The US Dollar Index Futures Contract

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1 The S Dollar Inde uures Conrac I. Inroducon The S Dollar Inde uures Conrac Redfeld (986 and Eyan, Harpaz, and Krull (988 presen descrpons and prcng models for he S dollar nde (SDX fuures conrac. Ths arcle updaes he orgnal Redfeld (986 and Eyan, Harpaz, and Krull (988 sudes o ncorporae changes n he conrac specfcaons. The SDX fuures conrac has wo feaures ha nfluence s prcng and s use. rs, he SDX nde s a geomerc average, raher han an arhmec average, of he consuen currences. Second, he foregn echange (X raes n he SDX nde (n S dollars per foregn echange rae are n he denomnaor of he nde, mplyng ha a dollar apprecaon leads o a hgher nde level. Boh he geomerc averagng and he use of quong convenon have mplcaon for he use of he SDX fuures conrac n hedgng a foregn echange eposure. Ths arcle begns by provdng updaed coverage of he SDX fuures conrac, whch has changed conrac specfcaons snce hese earler publcaons. Ne, hs arcle presens an nuve eplanaon of why he sandard cos-of-carry fuures prcng formula s napproprae for he SDX fuures conrac, whch mus be adjused by an adjusmen facor (somemes referred o as a volaly premum. Ths arcle hen noes he facors ha would play a role n deermnng he magnude of he adjusmen facor. II. Hsory and Curren Saus of he SDX uures Conrac The SDX fuures conrac began radng on November 2, 985 on he nancal Insrumens Echange, a dvson of he New York Coon Echange, whch s now par of he New York Board of Trade (NYBOT. The SDX nde was orgnally a geomercally weghed average of en dfferen currences, wh each currency represenng a counry ha was a major radng parner wh he ned Saes. Wh he nroducon of he Euro, he SDX nde became a geomercally weghed average of s currences, whch represen fve major.s. radng parners and he Euro. Append A descrbes he curren conrac specfcaons for he SDX fuures conrac. Inde ormula The formula for he nde level on dae s he produc of he s currences spo raes, each rased a power relaed o a currency-specfc wegh. The general formula for he nde can be wren as SDX K N ( X w, Page of

2 The S Dollar Inde uures Conrac where SDX s he calculaed level of he SDX nde on dae, X, s he foregn echange rae (.S. dollars per foregn currency un for currency on dae, w s he wegh assocaed wh currency (he weghs are deermned by N he conrac specs and sum o one,.e., w ; N s he number of currences n he nde (for he SDX nde, N s currenly s and was formerly en; and K s a consan. nder he curren SDX fuures conrac specs, he SDX nde s equal o SDX ( Euro.576 ( SwedshKroner ( Yen ( Swssranc ( Serlng.36.9 ( CanadanDollar.9 Table shows he X raes of he consuens currences on July 3, 22 (from Bloomberg, he weghs appled o he X raes, he X rae rased o (he negave of ha wegh, and he produc of he X raes rased o her wegh. Ths produc, when mulpled by he nde consan of , yelds he nde level of TABLE as of July 3, 22 oregn X rae Wegh (X rae -Wegh currency (X Euro Yen Serlng Canadan Dollar Swedsh Kroner Swss ranc SM: -. - PRODCT: INDEX LEVEL* 7.3 * Inde level s PRODCT mes he nde consan of gure shows he monhly levels of he SDX nde from he SDX fuures conrac from Aprl 986, a a level of 2.4, hrough July 3, 22, when offcally closed a 7.4. The consan, currenly equal o , was se back a he naon of he nde n March 973, n order o make he nde equal o. a ha me. Page 2 of

3 The S Dollar Inde uures Conrac gure :.S. Dollar Inde (Monhly Source: Bloomberg III. Deermnng he ar uures Prce of he SDX uures Conrac Two feaures of he SDX nde formula are mporan n deermnng he far fuures prce for a SDX nde fuures conrac. rs, he SDX nde s consruced such ha, as he.s. dollar apprecaes, he nde level rses. Therefore, a poson ha s long he foregn currences (and wll herefore lose value f he.s. dollar apprecaes wll need a long poson n he SDX fuures conrac o hedge hs eposure. Ths suaon s somemes referred o as a long-long hedge. Second, he SDX nde level s consruced as a geomerc mean (where he X raes are mulpled ogeher, raher han as an arhmec mean, of he consuen X raes. The use of he geomerc mean mples ha he sandard cos-of-carry model o arrve a he fuures prce s no applcable for hs conrac. As hs secon demonsraes, each of hese convenons whn he SDX formula play a par n deermnng how o farly value a fuures conrac based on hs nde. A. Prcng he SDX fuures conrac when here s no uncerany n foregn echange raes In order o undersand how o prce he SDX fuures conrac, frs consder a suaon where here s no uncerany abou he level of foregn echange raes n he fuure. In hs suaon, we could calculae he forward raes n each of he s currences ha consue he SDX nde. Table shows he foregn echange raes, he local rsk-free raes (as measured by an esmae of he local neres raes, from Bloomberg, and he resulng forward raes for each of he s currences, as of July 3, 22. Page 3 of

