Recovering Market Expectations of FOMC Rate Changes with Options on Federal Funds Futures

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1 w o r k i n g p a p e r 5 7 Recovering Marke Expecaions of FOMC Rae Changes wih Opions on Federal Funds Fuures by John B. Carlson, Ben R. Craig, and William R. Melick FEDERAL RESERVE BANK OF CLEVELAND

2 Working papers of he Federal Reserve Bank of Cleveland are preliminary maerials circulaed o simulae discussion and criical commen on research in progress. They may no have been subjec o he formal ediorial review accorded official Federal Reserve Bank of Cleveland publicaions. The views saed herein are hose of he auhors and are no necessarily hose of he Federal Reserve Bank of Cleveland or of he Board of Governors of he Federal Reserve Sysem. Working papers are now available elecronically hrough he Cleveland Fed s sie on he World Wide Web:

3 Working Paper 5-7 July 25 RECOVERING MARKET EXPECTATIONS OF FOMC RATE CHANGES WITH OPTIONS ON FEDERAL FUNDS FUTURES By John B. Carlson, Ben R. Craig, and William R. Melick This paper demonsraes how opions on federal funds fuures, which began rading in March 23, can be used o recover he implied probabiliy densiy funcion (PDF) for fuure Federal Open Marke Commiee (FOMC) ineres rae oucomes. The discree naure of he choices made by he FOMC allows for a very sraighforward recovery of he implied PDF using ordinary leas squares (OLS) esimaion. This simple recovery mehod sands in conras o he relaively complicaed PDF recovery echniques developed for opions wrien on asses such as equiies, foreign exchange, or commodiy fuures where he underlying prices are mos appropriaely modeled as being drawn from coninuous disribuions. The OLS esimaion is used o recover PDFs for single FOMC meeings as well as PDFs for join esimaion of muliple FOMC meeings, and allows for he imposiion of resricions on he recovered probabiliies, boh wihin and across FOMC meeings. Finally, recovered probabiliies are used o assess he impac of daa releases and Fed communicaion on he perceived likelihood of acual policy oucomes. John B. Carlson is a he Federal Reserve Bank of Cleveland and may be reached a or (26) Ben Craig is a he Federal Reserve Bank of Cleveland and may be reached a or (26) William R. Melick is a Kenyon College and may be reached a

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5 Opions on CBOT fed funds fuures are quie possibly he bes means available o express marke opinions abou wha he Fed migh or migh no do a he upcoming meeings. Chicago Board of Trade Meeings of he Federal Open Marke Commiee (FOMC) have always commanded a grea deal of aenion. Many analyss use he price of a near-o-expiraion federal funds fuures conrac o recover he probabiliy ha a change in he arge federal funds rae will be announced a he conclusion of he upcoming FOMC meeing. i Of course, his simple mehod of esimaion has severe limiaions. In paricular, probabiliies can only be idenified under he assumpion ha he FOMC will choose beween jus wo arge raes. ii I has long been recognized (Breeden and Lizenberger (978)) ha opion prices can be used o avoid he resricive assumpion inheren in a fuures price o recover he enire risk-neural probabiliy densiy funcion (PDF) for an underlying asse s price. iii However, unil March 4, 23, opions on federal funds fuures were no raded. Wih he inroducion of hese opions by he Chicago Board of Trade (CBOT), i is now possible o recover he enire PDF for he arge federal funds rae ahead of an FOMC meeing. Moreover, he fac ha he FOMC always changes he arge federal funds rae in 25 basis poin incremens makes he PDF recovery much more sraighforward han in he case of asses such as equiies, foreign exchange or commodiies wih prices ha are bes modeled as realizaions drawn from coninuous disribuions. The main conribuion of his paper is o demonsrae how he recovery of he PDF for he arge federal funds rae can be cas in an ordinary leas squares (OLS) framework. Wih several acive conracs in he federal funds fuures opions marke, i is possible o joinly recover he PDF for he arge federal funds rae for several poins in ime. Thus, he opions marke can be used o esimae he probabiliies associaed wih several possible pahs for he arge federal funds rae over he nex several FOMC meeings. There hen arise naural resricions on he probabiliies of arge rae oucomes across he pahs for hese joinly esimaed FOMC meeings, resricions ha are easily incorporaed in he OLS framework. Oher recen research on FOMC decision making describes he arge federal funds rae wih a parameric saisical model. For example, Hamilon and Jorda (22) esimae a model in which he

6 FOMC decides wheher o change he arge rae according o an auoregressive condiional duraion specificaion. Once a change rigger has been reached, he FOMC makes an ordered-probi decision ha deermines he change in he arge rae, according o 25 basis poin incremens. Similarly, Hu and Phillips (24) employ a discree choice model o esimae no only he magniude of a arge rae change, bu also he iming of he change. Roberson and Tallman (2) use a series of VAR models o forecas he one- and wo-monh ahead federal funds rae. iv These papers differ from ours in several respecs. In conras o he parameric models, where mos of he parameer s underlying he model are unchanging hrough ime, our srucure can change daily. All ha is required is ha he basic premises underlying our esimaion hold each day: he federal funds arge rae will change only on he meeing dae; he changes will be in incremens ha are divisible by 25 basis poins; and he possible changes are spanned by he opions marke. Furher, wha is of ineres o our paper is he daily densiy of he arge rae changes, whereas much of he preceding lieraure s focus is on he parameer esimaes hemselves, and no on he forecas densiy implied by hese esimaes. The paper is organized as follows. The firs secion describes he federal funds marke, how i is influenced by FOMC decisions, and he naure of he federal funds fuures and opions conracs. The second secion explains how opion prices can be used in an OLS regression o recover he PDF for he arge federal funds rae, boh for a single upcoming FOMC meeing and joinly for several upcoming FOMC meeings. This secion also shows how resricions can be imposed on he recovered probabiliies. The hird secion presens recovered probabiliies, boh for single and join FOMC esimaions. A comparison of probabiliies recovered from opions prices o probabiliies recovered from fuures prices is found in he fourh secion. The fifh secion demonsraes a simple regression echnique aimed a undersanding how marke expecaions for fuure arge federal funds raes respond o new informaion such as daa releases and commenary from Federal Reserve officials. The sixh secion concludes. 2

