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 Soc 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 July 31, 2006 **Preliminary and Incomplee: Please do no quoe or circulae. * Deparmen of Finance, College of Busines Universiy of Illinois a Urbana-Champaign, 340 Wohlers Hall, 1206 Souh Sixh Sree, Champaign, IL (Phone: (217) , phone: (217) , and phone: (217) , We han Joe Levin, Eileen Smih, and Dic Thaler for assisance wih he daa used in his paper. Qian Deng provided excellen research assisance. We bear full responsibiliy for any remaining errors.

2 Does Opion Trading Have a Pervasive Impac on Underlying Soc Prices? ABSTRACT The quesion of wheher and o wha exen opion rading impacs underlying soc 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 soc 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 soc reurn volailiy and ne purchased opion posiions of invesors who are liely o hedge. Hence, we provide he firs evidence for a subsanial and pervasive influence of opion rading on soc prices.

3 1. Inroducion Ever since individual equiy opions began rading in 1973, invesor exchange official and regulaors have been concerned ha underlying soc 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 soc price lile evidence has been produced ha opion rading influences he prices of underlying socs. Indeed, he only convincing evidence ha opion aciviy alers underlying socs involves soc price deviaions righ a opion expiraion. The presen paper invesigaes wheher opion mare aciviy has a subsanially more pervasive influence on underlying soc prices. A firs srand of research on he impac of opion rading on underlying socs examines wheher opion inroducion generaes a one-ime change in soc price level. Earlier papers by Conrad (1989) and Deemple and Jorion (1990) indicae ha opion inroducion produces an increase in he level of underlying soc prices. These finding however, do no appear o be robus. Sorescu (2002) and Ho and Liu (1997) show ha in a laer ime period soc prices decrease upon opion inroducion. Mos recenly, Mayhew and Mihov (2004) find ha he price level effecs disappear when benchmared agains he price changes of mached firms ha do no have opions inroduced. A second srand of research invesigaes wheher opion aciviy causes sysemaic changes in he prices of he underlying socs a opion expiraion daes. An early CBOE (1976) repor does no find evidence of abnormal underlying soc price behavior leading up o opion expiraion. Using small sample Klemosy (1978) documens negaive reurns on underlying socs in he wee leading up o expiraion and posiive reurns in he wee afer expiraion while 1 See Whaley (2003) for an accoun of he early period of exchange raded opions. 1

4 Cinar and Vu (1987) find ha he average reurn and volailiy of opioned socs on he Thursday o Friday of expiraion wee are largely he same as from he Thursday o Friday of non-expiraion wees. Ni, Pearson, and Poeshman (2005), on he oher hand, provide srong evidence ha he prices of opioned socs cluser a srie 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 soc prices movemens changes no limied o he imes ha opions are inroduced or expire. Bansal, Prui, and Wei (1989), Conrad (1989), and Sinner (1989) all find ha being opioned yields a decrease in he volailiy of underlying soc prices. However, Lamoureux and Paniah (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 lac opion inroducion. All in all, he lieraure conains lile evidence ha opion rading has a significan impac on underlying soc prices. The only compelling evidence ha soc prices are alered is limied o expiraion daes. Specifically, Ni, Pearson, and Poeshman (2005) documen ha he prices of opionable socs (i.e., socs wih exchange-raded opions) end o cluser a opion srie prices on opion expiraion dae and show ha soc rading underaen by opion mare paricipans in order o eep heir porfolios hedged as he delas of heir expiring opion posiions change rapidly as he remaining ime o expiraion shrins o zero is a major driver of his soc price clusering. 2 Avellaneda and Lipin (2003) model his mechanism, focusing on he role of he 2 The dela of an equiy opion is he change in is value per uni increase in he value of he underlying soc. 2

5 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 derivaive as ime passes dela-hedgers who have ne purchased (wrien) opion posiions will sell (buy) soc when he soc price is above he opion srie price and buy (sell) soc when he price is below he srie price, ending o drive he soc price oward he opion srie price. As documened by Ni, Pearson, and Poeshman (2005), his causes clusering (declusering) a opion expiraion when dela-hedging opion mare paricipans have ne purchased (wrien) posiions in opions on an underlying soc. The finding ha re-hedging of opion posiions jus before expiraion produces measurable deviaions in soc price pahs leads naurally o he quesion of wheher re-hedging away from expiraion also changes soc 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 ey 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 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 3