4 The S Dollar Inde uures Conrac TABLE 2 oregn oregn currency X rae currency 3-monh neres rae S dollar 3-monh neres rae Impled orward rae Euro %.824%.975 Yen.8.68 %.824 %.8 Serlng %.824 %.555 Canadan dollar %.824 %.63 Swedsh Kroner %.824 %.5 Swss ranc %.824 %.675 Gven hese X raes and neres raes, he curren SDX nde level s equal o 7.36, as derved n Secon II. The assumpon of no uncerany n echange raes mples ha he curren 3-monh forward X rae wll eacly equal he realzed X rae n hree monhs. Therefore, he SDX nde n hree monhs wll be equal o forward SDX ( (.5 ( ( ( (.63 Therefore, he far fuures value whou uncerany would be equal o Now le s consder he more realsc case wh uncerany n he level of fuure echange raes. B. When he spo rae equals he forward rae on average, bu wh volaly: The effec of geomerc averagng Consder as a base case he echange raes as lsed n Table. Suppose ha he X rae for he pound serlng apprecaes, bu he X rae for he Canadan dollar deprecaes, so ha he resulng SDX nde rae s unchanged. In parcular, he pound serlng apprecaes by % and he Canadan dollar deprecaes by 59.6%. Suppose an nvesor s long $ mllon n he s consuen currences of he SDX nde, n he proper weghs (57.6% n euro, 3.5% n yen, ec.. 2 Invesors who are long he foregn currences and use he SDX fuures conrac o hedge her eposure wll go long $ mllon of he fuures conracs. If he echange raes change such ha he pound serlng apprecaes by % whle he Canadan dollar deprecaes by 59.6%, he hedged nvesor wll eperence an unepeced P&L from he combned 2 The eamples n hs paper gnore mporan real-world consderaons such as he bd-ask spread, brokerage commssons, and fuures mark-o-marke margn paymens. Page 4 of

5 The S Dollar Inde uures Conrac cash-marke and fuures-marke poson. In parcular, he fuures poson wll no change n value a all, snce he nde level would no changed, bu he cash poson wll change. The pound serlng poson wll change by $9, (he amoun of he change (% mes he sze of he poson (.9% of $ mllon whle he Canadan dollar poson wll change by $54,236 (he amoun of he change (59.6% mes he sze of he poson (9.% of $ mllon. Why s he dollar change n he nvesors posons no refleced n he SDX nde, and herefore no n he DSX nde fuures conrac? The dfference sems from fac ha he value of he poson s deermned by he (weghed sum of he ndvdual componens of he radng sraegy, whle he nde s deermned by he (weghed produc of he ndvdual componens of he nde. In oher words, he porfolo performance s an arhmec average, whle he nde performance (and herefore he fuures conrac performance s a geomerc average. Mahemacally, for any gven change n he componen echange raes, he geomerc average wll always be less han he arhmec average. Ths dfference beween arhmec and geomerc averagng s he source of he dvergence beween he nde (and herefore fuures conrac performance and he porfolo performance. The larger he dvergence of performance of he dfferen currences, he larger he dvergence beween he geomerc average and he arhmec average. Therefore, as Eyan, Harpaz, and Krull (988 pon ou, he dvergence beween he geomerc and arhmec averages depend on boh he volales of he ndvdual currences and her co-movemens (somemes referred o as her correlaons. In fac, he only me ha he geomerc average and he arhmec average yeld he same reurn s f all of he ndvdual componens of he nde change by he same amoun. However, he second feaure of he SDX nde where he SDX nde s quoed such ha a dollar apprecaon leads o an ncrease n he nde prevens he sandard cos-of-carry model from prcng he SDX nde fuures conrac, even when all of he ndvdual currences change by he same proporon. We urn o hs eample ne. C. Prcng he SDX fuures conrac wh a smple srucure o he uncerany: The effec of prcng n European erms Now consder a suaon where all of he echange raes are perfecly correlaed. Specfcally, suppose ha we knew ha all echange raes shfed by he same percenage, denoed. In hs suaon, consder how an nvesor who has a long cash poson n he foregn currences would hedge ha eposure. Suppose ha he nvesor were long Z uns of currency and he echange raes (n S dollar per foregn currency uns were denoed X, so ha he curren value of her holdngs (n.s. dollars would be Page 5 of

6 The S Dollar Inde uures Conrac Page 6 of 6 X Z. We denoe as he value of he holdngs of he fuures conrac, so ha, for a hedged porfolo,. Le us consder he value of foregn-echange holdngs and he fuures conrac f all s currences have a proporonal shf of. In hs scenaro, he value of he foregn echange holdngs, denoed, wll be equal o 6 6 ( ( ( Z X Z X Wh he shf n echange raes of, he nde level, and herefore he fuures conrac, wll also shf by he amoun, : /( ( ( ( ( ( ( ( X K X K X K N w N w w N w In hs suaon, he change n he value of he cash holdngs would be ( and he change n he value of he fuures poson would be The combned gan or loss would hen be he sum of hese values: ( (

