EURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN HJM ONE-FACTOR MODEL

Transcription

1 EURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN HJM ONE-FACTOR MODEL MARC HENRARD Absrac. In his noe we give pricing formlas for differen insrmens linked o rae fres ero-dollar fres. We provide he fre price inclding he convexiy adjsmen he exac daes. Based on ha resl we price opions on fres, inclding he mid-crve opions.. Inrodcion This noe is dedicaed o ero-fres in he HJM framework wih deerminisic volailiy. The fres we inend o price are he Libor-fres as raded on CME for USD on LIFFE for EUR. The daes relaed o hose fres are based on he hird Wednesday of he monh, which is he sar dae of he Libor rae nderlying he fre. We denoe ha rae by L. This rae is fixed a a spo lag prior o ha dae. In EUR USD his lag is wo bsiness days he fixing ake place on he Monday. The mariy dae of he Libor rae is hree monh afer he sar dae. On he fixing dae a he momen of he pblicaion of he Libor raes he fre price is L. Before ha momen, he price evolves wih dem offer. Every day he closing price is sed for margining. The margining process consiss in receiving he difference in price beween he closing price of he day he closing price of he previos day or he ransacion price on he rade dae. The price are mliply by he nominal divided by for. The one forh represen he hree monh period as par of he year, as he Libor raes are qoed on annal basis. The firs ask of his noe is o compe he fair fres price in he arbirage free sense from his descripion. This qesion is cerainly no new answers are abndan also. We refer o [4, Secion.4] [6, Secion.5] for he heoreical framework, [5] for a formla in he exended Vasicek model [7] for a more sophisicaed approach inclding he volailiy smile. The specificiy of his noe is o provide explici formla for he general HJM framework wih deerminisic volailiy han can be fond in [7] [6] nder he name of Gassian HJM o inclde he exac fres daes. All he formlas in he menioned references do no differeniae beween he fixing dae he sar dae. The second ask will be o price he opions on fres as raded on CME. Those opions are no sbjec o margining hem-self. The premim is paid p-fron. If he opion is exercised, one ener ino a fre rade a a price eqal o he srike no cash is exchanged. The vale resling from he exercise comes hrogh he margining of he fres. The exercise dae is se before or on he fixing dae of he fre. Only he sard qarerly fres March, Jne, Sepember December are sed as nderlying of he opions. The mos poplar opions, called qarerly opions, have heir exercise dae se on he fixing dae. Anoher ype of opions, called serial opions, have heir expiry one or wo monhs before he fixing dae. The las ype of opions, called mid-crve opions, have heir expiry one or wo year before he fixing dae. Those characerisics creae a echnically qie complex prodc. Even if we don ake ino accon he nderlying fre convexiy adjsmen, here is a second convexiy adjsmen. he Dae: Firs version: March 4, 4; his version: March 6, 5. Key words phrases. Ineres rae fres, opions on fres, HJM, one facor model. JEL classificaion: G3, E43. AMS mahemaics sbjec classificaion: 9B8, 9B4, 9B7. All daes menioned in his noe are adjsed in some way in he case hey fall on a non-good bsiness day. Fres exiss also on he one monh Libor, b he mos poplar are on hree monh.

2 M. HENRARD opion is on a forward rae no on a deposi. The pay-off is no paid a he deposi mariy, no even a is sar or fixing b a he opion expiry ha can be several years before. For hose opions, we also provide explici formlas in he HJM framework. We don know any reference wih similar formlas. All he exac daes of he opions are also sed in his case expiry dae, fixing dae, sar dae end dae. In [6] a general heoreical formla is provided when he opion exercise dae is eqal o he fre fixing dae is eqal o he sar of he nderlying deposi. We specialize all he resls for he exended Vasicek or Hll-Whie model.. Model hypohesis We model ineres rae prodcs. The base asses are P,, he price in of he zero-copon bond paying in. We describe hem for all, T, where T is some fixed consan. We work in a Heah-Jarrow-Moron [] one facor model framework see for example he chaper Dynamical erm srcre model in [4]. By his we mean we have a model wih he following properies. The fncion P is posiive reglar enogh so ha i can be wrien as P, = exp f, sds. Le A = {s, R : [, T ] s [, ]}. We work in a filered probabiliy space Ω, {F }, F, P real. The filraion F is he agmened filraion of a one-dimensional sard Brownian moion W real T. H: There exiss σ : [, T ] R + measrable bonded 3 wih σ = on [, T ] \A sch ha for some process r s T, N = exp r sds forms wih some measre N a nmeraire pair 4 wih Brownian moion W, r = f,. df, = σ, dp N, = P N, σ, sds d σ, dw σ, sds dw The noaion P N, s designaes he nmeraire rebased vale of P, i.e. P N, s = N P, s. To simplify he wriing in he res of he paper, we will se he noaion ν, = σ, sds. Noe ha ν is increasing in, measrable bonded. Moreover for >, ν, =. In he case of he exended Vasicek model, he volailiy fncion is given by νs, = σ a exp a s. We will analyse his model for σ consan ime-dependen. In he imedependen version, he volailiy parameer is piecewise consan wih σs = σ i for s [s i, s i where = s < s < < s i < < T. 3. Preliminary resls We now sae wo echnical lemmas, he proof of which can be fond in [3]. Similar formlas can be fond in [, 3.3,3.4] in he framework of coheren ineres-rae models. Lemma. Le v. In a HJM one facor model, he price of he zero copon bond can be wrien has, P, v P, v = P, exp νs, v νs, dw s ν s, v ν s, ds. 3 Bonded is oo srong for he proof we se, some L L condiions are enogh, b as all he examples we presen are bonded, we se his condiion for simpliciy. 4 See [4] for he definiion of a nmeraire pair. Noe ha here we reqire ha he bonds of all mariies are maringales for he nmeraire pair N, N.