6 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 soc of dela-hedging invesors is posiive (negaive), hedge re-balancing will reduce (increase) he volailiy of he soc. This predicion has no ye been empirically esed. 3 We invesigae wheher he ne gamma of dela-hedging invesors is indeed negaively relaed o he volailiy of he underlying soc by using a daase ha allows us o compue on a daily basis for each underlying soc he gamma of he ne opion posiion of liely dela hedgers. We indeed find a highly significan negaive relaionship beween he gamma of he ne opion posiion of liely dela-hedgers and he absolue reurn of he underlying soc. The finding is robus o conrolling for persisence in soc volailiy and also for he possibiliy ha he opion posiions of liely dela-hedgers are changed as he resul of invesors rading opions o profi from informaion abou he fuure volailiy of underlying socs. In addiion, he finding is presen for large and small underlying soc in he firs and second half of our sample period, when we define liely dela hedgers o include firm proprieary raders in addiion o mare maer and when we exclude he wee of opion expiraion from our analysis. Hence, we provide evidence ha opion mare aciviy has a pervasive impac on he price pahs of underlying socs. In paricular, he impac is no limied o imes very close o opion expiraion. Furhermore, he effec is economically significan. The average daily absolue reurn of he socs in our sample is 310 basis poins and a one sandard deviaion shoc 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 3 Cein, Jarrow, Proer, and Waracha (2006) carry ou empirical wor examining he effecs of soc illiquidiy on opion prices for five differen soc bu do no address he impac of opion hedging on soc prices. 4

7 of opioned socs can be accouned for by opion mare paricipans re-balancing he hedges of heir opion posiions. In order for i o be plausible ha he soc rading from hedge re-balancing has a nonnegligible influence on underlying soc price pah he volume of his rading should have a non-rivial impac on oal soc volume. Thu as a chec on he reasonableness of our resuls we invesigae he relaionship beween hese volumes by regressing oal daily soc volume on a measure of he soc volume due o hedge re-balancing and a number of conrol variables. We find ha he coefficien on he hedge re-balancing volume is significanly posiive and ha a one sandard deviaion shoc o his volume is associaed wih a change in oal volume equal o 14 percen of is average value. Hence, i seems quie possible ha he soc volume associaed wih hedge re-balancing is large enough o produce non-rivial soc price changes. 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 socs. As noed above, his lieraure finds ha wih proper benchmaring no overall increase or decrease in volailiy is deecable. We show, by conra ha volailiy increases or decreases depending upon he sign of he ne gamma of dela-hedging invesors. Consequenly, even hough opion rading does change he variabiliy of underlying soc 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 resul 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 mare, 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 (posiive or negaive 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 socs. 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 mare maer 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 soc o be zero. Because he opion posiion consiss of wrien conrac is gamma is negaive, and o mainain dela-neuraliy he mare maer mus buy he underlying soc when is price increases and sell i when is price decreases. Similarly, he rading sraegy o dela-hedge a posiive-gamma opions 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 mare maers and oher dela-hedgers is negaive, hen he rading due o hedge rebalancing (buying if he soc price increase and selling if i decreases) will have he effec of increasing he volailiy of he underlying soc. Conversely, if he gamma of he aggregae posiion of mare maers and oher dela-hedgers is posiive, hen he rading due o 6

9 hedge rebalancing (selling if he soc price increase and buying if i decreases) will have he effec of reducing he volailiy of he underlying soc. This reasoning predics ha he volailiy of he underlying soc will be negaively relaed o he gamma of he aggregae opion posiion of he opion mare maers and any oher dela hedgers. As briefly menioned in he inroducion, he possible effecs of he soc 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 mare 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 ey role. Anoher benefi of looing a hese explici formulas is ha hey also provide guidance for he empirical wor 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 Blac-Scholes-Meron analysis. When here are dela hedger 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 Blac-Scholes-Meron model in he special case of no program rader 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 soc ousanding, S is he price per share, V(, S) is he value of he opions posiion of he dela-hedger and Δ = V(, S)/ S and Γ = 2 V(, S)/ S 2 are he dela and gamma of he dela-hedgers aggregae opion posiion. Plaen and Schweizer (1998) describe a similar model in which he scaling funcion is 5 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. 6 Maing his assumpion, he scaling funcion in (2) becomes 4 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.) 5 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 soc 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. Doing hi 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 socs. In paricular, here will be a negaive relaionship beween he ne gamma of dela-hedging invesors opion posiions on an underlying soc and he variabiliy of he soc s reurn. Noably, in all models Γ(, S) is eiher he ey or (excep for he parameers) 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)). 7 For hese reason our empirical analysis below focuses on he relaion beween gamma and soc reurn volailiy using he scaled gamma (S/M) Γ(, S). In he empirical wor below we use he Blac-Scholes model o compue he ne gamma of he hedge rebalancer s opion posiion on an underlying soc. We 6 The demand funcion is equaion (2.3) of Plaen and Schweizer (1998). 7 Dimensional analysis also suggess scaling Γ(, S) by he raio S/M. The unis of Δ, Γ, S, and M are share (shares) 2 /$, $/share, and share respecively, implying ha he raio (S/M) Γ(, S) is dimensionless. 9