7 I. Federal Funds and Federal Funds Fuures and Opions The federal funds rae is he ineres rae paid on overnigh loans made beween deposiory insiuions. I is commonly viewed as an anchor for all ineres raes, especially a shorer mauriies. Since he lae 98s, he Federal Reserve has implemened moneary policy by using open marke operaions o arge an inended federal funds rae. v Thus, he federal funds rae is no deermined by marke forces bu is effecively adminisered by he Federal Reserve. Alhough he federal funds rae may vary day o day in response o unconrollable marke facors, Federal Reserve acions are generally successful in achieving he FOMC s federal funds rae arge on average. The deviaion of he monhly average fed funds rae from is day-weighed average arge level is zero over he pas five years, wih a sandard deviaion of 5 basis poins. Since 99, he FOMC has always changed he arge federal funds rae in muliples of 25 basis poins. In 988, federal funds fuures began rading on he CBOT. Federal funds fuures are ineres rae fuures conracs ha are cash seled based on he average rae during he delivery monh. In simple erms one can hink of he conrac as specifying a predeermined average ineres rae for a given monh. vi Thus, a buyer (or seller) can lock in a cerain ineres rae on a borrowed (or loaned) amoun specified o be $5 million for each conrac. In pracice he loan is no exended; raher, he difference beween he marke rae and he fuures rae a he ime he fuures conrac was bough or sold is muliplied by he noional $5 million dollar loan and is seled in cash. Federal funds fuures conracs are lised on he CBOT for he curren monh and for each of he 24 monhs ha follow. The fuures price is calculaed as minus he average daily federal funds rae for he delivery monh. So, for example, he July 24 conrac selemen price on June 5, 24, of implies an average federal funds rae over July 24 of.275 percen. Thus, he June 5, 24, selemen price for he July 24 conrac means ha marke paricipans were facoring in some probabiliy of a rae increase a he June 3, 24, FOMC meeing from he hen-curren arge rae of. percen. I should be noed ha he correspondence beween he average federal funds rae implied by he fuures price and he expeced arge federal funds rae can be complicaed by he presence of a 3

8 risk-premium in he fuures marke. vii Nosal (2), Chernenko, Schwarz and Wrigh (24) and Piazzesi and Swanson (24) have found his risk premium o be relaively small a he shor horizons ha will be used in his paper, on he order of hree basis poins a he one-monh horizon and six basis poins a he wo-monh horizon. The fuures price on June 5, 24, also demonsraes he limiaions of using he fuures price o recover he probabiliy of a change in he arge federal funds rae. The implied fuures rae of.275 percen is more han 25 basis poins above he hen-curren arge rae of. percen. Therefore, for he June FOMC meeing, marke paricipans saw some chance eiher a 5 basis poin increase, a 25 basis poin increase, or no change in he arge rae. However, he fuures price alone canno recover he hree relevan probabiliies associaed wih hese hree possible oucomes. Opion prices can be used o surmoun his limiaion. Figure illusraes ha he marke response o he inroducion of rading in federal funds fuures was no immediaely overwhelming. However, rading volume, for reasons largely no widely undersood, picked up dramaically beginning in 2. This evenual success led he CBOT o inroduce rading in opions on federal funds fuures in March 23. Purchase of a federal funds fuures call (pu) opion gives he owner he righ o exercise and obain a long (shor) posiion in federal funds fuures. The opions are American, allowing for exercise a any poin in ime. Srike prices are inroduced around he previous day s selemen price for he fuures conrac. Around his price are 2 srike prices a 6.25 basis poin inervals, wih an addiional srike prices ouside his band (5 below and 5 above) a 2.5 basis poin inervals. A a minimum, opions a 3 srike prices will be available, alhough no all of hese opions will necessarily be raded or priced. Since he inroducion of opions on federal funds fuures, rading volume has been uneven, as can be seen in Figure 2. As would be expeced, volume increases around imes of uncerainy concerning possible changes in he sance of moneary policy. Volume increased noiceably during he lae spring of 23, when marke commenary suggesed he possibiliy of furher FOMC rae cus o comba fears of deflaion. Trading dropped dramaically afer he Augus 2 FOMC saemen indicaed ha policy 4

9 accommodaion could be mainained (i.e., lef unchanged) for a considerable period. The marke remained somewha inacive unil he early spring of 24, when fears of deflaion abaed, suggesing ha he FOMC would soon be ready o swich o a less accommodaive policy. II. Recovering he Implied PDF wih Opion Prices Alhough fed funds fuures opions are American opions, i is insrucive o examine he simpler problem of pricing European opions. The unique payoff srucure of European opions relaes heir price direcly o he risk-neural PDF from which he price of he underlying asse will be drawn. For example, a call opion gives is holder he righ o purchase he asse a he srike price. Therefore, he price of he call opion is a funcion of boh he probabiliy of he underlying asse price moving above he srike price, and he expecaion for he underlying asse price given ha i has moved above he srike price. This payoff srucure allows he price of he opion o be wrien in erms of he PDF. The recovery and inerpreaion of he risk-neural PDF for he arge federal funds rae is slighly complicaed by hree facors. Firs, he early exercise premium associaed wih American opions invalidaes he relaively sraighforward relaionship beween an opion s price and he underlying PDF. There are several ways o overcome his problem: using he raher igh upper and lower bounds ha can be pu on American opion prices insead of a single equaion ha relaes a European opion price o he PDF (Melick and Thomas (997)), adjusing each American opion price o creae an arificial European price (Bliss and Panigirzoglou (24)), or ignoring he American premium alogeher. We choose he las approach o make he presenaion of our OLS echnique as sraighforward as possible and because opions on federal funds fuures are almos never exercised early, suggesing ha he American premium is likely o be very small. viii The second complicaion is ha in he presence of a risk premium, he esimaed risk-neural PDF may no correspond o he real objec of ineres, he acuarial PDF ha has been purged of any disorions brough abou by marke paricipans who have over- or underpaid for opions on accoun of an aversion o risk. The problem is equivalen o an aemp o deduce he probabiliy of a fire from he price a 5

