rder Flows Dela Hedging and Exchange Rae Dynamics Bronka Rzepkowski # Cenre d Eudes rospecives e d Informaions Inernaionales (CEII) ABSTRACT This paper proposes a microsrucure model of he FX opions and spo markes. n boh marke segmens dealers receive cusomer order flows and use his privae informaion sraegically o speculae during inerdealer rounds. This non-payoff informaion is firs impounded in privae dealers invenories before affecing prices. Derivaive rading impacs he equilibrium exchange rae via feedback effecs of dela hedging sraegies followed by opion dealers o cover he FX risk embedded in heir opions porfolio. I is shown ha depending on he correlaion beween spo and opion order flows he volailiy of he exchange rae can eiher be amplified or reduced. JEL Classificaion: F3 D4 D8 D84 G3. Keywords: FX microsrucure model feedback effec dela hedging order flows implied volailiy. # CEII 9 rue Georges iard 7505 aris. Tel : 33 53 68 55 54 - rzepkowski@cepii.fr.
. Inroducion The FX microsrucure lieraure focuses exclusively on spo order flows o explain he changes in he exchange rae (Lyons 997; Lyons 00; Evans and Lyons 00). I excludes a priori he possible influence of he derivaive asses rading on he dynamics of he primary asse price. Such an approach would be warraned if markes were complee. The opion price would herefore be given by he replicaion cos of he synheic opion payoff wih a porfolio composed of he risky asse and he risk-free asse. In his framework opions are indeed redundan asses so ha neiher heir price nor heir order flows could provide relevan informaion abou he dynamics of he underlying asse price. Such an idealised environmen is described by a se of assumpions surrounding he Black and Scholes (973) (hereafer B&S) model or Garman and Kohlhagen (983) (G&K) for currency opions. However empirical evidence suggess ha markes are incomplee noably due o sochasic volailiy. Thus opion prices as well as opion order flows are likely o convey specific informaion no only abou he expeced reurn of he underlying asse bu also abou is fuure volailiy. The informaion may be all he more relevan ha he share of rading in currency opions markes in he oal FX rading is growing. The BIS riennial survey of foreign exchange aciviy (00) shows ha he marke average urnover in he FX opions segmen accouned for US$ 60 billions in April 00 i.e. 5.5% of he urnover in he spo segmen which is in consan decline (US$ 387 billions). The noional amouns ousanding of opions also provides an indicaion on he poenial impac of dynamic hedging sraegies on he spo marke. Indeed opion dealers use such sraegies o cover he FX risk embedded in opions porfolios which involves aking posiions in he spo marke. A he end of 00 his amoun was equal o US$ 3 38 billions (BRI 003). Alhough he FX microsrucure lieraure has ignored he ineracion beween opion rading and he spo dynamics research focusing on he sock markes has widely explored his area (Deemple and Selden 99; Back 993; John Koicha and Subramanyam 994; Easley Hara and Srinivas 998). The reason ress on he fac he sock microsrucure models are grounded on he assumpion ha sock raders have privae informaion abou he fuure payoff. Traders wih privae informaion may prefer o rade on opion markes due o lower coss and leverage effecs (Black 976; Mayhew Sarin and Shasri 995). If he privae informaion is relaed Three key hypoheses abou he underlying asse markes are posulaed in sandard derivaive pricing heory based on arbirage argumens. Markes are assumed o be complee fricionless and perfecly elasic. However he impac on he spo marke via dynamic hedging of opion dealers is more direcly relaed o he difference beween opions bough and sold. The noional amoun ousanding would hen be reduced o. billions of US dollars.
o fuure volailiy raders may only inervene on he derivaive marke (Back 993; Cherian and Jarrow 998). However his heoreical lieraure does no allow a consensus regarding he impac of opion rading on he underlying asse marke o be reached. 3 Whereas he exisence of privae informaion is a widely acceped assumpion when he sock markes are considered i is generally viewed as irrelevan for he foreign exchange marke perceived as he mos informaionally efficien marke. her approaches unrelaed o privae informaion analyse he influence of derivaive rading on he dynamics of he underlying asse price. The feedback effec lieraure highlighs he impac of dynamic hedging behavior of opion dealers on he underlying asse s equilibrium price and volailiy. These sraegies prove generally desabilising as opion dealers are supposed o be ne opion wriers so ha hey are buying in bullish spo marke and selling in bearish spo marke. In a parial equilibrium analysis laen and Schweizer (998) and Frey and Sremme (997) saring from a Black and Scholes economy show ha he parameers of he diffusion process for he price of he underlying asse become boh ime and price dependen. However his kind of feedback effec is raher indirec because only he coefficiens of he sochasic process followed by he underlying price reac o he dynamic hedging sraegies bu no he price iself. Garber and Spencer (996) assess he exen o which dynamic hedging sraegies may impinge he efficacy of ineres rae defence of fixed exchange rae regimes. Krueger (999) show ha during currency crisis he volailiy can reach sufficien high levels so ha he impac of ineres rae changes on dynamic hedgers is likely o be small. Genoe and Leland (990) evaluae he impac of such hedging sraegies assuming differences in informaion beween marke paricipans ha induce relaive illiquidiy in he sock marke. Due o is pro-cyclical impac he feedback effec lieraure leads 3 In a general equilibrium framework wih incomplee financial markes Deemple and Selden (99) show ha he inroducion of an opion may increase he equilibrium sock price and decrease is volailiy. Back (993) exends Kyle s (985) model of informed rading by inroducing he opions marke. The volailiy becomes sochasic bu is expeced average level does no change. The inuiion followed is similar o ha of Grossman (988). pion rading ransmis informaion ha would no be available if replaced wih opions synhesised by dynamic rading sraegies. Easley Hara and Srinivas (998) invesigae he informaional role of ransacions volume in opions markes in a model wih asymmeric informaion where raders can eiher rade in he equiy or in he opion markes. They find ha opion volumes conain informaion abou fuure sock prices. her approaches consider a marke ha is incomplee wihou he opions bu which becomes complee when he opions are inroduced. Brennan and Cao (996) show ha including opions does no change he price of he underlying asse. Cao s model (999) endogenizes he acquisiion of informaion so ha he increasing incenive o collec informaion leads o a higher sock price and a reducion in is volailiy. 3
o he predicion of an increase in he volailiy of he underlying asse price. Bu his conclusion does no necessarily fi he empirical evidence (Mayhew 000). Furhermore assuming ha opion dealers are always ne opion wriers is no confirmed by he daa repored by he BIS (003): here are almos as many opions bough as opions sold by dealers. The aim of his paper is o analyse he ineracion beween currency opion rading and he inra-daily exchange rae dynamics. The FX inerdealer microsrucure model of Evans and Lyons (00) is hus exended o include a derivaive marke segmen. Their model is based on cusomer order flows which dealers claim o be heir mos imporan source of informaion (Lyons 995; Yao 998). In his framework cusomer order flow is he source of asymmery among dealers. Boh spo and opion dealers are hus supposed o use heir privae informaion sraegically in ha heir speculaive demand will depend on he impac heir rade will have on subsequen prices. Hence he privae informaion is impounded in dealers invenories before being refleced in quoed prices. The ineracion beween he wo markes is capured by he effec of dela hedging behaviour on he dynamics of he underlying asse price. In a parial equilibrium framework i is shown ha he volailiy of he exchange rae could increase or decrease depending on he correlaion beween he spo and he currency opion order flows. The paper is organised as follows. Secion describes he main feaures of he inerdealer rading model of he spo and derivaives marke segmens. Secion 3 displays he opimal quoing and rading sraegies of spo and opion dealers. Secion 4 presens simulaion resuls considering differen correlaion coefficiens beween he spo and he currency opion order flows and differen mauriies for opions. The las secion concludes.. The inerdealer rading model of he FX opions and spo markes.. Environmen The decenralised FX marke is supposed o be composed of he spo and opion marke segmens and o operae in discree ime. Trading is simulaneous on he wo segmens. There are N spo dealers indexed by i M opions dealers indexed by q a coninuum of cusomers rading in he spo marke a coninuum of cusomers rading in he opions marke and dela hedgers. All agens on he spo and currency opions markes are raional and have he same risk aversion parameer γ. They are supposed o maximise a negaive exponenial uiliy funcion: [ ( W )] U E exp γ + () 4