12 also re-esimae he empirical models using opion gammas from he OpionMerics Ivy DB daabase for he opions for which hese are available. 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 soc rade boh a he CBOE and also a oher exchange he open ineres daa cover he opion series on he underlying soc from all exchanges. If equiy opions on an underlying soc 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 mare maer o each side of every ransacion, he CBOE daa encompass all non-mare maer 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 ban s own accoun is an example of a firm proprieary rader. Daily reurn closing price volume, and number of shares ousanding are obained for he underlying socs for which we have opion daa from he Cener for Research in Securiies Prices (CRSP). For some analyses we use opion gammas aen 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 soc price pahs and wheher soc volume from dela-hedging is a non-rivial par of oal soc volume, we need daily measures of he ne dela and ne gamma of he opion posiions of liely dela hedgers. This secion of he paper begins by defining hese measures and hen goes on o invesigae he wo quesions in urn Ne dela and gamma of liely dela-hedgers The number of purchased and wrien posiions on each opion series is necessarily idenical. Consequenly, a any poin in ime for any underlying soc, he ne dela and ne gamma of he opion posiions on each opion series (and, hence, on he opions on any underlying soc) from all invesors is zero. Some invesor however, are more liely han ohers o dela-hedge heir opion posiions. Cox and Rubinsein (1985) mainain ha mare maers are he opion mare acors who are mos liely o dela-hedge heir ne opion posiions on underlying socs. They wrie: many Mare Maers 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, Mare Maers do no. (p. 308) Hull (2003, pp. 299, 309) similarly mainains ha mare maers and firm proprieary raders bu no public cusomers are liely 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 soc. These invesors can dela-hedge heir enire porfolios wih a single ransacion 11

14 in he underlying soc 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 asse, on he oher hand, is exremely expensive. McDonald (2006) devoes an enire chaper of his exboo o Mare maing and Dela-Hedging. Based on he widely held view ha nonpublic invesors are he predominan dela-hedgers in he opion mare, our ess assume ha dela-hedging is concenraed eiher in mare maers or in mare maers plus firm proprieary raders. We denoe by nedela he ne dela of invesor group s opion posiions on an underlying soc s a he close of rade dae. The invesor group is eiher mare maers (MM) or mare maers plus firm proprieary raders (MM+Firm Prop), who ogeher comprise all non-public raders. Alhough we do no have daa on mare maer open inere 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 mare maer, public cusomer, and firm proprieary rader open ineres on any opion series a any poin of ime mus be zero o define nedela by N s, Purchased Firm Prop Wrien Firm Prop s, 100 1= MM j, j, j= 1,, ( ) nedela OpenIneres OpenIneres + OpenIneres OpenIneres Δ Purchased, Public Wrien, Public j, j, j,. (6) where N is he number of differen opions lised on soc s on rade dae, OpenIneres is he number of conracs of open ineres of ype x (i.e., purchased or wrien) by invesor class y on he jh of he N opions on underlying soc s on rade dae, and Δ j, is he dela of he jh opion on underlying soc s on rade dae. The indicaor funcion 1 aes he value 1 if = MM x, y j, = MM and zero oherwise. The facor of 100 appears because each opion conrac is for 100 shares of soc. 12