10 homeowner pays for fire insurance. Clearly, a risk-averse homeowner will pay more han he acuarially fair price for fire insurance. Thus an esimae of he probabiliy of a fire based on he price paid for fire insurance will oversae he rue probabiliy of a fire. However, as noed earlier, risk premiums in he federal funds fuures marke are quie small a horizons of one or wo monhs. For his reason, and again o avoid addiional complexiy, we will inerpre he esimaed risk-neural PDFs as good characerizaions of he acuarial PDFs, especially when comparing changes in he PDFs from day o day (see Piazzesi and Swanson (24) on he noion of differencing ou risk premia.) The hird complicaion is ha he arge federal funds rae is an ineres rae, bu he fuures and opions prices are quoed in erms of an index ha equals minus he federal funds rae, conforming wih he convenion adoped for Eurodollar fuures and opions. However, as shown in Appendix, subracing each opion s srike price from and reclassifying calls as pus and vice versa ransforms opions wrien on minus he federal funds fuures rae o opions ha are wrien direcly on a federal funds fuures rae. Wih hese adjusmens, federal funds fuures opions prices can be expressed mahemaically in erms of he arge-rae PDF as follows. For European opions, he price of he opion can be expressed as he discouned value of he opion s pay-off. For a call opion ha finishes in he money, he pay-off is he difference beween he underlying asse s price a expiraion and he srike price. If he call opion finishes ou of he money, he pay-off is zero. For a pu opion ha finishes in he money, he pay-off is he difference beween he srike price and he underlying asse s price a expiraion. If he pu opion finishes ou of he money, he pay-off is zero. Usually, he fuures price, F, is assumed o be a coninuous random variable wih densiy funcion, f ( ) F T. In his sandard case, and in a risk-neural world, discouned call and pu prices can be wrien as 6

11 C P (, T, X, FT ) ( ) ( ( ) = max, FT X df r T e (, T, X, F ) e where r ( T ) T = max (, X F ) df ( = he dae on which he opion price is observed T = he dae on which he opion expires X = he opion's srike price F T = r = he risk - free rae of ineres. T he price of he underlying asse (fuures rae) a dae T F F T T ) ), However, in he case of opions wrien on federal funds fuures, he price of he underlying asse is mos appropriaely assumed o be a discree random variable. The price of he underlying asse is he average federal funds rae for he conrac monh, and his average is almos perfecly deermined by inervenions in he federal funds marke underaken by he Trading Desk a he Federal Reserve Bank of New York. Such inervenions are aimed a keeping he rae equal o he FOMC s arge level. Though he Trading Desk may miss he arge on any given day, is average monhly deviaion is essenially zero. Imporanly, arge raes are generally specified in incremens of 25 basis poins. Moreover, he number of possible federal funds arge alernaives ha are likely o be considered a any paricular FOMC meeing is usually small mos ofen hree or fewer. For example, i is someimes he case ha he FOMC chooses beween mainaining he arge federal funds rae a is curren level, or raising he arge level eiher 25 or 5 basis poins. In such a case, he hree discree oucomes would accoun for percen of he associaed probabiliies. In monhs which include FOMC meeings, he number of possible oucomes would imply he same number of corresponding monhly average oucomes, each based on he day-weighed average of he arge rae in place before he meeing and he arge rae chosen a he meeing. Of course, we assume here ha he arge rae before he meeing is known wih cerainy. This assumpion would be violaed if 7

12 policy changes beween meeing daes, or if he horizon is exended ou furher han one meeing. Concerning he firs case, i has become exremely rare ha policy acions ake place beween meeings. We hus argue ha i is reasonable o assume ha rae changes occur only a meeing daes. We deal wih he second case in urn. If here are N possible arge raes, each wih an associaed probabiliy ha he FOMC will selec ha arge rae, he price of call and pu opions can hen be wrien as C P (, T, X, F ) e r N T ( ) = ( ) T i max, FT, i X (, T, X, F ) e r i= N T ( ) = ( ) T i max, X FT, i i= where is he probabiliy ha he FOMC a is nex meeing will selec arge federal funds rae T, in i F T, i which case he average federal funds rae for he monh will ake on he value. On any given day for any opions conrac, here will be many opions ha are rading: several calls wih differen srike prices and several pus wih differen srike prices. I is sraighforward o express he discouned prices of hese opions in marix noaion. As an example, consider a rading day i wih five opions ha are acively raded, hree call opions and wo pu opions. ix For generaliy, assume ha each opion has a differen srike price. To make he example concree, suppose he paricipans in he upcoming FOMC meeing are expeced o choose from among hree possible arges for he federal funds rae. x For his example, he opion pricing equaions in marix form are given by C C C P P (, T, X, F ) e r ( T ) (, T, X, F ) e r ( T ) (, T, X, F ) e r ( T ) (, T, X, F ) e r ( T ) (, T, X, F ) e r ( T ) T T T T T max max = max max max (, FT, X ) max(, FT,2 X ) max(, FT,3 X ) (, FT, X 2 ) max(, FT,2 X 2 ) max(, FT,3 X 2 ) (, FT, X 3 ) max(, FT,2 X 3 ) max(, FT,3 X 3 ) (, X 4 FT, ) max(, X 4 FT,2 ) max(, X 4 FT,3 ) (, X F ) max(, X F ) max(, X F ) 5 T, 5 T,2 5 T,

13 These marices lend hemselves o ordinary leas squares (OLS) esimaion. In more compac noaion we have ( ). ' ' ˆ Y Z Z Z Z Y = = However, his sandard OLS esimaion does no impose he resricion ha he probabiliies sum o one,. This resricion can be wrien in marix form as = = N i i res sub r r = + 2, where sub is a subse of he probabiliies. Noice ha we do no impose he resricion ha he probabiliies mus lie beween zero and one. Such a resricion would require a much more complicaed esimaion procedure. For he example oulined above, he resricion ha he probabiliies sum o one would be wrien in marix noaion as = + = res sub r r. This resricion can be imposed on he OLS esimaors as follows xi. ( ) ( ) ( ) ( ) ( ) ( ). ˆ ˆ and ' ' ˆ The esimaed probabiliies are hen found as. which simplifies o, sub res sub sub sub sub res r r r Z Y r Z r Z r Z r Z r Z Y r Z r Z r r Z Z Y + = = = + = + = = Therefore, in mos cases i will be possible o obain boh an unresriced and a resriced esimae of he probabiliies associaed wih he possible federal funds arges. This will be rue so long as a possible federal funds arge rae falls in beween he srike prices for all possible pairs of opion prices. In some insances, however, i may be he case ha wo or more of he opions ha are raded would finish in and ou of he money for exacly he same possible federal funds arge raes. In his case, wo or 9

