Hedging in bond markets by the Clark-Ocone formula

Transcription

1 Communicaions on Sochasic Analysis Volume 8 Number 2 Aricle Hedging in bond markes by he Clark-Ocone formula Nicolas Privaul imohy Robin eng Follow his and addiional works a: hps://digialcommons.lsu.edu/cosa Par of he Analysis Commons, and he Oher Mahemaics Commons Recommended Ciaion Privaul, Nicolas and eng, imohy Robin (214) "Hedging in bond markes by he Clark-Ocone formula," Communicaions on Sochasic Analysis: Vol. 8 : No. 2, Aricle 8. DOI: /cosa Available a: hps://digialcommons.lsu.edu/cosa/vol8/iss2/8

2 Communicaions on Sochasic Analysis Vol. 8, No. 2 (214) Serials Publicaions HEDGING IN BOND MARKES BY HE CLARK-OCONE FORMULA NICOLAS PRIVAUL* AND IMOHY ROBIN ENG Absrac. Hedging sraegies in bond markes are compued by maringale represenaion and he choice of a suiable of numeraire, based on he Clark- Ocone formula in a model driven by he dynamics of bond prices. Applicaions are given o he hedging of swapions and oher ineres rae derivaives and we compare our approach o dela hedging when he underlying swap rae is modeled by a diffusion process. 1. Inroducion he pricing of ineres rae derivaives is usually performed by he change of numeraire echnique under a suiable forward measure ÎP. On he oher hand, he compuaion of hedging sraegies for ineres rae derivaives presens several difficulies, in paricular, hedging sraegies appear no o be unique and one is faced wih he problem of choosing an appropriae enor srucure of bond mauriies in order o correcly hedge mauriy-relaed risks, see e.g. 2 in he jump case. In his paper we consider he applicaion of he change of numeraire echnique o he compuaion of hedging sraegies for ineres rae derivaives. he payoff of an ineres derivaive is usually based on an underlying asse priced ˆX a ime (e.g. a swap rae) which is defined from a family (P ( i )) i of bond prices wih mauriies ( i ) i. We will disinguish beween wo differen modeling siuaions. (1) Modeling ˆX as a Markov diffusion process d ˆX ˆσ ( ˆX )dŵ (1.1) where (Ŵ) R+ is a Brownian moion under he forward measure ÎP. In his case dela hedging can be applied and his approach has been adoped in 7 o compue self-financing hedging sraegies for swapions based on geomeric Brownian moion. In Secion 4 of his paper we review and exend his approach. Received ; Communicaed by he ediors. 21 Mahemaics Subjec Classificaion. Primary 91B28; Secondary 6H7, 6H3, 46N1. Key words and phrases. Bond markes, hedging, forward measure, Clark-Ocone formula under change of measure, bond opions, swapions. * Research suppored by he NU ier 1 Gran RG19/

3 27 NICOLAS PRIVAUL AND IMOHY ROBIN ENG (2) Modeling each bond price P ( ) by a sochasic differenial equaion of he form dp ( ) r P ( )d + P ( )ζ ( )dw, (1.2) where W is a sandard Brownian moion under he risk-neural measure IP. In his case he process ˆX may no longer have a simple Markovian dynamics under ÎP (cf. Lemma 3.2 or (3.17) below) and we rely on he Clark-Ocone formula which is commonly used for he hedging of pah-dependen opions. Precisely, due o he use of forward measures we will apply he Clark-Ocone formula under change of measure of 9. his approach is carried ou in Secion 3. We consider a bond price curve (P ) R+, valued in a real separable Hilber space G, usually a weighed Sobolev space of real-valued funcions on R +, cf. 4 and of 1, and we denoe by G he dual space of coninuous linear mappings on G. Given µ G a signed finie measure on R + wih suppor in, ), we consider P (µ) : µ, P G,G P (y)µ(dy), which represens a baske of bonds whose mauriies are beyond he exercise dae > and disribued according o he measure µ. he value of a porfolio sraegy (ϕ ), is given by V : ϕ, P G,G P (y)ϕ (dy) (1.3) where he measure ϕ (dy) represens he amoun of bonds wih mauriy in y, y + dy in he porfolio a ime,. Given ν G anoher posiive finie measure on R + wih suppor in, ), we consider he generalized annuiy numeraire P (ν) : ν, P G,G and he forward bond price curve ˆP P (y)ν(dy), P P (ν),, which is a maringale under he forward measure ÎP defined by IE dîp F S e S r sds P S(ν) d IP P (ν), (1.4) where he mauriy S is such ha S. In pracice, µ(dy) and ν(dy) will be finie poin measures, i.e. sums j α k δ k (dy) ki of Dirac measures based on he mauriies i,..., j of a given a enor srucure, in which α k represens he amoun allocaed o a bond wih mauriy