where E is he expecaions operaor condiional on agens informaion and + W is he wealh a he end of day +. There are hree asses in he model one riskless one risky and a call opion. The daily ineres rae on he risk free asse is denoed r. The daily payoff on he risky foreign asse a ime denoed observed publicly a he beginning of he rading day. I is composed of a series of incremens R is realised and R so ha: R R s s The payoff incremens are i.i.d. N ( ) R 0 R σ and represen he flow of macroeconomic informaion publicly () available such as he change in ineres raes. Each day is characerised by hree rading rounds (Table ). Table - Three rading rounds in he currency opion and spo markes Spo marke Currency opions marke Round The payoff R is realised Spo dealers quoe Cusomers rade wih spo dealers Round Spo dealers quoe Spo dealers rade wih spo dealer Spo and opions inerdealer order flows revealed Round 3 Spo dealers quoe Spo dealers rade wih he public pion dealers quoe Cusomers rade wih opion dealers pion dealers quoe pion dealers rade wih opion dealers pion and spo inerdealer order flows revealed pion dealers rade wih dela hedgers Dela hedger rade wih he public on he spo marke.. Quoes on he spo and derivaive markes The quoing rules described in Lyons (997) prevail. Quoing is simulaneous independen and required. Hence a dealer s quoe on he spo or on he opion marke canno be condiioned on he quoes of oher spo and opion dealers wihin a given round. Quoes are observable and available o all paricipans and each quoe is a single price a which each dealer agrees o buy and sell any amoun. Finally no arbirage requires ha quoes are common o all dealers in all rounds so ha hey can only be condiioned on public informaion. Spo dealers quoe simulaneously and independenly he price of he foreign exchange rae denoed i k where k corresponds he k-round price and k3. n he derivaives segmen simulaneously and independenly opion dealers quoe he Black and Scholes (973) a-he-money-forward (ATMF) implied 5
volailiy. In he FX over-he-couner marke currency opions are de faco quoed in B&S volailiy raher han in opion prices and ATMF opions are also he mos raded. The average expeced volailiy of he underlying asse price for a given mauriy τ is denoed V q k. I is hus a single poin on he volailiy erm srucure. This volailiy is updaed during a rading day wih he arrival of new public informaion. Implied volailiy herefore flucuaes in an unpredicable way. Boolen and Whaley (003) show ha an imporan facor driving he change in implied volailiy quoes is he ne buying pressure from public order flow. Hence in our microsrucure framework he implied volailiy will vary wih he sochasic ne order flow from opion cusomers. In his conex of ime-varying volailiy fair opion value and perfec hedge canno be deermined wih cerainy. Indeed markes are incomplee due o he volailiy risk in opion rading. Engle and Rosenberg (000) propose he following approximae valuaion formula o price a-he-money-forward calls which is referred o as he Black-Scholes-plug-in formula (hereafer BS): ( E [ V ( )] τ ) C (3) k BS k k k where C k sands for he price of ATMF calls wih mauiry τ in round k. As in Engle and Rosenberg (00) dependence on he srike price and on domesic and foreign ineres raes are ignored. Implicily i is supposed ha he mos imporan variables of he call price are he price of he foreign currency and he implied volailiy for a given mauriy. As opions are ATMF he BS price is relaively accurae because he BS model relies on he approximae lineariy of he B&S formula in he volailiy parameer for ATMF opions. 4 Hence updaing expeced volailiy based on dealers expanded informaion se during he day will resul in raional changes in he call price. Furhermore as volailiy quoes are consan and common o all opion dealers wihin each round o avoid arbirage opporuniies opion dealers have no incenive o deviae from he BS model wihin each round once hey believe i is he correc pricing model. 5 Indeed as he opion price is equivalen o he replicaion cos of a synhesised opion using anoher model would necessary give rise o a differen opion price and hus o arbirage opporuniies. 4 As noed by Engle and Rosenberg (000) he effec of he volailiy risk premium should be small because average volailiy is used raher han average volailiy under a risk-neural measure. 5 Cherian and Jarrow (998) show ha when dealers share he same beliefs ha he B&S formula is he correc opion pricing model his resuls in a self-fulfilling raional equilibrium opion price. 6
.3. Trading rounds.3. Round Afer he realisaion of he payoff of holding foreign exchange each spo dealer i quoes i a which he agrees o sell or buy any amoun whereas simulaneously and independenly each opions dealer q quoes a-hemoney-forward implied volailiy V q. Each spo dealer i receives a ne order flow denoed i which is x normally disribued wih zero mean and known sandard deviaion σ x. Each opion dealer q receives a ne order flow denoed y q which is also normally disribued wih mean 0 and known sandard deviaion σ y. Whereas he spo order flow refers o he difference beween he buying and selling posiion in foreign exchange received by he dealer he opion order flow denoes he difference beween he number of calls bough and sold by opion cusomers. Spo and derivaive flows are supposed o be independen of he payoff incremen R. Each spo order flow x i is execued a he quoed price i where posiive (negaive) i denoes a buy x (sell) order from cusomers. Each opion order flow T i C is execued a price q according o equaion 3 where posiive (negaive) D i means ha cusomers are nes buyers (sellers) of calls. Boh order flows on he spo and derivaive marke segmen are liquidiy demand shocks from he public which are privae informaion for he dealer who receives i..3. Round Round is an inerdealer rading round. Simulaneously and independenly each spo dealer quoes i and each opions dealer quoes V q. These quoes are observable and available o all dealers on boh markes. During his round dealers aemp o reduce heir risk exposure semming from heir rade wih cusomers bu hey also use heir privae informaion o speculae. This privae informaion will be firs impounded in each dealer round invenory before affecing he round 3 quoes. Le E T i i spo dealer i and [ ] ougoing order for dealer i D i be he speculaive demand from he he expecaion of dealer i incoming flows from oher dealers hen he ne T i will be: [ T ] T i Di + xi + E i i (4) 7