15 We measure he ne gamma of invesor group s opion posiions on an underlying soc s a he close of rade dae similarly o he way ha we measure he ne dela, excep ha we normalize all ne gamma variables by muliplying hem by he rade day s underlying soc price and dividing by he number of shares ousanding in order o mae he coefficien esimaes comparable across underlying socs. In paricular, negamma 100 Ns, Purchased, Firm Prop Wrien, Firm Prop ( S / M ) [ 1 MM ( OpenIneres j, OpenIneres j, ) j= 1 = (7) + OpenIneres Purchased, Public j, OpenIneres Wrien, Public j, ] Γs, j, where Γ j, is he gamma of he jh opion on underlying soc s on rade dae, S is he price of soc s a ime, and M is he number of shares ousanding. We will also need o measure he ne gamma a ime of invesor group s ime τ opion posiions under he assumpion ha he soc price did no change from is ime τ value,, and also under he assumpion S τ ha he soc price changed o is acual ime value,. S We define he variable ha measures hese quaniies by negammapriorpos 100, τ ( S ) > N s, τ Purchased, Firm Prop Wrien, Firm Prop ( S / M ) [ 1 = MM ( OpenIneres j, τ OpenIneres j, τ ) j= 1 u + OpenIneres Purchased, Public j, τ OpenIneres Wrien, Public j, τ ] Γ ( S ) j, u (8) where > Γ j, ( S u ) is he gamma a ime of he jh of he N τ opions on underlying soc s available on rade dae τ ha expire afer under he assumpion ha he ime soc price is S u, u {, τ}. When compuing Γ S ) all quaniies oher han possibly he soc price j, ( u (i.e., he ime o expiraion of he jh opion, he ris free rae, and he volailiy and dividend raes of he underlying soc s) are a heir ime values. 13

16 j, In he empirical wor below, we use Blac-Scholes delas and gammas as proxies for Δ and Γ.,, When compuing he Blac-Scholes delas and gamma he risfree rae is se s j o day s coninuously compounded, annualized 30 day LIBOR rae, he volailiy of he underlying asse is se o he annualized sample volailiy 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 expiraion of he opion equal o he presen value of he acual dividends paid over he inerval. The assumpions of he Blac-Scholes model are violaed in a number of ways (e.g., he volailiies of he underlying socs are no consan, here may well be jumps in he underlying soc reurn proces and he opions are American raher han European.) We believe he Blac- Scholes model provides adequae approximaions o dela and gamma for our purposes. Any noise in our esimaes of dela and gamma should bias agains finding significan resuls. Noneheles as a robusness chec we will use opion gammas aen from he Ivy DB daabase from OpionMerics LLC in order o verify ha our resuls are no being affeced in any imporan way by our use of he Blac-Scholes model Impac of opions on underlying soc price pahs Figure 1 is a bar char ha depics average absolue soc reurn on day +1 as a funcion of mare maer ne opion gamma on he underlying soc a he close of day. We consruc Figure 1 in he following way. Fir for each underlying soc for which here are daa available for a leas 200 rade day we use equaion (7) o obain a he end of each rade day he mare maer ne opion gamma. As discussed above, we hen normalize his mare maer ne gamma by muliplying by he rade day s closing soc price and dividing by he number of shares 14

17 ousanding. Nex, we sor he soc s daily normalized mare maer ne gamma ino en equally sized bins and compue for each bin he soc s average nex day absolue reurn. The heigh of each bin in he figure is he average of his quaniy across underlying socs. Figure 1 maes i clear ha here is a negaive relaionship beween mare maer ne opion gamma and he variabiliy of soc reurns. Indeed, he negaive relaionship is monoonic and economically meaningful: he average daily absolue reurn of he low ne mare maer gamma group is 100 basis poins greaer han he average absolue reurn for he high ne mare maer gamma group. 8 In addiion, he resuls are very srong saisically. We do no, however, repor he resuls of saisical es because we recognize ha here is a possible alernaive explanaion for he negaive relaionship. In paricular, if invesors rade on volailiy informaion in he opion mare, hen we would expec hem o buy (sell) opions when hey have informaion ha he variabiliy of underlying socs is going o increase (decrease). As a resul, mare maers will sell (buy) opions and, herefore decrease (increase) he ne gamma of heir posiions before volailiy increases (decreases). Our concern abou his mechanism is miigaed by Laonisho, Lee, Pearson, and Poeshman s (2006) finding ha explici volailiy rading hrough sraddle srangle and buerflies consiues a small fracion of opion mare aciviy. Noneheles he evidence in Ni, Pan, and Poeshman (2006) ha volailiy informaion rading is deecable from oal opion mare demand for volailiy leads us o develop a specificaion ha recognizes he possibiliy of informed volailiy rading in he opion mare. The ey o his specificaion is he idenificaion of changes in ne opion gamma ha do no resul from invesors buying or selling opions on he basis of volailiy informaion. We isolae such changes by recognizing ha par of he change in ne opion gamma of an invesor 8 The figure is similar if he mare maer ne gamma is no normalized or if mare maer plus firm proprieary ne gamma is used in place of mare maer ne gamma. 15