4 HEDGING IN BOND MARKES BY HE CLARK-OCONE FORMULA 271 k, k i,..., j. In his case we are ineresed in finding a hedging sraegy ϕ (dy) of he form j ϕ (dy) α k ()δ k (dy) in which case (1.3) reads V ki j α k ()P ( k ),, ki and similarly for P (µ) and P (ν) using µ(dx) and ν(dx) respecively. Lemma 2.1 below shows how o compue self-financing hedging sraegies from he decomposiion ˆξ ÎEˆξ + ϕ s, d ˆP s G,G, (1.5) of a forward claim payoff ˆξ ξ/p S (ν), where (ϕ ), is a square-inegrable G - valued adaped process of coninuous linear mappings on G. he represenaion (1.5) can be obained from he predicable represenaion ˆξ ÎEˆξ + ˆα, dŵ H, (1.6) where (Ŵ) R+ is a Brownian moion under ÎP wih values in a separable Hilber space H, cf. (2.7) below, and (ˆα ) R+ is an H-valued square-inegrable F -adaped process. In case he forward price process ˆP P /P (ν), R +, follows he dynamics d ˆP ˆσ dŵ, (1.7) where (ˆσ ) R+ is an L HS (H, G)-valued adaped process of Hilber-Schmid operaors from H o G, cf. 1, and ˆσ : H G is inverible,, Relaion (1.7) shows ha he process (ϕ ) R+ in Lemma 2.1 is given by ϕ (ˆσ ) 1 ˆα,. (1.8) However his inveribiliy condiion can be oo resricive in pracice. On he oher hand he inveribiliy of σ : G H as an operaor is no required in order o hedge he claim ξ. As an illusraive example, when H R we have ˆξ IEˆξ + ˆα dŵ IEˆξ + n c i i1 ˆα ˆσ ( i ) d ˆP ( i ), where { 1,..., n } R + is a given enor srucure and c 1,..., c n R + saisfy c c n 1, and we can ake n ˆα ϕ c i ˆσ ( i ) δ i. i1 Such a hedging sraegy (ϕ ), depends as much on he bond srucure (hrough he volailiy process σ (x)) as on he claim ξ iself (hrough α ), in connecion wih he problem of hedging mauriy-relaed risks.

5 272 NICOLAS PRIVAUL AND IMOHY ROBIN ENG he predicable represenaion (1.6) can be compued from he Clark-Ocone formula for he Malliavin gradien ˆD wih respec o (Ŵ) R+, cf. e.g. Proposiion 6.7 in of 1 when he numeraire is he money marke accoun, cf. also 11 for examples of explici calculaions in his case. his approach is more suiable o a non-markovian or pah-dependen dynamics specified for ( ˆP ) R+ as a funcional of (Ŵ) R+. However his is no he approach chosen here since he dynamics assumed for he bond price is eiher Markovian as in (1.1), cf. Secion 4, or wrien in erms of W as in (1.2), cf. Secion 3. In his paper we specify he dynamics of (P ) R+ under he risk-neural measure and we apply he Clark-Ocone formula under a change of measure 9, using he Malliavin gradien D wih respec o W, cf. (2.1) below. In Proposiion 3.1 below we compue self-financing hedging sraegies for coningen claims wih payoff of he form ξ P S (ν)ĝ (P (µ)/p (ν)). his paper is organized as follows. Secion 2 conains he preliminaries on he derivaion of self-financing hedging sraegies by change of numeraire and he Clark-Ocone formula under change of measure. In Secion 3 we use he Clark- Ocone formula under a change of measure o compue self-financing hedging sraegies for swapions and oher derivaives based on he dynamics of (P ) R+. In Secion 4 we compare he above resuls wih he dela hedging approach when he dynamics of he swap rae ( ˆX ) R+ is based on a diffusion process. 2. Preliminaries In his secion we review he hedging of opions by change of numeraire, cf. e.g. 5, 12, in he framework of 1. We also quoe he Clark-Ocone formula under change of measure. Hedging by change of numeraire. Consider a numeraire (M ) R+ under he riskneural probabiliy measure IP on a filered probabiliy space (Ω, (F ) R+, IP), ha is, (M ) R+ is a coninuous, sricly posiive, F -adaped asse price process such ha he discouned price process e rsds M is an F -maringale under IP. Recall ha an opion wih payoff ξ, exercise dae and mauriy S, is priced a ime as IE e S rsds ξ F M ÎEˆξ F,, (2.1) under he forward measure ÎP defined by IE dîp F S e S rsds M S, (2.2) d IP M S, where ˆξ ξ L 1 M (ÎP, F S) S denoes he forward payoff of he claim ξ. In he framework of 1, consider (W ) R+ a cylindrical Brownian moion aking values in a separable Hilber space H wih covariance EW s (h)w (k) (s ) h, k H, h, k H, s, R +,

6 HEDGING IN BOND MARKES BY HE CLARK-OCONE FORMULA 273 and generaing he filraion (F ) R+. Consider a coninuous F -adaped asse price process (X ) R+ aking values in a real separable Hilber space G, and assume ha boh (X ) R+ and (M ) R+ are Iô processes in he sense of of 1. he forward asse price ˆX : X M,, is a maringale in G under he forward measure ÎP, provided i is inegrable under ÎP. he nex lemma will be key o compue self-financing porfolio sraegies in he asses (X, M ) by numeraire invariance, cf. 12, 6 for he finie dimensional case. We say ha a porfolio (ϕ, η ), wih value is self-financing if ϕ, X G,G + η M,, dv ϕ, dx G,G + η dm. (2.3) he porfolio (ϕ, η ), is said o hedge he claim ξ M S ˆξ if ϕ, X G,G + η M IE e S rsds M S ˆξ F,. Lemma 2.1. Assume ha he forward claim price ˆV : ÎEˆξ F has he predicable represenaion ˆV ÎEˆξ + ϕ s, d ˆX s G,G,, (2.4) where (ϕ ), is a square-inegrable G -valued adaped process of coninuous linear mappings on G. hen he porfolio (ϕ, η ), defined wih and priced as η ˆV ϕ, ˆX G,G,, (2.5) V ϕ, X G,G + η M,, is self-financing and hedges he claim ξ M S ˆξ. Proof. For compleeness we provide he proof of his lemma, alhough i is a direc exension of classical resuls. In order o check ha he porfolio (ϕ, η ), hedges he claim ξ M S ˆξ i suffices o noe ha by (2.1) and (2.5) we have ϕ, X G,G + η M M ˆV IE e S rsds M S ˆξ F,. he porfolio (ϕ, η ), is clearly self-financing for ( ˆX, 1) by (2.4), and by he semimaringale version of numeraire invariance, cf. e.g. page 184 of 12, and 6, i is also self-financing for (X, M ), cf. also 3.2 of 8 and references herein. For compleeness we quoe he proof of he self-financing propery, as follows: dv d(m ˆV ) ˆV dm + M d ˆV + dm d ˆV ˆV dm + M ϕ, d ˆX G,G + dm ϕ, d ˆX G,G