Le D q be he speculaive demand from he opion dealer q and E [ q q ] he expecaion of dealer q incoming flows from oher opion dealers hen he ne ougoing order for dealer q [ q q ] q Dq + yq + E q will be: (5) A he end of round spo and opions inerdealer order flows are revealed o all marke paricipans and hus become public informaion. A ne posiive spo inerdealer order flow indicaes ha buy orders are greaer han sell orders whereas a posiive opions order flow denoes ha dealers are ne buyers of call opions. n X T i i and m q Y q (6).3.3. Round 3 In round 3 opion dealers rade wih dela hedgers so as o hold no ne posiion in opions a he end of he rading day. Disinguishing wo classes of agens in he derivaive marke is quie arificial because opion dealers use de faco dela hedging sraegies o cover heir exchange rae risk embedded in heir opions porfolio. Bu his allows he logic of dela hedging o be presened in a simplified manner. pion rading is indeed quie differen from underlying asse rading in ha here is no need o keep real invenories of opions because in equilibrium he poin is o decide which opimal hedging sraegy o use. Furhermore due o ransacion coss dela hedging canno operae in coninuous ime and rebalancing he posiion in he underlying currency can only ake place in discree ime. Hence i is supposed ha dela hedgers will have o deermine and implemen he bes hedging sraegy on a daily frequency in round 3. The purpose of dela hedging is o eliminae he firs order sensiviy of he opions porfolio value wih respec o he price of he underlying asse (he foreign currency). To do so dela hedgers replicae he payoff of he opions sold or bough by a self-financing porfolio composed of bonds and of he foreign asse. Le consider he hedge of a shor posiion in one call ha gives he righ o is holder o buy an amoun z of foreign currency a mauriy if he opion is exercised. Dela hedging his posiion in round 3 involves o buy foreign currency a he price r 3 and o borrow e ( C3 3 3 ) 3 unis of he on he domesic marke. If he foreign currency appreciaes he increase in he value of he spo posiion will be exacly compensaed by he decrease in he value of he porfolio composed of one call and conversely if he exchange rae depreciaes. The value of 8
3 is equal o z he quaniy of foreign currency likely o be bough a mauriy specified in he opion conrac imes he dela of he call opion. The dela of a call opion is given by he firs derivaive of he BS formula wih respec o level of volailiy: a he curren υ (7) BS 3 3 + 3 V 3 / 3 where BS 3 is he dela of he B&S formula and 3 υ is he vega of he call i.e. he firs derivaive of he call price wih respec o volailiy. Equaion 7 indicaes ha a change in he underlying price affecs he call price no only hrough he dela under consan volailiy bu also hrough he vega of he call opion and a shif in he average expeced volailiy. However in order o simplify calculaions i is supposed ha dela hedgers when covering heir exchange rae risk on he spo marke use a dela deriving from he sandard B&S model. 6 Tha amouns o neglecing he change in average expeced volailiy due o a firs order change in he price of he foreign exchange. I is supposed ha he dela formula is known by all marke paricipans. In round 3 dela hedgers will rade wih he public o be immunised agains he exchange rae risk a he end of he day. However dela hedging does no allow all he risk o be removed. The change in value of a dela hedged porfolio due o ineres raes change ime decay and o he gamma are ignored. Their effec especially on longerm opions can be considered of second order. However he volailiy risk for dela hedgers remains significan. When quoing he round 3 implied volailiy opion dealers have herefore o know no only he size of he oal opions flow ha will be absorbed by dela hedgers bu also heir volailiy risk bearing capaciy. Simulaneously and independenly hey quoe a volailiy so ha dela hedgers accep o absorb heir opions invenory imbalances. n he spo marke dealers rade wih he public o have a zero ne posiion a he end of he day. The public encompasses non-dealer agens whose risk bearing capaciy is greaer han ha of spo dealers. The number of cusomers is indeed supposed o be large relaive o he N dealers (in a convergence sense). Their capaciy is however limied so ha he aggregae demand of liquidiy suppliers is no perfecly elasic. They will herefore 6 The dela of ATMF opions is ypically equal o 0.50. If he price of he foreign currency increases in round 3 wih respec o he round he call becomes in-he-money and he value of he dela is hen greaer han 0.50. Conversely if he exchange rae depreciaes he call becomes ou-of-he-money and is dela becomes inferior o 0.50. 9
ask for a lower price o accep o hold larger posiions in foreign currency. Their moive o rade in round 3 is non-sochasic bu purely speculaive. The raional round 3 exchange rae quoe should be se so ha liquidiy suppliers accep o absorb he invenories imbalance on he spo marke and he flow arising from dela hedging behaviour on he derivaive marke. Spo dealers are hus supposed o know he aggregae demand from he spo segmen and he risk bearing capaciy of he public. Bu hey also are supposed o perfecly infer he ne demand from dela hedgers. To do so spo dealers mus know he dela formula and be able o deduce he round 3 implied volailiy from he inerdealer opion order flow. This implies ha he opimal rading rule followed by opion dealers is known by spo dealers as well as he volailiy risk bearing capaciy of dela hedgers..4. bjecive funcion and demands.4. bjecive funcion of spo dealers Each spo dealer deermines exchange rae quoes and his speculaive demand by maximising a negaive exponenial uiliy funcion. Spo dealers are supposed o have a closed posiion a he end of each day. Max i i i 3 Di where W E [ exp( W ) ] i 3 W γ (8) + x i + T T s.. ( ) i 3 i0 i i i i i i i i i i3 W i 0 is he wealh of dealer i a he beginning of he firs rading round and a denoes a rade or a quoe received by dealer i from oher dealers. x + T T.4.. bjecive funcion of opion dealers The opimisaion problem for each opions dealer is also defined over four variables he hree implied volailiy quoes V q V q V q 3 and is speculaive demand in round D q. pions dealers are supposed o hold zero invenory a he end of he day. Max Vq Vq Vq 3 Dq E [ exp( Wq 3 ) q ] γ (9) s.. W W + y C + C ( y + ) C q 3 q0 q q q q q q q q q 3 C where W q 0 is he wealh of dealer q a he beginning of he firs rading round and a denoes a rade or a quoe received by dealer q from oher opion dealers. The informaion ses of boh classes of dealers a each round are summarised in Appendix A. 0
.4.3 The public The public s oal demand for he risky asse is given by maximising he expeced uiliy in equaion () subjec o he following budge consrain: ( + R ) W W + h3 3 + + 3 where + (0) W is he wealh a he end of ime. The demand for he foreign currency in round 3 denoed hus a linear funcion of is expeced reurn condiional on public informaion. ( ) ( E[ + R ] ) h 3 3 3 + + 3 3 h 3 is θ () θ and γσ + R where he posiive coefficien θ capures he risk bearing capaciy of he public wih ( ) denoes he condiional variance of + 3 + R+. σ +R.4.4 The dela hedgers n he assumpion ha he change in he value of a dela hedged porfolio only depends on he variaion of he volailiy beween and + he budgeary consrain of dela hedgers simplifies o: 7 ( V V ) W W + h3 3 3 + 3 + υ () where υ 3 is he vega of he call opion ha is he firs derivaive of he call price wih respec o a small change * r in he volailiy: υ τ e N ( d ) 3 3 3. Maximising expeced uiliy in equaion () subjec o his budgeary consrain will deermine he demand of dela hedgers for call opions in round 3 ( V ) υ ( E[ V ] V ) h 3 3 3 3 + 3 3 h 3. θ (3) where he posiive parameer θ capures he volailiy risk bearing capaciy of dela hedgers wih θ and ( γσ C ) σ C denoing he condiional variance of he call price condiioned on public informaion in round 3 a ime. Dela hedgers will hen accep o buy or o sell opions from opion dealers a a price ha depends on he expeced change of implied volailiy beween round 3 of period and + and on he sensiiviy 7 The fac ha he ousanding amoun of opions hold by dela hedgers is no more dela hedged due o he change in he price of he foreign exchange in round is no aken ino accoun in his specificaion. This amouns o neglecing he impac of rebalancing opions porfolio on he round 3 implied volailiy and o ignoring he effec of dynamic hedging he sock of opions on he equilibrium exchange rae.