14 more of he columns of he marix Z would be linear combinaions of each oher because he values in he wo columns would be equal o each oher plus he difference in srike prices beween he wo opions. Thus, he Z marix would no be of full rank, and he OLS esimaes could no be recovered. The resriced probabiliies could sill be esimaed, so long as only wo of he columns were linear combinaions of each oher. I is also possible o joinly esimae probabiliies for wo or more ses of federal funds arge raes coming from wo or more FOMC meeings. Coninuing wih a relaively simple example, suppose we have five opion prices from one conrac where here are wo expeced arge rae oucomes, and we have four opion prices from a second conrac (laer in he year) where here are hree expeced arge rae oucomes. We mus inroduce a second subscrip ha denoes wheher a arge rae is chosen a he firs FOMC meeing or he second FOMC meeing. Thus F 2, is he value aken on by he monhly average federal funds rae if T is chosen a he second FOMC meeing. Suppose ha he researcher imposes he 2, following srucure: If he firs arge rae is chosen a he firs meeing, hen here is only one possible choice for he arge rae a he second meeing, while if he second arge rae is chosen a he firs meeing, hen eiher of wo arge raes can be chosen a he second meeing. This srucure can be shown diagrammaically: Firs FOMC Second FOMC T, T 2, T,2 T 2,2 T 2,3 Given his srucure, he opion pricing equaions for all nine opions (five from he firs conrac and four from he second conrac) are wrien in marix noaion as

15 (, T, X, F ) C, T r( T ) C(, T e, X,2, FT ) r( T ) C(, T e, X,3, F ) T m F ( ) r T m F P(, T e, X,4, F ) T m F r( T ) (,,,5, ) m X P T e X FT = ( ) m X r T,5 e (,,, ) C T2 X 2, FT 2 r( T ) 2 e C(, T,, ) 2 X 2,2 F 2 T ( 2 ) r T e P(, T2, X 2,3, F 2 ) T r( T2 ) e P(, T2, X 2,4, FT 2 ) r( T2 ) e where, in order o save space, m ( F, X ) max( F X ) =.,,,, (,, X, ) m( F,2, X, ) (,, X,2 ) m( F,2, X,2 ) (,, X,3 ) m( F,2, X,3 ) (,4, F, ) m( X,4, F,2 ) (, F ) m( X, F ),,,5,2 m m ( 2,, X 2, ) m( F 2,2, X 2, ) m( F 2,3, X 2, ) ( 2,, X 2,2 ) m( F 2,2, X 2,2 ) m( F 2,3, X 2,2 ) ( X 2,3, F 2, ) m( X 2,3, F 2,2 ) m( X 2,3, F 2,3 ) ( X, F ) m( X, F ) m( X, F ) m F m F 2,4 2, 2,4 2,2 2,4 2,3,,2 2, 2,2 2,3, Esimaing his sysem via OLS is sraighforward and will yield resuls ha are idenical o a separae esimaion for each of he wo FOMC meeings. However, here are hree resricions ha can be imposed, given he relaionship beween he arge federal funds raes across he wo meeings. Firs, he probabiliy of he FOMC selecing he arge rae T and he probabiliy of he FOMC selecing arge rae T mus sum o one: ha is,,2, +,2 =,. Second, he srucure imposed by he researcher (perhaps based on marke commenary) implies ha if T is chosen a he firs FOMC meeing, hen T will be chosen a he second meeing, so i mus be ha =. Finally, he sum of he probabiliies, 2,, ha arge raes T and T will be chosen mus equal he probabiliy ha arge rae T is chosen a 2,2 2, 3 he firs meeing. Tha is, + 2 =. Noice ha hese hree resricions also guaranee ha 2,2,3 + 2 =. The resricions are wrien in marix noaion as 2, 2,2 +, 3,2,2 2,

16 r + r + 2 sub, 2,2 = res,, =, 2,2, 2,2 wih he resriced esimaion proceeding as above. III. Esimaes of he Implied PDFs Daa on daily selemen prices for federal funds fuures and opions were obained from he CBOT for he period January, 23, hrough Sepember 22, 24. Selemen prices are used o avoid sale and asynchronous quoes. This daa se conained 5,92 fuures prices and 28,4 opion prices. Opion prices were hen checked for violaions of opions selling for less han heir exercise value, monooniciy and concaviy when ploing opion price agains he srike price, changes in opion prices ha were greaer han changes in he srike price when comparing opions wih differen srike prices for a given conrac on a single day, and pu-call pariy. All old, 28 opions were found o violae hese condiions, so roughly.8 percen of he daa were eliminaed. The risk-free rae of ineres was se equal o he yield on he Treasury bill ha maured as few days as possible afer he expiraion of each opion conrac. xii Single FOMC Meeing Esimaion Each opions conrac ha expired before November 24 (May 23 hrough Ocober 24) was used o recover he PDF for he arge federal funds rae ahead of a single FOMC meeing. Tha is, each conrac was assigned a single FOMC meeing oucome ha i was o predic. A PDF was recovered each day for each conrac, alhough days were deleed under he following condiions. Firs, a day was deleed if i had fewer han five opions wih prices ha were above he minimum price of one-quarer of a basis poin. xiii Second, days were deleed ha fell afer he FOMC meeing ha he conrac is 2