3 FUTURES & CIE. 3 Lemma. Le v. In he HJM one facor model, we have v v N Nv = exp r s ds = P, v exp νs, vdw s 4. Fres v ν s, vds. The fre fixing dae is denoed. The fixing is on he rae beween. The fixing rae is denoed L. If he accral facor for he period is δ, he fixing is linked o he yield crve by + δl = P, P,. The fres price is Φ. On he fixing dae, he relaion beween he price he rae is Φ = L. The fres margining is done on he fres price mliply by he noional divided by 4. Theorem. Le. In he HJM one-facor model, he price of he fres fixing on for he period wih accral facor δ is given by Φ = P, δ P, γ where γ = exp νs, νs, νs, ds. Proof. Using he generic pricing fre price process heorem [4, Theorem.6], Φ = E N [ L F ]. In L, he only non consan par is he raio of discon facors. Using Lemma wice, we obain P, P, = P, P, exp ν s, ν s, ds + νs, νs, dw s. Only he second inegral conains a sochasic par. This inegral is normally disribed of variance νs, νs, ds. So he expeced vale of he raio of discon facors is redced o P, P, exp ν s, ν s, ds + νs, νs, ds we have he annonced resl. In he case of he exended Vasicek model, he adjsmen facor can be wrien explicily ln γ = σ a 3 exp a exp a expa expa exp a exp a. When he sar dae of he nderlying rae is eqal o he fixing dae = like in he Serling marke, his las formla is eqivalen o he one of Kirikos Novak [5]. In he ime-dependen case, if we ake = s n =, we obain ln γ = a 3 exp a exp a σi expas i expas i exp a s i exp a s i. i= The impac on differeniaing beween he fixing dae he sar dae is he inegral in he gamma beween. For normal volailiy level his is below. basis poins. The dae impac is obviosly larger for he opions reaed in he nex secion.

4 4 M. HENRARD 5. Opions on fres We se he noaion X for he srike rae, K = X for he srike price. Theorem. Le θ 3, X, σ = νs, νs, ds, σ = νs, θ νs, νs, ds, σ = ν s, θds Σ be he marix defined by σ Σ = σ σ σ. In he a HJM one-facor model wih deerminisic volailiy, if he marix Σ is inverible, he price of he call opion wih expiry srike rae X on he fre wih fixing paymen dae on he rae beween 3 is given in by C = P, θ where κ is defined by X + /δn The price of he p is given by γθ P = P, θ δ κ + σ σ γθ δ P, δp, expα σ N X + /δ = P, δp, exp σ κ σ + α, α = P, P, expα σ N νs, νs, νs, ds κ σ σ σ κ + σ σ X + /δn κ σ. σ σ Proof. Using he generic pricing heorem [4, Theorem ] we have C = N E N Φθ K + N θ = E N X γθ P, δ P, + δ + N θ This expeced vale can be comped explicily sing sard decomposiion compaion of normal disribion. Here hose compaion are a lile bi more involved reqired some exra noaions. Le X = νs, νs, dw s, X =. νs, θdw s. The rom variables X X are joinly normally disribed [8, Theorem 3., p. 6] wih covariance marix Σ. Using Lemma Lemma, we have P θ, P θ, = P, P, exp X σ + α N θ = P, θ expx σ. In he expecaion, he parenhesis is posiive when X + /δ > P, δp, 3 exp X σ + α or when X > σ κ. Wih hese resls, he expeced vale becomes A P, θ X + /δ γ P, δ P, exp x σ + α expx σ exp xt Σ xdx x >σ κ

