Demand-Based Option Pricing

Transcription

1 Demand-Based Opion Pricing Nicolae Gârleanu Universiy of California a Berkeley, CEPR, and NBER Lasse Heje Pedersen New York Universiy, CEPR, and NBER Allen M. Poeshman Universiy of Illinois a Urbana-Champaign We model demand-pressure effecs on opion prices. The model shows ha demand pressure in one opion conrac increases is price by an amoun proporional o he variance of he unhedgeable par of he opion. Similarly, he demand pressure increases he price of any oher opion by an amoun proporional o he covariance of he unhedgeable pars of he wo opions. Empirically, we idenify aggregae posiions of dealers and end-users using a unique daase, and show ha demand-pressure effecs make a conribuion o wellknown opion-pricing puzzles. Indeed, ime-series ess show ha demand helps explain he overall expensiveness and skew paerns of index opions, and cross-secional ess show ha demand impacs he expensiveness of single-sock opions as well. One of he major achievemens of financial economics is he no-arbirage heory ha deermines derivaive prices independenly of invesor demand. Building on he seminal conribuions of Black and Scholes (1973) and Meron (1973), a large lieraure develops various parameric implemenaions of he heory. This lieraure is surveyed by Baes (2003), who emphasizes ha i canno fully capure much less explain he empirical properies of opion prices and concludes ha here is a need for a new approach o pricing derivaives ha focuses on he financial inermediaion of he underlying risks by opion marke-makers (Baes 2003, p. 400). We are graeful for helpful commens from David Baes, Nick Bollen, Oleg Bondarenko, Menachem Brenner, Andrea Buraschi, Josh Coval, Domenico Cuoco, Apoorva Koicha, Sophie Ni, Jun Pan, Neil Pearson, Josh Whie, and especially from Seve Figlewski, as well as from seminar paricipans a he Caesarea Cener Conference, Columbia Universiy, Cornell Universiy, Darmouh Universiy, Harvard Universiy, HEC Lausanne, he 2005 Inquire Europe Conference, London Business School, he 2005 NBER Behavioral Finance Conference, he 2005 NBER Universiies Research Conference, he NYU-ISE Conference on he Transformaion of Opion Trading, New York Universiy, Nomura Securiies, Oxford Universiy, Sockholm Insiue of Financial Research, Texas A&M Universiy, Universiy of California a Berkeley, Universiy of Chicago, Universiy of Illinois a Urbana- Champaign, Universiy of Pennsylvania, Universiy of Vienna, and he 2005 WFA Meeings. Send correspondence o Nicolae Gârleanu, Haas School of Business, Universiy of California, Berkeley, CA [email protected]. C The Auhor Published by Oxford Universiy Press on behalf of The Sociey for Financial Sudies. All righs reserved. For Permissions, please [email protected]. doi: /rfs/hhp005 Advance Access publicaion February 25, 2009

2 The Review of Financial Sudies / v 22 n We ake on his challenge. Our model depars fundamenally from he noarbirage framework by recognizing ha opion marke-makers canno perfecly hedge heir invenories, and, consequenly, opion demand impacs opion prices. We obain explici expressions for he effecs of demand on opion prices, provide empirical evidence consisen wih he demand-pressure model using a unique daase, and show ha demand-pressure effecs can play a role in resolving he main opion-pricing puzzles. The saring poin of our analysis is ha opions are raded because hey are useful and, herefore, opions canno be redundan for all invesors (Hakansson 1979). We denoe he agens who have a fundamenal need for opion exposure as end-users. Inermediaries such as marke-makers provide liquidiy o end-users by aking he oher side of he end-user ne demand. If compeiive inermediaries can hedge perfecly as in a Black-Scholes-Meron economy hen opion prices are deermined by no-arbirage and demand pressure has no effec. In realiy, however, even inermediaries canno hedge opions perfecly ha is, even hey face incomplee markes because of he impossibiliy of rading coninuously, sochasic volailiy, jumps in he underlying, and ransacion coss (Figlewski 1989). 1 In addiion, inermediaries are sensiive o risk, e.g., because of capial consrains and agency. In ligh of hese facs, we consider how opions are priced by compeiive risk-averse dealers who canno hedge perfecly. In our model, dealers rade an arbirary number of opion conracs on he same underlying a discree imes. Since he dealers rade many opion conracs, cerain risks ne ou, while ohers do no. The dealers can hedge par of he remaining risk of heir derivaive posiions by rading he underlying securiy and risk-free bonds. We consider a general class of disribuions for he underlying, which can accommodae sochasic volailiy and jumps. Dealers rade opions wih endusers. The model is agnosic abou he end-users reasons for rade, which are irrelevan for our resuls and heir empirical implemenaion. We compue equilibrium prices as funcions of demand pressure, ha is, he prices ha induce he uiliy-maximizing dealers o supply precisely he opion quaniies ha he end-users demand. We show explicily how demand pressure eners ino he pricing kernel. Inuiively, a posiive demand pressure in an opion increases he pricing kernel in he saes of naure in which an opimally hedged posiion has a posiive payoff. This pricing-kernel effec increases he price of he opion, which enices he dealers o sell i. Specifically, a marginal change in he demand pressure in an opion conrac increases is price by an amoun proporional o he variance of he unhedgeable par of he opion, where he variance is compued under a cerain probabiliy measure depending on he demand. Similarly, demand pressure increases he price of any 1 Opions may also be impossible o replicae due o asymmeric informaion (Back 1993; Easley, O Hara, and Srinivas 1998). 4260

3 Demand-Based Opion Pricing oher opion by an amoun proporional o he covariance of heir unhedgeable pars. Hence, while demand pressure in a paricular opion raises is price, i also raises he prices of oher opions on he same underlying. Our main heoreical resuls relaing opion-price effecs o he variance or covariance of he unhedgeable par of opion-price changes hold regardless of he source of marke incompleeness. The magniudes of he variances and covariances, and hence of he demand-based opion-price effecs, depend upon he paricular source of marke incompleeness. Empirically, we es he specific predicions of he model under he assumpions ha marke incompleeness sems from discree rading, sochasic volailiy, or jumps. We use a unique daase o idenify aggregae daily posiions of dealers and end-users. In paricular, we define dealers as marke-makers and end-users as proprieary raders and cusomers of brokers. 2 We are he firs o documen ha end-users have a ne long posiion in S&P 500 index opions wih large ne posiions in ou-of-he-money (OTM) pus. 3 Since opions are in zero ne supply, his implies ha dealers are shor index opions. 4 We esimae ha hese large shor dealer posiions lead o daily dela-hedged profis and losses varying beween $100 million and $-100 million, and cumulaive dealer profis of approximaely $800 million over our six-year sample. Hence, consisen wih our framework, dealers face significan unhedgeable risk and are compensaed for bearing i. Inroducing enry of dealers ino our model, we show ha he oal risk-bearing marke-making capaciy increases wih he demand pressure, and we esimae empirically ha he marke-maker s risk-reurn profile is consisen wih equilibrium enry. The end-user demand for index opions can help o explain he wo puzzles ha index opions appear o be expensive, and ha low-moneyness opions seem o be especially expensive (Rubinsein 1994; Longsaff 1995; Baes 2000; Jackwerh 2000; Coval and Shumway 2001; Bondarenko 2003; Amin, Coval, and Seyhun 2004; Driessen and Maenhou 2008). In he ime series, he model-based impac of demand for index opions is posiively relaed o heir expensiveness, measured by he difference beween heir implied volailiy and he volailiy measure of Baes (2006). Indeed, we esimae ha on he order of one-hird of index-opion expensiveness can be accouned for by demand effecs. 5 In addiion, he link beween demand and prices is sronger following recen dealer losses, as would be expeced if dealers are more risk averse a such imes. Likewise, he seepness of he smirk, measured by he difference 2 The empirical resuls are robus o classifying proprieary raders as eiher dealers or end-users. 3 These posiions are consisen wih he end-users suffering from crashophobia, as suggesed by Rubinsein (1994). 4 Opion raders recognize he imporance of demand effecs; e.g., Vanessa Gray, direcor of global equiy derivaives, Dresdner Kleinwor Benson, saes ha opion-implied volailiy skew is heavily influenced by supply and demand facors. 5 Premiums for sochasic volailiy or jump risk, as well as premiums for oher risk facors, in all likelihood are also conribuing o opion expensiveness. 4261

4 The Review of Financial Sudies / v 22 n beween he implied volailiies of low-moneyness opions and a-he-money (ATM) opions, is posiively relaed o he skew of opion demand. Anoher opion-pricing puzzle is he significan difference beween indexopion prices and he prices of single-sock opions, despie he relaive similariy of he underlying disribuions (e.g., Bakshi, Kapadia, and Madan 2003; Bollen and Whaley 2004). In paricular, single-sock opions appear cheaper and heir smile is flaer. Consisenly, we find ha he demand paern for single-sock opions is very differen from ha of index opions. For insance, end-users are ne shor single-sock opions no long, as in he case of index opions. Demand paerns furher help o explain he cross-secional pricing of single-sock opions. Indeed, individual sock opions are relaively cheaper for socks wih more negaive demand for opions. The aricle is relaed o several srands of lieraure. Firs, a large lieraure documens opion puzzles relaive o he exising models (cied above). 6 Second, he lieraure on opion pricing in he conex of rading fricions and incomplee markes derives bounds on opion prices (Soner, Shreve, and Cvianic 1995; Bernardo and Ledoi 2000; Cochrane and Saa-Requejo 2000; Consaninides and Perrakis 2002; Consaninides, Jackwerh, and Perrakis forhcoming). Raher han deriving bounds, we compue explici prices based on he demand pressure by end-users. We furher complemen his lieraure by aking porfolio consideraions ino accoun, ha is, he effec of demand for one opion on he prices of oher opions. Third, he general idea of demand pressure effecs goes back a leas o Keynes (1923) and Hicks (1939), who considered fuures markes. Our model is he firs o apply his idea o opion pricing and o incorporae he imporan feaures of opions markes, namely dynamic rading of many asses, hedging using he underlying and bonds, sochasic volailiy, and jumps. The generaliy of our model also makes i applicable o oher siuaions e.g., sock-index addiions (Shleifer 1986, Wurgler and Zhuravskaya 2002, Greenwood 2005), he fixed-income marke (Newman and Rierson 2004), morgage-backed securiy markes (Gabaix, Krishnamurhy, and Vigneron 2007), fuures markes (de Roon, Nijman, and Veld 2000), and opion end-users over-reacion o changes in implied volailiy (Sein 1989; Poeshman 2001). Fourh, he lieraure on uiliy-based opion pricing derives he price of he firs marginal opion ha would make an agen indifferen beween buying he opion and no buying i (Rubinsein 1976; Brennan 1979; Sapleon and Subrahmanyam 1984; Hugonnier, Kramkov, and Schachermayer 2005, and references herein), and we show how opion-prices change when demand is nonrivial. A closely relaed paper is Bollen and Whaley (2004), which demonsraes ha changes in implied volailiy are correlaed wih signed opion volume. These 6 In conras o hese papers, Benzoni, Collin-Dufresne, and Goldsein (2005) find ha opion prices can be raionalized by cerain preferences combined wih persisen long-run jump risk. 4262

