke makers plus firm proprieary raders Marke Maker + Firm Proprieary). The descripive saisics are firs calculaed for each underlying sock and hen he averages across he underlying socks are repored. The average mean and median absolue reurns are or 3.1% and or 2.2%, respecively, and he average minimum and maximum values are zero and %). For marke makers he average mean value of he normalized ne posiion gamma is and he average sandard deviaion is The average means and sandard deviaions of he corresponding unnormalized variables are 9,993 and 19,058, respecively. For marke makers plus firm proprieary raders, he average mean and 22

25 sandard deviaion are slighly larger. For marke makers, he average minimum and maximum values for he normalized negamma variable are, respecively, shares and , while he corresponding quaniies for he unnormalized ne posiion gamma are 56,690 and 128,513. As one migh expec, for marke makers plus firm proprieary raders he average minimum and maximum values are slighly more exreme. Table 2 repors he resuls of esimaing model 11) for he case when marke makers are he likely dela hedgers and τ = 5 rade days. The able repors averages across underlying socks of poin esimaes and -saisics for he averages, where he -saisics are consruced from sandard errors based on clusering by dae as described above. Hence, he -saisics accoun for any cross-secional correlaion in he daa. The average of he coefficien esimaes on he key righ-hand side variable negamma τ, S ) negamma τ, S ) is equal o and highly significan, wih τ a -saisic of The negaive average coefficien indicaes ha here is a negaive relaionship beween marke maker ne gamma and he variabiliy of he underlying sock price ha is no rooed in volailiy informaion rading. Hence, he main predicion from above is confirmed, and here is evidence ha opion marke aciviy has a pervasive influence on underlying sock price pahs. Furhermore, he effec appears o be economically meaningful. The average daily absolue reurn of he socks in our sample is 310 basis poins and from Table 1 he sandard deviaion of he marke maker normalized ne posiion gamma is Thus, a one sandard deviaion shock o he marke maker ne posiion gamma is associaed wih a = 36.8 basis poins change in absolue reurn. Consequenly, we esimae ha on he order of 11.8 percen =36.8/310) of he daily absolue reurn of opioned socks can be accouned for by opion marke paricipans re-balancing he hedges on heir opion posiions. 23

26 The average coefficiens on he variables negamma τ, S ) and τ negamma negamma τ, S ) are also negaive and significan. In boh cases, he negaive esimaes may come from he marke makers dela hedging heir opion posiions, volailiy informaion rading of non-marke makers, or some combinaion of he wo. The curren and lagged absolue sock reurn variables all have posiive and significan coefficien esimaes, which is consisen wih he well-known phenomenon of volailiy clusering in sock reurns. Finally, we noe ha in unrepored resuls he esimaes on he hree ne gamma variables are similar when he absolue sock reurn variables are omied from he model. Since he absolue reurn conrol variables capure he firs-order feaures of he volailiy process, he fac ha omiing hem does no change our findings suggess ha our resuls are unlikely o change much if alernaive echniques are used o conrol for he ime-series properies of volailiy. The absolue reurns also provide a conrol for he overall level of volailiy. Here again, he fac ha omiing hem all ogeher does no change our findings suggess ha our resuls would no change wih alernaive conrols for he overall level of volailiy. 10 The fourh and fifh columns of Table 2 he columns headed Marke Maker plus Firm Proprieary Posiions ) are based on he alernaive assumpion ha boh marke makers and firm proprieary raders dela-hedge heir opion posiions. Thus, he hree gamma variables in his specificaion are compued using he combined opion posiion of he marke makers and firm proprieary raders. As wih he resuls using he marke maker gammas, we esimae a imeseries equaion for each of he 2,308 underlying socks for which here are a leas 200 rade days 10 I is unclear ha here is any need o conrol for he overall level of volailiy. While ne gamma may vary sysemaically wih he level of volailiy, here is no reason o believe ha he differenced ne gamma variable ha is he main objec of ineres is lower higher) when overall volailiy is higher lower). 24