7 274 NICOLAS PRIVAUL AND IMOHY ROBIN ENG ϕ, ˆX G,GdM + M ϕ, d ˆX G,G + dm ϕ, d ˆX G,G +( ˆV ϕ, ˆX G,G)dM ϕ, d(m ˆX ) G,G + ( ˆV ϕ, ˆX G,G)dM ϕ, dx G,G + η dm. Lemma 2.1 yields a self-financing porfolio (ϕ, η ), wih value V V + η s dm s + ϕ s, dx s G,G,, (2.6) given by (2.3), which hedges he claim wih exercise dae and random payoff ξ. Clark formula under change of measure. Recall ha by he Girsanov heorem, cf. heorem 1.14 of 3 or heorem 4.2 of 1, he process (Ŵ) R+ defined by dŵ dw 1 M dm dw, R +, (2.7) is a H-valued Brownian moion under ÎP. Le D denoe he Malliavin gradien wih respec o (W ) R+, defined on smooh funcionals of Brownian moion, f C b (R n ), as D ˆξ n k1 ˆξ f(w 1,..., W n ) 1,k () f x k (W 1,..., W n ), R +, and exended by closabiliy o is domain Dom (D). he proof of Proposiion 3.1 relies on he following Clark-Ocone formula under a change of measure, cf. 9, which can be exended o H-valued Brownian moion by sandard argumens. Lemma 2.2. Le (γ ) R+ denoe a H-valued square-inegrable F -adaped process such ha γ Dom (D), R +, and Le ˆξ Dom (D) such ha and Ê ˆξ dw γ d + dŵ. 2 Ê D ˆξ H d < (2.8) hen he predicable represenaion ˆξ ÎEˆξ + D γ s dŵs 2 H ˆα, dŵ H d <. (2.9)

8 HEDGING IN BOND MARKES BY HE CLARK-OCONE FORMULA 275 is given by ˆα D ÎE ˆξ + ˆξ D γ s dŵsf,. (2.1) 3. Hedging by he Clark-Ocone Formula In his secion we presen a compuaion of hedging sraegies using he Clark- Ocone formula under change of measure and we assume ha he dynamics of (P ) R+ is given by he sochasic differenial equaion dp r P d + P ζ dw, (3.1) in he Sobolev space G which is assumed o be an algebra of real-valued funcions on R +. he process (r ) R+ represens a shor erm ineres rae process adaped o he filraion (F ) R+ generaed by (W ) R+, and (ζ ) R+ is an L HS (H, G)- valued deerminisic funcion. he aim of his secion is o prove Proposiion 3.1 below under he nonresricive inegrabiliy condiions and ζ (y) 2 HÎE ˆP 2 (y)µ(dy)d < (3.2) ζ (y) 2 HÎE ˆP (µ) 2 ( ˆP 2 (y) + ˆP 2 (y))ν(dy)d <. (3.3) which are respecively derived from (2.8) and (2.9). he nex proposiion provides an alernaive o Proposiion 3.3 in 11 by applying o a differen family of payoff funcions. I coincides wih Proposiion 3.3 of 11 in case S and ν δ. Proposiion 3.1. Consider he claim wih payoff ( ) P (µ) ξ P S (ν)ĝ, P (ν) where ĝ : R R is a Lipschiz funcion. hen he porfolio ˆP ϕ (dy) : ÎE (y) ˆP (y) ĝ ( ˆP (µ)) F µ(dy) +ÎE (ĝ( ˆP (µ)) ˆP (µ)ĝ ( ˆP (µ))) ˆP (y) F ν(dy) (3.4) ˆP (y), is self-financing and hedges he claim ξ. Before proving Proposiion 3.1 we check ha he porfolio ϕ hedges he claim ξ P S (ν)ĝ( ˆP (µ)) by consrucion, since we have ϕ, P G,G P (y)ϕ (dy) ˆP (y) ÎE ˆP (y) ĝ ( ˆP (µ)) F P (y)µ(dy)