18 group from ime τ o ime comes from changes in he gammas of he opion posiions held by he invesor group a τ. Specifically, we recognize ha he ne gamma a ime can be decomposed ino he hree componens, τ, τ negammas, = negammapriorposs, ( S) negammapriorposs, ( S τ ) + negammapriorpos S + negamma negammapriorpos S, τ, τ s, ( τ ) s, s, ( ), (9), τ, τ and include he hree componens negammapriorpos ( S ) negammapriorpos ( S ),, τ, τ negammapr iorpos s ( S ), and negamma negammapriorpos ( S ) separaely as, τ independen variables in our regression specificaions. τ The firs componen represens he change in he ne opion gamma of he posiions ha were held by he invesor group a τ ha is due o changes in he soc price from ime o ime. Variaion in his variable comes from he fac ha he gamma of an opion is greaes (or smalle for a wrien opion) when he soc price is close o he opion srie price, and close o zero when he soc price is disan from he srie. Because a cusomer group s ne opion posiion will be differen a differen srie movemen of he soc price oward or away from a srie, or from he neighborhood of one srie o he neighborhood of anoher, leads o variaion in he variable negammapriorpos ( S ) negammapriorpos ). This variaion allows us o idenify he effec of hedge rebalancing on volailiy, as follows. ( S, τ, τ τ Fir he opion posiions ha exised a τ canno have been esablished based on volailiy informaion rading subsequen o he close of rading a day τ. Hence, he change in he invesor group s ne gamma due o he changes in he gammas of hese opions canno resul from volailiy informaion rading beween informaion rading prior o τ τ and. Furhermore, alhough volailiy τ may be responsible for some of he opion posiions held a τ, such volailiy informaion rading is highly unliely o induce a negaive correlaion 16

19 beween he change in he gammas of he opion posiions beween τ and,, τ, τ negammapr iorpos ( S ) negammapriorpos ( S ), and he absolue reurn r +1. In τ order for some par of he correlaion beween r +1 and he variable negammapriorpos ( S ) negammapriorpos ( S, τ, τ τ ) o be due o volailiy informaion rading abou r+1 carried ou on or prior o τ i mus be ha some par of he privae informaion abou volailiy is realized prior o dae (and hus conribues o he changes S S τ and negammapriorpos ( S ) negammapriorpos ( S, τ, τ τ ) ) and some par of he privae volailiy informaion is realized in he reurn r +1, and his dependence beween S S τ and r +1 is no capured by he lagged absolue reurns used as conrols. While his possibiliy canno be ruled ou a priori, he scenarios ha seem mos liely sugges ha he correlaion beween privae signals abou volailiy and negammapriorpos ( S ) negammapriorpos ( S, τ, τ τ ) will be posiive, ending o bias he esimaed coefficien on his variable oward zero and agains finding evidence ha hedge rebalancing affecs soc reurn volailiy. beween For hese reason he main variable in our specificaion is he variable negammapriorpos, τ ( S ) negammapriorpos ( S, τ τ 9 ), ha is he change in he ne gamma τ and of opion posiions held by invesor group a ime τ ha resuls from he change in he underlying soc price from S τ o S. Our specificaion has one ime-series 9 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. Mare maers will wrie hese opion and he gamma of he ne mare maer posiion will be negaive. Then if he underlying soc price changes from S τ o S he change in he gamma will liely be posiive, so ha he change in gamma negammapriorpos S negammapriorpos S, τ, τ s, ( ) s, ( τ ) will be posiively relaed o he cusomers privae informaion abou r s, +1. Conversely, if a cusomer obains privae informaion ha volailiy will decrease he or she will wrie opion he ne mare maer gamma will be posiive, and he change in gamma due o a soc price change from S τ o S. will liely be negaive and hus posiively correlaed wih he (negaive) privae informaion abou r s,