5 Demand-Based Opion Pricing empirical resuls se he sage for our analysis by showing ha changes in opion demand lead o changes in opion prices while leaving open he quesion of wheher he level of opion demand impacs he overall level (i.e., expensiveness) of opion prices or he overall shape of implied-volailiy curves. 7 We complemen Bollen and Whaley (2004) by providing a heoreical model, by invesigaing empirically he relaionship beween he level of end-user demand for opions and he level and overall shape of implied-volailiy curves, and by esing precise quaniaive implicaions of our model. In paricular, we documen ha end-users end o have a ne long SPX opion posiion and a shor equiy-opion posiion, hus helping o explain he relaive expensiveness of index opions. We also show ha here is a srong downward skew in he ne demand of index bu no equiy opions, which helps o explain he difference in he shapes of heir overall implied-volailiy curves. In addiion, we demonsrae ha opion prices are beer explained by model-based raher han simple nonmodel-based use of demand. 1. A Model of Demand Pressure We consider a discree-ime infinie-horizon economy. There exis a risk-free asse paying ineres a he rae of R f 1 per period and a risky securiy ha we refer o as he underlying securiy. A ime, he underlying has an exogenous sricly posiive price 8 of S, dividend D, and an excess reurn of R e = (S + D )/S 1 R f. The disribuion of fuure prices and reurns is characerized by a Markov sae variable X, which includes he curren underlying price level, X 1 = S, and may also include he curren level of volailiy, he curren jump inensiy, ec. We assume ha (R e, X ) saisfies a Feller-ype condiion (made precise in he Appendix) and ha X is bounded for every. The economy furher has a number of derivaive securiies, whose prices are o be deermined endogenously. A derivaive securiy is characerized by is index i I, where i collecs he informaion ha idenifies he derivaive and is payoffs. For a European opion, for insance, he srike price, mauriy dae, and wheher he opion is a call or pu suffice. The se of derivaives raded a ime is denoed by I, and he vecor of prices of raded securiies is p = (p i) i I. The payoffs of he derivaives depend on X. We noe ha he heory is compleely general and does no require ha he derivaives have payoffs ha depend on he underlying price. In principle, he derivaives could be any securiies in paricular, any securiies whose prices are affeced by demand 7 Indeed, Bollen and Whaley (2004) find ha a nonrivial par of he opion-price impac from day signed opion volume dissipaes by day All random variables are defined on a probabiliy space (,F, Pr), wih an associaed filraion {F : 0} of sub-σ-algebras represening he resoluion over ime of informaion commonly available o agens. 4263

6 The Review of Financial Sudies / v 22 n pressure. Furher, i is sraighforward o exend our resuls o a model wih any number of exogenously priced securiies, as discussed in Secion 2.4. While we use he model o sudy opions in paricular, we hink ha he generaliy helps o illuminae he driving forces behind he resuls and, furher, i allows fuure applicaions of he heory in oher markes. The economy is populaed by wo kinds of agens: dealers and endusers. Dealers are compeiive and here exiss a represenaive dealer who has consan absolue-risk aversion, ha is, his uiliy for remaining life ime consumpion is [ ] U(C, C +1,...) = E ρ v u(c v ), (1) where u(c) = 1 γ e γc and ρ < 1 is a discoun facor. A any ime, he dealer mus choose he consumpion C, he dollar invesmen in he underlying θ, and he number of derivaives held q = (q i ) i I, so as o maximize his uiliy while saisfying he ransversaliy condiion lim E[ρ e kw ] = 0. The dealer s wealh evolves as v= W +1 = (W C )R f + q (p +1 R f p ) + θ R e +1. (2) In he real world, end-users rade opions for a variey of reasons, such as porfolio insurance, agency reasons, behavioral reasons, insiuional reasons, ec. Raher han rying o capure hese various rading moives endogenously, we assume ha end-users have an exogenous aggregae demand for derivaives of d = (d i ) i I a ime. The disribuion of fuure demand is characerized by X. We also assume, for echnical reasons, ha demand pressure is zero afer some ime T, ha is, d = 0for > T. Derivaive prices are se hrough he ineracion beween dealers and endusers in a compeiive equilibrium. Definiion 1. A price process p = p (d, X ) is a (compeiive Markov) equilibrium if, given p, he represenaive dealer opimally chooses a derivaive holding q such ha derivaive markes clear, i.e., q + d = 0. We noe ha he assumpion of inelasic end-user demand is made for noaional simpliciy only, and is unimporan for he resuls we derive below. The key o our asse-pricing approach is he insigh ha, by observing he aggregae quaniies held by dealers in equilibrium, one can deermine he derivaive prices consisen wih he dealers uiliy maximizaion, ha is, inver prices from quaniies. Our goal is o deermine how derivaive prices depend on he demand pressure d coming from end-users. All ha maers is ha end-users have demand curves ha resul in dealers choosing o hold, a he marke prices, a posiion of q = d ha we observe in he daa. 4264

7 Demand-Based Opion Pricing To deermine he represenaive dealer s opimal behavior, we consider his value funcion J(W ;, X), which depends on his wealh W, he sae of naure X, and ime. Then, he dealer solves he following maximizaion problem: max 1 C,q,θ γ e γc + ρe [J(W +1 ; + 1, X +1 )] (3) s.. W +1 = (W C )R f + q (p +1 R f p ) + θ R e +1. (4) The value funcion is characerized in he following lemma. Lemma 1. If p = p (d, X ) is he equilibrium price process and k =, hen he dealer s value funcion and opimal consumpion are given by γ(r f 1) R f J(W ;, X ) = 1 k e k(w +G (d,x )) (5) C = R f 1 (W + G (d, X )) (6) R f and he sock and derivaive holdings are characerized by he firs-order condiions [ 0 = E e k(θ R+1 e +q (p +1 R f p )+G +1 (d +1,X +1 )) R+1 e ] (7) 0 = E [ e k(θ R e +1 +q (p +1 R f p )+G +1 (d +1,X +1 )) (p +1 R f p ) ], (8) where, for T,G (d, X ) is derived recursively using (7), (8), and e kr f G (d,x ) = R f ρe [e k(q (p +1 R f p )+θ R e +1 +G +1(d +1,X +1 )) ] (9) and for > T, he funcion G (d, X ) = Ḡ(X ), where (Ḡ(X ), θ(x )) solves e kr f Ḡ(X ) = R f ρe [e k( θ R+1 e +Ḡ(X +1 )) ] (10) [ ] 0 = E e k( θ R+1 e +Ḡ(X +1 )) R e +1. (11) The opimal consumpion is unique. The opimal securiy holdings are unique, provided ha heir payoffs are linearly independen. While dealers compue opimal posiions given prices, we are ineresed in invering his mapping and compue he prices ha make a given posiion opimal. The following proposiion ensures ha his inversion is possible. Proposiion 1. Given any demand pressure process d for end-users, here exiss a unique equilibrium price process p. 4265

8 The Review of Financial Sudies / v 22 n Before considering explicily he effec of demand pressure, we make a couple of simple pariy observaions ha show how o rea derivaives ha are linearly dependen, such as European pus and calls wih he same srike and mauriy. For simpliciy, we do his only in he case of a nondividend paying underlying, bu he resuls can easily be exended. We consider wo derivaives i and j such ha a nonrivial linear combinaion of heir payoffs lies in he span of exogenously priced securiies, i.e., he underlying and he bond: Proposiion 2. Suppose ha D = 0 and pt i = p j T + α + βs T. Then: (i) For any demand pressure, d, he equilibrium prices of he wo derivaives are relaed by p i = p j (T ) + αr f + βs. (12) (ii) Changing he end-user demand from (d i, d j ) o (d i + a, d j a), for any a R, has no effec on equilibrium prices. The firs par of he proposiion is a general version of he well-known pu-call pariy. I shows ha if payoffs are linearly dependen, hen so are prices. The second par of he proposiion shows ha linearly dependen derivaives have he same demand-pressure effecs on prices. Hence, in our empirical exercise, we can aggregae he demand of calls and pus wih he same srike and mauriy. Tha is, a demand pressure of d i calls and d j pus is he same as a demand pressure of d i + d j calls and 0 pus (or vice versa). 2. Price Effecs of Demand Pressure To see where we are going wih he heory, consider he empirical problem ha we ulimaely face: On any given day, around 120 SPX opion conracs of various mauriies and srike prices are raded. The demands for all hese differen opions poenially affec he price of, say, he one-monh ATM SPX opion because all of hese opions expose he marke-makers o unhedgeable risk. Wha is he aggregae effec of all hese demands? The model answers his quesion by showing how o compue he impac of demand d j for any one derivaive on he price p i of he one-monh ATM opion. The aggregae effec is hen he sum of all of he individual demand effecs, ha is, he sum of all he demands weighed by heir model-implied price impacs p i/ d j. We firs characerize p i/ d j in complee generaliy, as well as oher general demand effecs on prices (Secion 2.1). We hen show how o compue p i/ d j specifically when unhedgeable risk arises from, respecively, discreeime hedging, jumps in he underlying asse price, and sochasic-volailiy risk (Secion 2.2). Secion 2.3 sudies he price effec wih an endogenous number of dealers, which gives rise o esable implicaions relevan for our 4266

9 Demand-Based Opion Pricing cross-secion of equiy opions. Finally, Secion 2.4 generalizes o muliple underlying securiies. 2.1 General resuls We hink of he price p, he hedge posiion θ in he underlying, and he consumpion funcion G as funcions of d j and X. Alernaively, we can hink of he dependen variables as funcions of he dealer holding q j and X, keeping in mind he equilibrium relaion ha q = d. For now we use his laer noaion. A mauriy dae T, an opion has a known price p T. A any prior dae,he price p can be found recursively by invering (8) o ge [ ] E e k(θ R+1 e +q p +1 +G +1) p+1 p = [ ], (13) R f E e k(θ R+1 e +q p +1 +G +1) where he hedge posiion in he underlying, θ,solves 0 = E [ e k(θ R e +1 +q p +1 +G +1) R e +1 ], (14) and where G is compued recursively as described in Lemma 1. Equaions (13) and (14) can be wrien in erms of a demand-based pricing kernel: Theorem 1. Prices p and he hedge posiion θ saisfy ( ) p = E m d 1 +1 p +1 = E d R (p +1) f (15) ( 0 = E m d +1 R+1 e ) 1 = E d ( ) R e R +1, f (16) where he pricing kernel m d is a funcion of demand pressure d: m d +1 = = e k(θ R e +1 +q p +1 +G +1) R f E [ e k(θ R e +1 +q p +1 +G +1) ] (17) e k(θ R e +1 d p +1 +G +1) [ ], (18) R f E e k(θ R+1 e d p +1 +G +1) and E d is expeced value wih respec o he corresponding risk-neural measure, i.e., he measure wih a Radon-Nikodym derivaive wih respec o he objecive measure of R f m d +1. To undersand his pricing kernel, suppose for insance ha end-users wan o sell derivaive i such ha d i < 0, and ha his is he only demand pressure. In equilibrium, dealers ake he oher side of he rade, buying q i = d i >

10 The Review of Financial Sudies / v 22 n unis of his derivaive, while hedging heir derivaive holding using a posiion θ in he underlying. The pricing kernel is small whenever he unhedgeable par q p +1 + θ R+1 e is large. Hence, he pricing kernel assigns a low value o saes of naure in which a hedged posiion in he derivaive pays off profiably, and i assigns a high value o saes in which a hedged posiion in he derivaive has a negaive payoff. This pricing-kernel effec decreases he price of his derivaive, which is wha enices he dealers o buy i. I is ineresing o consider he firs-order effec of demand pressure on prices. In order o do so, we firs define he unhedgeable par of he price changes of a securiy. Definiion 2. The unhedgeable price change p +1 k of any securiy k is defined as is excess reurn p+1 k R f p k opimally hedged wih he sock posiion Cov d (pk +1,Re +1 ) Var d (Re +1 ) : p k +1 = R 1 f ( ( ) p+1 k R f p k Covd p k +1, R+1) e ( ) Var d R e R e +1. (19) +1 We prove in he Appendix he following resul. Theorem 2. The sensiiviy of he price of securiy i o demand pressure in securiy j is proporional o he covariance of heir unhedgeable risks: p i d j = γ(r f 1)E d ( p i +1 p +1) j = γ(r f 1)Cov d ( p i +1, p +1) j. (20) This resul is inuiive: i saes ha he demand pressure in an opion j increases he opion s own price by an amoun proporional o he variance of he unhedgeable par of he opion and he aggregae risk aversion of dealers. We noe ha since a variance is always posiive, he demand-pressure effec on he securiy iself is naurally always posiive. Furher, his demand pressure affecs anoher opion i by an amoun proporional o he covariance of heir unhedgeable pars. Under he condiion saed below, we can show ha his covariance is posiive, and herefore ha demand pressure in one opion also increases he price of oher opions on he same underlying. Proposiion 3. Demand pressure in any securiy j: (i) increases is own price, ha is, p j 0; d j p (ii) increases he price of anoher securiy i, ha is, i 0, provided d j ha E d [pi +1 S +1] and E d [p j +1 S +1] are convex funcions of S +1 and Cov d (pi +1, p j +1 S +1)

