The Dothan pricing model revisited

Transcription

1 The Dohan pricing model revisied Caroline Pinoux Laboraoire de Mahémaiques Universié de Poiiers Télépor - BP Chasseneuil Cedex France Nicolas Privaul Deparmen of Mahemaics Ciy Universiy of Hong Kong Ta Chee Avenue Kowloon Tong Hong Kong Augus 4, 17 Absrac We compue zero coupon bond prices in he Dohan model by solving he associaed PDE using inegral represenaions of hea kernels and Harman- Wason disribuions. We obain several inegral formulas for he price P, T a ime > of a bond wih mauriy T > ha complee hose of he original paper [7], which are shown no o always saisfy he boundary condiion P T, T = 1. Key words: Ineres rae models, Dohan model, PDE, hea kernel, opion pricing, Harman-Wason disribuion, Bessel funcions. MSC : 91B4, 6J65, 6H3, 81S4, 33C1, 35A. 1 Inroducion In he Dohan [7] model, he shor erm ineres rae process r IR+ according o a geomeric Brownian moion is modeled dr = λr d + r db, 1.1 where he volailiy > and he drif λ IR are consan parameers and B IR+ is a sandard Brownian moion. In he Dohan model, he shor erm ineres rae r remains always posiive, while he proporional volailiy erm r accouns for he caroline.pinoux@mah.univ-poiiers.fr nprivaul@ciyu.edu.hk 1

2 sensiiviy of he volailiy of ineres rae changes o he level of he rae r. On he oher hand, he Dohan model is he only lognormal shor rae model ha allows for an analyical formula for he zero coupon bond price P, T = IE [e T r sds ] F, T, cf. [7], and i is commonly referenced in he bond pricing lieraure, cf. e.g. [5]. Oher lognormal ineres rae models include he BGM [4] model. For convenience of noaion we le p = 1 λ/ and wrie he soluion of 1.1 as r = r exp B p /, IR +, where p/ idenifies o he marke price of risk, cf. e.g. [13], Secion 4.. By he Markov propery of r IR+, he bond price P, T is a funcion F τ, r of r and of he ime o mauriy τ = T : P, T = F τ, r = IE [e T r sds ] r, T. 1. In addiion, by a sandard arbirage argumen, F τ, r saisfies he PDE F τ τ, r = 1 r F F τ, r + λr τ, r rf τ, r r r F, r = 1, r IR The zero coupon bond price given in [7], page 64, cf. also [5] page 63, is F τ, r = x p/ π e τp /8 + xp/ Γp K p x, sin x sinh a u sinuae u τ/8 cosh πu Γ p + iu duda 1.4 wih x = r/, where Γz = funcion and + z 1 e d, z C, Rz >, is he Gamma K w x = e x cosh z coshwzdz = 1 + e x cosh z+wz dz, x IR, 1.5

3 is he modified Bessel funcion of he second kind of order w C, cf. page 376 of [1] or page 181 of [14]. A proof of 1.4 is given in [7] in case p = 1, while he argumen given herein is no complee in he general case p IR. We will show in paricular ha 1.4 does no saisfy he correc iniial condiion F, r = 1, r >, for all values of he parameer p, alhough i saisfies F τ, = 1 for all p and τ. In Secion we obain a bond pricing formula based on he join probabiliy densiy of [15]. As an example, Figure 1 provides a numerical comparison beween he resul of Corollary.3 below and Relaion 1.4 as funcions of τ > wih r =.6, =.5 and p =.8441, in which he bond price given by 1.4 appears o be an underesimae ha can become negaive and does no mach he erminal condiion P T, T = 1. Figure 1: Comparison beween Relaions.9 sraigh line and 1.4 doed line. As can be expeced from 1., racable expressions for he bond price P, T are more difficul o obain for large values of p IR. We also derive an analyical formula for he pricing of bond opions in Proposiion.4. Noe ha relaed echniques have been applied o he pricing of Asian opions, cf. e.g. [], [6], [8], and references herein. In Secion 3 we presen oher expressions for P, T, which are closer o he original formula 1.4, by solving he PDE 1.3 using a hea kernel represenaion and Gamma funcions, and in Corollary 3.3 we obain anoher probabilisic inerpreaion 3