of heir opions porfolio wih respec o a small change in he implied volailiy. The higher he quaniy dela hedgers have o sell (buy) he higher (lower) he implied volailiy ha opion dealers have o quoe in round 3..5. Marke clearing n he derivaive marke he round 3 implied volailiy quoe should be such ha he dela hedgers demand maches he quaniy of opions liquidaed by opion dealers. As here is no leakage during inerdealer rading rounds he amoun of opions o be sold or bough by opion dealers corresponds exacly o he ne selling or buying posiion of cusomers in round. h m ( V3 ) y 3 q (4) q n he spo marke in round 3 he liquidiy suppliers will have o absorb no only he demand of spo dealers which is exacly he ne aggregae cusomers order flows bu also he flow arising from dela hedgers who cover heir exchange rae risk on he spo marke according o he dela of heir porfolio m y q 3 q. The round 3 marke-clearing price of he foreign currency should herefore saisfy: n m ( ) + [ ( )] 3 xi y q 3 3 V3 h3 (5) i q These wo marke clearing condiions on he derivaive and spo markes lead o he following equilibrium sysem (see Appendixes B and C for deails): V 3 3 E E [ 3 + + R+ 3 ] + λx + λ Y 3 ( 3 V3 ) [ V ] + λ Y / υ ( V ) 3 + 3 3 3 3 λ αθ ( β θ ) where ( ) λ and λ ( β θ ) wih α and respecively he spo and opions dealers. As ( V ) and ( V ) 3 3 3 β enering he opimal rading rule of υ are non-linear funcions in he 3 3 3 round 3 price of he foreign currency and in implied volailiy here is no closed-form soluion for he simulaneous equilibrium on he wo markes. However as he change in dela and in vega beween round and round 3 only depends on he change of V and using he Taylor firs order linearisaion allows an approximae analyical soluion o be provided. 3. Equilibrium and implicaions
As in Evans and Lyons (00) he equilibrium concep used refers o he Bayesian-Nash Equilibrium (BNE). Under his equilibrium concep agens updae beliefs according o Bayes rule and given hese beliefs quoing and rading sraegies followed by dealers are sequenially raional. Deails on he proofs are provided in he Appendixes B and C. 3.. Equilibrium quoing sraegies The assumpion of no arbirage wihin each round implies ha all dealers quoe a common price (he exchange rae on he spo marke and he implied volailiy on he derivaive segmen). As in a given marke he quoed price is common o all dealers i is herefore condiioned on public informaion only. The common informaion shared by all dealers a he beginning of round includes he payoff incremens he round 3 quoe of he foreign exchange and of he implied volailiy a ime -. 3.. The opion marke In he derivaive marke i is supposed ha he innovaion in payoff does no modify he average expeced volailiy of opion dealers. Hence he only relevan informaion for opion dealers a he beginning of round is he round 3 volailiy quoe of he previous day. Inerdealer FX opion order flows are observed a he end of round. This informaion will be refleced in he round 3 implied volailiy quoe. roposiion : A quoing sraegy on he opion marke is consisen wih symmeric BNE only if he implied volailiy quoes in rounds and are common across dealers and equal o: V (6) V V3 where V 3 is he round 3 implied volailiy quoe from he previous day. roposiion : A quoing sraegy is consisen wih symmeric BNE only if he common round 3 implied volailiy quoe is equal o: V 3 λ Y V + (7) τ π 3 roposiion 3: If dela hedgers hold raional expecaions and opion dealers quoe according o proposiions and he change in he volailiy quoe from one day o he oher is given by: λ Y V3 (8) τ π 3 3
where λ is posiive. roposiions and 3 show how he change in implied volailiy quoe is relaed o he inerdealer FX opion order flows and o he sensiiviy of he call price relaive o a small change in volailiy. This sensiiviy is capured by τ / π which sems from he firs order linearisaion of he vega. The 3 higher his sensiiviy he lower he change in implied volailiy needed so ha dela hedgers accep o absorb he aggregae cusomers porfolio shif where λ Y λ q q λ β y q q. 3.. The spo marke roposiion 4: A quoing sraegy on he spo marke is consisen wih symmeric BNE only if he quoed prices in rounds and are common across dealers and se o: 3 + R (9) where 3 is he round 3 price of he previous day and R is he payoff innovaion observed a he beginning of round. roposiion 5: A quoing sraegy on he spo marke is consisen wih symmeric BNE only if he common round 3 price is se o: 3 λx + + λ Y z ( + ( V3 V ) τ / π ) ( λ Y zγ ) (0) where and Γ are respecively he dela and he gamma of each call opion held by opion dealers in round and z is he amoun of foreign exchange ha he holder of one call is allowed o buy if he opion is exercised a mauriy. roposiion 6: In a BNE where he public holds raional expecaions he change in he round 3 price of he foreign exchange rae beween period - and is given by: λx R + + λ Y z ( + ( V3 V ) τ / π ) ( λ Y zγ ) () The change in he price of he foreign exchange is a funcion of he payoff incremen of he spo and opion inerdealer order flows and of he inra-daily change in he implied volailiy. The variaion from period - o is herefore relaed o marke condiions prevailing in round on he spo and derivaive segmens. Finally equilibrium quoes in round 3 on boh marke segmens are inerdependen and simulaneous. Indeed he equilibrium on he opion marke depends on he round 3 price quoed on he spo segmen and he equilibrium round 3 exchange rae is a funcion of he change in volailiy quoe during he day. 4
3.. Equilibrium rading sraegies The appendix C shows ha he opimal rading rule on he spo and derivaive markes can be expressed as a linear funcion of he order flows received by dealers on heir own marke in round. Alhough dealers receive cusomers orders ha are differen in magniude he relaion beween he size of hese orders and he dealer ne ougoing orders is he same across all dealers in a given marke. However he opimal rading rule is no idenical in he spo and in he derivaive marke segmens. 8 roposiion 7: An opimal rading sraegy on he opion marke ha conforms o he BNE is given by: q β yq for q. () where β >. roposiion 8: An opimal rading sraegy on he spo marke ha conforms o he BNE is given by: T i αxi for i. (3) where α >. In boh marke segmens dealers are hus able o infer he aggregae porfolio shif in round from he inerdealer order flows revealed a he end of round. They also know ha he public in he spo marke and he dela hedgers on he derivaive marke will have o absorb hese porfolio shifs and ha his will induce prices o change accordingly. In his framework opions are no redundan asses as derivaive rading will impac on he opion price hrough he change in implied volailiy and finally on he dynamics of he underlying asse hrough dela hedging behaviour. Speculaive demand on he opions marke will indeed influence he oal inerdealer order flow a he end of round and hus he round 3 implied volailiy quoe. The larger he posiive FX opion order flow he higher he implied volailiy in round 3 he greaer he flows from dela hedging behaviour of opion dealers and he higher he price of he foreign exchange. Bu he exchange rae shif beween round and round 3 may be dampened if currency opions order flows are negaively correlaed wih spo flows. 4. Simulaion resuls 8 Whereas he opimal rading rule in he spo marke involves a consan parameer α he opimal rading rule in he derivaive segmen is relaed o marke condiions prevailing in round and especially o he round price of he foreign currency (see Appendix C). 5