11 Demand-Based Opion Pricing The condiions imposed in par (ii) are naural. Firs, we require ha prices inheri he convexiy propery of he opion payoffs in he underlying price. Convexiy lies a he hear of his resul, which, informally speaking, saes ha higher demand for convexiy (or gamma, in opion-rader lingo) increases is price, and herefore hose of all opions. Second, we require ha Cov d (pi +1, p j +1 S +1) 0, ha is, changes in he oher variables have a similar impac on boh opion prices for insance, boh prices are increasing in he volailiy or demand level. Noe ha boh condiions hold if boh opions maure afer one period. The second condiion also holds if opion prices are homogenous (of degree 1) in (S, K ), where K is he srike, and S is independen of X 1 (X 2,...,X n ). I is ineresing o consider he oal price ha end-users pay for heir demand d a ime. Vecorizing he derivaives from Theorem 2, we can firs-order approximae he price around zero demand as p p (d = 0) + γ(r f 1)E d ( p +1 p +1 )d. (21) Hence, he oal price paid for he d derivaives is d p = d p (d = 0) + γ(r f 1)d Ed ( p +1 p +1 )d (22) = d p (d = 0) + γ(r f 1)Var d (d p +1). (23) The firs erm d p (d = 0) is he price ha end-users would pay if heir demand pressure did no affec prices. The second erm is oal variance of he unhedgeable par of all of he end-users posiions. While Proposiion 3 shows ha demand for an opion increases he prices of all opions, he size of he price effec is, of course, no he same for all opions. Nor is he effec on implied volailiies he same. Under cerain condiions, demand pressure in low-srike opions has a larger impac on he implied volailiy of low-srike opions, and conversely for high-srike opions. The following proposiion makes his inuiively appealing resul precise. For simpliciy, he proposiion relies on unnecessarily resricive assumpions. We le p(p, K, d), respecively p(c, K, d), denoe he price of a pu, respecively a call, wih srike price K and one period o mauriy, where d is he demand pressure. I is naural o compare low-srike and high-srike opions ha are equally far ou of he money. We do his by considering an OTM pu wih he same price as an OTM call. Proposiion 4. Assume ha he one-period risk-neural disribuion of he underlying reurn is symmeric and ha X 1 +1 is independen of S +1. Consider demand pressure d >0 in an opion wih srike K < R f S ha maures afer one rading period. Then here exiss a value K such ha, for all K K and K such ha p(p, K, 0) = p(c, K, 0), i holds ha p(p, K, d) > p(c, K, d). Tha is, he price of he OTM pu p(p, K, ) is more affeced by he demand 4269

12 The Review of Financial Sudies / v 22 n pressure han he price of OTM call p(c, K, ). The reverse conclusion applies if here is demand for a high-srike opion. Fuure demand pressure in a derivaive j also affecs he curren price of derivaive i. As above, we consider he firs-order price effec. This is slighly more complicaed, however, since we canno differeniae wih respec o he unknown fuure demand pressure. Insead, we scale he fuure demand pressure, ha is, we consider fuure demand pressures d j s = ɛd j q j s = ɛq j s )forsomeɛ R, s >, and j. s for fixed d (equivalenly, Theorem 3. Le p (0) denoe he equilibrium derivaive prices wih 0 demand pressure. Fixing a process d wih d = 0 for all > T and a given T, he equilibrium prices p wih a demand pressure of ɛd is [ p = p (0) + γ(r f 1) E 0 ( p+1 p +1 ) d + s> R (s ) f E 0 ( ps+1 p s+1 d ) ] s ɛ + O(ɛ 2 ). (24) This heorem shows ha he impac of curren demand pressure d on he price of a derivaive i is given by he amoun of hedging risk ha a marginal posiion in securiy i would add o he dealer s porfolio, ha is, i is he sum of he covariances of is unhedgeable par wih he unhedgeable par of all he oher securiies, muliplied by heir respecive demand pressures. Furher, he impac of fuure demand pressures d s is given by he expeced fuure hedging risks. Of course, he impac increases wih he dealers risk aversion. Nex, we specialize he seup o several differen sources of unhedgeable risk o show how o compue hese covariances, and herefore he price impacs, explicily. 2.2 Implemenaion: specific cases We consider now hree examples of unhedgeable risk for he dealers, arising from (i) he inabiliy o hedge coninuously, (ii) jumps in he underlying price, and (iii) sochasic-volailiy risk, respecively. We focus on small hedging periods and derive he resuls informally while relegaing a more rigorous reamen o he Appendix. The coninuously compounded risk-free ineres rae is denoed by r, i.e., he risk-free reurn over one ime period is R f = e r. We are ineresed in he price p i = p i(d, X ) of opion i as a funcion of demand pressure d and he sae variable X. (Remember ha S = X 1.) We denoe he opion price wihou demand pressure by f, ha is, f i (, X ):= p i(d = 0, X ), and assume hroughou ha f is smooh for < T. We use he noaion f i = f i (, X ), f i = f i (, X ), fs i = S f i (, X ), fss i = 2 f i (, X S 2 ), S = S +1 S, and so on. 4270

13 Demand-Based Opion Pricing Case 1: Discree-ime rading. To focus on he specific risk due o discreeime rading (raher han coninuous rading), we consider a sock price ha is a diffusion process driven by a Brownian moion 9 wih no oher sae variables (i.e., X = S). In his case, markes would be complee wih coninuous rading, and, hence, he dealer s hedging risk arises solely from his rading only a discree imes, spaced ime unis apar. The change in he opion price evolves approximaely according o p i +1 = f i + f i S S f i SS ( S)2 + f i, (25) while he unhedgeable opion-price change is e r p +1 i = pi +1 er p i f S i ( S+1 e r ) S (26) = r f i + f i + r f i S S f i SS ( S)2. (27) The covariance of he unhedgeable pars of wo opions i and j is ( Cov e r p +1 i, er p j ) = f SS i f j SS Var (( S) 2 ), (28) so ha, by Theorem 2, we conclude ha he effec on he price of demand a d = 0is p i d j = γrvar (( S) 2 ) 4 fss i f j SS + o( 2 ) and he effec on he Black-Scholes implied volailiy ˆσ i is (29) ˆσ i d j = γrvar (( S) 2 ) 4 f i SS ν i f j SS + o( 2 ), (30) where ν i is he Black-Scholes vega. 10 Ineresingly, he raio of he Black- Scholes gamma o he Black-Scholes vega, fss i /νi, does no depend on moneyness, so he firs-order effec of demand wih discree rading risk is o change he level, bu no he slope, of he implied-volailiy curves. Inuiively, he impac of he demand for opions of ype j depends on he gamma of hese opions, f j SS, since he dealers canno hedge he nonlineariy of he payoff. The calculaions above show ha he effec of discree-ime rading is small if hedging is frequen. More precisely, he effec is of he order of Var (( S) 2 ), 9 Sricly speaking, we need all price processes o be bounded, e.g., runcaed. 10 Even hough he volailiy is consan wihin he Black-Scholes model, we follow he sandard convenion ha defines he Black-Scholes implied volailiy as he volailiy ha, when fed ino he Black-Scholes model, makes he model price equal o he opion price, and he Black-Scholes vega as he parial derivaive measuring he change in he opion price when he volailiy fed ino he Black-Scholes model changes. 4271

14 The Review of Financial Sudies / v 22 n namely 2. Hence, adding up T/ erms of his magniude corresponding o demand in each period beween ime 0 and mauriy T resuls in a oal effec of order, which approaches zero as approaches zero. This is consisen wih he Black-Scholes-Meron resul of perfec hedging in coninuous ime. As he nex examples show, he risks of jumps and sochasic volailiy do no vanish for small (specifically, hey are of order ). Case 2: Jumps in he underlying. Suppose now ha S isajumpdiffusion wih i.i.d. jump size and jump inensiy π (i.e., jump probabiliy over a period of approximaely π ). The unhedgeable price change is e r p +1 i = r f i + f i + r fs i S + ( fs i S θ i) S1 (no jump) + κ i 1 (jump), (31) where κ i = f i (S + η) f i θ i η (32) is he unhedgeable risk in case of a jump of size η. I hen follows ha he effec on he price of demand a d = 0is p i d j = γr [( f i S S θ i)( f j S S θ j) Var ( S) + π E ( κ i κ j)] + o( ) (33) and he effec on he Black-Scholes implied volailiy ˆσ i is ˆσ i d j = γr[( f i S S θ i)( f j S S θ j) Var ( S) + π E ( κ i κ j)] ν i + o( ). (34) The erms of he form f i S S θ i arise because he opimal hedge θ differs from he opimal hedge wihou jumps, f i S S, which means ha some of he local noise is being hedged imperfecly. If he jump probabiliy is small, however, hen his effec is small (i.e., i is second order in π). In his case, he main effec comes from he jump risk κ (kappa). We noe ha, while convenional wisdom holds ha Black-Scholes gamma is a measure of jump risk, his is rue only for he small local jumps considered in Case 1. Large jumps have qualiaively differen implicaions capured by kappa. For insance, a far-otm pu may have lile gamma risk, bu, if a large jump can bring he opion in he money, he opion may have kappa risk. I can be shown ha his jump-risk effec (34) means ha demand can affec he slope of he implied-volailiy curve o he firs order and generae a smile Of course, he jump risk also generaes smiles wihou demand-pressure effecs; he resul is ha demand can exacerbae hese. 4272

15 Demand-Based Opion Pricing Anoher imporan source of unhedgeabiliy, iself an imporan saisical propery of underlying prices and herefore playing a significan role in modern opion-pricing models, is sochasic volailiy. We illusrae below how o calculae he price impac of demand in is presence. Case 3: Sochasic-volailiy risk. We now le he sae variable be X = (S, σ ), where he sock price S is a diffusion wih volailiy σ, which is also a diffusion, driven by an independen Brownian moion. The opion price p i = f i (, S, σ ) has unhedgeable risk given by e r p i +1 = pi +1 er p i θi R e +1 (35) = r f i + f i + f i S S r + f i σ σ +1, (36) so ha he effec on he price of demand a d = 0is p i d j = γrvar( σ) f i σ f j σ + o( ) (37) and he effec on he Black-Scholes implied volailiy ˆσ i is ˆσ i d j = γrvar( σ) f i σ ν i f j σ + o( ). (38) Inuiively, volailiy risk is capured o he firs order by f σ. This derivaive is no exacly he same as Black-Scholes vega, since vega is he price sensiiviy o a permanen volailiy change, whereas f σ measures he price sensiiviy o a volailiy change ha may decay. If volailiy mean revers a he rae φ, hen, for an opion wih mauriy a ime + T,wehave fσ i = ν i E σ ( +T σ s ds T ) σ = ν i 1 e φt φt. (39) Hence, combining (39) wih (38) shows ha sochasic-volailiy risk affecs he level, bu no he slope, of he implied-volailiy curves o he firs order. We make use of each of hese hree explicily modeled sources of unhedgeable risk in our empirical work, where we base he model-implied empirical measures of demand impac on he formulae (30), (34), and (38), respecively. Our resuls could be generalized by inroducing a ime-varying jump inensiy, jumps in he volailiy (mahemaically, his would be similar o our analysis of jumps in he underlying), or a more complicaed correlaion srucure for he sae variables. While such generalizaions would add realism, we wan o es he effec of he demand in he presence of he mos basic sources of unhedgeable risk considered here. 4273