4 for P, T using hyperbolic cosine random variables. Probabilisic approach In [15], Proposiion, he join probabiliy densiy of τ e Bs ps/ ds, B τ, τ >, has been compued in he case =, cf. also [1]. Applying Brownian rescaling, his densiy can be wrien for an arbirary variance parameer as τ P e Bs ps/ ds du, B τ pτ/ dy = τ/8 e py/ p exp 1 + ey 4e y/ θ u u, τ du 4 u dy, u >, y IR, τ >, where θv, = veπ / π3.1 e ξ / e v cosh ξ sinhξ sin πξ/ dξ, v, >.. The following resul is obained by applying.1 o he compuaion of he condiional expecaion 1.. Proposiion.1 The zero-coupon bond price P, T = F T, r is given for all p IR by F τ, r = e p τ/8 Proof. We have e ur exp 1 + z 4z θ u u, τ du 4 u dz..3 zp+1 F T, r = P, T [ = IE exp T [ T = IE exp r [ = IE exp r [ = IE exp r T T F ] r s ds F ] e Bs B ps / ds ] e Bs B ps / ds ] e B +s B ps/ ds r=r r=r.4 4

5 [ = IE exp r = T ] e Bs ps/ ds r=r τ e ru P e Bs ps/ ds du, B τ dy, wih τ := T, and he conclusion follows from he change of variable z = e y/, using.1. The above formula involves a riple inegral which can be difficul o evaluae in pracice. Nex we presen alernaive represenaion formulas under some inegrabiliy condiions ha involve only double inegrals and special funcions as in [7], and are more appropriae for numerical compuaion. Corollary. The zero-coupon bond price P, T = F T, r is given for all p IR by F τ, r = 8 r π 3 τ e p τ/8+π / τ e ξ / τ sinhξ sin 4πξ/ τ K 1 8r 1 + z cosh ξ + z / 1 + z cosh ξ + z Proof. Relaion.3 can be rewrien as F τ, r = e p τ/8 exp = eπ / τ π3 τ/ 4zr v v 1 + z z θ v, τ dv 4 v.5 dξ dz z p. dz,.6 zp+1 afer he change of variable v = 4z/ u. Now, applying he Fubini heorem we have exp 4zr v v 1 + z θ v, τ dv z 4 v e ξ / 4πξ τ sin sinhξ exp 4rz 1 + z cosh ξ + z v τ v z since he inegrand in.7 belongs o L 1 IR + as i is bounded by Nex we have exp ξ, v 1 ξ / τ π3 τ/ eπ sinhξ exp v 1 + z 4zr. z v 4rz 1 + z cosh ξ + z v v z dv = 4rz.7 dvdξ, 1 + z cosh ξ + z du exp u r u u 5

6 = 4z r where we used he ideniy 8r K z cosh ξ + z /, 1 + z cosh ξ + z K ν z = zν ν+1 exp u z du, ν IR,.8 4u uν+1 cf. [14] page 183, provided Rz >. The nex corollary provides an alernaive expression for he bond price using a double inegral, which is however valid only for p < 1. Corollary.3 The zero-coupon bond price P, T = F T, r is given for all p < 1 by F τ, r = e p τ/8 v + 8r/ p/ θ v, τ 4 Proof. From Relaions.6 and.8 we ge F τ, r = e p τ/8 θ v, τ exp 4 = e p τ/8 θ v, τ v + 8r 4 where, leing C := eπ / π3 dv K p v + 8r/..9 vp+1 v v z z + 4r v p/ K p e ξ / sinhξdξ = 1 π 3 e/+π / >, we have applied he Fubini heorem as θ v, τ exp v 4 z 1 + z 4rz dv v v C τ/4 exp v z 1 + z 4rz v = 4 r C τ/4 <, z K 1 8r 1 + z / z + 1 v + 8r dz dv z p+1 v dv v p+1, / e x dx <, dz z p+1 dv dz z p+1 dz z p+1 for all p < 1, since from [1] page 378 we have K 1 y π y y e y. 6