17 predicing. For example, days afer June 25, 23, were deleed for he July 23 conrac, since he July conrac was used only o predic he oucome of he June 25, 23, FOMC meeing. Third, days were deleed if hey had one (or more) FOMC meeings before he FOMC meeing ha he conrac is predicing. For example, days before he May 6, 23, FOMC meeing for he July 23 conrac were deleed, again since he July 23 conrac was used only o recover a PDF for he oucome of he June 25, 23, FOMC meeing. Finally, he June 24 conrac was no used because he second day of he June 24 FOMC meeing fell on he las day of he monh, meaning ha he June 24 FOMC decision would have almos no effec on he monhly average federal funds rae. These deleions lef a daa se of 434 days (and fuures prices) and 6,377 opions prices across 4 conracs. Table presens descripive saisics for his daa se by conrac, including deails on which FOMC meeing was assigned o which conrac and he number of daily PDFs recovered for each conrac. Each conrac was used for abou six weeks ahead of he FOMC meeing i was predicing, so each conrac generaed abou 3 PDFs. Almos 46 percen of he opions used in esimaion were calls. The Sepember 24 conrac had he mos opions raded per day. Srike prices for he opions ranged from a low of.25 percen o a high of 2. percen. Beween he conclusion of he June 23 FOMC meeing and he June 24 FOMC meeing, here was lile expecaion of anoher change in he FOMC arge rae, wih lile variabiliy in he fuures price for he Sepember 23 hrough May 24 conracs. Figure 3 conains a recovered PDF for a rading day in which probabiliies for five policy oucomes were esimaed, alhough wo of he oucomes appear o be irrelevan. According o he esimaes, as of May 27, 23, marke paricipans saw a roughly 2 percen chance ha he FOMC would reduce he arge federal funds rae from.25 o.75 percen on June 25, 23, a he conclusion of is wo-day meeing. Marke paricipans saw a 3 percen chance ha he FOMC would reduce he arge rae from.25 o. percen and a 5 percen chance ha he arge federal funds rae would remain unchanged a.25 percen. Consisen wih marke commenary a he ime, marke paricipans saw essenially no chance of a 75 basis poin reducion or 25 basis poin increase in he arge rae. 3

18 Esimaes like hose in Figure 3 were obained for he 434 conrac days summarized in Table. For each day, he call wih he highes srike and he pu wih he lowes srike were used o assign he possible arge rae oucomes. The highes srike price for he calls was rounded up o he neares muliple of 25 basis poins, and he lowes srike price for he pus was rounded down o he neares muliple of 25 basis poins. This algorihm esablishes he range for he arge rae oucomes, wih all inermediae 25 basis poin muliples filled in. Taking Figure 3 as an example, on May 27, 23, he call wih he highes srike was.25 and he pu wih he lowes srike was.625. Thus, he esimaed PDF has five possible oucomes, ranging from.5 o.5 in 25 basis poin incremens. Summary saisics for all of he 434 esimaed PDFs are presened in Table 2. The op panel of he able repors resuls for unresriced esimaion of he probabiliies while he boom panel repors resuls when imposing he resricion ha he probabiliies sum o one. The 2 R saisics are uniformly high, wih lile drop-off in goodness of fi when he resricion is imposed. Under he assumpion of riskneuraliy, i should be he case ha F n = ˆ, ha is he fuures price should be equal o he mean i FT, i i= of he PDF. In fac, his resricion could also be imposed on he OLS esimaes. We chose o save he fuures price for he above ou-of-sample comparison o he mean of he PDF. For he esimaes ha do no impose he resricion ha he probabiliies sum o one, he mean of he PDF is quie close o he fuures price, usually well wihin wo basis poins. The relaively large misses for he May 23 conrac are probably no due o pricing anomalies associaed wih he inroducion of he new opions conrac, since he mean for he resriced esimaes for May (boom panel of he char) are quie close o he fuures price. For all bu wo of he eleven conracs, he mean of he PDF from he resriced esimaes does a beer job, on average, of maching he fuures price. These means from he resriced esimaes are almos always wihin / h of basis poin of he fuures price. Figure 4 summarizes he evoluion over ime of esimaes of he probabiliies associaed wih alernaive June 23 FOMC meeing oucomes. The procedure allows for five possible oucomes: no change; rae cus of 25, 5, and 75 basis poins each; a rae hike of 25 basis poins. Immediaely afer he 4

19 May meeing, opions prices indicaed ha probabiliy of he no-change oucome exceeded 6 percen. Uncerainies surrounding he war in Iraq had been diminishing, and forecasers anicipaed an acceleraion in economic aciviy. As he summer approached, however, incoming daa failed o clearly confirm such expecaions, and he prospec of a rae cu became more likely. Indeed, markes began o enerain a remoe possibiliy of a 75 basis poin rae reducion. The prospec of a rae increase was never given much of a chance. Around mid-may he probabiliy of no cu fell below 5 percen. However, Chairman Greenspan s May 9 esimony before Congress was inerpreed as being relaively upbea, and he rend reversed some, sabilizing around even odds for a rae reducion versus no change. By early June, relaively disappoining economic news raised concerns ha FOMC migh need o reduce he federal funds rae furher as an insurance measure o avoid a small probabiliy ha a general deflaion migh emerge. The perceived one of a June 3 saellie speech by Chairman Greenspan o foreign cenral bankers heighened such concerns, as did a subsequen speech by Governor Kohn. Wihin a week or so, he prospec of a furher cu was esimaed o be virually cerain. The only issue was wheher he cu would be 25 or 5 basis poins, wih he opions pricing srucure giving boh oucomes abou equal probabiliy. I is worhwhile o noe ha he May-June 23 inermeeing period highlighs he advanage ha our echnique provides over using a simple approach based only on he federal funds fuures price. For example, consider he fuures price on May 6, which implied a yield of. percen, or more han half way beween no change and a 25 basis poin cu. Assuming ha hose were he only wo possible oucomes, one is led o believe ha a rae cu was likely. The srucure of opions prices, however, suggesed ha a 5 basis poin change was also a possible oucome. Indeed, is esimaed probabiliy was even more likely han ha of a 25 basis poin change. Our approach allows he daa o speak, and hey indicae ha alhough he federal funds fuures price was more han halfway oward a rae cu, he prospec of no cu was more likely. 5