9 276 NICOLAS PRIVAUL AND IMOHY ROBIN ENG by (2.1). Hence + ÎE (ĝ( ˆP (µ)) ˆP (µ)ĝ ( ˆP (µ))) ˆP (y) F P (y)ν(dy) ˆP (y) P ĝ( (ν)îe ˆP (µ)) F P (ν) ÎE ˆP (y)ĝ ( ˆP (µ)) F µ(dy) +P (ν) ÎE ˆP (µ)ĝ ( ˆP (µ)) ˆP (y) F ν(dy) P ĝ( (ν)îe ˆP (µ)) F IE e S rsds ξ F. ϕ, ˆP G,G ĝ( ÎE ˆP (µ)) F ˆV (3.5) he ideniy (3.5) will also be used in he proof of Lemma 3.5 below. Before moving o he proof of Proposiion 3.1 we consider some examples of applicaions of he resuls of Proposiion 3.1, in which he dynamics of (P ) R+ is given by (1.2). Exchange opions. In he case of an exchange opion wih S and payoff (P (µ) κp (ν)) +, Proposiion 3.1 yields he self-financing hedging sraegy ϕ (dy) 1 ÎE ˆP (y) { ˆP (µ)>κ} F µ(dy) 1 ˆP (y) κîe ˆP (y) { ˆP (µ)>κ} F ν(dy) ˆP (y) 1 ÎE ˆP (y) { ˆP (µ)>κ} F (µ(dy) κν(dy)). ˆP (y) Bond opions. In he case of a bond call opion wih S and payoff (P (U) κ) + and µ δ U, ν δ, his yields ϕ (dy) P ( ) P (U)ÎE 1 ˆP { ˆP (U)>κ} (U) F δ U (dy) 1 κîe { ˆP (U)>κ} F δ (dy). (3.6) his paricular seing of bond opions can be modeled using he diffusions of Secion 4 since in ha case ˆP (µ) P (U)/P ( ) is a geomeric Brownian moion under ÎP wih volailiy ˆσ() ζ (U) ζ ( ) (3.7) given by (3.12) below, in which case he above resul coincides wih he dela hedging formula (4.1) below. Caples on he LIBOR rae. In he case of a caple wih payoff on he LIBOR rae (S )(L(,, S) κ) + (P (S) 1 (1 + κ(s ))) +, (3.8) L(,, S) P ( ) P (S), < S, (3.9) (S )P (S)

10 HEDGING IN BOND MARKES BY HE CLARK-OCONE FORMULA 277 and µ δ, ν δ S, Proposiion 3.1 yields ϕ (dy) P (S) 1 P ( )ÎE P (S) 1 {P (S)<1/(1+κ(S ))} F δ (dy) (3.1) (1 + κ(s 1 ))ÎE {P (S)<1/(1+κ(S ))} F δ S (dy) In his case, ˆP (µ) P ( )/P (S) is modeled by a geomeric Brownian moion wih volailiy ˆσ() ζ ( ) ζ (S) as in Secion 4 and he above resul coincides wih he formula (4.11) below. Swapions. In his case he modeling of he swap rae differs from he diffusion model of Secion 4. For a swapion wih S and payoff on he LIBOR, where (P ( i ) P ( j ) κp (ν)) + j 1 µ(dy) δ i (dy) δ j (dy) and ν(dy) τ k δ k+1 (dy), wih τ k k+1 k, k i,..., j 1, we obain ϕ (dy) 1 ÎE ˆP i ( i ) { ˆP (µ)>κ} F δ i (dy) ˆP ( i ) (1 + κτ j 1 1 )ÎE ˆP i ( j ) { ˆP (µ)>κ} F δ j (dy) ˆP ( j ) j 1 ˆP i ( k ) κ τ k 1 ÎE 1 { ˆP (µ)>κ} F δ k (dy). (3.11) ˆP ( k ) ki+1 he above consequence of Proposiion 3.1 differs from (4.13) in Secion 4 because of differen modeling assumpions. Moreover, in his case he volailiy of ( ˆP (µ)), may no be deerminisic, cf. (3.14), (3.17) below. Proof of Proposiion 3.1. By Lemma 3.5 below he forward claim price ˆV has he predicable represenaion ˆV ÎEˆξ + Hence by Lemma 2.1 he porfolio priced as ki ϕ s, d ˆP s G,G,. V ϕ, P G,G,, is self-financing and i hedges he claim ξ P S (ν)ĝ(p (µ)/p (ν)), since η by (2.5) and (3.5). he nex lemma, which will be used in he proof of Lemma 3.4 below, shows in paricular ha for fixed U >, ( ˆP (U)) R+ is usually no a geomeric Brownian

11 278 NICOLAS PRIVAUL AND IMOHY ROBIN ENG moion, excep in he case of bond opions wih µ(dy) δ U (dy) and ν(dy) δ (dy), where we ge and Lemma 3.2. For all y R + we have where d P (U) P ( ) P (U) P ( ) (ζ (U) ζ ( ))dŵ, ˆσ() ζ (U) ζ ( ),. (3.12) d ˆP (y) ˆσ ( ˆP, y)dŵ,, y R +, (3.13) ˆσ ( ˆP, y) : ˆP (y) ˆP (z)(ζ (y) ζ (z))ν(dz),, y R +. (3.14) Proof. Defining he discouned bond price P by ( ) P exp r s ds P, R +, (3.15) we have ( ) d ˆP P (y) (y) d P (ν) d P ( ) ( ) (y) P (ν) + P 1 (y)d + d P P 1 (y) d (ν) P (ν) d P (y) P (ν) + P (y) d P ( (ν) P (ν) P (ν) + d P ) 2 (ν) d P (y) P (ν) P (ν) d P (ν) P (ν) d P (y) P (ν) ˆP (y) d P (ν) P (ν) + ˆP (y) ζ (y) ˆP (y) ˆP (s) ˆP (y)ζ (y)dw ˆP (y) ˆP (y) ˆP (y) ˆP (y) ˆP (y) ˆP (s) ˆP (z)ζ (z)ζ (s)ν(dz)ν(ds)d ˆP (z)ζ (z)ν(dz)d ˆP (z)ζ (z)ν(dz)dw ˆP (z)(ζ (y) ζ (z))ζ (s)ν(dz)ν(ds)d ˆP (z)(ζ (y) ζ (z))ν(dz)dw ˆP (z)(ζ (y) ζ (z)) ˆP (s)ζ (s)ν(ds)ν(dz)d ˆP (z)(ζ (y) ζ (z))ν(dz)dŵ,