7 Figure provides a numerical comparison beween he resul.9 of Corollary.3 and 1.4 as funcions of r > wih T = 1.8, =.6, and p =.48. Here he bond price given by 1.4 may also become negaive and does no mach he erminal condiion F, r = 1. In addiion i is numerically less sable han.9, given ha he same numerical algorihm has been used for he discreizaion of inegrals. We close his secion wih an analyical formula for he price of a bond opion, obained from he probabiliy densiy funcion.1 and he same argumen as in Corollary.. Proposiion.4 The price of a bond opion wih payoff funcion hx is given by [ T ] IE exp r s ds hf S T, r T F = 8 r π 3 τ eπ /τ p τ/8 e ξ / τ sinhξ sin 4πξ/ τ z + eξ z + e ξ K 1 z p 1 hf S T, r e pτ/ z 8r z + eξ z + e ξ dξdz. Figure : Comparison beween Relaions.9 sraigh line and 1.4 doed line. 3 PDE approach In his secion we derive anoher inegral represenaion for he soluion of he bond pricing PDE for p,, which is closer o Dohan s original formula 1.4. The 7

8 Dohan PDE 1.3 can be ransformed ino he simpler equaion U s, y = H + p Us, y s U, y = e py, y IR, 3.1 under he change of variable 3/ p r F τ, r := U τ 4, log 3/ r, where H := 1 y + 1 ey is a Hamilonian operaor wih Surm-Liouville poenial, cf. [9], hence he soluion Us, y of 3.1 is given by Us, y = e sp / q s y, xe px dx, 3. where he kernel q s x, y of e sh s IR+ can be expressed as q x, y = ue u/ sinhπuk π iu e y K iu e x du, 3.3 >, x, y IR, cf. [3], page 115, and [9] page 8. As a consequence he zero-coupon bond price P, T = F T, r is given for all p IR by 3/ p r τ 3/ F τ, r = U 4, log r = p r p/ exp p τ e py r q p 8 τ/4 log, y dy. Using he inegral represenaion 3.3 of he kernel q x, y we obain he following resul which clearly does no coincide wih Dohan s formula 1.4 when p <, due o he absence of he Bessel funcion erm xp/ Γp K p x in 3.5 below. Proposiion 3.1 The zero-coupon bond price P, T = F T, r is given for all p IR by F τ, r = p+1 r p π p e p τ/8 r >, τ >. e py u sinhπue u τ/8 K iu 8r/K iu e y dudy, 3.4 8

9 From a compuaional poin of view he above formula acually involves a riple inegral of a Bessel funcion, which can be simplified o a double inegral of a Gamma funcion under some addiional condiions on p. Corollary 3. The zero-coupon bond price P, T = F T, r is given for all p < by F τ, r = x p/ π e p τ/8 sin x sinh a r >, τ >, wih x = r/. Proof. ue u τ/8 cosh πu Γ p + iu sinuaduda, 3.5 Firs, from 3.4, afer he change of variable z = e y, we noice ha p/ 1+3p/ F τ, r = r π p e p τ/8 and we noe ha 1 z p+1 C ue u τ/8 sinhπuk iu zk iu 8r/ du dz z p+1, ue u τ/8 dudz sinhπuk iu zk iu 8r/ ue u τ/8 sinhπudu sup w IR + z p 1 Kiw z dz <, where C > is a consan. From his bound we can apply he Fubini heorem, hence from Proposiion 3.1 and he relaion we ge Γ p iw 1 = p F τ, r = p+1 r p π p e p τ/8 = r p π p e p τ/8 = xp/ π e p τ/8 K iu e y e py dydu, ω IR, u sinhπuk iu 8r/e u τ/8 K iu e y e py dydu u sinhπuk iu 8r/e u τ/8 Γ p + iu du, 3.6 ue u τ/8 cosh πu sin x sinh a sinuada Γ which yields 3.5 for all p <, where x = r/ and we used he inegral represenaion K iµ z = sinz sinh sinµd, z IR +, µ IR, ha can be 1 + sinh πµ/ derived from he relaions on pages of [14]. 9 p + iu du,

