INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying an up-fron premium and hose wih fuure-like daily margining. 1. Inroducion The pricing is done in a muli-curve framework, wih discouning curve denoed and forward or esimaion curves denoed P D (s, ). P j (s, ) wih j he enor of he relevan Ibor index or j = O for he overnigh index. Mos of he fuures are linked o hree monh Ibor indices, bu some are linked o he one monh Ibor index or an overnigh index. The (Ibor or overnigh) fixing aking place on 0 for he period [ 1 ; 2 ] wih accrual facor δ is linked o he curve by 1 + δl j 0 = P j ( 0, 1 ) P j ( 0, 2 ). 2. Ibor-like Ineres Rae Fuures The fuures ype described here are he Ibor fuures as raded on CME 1 for USD and JPY, on NYSE-Euronex-Liffe 2 for EUR, GBP, CHF and USD and on EUREX 3 for EUR. The daes relaed o hose fuures are based on he hird Wednesday of he monh 4, which is he sar dae of he Ibor rae underlying he fuure. The rae is fixed a a spo lag prior o ha dae; he fixings usually ake place on he Monday or on he Wednesday iself. The fixing dae is also he las rading dae for he fuure. The end dae of he Ibor rae period is hree monhs 5 afer he sar dae (using he convenions associaed o he relevan Ibor-index). The margining process works in he following way. For a given closing price (as published by he exchange), he daily margin paid is ha price minus he reference price muliplied by he noional and by he accrual facor of he fuure. Equivalenly, i is he price difference muliplied by one hundred and by he poin value, he poin value being he margin associaed wih a one (percenage) poin change in he price, he reference price he rade price on he rade dae, or he previous closing price on subsequen daes. Dae: Firs version: 18 May 2011; his version: 23 February 2012. Version 1.3. 1 www.cmegroup.com 2 www.euronex.com 3 www.eurexchange.com 4 When he day no a business day, i is adjused o he following day. 5 Fuures on he one monh Ibor also exis, hose on hree monh have higher volume. 1

2 OPENGAMMA The price of he fuure on is denoed Φ j. On he fixing dae a he momen of he publicaion of he Ibor raes L j, he fuure price is Φ j = 1 L j. Before ha momen, he price evolves wih supply and demand. For hree monh fuures, he nominal is 1,000,000 (500,000 for GBP and 100,000,000 for JPY) and he accrual facor is 1/4. For one monh fuures, he nominal is 3,000,000 and he accrual facor is 1/12. In boh cases, he nominal muliplied by he accrual facor is he equal (250,000 for USD, 125,000 for GBP and 25,000,000 for JPY). The value of one (percenage) poin is 2,500 currency unis for USD, 1,250 for GBP and 250,000 for JPY. The ick value is he value of he smalles permissible incremen in price. The prices usually change in 1.0 or 0.5 basis poin incremens. The fuures are designaed by characer codes. The firs par depends on he daa provider and is usually wo or four characers. The main codes are given in Table 1. The second par describes he monh, wih he code given in Table 2, and he year by is las digi. As ineres rae fuures are only quoed up o 10 years in he fuure, here is no ambiguiy in using only one figure for he year. Noe also ha i means ha when a fuure reaches is las rading dae, a new one is creaed a couple of days laer wih he same name bu for a mauriy 10Y in he fuure. Currency Tenor Exchange Underlying Noional USD 3M CME LIBOR 1,000,000 USD 1M CME LIBOR 3,000,000 EUR 3M Eurex EURIBOR 1,000,000 EUR 3M Liffe EURIBOR 1,000,000 GBP 3M Liffe LIBOR 1,000,000 CHF 3M Liffe LIBOR 1,000,000 JPY 3M SGX/CME TIBOR 1,000,000 JPY 3M SGX LIBOR 1,000,000 Table 1. Fuures deails and codes. Monh Code Monh Code Monh Code January F February G March H April J May K June M July N Augus Q Sepember U Ocober V November X December Z Table 2. Rae fuures monh codes. The fuure fixing dae is denoed 0. The fixing is on he Ibor rae beween 1 = Spo( 0 ) and 2 = 1 + j monh and he accrual facor for he period [ 1, 2 ] is denoed δ. The generic fuure price process heorem (see [Hun and Kennedy, 2004, Theorem 12.6]), saes ha, in he cash accoun numeraire wih probabiliy N, he fuure price is given by (1) Φ = E N [Φ 0 F ]. 2.1. Pricing by discouning. In his approach he margining deails of he fuure are no aken ino accoun. The fuure is priced like an Ibor coupon. The fuure rae is compued as he forward rae F j 0 = 1 ( P j ) (0, 1 ) δ P j (0, 2 ) 1.