The aim of his secion is wofold. Firs i is devoed a assessing how FX opions rading may impac he equilibrium exchange rae. As he equilibrium is simulaneous on he spo and on he derivaive segmen he implied volailiy quoe ha clears he opions marke also depends on he equilibrium price of he foreign exchange. In order o analyse how he wo markes inerac a sensiiviy analysis is conduced according o he amoun of FX opions order flows is posiive or negaive correlaion wih spo flows and wih respec o he mauriy of raded opions. rediced equilibrium exchange rae and implied volailiy however sem from an approximae model ha ress on hree simplificaions necessary o ge a closed-form soluion. Hence his secion is also direced a measuring he disance beween he approximaed equilibrium and he numerical equilibrium soluion ha does no rely on any simplificaion. The scope of validiy of he approximae equilibrium equaions presened in secion 3 is hen evaluaed. 4.. Equilibrium and FX opion order flows The sensiiviy analysis of he change in equilibrium exchange rae and implied volailiy beween round and round 3 is based on he following benchmark scenario. The round price of he foreign exchange is supposed o be equal o he round implied volailiy o 0% he domesic ineres rae o % he foreign ineres rae is posulaed o be null. The risk aversion γ is equal o 3 he parameers α and β caching he speculaive behaviour of dealers are se o.. 9 The daily volailiy of he exchange rae σ is % and ha of he call opion σ C is.%. Given he posulaed value for he parameers he spo order flow is calibraed o yield a % change in he exchange rae when no derivaive rading is accouned for. X is hus supposed o equal 40. This provides a benchmark o sudy how he inroducion of FX opions order flows will impac on he equilibrium. I is furhermore supposed ha he amoun of foreign currency ha he holder of a call is allowed o buy z is equal o. FX opions urnover a presen accouns for around 5% of spo rading. Given he growing developmen of he derivaive segmen simulaions will consider a range for opions order flows up o half ha prevailing on he spo marke. Furhermore he impac on he equilibrium exchange rae does no only depend on he quaniy of raded opions bu also on he ne aggregae buying or selling posiion of cusomers. Hence simulaions should consider posiive and negaive correlaions beween he spo and opion order flows. Finally as he value of he 9 Dealers on he opions marke expecing for example an increase in round 3 implied volailiy because hey have received ne posiive call order flows from heir cusomers will ry o be ne buyers of volailiy a he end of round (β>). In round 3 if implied volailiy increases hey will make a profi from he sale of heir opions porfolio. 6
dela used o cover foreign exchange risk depends on he mauriy of opions he sensiiviy of he equilibrium o his parameer has also o be evaluaed. Figure displays he equilibrium exchange rae relaive o he amoun of FX opion order flows wih respec o spo order flows when he laer are posiive. osiive spo order flows indicae ha cusomers have a ne aggregae buying posiion of he foreign exchange. Wih no derivaive rading he benchmark scenario predics a % increase in he price of he foreign exchange. When he currency opions order flow is posiive meaning ha cusomers are ne buyers of calls he exchange rae response is amplified. Indeed for derivaive order flows equal o half he spo flows he change in he exchange rae ranges from.6% for he five-years mauriy o.30% for he hree-monhs ime o expiraion. The exchange rae variaion is hen all he higher ha he mauriy is shor. The dela is indeed a decreasing funcion of he ime o mauriy when opions are in-hemoney i.e. wih a dela superior o 0.5. As he price of he foreign exchange rises due o posiive spo order flows opions ha were previously a-he-money-forward become in-he-money. Hence he dela of he opions porfolio decreases wih longer mauriy so ha he addiional flows from dela hedging behaviour are limied and he exchange rae response is lower for he five years mauriy han for he hree-monh mauriy. The difference in exchange rae response wih respec o he mauriy of raded opions appears however raher sligh. Besides when FX opion order flows are negaive dela hedging behaviour will end o reduce he equilibrium value of he exchange rae relaive o is iniial value of.0 because selling flows from dela hedgers will have o be absorbed in a buying spo marke. Boh he relaive decline in he price of he foreign exchange and he decrease in equilibrium implied volailiy will feedback ino a reducion of he dela of he opions porfolio. The downurn in he equilibrium price is hus slighly limied by he decrease of he dela. Moreover his fall is all he more pronounced he shorer he ime o expiraion of opions. Indeed he shorer he mauriy he higher he dela and he greaer he selling flows from dela hedging on he spo marke. The change in he exchange rae is hen dampened o 0.7% relaive o he % increase wih no derivaive rading (-0.7% compared wih he iniial.0 exchange rae) when he hree monhs mauriy is concerned and added up o 0.75% for he five years mauriy (-0.5%). Hence he exchange rae response is no linear wih respec o he amoun of FX opion order flows. For a given mauriy and a given amoun of opions order flows he magniude of he exchange rae change is sronger for posiive han for negaive opion flows because he value of he dela is also lower for negaive han for posiive FX opion order flows. 7
Figure displays he equilibrium exchange rae pah wih respec o FX opions order flows when spo order flows are negaive. Wih no derivaive rading he price of he foreign exchange decreases by %. Compared wih he siuaion where spo flows are posiive he change in he exchange rae is no symmerical and is of smaller size. Indeed opions ha were iniially a-he-money-forward hen become ou-of-he-money i.e. wih a dela inferior o 0.5. So for a given amoun of FX opion order flows he sensiiviy of he exchange rae response is weaker. Besides The dela of ou-of-he-money opions is an increasing funcion of he ime o expiraion. Thus when FX opions order flows are negaive he exchange rae response is all he more magnified ha he mauriy is far away. In fac he change in he equilibrium price is.0% when he hree-monhs mauriy is considered whereas i urns o be.4% for he five-years mauriy. By conras posiive FX opion order flows dampen he iniial % decrease of he exchange rae o 0.78% for he hree-monhs mauriy o 0.75% for he five-years mauriy. So he exchange rae response o dela hedging sraegies is asymmeric regarding he posiive or negaive spo order flows. Hence he model predics a desabilising effec of derivaive rading on he spo dynamics when he FX opions order flow is of he same sign as ha of he spo order flow. Indeed dela hedging behaviours exacerbae he exchange rae change. This magnifying effec is more imporan when shor-erm opions are considered and when spo order flows are posiive. By conras when spo order flows are negaive he amplified response of he price of he foreign exchange is observed for long-erm opions. Bu when derivaive and spo order flows are negaively correlaed he impac on he exchange rae proves o be sabilising as i reduces he daily volailiy. This sabilising effec is all he more significan he longer he opions erm whaever he ne posiion on he spo marke. Figures 3 and 4 presen he equilibrium implied volailiy relaive o FX opion order flows when spo order flows are respecively posiive and negaive. Figure 3 shows ha implied volailiy increases when FX opions order flows are posiive from an iniial level of 0% up o levels ranging from 0.80% for he five-years mauriy o 3.56% for he hree-monhs mauriy. Tha resul direcly sems from he equilibrium implied volailiy equaion. I is a decreasing funcion of he opions mauriy so ha he volailiy response is all he more dampened ha opions have a long mauriy. Conversely implied volailiy decreases when opions order flows are negaive urning o levels ranging from 6.4% for he hree-monhs mauriy o 9.0% for he fiveyears mauriy. The decline in volailiy is hen all he higher ha he mauriy is shor. Figure 4 shows ha equilibrium implied volailiy is no sharply differen if spo order flows are negaive especially for long 8
mauriies. Indeed equilibrium quoes range from 0.8% for he five-years mauriy o 3.64% for he hreemonhs ime o expiraion when FX opion order flows are posiive. These figures respecively equal 6.35% and 9.8% when derivaive flows indicae a ne aggregae selling posiion. The raher limied difference in implied volailiy wheher spo order flows are posiive or negaive can be explained as follows. Firs is influence is only indirec via he equilibrium exchange rae. When spo order flows are posiive he price of he foreign exchange increases. This ends o limi he increase (decrease) in implied volailiy when posiive (negaive) FX opion order flows are accouned for. By conras negaive spo order flows lead o a decline in he exchange rae which ends o amplify he reacion of he implied volailiy: i increases (decreases) more when FX opions order flows are posiive (negaive). This phenomenon is more marked for shor-erm mauriy because he volailiy equilibrium is a decreasing funcion of he ime o expiraion. Hence he impac of exchange rae on he implied volailiy equilibrium is higher when raded opions have a shor-erm mauriy. Bu for long-erm opions he impac of he spo marke on he equilibrium in he derivaive marke is almos insignifican. In his case he implied volailiy response o FX opion order flows is indeed almos linear wheher spo order flows are posiive or negaive. The equilibrium exchange rae is very sensiive o he sign of spo order flows and o a lesser exen o he sign of he FX opion order flows. The influence of he equilibrium implied volailiy on he equilibrium exchange rae is only indirec via he value of he dela. Insofar as his value depends prominenly on he change in he exchange rae raher han on a change in implied volailiy is impac on he spo marke is really limied. The equilibrium implied volailiy is sensiive o he FX opion order flows and o a far lesser exen o he equilibrium exchange rae. Alhough he price of he foreign exchange eners he equilibrium implied volailiy equaion is effec is quie limied. The influence of spo order flows is only indirec hrough he equilibrium price of he foreign exchange. 4.. Accuracy of he approximae equilibrium These predicions regarding he behaviour of he equilibrium exchange rae and implied volailiy come from a model ha resors o hree approximaions. The equilibrium equaions displayed in secion 3 are based on he B&S dela formula raher han he BS dela ha is obained by differeniaing equaion 3 wih respec o he underlying asse price. The change in he average expeced volailiy due o a firs order change in he exchange rae is hus ignored. Second hey depend on he firs-order linearisaion of he vega formula ha eners he equaion of opions demand from dela hedgers. This linearisaion involves ha he round 3 approximae vega is 9