12 HEDGING IN BOND MARKES BY HE CLARK-OCONE FORMULA 279 by he relaion dŵ dw ˆP (s)ζ (s)ν(ds)d, R +, (3.16) which follows from (2.7). In he case of a swapion wih j 1 µ(dy) δ i (dy) δ j (dy) and ν(dy) τ k δ k+1 (dy), ˆP (µ) becomes he corresponding swap rae and Lemma 3.2 yields d P (µ) P (ν) P (µ) P (ν) which shows ha ( P ( j ) P (µ) (ζ j 1 ( i ) ζ ( j )) + ki ki ) P ( k+1 ) τ k (ζ ( i ) ζ ( k+1 )) dŵ, P (ν) ˆσ() P ( j ) P (µ) (ζ j 1 P ( k+1 ) ( i ) ζ ( j )) + τ k (ζ ( i ) ζ ( k+1 )), (3.17) P (ν) ki, and coincides wih he dynamics of he LIBOR swap rae in Relaion (1.28), page 17 of 13. Lemma 3.3 has been used in he proof of Proposiion 3.1. Lemma 3.3. We have D ˆPu (y) ˆσ ( ˆP u, y), u, y R +, (3.18) where ˆσ ( ˆP u, y) ˆP u (y) ˆP u (z)(ζ (y) ζ (z))ν(dz), (3.19) u, y R +. Proof. he discouned bond price P defined in (3.15) saisfies he relaion ( u P u (y) P (y) exp ζ (y)dw 1 u ) ζ (y) 2 d, y R +, 2 wih Hence we ge D u P (y) P (y)ζ u (y), u, y R +. D ˆPu (y) D Pu (y) P u (ν) D P u (y) P u (y) P u (ν) P u (ν) P ( u (y) ζ (y) P u (ν) D Pu (ν) P u (ν) ζ (z) P u (z) P u (ν) ν(dz) )

13 28 NICOLAS PRIVAUL AND IMOHY ROBIN ENG ˆP u (y) ˆP u (z)(ζ (y) ζ (z))ν(dz) ˆσ ( ˆP u, y), u, y R +. he following lemma has been used in he proof of Lemma 3.5. Lemma 3.4. aking ˆξ ĝ( ˆP (µ)), he process in Lemma 2.2 is given by ˆα ÎE ĝ ( ˆP (µ)) ˆP (y) F ζ (y)µ(dy) + ÎE ˆP (µ)ĝ ( ˆP (µ)) ˆP (y) F ζ (y)ν(dy) ÎE ĝ( ˆP (µ))( ˆP (y) ˆP (y)) F ζ (y)ν(dy) Proof. By (3.16), he process (γ ) R+ in (2.1) is given by γ aking ˆξ ĝ( ˆP (µ)), Lemma 2.2 yields where ˆV ÎEĝ( ˆP (µ)) + ˆα s ÎE D s ĝ( ˆP (µ)) + ĝ( ˆP (µ)) ˆP (s)ζ (s)ν(ds) H, R +. s ˆα s, dŵs H,, D s ˆP u (y)ζ u (y)ν(dy)dŵuf s, (3.2) s. By inegraion wih respec o µ(dy) in (3.18) we ge D ˆP (µ) ζ (y) ˆP (y)µ(dy) ˆP (µ) ζ (y) ˆP (y)ν(dy), which allows us o compue D ĝ( ˆP (µ)) ĝ ( ˆP (µ))d ˆP (µ) in (3.2),. On he oher hand, by Relaions (3.14) and (3.19) in Lemmas 3.2 and Lemma 3.3 we have ˆσ ( ˆP u, y)ζ u (y)ν(dy) ˆP u (y) ˆP u (z)(ζ (y) ζ (z))ν(dz)ζ u (y)ν(dy) ˆP u (y) ζ (y)ˆσ u ( ˆP u, y)ν(dy), ˆP u (z)ζ (y)(ζ u (y) ζ u (z))ν(dz)ν(dy) hence from Relaions (3.13) and (3.18) he second erm in (3.2) can be compued as D ˆP u (y)ζ u (y)ν(dy)dŵu D ˆPu (y)ζ u (y)ν(dy)dŵu