20 equaion for each underlying soc, and his main variable is he firs one on he righ hand side of he following equaion: τ ( ) ( ) r = a + b negammapriorpos S negammapriorpos S, τ, s, + 1 s, s, τ τ ( ) Pos ( S ) + c negammapriorpos S + d negamma negammaprior,, τ, s, τ s, s + er + f r + gr + hr + ir + jr + r + lrs, 7 + m rs, 8 s, s, 1 s, 2 s, 3 s, 4 s, 5 s, 6 S + nr, 9 + or, 10 + ε,, s= 1,..., N ; = 1,..., T. s s s (10) We will esimae he model (10) wih τ se equal o 3, 5, and 10 rade daes. Our primary predicion is ha he b coefficiens are negaive. The second independen variable measures invesor group s underlying soc s ne gamma τ 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 soc variabiliy, because he volailiy informaion effec will also end o mae 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 day a negaive c coefficien does in fac indicae ha dela-hedging effecs soc price variabiliy. We canno, however, be cerain of he horizon of volailiy changes prediced by volailiy informaion rading. The hird independen variable measures he change in ne gamma from posiion on underlying soc s from τ o ha resuls from he change in invesor group s opion τ o. Since boh he dela re-hedging and volailiy informaion sories predic a negaive coefficien for his variable, a negaive coefficien esimae does no provide sraighforward evidence for eiher. These second and hird independen variables also serve o conrol for volailiy rading based on privae informaion. The curren and en pas daily lags of absolue reurns conrol for well nown clusering effecs (i.e., GARCH effecs) in soc reurn variabiliy. 18

21 We esimae all 2,308 equaions simulaneously in a saced regression, allowing coefficiens in each equaion o be independenly deermined. We exclude socs 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 coefficien clusered by dae. We hen derive he sandard error for he average direcly from his covariance marix as ΞV Ξ', where Ξ is chosen o consruc he arihmeic average of individual equaion coefficiens from he saced coefficien vecor. An advanage of his approach is ha sandard errors are robus o he cross-secional covariance srucure of he individual equaion regression error which is of unnown srucure. Table 1 conains descripive saisics on he absolue reurn variables r and he normalized ne posiion gamma negammapo, for he wo groups of liely dela hedger = MM and = MM + Firm Prop. The descripive saisics are firs calculaed for each underlying soc and hen he averages across he underlying socs 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 0.31 (31%). For mare maers 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 s s 9,993 and 19,058, respecively. For mare maers plus firm proprieary rader he average mean and sandard deviaion are slighly larger. The average minimum and maximum values for he negammapo, MM s s 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 mare maers plus firm proprieary raders he average minimum and maximum values are slighly more exreme. 19

22 Table 2 repors he resuls of esimaing model (10) for he case = MM and τ = 5 rade days. An equaion is included for each of he 2,308 underlying socs for which here are a leas 200 rade days on which observaions on all of he variables are available. The able repors averages across underlying socs of poin esimaes and -saisics for he average 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 ey righ-hand side variable τ ( ) ( ) negammapriorpos S negammapriorpos S, τ, s, s, τ is equal o and highly significan, wih a -saisic of The negaive average coefficien indicaes ha here is a negaive relaionship beween mare maer ne gamma ha is no rooed in volailiy informaion rading and he variabiliy of he underlying soc price. Hence, he main predicion from above is confirmed, and here is evidence ha opion mare aciviy has a pervasive influence on underlying soc price pahs. Furhermore, he effec appears o be economically significan. The average daily absolue reurn of he socs in our sample is 310 basis poins and from Table 1 he sandard deviaion of he mare maer normalized ne posiion gamma is Thu a one sandard deviaion shoc o he mare maer 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 socs can be accouned for by opion mare paricipans re-balancing he hedges on heir opion posiions., τ The average coefficiens on he variables negammapriorpos ( S ), τ, ( ) s, τ and negamma negammapriorposs S are also negaive and significan. In boh case he negaive esimaes may come from he mare maers dela hedging heir opion posiion 20

23 volailiy informaion rading of non-mare maer or some combinaion of he wo. Finally, he curren and lagged absolue soc reurn variables all have posiive and significan coefficien esimae which is consisen wih he well-nown phenomenon of volailiy clusering in soc reurns. The fourh and fifh columns of Table 2 (he columns headed Mare Maer plus Firm Proprieary Posiions ) are based on he alernaive assumpion ha boh mare maers and firm proprieary raders dela-hedge heir opion posiions. Thu he hree gamma variables in his specificaion are compued using he combined opion posiion of he mare maers and firm proprieary raders. As wih he resuls using he mare maer gamma we esimae a imeseries equaion for each of he 2,308 underlying socs for which here are a leas 200 rade days 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 mare maer gamma variable 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 τ ( ) ( ) negammapriorpos S negammapriorpos S, τ, s, s, τ is (wih -saisic 6.861) raher han There are similar small differences in he average coefficien esimaes on he oher wo gamma variable 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 dela-hedge and hus including heir posiions in he compuaion of he gamma variables inroduces some measuremen error. Regardles hese resuls also indicae ha here is a negaive relaion beween gamma and volailiy ha is no due o volailiy informaion rading. 21