equal o a vega of a-he-money-forward opions porfolio. Bu as he round 3 price of he foreign exchange and implied volailiy change relaive o heir round value he opions in round 3 are of course no more a-hemoney-forward. Third a firs-order linearisaion of he B&S dela formula is used o provide a closed-form equilibrium exchange rae. Hence he accuracy of he approximae equilibrium has o be evaluaed and compared wih simulaions resing on a model wihou any simplificaion o provide he scope of validiy of he projeced pahs. Simulaions wih respec o he amoun of FX opion order flows o heir correlaion wih spo flows and according he opions mauriy have been conduced o gauge he forecas errors. Figures 5 and 6 presen he difference in equilibrium exchange rae pah wih respec o currency opions order flows when he spo flow is respecively posiive and negaive. The curves represen he difference beween he numerical and he approximae soluions. In mos cases he approximae model leads o overesimae he exchange rae response when FX opions rading is inroduced wheher spo order flows are posiive or negaive. The approximae dela ends indeed o overesimae he numerical dela when FX opion order flows are posiive whereas i underesimaes i when opions flows are negaive. 0 This disance is decreasing wih mauriy. However in all he cases he percenage error is less han %. The highes error is equal o 0.008% for he hree-monhs mauriy when spo and FX opion order flows are posiive. When hey are boh negaive he highes forecas errors grow o 0.04% for he greaes amoun of opion flows. The accuracy of he approximae model for his range of derivaive rading appears o be quie good. However for shorer mauriy he errors could poenially grow and become significan. Figures 7 and 8 display he difference beween he numerical equilibrium implied volailiy and ha coming from he approximae closed form soluion. When spo order flows are posiive he percenage deviaion is almos always less han %. The difference is all he lower ha he mauriy of opions is long. The numerical equilibrium volailiy is higher when FX opion order flows are posiive and lower when derivaive flows are negaive. This is due o he firs order linearisaion of he vega ha amouns o posulaing ha he value of he normal densiy funcion in round 3 is a is highes level whereas any variaion of he exchange rae or of he implied volailiy in round 3 reduces his value as is argumen is no more equal o zero. However errors slighly 0 For he hree-monhs mauriy however when spo flows are posiive and when FX opion flows range beween one quarer o one half of spo flows he approximae dela is higher han he numerical dela. So he equilibrium exchange rae coming from he approximae model is lower han he numerical equilibrium soluion. 0
superior o % are observed when negaive FX opions order flows amoun o around one half of spo flows for he hree-monhs mauriy. When boh spo and FX opion order flows are negaive forecas errors urn o be significan for hree-monhs opions flows ranging from one quarer o one half of spo flows. The error is equal o 6.% for he highes amoun of FX opions flows a a hree-monhs mauriy and.% for he six-monhs mauriy when FX opion flows are negaive. These simulaions have shown ha he mos problemaic approximaion is relaed o he linearisaion of he vega. Indeed i inroduces significan differences in he equilibrium implied volailiy a leas for shor-erm opions when FX opion order flows amoun o half ha of spo flows. However o he exen ha implied volailiy does no significanly aler he equilibrium on he spo marke his valuaion error does no reflec in he equilibrium exchange rae. Hence he approximae model provides accepable equilibrium pahs of he price of he foreign exchange a leas for he mauriies considered and he amoun of FX opion order flows. I seems however ha for shorer mauriy (less han hree monhs) and imporan amouns of derivaive rading errors could become significan. Bu if currenly observed FX opion order flows relaive o spo flows are considered even for shorer mauriy he approximae model provides accurae predicions regarding he behaviour of he equilibrium exchange rae. 5. Conclusion This paper proposes a FX microsrucure model of inerdealer rading on he spo and opions markes ha explicily accouns for ineracions beween he wo marke segmens. Boh prices are direcly relaed o inerdealer order flows in heir own marke. Bu he equilibrium on he derivaive marke is also influenced by he price of he foreign exchange and he equilibrium exchange rae depends on he FX opion order flows and indirecly on he equilibrium implied volailiy hrough is impac on he value of he dela. The equilibrium on he wo marke segmens is hus simulaneous. I is shown ha depending on he correlaion beween spo and opion order flows he daily volailiy of he exchange rae can eiher be amplified or reduced. When he order flows on he wo marke segmens are posiively correlaed dela hedging sraegies amplify he exchange rae response. For posiive spo order flows indicaing ha cusomers are ne buyers of he foreign currency he increase in he exchange rae is all he higher ha he mauriy of opions is shor. By conras when spo order flows are negaive he response is all he more pronounced ha he ime o expiraion of opions
is far away. Furhermore when order flows on he wo markes are negaively correlaed dela hedging has a sabilising impac on he exchange rae o he exen ha i limis is variaion. This effec grows when he mauriy of opions is long. Finally simulaions show ha depending on he sign of spo order flows he exchange rae response is asymmeric o FX opion order flows. Is impac is indeed more pronounced when spo flows are posiive his influence being all he higher ha raded opions have a shor mauriy. As far as he equilibrium implied volailiy is concerned is change is mainly driven by FX opion order flows and he effec of he equilibrium exchange rae is very limied. Volailiy increases (decreases) wih posiive (negaive) opion flows he rise (fall) being all he higher han he mauriy of opions is shor. Moreover he equilibrium equaions for he spo and derivaive markes depend on some firs-order approximaions. Compared wih numerical soluions ha do no rely on any simplificaion hese approximaions proved o be accurae regarding he equilibrium exchange rae whereas differences may be significan for he equilibrium volailiy when he amoun FX opion order flows are imporan relaive o spo flows. Finally he proposed model does no accoun for he poenial addiional effec on he equilibrium exchange rae of dynamic hedging of he risk embedded in older ousanding amoun of opions porfolio. Indeed only he order flows of he curren period are considered. A possible exension could hen include he impac of hedging he previous sock ha are no more perfecly dela hedged wih he change in he equilibrium exchange rae a he beginning of he day. References Back K. 993. Asymmeric Informaion and pions Review of Financial Sudies 6 435-7. Bank for Inernaional Selemen 00. Triennial Cenral Bank Survey of Foreign Exchange and Derivaives Marke Aciviy Basle. Bank for Inernaional Selemen 003. TC derivaives marke aciviy in he second half of 00. Black F. Scholes M. 973. The pricing of opions and corporae liabiliies Journal of oliical Economy 8 637-654. Black F. 976. Sudies of sock price volailiy changes roceedings of he 976 Meeings of he Business and Economic Saisics Secion American Saisical Associaion 77-8. Boolen N..B. Whaley R.E. 003. Does ne buying pressure affec he shape of implied volailiy funcions Journal of Finance forhcoming.