RATE FUTURES AND OPTIONS 3 The fuure price is Φ j 0 = 1 F j 0. 2.2. Pricing in he Hull-Whie one facor model. The model used is a Hull-Whie one-facor model, or exended Vasicek model. The muli-curve pricing formula presened here is from Henrard [2010]. I uses he deerminisic spread hypohesis beween he curves. The model volailiy is σ(, s) and we denoe ν(, u) = u σ(, s)ds. Theorem 1. Le 0 0 1 2. In he HJM one-facor model on he discoun curve in he muli-curve framework and wih he independence hypohesis beween spread and discoun facor raio SI proposed in Henrard [2010], he price of he fuures fixing a 0 for he period [ 1, 2 ] wih accrual facor δ is given by (2) where Φ j = 1 1 δ ( P j ) (, 1 ) P j (, 2 ) γ() 1 = 1 γ()f j + 1 (1 γ()) δ ( 0 ) γ() = exp ν(s, 2 )(ν(s, 2 ) ν(s, 1 ))ds. The rae adjusmen (γ()f j 0 1 δ (1 γ()) F j 0 ) is given in Figure 1 for several volailiy levels and fuures up o 5 years. The Hull-Whie volailiy used are consan wih level σ = 0.005, 0.01 and 0.02. 45 40 =0.005 =0.01 =0.02 35 Adjusmen (bps) 30 25 20 15 10 5 0 0 1 2 3 4 5 Mauriy (in years) Figure 1. Rae adjusmens for differen level of volailiies and fuure up o 5 years. 3. Ineres Rae Fuures Opions: Premium An opion on fuure is described by he underlying fuure, an opion expiraion dae θ, a srike K and an opion ype (Pu or Call). The expiraion is before or on he fuure las rading dae: θ < 0. The opion on fuures deal wih in his secion are American ype and pay he premium upfron a he ransacion dae. There is no margining process for he opion. This ype of opion is raded on he CME exchange for eurodollar fuures (one and hree monhs).

4 OPENGAMMA There are hree ype of opions: quarerly opions, serial opions and mid-curve opions. Quarerly opions expire on he las rading dae of he underlying fuure, i.e. θ = 0. Serial and mid-curve opions expire before he fuure las rading dae. For serial opions, he delay is one or wo monhs (plus one week-end). For mid-curve opions, he delay is one, wo or four years. The quoed price for he opions follow he same rule as he fuure. For a quoed price, he amoun paid is he price muliplied by he noional and by he accrual facor of he underlying fuure. 4. Ineres Rae Fuures Opions: Margin An opion on fuure is described by he underlying fuure, he opion expiraion dae θ, he srike K and an opion ype (Pu or Call). The expiraion is before or on he fuure las rading dae: θ 0. The opion on fuures deal wih in his secion are American ype and have a fuure-like margining process. This ype of opion is raded on he Liffe for EUR, GBP, CHF and USD fuures (hree monhs) and Eurex for EUR (hree monhs). Noe ha here are wo margin processes involved in his insrumen: one on he underlying fuure and one on he opion iself. The quoed price for he opions follows he same rule as for he fuure. For a quoed price, he daily margin paid is he curren closing price, minus he reference price, muliplied by he noional and he accrual facor of he underlying fuure. The reference price is he rade price on he rade dae, and he previous closing price on subsequen daes. Currency Tenor Exchange Underlying Type USD 3M LIFFE LIBOR Opion on fuure USD 3M LIFFE LIBOR Mid-Curve Opions EUR 3M LIFFE EURIBOR Opion on fuure EUR 3M LIFFE EURIBOR Mid-Curve Opions EUR 3M LIFFE EURIBOR 2 year Mid-Curve Opions GBP 3M LIFFE LIBOR Opion on fuure GBP 3M LIFFE LIBOR Mid-Curve Opions GBP 3M LIFFE LIBOR 2 year Mid-Curve Opions CHF 3M LIFFE LIBOR Opion on fuure Table 3. Ineres rae fuures opions codes. 