20 Join FOMC Meeing Esimaion As noed in Secion II, a join esimaion of he PDFs for wo or more FOMC meeings requires he researcher o impose a srucure on he poenial pahs ha he FOMC migh follow when seing he arge federal funds rae. We have no ye developed an algorihm for generaing hese srucures ha would enable us o conduc a join esimaion for every day in he daa se. Insead, we simply presen a single day s esimaion in order o provide an example of a join esimaion. Ahead of he June 25, 23, FOMC meeing he arge federal funds rae sood a.25 percen. To perform he esimaion, we imposed he following srucure on he possible pahs for he arge rae from he June 25, 23, o he Augus 2, 23, FOMC meeings. Noe ha we ruled ou he possibiliy of a 5 basis poin rae cu in Augus if here had been no change in he arge rae a he June meeing. June FOMC Augus FOMC Selemen opions prices from May 27, 23, for he July 23 and Augus 23 conracs were used o esimae boh he unresriced and resriced probabiliies for he pahs, wih resuls shown in Table 3, and he resriced resuls ploed in Figures 5a and 5b. The joinly esimaed resuls for he June 25, 23, FOMC meeing are slighly differen han he resuls presened in Figure 3, as five possible oucomes for he June FOMC meeing were allowed for in Figure 3, while only he hree mos likely oucomes were allowed for in he resuls shown in Table 3 and Figures 5a and 5b. Neverheless, i is reassuring ha he join esimaes are no much differen han he 6

21 single esimaes, boh indicaing roughly a 5 percen chance of he arge rae remaining a.25 percen in June, roughly a 3 percen chance of he arge rae being reduced o. percen, and roughly a 2 percen chance of he arge rae being reduced o.75 percen. Noice from he hird column of Table 3 ha i is possible o obain a negaive esimae for one of he probabiliies (in his case he probabiliy ha he arge rae would remain a.25 percen in June and be cu o. percen in Augus.) There is nohing in he OLS esimaion ha prevens a negaive probabiliy. The las column of Table 3 presens he resriced esimaes, where a negaive probabiliy is no longer a problem. Our esimaion procedure imposes several resricions. Firs, he probabiliies for he hree possible oucomes from he June meeing mus sum o one. Second, he probabiliies for he wo oucomes for he Augus meeing, condiional on a choice of.25 percen for he arge rae a he June meeing, mus sum o he uncondiional probabiliy of a arge rae of.25 percen being chosen a he June meeing. This is indeed he case, as.5422 plus.38 equals.546. Third, he probabiliies for he hree oucomes for he Augus meeing, condiional on a choice of. percen for he arge rae a he June meeing, mus sum o he uncondiional probabiliy of a arge rae of. percen being chosen a he June meeing. Finally, he probabiliies for he wo oucomes for he Augus meeing, condiional on a choice of.75 percen for he arge rae a he June meeing, mus sum o he uncondiional probabiliy of a arge rae of.75 percen being chosen a he June meeing. All of hese resricions also imply ha he probabiliies for all of he Augus oucomes will sum o one. IV. Comparing Opions-Based PDFs wih Fuures-Based Probabiliies As noed in he inroducion, i is quie common o use prices from federal funds fuures conracs o recover he probabiliies associaed wih arge federal funds raes ha migh be chosen by he FOMC a upcoming meeings. Of course, his procedure is limied, since a single fuures price can be used o recover only wo probabiliies. However, if he FOMC is believed o be choosing from only wo alernaives, he procedure is appropriae. xvi In any even, i is insrucive o compare he probabiliies esimaed from opions prices o hose recovered from fuures prices. If he probabiliies are he same, or 7

22 if i can be shown ha he fuures-based probabiliies are in some sense superior, hen analyss should no spend he exra ime and effor needed o esimae he probabiliies using opions prices. To recover probabiliies from fuures prices we solved he following equaion Fuures where TFFL DB = TD CTFF CTFF = CTFF CTFF y CTFF + 25 bp - 25 bp - 5 bp if if if DA + TD y oherwise L H ( TFFL + TFFH ), DB DAi, Fuures > CTFF, + i TD TD DB DAi, CTFF < Fuures CTFF, + i TD TD DAi, DB ( CTFF, 25 bp) +, < i CTFF i Fuures TD TD ( CTFF + 25 bp) ( CTFF + 25 bp) CTFF TFFH = TFFL + 25 bp, subjec o he resricion ha =, and where Fuures DB DA TD L H CTFF TFFL TFFH L + H ( - he fuures price for conrac i on day ) = days in he monh before he FOMC meeing prediced by conrac i on day = days in he monh afer he FOMC meeing prediced by conrac i on day = oal days in he monh of = curren arge federal = = = lowes arge federal funds rae ha migh be seleced a he upcoming FOMC meeing for conrac i on day highes arge federal funds rae ha migh be seleced a he upcoming FOMC meeing for conrac i on day = he probabiliy associaed wih TFFL = he probabiliy associaed wih TFFH he FOMC meeing prediced by conrac i on day funds rae (i.e., ha is, before he FOMC meeing) for conrac for conrac i on day for conrac i on day. i on day L H The probabiliies and can hen be compared o he probabiliies esimaed from opions prices for he same alernaive arge federal funds raes. Table 4 conains he resuls of simple -ess for 8

23 a difference in means across wo populaions when comparing he probabiliies esimaed from opions prices o hose recovered from fuures prices. Comparisons are made by individual conrac as well as for all conracs reaed as a whole. For he comparison by conrac, in only 6 ou of he 28 possible comparisons do we fail o rejec a he five percen level of significance he null hypohesis ha he probabiliies are drawn from he same populaion. Taking all conracs ogeher, he null of he same populaion is rejeced a any convenional level of significance. I is apparen ha he wo mehods are reurning probabiliies ha are saisically differen from each oher. Noice ha here is a discernable ime paern o he size of he differences beween he opions probabiliies and he fuures probabiliies. In paricular, he probabiliies are mos differen for he May, June, and July 23 conracs, a period in which i was likely ha here were more han wo possible arge federal funds raes in play. For laer conracs, when he direcion and magniude of fuure arge rae changes was much clearer, here is lile difference beween he probabiliies. For conracs afer July 23, he difference usually amouns o only around five percenage poins. Of course, he quesion remains wheher one se of probabiliies is in some sense superior o he oher. To shed some ligh on his issue, we compare he forecasing performance of he wo probabiliies. Resuls of an exremely simple-minded comparison are found in Table 5, which conains he convenional calculaion of roo mean squared forecas errors (RMSE). Using RMSE, he fuures probabiliies would be judged as superior o hose from opions. As discussed in Appendix 2, however, he RMSE crierion is an inappropriae means o judge densiies. By design, i heavily favors any densiy ha is based on he assumpion ha he FOMC will choose from only wo alernaives. Thus, he fuures-based forecass will enjoy a considerable advanage. We hence view he resuls of Table 5 as misleading and conclude ha hey should no be used o argue ha he fuures-based probabiliies are superior. We recognize ha he FOMC mos ofen considers only wo alernaive arge raes. Thus, probabiliies from he fed funds fuures conrac should usually offer superior predicive conen. However, we would emphasize ha when he wo esimaes differ significanly, opions-based forecass can be paricularly insrucive. This siuaion occurs when here are more han wo oucomes in play as 9