14 HEDGING IN BOND MARKES BY HE CLARK-OCONE FORMULA 281 ζ (y) ˆσ ( ˆP u, y)ζ u (y)ν(dy)dŵu ˆσ u ( ˆP u, y)ζ (y)ν(dy)dŵu ζ (y)ˆσ u ( ˆP u, y)dŵuν(dy) d ˆP u (y)ν(dy) ( ˆP (y) ˆP (y))ζ (y)ν(dy), where ˆσ ( ˆP u, y) is given by (3.19) above. Hence we have D ĝ( ˆP (µ)) + ĝ( ˆP (µ)) ĝ ( ˆP (µ))d ˆP (µ) + ĝ( ˆP (µ)) ĝ ( ˆP (µ)) + D ˆP u (y)ζ (y)ν(dy)dŵu D ˆP u (y)ζ (y)ν(dy)dŵu ζ (y) ˆP (y)µ(dy) ˆP (µ)ĝ ( ˆP (µ)) ĝ( ˆP (µ))( ˆP (y) ˆP (y))ζ (y)ν(dy), ζ (y) ˆP (y)ν(dy) which is square-inegrable by Condiions (3.2) and (3.3). By (3.2), his yields ˆα ÎE ĝ ( ˆP (µ)) ˆP (y) F ζ (y)µ(dy) + ÎE ˆP (µ)ĝ ( ˆP (µ)) ˆP (y) F ζ (y)ν(dy) ÎE ĝ( ˆP (µ))( ˆP (y) ˆP (y)) F ζ (y)ν(dy) he nex lemma has been used in he proof of Proposiion 3.1. Lemma 3.5. he process ϕ in he predicable represenaion ˆV ÎEˆξ + ϕ s, d ˆP s G,G,, of he forward claim price ˆV : ÎEˆξ F, cf. (2.4), is given by ˆP ϕ (dy) ÎE (y) ˆP (y) ĝ ( ˆP (µ)) F µ(dy) +ÎE (ĝ( ˆP (µ)) ˆP (µ)ĝ ( ˆP (µ))) ˆP (y) F ν(dy), ˆP (y),

15 282 NICOLAS PRIVAUL AND IMOHY ROBIN ENG Proof. By Lemma 3.4 above we have, since ˆP (ν) ˆα, dw H + ÎE ĝ ( ˆP (µ)) ˆP (y) + F ζ (y)µ(dy)dw P (y) ν(dy) 1, P (ν) ÎE ˆP (µ)ĝ ( ˆP (µ)) ˆP (y) F ζ (y)ν(dy)dw ÎE ĝ( ˆP (µ))( ˆP (y) ˆP (y)) F ζ (y)ν(dy)dw ÎE ĝ ( ˆP (µ)) ˆP ( ) dp (y) (y) F µ(dy) P (y) r d ÎE ˆP (µ)ĝ ( ˆP (µ)) ˆP ( ) dp (y) (y) F ν(dy) P (y) r d ÎE ĝ( ˆP (µ))( ˆP (y) ˆP ( dp (y) (y)) F ν(dy) P (y) r d + ÎE ĝ ( ˆP (µ)) ˆP (y) F µ(dy) dp (y) P (y) ÎE ˆP (µ)ĝ ( ˆP (µ)) ˆP (y) F + ÎE ĝ( ˆP (µ))( ˆP (y) ˆP (y)) F ÎE ˆP (y)ĝ ( ˆP (µ)) F µ(dy) dp (y) P (y) ν(dy) dp (y) P (y) ν(dy) dp (y) P (y) ÎE ˆP (y)(ĝ( ˆP (µ)) ˆP (µ)ĝ ( ˆP (µ))) F ĝ( ÎE ˆP dp (ν) (µ)) F P (ν) 1 M ϕ, dp G,G ˆV dp (ν) P (ν), and by (2.7) and (3.5) we have ) ν(dy) dp (y) P (y) ˆα, dŵ H ˆα, dw H 1 dm ˆα, dw H M ˆα, dw H 1 ( 1 dm ϕ, dp G,G 1 ) ˆV dm M M M ˆα, dw H 1 ( dm ϕ, d M ˆP G,G + 1 ϕ, M ˆP G,GdM + 1 dm ϕ, d M ˆP G,G 1 ) ˆV dm M ˆα, dw H 1 M dm ( ϕ, d ˆP G,G + 1 ) dm ϕ, d M ˆP G,G

16 HEDGING IN BOND MARKES BY HE CLARK-OCONE FORMULA 283 since 1 M ϕ, dp G,G 1 M ˆV dm 1 M dm ϕ, d ˆP G,G ϕ, d ˆP G,G, (3.21) dp M d ˆP + ˆP dm + dm d ˆP. When he forward price process ( ˆP ) R+ follows he dynamics (1.7), Relaion (3.21) above shows ha we have he relaion which shows ha and recovers (1.8). ˆα, dŵ H ϕ, d ˆP G,G ϕ, ˆσ dŵ G,G, ˆα ˆσ ϕ, 4. Dela Hedging In his secion we consider a G-valued asse price process (X ) R+ and a numeraire (M ) R+, and we assume ha he forward asse price ˆX : ˆX /M, R +, is modeled by he diffusion equaion d ˆX ˆσ ( ˆX )dŵ, (4.1) under he forward measure ÎP defined by (2.2), where x ˆσ (x) L HS (H, G) is a Lipschiz funcion from G ino he space of Hilber-Schmid operaors from H o G, uniformly in R +, Vanilla opions. In his Markovian seing a Vanilla opion wih payoff ξ M S ĝ( ˆX ) is priced a ime as IE e S rsds M S ĝ( ˆX ) F M ÎE ĝ( ˆX ) F M Ĉ(, ˆX ), (4.2) for some measurable funcion Ĉ(, x) on R + G, and Lemma 2.1 has he following corollary. Corollary 4.1. Assume ha he funcion Ĉ(, x) is C2 on R + G, and le η Ĉ(, ˆX ) Ĉ(, ˆX ), ˆX G,G,. hen he porfolio ( Ĉ(, ˆX ), η ), wih value V η M + Ĉ(, ˆX ), X G,G,, is self-financing and hedges he claim ξ M S ĝ( ˆX ). Proof. By Iô s formula, cf. heorem 4.17 of 3, and he maringale propery of ˆV under ÎP, he process (ϕ ), in he predicable represenaion (2.4) is given by ϕ Ĉ(, ˆX ),.