16 The Review of Financial Sudies / v 22 n Equilibrium number of dealers The number of dealers and heir aggregae risk-bearing capaciy are deermined in equilibrium by dealers radeoff beween he coss and benefis of making markes. This secion shows ha he aggregae dealer risk aversion γ can be deermined as he oucome of equilibrium dealer enry and, furher, provides some naural properies of γ. We consider an infiniesimal agen wih risk aversion γ, who could become a dealer a ime = 0aacosofM dollars. There is a coninuum of available dealers indexed by i [0, ) wih risk aversion γ (i) increasing in i. The disribuion of i has no aoms, and is denoed by μ. We assume ha 0 γ (i) 1 dμ(i) =. Paying he cos M allows he dealer o rade derivaives a all imes. This cos could correspond o he cos of a sea on he CBOE, he salaries of raders, he cos of running a back office, ec. In Secion 3.3, we esimae he coss and benefis of being a marke-maker o be of he same magniude, consisen wih his equilibrium condiion. In he Appendix we show ha here exiss an equilibrium o he dealer enry game and ha, naurally, he leas risk-averse dealers i [0, ī]forsomeī R ener he marke o profi from price responses o end-user demand. Furher, he aggregae dealer demand is he same as ha of a represenaive dealer wih risk aversion γ given by γ 1 = ī 0 γ (i) 1 dμ(i). Proposiion 5. Equilibrium enry of dealers a ime 0 implies he following: 1. Suppose ha he end-user demand is d = ɛd for some demand process d. a. A higher expeced end-user demand leads o more enry of dealers. Specifically, he equilibrium number of dealers increases in ɛ and he equilibrium dealer risk aversion γ decreases in ɛ. b. If poenial dealers have differen risk aversions, hen prices are more disored by demand if demand is larger. Rigorously, he absolue price deviaion of any unhedgeable opion from is zero-demand value increases sricly in ɛ on [0, ɛ],forsome ɛ > 0. c. If all poenial dealers have he same risk aversion, hen derivaive prices are independen of ɛ. Neverheless, derivaive prices vary wih demand in he ime series. 2. The equilibrium number of dealers decreases wih he cos M of being a dealer. Hence, he aggregae dealer risk aversion γ increases wih he (opporuniy) cos M. Par 1(b) of he proposiion saes he naural resul ha increased demand leads o larger price deviaions. Indeed, while larger demand leads o enry of dealers, hese dealers are increasingly risk averse, leading o he increased demand effec. Par 1(c) gives he surprising resul ha he overall level of demand does no affec opion prices when all dealers have he same risk aversion because of 4274

17 Demand-Based Opion Pricing he enry of dealers. Noe, however, ha even in his case demand affecs prices in he ime series, ha is, a imes wih more demand, prices are more affeced. This ime-series effec would be reduced if dealers could ener a any ime. The ime 0 enry capures ha he decision o se up a rading capabiliy is made only rarely due o significan fixed coss, alhough, of course, enry and exi does happen over ime in he real world. 2.4 Muliple underlying securiies So far, we have considered dealers who rade opions on he same underlying, bu our resuls exend o he case in which dealers rade opions on muliple underlying asses: Indeed, Theorems 1 2 and Proposiions 1 2 coninue o hold, subjec o reaing R e and θ as vecors and herefore he variances Var d (Re +1 ) and covariances Cov d (p j +1, Re +1 ) as marices. Hence, demand sill affecs prices hrough he unhedgeable risk, bu now he unhedgeable par is he residual risk afer hedging wih he muliple underlying securiies. Naurally, demand for an opion sill increases is own price (par (i) of Proposiion 3), and increases he price of oher opions on ha underlying under cerain assumpions (as in par (ii) of Proposiion 3). The effec of demand for, say, an IBM opion on he price of a Microsof opion is, however, more suble. We wan o deermine his cross-effec when unhedgeable risk is driven by eiher (i) discree-ime rading, or (ii) sochasic volailiy (defined as in Secion 2.2). For case (i), we suppose ha S 1 and S 2 are geomeric Brownian moions wih insananeous correlaion ρ and compue he covariance of he unhedgeable pars o be p i d j [ ] = γrcov 0 p + i, p j + [ 1 = γrcov 0 2 f SS i S2 1 + S 1 O( ) +O ( 2 = γr 4 f i SS f j SS Cov0 ), 1 2 f j SS S2 2 + S 2 O( ) + O ( 2 [ ( ) S 2 1, S2] 2 + O 5 2 = γr 2 f i SS f j SS Var [ S 1 ]Var [ S 2 ]ρ 2 + O ) ] ( ) 5 2. (40) Hence, since he opion gammas f SS are posiive, we see ha he cross-demand effec is posiive regardless of he sign of he price correlaion. The inuiion for his surprising resul is ha he opion dealer is hedged and, herefore, has profis or losses depending on he magniude of he price changes, no heir 4275

18 The Review of Financial Sudies / v 22 n direcion. Since he absolue price changes are posiively correlaed in his model, he unhedgeable risk is posiively relaed, explaining he resul. For case (ii), we assume ha asses have correlaed sochasic volailiies and, for simpliciy, ha he sochasic volailiies are independen of he underlying prices. The price effec of demand is compued o be p i d j = γrcov ( σ 1, σ 2 ) f i σ f j σ + o( ). (41) Making he reasonable assumpion ha f i σ > 0 and f j σ > 0, he sign of he price impac is he same as ha of he correlaion beween σ 1 and σ 2. Hence, in his case, he demand for IBM opions increases he price of Microsof opions if heir volailiies are posiively correlaed and oherwise decreases he price of Microsof opions. We could furher exend he equilibrium deerminaion of he number of dealers o he case in which hey make markes in muliple underlyings (and possibly endogenize he number of underlyings dealers make markes in). Naurally, dealers enjoy he benefis of diversificaion and may addiionally have economies of scale, a leas up o a cerain poin. This would increase he equilibrium number of dealers, hus reducing he effec of demand pressure. 3. Empirical Resuls The main focus of his aricle is he impac of ne end-user opion demand on opion prices. We explore his impac empirically boh for S&P 500 index opions and for equiy (i.e., individual sock) opions. 3.1 Daa We acquire he daa from hree differen sources. Daa for compuing ne opion demand were obained direcly from he Chicago Board Opions Exchange (CBOE). These daa consis of a daily record of closing shor and long open ineres on all SPX and equiy opions for public cusomers and firm proprieary raders from he beginning of 1996 o he end of We compue he ne demand of each of hese groups of agens as he long open ineres minus he shor open ineres. We focus our analysis on non-marke-maker ne demand defined as he sum of he ne demand of public cusomers and proprieary raders, which is equal o he negaive of he marke-maker ne demand (since opions are in zero ne supply). Hence, we assume ha boh public cusomers and firm proprieary raders ha is, all non-marke-makers are end-users. We acually believe ha proprieary raders are more similar o marke-makers, and, indeed, heir posiions are more correlaed wih marke-maker posiions (he ime-series correlaion is 0.44). Consisen wih his fac, our resuls are indeed sronger when we reclassify proprieary raders as marke-makers (i.e., assume ha 4276

19 Demand-Based Opion Pricing end-users are he public cusomers). However, o be conservaive and avoid any sample selecion in favor of our predicions, we focus on he slighly weaker resuls. We noe ha, since proprieary raders consiue a relaively small group in our daa, none of he main feaures of he descripive saisics presened in his secion or he resuls presened in he nex secion change under his alernaive assumpion. The SPX opions rade only a he CBOE while he equiy opions someimes are cross-lised a oher opion markes. Our open ineres daa, however, include aciviy from all markes a which CBOE-lised opions rade. The enire opions marke is comprised of public cusomers, firm proprieary raders, and markemakers so our daa are comprehensive. For he equiy opions, we resric aenion o hose underlying socks wih sricly posiive opion volume on a leas 80% of he rade days over he period. This resricion yields 303 underlying socks. The oher main source of daa for his aricle is he Ivy DB daase from OpionMerics LLC. The OpionMerics daa include end-of-day volailiies implied from opion prices, and we use he volailiies implied from SPX and CBOE-lised equiy opions from he beginning of 1996 hrough he end of SPX opions have European-syle exercise, and OpionMerics compues implied volailiies by invering he Black-Scholes formula. When performing his inversion, he opion price is se o he midpoin of he bes closing bid and offer prices, he ineres rae is inerpolaed from available LIBOR raes so ha is mauriy is equal o he expiraion of he opion, and he index dividend yield is deermined from pu-call pariy. The equiy opions have Americansyle exercise, and OpionMerics compues heir implied volailiies using binomial rees ha accoun for he early exercise feaure and he iming and amoun of he dividends expeced o be paid by he underlying sock over he life of he opions. Finally, we obain daily reurns on he underlying index or socks from he Cener for Research in Securiy Prices (CRSP). Definiions of variables: We refer o he difference beween implied volailiy and a reference volailiy esimaed from he underlying securiy as excess implied volailiy. This measures he opion s expensiveness, ha is, is risk premium. The reference volailiy ha we use for SPX opions is he filered volailiy from he sae-of-he-ar model by Baes (2006), which accouns for jumps, sochasic volailiy, and he risk premium implied by he equiy marke, bu does no add exra risk premiums o (over-)fi opion prices. 12 By subracing he volailiy from he Baes (2006) model, we accoun for he direc effecs of jumps, sochasic volailiy, and he risk premium implied by he equiy marke. 12 We are graeful o David Baes for providing his measure. 4277

20 The Review of Financial Sudies / v 22 n Hence, excess implied volailiy is he par of he opion price unexplained by his model, which, according o our model, is due o demand pressure (and esimaion error). The reference volailiy ha we use for equiy opions is he prediced volailiy over heir lives from a GARCH(1,1) model esimaed from five years of daily underlying sock reurns leading up o he day of opion observaion. 13 (Alernaive measures using hisorical or realized volailiy lead o similar resuls.) We conduc several ess on he ime series ExcessImplVolATM of approximaely ATM opions wih approximaely one monh o expiraion. Specifically, for SPX, ExcessImplVolATM is he average excess implied volailiy of opions ha have a leas 25 conracs of rading volume, beween 15 and 45 calendar days o expiraion, 14 and moneyness beween 0.99 and (We compue he excess implied volailiy variable only from reasonably liquid opions in order o make i less noisy in ligh of he fac ha i is compued using only one rade dae.) For equiy opions, ExcessImplVolATM is he average excess implied volailiy of opions wih moneyness beween 0.95 and 1.05, mauriy beween 15 and 45 calendar days, a leas five conracs of rading volume, and implied volailiies available on OpionMerics. For he SPX, we also consider he excess implied-volailiy skew ExcessImplVolSkew, defined as he implied-volailiy skew over and above he skew prediced by he jumps and sochasic volailiy of he underlying index. Specifically, he implied-volailiy skew is defined as he average implied volailiy of opions wih moneyness beween 0.93 and 0.95 ha rade a leas 25 conracs on he rade dae and have more han 15 and fewer han 45 calendar days o expiraion, minus he average implied volailiy of opions wih moneyness beween 0.99 and 1.01 ha mee he same volume and mauriy crieria. In order o eliminae he skew ha is due o jumps and sochasic volailiy of he underlying, we consider he implied-volailiy skew ne of he similarly defined volailiy skew implied by he objecive disribuion of Broadie, Chernov, and Johannes (2007), where he underlying volailiy is ha filered from he Baes (2006) model. 15 We consider four differen demand variables for SPX opions based on he aggregae ne non-marke-maker demand for opions wih calendar days o expiraion and moneyness beween 0.8 and Firs, NeDemand 13 In paricular, we use he GARCH(1,1) parameer esimaes for he rade day (esimaed on a rolling basis from he pas five years of daily daa) o compue he minimum mean square error volailiy forecas for he number of rade days lef in he life of he opion. We annualize he volailiy forecas (which is for he number of rade days lef unil he opion maures) by muliplying by he square roo of 252 and dividing by he square roo of he number of rade days remaining in he life of he opion. 14 On any give rade day, hese are he opions wih mauriy closes o one monh. Alernaively, for each monh we could include in our es only he day ha is precisely one monh before expiraion. This approach yields similar resuls. 15 The model-implied skew is evaluaed for one-monh opions wih moneyness of, respecively, 0.94 and 1. We hank Mikhail Chernov for providing his ime series. 4278