27 on which observaions on all of he variables are available and repor in he able he means of he 2,038 coefficien esimaes and he associaed -saisics. These resuls are very similar o hose using he marke maker gamma variables, wih he principal difference being ha he magniudes of he average coefficien esimaes on he hree gamma variables are slighly smaller. For example, he average coefficien on he variable negamma τ, S ) negamma τ, S ) is wih -saisic 6.861) raher han τ There are similar small differences in he average coefficien esimaes on he oher wo gamma variables, while he average coefficien esimaes on he lagged absolue reurn variables are almos unchanged. The small decreases in he magniudes of he coefficien esimaes on he gamma variables are consisen wih he hypohesis ha no all of he firm proprieary raders dela-hedge and hus including heir posiions in he compuaion of he gamma variables dampens he effec. Regardless, hese resuls also indicae ha here is a negaive relaion beween gamma and volailiy ha is no due o volailiy informaion rading. The exen o which re-balancing by dela-hedgers impacs he frequency of large sock price moves is also of ineres. We assess he impac on large sock price movemens by reesimaing equaion 11) wih he dependen variable r +1 replaced by one of wo indicaor variables: he firs akes he value one when r +1 is greaer han 3% and oherwise is zero, and he second akes he value one when r +1 is greaer han 5% and oherwise is zero. The esimaion is carried ou in he same way as before, and he resuls are repored in Table 3. The resuls indicae ha he average coefficien on he negamma τ, S ) negamma τ, S ) τ variable is negaive and significan for boh he 3% and he 5% dependen indicaor variables. Consequenly, here is evidence ha he re-balancing of dela hedges of opion posiions impacs he probabiliy of large absolue sock reurns on underlying socks. In order o undersand he 25

28 economic imporance of he hedge re-balancing effec on large absolue sock reurns, noe ha in our sample he uncondiional probabiliy ha a daily absolue reurn will be greaer han 3% is A one sandard deviaion movemen in he negamma variable is As a resul, a one sandard deviaion shock o his variable reduces he probabiliy ha he daily absolue reurn on he underlying sock is greaer han or equal o 3% by = ). This change in probabiliy corresponds o an 11% reducion from o A similar argumen indicaes ha for daily absolue reurns greaer han 5%, he probabiliy is reduced by 18.5% from o Hence, dela hedge rebalancing by marke makers appears o have an imporan impac on large sock price movemens. Alhough i is unclear how microsrucure effecs such as bid-ask bounce could bias oward our findings in he firs place, he fac ha he main effec is presen for large absolue reurns suggess ha microsrucure phenomena are unlikely o provide an alernaive explanaion for our resuls. 4.4 Analysis of subsamples Ni, Pearson, and Poeshman 2005) presen evidence ha sock rading o rebalance opion marke makers dela hedges of heir opion posiions conribues o sock price clusering on he opion expiraion Friday and he preceding Thursday, bu find no evidence of any effec prior o he expiraion week. This raises he possibiliy ha he negaive relaion beween volailiy and gamma documened above is no pervasive bu raher is driven by he observaions from opion expiraion daes or he immediaely preceding rading days. This concern is exacerbaed by he fac ha he gammas of opions ha are very close-o-he-money become large as he remaining ime o expiraion shrinks o zero, implying ha dela hedgers wih 26

29 posiions in such opions may need o engage in considerable sock rading jus prior o expiraion in order o mainain heir hedges. Table 4 addresses his issue by presening resuls for a sub-sample ha excludes he daa from he expiraion week. The regression specificaions are idenical o hose ha were used for he resuls repored in Table 2, and he sample is idenical excep ha observaions for which he rade dae was from an opion expiraion week were dropped. This resuled in eliminaing slighly less han 25 percen of he observaions. Following he forma of Table 2, Table 4 repors he averages across firms of he coefficien esimaes of he ime-series regressions for he underlying socks. The resuls in Table 4 are almos idenical o hose in Table 2. When he gammas are compued using only marke maker opion posiions he mean coefficien esimae for he key variable negamma τ, S ) negamma τ, S ) is wih -saisic 5.451) τ insead of he average of saisic 7.624) repored in Table 2. We expec he reducion in he -saisic, because approximaely 25% of he daa have been eliminaed. The average coefficien esimaes for he oher wo gamma variables are also nearly unchanged. When he gammas are compued using he posiions of marke makers plus firm proprieary raders he siuaion is he same he average coefficien esimaes on he posiion gamma variables repored in Table 4 are only very slighly differen from he corresponding averages in Table 2. The average coefficien esimaes on he lagged absolue reurn variables also are lile changed. These resuls indicae ha he relaion beween he gamma of dela hedgers opion posiions and sock reurn volailiy is pervasive and no limied o opion expiraion weeks. Table 5 presens resuls for sub-samples based on a differen ime pariion. In paricular, he second and hird columns presen he average coefficien esimaes and associaed -saisics 27

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