10 Finally we derive a probabilisic represenaion of he bond price ha can be useful for Mone Carlo esimaion, using he hyperbolic cosine disribuion wih characerisic funcion u cosh u p, p <, cf. [1]. Corollary 3.3 For all p < we have F τ, r = Γ p 3p/ r p/ π p [ e τp /8 IE Z p e τzp /8 sinhπz p K izp ] 8r/, r >, τ >, where Z p is an hyperbolic cosine random variable wih parameer p. Proof. We use Relaion 3.6 above and he fac ha he densiy of Z p is given by u 1 π + e iuy cosh y p dy = p Γ p πγ p iu, u IR, cf. [1], page 3. Noe added in proof The resul of Corollary 3. can be exended o all p IR using specral expansions for he Fokker-Planck equaion, cf. [11] and he references herein. References [1] M. Abramowiz and I.A. Segun. Handbook of mahemaical funcions wih formulas, graphs, and mahemaical ables. Dover Publicaions, New York, [] P. Barrieu, A. Rouaul, and M. Yor. A sudy of he Harman-Wason disribuion moivaed by numerical problems relaed o he pricing of Asian opions. J. Appl. Probab., 414: , 4. [3] A. N. Borodin and P. Salminen. Handbook of Brownian moion Facs and formulae. Probabiliy and is Applicaions. Birkhäuser Verlag, Basel, [4] A. Brace, D. Gaarek, and M. Musiela. The marke model of ineres rae dynamics. Mah. Finance, 7:17 155, [5] D. Brigo and F. Mercurio. Ineres rae models heory and pracice. Springer Finance. Springer- Verlag, Berlin, second ediion, 6. [6] P. Carr and M. Schröder. Bessel processes, he inegral of geomeric Brownian moion, and Asian opions. Theory Probab. Appl., 483:4 45, 4. [7] L.U. Dohan. On he erm srucure of ineres raes. Jour. of Fin. Ec., 6:59 69, [8] D. Dufresne. Laguerre series for Asian and oher opions. Mah. Finance, 14:47 48,. [9] C. Grosche and F. Seiner. Handbook of Feynman pah inegrals, volume 145 of Springer Tracs in Modern Physics. Springer-Verlag, Berlin,

11 [1] H. Masumoo and M. Yor. Exponenial funcionals of Brownian moion. I. Probabiliy laws a fixed ime. Probab. Surv., : elecronic, 5. [11] C. Pinoux and N. Privaul. A direc soluion o he Fokker-Planck equaion for exponenial Brownian funcionals. Analysis and Applicaions, 83:87 34, 1. [1] J. Piman and M. Yor. Infiniely divisible laws associaed wih hyperbolic funcions. Canad. J. Mah., 55:9 33, 3. [13] N. Privaul. An Elemenary Inroducion o Sochasic Ineres Rae Modeling. Advanced Series on Saisical Science & Applied Probabiliy, 1. World Scienific Publishing Co., Singapore, 8. [14] G. N. Wason. A reaise on he heory of Bessel funcions. Cambridge Universiy Press, Cambridge, Reprin of he second 1944 ediion. [15] M. Yor. On some exponenial funcionals of Brownian moion. Adv. in Appl. Probab., 43:59 531,