21 Demand-Based Opion Pricing is simply he sum of all ne demands, which provides a simple aheoreical variable. The oher hree independen variables correspond o weighing he ne demands using he models based on he marke-maker risks associaed wih, respecively, discree rading, jumps in he underlying, and sochasic volailiy (Secion 2.2). Specifically, DiscTrade weighs he ne demands by he Black- Scholes gamma as in he discree-hedging model, JumpRisk weighs by kappa compued using equally likely up and down moves of relaive sizes 0.05 and 0.2 as in he jump model, and SochVol weighs by mauriy-adjused Black- Scholes vega as in he sochasic volailiy model. The Appendix provides more deails on he compuaion of he model-based weighing facors. For equiy opions, we use jus he NeDemand variable in he empirical work. As a measure of skew in SPX-opion demand, we use JumpRiskSkew, he excess implied-volailiy skew from he demand model wih underlying jumps described in Secion 2.2. (We do no consider he models wih discree rading and sochasic volailiy, since hey do no have firs-order skew implicaions, as explained in Secion 2.2. We obain similar, alhough weaker, resuls using an aheoreical measure based on raw demand.) Furhermore, o es he robusness of our resuls, we also consider several conrol variables, which we moivae when we run he ess, in subsecions 3.4 and 3.5. For he case of he SPX opions, we consider hree such variables. The firs is he ineracion beween dealer profis (P&L) over he previous calendar monh, calculaed using dealer posiions and assuming daily dela-hedging as deailed in Secion 3.3 and he measure of demand pressure. The oher wo variables are he curren S&P 500 volailiy filered by Baes (2006) and he S&P 500 reurns over he monh leading up o he observaion dae. For he case of equiy opions, he firs conrol variable is he ineracion beween he number of opion conracs raded on he underlying sock over he pas six monhs, OpVolume and he measure of demand pressure. The oher variables are he curren volailiy of he underlying sock measured from he pas 60 rade days of underlying reurns, he reurn on he underlying sock over he pas monh, as well as OpVolume on is own. 3.2 Descripive saisics on end-user demand Even hough he SPX and individual equiy opion markes have been he subjec of exensive empirical research, here is no sysemaic informaion on end-user demand in hese markes. Panel A of Table 1 repors he average daily non-marke-maker ne demand for SPX opions broken down by opion mauriy and moneyness (defined as he srike price divided by he underlying index level). Since our heoreical resuls indicae ha he demand from a pu or a call wih he same srike price and mauriy should have idenical price impac, his able is consruced from he demands for pus and calls of all moneyness and mauriy. For insance, he moneyness range consiss of pu opions ha are up o 5% in-he-money and call opions ha are up o 5% OTM. Panel A indicaes ha 39% of he ne demand comes from conracs 4279

22 4280 Table 1 Ne demand for opions by end-users Moneyness range (K/S) Ma. range (cal. days) All Panel A: SPX opion non-marke-maker ne demand 1 9 6,014 1,780 1,841 2,357 2,255 1, , ,953 1,300 1,115 6,427 2,883 2, , , ,679 7,296 1, ,038 1,092 20, ,536 1,108 2,287 2,420 1, , ,011 2,813 2,689 2, , ,406 18, ,630 3,096 2,335 1, , , ,673 1,340 1, ,158 5,831 All 32,519 11,785 14,621 20,530 9,987 6,457 3,074 4, ,260 The Review of Financial Sudies / v 22 n Panel B: Equiy opion non-marke-maker ne demand All Average non-marke-maker ne demand for pu and call opion conracs for SPX and individual equiy opions by moneyness and mauriy, Equiy-opion demand is per underlying sock.

23 Demand-Based Opion Pricing Non Marke Maker Ne Demand (Conracs) SPX Opion Ne Demand and Excess Implied Volailiy ( ) 3.5 x Excess Implied Vol Moneyness (K/S) Figure 1 Index opions: demand and expensiveness, measured in erms of implied volailiy The bars show he average daily ne demand for pus and calls from non-marke-makers for SPX opions in he differen moneyness caegories (lef axis). The op par of he lefmos (righmos) bar shows he ne demand for all opions wih moneyness less han 0.8 (greaer han 1.2). The line is he average SPX excess implied volailiy, ha is, implied volailiy minus he volailiy from he underlying securiy filered using Baes (2006), for each moneyness caegory (righ axis). The daa cover Implied Vol. Minus Baes ISDHa. wih fewer han 30 calendar days o expiraion. Consisen wih convenional wisdom, he good majoriy of his ne demand is concenraed a moneyness where pus are OTM (i.e., moneyness <1.) Panel B of Table 1 repors he average opion ne demand per underlying sock for individual equiy opions from non-marke-makers. Wih he excepion of some long mauriy opion caegories (i.e., hose wih more han one year o expiraion and in one case wih more han six monhs o expiraion), he non-marke-maker ne demand for all of he moneyness/mauriy caegories is negaive. Tha is, non-markemakers are ne suppliers of opions in all of hese caegories. This sands in a sark conras o he index-opion marke in Panel A where non-marke-makers are ne demanders of opions in almos every moneyness/mauriy caegory. Figure 1 illusraes he SPX opion ne demands across moneyness caegories and compares hese demands o he expensiveness of he corresponding opions. The line in he figure plos he average SPX excess implied volailiy for eigh moneyness inervals over he period. In paricular, on each rade dae he average excess implied volailiy is compued for all pus and calls in a moneyness inerval. The line depics he means of hese daily averages. The excess implied volailiy inheris he familiar downward sloping smirk in SPX opion-implied volailiies. The bars in Figure 1 represen he average daily ne demand from non-marke-maker for SPX opions in he moneyness caegories, where he op par of he lefmos (righmos) bar shows he ne demand for all opions wih moneyness less han 0.8 (greaer han 1.2). 4281

24 The Review of Financial Sudies / v 22 n SPX Opion Ne Demand and Dollar Expensiveness ( ) 3.5 x 104 Dollar Expensiveness. 10 Non Marke Maker Ne Demand (Conracs) Dollar Expensiveness Moneyness (K/S) Figure 2 Index opions: demand and expensiveness, measured in erms of dollars The bars show he average daily ne demand for pus and calls from non-marke-makers for SPX opions in he differen moneyness caegories (lef axis). The op par of he lefmos (righmos) bar shows he ne demand for all opions wih moneyness less han 0.8 (greaer han 1.2). The line is he average SPX price expensiveness, ha is, he opion price minus he fair price implied by Baes (2006) using filered volailiy from he underlying securiy and equiy risk premiums, for each moneyness caegory (righ axis). The daa cover The firs main feaure of Figure 1 is ha index opions are expensive (i.e., have a large risk premium), consisen wih wha is found in he lieraure, and ha end-users are ne buyers of index opions. This is consisen wih our main hypohesis: end-users buy index opions and marke-makers require a premium o deliver hem. The second main feaure of Figure 1 is ha he ne demand for low-srike opions is greaer han he demand for high-srike opions. This could help explain he fac ha low-srike opions are more expensive han high-srike opions (Proposiion 4). The shape of he demand across moneyness is clearly differen from he shape of he expensiveness curve. This is expeced for wo reasons. Firs, our heory implies ha demand pressure in one moneyness caegory impacs he implied volailiy of opions in oher caegories, hus smoohing he impliedvolailiy curve and changing is shape. Second, our heory implies ha demands (weighed by he variance of he unhedgeable risks) affec prices, and he price effec mus hen be ranslaed ino volailiy erms. I follows ha a lef-skewed hump-shaped price effec ypically ranslaes ino a downward sloping volailiy effec, consisen wih he daa. In fac, he observed average demands can give rise o a paern of expensiveness similar o he one observed empirically when using a version of he model wih jump risk. I is helpful o link hese demands more direcly o he predicions of our heory. Our model shows ha every opion conrac demanded leads o an increase in is price in dollar 4282

25 Demand-Based Opion Pricing Equiy Opion Ne Demand and Excess Implied Volailiy ( ) Excess Implied Vol. Non Marke Maker Ne Demand (Conracs) Implied Vol. Minus GARCH(1,1) Forecas Moneyness (K/S) 0.03 Figure 3 Equiy opions: demand and expensiveness, measured in erms of implied volailiy The bars show he average daily ne demand per underlying sock from non-marke-makers for equiy opions in he differen moneyness caegories (lef axis). The op par of he lefmos (righmos) bar shows he ne demand for all opions wih moneyness less han 0.8 (greaer han 1.2). The line is he average equiy opion excess implied volailiy, ha is, implied volailiy minus he GARCH(1,1) expeced volailiy, for each moneyness caegory (righ axis). The daa cover erms proporional o he variance of is unhedgeable par (and an increase in he price of any oher opion proporional o he covariance of he unhedgeable pars of he wo opions). Hence, he relaionship beween raw demands (ha is, demands no weighed according o he model) and expensiveness is more direcly visible when expensiveness is measured in dollar erms, raher han in erms of implied volailiy. This fac is confirmed by Figure 2. Indeed, he price expensiveness has a similar shape o he demand paern. Because of he cross-opion effecs and he absence of he weighing facor (he covariance erms), we do no expec he shapes o be idenical. 16 Figure 3 illusraes equiy opion ne demands across moneyness caegories and compares hem o heir expensiveness. The line in he figure plos he average equiy opion excess implied volailiy (wih respec o he GARCH(1,1) volailiy forecas) per underlying sock for eigh moneyness inervals over he period. In paricular, on each rade dae for each underlying sock he average excess implied volailiy is compued for all pus and calls in a moneyness inerval. These excess implied volailiies are averaged across underlying socks on each rade day for each moneyness inerval. The line depics he means of hese daily averages. The excess implied volailiy line is downward sloping bu only varies by abou 5% across he moneyness caegories. 16 Even if he model-implied relaionship involving dollar expensiveness is more direc, we follow he lieraure and concenrae on expensiveness expressed in erms of implied volailiy. This can be hough of as a normalizaion ha eliminaes he need for explici conrols for he price level of he underlying asse. 4283

26 The Review of Financial Sudies / v 22 n Daily Marke Maker Profi/Loss P/L ($1,000,000s) /04/96 12/18/97 12/08/99 12/31/ Cumulaive Marke Maker Profi/Loss P/L ($1,000,000s) /04/96 12/18/97 12/08/99 12/31/01 Dae Figure 4 Opion marke-makers esimaed profis and losses from posiion aking The op panel shows he marke-makers daily profis and losses (P&L) assuming hey dela-hedge heir opion posiions once per day. The boom panel shows he corresponding cumulaive P&Ls. By conras, for he SPX opions he excess implied volailiy line varies by 15% across he corresponding moneyness caegories. The bars in he figure represen he average daily ne demand per underlying sock from non-markemakers for equiy opions in he moneyness caegories. The figure shows ha non-marke-makers are ne sellers of equiy opions on average. Consisen wih heir differen demand paern, equiy opions do no appear expensive on average like index opions. 3.3 Marke-maker profis and losses To illusrae he magniude of he ne demands, we compue approximae daily profis and losses (P&Ls) for he S&P 500 marke-makers hedged posiions assuming daily dela-hedging. The daily and cumulaive P&Ls are illusraed in Figure 4, which shows ha he group of marke-makers faces subsanial risk ha canno be dela-hedged, wih daily P&L varying beween ca. $100 million and $ 100 million. Furher, he marke-makers make cumulaive profis of ca. $800 million over he six-year period on heir posiion aking. 17 Wih jus over a hundred SPX marke-makers on he CBOE, his corresponds o a profi of approximaely $1 million per year per marke-maker. Hence, consisen wih he premise of our model, marke-makers face subsanial risk and are compensaed on average for he risk ha hey ake. Furher, 17 This number does no ake ino accoun he coss of marke-making or he profis from he bid-ask spread on round-rip rades. A subsanial par of marke-makers profi may come from he laer. 4284