4.1. SABR on fuure rae. The fuure price is a maringale in he cash-accoun numeraire. The fuure rae R = 1 Φ is also a maringale. Due o he margining process on he opion, he price of he opion wih margining is (see [Hun and Kennedy, 2004, Theorem 12.6]) E N [ (Φ θ K) +] = E N [ ((1 K) R θ ) +]. The fuure rae can be modelled hrough any process ha leads o a maringale. We choose o model i wih a SABR process wihou drif. Le SABR(S, M, T ) be he price of an opion of srike M and ype T when he underlying is S. In he SABR on fuure rae framework, he fuure opion will be priced by SABR(1 Φ j 0, 1 K,!T ) where! Call = Pu and! Pu = Call (a call on he price is a pu on he rae). In his implemenaion, he fuure price is calculaed wihou convexiy adjusmen.

RATE FUTURES AND OPTIONS 5 4.2. SABR on fuure rae: curve sensiiviy. The curve sensiiviy is obained from he above formula by composiion. The derivaive of he SABR formula wih respec o he underlying is known and he derivaive of he fuure price wih respec o he raes is also known. 4.3. American opion. The opions on fuures are acually American opions, even if we have reaed hem as European above. For he explanaion why for margined opions on margined fuures, here is no difference in valuaion beween American and European opions (i.e. i is never opimal o exercise early), see Chen and Sco [1993]. The American opion has he same price as he European opion. The argumen proceeds as follows. Le f(x) be he opion pay-off funcion (i.e. f(x) = (x K) + for a call and f(x) = (K x) + for a pu). Those wo funcions are convex. The Jensen s inequaliy can be applied ( ) E N [f(φ θ )] f E N [Φ θ ] = f(φ 0 ). where he fuure price equaion (1) and is maringale propery is used in he las line. The lef hand side of he inequaliy is he European opion value now and he righ hand side is he pay-off for early exercise. This proves ha i is never opimal o exercise before expiry. 5. Fed fund fuures The 30-Day Federal Funds Fuures (simply called Fed Funds fuures) are based on he monhly average of overnigh Fed Funds rae for he conrac monh. The noional is 5,000,000 USD. The conrac monhs are he firs 36 calendar monhs. They are quoed on CBOT 6 for USD. In pracice, even if he Ibor and Fed Fund curves are differen, he echnical adjusmen are similar. In he laer case he underlying is an arihmeic average and he adjusmen has o be done several imes (one for each day, on average). Le 0 < 0 < 1 < < n < n+1 be he relevan dae for he Fed Funds fuures, wih 1 he firs business day of he reference monh, i+1 he business day following i and n+1 he firs business day of he following monh. Le δ i be he accrual facor beween i and i+1 (1 i n) and δ he accrual facor for he oal period [ 1, n+1 ]. The day coun convenion for he USD overnigh is ACT/360. The overnigh raes beween i and i+1 are given in i by Fi O wih 1 + δ i Fi O 1 = P O ( i, i+1 ). The fuure price on he final selemen dae n+1 is ( Φ n+1 = 1 1 n ) δ i Fi O. δ The margining is done on he price muliplied by he noional and divided by he one monh accrual fracion (1/12). The model used is a Hull-Whie one-facor model or exended Vasicek model. The resul uses he deerminisic spread hypohesis beween he curves. The model volailiy is σ(, s) and we denoe ν(, u) = u σ(, s)ds. Theorem 2. In he HJM one-facor model on he discoun curve in he muli-curve framework and wih he deerminisic hypohesis beween spread and discoun facor raio S0 proposed in Henrard [2010], he price of he average overnigh fuure for he period [ 1, n+1 ] is given for j < j+1 (j {0, 1,..., n}) by Φ = 1 1 j δ i Fi O + δ i=1 n i=j+1 i=1 ( P O ) (, i ) δ i P O (, i+1 ) γ i 1 6 www.cmegroup.com