5 FUTURES & CIE. 5 where A =. Like in he proof of [3, Theorem 8], we have π Σ π R exp x Σ xt Σ x dx = exp x σ σ x σ Σ. Noe ha x x σ Σ + σσ can be wrien as x σ x x σ Σ + x σ + σ 4 + σσ can be wrien as x + σ σ + σ σ. So he doble inegral in is eqal o P, θ X + /δ exp π x >σ κ σ x σ dx σ γ P, δ P, expα σ exp π x >σ κ σ x + σ σ dx σ Wriing he inegrals as normal cmlaive disribions gives he resl. For opion on fres, i is very imporan o differeniae beween he opion expiry dae he fixing of he fre, specialy for mid-crve opions. In he exended Vasicek model, he coefficiens can be wrien explicily. When he volailiy is consan he resls are σ σ = σ a 3 exp a exp a expaθ, = σ aθ exp aθ exp aθ, a3 σ = σ a 3 exp a exp a exp aθ + expaθ α = σ a 3 exp a exp a expaθ exp a θ + exp a. When he volailiy is piecewise consan he formlas are a lile bi more involved inclde a sm b are similar. We wrie he formlas wih = s θ = s n. σ σ = = a 3 exp a exp a a 3 i= σ i σi expas i expas i, i= as i s i exp aθ s i exp aθ s i 4 exp aθ s i exp aθ s i, σ = a 3 exp a exp a σi expas i expas i exp aθ s i exp aθ s i i= α = a 3 exp a exp a σi expas i expas i exp a s i exp a s i. i= 6. Conclsion We review he pricing of ero-dollar fres in he HJM framework wih deerminisic volailiy. Even if his noe is no he firs one o deal wih his convexiy adjsmen problem, i improves exising resls by aking care of all he relevan daes, inclding he spo-lag beween he fixing he nderlying sar dae. The formla obained is specialized for he case of he ime-dependen exended Vasicek model wih piecewise consan volailiy parameer.

6 6 M. HENRARD The noe also provide explici formlas for exchange raded opions on fres. In a similar way all he relevan dae are aken ino accon, inclding he expiry, fixing sar dae. The explici resls are also specialized for he exended Vasicek model. To or knowledge his is he firs noe ha gives explici formla for hose prodcs. The resling formla is somehow involved in is formlaion. his is de o he margining characerisic of he nderlying fre ha reqires an convexiy adjsmen facor. On op of i, he opion is on he rae, no on he deposi which creaes a second differen convexiy adjsmen. Disclaimer: The views expressed here are hose of he ahor no necessarily hose of he Bank for Inernaional Selemens. References [] D. C. Brody L. P. Hghson. Chaos coherence: a new framework for ineres-rae modelling. Proc. R. Soc. Lond. A., 46:85, 4. [] D. Heah, R. Jarrow, A. Moron. Bond pricing he erm srcre of ineres raes: a new mehodology for coningen claims valaion. Economerica, 6:77 5, Janary 99. [3] M. Henrard. A semi-explici approach o canary swapions in HJM one-facor model. Applied Mahemaical Finance, To appear 5. Preprin in Economics Working Paper Archive, Ewp-fin 38, 3., 5 [4] P. J. Hn J. E. Kennedy. Financial Derivaives in Theory Pracice. Wiley series in probabiliy saisics. Wiley, second ediion, 4.,, 3, 4 [5] G. Kirikos D. Novak. Convexiy conndrms. Risk, pages 6 6, March 997., 3 [6] M. Msiela M. Rkowski. Maringale Mehods in Financial Modelling, volme 36. Springer, second ediion, 4., [7] V. Pierbarg M. Renedo. Erodollar fres convexiy adjsmens in sochasic volailiy models. Technical repor, Bank of America, 4. [8] R. Rebonao. Modern pricing of ineres-rae derivaives: he LIBOR marke model Beyond. Princeon Universiy Press, Princeon Oxford,. 4 Conens. Inrodcion. Model hypohesis 3. Preliminary resls 4. Fres 3 5. Opions on fres 4 6. Conclsion 5 References 6 Derivaives Grop, Banking Deparmen, Bank for Inernaional Selemens, CH-4 Basel Swizerl address: Marc.Henrard@bis.org