27 Demand-Based Opion Pricing Average Monhly Sharpe Raios (annualized) Marke makers and proprieary raders Marke makers Number of equiies in porfolio Figure 5 Opion marke-makers esimaed Sharpe raio from posiion aking The figure shows he annualized average monhly Sharpe raios of he profis o making markes in opions on a given number of individual equiies (he horizonal axis). The profis are calculaed using eiher he marke-maker posiions, or he combined marke-maker and proprieary-rader posiions. consisen wih he equilibrium enry of marke-makers of Secion 2.3, he marke-makers profis of abou $1 million per year per marke-maker appear o be of he same order of magniude as heir cos of capial ied up in he rade, rader salaries, and back office expenses. In fac, he profi number appears low, bu one mus remember ha marke-makers likely make subsanial profis from he bid-ask spread, an effec ha we do no include in our profi calculaions. Anoher measure of he marke-makers compensaion for accommodaing SPX opion demand pressures is he annualized Sharpe raio of heir profis or losses. This measure is 0.41 when compued from he daily P&L, an unimpressive risk/reward radeoff comparable o ha of a passive invesmen in he overall (sock) marke. Since he daily P&L is negaively auocorrelaed, he annualized Sharpe raio increases o 0.85 when compued from monhly P&L (and hardly increases if we aggregae over longer ime horizons). This Sharpe raio reflecs compensaion for he risk ha marke-makers bear, as well as he commied capial ha has alernaive producive use and he dealers effor and skill. This Sharpe raio is of a magniude consisen wih equilibrium enry of marke-makers as modeled in Secion 2.3. To illusrae he profis from making markes in individual-equiy opions, as well as he value from diversifying across a number of differen underlying equiies, we consruc he annualized average monhly Sharpe raios for hypoheical marke-makers ha deal opions on n differen socks, wih n varying from 1 o 300. Figure 5 shows he resuls when using boh only he markemaker, and he combined marke-maker and proprieary-rader posiions as measures of he demand. The Sharpe raios are, as expeced, posiive and increasing in he number of socks in which a marke-maker deals opions. This provides furher evidence ha marke-makers are compensaed for providing 4285

28 The Review of Financial Sudies / v 22 n liquidiy, and ha his liquidiy provision is associaed wih a significan risk, par of which is diversifiable. 3.4 Ne demand and expensiveness: SPX index opions Theorem 2 relaes he demand for any opion o a price impac on any opion. Since our daa conain boh opion demands and prices, we can es hese heoreical resuls direcly. Doing so requires ha we choose a reason for he underlying asse and risk-free bond o form a dynamically incomplee marke, and, hence, for he weighing facors Cov ( p +1 i, p j +1 ) o be nonzero. As sources of marke incompleeness, we consider discree-ime rading, jumps, and sochasic volailiy using he resuls derived in Secion 2.2. We es he model s abiliy o help reconcile he wo main puzzles in he opion lieraure, namely he drivers of he overall level of implied volailiy and is skew across opion moneyness. The firs se of ess invesigaes wheher he overall excess implied volailiy is higher on rade daes where he demand for opions aggregaed according o he model is higher. The second se of ess invesigaes wheher he excess implied-volailiy skew is seeper on rade daes where he model-implied demand-based skew is seeper. Level: We invesigae firs he ime-series evidence for Theorem 2 by regressing a measure of excess implied volailiy on one of various demand-based explanaory variables: ExcessImplVolATM = a + b Demand Variable + ε. (42) We run he regression on a monhly basis by averaging demand and expensiveness over each monh. We do his o avoid day-of-he-monh effecs. (Our resuls are similar in an unrepored daily regression.) The resuls are shown in Table 2. We repor he resuls over wo subsamples because here are reasons o suspec a srucural change in The change, apparen also in he ime series of open ineres and marke-maker and publiccusomer posiions (no shown here), sems from several evens ha alered he marke for index opions in he period from lae 1996 o Ocober 1997, such as he inroducion of S&P 500 e-mini fuures and fuures opions on he compeing Chicago Mercanile Exchange (CME), he inroducion of Dow Jones opions on he CBOE, and changes in margin requiremens. Our resuls are robus o he choice of hese sample periods. 18 We see ha he esimae of he demand effec b is posiive bu insignifican over he firs subsample, and posiive and saisically significan 18 We repeaed he analysis wih he sample periods lenghened or shorened by wo or four monhs. This does no change our resuls qualiaively. 4286

29 Demand-Based Opion Pricing Table 2 Index-opion expensiveness explained by end-user demand Before srucural changes Afer Srucural changes 1996/ / / /12 Consan (0.004) (0.30) (0.17) (0.93) (7.28) (4.67) (7.7) (7.4) NeDemand (0.87) (1.55) DiscTrade (0.91) (3.85) JumpRisk (0.79) (3.68) SochVol (0.27) (2.74) Adj. R 2 (%) N The SPX excess implied volailiy (i.e., observed implied volailiy minus volailiy from he Baes (2006) model) is regressed on he SPX non-marke-maker demand pressure. The demand pressure is eiher (i) equal weighed demand across conracs (NeDemand), or demand weighed using our model in which marke-maker risk is due o (ii) discree-ime rading risk (DiscTrade), (iii) jump risk (JumpRisk), or (iv) sochasic-volailiy risk (SochVol). -saisics compued using Newey-Wes are in parenheses. over he second, longer, subsample for all hree model-based explanaory variables. 19 The expensiveness and he fied values from he jump model are ploed in Figure 6, which clearly shows heir comovemen over he laer sample. The fac ha he b coefficien is posiive indicaes ha, on average, when SPX ne demand is higher (lower), SPX excess implied volailiies are also higher (lower). For he mos successful model, he one based on jumps, changing he dependen variable from is lowes o is highes values over he lae subsample would change he excess implied volailiy by abou 5.6 percenage poins. A onesandard-deviaion change in he jump-based demand variable resuls in a onehalf-sandard-deviaion change in excess implied volailiy (he corresponding R 2 is 26%). The model is also successful in explaining a significan proporion of he level of he excess implied volailiy. Over he lae subsample, demand explains on average 1.7 percenage poins of excess implied volailiy given he average demand and he regression coefficien, more han a hird of he esimaed average level of excess implied volailiy. We noe ha, in addiion o he CBOE demand pressure observed in our daa, here is over-he-couner demand for index opions, for insance, via such producs for individual invesors as index-linked bonds. These securiies give end-users essenially a risk-free securiy in combinaion wih a call opion on he index (or he index plus a pu opion), which leaves Wall Sree shor index opions. Of course, his demand also conribues o he excess implied volailiy. 19 The model-based explanaory variables work beer han jus adding all conracs (NeDemand), because hey give greaer weigh o near-he-money opions. If we jus coun conracs using a more narrow band of moneyness, hen he NeDemand variable also becomes significan. 4287

30 The Review of Financial Sudies / v 22 n Expensiveness Expensiveness Fied values, early sample Fied values, lae sample Monh Number Figure 6 Acual and model-implied expensiveness of index opions The solid line shows he expensiveness of SPX opions, ha is, implied volailiy of one-monh a-he-money opions minus he volailiy measure of Baes (2006), which akes ino accoun jumps, sochasic volailiy, and he risk premium from he equiy marke. The dashed lines are, respecively, he fied values of demand-based expensiveness using a model wih underlying jumps, before and afer cerain srucural changes (1996/ /10 and 1997/ /12). Hence, our esimae ha demand pressure explains on average 1.7 percenage poins (and 26% of he variaion) of he excess implied volailiy can be viewed as a conservaive lower bound. Furher suppor for he hypohesis ha he supply for opions is upward sloping comes from he comparison beween he esimaed supply-curve slopes following marke-maker losses and, respecively, gains. If marke-maker risk aversion plays an imporan role in pricing opions, hen one would expec prices o be less sensiive o demand when marke-makers are well funded following profiable periods. This is exacly wha we find. Breaking he daily sample 20 ino wo subsamples depending on wheher he hedged marke-maker profis over he previous 20 rading days is posiive or negaive, 21 we esimae he regression (42) for each subsample and find ha, following losses, he b coefficien is approximaely wice as large as he coefficien obained in he oher subsample. For insance, in he jump model, he regression coefficien following losses is 2.6E-05 wih a -saisic of 3.7 (330 observaions), compared o a value of 1.1E-5 wih a -saisic of 6.1 in he complemenary subsample (646 observaions). This effec of recen profis and losses (P&L) can also be sudied in a imeseries regression wih he recen P&L on he righ-hand side ineraced wih 20 Because of he srucural changes discussed above, we resric our aenion o he period saring on Ocober 1, Similar resuls obain if he breaking poin is he mean or median daily profi. 4288

31 Demand-Based Opion Pricing Table 3 Index-opion expensiveness explained by end-user demand wih conrol variables Non-marke-maker demand Consan (2.75) (2.79) (1.61) Demand (4.20) (3.86) (4.44) P&L Demand ( 1.22) ( 1.73) Volailiy ( 0.88) S&P Reurn (0.23) Adj. R 2 (%) The SPX excess implied volailiy is regressed on non-marke-maker demand as well as conrol variables, 1997/ /12. The demand across conracs is weighed using our model wih jump risk. The conrols are (i) he produc beween lagged monhly marke-maker profi and demand, (ii) curren S&P 500 volailiy, and (iii) he lagged monhly S&P 500 reurn. -saisics compued using Newey-Wes are in parenheses. Demand has a posiive effec on implied volailiy, and he negaive coefficien on he ineracion beween marke-maker profis and demand pressure means ha he effec of demand is larger following marke-maker losses. he magniude of demand. Table 3 repors he resul of including his and oher conrol variables, namely he volailiy of he underlying and he pas reurn on he S&P. In order o save space, he only mehod of weighing demand ha we presen is he one given by he jump model and we resric aenion o he longer subsample. Firs, he able makes i clear ha, regardless of he addiional conrols used, demand pressure has a significan posiive effec on he level of he impliedvolailiy curve. Furher, here is a negaive coefficien on he ineracion beween marke-maker profis and demand pressure. This means ha he effec of demand is larger following marke-maker losses (i.e., negaive profis) and smaller following posiive profis. As explained above, his is consisen wih he idea ha following losses, marke-makers are more risk averse and herefore he opion prices are more sensiive o demand. Hence, he negaive coefficien offers furher suppor for our demand-based opion pricing. Skew: We nex invesigae he explanaory power of he model for he skew of he implied-volailiy curve by regressing a measure of he seepness of he implied-volailiy skew on our demand-based explanaory variable: ExcessImplVolSkew = a + b JumpRiskSkew + ε. (43) Column 1 of Table 4 repors he monhly OLS esimaes of his skewness regression. As expeced, he model-implied effec of demand on he impliedvolailiy skew has a posiive coefficien. To illusrae he magniude of he effec, we noe ha a one-sandard-deviaion move in he independen variable resuls in a change in he dependen variable of 0.53 sandard deviaions. Furher, Figure 7 illusraes graphically he demand effec on he volailiy skew. (As discussed above, we divide he sample ino wo subsamples because 4289

32 The Review of Financial Sudies / v 22 n Table 4 Index opion skew explained by end-user demand Non-marke-maker demand Consan (13.23) (11.10) (0.50) Demand (2.99) (2.73) (3.40) P&L Demand ( 2.96) ( 1.76) Volailiy (2.80) S&P Reurn (2.96) Adj. R 2 (%) The acual SPX implied-volailiy skew is regressed on demand-based skew implied by he non-marke-maker demand and he jump-risk model, in he presence of conrol variables. The conrols are (i) he produc beween lagged monhly marke-maker profi and demand, (ii) curren S&P 500 volailiy, and (iii) he lagged monhly S&P 500 reurn. -saisics compued using Newey-Wes are in parenheses IV Skew Fied values, early sample Fied values, lae sample Skew Monh Number Figure 7 Acual and model-implied implied-volailiy skew of index opions The solid line shows he implied-volailiy skew for SPX opions. The dashed lines are, respecively, he fied values from he skew in demand before and afer cerain srucural changes (1996/ /10 and 1997/ /12). of srucural changes and, while he figure shows boh subsamples, he able focuses on he longer subsample o save space.) Columns 2 and 3 of Table 4 include conrol variables. We see ha he model-implied effec of demand on he implied-volailiy skew has a posiive coefficien in all specificaions in confirmaion of he model. Ineresingly, he ineracion of marke-maker profis wih he demand effec has a negaive 4290