24 illusraed in Figure 4. As he meeing dae approached, opions-based PDFs revealed a fairly diverse and dynamic change of marke opinion. Moreover, he esimaed PDF on May 27, 23, revealed in Figure 3 indicaes ha he no-policy-change alernaive (keeping he arge rae a.25) was he mos likely oucome. The fed funds rae fuures esimae, on he oher hand, indicaed ha a 25 basis poin rae cu oucome was near cerain. We hus conclude ha marke paricipans, a a minimum, will wan o confirm ha he simpler fuures-based probabiliies are appropriae by comparing hem o opions-based probabiliies. No maer wha comparison mehod is used, however, we have only a very small sample of 4 conracs a our disposal. Given his sample size, we are unable o make any srong saemens abou he superioriy of opions or fuures based probabiliies. xvii We can say ha hey are differen, and hose differences are larges a imes when marke commenary suggess ha he FOMC is seen o be choosing from more han wo possible arge federal funds raes. V. The Impac of Informaion and Federal Reserve Communicaion on Esimaed Probabiliies The preceding secions of he paper have demonsraed ha opions on federal funds fuures can be used o recover he marke s assessmen of he probabiliies associaed wih possible oucomes for upcoming FOMC meeings. Obviously, hese recovered probabiliies conain useful informaion abou marke expecaions, bu hey also provide a unique measure ha can idenify wha informaion shapes marke expecaions and quanify he qualiy ( ransparency, in policy jargon) of communicaion beween he FOMC and he marke. Since he early 99s he Federal Open Marke Commiee has adoped several changes in is procedures and communicaions designed o improve ransparency (see Greenspan [2]). Swanson (24) shows ha since he 98s, U.S. financial markes and privae secor forecasers have become beer able o forecas he federal funds rae a horizons ou o several monhs and have been less surprised by Federal Reserve announcemens. A key implicaion of his analysis is ha financial marke forecasers 2

25 undersand wha informaion is likely o guide acual policy acions, and hey reac o i in a consisen way. I has become widely known ha he FOMC reacs sysemaically (hough no exclusively) o informaion abou inflaion and economic aciviy in a manner idenified by John Taylor. In simple erms, his relaionship he so-called Taylor rule is characerized by an equaion ha implies ha he fed funds rae ends o increase (decrease) wih informaion indicaing a srenghening (weakening) in economic aciviy relaive o poenial and o decrease (increase) wih he difference beween inflaion and some implied arge rae. A any poin in ime fuures and opions prices migh be expeced o fully reflec all informaion abou he economy and inflaion. We should hus expec surprises in daa releases on employmen and inflaion o be associaed wih immediae changes in he probabiliy disribuion of alernaive oucomes of fuure FOMC meeings. The esimaed probabiliies for alernaive June 24 oucomes in Figure 6 provide a clear illusraion. During 23, unusually slow employmen growh raised doubs abou he susainabiliy of he economic expansion. Moreover, i was hough ha if employmen growh did no begin o rise soon, he economy migh be a risk of deflaion. As a consequence, he arge fed funds rae had been reduced o one percen, a level viewed as accommodaive. A he Augus 23 meeing, he Commiee adoped language ha indicaed ha he FOMC would no likely remove is accommodaive sance for a considerable period. The marke for opions of fed funds fuures was largely inacive, offering few opions a alernaive srike prices. Esimaes of he probabiliy of no change in policy for he upcoming meeing ranged beween 9 and percen from lae summer hrough early spring. As illusraed in Figure 6, hings began o change wih he release of he April 24 Employmen Siuaion. Esimaes of he probabiliy disribuion of possible oucomes for he June FOMC meeing revealed an increasing likelihood of a rae hike as a series of wo srong employmen repors indicaed ha he recovery was gaining racion. When he May employmen repor was released in early June, he esimaed probabiliy for no change fell o around 5 percen less han he esimaed probabiliy for a 5 basis poin rae hike. 2

26 I is quie clear ha, a imes, he release of new economic daa (such as he monhly Employmen Siuaion or Consumer Price Index repors) can change marke views of he arge federal funds rae likely o be seleced a he upcoming FOMC meeing. I could also be he case ha marke views are influenced by speeches and esimony given by Federal Reserve officials. The above descripions of reacions o he arrival of informaion raise he quesion of wheher or no here is a sysemaic relaionship beween marke views and he arrival of new informaion. The probabiliy esimaes recovered from opions prices provide a means of exploring he exisence of such a sysemaic relaionship. To address his issue, we define as he ex ane probabiliy (esimaed from opions conrac i on day ) ha he ex-pos acual FOMC arge rae would be chosen a he upcoming FOMC meeing. Thus, A A is he probabiliy, as of day, ha he marke placed on he FOMC choosing he arge rae ha acually was chosen. We posi he following regression: A A = α α DaysBefFOMC 9 6 α ( CP CumEMP ) + α ( CP CumFedCom ) + µ + ε, + α CumCPI 2 + α CumEMP α DaysBefFOMC + α CumFedTes 4 A + α 8 i ( CP CumCPI ) + α CumFedCom where CumCPI CP CumEMP CumFedTes = he cumulaive number of CPI repors released during conrac i' s period of observaion up o day. = CumFedCom he cumulaive number of employmen repors released during conrac of observaion up o day. = DaysBefFOMC he cumulaive number of esimonies on economic condiions by he Federal Reserve chairman during conrac i' = he cumulaive number of speeches on economic condiions by he Federal Reserve chairman during conrac i' s period of s period of observaion up o day. observaion up o day. ' s period = he number of days unil he FOMC meeing prediced by conrac i on day. = a dummy variable ha equals one if day for conrac i falls in he period during which he FOMC press release used he language"considerable period"(augus 2, 23, hrough January 27, 24). i 22