6 OPENGAMMA where ( i ) γ i = exp ν(s, i+1 )(ν(s, i+1 ) ν(s, i ))ds. The formula is divided ino wo pars o cope wih he case where he averaging period has sared already. In hose cases he convexiy adjusmens γ i are very small. 6. Bank bill fuures (AUD syle) AUD bill fuures are physically seled. A expiry, differen bills can be delivered. The bills eligible for delivery are bills wih beween 85 and 95 days o mauriy a he selemen dae. The issuers of he bills are he one in he eligible counerparies. The pary shor of he fuure chooses he bill o be delivered 7. The shor pary has an opion and he will, of course, choose o deliver he cheapes bill. This is a siuaion similar o he one for bond fuures in oher currencies. Le θ be he expiry (or announcemen) dae and 0 be he selemen dae. In pracice, hose daes are he second Friday of he fuure monh and he nex business day (Monday). Le i (1 i N) denoe he possible mauriy daes of he bills 8. A selemen, he price received for he bill will depend of he las quoed fuure index ha we denoe F θ. The yield associaed o his index is R θ = 1 F θ. The price paid is 1 1 + δ i R θ where δ i is he accrual facor associaed o he daes 0 and i. For AUD bill fuures his facor is he number of calendar days beween he wo daes divided by 365. In exchange for he price he shor pary gives he bill wih a noional equal o he noional of he fuure 9. 7. Implemenaion The fuure securiy descripion is in he class IneresRaeFuure. The implemenaion of he fuure pricing by discouning is in he mehod IneresRaeFuureTransacionDiscouningMehod. The implemenaion for he Hull-Whie one facor model is in he mehod IneresRaeFuureTransacionHullWhieMehod. The implemenaion of he opion on fuure wih margin is in he mehod IneresRaeFuureOpionMarginTransacionSABRMehod. The SABR parameers are represened by surfaces. The firs dimension is he ime o expiraion; he second dimension is he delay beween opion expiry and fuure las rading dae. In his implemenaion, he fuure price is compued wihou convexiy adjusmen (discouning mehod). This is sufficien for shor daed fuures bu probably no for long-daed mid-curve opions. References R. R. Chen and L. Sco. Pricing ineres rae fuures opions wih fuures-syle margining. The Journal of Fuures Marke, 13(5):15 22, 1993. 5 M. Henrard. The irony in he derivaives discouning par II: he crisis. Wilmo Journal, 2(6): 301 316, December 2010. Preprin available a SSRN: hp://ssrn.com/absrac=1433022. 3, 5 7 Acually, for each conrac he can choose up o 10 differen bills of AUD 100,000 each. 8 In pracice here are nine possible daes aking he week-end ino accoun 9 The noional of he bill fuures is AUD 1,000,000. This noional can be spli ino several physical bills, up o 10 pieces of AUD 100,000.

RATE FUTURES AND OPTIONS 7 P. J. Hun and J. E. Kennedy. Financial Derivaives in Theory and Pracice. Wiley series in probabiliy and saisics. Wiley, second ediion, 2004. ISBN 0-470-86359-5. 2, 4 Conens 1. Inroducion 1 2. Ibor-like Ineres Rae Fuures 1 2.1. Pricing by discouning 2 2.2. Pricing in he Hull-Whie one facor model 3 3. Ineres Rae Fuures Opions: Premium 3 4. Ineres Rae Fuures Opions: Margin 4 4.1. SABR on fuure rae 4 4.2. SABR on fuure rae: curve sensiiviy 5 4.3. American opion 5 5. Fed fund fuures 5 6. Bank bill fuures (AUD syle) 6 7. Implemenaion 6 References 6 Marc Henrard, Quaniaive Research, OpenGamma E-mail address: quan@opengamma.com