24 4.3 Analysis of subsamples Ni, Pearson, and Poeshman (2005) presen evidence ha soc rading o rebalance opion mare maers dela hedges of heir opion posiions conribues o soc price clusering on he opion expiraion Friday and he preceding Thursday, bu find no evidence of any effec prior o he expiraion wee. 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 shrins o zero, implying ha dela hedgers wih posiions in such opions may need o engage in considerable soc rading in order o mainain heir hedges. Table 3 addresses his issue by presening resuls for a sub-sample ha excludes he daa from he expiraion wee. The regression specificaions are idenical o hose ha were used for he resuls repored in Table 2, and he sample is idenical excep ha he observaions for which he rade dae was from an opion expiraion wee were dropped. This resuled in dropping slighly less han 25 percen of he observaions. Following he forma of Table 2, Table 3 repors he averages across firms of he coefficien esimaes of he ime-series regressions for he underlying socs. The resuls in Table 3 are almos idenical o hose in Table 2. When he gammas are compued using only mare maer opion posiions he mean coefficien esimae for he ey, τ, τ variable negammapriorpos ( S ) negammapriorpos ( S ) s, s, τ is (wih -saisic 5.451) insead of he average of (-saisic 7.624) repored in Table 2. The 22

25 average coefficien esimaes for he oher wo gamma variables are also nearly unchanged. When he gammas are compued using he posiions of mare maers plus firm proprieary raders he siuaion is he same he average coefficien esimaes on he posiion gamma variables repored in Table 3 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 soc reurn volailiy is pervasive and no limied o opion expiraion wees. Table 4 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 from ime-series regressions for each soc using daa from he firs half of he sample period , while he fourh and fifh columns presen he average coefficien esimaes and associaed -saisics from he second half of he sample period For boh subsamples and all hree ne gamma variables he average coefficiens are significanly differen from zero, consisen wih he resuls in previous ables. However, he magniudes of he coefficien esimaes from he sub-sample are maredly smaller han hose from he enire sample period repored in Table 2, while hose from he sub-sample are slighly larger han he corresponding coefficiens in Table A similar paern in observed in he average coefficien esimaes on he lagged absolue reurn variables he esimaes from he sub-sample are smaller han hose for he enire sample period, while some of he esimaes for he sub-sample are a bi larger han hose for he enire sample period. The differences beween he resuls for he wo sub-samples migh arise eiher because he period 10 The finding ha ha esimaes based on he enire sample period are no close o a simple average of hose from he wo subperiods should no be surprising. More socs were opionable during he ime period han during he period, so he compuaion of he mean coefficien esimaes across firms has he effec of placing more weigh on he period. 23

26 was one of generally low volailiy, or because he characerisics of opionable socs changed due o growh in he number of opionable socs during he early 1990 s. Regardles i remains he case ha he average coefficiens for all hree posiion gamma variables are significanly differen from zero for boh sub-samples. Table 5 presens he resuls of esimaing model (10) expressing he absolue reurn r s, +1 in erms of he componens of he normalized ne posiion gammas and lagged reurns for he subsamples of large firms and oher firm where he prior posiions are hose ha were held τ = 5 days prior o dae. In each year a large firm is defined o be a firm ha was among he 250 opionable socs wih greaes soc mare capializaion as of December 31 of he previous year. In he second column he average coefficien esimae for he ey variable is (saisic 3.728), similar o he corresponding average coefficien esimaes in Tables 2 4. Ineresingly, he magniude of he average coefficien esimae on he variable, τ, ( ) negamma negammapriorposs S measuring he change in posiion gamma semming from new opion posiions is now smaller, consisen wih he hypohesis ha here is less volailiy informaion rading in large soc hough his may well be over-inerpreing he differences in he poin esimaes. Turning o he resuls for he oher firms in he fourh column, one can see ha he magniudes of he average coefficien esimaes for he firs and second posiion gamma variables are similar o he corresponding averages for he large firms. However, he magniude of he average coefficien esimae on he variable, τ, ( ) negamma negammapriorposs S measuring he change in posiion gamma semming from new opion posiions is now larger, consisen wih he hypohesis ha here is more volailiy informaion rading in smaller socs. However, again his may be over-inerpreing 24