27 Descripive saisics on hese variables are repored in Table 6. The presumpion is ha new informaion should increase he ex ane probabiliy associaed wih he arge rae ha will acually be chosen by he FOMC. In erms of he equaion, if new informaion is revealed ha is perinen o fuure FOMC decisions, hen coefficiens α 2 hrough α 5 should be posiive. This new informaion can ake he form of daa releases and esimony or commenary from he Federal Reserve chairman. The passage of ime migh also affec he ex ane probabiliy. As an FOMC meeing approaches, he marke may become more informed abou he inenions of he FOMC, hrough a variey of sources no considered above. Because he righ-hand-side variables do no capure all of he informaion flowing o marke paricipans, informaion gained from oher sources would be revealed over ime he so-called secular effec. We measured ime as days before he FOMC meeing, so as we approach he meeing, his variable is aking on smaller values. Hence, if he passage of ime allows marke paricipans o beer undersand FOMC inenions, α 6 should be negaive, since he probabiliy of he acual oucome should increase as he days before he meeing diminish. We also inerac he passage of ime wih he iniial ex ane probabiliy from he firs day of he conrac ( A ) o allow ime o have differen effecs ha depend on how far marke expecaions are away from he arge ha will evenually be chosen. The coefficien on his ineracion erm, α 7, should also be negaive. Finally, when i is clear ha he FOMC has signaled ha i does no plan any policy changes a an upcoming meeing, he arrival of new informaion may be of no value for he esimaed probabiliy disribuion. Such migh have been he case beween mid-augus 23 and lae January 24, when FOMC saemens included he senence In hese circumsances, he Commiee believes ha policy accommodaion can be mainained for a considerable period. The period leading up o he December 23 meeing illusraes he poin (see Figure 7). Markes expeced no acion by FOMC during he whole inermeeing period, effecively ignoring informaion ha would have oherwise affeced he policy oucome. 23

28 To accoun for he effecs of his kind of seering language, we creae a dummy variable ha equals one during his period and zero oherwise and hen inerac ha dummy wih our informaion variables. xviii If new informaion was of no value when he considerable period language was in effec, hen i should be he case ha α 2 = α 8, α 3 = α 9, α 4 = α ; hence, effecively shuing off normal sysemaic effecs. Fixed-effecs-esimaion resuls for he regression model are found in Table 7. We presen hree ses of OLS resuls; he firs wo columns do no include he lagged dependen variable as an explanaory variable. xix The hird column is presened o assuage fears ha our resuls are sensiive o a more dynamic specificaion. Looking across he hree columns, he CPI and employmen variables are always posiive, as expeced, and significan a sandard levels. They also appear o be economically significan. Using he resuls in he second column, boh a new CPI release and a new employmen release are associaed wih a roughly 4 percenage poin increase in he ex ane probabiliy associaed wih he arge rae acually chosen by he FOMC. Ineresingly, Congressional esimony presened by he chairman in wo of he hree regressions does no have a saisically significan effec on he ex ane probabiliy. I may be ha he chairman s esimonies, which are scheduled well in advance, offer relaively lile new informaion o he marke. In conras, he posiive and significan coefficien on he chairman s speeches in all hree specificaions indicaes ha his informaion moves he marke closer o he arge rae ha will be chosen by he FOMC. Moreover, he speeches have an economically meaningful effec. Each speech by he chairman is associaed wih a leas a 5 percenage poin increase in he ex ane probabiliy associaed wih he arge rae acually chosen by he FOMC. We also find evidence of a secular rend effec. The passage of ime eners significanly in he firs wo regressions, while he lagged dependen variable does so in he hird. Jus as one migh expec, informaion no measured by he explici informaion explanaory variables finds is way ino our esimaes of he PDFs. As an upcoming FOMC meeing approaches, he probabiliy associaed wih he 24

29 choice ulimaely made by he FOMC increases. The ineracion erm in he second regression indicaes ha he effec of he passage of ime is larger when marke expecaions iniially aach a low probabiliy o he arge rae ulimaely seleced by he FOMC. For all of hese resuls, i should no be forgoen ha hese daa only cover he period from May 23 o Sepember 24, quie obviously a small sample, and ye he observed informaion affecs he probabiliies so significanly. The considerable period language appears o change he way in which he marke responds o new informaion. As expeced, he coefficiens on he considerable period ineracion erms are of he opposie sign and of roughly equal magniude of he informaion variables, alhough usually only significan a marginal levels. Taken ogeher, he coefficiens on he ineracion variables sugges ha when he FOMC was using he considerable period language, he arrival of new informaion had no effec on marke views of he arge rae o be chosen by he FOMC. Tha is, he considerable period language seemed o urn off he normal incremenal effec of informaion. We also esimaed an alernaive specificaion ha used he absolue deviaion of he ex ane mean of he PDF from he ex pos arge rae as he dependen variable. In his specificaion, he CPI and employmen variables remained saisically significan, alhough heir signs changed as a daa release narrowed he difference beween he ex ane mean and he ex pos arge rae, as opposed o raising he probabiliy associaed wih he ex pos arge rae. In his regression, a new daa release brough he ex ane mean abou one basis poin closer o he ex pos arge rae. We repor he specificaion wih he ex ane probabiliy as he dependen variable since here can be days where here is a dramaic change in he ex ane probabiliy bu lile change in he mean of he PDF. xx Thus, he specificaion wih he absolue deviaion as he dependen variable can miss insances where new informaion has a dramaic impac on marke views. Alhough covering a very shor period of ime, hese resuls sugges ha he marke expecs he FOMC o reac o inflaion and employmen in a manner consisen wih he Taylor rule. In addiion, he marke pays considerable aenion o, and correcly inerpres, commenary by he chairman. We would sress, however, ha hese resuls are very preliminary. Neverheless, we believe hey offer some measure 25

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