17 284 NICOLAS PRIVAUL AND IMOHY ROBIN ENG and When X P (µ) : µ, P G,G M P (ν) ν, P G,G P (y)µ(dy), P (y)ν(dy), Corollary 4.1 shows ha he porfolio ( ) ϕ (dy) Ĉ x (, ˆX )µ(dy) + Ĉ(, ˆX ) ˆX Ĉ x (, ˆX ) ν(dy), (4.3), where Ĉ(, x) is defined in (4.2), is a self-financing hedging sraegy for he claim ( ) P (µ) ξ P S (ν)ĝ, P (ν) wih M P (ν), R +. When G R and ( ˆX ) R+ is a geomeric Brownian moion wih deerminisic volailiy H-valued funcion (ˆσ()) R+ under he forward measure ÎP, i.e. he exchange call opion wih payoff d ˆX ˆX ˆσ ()dŵ, (4.4) M S ( ˆX κ) +, is priced by he Black-Scholes-Margrabe 1 formula IE e S rsds M S ( ˆX κ) + F X Φ +(, κ, ˆX ) κm Φ (, κ, ˆX ), R +, where Φ +(, κ, x) Φ and ( log(x/κ) v(, ) (4.5) + v(, ) ) ( log(x/κ), Φ 2 (, κ, x) Φ v(, ) ), v(, ) 2 (4.6) v 2 (, ) By Corollary 4.1 and he relaion ( Ĉ log(x/κ) (, x) Φ x v(, ) his yields a self-financing porfolio ˆσ 2 (s)ds. + v(, ) ) Φ 2 +(, κ, x), (Φ +(, κ, ˆX ), κφ (, κ, ˆX )), in (X, M ) ha hedges he claim ξ M S ( ˆX κ) +. In paricular, when he shor rae process (r ) R+ is a deerminisic funcion and M e rsds,, (4.5) is Meron s zero ineres rae version of he Black-Scholes formula, a propery which has been used in 7 for he hedging of swapions.

18 HEDGING IN BOND MARKES BY HE CLARK-OCONE FORMULA 285 In paricular, from (4.5) we have IE e S rsds P S (ν)( ˆX κ) + F P (ν)ĉ(, ˆX ) (4.7) P (µ)φ +(, κ, ˆX ) κp (ν)φ (, κ, ˆX ), and he porfolio ϕ (dy) Φ +(, κ, ˆX )µ(dy) κφ (, κ, ˆX )ν(dy),, (4.8) is self-financing, hedges he claim P S (ν)( ˆX κ) +, and is evenly disribued wih respec o µ(dy) and o ν(dy). As applicaions of (4.3) and (4.7), we consider some examples of dela hedging, in which he asse allocaion is uniform on µ(dy) and ν(dy) wih respec o he bond mauriies y, ). Bond opions. aking S, he bond opion wih payoff belongs o he above framework wih ξ M ĝ(p (U)), U, µ(dy) δ U (dy) and ν(dy) δ (dy), hence M P (ν) P ( ) and when ˆX P (U)/P ( ) is Markov as in (4.1), he self-financing hedging sraegy is given from (4.3) by ( ) ϕ (dy) Ĉ x (, ˆX )δ U (dy) + Ĉ(, ˆX ) ˆX Ĉ x (, ˆX ) δ (dy). (4.9) Furhermore, when ( ˆX ) R+ is a geomeric Brownian moion given by (4.4) under ÎP, he bond call opion wih payoff (P (µ) κp (ν)) + (P (U) κ) + is priced as IE e rsds (P (U) κ) + F P (U)Φ +(, κ, ˆX ) κp ( )Φ (, κ, ˆX ), and he corresponding hedging sraegy is herefore given by ϕ (dy) Φ +(, κ, ˆX )δ U (dy) κφ (, κ, ˆX )δ (dy), (4.1) from (4.8). When he dynamics of (P ) R+ is given by (3.1) where ζ (y) is deerminisic, ˆσ() is given from (3.12) and Lemma 3.2 as ˆσ() ζ (U) ζ ( ), U, and we check ha (4.1) coincides wih he resul (3.6) obained in Secion 3, cf. also page 27 of 11.