27 he differences in he poin esimaes. Regardles he resuls in Table 5 indicae ha he effec of hedge rebalancing of soc reurn volailiy is found in boh large and small firms. Secion 4.4 Robusness o choice of lag lengh τ and use of he Blac-Scholes gammas The primary resuls in Table 2 are based on a choice of τ = 5 days in consrucing he prior opion posiions. Such a choice is inherenly somewha arbirary. Those resuls also are based on opion gammas from he Blac-Scholes model, a simplificaion. This subsecion presens evidence ha he resuls are robus o differen choices. Table 6 repors he resuls of re-esimaing he regressions for which resuls are shown in Table 2, bu now defining he prior opion posiions o be hose ha exised τ = 10 days previously. Following he forma of Table 2, he second and hird columns headed Mare Maer Posiions presen he averages of he coefficien esimaes from he soc ime-series regressions and he corresponding sandard errors assuming mare maers are he dela hedger while he fourh and fifh columns head Mare Maer plus Firm Proprieary Posiions provide he resuls assuming ha boh mare maers and firm proprieary raders dela hedge heir opions posiions. Comparing he average coefficien esimaes for he posiion gamma variables shown in Table 6 o he corresponding averages in Table 2, one can see ha he resuls are very similar. For example, in he second columns he average coefficien on he ey variable negammapriorpos ( S ) negammapriorpos ( S, τ, τ τ ) changes from (-saisic = 7.624) o (-saisic = 8.052), while in he fourh columns he average coefficien on his variable changes from (-saisic = 6.861) o (-saisic = 7.359). In addiion, he average coefficien esimaes for he absolue reurn variables are 25

28 almos unchanged. Unrepored resuls based on a lag lengh of τ = 3 days are also similar o hose for he lag lengh of τ = 5 days repored in Table 2. The averages of he coefficiens on he second gamma variable, τ negammapriorpos s ( S, τ ) are virually unchanged, going from and in he second and fourh columns of Table 2 o and in he second and fourh columns of Table 6, respecively. This lac of change in he coefficien esimaes when he lag τ is increased from 5 o 10 days suggess ha opions posiions esablished beween 10 and 5 conain lile privae informaion abou r +1. Among he posiion gamma variables he larges change occurs in he average coefficien esimae on he variable negammapos negammapriorpos, τ ( S ), which changes from o for he case of Mare Maer Posiions and from o for he case of Mare Maer plus Firm Proprieary Posiions. This variable measures he componen of he ne gamma on day ha is due o opion posiions esablished afer τ, and he average esimaed coefficien reflecs he fac ha raders wih informaion abou r+1 migh open new opion posiions during he period beween τ and. The reducion in he magniude of he average esimaed coefficien when he lag τ is increased from five o en days also suggess ha opion rades beween 10 and 5 conain much less informaion abou r +1 han do opion rades beween 5 and. Regardless of hese inerpreaions abou he informaion conained in he second and hird gamma variable he imporan finding in Table 6 is ha he average coefficien esimae on he firs posiion gamma variable is lile affeced by increasing he lag τ from five o en days. 26

29 As menioned above, he opion posiion gammas ha underlie he resuls in Tables 2 6 were compued using Blac-Scholes gammas for he opions ha comprise he posiions. Table 7 addresses he issue of wheher he resuls are robus o using differen esimaes of individual opion gammas in compuing he posiion gammas. The regressions for which resuls are repored in Table 7 use posiion gammas ha are compued using opion gammas aen from he OpionMerics Ivy DB daabase when hey are available, and Blac-Scholes gammas when OpionMerics gammas are no available. OpionMerics compues gammas using sandard indusry pracices: i uses he binomial model o capure he possibiliy of early exercise of American opion he acual implied volailiy of he opion for which he gamma is being compued, he erm srucure of ineres rae and esimaes of he dividend yield on he underlying soc and he fuure ex-dividend daes (OpionMerics LLC 2005, pp ). Thus he OpionMerics gammas capure boh he American feaure of exchange-raded individual equiy opions and he dependence of opion implied volailiies on he opion srie price and ime o expiraion. A limiaion of he OpionMerics gammas is ha hey are no always available. Fir opions ha are well away-from-he-money frequenly have quoed prices ha violae elemenary arbirage bounds. In such cases (specifically, when he bid-as average violaes elemenary arbirage bounds) OpionMerics is unable o compue he implied volailiy, and hus is unable o compue he opion gamma. For our purposes his problem is no imporan because he gammas of away-from-he-money opions end o be small regardless of he opionpricing model used o compue hem, and we can safely use Blac-Scholes gammas in such cases. Second, he OpionMerics daa begin only in 1996, and hus are no available during he firs half of our sample period of However, his problem is no as severe as i migh 27

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