33 Demand-Based Opion Pricing Table 5 Equiy opion expensiveness explained by end-user demand Non-marke-maker demand 1/96-12/01 1/96-6/99 10/99-12/01 Consan (13.58) (12.34) (7.84) NeDemand (7.66) (11.64) (4.62) NeDemand OpVolume ( 3.91) ( 5.97) ( 1.34) Volailiy ( 14.23) ( 13.73) ( 10.51) Reurn ( 11.57) ( 12.09) ( 6.16) OpVolume (1.98) (2.11) (0.89) Adj. R 2 (%) The relaionship beween excess-implied expensiveness i.e., implied volailiy minus GARCH volailiy and ne demand for equiy opions on 303 differen underlying sock (Equaion 44), conrolling for volailiy, pas reurn, opion volume, and he ineracion beween ne demand and opion volume. We run monhly Fama- MacBeh cross-secional regressions and repor -saisics adjused for serial correlaion via he Newey-Wes mehod. coefficien. This is consisen wih he idea ha demand affecs prices more afer marke-maker losses, when heir risk aversion is likely o be higher Ne demand and expensiveness: equiy opions We nex invesigae he equiy opion evidence for Theorem 2 by performing a cross-secional Fama-MacBeh regression of a measure of excess implied volailiy on he NeDemand explanaory variable: ExcessImplVolATM i = a + b NeDemand i + ε i. (44) We run he Fama-MacBeh regression on he 303 underlying socks discussed above, namely hose wih sricly posiive opion volume on a leas 80% of he rade days from he beginning of 1996 hrough he end of As in he case of he SPX opions, we run he regressions on a monhly basis by averaging demand and expensiveness for each underlying sock for each monh. (Once again, he resuls are similar for an unrepored daily analysis.) Coefficien esimaes and -saisics (adjused for serial correlaion via Newey-Wes) are repored in Table 5. The average R 2 s for he individual monhly cross-secional regressions are also repored. The firs column of Table 5 shows he analysis for he enire sample period from he beginning of 1996 hrough he end of We see ha he coefficien on he demand variable is posiive and highly significan, conrolling for volailiy, pas reurn, and opion volume. In addiion, he coefficien on he 22 As a furher robusness check, we confirm ha he demand effec on he skew is also presen when we use an aheoreical demand-skew measure based on aggregaing he raw demands for low srikes, minus hose wih srikes close o 1. (This regression is no repored for breviy.) 4291

34 The Review of Financial Sudies / v 22 n ineracion of demand wih he level of opion volume on he underlying sock is significanly negaive. This resul indicaes ha he demand effec is weaker when here is more opion aciviy on a sock. This finding provides evidence in favor of our hypohesis, because greaer opion aciviy should be posiively correlaed wih more capial being devoed o opion marke-making and, consequenly, a smaller price impac per uni of opion demand. This conclusion follows direcly from our Proposiion 5: par 1(a) shows ha more expeced demand leads o enry of more dealers, lowering he demand effecs. Likewise, par 2 of he proposiion shows ha lower cos of being a dealer leads o more dealers, and addiional volume leads o lower opporuniy cos (denoed M in he proposiion) because of addiional profis o be made from earning he bid-ask spread on round-rip rades. Columns 2 and 3 of Table 5 rerun he analysis separaely on he subsamples before and afer he summer of We perform he analysis on hese subsamples because mos opions were lised only on one exchange before he summer of 1999, bu many were lised on muliple exchanges afer his period. Hence, here was poenially a larger oal capaciy for risk aking by marke-makers afer he cross lising. See, for insance, De Fonnouvelle, Fishe, and Harris (2003) for a deailed discussion of his well-known srucural break. We follow De Fonnouvelle, Fishe, and Harris (2003) in he specific choice of he saring and ending daes for our subsamples, bu we have confirmed ha our resuls are robus o variaions in his choice. For example, if he breakpoin for eiher subsample is moved by wo monhs in eiher direcion, here are only very small changes o he numbers repored in Table 5. The main resuls are presen in boh he earlier and he laer subperiod, alhough weaker in he laer sample (indeed, he negaive coefficien on he ineracion erm is no longer significan in he second par of he sample). 23 The demand coefficien is smaller in he second subsample when here was a greaer capaciy for risk aking by he marke-makers due o cross-lising, anoher confirmaion of he demand-based heory of opion pricing. Finally, in unrepored resuls we have run analogous Fama-MacBeh skew regressions on he equiy opions and also ge significan resuls in he expeced direcion. 4. Conclusion Relaive o he Black-Scholes-Meron benchmark, index and equiy opions display a number of robus pricing anomalies. A large body of research has aemped o address hese anomalies, in large par by generalizing he Black- Scholes-Meron assumpions abou he dynamics of he underlying asse. While hese effors have me wih undeniable success, nonrivial pricing puzzles remain. Furher, i is no clear ha his approach can yield a saisfacory descripion of opion prices. For example, index and equiy opion prices display very 23 When inerpreing he -saisics for he second subperiod, i should be borne in mind ha i consiss of only 27 monhly cross-secional regressions. 4292

35 Demand-Based Opion Pricing differen properies, even hough he dynamics of heir underlying asses are quie similar. This aricle akes a differen approach o opion pricing. We recognize ha, in conras o he Black-Scholes-Meron framework, in he real world opions canno be hedged perfecly. Consequenly, if inermediaries who ake he oher side of end-user opion demand are risk averse, end-user demand for opions will impac opion prices. The heoreical par of he aricle develops a model of compeiive risk-averse inermediaries who canno perfecly hedge heir opion posiions. We compue equilibrium prices as a funcion of ne end-user demand and show ha demand for an opion increases is price by an amoun proporional o he variance of he unhedgeable par of he opion and ha i changes he prices of oher opions on he same underlying asse by an amoun proporional o he covariance of heir unhedgeable pars. The empirical par of he aricle measures he expensiveness of an opion as is Black-Scholes implied volailiy minus a proxy for he expeced volailiy over he life of he opion. We show ha on average index opions are quie expensive by his measure, and ha hey have high posiive end-user demand. Equiy opions, on he oher hand, do no appear expensive on average and have a small negaive end-user demand. In accordance wih he predicions of our heory, we find ha index opions are overall more expensive when here is more end-user demand and ha he expensiveness skew across moneyness is posiively relaed o he skew in end-user demand across moneyness. In addiion, demand effecs are sronger for index opions afer recen markemaker losses han afer marke-maker gains. For equiy opions, we find a posiive cross-secional relaionship beween opion expensiveness and enduser demand. In addiion, he relaionship is sronger when here is less opion aciviy and herefore presumably less opion marke-maker capaciy for riskaking on an underlying sock. Appendix Appendix: Proofs Proof of Lemma 1. We sar by imposing he following echnical condiions. Firs, d and X have compac suppors. Second, he ransiion funcion π of X has he following regulariy propery. Assumpion 1. Whenever x n x, π(, x n ) π(, x) uniformly under he oal-variaion norm. Finally, we require ha R+1 e be bounded, o so ha all expecaions are well defined. The Bellman equaion is J(W ;, X ) = 1 +G (d,x )) e k(w k = max C,q,θ { 1 γ e γc + ρe [ J(W+1 ; + 1, X +1 ) ]}. (A1) 4293

36 The Review of Financial Sudies / v 22 n Given he sric concaviy of he uiliy funcion, he maximum is characerized by he firs-order condiions (FOCs). Using he proposed funcional form for he value funcion, he FOC for C is which, ogeher wih (A1), yields ha is, 0 = e γc + kr f ρe [ J(W+1 ; + 1, X +1 ) ], (A2) 0 = e γc + kr f [ J(W ;, X ) + 1 γ e γc ], (A3) e γc = e k(w +G (d,x )), (A4) implying (6). The FOCs for θ and q are (7) and (8). We derive G recursively as follows. Firs, we le G( + 1, ) be given. Then, θ and q are given as he unique soluions o Equaions (7) and (8). Clearly, θ and q do no depend on he wealh W. Furher, (A3) implies ha [ ( )] 0 = e γc R f ρe e k (W C )R f +q (p +1 R f p )+θ R+1 e +G +1(d +1,X +1 ), (A5) ha is, [ ( )] e γc kr f C +kr f W = R f ρe e k q (p +1 R f p )+θ R+1 e +G +1(d +1,X +1 ), (A6) which, using (6), yields he equaion ha defines G (d, X )(sincex is Markov): [ ( )] e kr f G (d,x ) = R f ρe e k q (p +1 R f p )+θ R+1 e +G +1(d +1,X +1 ). (A7) In he online appendix we furher prove he exisence of a saionary soluion a = T. Proof of Proposiion 1. Given a posiion process from dae onward and a price process from dae + 1 onward, he price a ime is deermined by (8). I is immediae ha p is measurable wih respec o ime- informaion. Proof of Proposiion 2. (i) is immediae, since prices are linear. Par (ii) follows because, for any a R, he pricing kernel is kep exacly he same by he offseing change in (q, θ). Proof of Theorem 2. We sar by calculaing explicily he sensiiviy of he prices of a derivaive p i j wih respec o he demand pressure of anoher derivaive d. We can iniially differeniae wih respec o q raher han d since q i = d i. For his, we firs differeniae he pricing kernel: 24 m d +1 q j using he facs ha G(+1,X +1;q) q j differeniae (16) o ge 0 = E (m d +1 = km d +1 ( = 0and p +1 q j ( ) p j +1 R f p j + θ q j R+1 e, (A8) = 0. Wih his resul, i is sraighforward o ) ) p j +1 R f p j + θ q j R+1 e R+1 e, (A9) 24 We suppress he argumens of funcions. We noe ha p, θ,andg are funcions of (d, X, ), and m d +1 is a funcion of (d, X, d +1, X +1, y +1, R e +1, ). 4294

37 Demand-Based Opion Pricing which implies ha he marginal hedge posiion is θ q j = E ( ( m d j +1 p +1 R f p j ) ) R e +1 ( ( ) E m d +1 R e 2 ) +1 Similarly, we derive he price sensiiviy by differeniaing (15): p i q j = ke [m d +1 ( ( = Covd j p Var d ) ] p j +1 R f p j + θ q j R+1 e p+1 i +1, ) Re +1 ( ). (A10) R e +1 [( = k ( E d p j +1 R R f p j Covd j p +1, ) ) ] Re +1 f Var d ( ) R e R e +1 p+1 i +1 = γ(r f 1)E d [ j ] p +1 pi +1 = γ(r f 1)Cov d [ j p +1, +1] pi, (A11) where p +1 i j and p +1 are he unhedgeable pars of he price changes defined in he ex. Proof of Proposiion 3. Par (a) is immediae since a variance is always posiive. The proof of (b) is based on he following resul, which is proved in he online appendix. Lemma 2. Given h 1 and h 2 convex funcions on R, β < 0, α, γ R, α, γ R such ha h 1 (x) α x γ h 1 (x) αx βh 2 (x) γ (A12) x R. Consequenly, under any disribuion, regressing h 1 on h 2 and he ideniy funcion resuls in a posiive coefficien on h 2. Leing p +1 = p +1 E d [ ] p+1 and suppressing subscrips, consider he expression = E d [ p i p j ] Var( R e ) E d [ p i R e] E d [ p j R e], (A13) which we wan o show o be posiive. Leing ˆp i = E d [ p i S]and ˆp j = E d [ p j S], we wrie = E d [ Cov ( p i, p j S ) Var( R e ) + ˆp i ˆp j Var( R e ) E d [ ˆp i R e] E d [ ˆp j R e]] = E d [ Cov ( p i, p j S ) Var( R e ) ] + E d [ ˆp i ˆp j Var( R e ) E d [ ˆp i R e] E d [ ˆp j R e]]. (A14) The firs erm is posiive by assumpion, while he second is posiive because ˆp i and ˆp j are convex and herefore Lemma 2 applies. Proof of Theorem 3. We compue he sensiiviy of curren prices o a deviaion in fuure posiions from 0 in he direcion of demand d s = ɛ s d s a ime s by differeniaing wih respec o ɛ s = ɛ (evaluaed a ɛ = 0). We hen aggregae he demands a all imes o compue he oal effec: p ɛ = s 0 p ɛ s ɛ s ɛ = p. ɛ s 0 s (A15) To compue he price effec of expeced demand a any ime s, we noe ha i follows from he dealer s problem ha p = E [ ρ s e γ(cs C ) p s ], (A16) 4295