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Appendix A. Informaion ses of dealers The firs hree informaion ses for each marke segmen are available o individual dealers a he ime of quoing in each round whereas he las hree are public informaion for each round. Spo dealers informaion ses n { { } { } x V } n { { } { } x V V } n { { R } { } x V V V T T X Y } i s R s i i i i s R s i i i i i 3 s s i i i i 3 i 3 i i { { } n { } V } { { } n{ } V V } { { R } n { } V V V X Y } s R s i i s R s i i i 3 s s i i i i 3 3 pion dealers informaion ses m { { R s} yq { Vq } m { { R s } yq { Vq Vq } m { { R s} 3 yq { V q V q V q 3 } q q Y X } q s q q s q q 3 s q m { { R s } { Vq } m { { R s} { Vq Vq } m { { R s} 3 { V q V q V q 3 } Y X } s q s q 3 s q Appendix B - roofs of he opimal quoing sraegies Reurns are independen across each rading round he sochasic environmen dealers face being unchanged. Thus he maximisaion problem for each dealer reduces o deermine an independen opimal quoing and rading sraegy for each round. Spo and opion dealers maximise a negaive exponenial uiliy funcion U E [ ( γ W )] and he erminal wealh is disribued ( µσ ) exp + [ ] E [ U ( W )] exp γ ( µ θσ / ) is equivalen o maximise ( γσ / ) µ. N. Hence maximising 5
roof of proposiions and 3: price deerminaion on he opions marke The assumpion of no arbirage opporuniy requires ha implied volailiy quoes mus be observable o all dealers and also common across dealers he quoe being a single price a which each dealer engaged o absorb any quaniy. Hence common quoes imply ha prices can only be condiioned on public informaion. In rounds and he public informaion includes he round 3 implied volailiy and exchange rae quoes of he previous day as well as he value of he payoff incremen opimal dealers and dela hedgers rading rules. [ y ] + E[ D ( V ) ] 0 q q R. The equilibrium level of implied volailiy resuls from he E (4) [ y ] + E[ D ( V ) ] 0 E (5) q q [ ( V ) ] 0 m E yq 3 + E h3 3 3 (6) q Equaions 4 and 5 sae ha he demand of each opions dealer is expeced o absorb he demand from cusomers a he quoed prices ha clear he marke in round and. There is nohing in he informaion se ha allows he cusomers order flows o be prediced so ha E [ ] 0 ha is consisen wih D q ( V ) y q E [ ] 0 is herefore: V E[ V3 ] V. The only round quoe. Hence he raional round implied volailiy quoe condiioned on public informaion is V V3. (I is supposed ha he payoff informaion revealed a he beginning of round does no impac on he implied volailiy quoe.) Wih he same logic a he beginning of round he iner-dealer order flows do no ener he informaion se so E [ D q ( V ) ] 0 and E D q ( V ) [ ] 0 inerdealer rading round is [ 3 ] rounds and. V on public informaion. V. The only volailiy quoe ha clears he marke during he V E V V as here is no new public informaion revealed beween and are herefore unbiased predicors of he fuure round volailiy quoe condiional The hird equaion (6) esablishes ha he round 3 implied volailiy quoe should be se so ha dela hedgers accep o absorb he oal iniial order flows from opion cusomers. The volailiy quoe should adjus o ensure 6
ha: h ( V3 ) 3 yq 0. Given he posulaed opimal rading rule on he opion marke (which is proved + m q in appendix C) and since iner-dealer opion order flows are included in 3 we have E m q y Y q 3. β Furhermore he maximisaion problem of dela hedgers can be wrien as follows: Max V3 where h 3 h 3 γ ( E[ C ] C ) ( h ) σ 3 + 3 3 3 C (7) σ C is he sandard deviaion of he call price. The firs order condiion yields: [ ] E C 3 + γσ 3 C C 3. Ignoring he impac of ime decay of ineres raes changes and of second-order facors on he change in he call price he change in he value of a dela hedged porfolio only depends on he variaion of volailiy: dc υdv where υ is he vega of he call opion. Hence he demand from dela hedgers can be rewrien as h 3 υ 3 ( E[ V ] V ) 3 + he ex. The round 3 marke-clearing quoe is herefore given by: γσ C 3 3 θ γσ C. Le ( ) his leads o he equaion 3 in V 3 [ ] λ Y E V3 + 3 + (8) υ3 wih λ ( β θ ) and υ τ N ( d ) where N ( d ) { ( d ) } funcion of V 3 3 3 3 exp π 3 υ is a non-linear. 3 3. In order o have a closed-form soluion υ 3 is linearised around is round value. υ υ υ + (9) ( ) + ( V V ) 3 υ 3 3 V As opions in round and are a-he-money-forward heir dela e r* τ N( d ) is equal o 0.5. Neglecing he discouning facor r*τ e his enails ha d enering he cumulaive normal disribuion is 0. Hence ( d ) / π N so ha υ τ υ 0 π V and υ τ. π So he round 3 approximae implied volailiy can be rewrien as: V E[ V ] E V + equilibrium [ 3 3 ] 3 V λ Y 3 3 + 3 +. As in τ he raional expecaions soluion of his equaion leads o equaion 8 in he π 3 7
ex and ogeher wih proposiion o equaion 7. The round 3 volailiy is he sum of he risk premium deermined by cumulaive porfolio shifs divided by he round 3 price of he foreign exchange rae. roof of proposiions 4 5 and 6: price deerminaion on he spo marke [ x ] + E[ D ( ) ] 0 E (30) i i [ x ] + E[ D ( ) ] 0 E (3) i i [ ] 0 n m E xi 3 + E y q 3 ( 3 V3 ) 3 + E h3 ( 3 ) 3 (3) i q Equaion 30 and 3 sae ha he demand of spo dealers is expeced o absorb he cusomers demand a prices ha clear he marke in rounds and. There is nohing in and ha can help o predic he price of he [ ] 0 foreign exchange in he subsequen round. Hence E D i ( ) and E D i ( ) round equilibrium quoe on he spo marke is herefore E[ ] [ ] 0. The. As he round 3 quoe of he previous period and he payoff incremen are included in he informaion se [ ] 3 + R E. As no new public informaion is revealed beween rounds and he raional E. Equaion 3 saes ha in expecaion he liquidiy suppliers will quoe in round is [ 3 ] have o absorb no only he demand from cusomers on he spo marke bu also he flow arising from dela hedgers who cover heir exchange rae risk. Since he specificaion of he liquidiy suppliers is: ( ) ( E[ + R ] ) h 3 3 3 + + 3 3 θ and given he opimal rading rules on he wo FX marke segmens n X E xi 3 and i α E m q y Y q 3 he β round 3 equilibrium price ha clears he marke mus saisfy: X Y [ + R ] + + ( V ) 3 E 3 + + 3 3 3 3 αθ β θ (33) where and z being he amoun of foreign exchange ha he holder of one call is allowed o buy if 3 z 3 he opion is exercised a mauriy. The dela formula is non-linear in he exchange rae and in he implied volailiy. In order o have a closed form soluion he firs order linear Taylor approximaion for he dela value 8