19 286 NICOLAS PRIVAUL AND IMOHY ROBIN ENG Caples. Here we ake < S, X P (µ) P ( ), M P (ν) P (S), wih µ(dy) δ (dy) and ν(dy) δ S (dy), and we consider he caple wih payoff (3.8) on he LIBOR rae (3.9), i.e. ξ (S )(L(,, S) κ) + ( ˆX (1 + κ(s ))) +. Assuming ha ˆX P ( )/P (S) is a (drifless) geomeric Brownian moion under ÎP wih ˆσ() a deerminisic funcion, his caple is priced as in (4.7) as (S ) IE e S rsds (L(,, S) κ) + F M ÎE ( ˆX (1 + κ(s ))) + F P ( )Φ +(, 1 + κ(s ), ˆX ) (1 + κ(s ))Φ (, 1 + κ(s ), ˆX )P (S), since P S (ν) 1, and he corresponding hedging sraegy is given as in (4.8) by ϕ (dy) Φ +(, 1+κ(S ), ˆX )δ (dy) (1+κ(S ))Φ (, 1+κ(S ), ˆX )δ S (dy). (4.11) When he dynamics of (P ) R+ is given by (3.1), where ζ (y) in (3.1) is deerminisic, Lemma 3.2 shows ha ˆσ() in (4.4) can be aken as ˆσ() ζ ( ) ζ (S), S, and in his case (4.11) coincides wih Relaion (3.1) above. Hedging sraegies for caps are easily compued by summaion of hedging sraegies for caples. Swapions on LIBOR raes. Consider a enor srucure { i,..., j } and he swapion on he LIBOR rae wih payoff ( ) P ( i ) P ( j ) ξ P (ν)ĝ, (4.12) P (ν) where ˆX P (µ) P (ν) P ( i ) P ( j ),, P (ν) is he swap rae, which is a maringale under ÎP, in which case we have and j 1 µ(dy) δ i (dy) δ j (dy) and ν(dy) τ k δ k+1 (dy) is he annuiy numeraire. j 1 M P (ν) τ k P ( k+1 ) ki ki

20 HEDGING IN BOND MARKES BY HE CLARK-OCONE FORMULA 287 When ( ˆX ) R+ is Markov as in (4.1), he self-financing hedging sraegy of he swapion wih payoff (4.12) is given by (4.3) as ( ) ϕ (dy) Ĉ x (, ˆX )δ i (dy) + Ĉ(, ˆX ) ˆX Ĉ j 1 x (, ˆX ) τ k 1 δ k (dy) ki+1 ( ) + τ j 1 Ĉ(, ˆX ) (1 + τ j 1 ˆX ) Ĉ x (, ˆX ) δ j (dy),. Finally we assume ha he swap rae ˆX : P ( i ) P ( j ) j 1 ki τ kp ( k+1 ),, is modeled according o a drifless geomeric Brownian moion under he forward swap measure ÎP deermined by M j 1 : τ k P ( k+1 ), R +, wih (ˆσ()), a deerminisic funcion. In his case he swapion wih payoff ki (P (µ) κp (ν)) + (P ( i ) P ( j ) κp (ν)) +, priced from (4.7) as IE e rsds (P ( i ) P ( j ) κp (ν)) + F (P ( i ) P ( j ))Φ +(, κ, ˆX ) κp (ν)φ (, κ, ˆX ) has he self-financing hedging sraegy ϕ (dy) Φ +(, κ, ˆX )δ i (dy) (Φ +(, κ, ˆX ) + κτ j 1 Φ (, κ, ˆX ))δ j (dy) κφ (, κ, ˆX ) j 1 ki+1 τ k 1 δ k (dy), (4.13) by (4.8). his recovers he self-financing hedging sraegy of 7. he above hedging sraegy (4.13) also shares he same mauriy daes as (3.11) above, alhough i is saed in a differen model. References 1. Carmona, R. A. and ehranchi, M. R.: Ineres rae models: an infinie dimensional sochasic analysis perspecive. Springer Finance. Springer-Verlag, Berlin, Corcuera, J. M.: Compleeness and hedging in a Lévy bond marke. In A. Kohasu-Higa, N. Privaul, and S.J. Sheu, ediors, Sochasic Analysis wih Financial Applicaions (Hong Kong, 29), volume 65 of Progress in Probabiliy, pages Birkhäuser, Da Prao, G. and Zabczyk, J.: Sochasic equaions in infinie dimensions. Encyclopedia of Mahemaics and is Applicaions. Cambridge Universiy Press, Cambridge, Filipović, D.: Consisency problems for Heah-Jarrow-Moron ineres rae models, volume 176 of Lecure Noes in Mahemaics. Springer-Verlag, Berlin, Geman, H., El Karoui, N., and Roche, J.-C: Changes of numéraire, changes of probabiliy measure and opion pricing. J. Appl. Probab., 32(2): , Huang, C.-F.: Informaion srucures and viable price sysems. Journal of Mahemaical Economics, 14:215 24, 1985.

21 288 NICOLAS PRIVAUL AND IMOHY ROBIN ENG 7. Jamshidian, F.: Soring ou swapions. Risk, 9(3):59 6, Jamshidian, F.: Numeraire invariance and applicaion o opion pricing and hedging. MPRA Paper No. 7167, Karazas, I. and Ocone, D. L.: A generalized Clark represenaion formula wih applicaion o opimal porfolios. Sochasics and Sochasics Repors, 34:187 22, Margrabe, W.: he value of an opion o exchange one asse for anoher. he Journal of Finance, XXXIII(1): , Privaul, N. and eng,.-r.: Risk-neural hedging in bond markes. Risk and Decision Analysis, 3:21 29, Proer, P.: A parial inroducion o financial asse pricing heory. Sochasic Process. Appl., 91(2):169 23, Schoenmakers, J.: Robus LIBOR modelling and pricing of derivaive producs. Chapman & Hall/CRC Financial Mahemaics Series. Chapman & Hall/CRC, Boca Raon, FL, 25. Nicolas Privaul: School of Physical and Mahemaical Sciences, Nanyang echnological Universiy, SPMS-MAS, 21 Nanyang Link, Singapore address: nprivaul@nu.edu.sg URL: hp:// imohy Robin eng: Deparmen of Mahemaics, Aeneo de Manila Universiy, Loyola Heighs, Quezon Ciy, Philippines address: eng@aeneo.edu