7 The S Dollar Inde uures Conrac Snce, hs can be smplfed o 2 ( Therefore, he proporonal change n he nal nvesmen of s 2 /(. gure 2 shows how he P&L from hs combned cash and fuures poson changes as a funcon of he level of. When s zero, as n he no-uncerany case of secon III.A, we have a zero P&L conssen wh a hedged poson. As devaes from zero, he combned cash and fuures poson always yelds a posve prof regardless of wheher s posve or negave. gure 2: Prof uncon: 2 /( 3 25 Prof (as a funcon of S dollar nvesmen % -4% -3% -2% -% % % 2% 3% 4% 5% Value of, he precen change n curency Therefore, f uncerany akes he form of proporonal shfs n all foregn echange raes, denoed by, hen he far fuures prce wll depend on he magnude of Page 7 of

8 The S Dollar Inde uures Conrac. The far fuures prce n hs case wll depend on he probably of a large move upward or a large movemen downward. 3 D. Prcng he SDX fuure conrac under more general uncerany condons The cases of no uncerany, revewed n Secon III.A, and of smple srucures of uncerany, revewed n Secon III.B and Secon III.C, are clearly mplausble models of realy. They do provde, however, he nuon for how a model for he SDX fuures conrac mus devae from he sandard cos-of-carry model. In general, he far fuures prce of he SDX fuures conrac depends on he foregn echange volaly srucure ha he marke (.e., he raders n he marke beleves. rom he prevous secons, hough, we do have some resrcons abou how an accurae prcng model wll prce he fuures conrac. rs, f he markes have no foregn echange uncerany, he SDX fuures conrac prcng model should provde he same resul as he smple cos of carry model. In fac, he lower he volaly n he marke, he closer he markedeermned fuures prce should be o he sandard cos-of-carry model of fuures prces. Second, he more dsperson ha markes epec n he performance of he dfferen currences n he nde, he more he marke-deermned fuures prce should devae from he sandard cos-of-carry model of fuures prces. Thrd, f he markes have only he smple uncerany assocaed wh proporonal shfs n all foregn echange raes, he prcng model should reurn a fuures prce ha declnes n he level of ha shf, whch we have parameerzed wh he varable. Those who are neresed n an eac prcng formula for he SDX fuures conrac, based on assumed dsrbuons of he SDX nde componen foregn echange raes, should consul Eyan, Harpaz, and Krull (988. The formulas gven n her paper need o be adjused for he change n he nde from en currences o s currences, wh he replacemen of fve European currences wh he Euro. 3 The dependence of he dfference beween he geomerc average and arhmec average on he volaly of he echange raes s lkely he sources of he erm volaly premum n descrbng he adjusmen facor n deermnng he far fuures prce of he SDX fuures conrac. Page 8 of

9 The S Dollar Inde uures Conrac Append A SDX uures Conrac Specfcaons.S. Dollar Inde (SDX uures Specfcaons (as of June 3, 22: Conrac sze: $ mes he SDX nde. Tradng hours: Conrac monhs: Tcker symbol: Prce quoaon: 3: a.m. o 8: a.m. and 8:5 a.m. o 3: p.m. March, June, Sepember, December DX The.S. dollar nde s quoed as a percen of s value as of March 973, calculaed o wo decmal places (e.g., on July 3, 22, he SDX nde offcally closed a 7.4 Mnmum prce flucuaon: The mnmum prce flucuaon, or ck sze for he SDX nde s. SDX pon, whch s equvalen o $. per fuures conrac. Lm on daly prce move: 2 cks above & below pror day's selemen, ecep durng las 3 mnues of any radng sesson when no lm apples. Should he prce reach he lm and reman whn cks of he lm for 5 mnues, hen new lms wll be esablshed 2 cks above and below he prevous prce lm Poson lms: Las day of radng: Selemen procedure: None 2 nd busness day pror o he 3 rd Wednesday of he eprng monh. On he las radng day, radng ceases a :6 a.m. Conracs held o epraon are seled n cash, based on he value of he SDX nde a am (New York me on he las day of radng for an eprng conrac. The SDX selemen value s compued by Reuers LTD, n accordance o New York Coon Echange regulaons. Page 9 of

10 The S Dollar Inde uures Conrac References Eyan, T. Hanan, Gora Harpaz, Seven Krull (988, The Prcng of Dollar Inde uures Conracs, Journal of uures Markes, Volume 8, Number 2, Aprl 988, pages Redfeld, Corey B. (986, A Theorecal Analyss of he Volaly Premum n he Dollar Inde Conrac, Journal of uures Markes, Volume 6, Number 4, Augus 986, pages Page of