is used. Ignoring he influence of ime decay and of ineres raes he round 3 dela value is linearised around is round value which is known. ( V ) ( V ) + ( ) ( V ) ( V V ) ( V ) 3 3 3 3 + 3 Hence he round 3 V dela will only depend on he change in he exchange rae and implied volailiy quoes beween rounds and 3. Ignoring he argumens of he dela funcion o simplify noaion and given ha V τ π for a-he-money forward calls he round 3 equilibrium quoe is given by: ( Y zγ ) [ + R ] + λx + λ Y z Γ + ( V V ) τ E 3 + 3 3 where λ π 3 + λ ( αθ ) λ ( β θ ) Γ and yield eq. in he ex and ogeher wih proposiion 4 o eq. 0.. Furhermore as in equilibrium E[ + 3 ] 3 + R 3 his Marke clearing on he spo and opion marke A he end of round boh spo and opion inerdealer order flows become public informaion. As spo dealers are supposed o know he opimal rading rule on he derivaive marke hey are able o perfecly forecas he subsequen implied volailiy quoe. Furhermore opion dealers are able o perfecly forecas he impac of dela hedging sraegies on he round 3 exchange rae quoe because hey know he risk bearing capaciy of he public and he opimal rading rule prevailing on he spo marke. The equilibrium on he spo and on he derivaive markes in round 3 is hus simulaneous and is given by he soluion of he following sysem. V 3 3 V λx + λ Y z + λ Y π + τ 3 ( + ( V3 V ) τ / π ) ( λ Y zγ ) The round 3 marke clearing price of he foreign exchange rae and ha of he implied volailiy are given by: * 3 * V3 V + ( λ Y zγ ) + λx + λ Y z + [ ( λ Y zγ ) + λx + λ Y z ] + 4λ λ z( Y ) ( λ Y zγ ) ( λ Y zγ ) τ ( λ Y zγ ) λ Y π ( λ Y zγ ) + λx + λ Y z + [ ( λ Y zγ ) + λx + λ Y z ] + 4zλ λ ( Y ) ( λ Y zγ ) (34) 9
Appendix C. roofs of he opimal rading sraegies n he spo marke during inerdealer rading each dealer will receive a share /(N-) of every oher inerdealer rade whereas on he derivaive marke each dealer will receive /(M-) of inerdealer opions rade. This is direcly relaed o he assumpion of common inerdealer quoe and corresponds respecively o he disurbance erm T and i in he maximisaion problem of dealers. During his round each dealer will speculae on q he subsequen price change on he basis of his privae informaion. roof of he opimal spo rading sraegy (proposiion 8): Ti αxi The opimisaion problem ha solves he speculaive demand for each spo dealer can be wrien as follows: Max Di D i γ ( E[ ] ) ( D ) σ 3 i i (35) { x V V } n where he informaion se is { } { } condiional variance of [ ] and where σ denoes he i s R s i i i i E 3 i. Spo dealers know ha he round 3 exchange rae quoe will clear he ne demand semming from he spo marke and he ne demand of dela hedgers who will cover heir foreign exchange risk on he spo marke (see roposiion 5). As he informaion se of spo dealers conains no i informaion abou he opion inerdealer order flows E[ λ ] Y 0 informaion o raionally predic E[ X ] E n [ λ X ] E λ T i λti i i exchange ha is proporional o is own rade. E i [ 3 i ] E[ λ X + λ Y 3 i ] λti i 3 i. Furhermore he only relevan λ is he ougoing order flow of he spo dealer. Hence. Hence each dealer expecs a change in he price of he foreign (36) Trade arising from cusomers is disribued ( ) N σ x and T i Di + xi. Hence he maximisaion problem becomes: 0 and independen across dealers so ha [ ] 0 E T i i Max Di D i λ ( D x ) ( D ) σ i γ + i i (37) The firs order condiion yields: i + i i λ D λx γσ D 0 (38) 30
The speculaive demand of spo dealers is herefore: D λ γσ λ i x i (39) The ougoing order flow from dealer i will herefore equal o: T i λ γσ λ + x i x i αxi (40) γσ λ γσ λ The condiion γσ λ > 0 ensures ha α >. roof of he opimal opions rading sraegy (proposiion 7): q β y q The opimisaion problem solving he speculaive demand of each opions dealer is given by: Max Dq D q [ 3 ( 3 V3 ) q ] C ( V ) γ ( E C ) ( D ) σ q C (4) where σ denoes he condiional variance of E[ C 3 ] C C. As nohing in he informaion se q allows he direcion of he round 3 exchange rae o be prediced opion dealers raionally expec i o be: [ 3 ] E q. Hence he only variable ha is expeced o influence he round 3 call price is he implied volailiy change from round o round 3 when ineres rae changes ime decay and second order facors are negleced. E [ C3 q ] C ( E[ V3 q ] V ) υ (4) where υ is he vega of he a-he-money-forward call opion. Each opions dealer uses sraegically his privae informaion relaed o he order flows he received from his cusomers o predic he change in implied volailiy quoe beween round and 3. E m [ V3 q ] V E[ λ Y q ] E λ q q λ q q (43) Wih he same logic as ha on he spo marke E [ ] 0 q q so ha he dealer q ougoing order flows are equal o q Dq + yq. The opions dealer maximisaion problem can hen be rewrien as: Max Dq D q υ λ γ ( D y ) ( D ) σ q + q q C (44) The firs order condiion yields: 3
q q C q υ λ D + υ λ y γσ D 0 (45) The speculaive demand from he opions dealer is hus equal o: D q υλ y γσ C υ λ q (46) As q Dq + yq he dealer q opimal rading sraegy will be: υ λ yq βyq (47) q + γσ C υ λ The condiion γσ υ λ > 0 ensures ha β >. C 3
FIGURES EQUILIBRIUM EXCHANGE RATE AND IMLIED VLATILITY Figure - Equilibrium exchange rae: posiive spo order flows 04 03 0 0 0 009 008 007 Y<0 Y>0 Y-X/ Y-X/4 Y-X/0 Y0 YX/0 YX/4 YX/ 3 monhs 6 monhs year 5 years Figure - Equilibrium exchange rae: negaive spo order flows 0993 099 099 099 0989 0988 Y<0 Y>0 0987 Y-X/ Y-X/4 Y-X/0 Y0 YX/0 YX/4 YX/ 3 monhs 6 monhs year 5 years 33
Figure 3 - Equilibrium implied volailiy: posiive spo order flows 04 03 0 0 0 009 008 007 Y<0 Y>0 006 Y-X/ Y-X/4 Y-X/0 Y0 YX/0 YX/4 YX/ 3 monhs 6 monhs year 5 years Figure 4 - Equilibrium implied volailiy: negaive spo order flows 05 04 03 0 0 0 009 008 007 006 Y>0 Y<0 005 Y-X/ Y-X/4 Y-X/0 Y0 YX/0 YX/4 YX/ 3 monhs 6 monhs year 5 years 34
Figure 5 - Difference in equilibrium exchange rae: posiive spo order flows 000006 000004 00000 0 Y-X/ Y-X/4 Y-X/0 Y0 YX/0 YX/4 YX/ -00000-000004 -000006-000008 Y<0 Y>0 3 monhs 6 monhs year 5 years Figure 6 - Difference in equilibrium exchange rae: negaive spo order flows 0 Y-X/ Y-X/4 Y-X/0 Y0 YX/0 YX/4 YX/ -000005-0000 -00005-0000 -00005 Y>0 Y<0 3 monhs 6 monhs year 5 years 35
Figure 7 - Difference in equilibrium implied volailiy: posiive spo order flows 00008 00006 00004 0000 0-0000 Y-X/ Y-X/4 Y-X/0 Y0 YX/0 YX/4 YX/ -00004-00006 -00008 Y<0 Y>0 3 monhs 6 monhs year 5 years Figure 8 - Difference in equilibrium implied volailiy: negaive spo order flows 00005 0-00005 Y-X/ Y-X/4 Y-X/0 Y0 YX/0 YX/4 YX/ -000-0005 -000-0005 -0003-00035 -0004 Y>0 Y<0 3 monhs 6 monhs year 5 years 36