38 The Review of Financial Sudies / v 22 n which implies p = E ɛ s where we use ha C ɛ s C s q j s [ ρ s e γ(cs C ) p ] s ɛ s = Cs ɛ s = k G(s, X s ; q) γ qs j = R (s ) f E 0 [ ] ps ɛ s = 0aq = 0. The equaliy Cs ɛ s [ = k2 R f ρ E 0 s p j s+1 γ R f p j = R (s ) f E 0 = 0 follows from [ ] p s q s j qs j, (A17) ] s + θ s qs j Rs+1 e = 0 (A18) and he oher equaliy follows from differeniaing he condiion ha marginal raes of subsiuion are equal: e γc = e ρ(s ) E [ e γc s ], (A19) which gives e γc C ɛ s = e ρ(s ) E [ e C ] γcs s = 0. (A20) ɛ s Finally, in he online appendix we show ha he price is a smooh (C ) funcion of ɛ. Proof of Proposiion 4. Consider an opimally hedged shor pu posiion wih srike price K < R f S. Wih x = S +1 R f S, he payoff from his posiion is (x) = d(k R f S x) + + θx. (A21) The opimaliy of he hedge means ha, under he risk-neural measure, [ ] E e k (x) x = 0. (A22) Noe ha, since K < R f S, (x) < 0forx > 0and (x) > 0forK R f S < x < 0. Consequenly, given he symmery of x around 0 and he zero-expecaion condiion above, wih ξ denoing he densiy of x, K R f S ( e k (x) x e k ( x) x ) K R f S ( ) ξ(x) dx = e k (x) x e k ( x) x ξ(x) dx < 0. (A23) 0 I immediaely follows ha i canno be rue ha ( x) (x) for all x > K R f S.In oher words, for some value x > K R f S, ( x) < (x), which hen gives d + θ > θ, or θ < 1 2 d : he payoff is more sensiive o large downward movemens in he underlying han o large upward movemens. Thus, here exiss K such ha, for all S +1 < K, (S +1 R f S ) < ( (S +1 R f S )), (A24) implying ha, whenever K < K and K = 2R f S K, p(p, K, d) > p(c, K, d) he second relaion being he resul of symmery. p(p, K, 0) = p(c, K, 0), (A25) (A26) Proof of Proposiion 5. Fix a demand process d and wrie G as G(γ, ɛ d, γ, X). I is hen easily seen ha G(γ, ɛ d, γ, X) = G(γ, d, ɛγ, X). I is equally clear ha, for a given disribuion of asse prices, he vecor γ (q(γ ), θ(γ )) is independen of he value of γ. This readily implies ha, if he 4296

39 Demand-Based Opion Pricing se of dealers I are in he marke, hen γ 1 = I γ (i) 1 dμ(i). I is also easily checked ha he value γ G(γ, ɛ d, γ, X) is independen of γ,soha G(γ, ɛ d, γ, X) = 1 γ G(1, ɛ d, γ, X). We now show ha G increases in ɛ. To ha end we wrie he value funcion as where J 0 = E 0 [ 1 γ T =0 W T +1 = (W 0 C 0 )R T +1 f + ] ρ e γ C 1 + k ρt +1 e k (W T +1 +G T +1 ), (A27) T =0 ( C+1 + θ R e +1 q (p +1 R f p ) ) R T f and k = γ r/(r 1), and proceed o calculae he derivaive J 0 ɛ. From he FOCs, [ J 0 ɛ = Ed 0 ρ T +1 T =0 q ( p+1 ɛ ) ] p R f R T f, ɛ wih p ɛ [ T ] = Ed R (s ) f p s+1 p s+1 d s. s= I follows ha [ T [ T J 0 ɛ Ed 0 q (E d =0 s=+1 [ T = E d 0 q E d =0 [ T = E d 0 =0 d E d R (s 1) f p s+1 p s+1 d s R f [ p+1 p +1] d R T +1 f [ p+1 p +1] d R T +1 f ] ] > 0. T s= R (s ) f p s+1 p s+1 d s ]) ] R T f if Consider now dealer enry. A given dealer wih risk aversion γ eners he business if and only 1 k e k (W 0 M+G(γ,d,γ,X)) 1 k e k (W 0 +Ḡ (γ )), or G(γ, d, γ, X) Ḡ (γ ) M, where from being able o inves in he underlying, bu having no access o opions. (This is he same as G when demand is idenically zero.) Using he resuls above, he enry condiion becomes G(1, d, ɛγ, X) Ḡ (1) γ M. (A28) 4297

40 The Review of Financial Sudies / v 22 n I is immediae ha, in any equilibrium, a dealer wih risk aversion γ > γ is in he marke if and only if all dealers wih risk aversion γ are. We may herefore assume ha any equilibrium is characerized by dealers in I = [0, ī]. Leing now ī vary from 0 o, he resuling aggregae risk aversion γ(ī) decreases from infiniy o 0. As G(1, ɛ d, γ(ī), X) Ḡ (1) decreases wih ī and ends o0asγ(ī) ends o 0, while γ (ī) increases, a unique dealer-enry equilibrium exiss. For par 1 (a), we use he fac ha G(γ (ī), d, ɛγ(ī), X) increases in ɛ, and herefore so does he equilibrium ī. Furhermore, if γ increases from one equilibrium value ī o anoher one, hen he righ-hand side of (A28) increases, which means ha so mus he lef-hand side, i.e., he produc ɛγ(ī) mus increase. Using Theorem 3, dp dɛ = d(γ(ɛ)ɛ) v, dɛ a ɛ = 0, for some vecor v. Since γ(ɛ)ɛ increases wih ɛ, he effec of increasing demand is o amplify price deviaions from zero-demand levels, a leas for ɛ close o 0. If γ(ɛ)ɛ is acually consan because γ (ī) is hen he wo equilibria are idenical from a pricing perspecive. This finishes he proof of par 1. Par 2 follows immediaely from he preceding analysis. References Amin, K., J. D. Coval, and H. N. Seyhun Index Opion Prices and Sock Marke Momenum. Journal of Business 77: Back, K Asymmeric Informaion and Opions. Review of Financial Sudies 6: Bakshi, G., N. Kapadia, and D. Madan Sock Reurn Characerisics, Skew Laws, and he Differenial Pricing of Individual Equiy Opions. Review of Financial Sudies 16: Baes, D. S Pos 87 Crash Fears in he S&P 500 Fuures Opions Marke. Journal of Economerics 94: Baes, D. S Empirical Opion Pricing: A Rerospecion. Journal of Economerics 116: Baes, D. S Maximum Likelihood Esimaion of Laen Affine Processes. Review of Financial Sudies 19: Benzoni, L., P. Collin-Dufresne, and R. S. Goldsein Can Sandard Preferences Explain he Prices of Ou of he Money S&P 500 Pu Opions? Working Paper, NBER. Bernardo, A. E., and O. Ledoi Gain, Loss and Asse Pricing. The Journal of Poliical Economy 108: Black, F., and M. S. Scholes The Pricing of Opions and Corporae Liabiliies. The Journal of Poliical Economy 81: Bollen, N. P., and R. E. Whaley Does Ne Buying Pressure Affec he Shape of Implied Volailiy Funcions? Journal of Finance 59: Bondarenko, O Why Are Pu Opions So Expensive? Working Paper, Universiy of Illinois a Chicago. Brennan, M The Pricing of Coningen Claims in Discree Time Models. Journal of Finance 34: Broadie, M., M. Chernov, and M. Johannes Model Specificaion and Risk Premia: Evidence from Fuures Opions. Journal of Finance 62: Cochrane, J. H., and J. Saa-Requejo Beyond Arbirage: Good-Deal Asse Price Bounds in Incomplee Markes. The Journal of Poliical Economy 108: Consaninides, G. M., J. C. Jackwerh, and S. Perrakis Mispricing of S&P 500 Index Opions. Review of Financial Sudies 22:

41 Demand-Based Opion Pricing Consaninides, G. M., and S. Perrakis Sochasic Dominance Bounds on Derivaives Prices in a Muliperiod Economy wih Proporional Transacion Coss. Journal of Economic Dynamics and Conrol 26: Coval, J. D., and T. Shumway Expeced Opion Reurns. Journal of Finance 56: De Fonnouvelle, P., R. P. H. Fishe, and J. H. Harris The Behavior of Bid-Ask Spreads and Volume in Opions Markes during he Compeiion for Lisings in Journal of Finance 58: de Roon, F., T. E. Nijman, and C. Veld Hedging Pressure Effecs in Fuures Markes. Journal of Finance 55: Driessen, J., and P. Maenhou An Empirical Porfolio Perspecive on Opion Pricing Anomalies. Review of Finance 11: Easley, D., M. O Hara, and P. S. Srinivas Opion Volume and Sock Prices: Evidence on Where Informed Traders Trade. Journal of Finance 53: Figlewski, S Opions Arbirage in Imperfec Markes. Journal of Finance 44: Gabaix, X., A. Krishnamurhy, and O. Vigneron Limis of Arbirage: Theory and Evidence from he Morgage-Backed Securiies Marke. Journal of Finance 62: Greenwood, R Shor-and Long-erm Demand Curves for Socks: Theory and Evidence on he Dynamics of Arbirage. Journal of Financial Economics 75: Hakansson, N The Fanasic World of Finance: Progress and he Free Lunch. The Journal of Financial and Quaniaive Analysis 14: Hicks, J. R Value and Capial. Cambridge: Oxford Universiy Press. Hugonnier, J., D. Kramkov, and W. Schachermayer On Uiliy Based Pricing of Coningen Claims in Incomplee Markes. Mahemaical Finance 15: Jackwerh, J Recovering Risk Aversion from Opion Prices and Realized Reurns. Review of Financial Sudies 13: Keynes, J. M Some Aspecs of Commodiy Markes. Mancheser Guardian Commercial, European Reconsrucion Series Secion 13: Longsaff, F. A Opion Pricing and he Maringale Resricion. Review of Financial Sudies 8: Meron, R The Theory of Raional Opion Pricing. Bell Journal of Economics and Managemen Science 4: Newman, Y., and M. Rierson Illiquidiy Spillovers: Theory and Evidence from European Telecom Bond Issuance. Working Paper, Sanford Graduae School of Business. Poeshman, A. M Underreacion, Overreacion, and Increasing Misreacion o Informaion in he Opions Marke. Journal of Finance 56: Rubinsein, M The Valuaion of Uncerain Income Sreams and Pricing of Opions. Bell Journal of Economics 7: Rubinsein, M Implied Binomial Trees. Journal of Finance 49: Shleifer, A Do Demand Curves for Socks Slope Down? Journal of Finance 41: Soner, H. M., S. E. Shreve, and J. Cvianic There Is No Nonrivial Hedging Porfolio for Opion Pricing wih Transacion Coss. The Annals of Applied Probabiliy 5: Sapleon, R. C., and M. G. Subrahmanyam The Valuaion of Mulivariae Coningen Claims in Discree Time Models. Journal of Finance 39: Sein, J Overreacions in he Opions Marke. Journal of Finance 44: Wurgler, J., and E. V. Zhuravskaya Does Arbirage Flaen Demand Curves for Socks? Journal of Business 75: