Three Ways to Solve for Bond Prices in the Vasicek Model
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1 JOURNAL OF APPLIED MATHEMATICS AND DECISION SCIENCES, 81, 1 14 Copyrigh c 24, Lawrence Erlbaum Associaes, Inc. Three Ways o Solve for Bond Prices in he Vasicek Model ROGEMAR S. MAMON Deparmen of Saisics, Universiy of Briish Columbia Vancouver, BC, Canada V6T 1Z2 roge@sa.ubc.ca Absrac. Three approaches in obaining he closed-form soluion of he Vasicek bond pricing problem are discussed in his exposiion. A derivaion based solely on he disribuion of he shor rae process is reviewed. Solving he bond price parial differenial equaion PDE is anoher mehod. In his paper, his PDE is derived via a maringale approach and he bond price is deermined by inegraing ordinary differenial equaions. The bond pricing problem is furher considered wihin he Heah-Jarrow-Moron HJM framework in which he analyic soluion follows direcly from he shor rae dynamics under he forward measure. Keywords: Bond pricing, Vasicek model, Maringales, HJM mehodology, Forward measure. 1. Inroducion Vasicek s pioneering work 1977 is he firs accoun of a bond pricing model ha incorporaes sochasic ineres rae. The shor rae dynamics is modeled as a diffusion process wih consan parameers. When he bond price is based on his assumpion, i has he feaure ha on a given dae, he raio of expeced excess reurn per uni of volailiy he marke price of risk is he same, regardless of bond s mauriy. Vasicek s model is a special version of Ornsein-Uhlenbeck O-U process, wih consan volailiy. This implies ha he shor rae is boh Gaussian and Markovian. The model also exhibis mean-reversion and is herefore able o capure moneary auhoriy s behavior of seing arge raes. Furhermore, hisorical experience of ineres raes jusifies he O-U specificaion. Given he pedagogical value of he Vasicek model in sochasic erm srucure modeling, he purpose of his paper is o presen alernaive derivaion of he bond price soluion. From he bond price he enire yield curve can be consruced a any given ime. Thus, in urn, he erm srucure dynamics is characerized by he evoluion of he shor rae. Requess for reprins should be sen o Rogemar S. Mamon, Deparmen of Saisics, Universiy of Briish Columbia, Vancouver, BC, Canada V6T 1Z2.
2 2 R. S. MAMON Vasicek model s racabiliy propery in bond pricing and he model s ineresing sochasic characerisics make his classical model quie popular. In his paper a review of shor rae s sochasic properies relevan o he derivaion of he closed-form soluion of he bond price wihin he Vasicek framework is presened. These properies become he basis for he firs mehod examined in secion 2. Under his echnique, he bond price is derived from he implicaions of he ineres rae s probabiliy disribuion. The developmen of he heory under his se-up follows from he ouline of Lamberon and Lapeyre The orginal derivaion of he explici formula for he bond price was based on solving he PDE ha mus be saisfied by he bond price. This is done by consrucing a locally riskless porfolio and using he no-arbirage argumens. Duffie and Kan 1996 provide a furher characerizaion of his PDE. They prove ha, if some Ricai equaions have soluions o he required mauriy, he bond price has an exponenial affine form. Vasicek s model belongs o his exponenial affine class because he specificaion of is drif and volailiy gives rise o a solvable se of equaions in accordance wih he Duffie-Kan descripions. The second approach discussed in secion 3 relies on he soluion of he bond price PDE. However, unlike he radiional approach, his paper presens a maringale-oriened derivaion of his PDE. This is moivaed by he equivalence of he no-arbirage pricing echnique and he risk-neural valuaion which is a maringale-based mehod. Recenly, Ellio and Van der Hoek 21 offer a new mehod of solving he problem sudied by Duffie and Kan. In heir paper, i is shown ha, when he shor rae process is given by Gaussian dynamics or square roo processes, he bond price is an exponenial affine funcion. Their echnique deermines he bond price by inegraing linear ODE and Ricai equaions are no needed. A similar idea is applied here o provide a soluion o he bond pricing problem in he Vasicek model. Secion 4 presens a hird alernaive ha considers he Heah-Jarrow- Moron HJM pricing paradigm. The equivalence beween he forward rae and he condiional expecaion of he shor rae under he forward measure is discussed. Elaboraing on he work of Geman, El Karoui and Roche 1995 using he bond price as a numéraire, he shor rae s dynamics is obained under he forward measure. Consequenly, he Vasicek forward rae dynamics is explicily deermined and herefore he analyic bond price follows immediaely from he HJM bond pricing formula.
3 THREE WAYS TO SOLVE FOR BOND PRICES 3 2. Bond Price Implied by he Shor Rae Disribuion In modeling he uncerainy of ineres raes, assume ha here is an underlying probabiliy space Ω, F, P equipped wih a sandard filraion {F }. Under he risk-neural measure P, he shor rae dynamics is given by dr = ab r d + σdw 1 where a, b and σ are all posiive consans. I can be verified using Iô s formula ha r = e a r + abe au du + σ e au dw u is a soluion o he sochasic differenial equaion SDE in 1. furher ha r = e r a + be a 1 + σe au dw u = µ + σ e au dw u, Noe where µ is a deerminisic funcion. Clearly, Er = µ. Observe furher ha r is a Gaussian random variable. This follows from n1 he definiion of he sochasic inegral erm, which is lim i= π eaui W ui+1 W ui and he incremen W ui+1 W ui N, u i+1 u i. In general, if δ is deerminisic i.e., a funcion only of, δudw u is Gaussian. While he expecaion follows immediaely from he soluion for r given above, Er can be deermined wihou necessarily solving explicily he SDE. Consider he inegral form of 1. Tha is, Hence, r = r + µ := Er = r + ab r u du + σdw u. ab Er u du. 2 From 2, d d µ = ab µ, which is a linear ordinary differenial equaion ODE. Consequenly, using he inegraing facor e a Er = e a r + be a 1 = µ. 3
4 4 R. S. MAMON In his model, b is some kind of level r is rying o aain. We call his he mean-revering level. Similarly, define σ 2 : = V arr = E σe a e au dw u 2 = σ 2 e 2a E = σ 2 1 e 2a 2a e 2au du by Iô s isomery. 4 Therefore, r Nµ, σ 2 wih mean and variance given in 3 and 4, respecively. Since normal random variables can become negaive wih posiive probabiliy, his is considered o be he weakness of he Vasicek model. Neverheless, he simpliciy and racabiliy of he model validae is discussion. Using he risk-neural valuaion framework, he price of a zero-coupon bond wih mauriy T a ime is T B, T = E exp r u du F. Wrie Xu = r u b. 5 Here, Xu is he soluion of he Ornsein-Uhlenbeck equaion dx = ax + σdw 6 wih X = r b. Applying Iô s lemma, he Xu process is given by Xu = e au X + u σe as dw s. 7 Clearly, Xu is a Gaussian process wih coninuous sample pahs. If Xu is Gaussian hen Xudu is also Gaussian. Using 7, we obain EXu = Xe au. Thus, E Xudu = X a 1 ea. 8
5 THREE WAYS TO SOLVE FOR BOND PRICES 5 Similarly, u CovX, Xu = σ 2 e au+ E e as dw s e as dw s Consequenly, V ar = σ 2 e au+ u Xudu = Cov Xudu, e 2as ds = σ2 2a eau+ e 2au 1. = E Xudu E Xudu = = Xsds EXu EXuXs EXsduds CovXu, Xsduds = Xsds E Xsds σ 2 2a eau+s e 2au s 1duds = σ2 2a 3 2a 3 + 4ea e 2a. 9 From 5, we have E r u du = E Therefore, ogeher wih equaion 8 E r u du Xu + bdu. = r b 1 e at bt. 1 a Furhermore, V ar r u du T = V ar Xudu = σ2 2a 3 2aT 3 + 4eaT e 2aT 11 by he resul from 9. From he Iô inegral represenaion of r, we also noe ha he defining process for he shor rae is also Markov. For proof, see Karazas and Shreve, p. 355.
6 6 R. S. MAMON Thus, B, T = E exp r u du F = E exp r u du r. We wrie B, T, r := E exp r u du r = E exp Tha is, r u is a funcion of r. Combining 1 and 11, he bond price is given by + 12 V ar where B, T, r = exp E r u r du r u r du r u r du = exp r b 1 e at bt a + σ2 4a 3 2aT 3 + 4eaT e 2aT 12 1 e at 1 e at = exp r + b T a a σ2 1 e at 2a 2 + σ2 σ2 1 2e at 2aT + e T a 2a2 4a a 2 = exp A, T r + ba, T bt σ2 2a 2 A, T + σ2 σ2 T A, T 2 = expa, T r 2a2 + D, T, 13 4a 1 eat A, T = and 14 a D, T = b σ2 2a 2 A, T T σ2 A, T a Since for all, he yield log B,T,r T obained from 13 is affine in r, equaion 13 is called an affine erm srucure model or an exponenial affine bond price..
7 THREE WAYS TO SOLVE FOR BOND PRICES 7 3. Soluion via Bond Price PDE Under his approach, he derivaion is based on he fac ha he r u process is Markov. In oher words, o deermine how r u evolves from we need know only he value of r, u. Thus, T B, T, r = E exp r u r du r and So r u = e au r + be au 1 + σ Wih r as a parameer, which is deerminisic. Also, B, T, r r r u r r = e au. u e av dw v. r u r T du = e au du = 1 r a 1 eat, T r u r T = E du exp r u r du r = 1 a 1 eat E = A, T B, T, r, where A, T is given as in 14. Thus, B r = AB. So, exp B, T, r = C, T expa, T r, for some funcion C independen of r. Consider exp r u r du T r u r du B, T, r = E exp r u du F.
8 8 R. S. MAMON Noe ha his is a P maringale by he ower propery. By Iô s lemma, we obain exp r u r du B, T, r = B, T, r exp exp u u exp u r u exp u r v dv Bu, T, r u du r v dv u Bu, T, r udu r v dv Bu, T, r u ab r u du + σdw u r u 2 r v dv ru 2 Bu, T, r u σ 2 du. Since his is a maringale, all he du erms mus sum o zero. So, r B, T, r + + σ2 2 B, T, r + B, T, r ab r r 2 r 2 B, T, r =. 16 Equaion 16 is he PDE for he bond price in he Vasicek model. Moreover, his is a backward parabolic equaion wih BT, T, r = 1 for every r. So far we know B, T, r = C, T expa, T r. Therefore, we ge he following parial derivaives. B = C expa, T r C A r expa, T r B r = ACexpA, T r 2 B r 2 = A 2 CexpA, T r So, subsiuing o he PDE in 16 we have r CexpAr + C expar C A r expar ACexpAr ab r + σ2 2 A2 CexpAr =.
9 THREE WAYS TO SOLVE FOR BOND PRICES 9 Therefore, r C + C C A r ACab r + σ2 2 A2 C =. Now, B, T, = C, T and by puing r = we ge C abac + σ2 2 A2 C =. Noing again ha we are solving a backward ODE wih CT, T = 1, we ge C, T = exp ab T 1 e at u du + σ2 a 2a 2 1 e at u 2 du = exp bt + ba 1 eat + σ2 T 2a2 + σ2 4a 3 1 e2at σ2 a 3 1 eat. Wrie D, T := log C, T. We see ha his reconciles wih he second o he las erms of equaion 12 and hence wih he expression of equaion 15. Under his approach, we have B, T, r = expa, T r + D, T where A, T is given by Bond Pricing by HJM Mehodology Following he erminology and noaion of Heah, Jarrow and Moron 1992, his pricing paradigm is based on he concep of forward rae. The insananeous forward rae a ime for dae T > is defined by f, T = log B, =T >. 17 This refers o he rae of ineres ha mus be paid beween and T. I is known a ime and herefore F measurable. Solving he differenial equaion in 17, yields B, T = exp f, udu. 18
10 1 R. S. MAMON The shor rae a ime, r, is he insananeous rae a ime, i.e., r = f, for every, T. From equaion 18, i is clear ha once f, T is compleely deermined he bond price immediaely follows. The dynamics of he forward rae and ha of he shor rae are relaed via he forward measure. Invoking he insighs of Geman, El Karoui and Roche 1995, he forward measure P T is defined on F T by seing dp T dp FT Consider he Radon-Nikodým process = Λ T := exp r udu. B, T, r Λ := EΛ T F := exp r udub, T,, T. B, T For any F T measurable random variable X we have E T X F = Λ 1 EX Λ T F = E Xexp T r u du B, T F. 19 Now, he bond price in erms of he shor rae is given by T B, T = E exp r u du F. Differeniaing wih respec o T, we ge B, T T = E r T exp r u du T F = E T r T F B, T, 2 where he las equaliy follows from 19 wih X = r T. The bond price in erms of he forward rae is given in equaion 18. Thus, differeniaing B, T wih respec o T, we obain B, T T = B, T f, T. 21 Comparing 2 and 21, in erms of he shor rae model, he forward rae is given by f, T = E T r T F 22 where E T denoes he expecaion under P T.
11 THREE WAYS TO SOLVE FOR BOND PRICES 11 Invoking he change of probabiliy measures and numéraire echnique, under he forward measure P T, he sochasic dynamics for r is given by dr = ab A, T σ 2 ar d + σdw T, 23 where W T is he P T Brownian moion defined by dw T = dw + σa, T d, and A, T is he funcion defined in equaion 14. See Appendix for he proof of 23. By Iô s lemma, for T, he soluion o 23 is given by r T = r e at + b σ2 a 2 1 e bt + σ2 2a 2 eat e 2aT + σ e at u dwu T. Thus, E u r u F = r e au + b σ2 2a 2 1 e au So, + σ2 2a 2 eau e 2au. E u r u F du = r e au T a b σ2 2a 2 + σ2 2a 3 + b σ2 2a 2 T 1 a eau T e au T σ2 4a 3 = r a 1 eat + b σ2 2a 2 b σ2 1 e at 2a 2 a σ2 e 2au T T + σ2 2a 3 1 eat 4a 3 1 e2at = r A, T + b σ2 2a 3 T + A, T 1 e +σ 2 at 2. a
12 12 R. S. MAMON Therefore, B, T, r = exp f, udu = expr A, T + D, T = exp E u r u F du and he A, T and D, T values are in agreemen wih ha of equaions 14 and 15, respecively. 5. Conclusion The pedagogical value of he Vasicek model is well-known in sochasic ineres rae modeling. This paper conribues o he developmen of he available mahemaical echniques in obaining he closed-form soluion of he bond price under he Vasicek framework. A discussion for each of he hree differen mehods was provided. The firs derivaion considers he disribuional properies of he shor rae proces r. The simple Gaussian srucure of r leads o a closed-form soluion of he bond price. The bond price backward PDE is also derived using a maringale-oriened mehodology. This PDE ogeher wih he Vasicek dynamics is he basis of he second mehod which inegraes ordinary differenial equaions o ge he bond price. Turning o he HJM pricing framework, he hird approach employs he dynamics of he forward rae o fully describe he bond price process. The forward rae is linked o he shor rae via he forward measure. When he shor rae dynamics is deermined under he forward measure, he HJM bond price is obained and his reconciles wih he prices compued from he oher wo approaches. Acknowledgmens The auhor wishes o hank an anonymous referee for many helpful suggesions. References 1. D. Duffie and R. Kan. A Yield-Facor Model of Ineres Raes. Mahemaical Finance, 64: , R.J. Ellio and J. van der Hoek. Sochasic flows and he forward measure. Finance and Sochasics, 5: , 21.
13 THREE WAYS TO SOLVE FOR BOND PRICES H. Geman, N. El Karoui and J. Roche. Changes of Numéraire, Changes of Probabiliy Measure and Opion Pricing. Journal of Applied Probabiliy, 32: , D. Heah, R. Jarrow and A. Moron. Bond Pricing and he Term Srucure of Ineres Raes: A New Mehodology. Economerica, 6: 7715, I. Karazas and S. Shreve. Brownian Moion and Sochasic Calculus. Springer Verlag, Berlin-Heidelberg-New York, D. Lamberon and B. Lapeyre. Inroducion o Sochasic Calculus Applied o Finance. Chapman & Hall, London, O. Vasicek. An Equilibrium characerizaion of he Term Srucure. Journal of Financial Economics, 5: , Appendix : Proof of Resul in Equaion 23 Le P be an equivalen maringale measure EMM for he numéraire H and Q an EMM for he numéraire J. Then for any V T L 2 Ω, F T, P and V T L 2 Ω, F T, Q V := E P V T H H T F = E Q V T J J T F. Assume ha P and Q are equivalen and denoe he Radon-Nikodým derivaive of Q wih respec o P by Γ. We hen have E Q J V T J T F = E P H V T Γ H T F. In paricular, Γ = H T H J J T for < T. Suppose ha he process under some measure P associaed wih numéraire H is given by dx = mx, d + σx, dw for some funcions mx, and σx,. We are ineresed on he process followed by X under anoher measure Q wih numéraire J. Consider Γ,T = H T H J J T. From Girsanov s heorem, if W Q is a Wiener process under Q, W Q = W P θ udu where dγ,t = Γ,T θ T dwt P and θ can be deermined. Moreover, condiional upon F, Γ,T is a process in T. Le H and J have dynamics under P given by dh = m H d + σ H dw P and dj = m J d + σ J dw P. Using hese dynamics and noing ha Γ is a maringale under P, i can be verified ha dγ T σj = σ H dwt P. Γ T J T H T
14 14 R. S. MAMON So, θ = σ J J σ H H. Applied o our curren siuaion, suppose P is he EMM under he bank accoun numéraire and Q is he forward measure wih bond as he associaed numéraire. For s < < T Γ := Γ s, = J Hs = B, T J s H T Bs, T exp s r u du. Under measure P, db,t B,T = r d + σ B dw for some funcion σ B. I is a sraighforward calculaion o show ha he process Γ = Γ s,, condiional upon F s, saifies dγ Γ = db, T B, T r d = σ B dw. This implies ha W Q = W P σ Budu. Hence, if under P we have he dynamics dx = mx, d + σx, dw P hen he Qprocess for X is dx = mx, + σ B σx, d + σx, dw Q. Equaion 23 follows from his resul wih X = r, Q = P T, σx, = σ, σ B = A, T σ and mx, = ab r.
15 Mahemaical Problems in Engineering Special Issue on Time-Dependen Billiards Call for Papers This subjec has been exensively sudied in he pas years for one-, wo-, and hree-dimensional space. Addiionally, such dynamical sysems can exhibi a very imporan and sill unexplained phenomenon, called as he Fermi acceleraion phenomenon. Basically, he phenomenon of Fermi acceleraion FA is a process in which a classical paricle can acquire unbounded energy from collisions wih a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanaion of he origin of he large energies of he cosmic paricles. His original model was hen modified and considered under differen approaches and using many versions. Moreover, applicaions of FA have been of a large broad ineres in many differen fields of science including plasma physics, asrophysics, aomic physics, opics, and ime-dependen billiard problems and hey are useful for conrolling chaos in Engineering and dynamical sysems exhibiing chaos boh conservaive and dissipaive chaos. We inend o publish in his special issue papers reporing research on ime-dependen billiards. The opic includes boh conservaive and dissipaive dynamics. Papers discussing dynamical properies, saisical and mahemaical resuls, sabiliy invesigaion of he phase space srucure, he phenomenon of Fermi acceleraion, condiions for having suppression of Fermi acceleraion, and compuaional and numerical mehods for exploring hese srucures and applicaions are welcome. To be accepable for publicaion in he special issue of Mahemaical Problems in Engineering, papers mus make significan, original, and correc conribuions o one or more of he opics above menioned. Mahemaical papers regarding he opics above are also welcome. Auhors should follow he Mahemaical Problems in Engineering manuscrip forma described a hp:// Prospecive auhors should submi an elecronic copy of heir complee manuscrip hrough he journal Manuscrip Tracking Sysem a hp:// ms.hindawi.com/ according o he following imeable: Gues Ediors Edson Denis Leonel, Deparameno de Esaísica, Maemáica Aplicada e Compuação, Insiuo de Geociências e Ciências Exaas, Universidade Esadual Paulisa, Avenida 24A, 1515 Bela Visa, Rio Claro, SP, Brazil ; edleonel@rc.unesp.br Alexander Loskuov, Physics Faculy, Moscow Sae Universiy, Vorob evy Gory, Moscow , Russia; loskuov@chaos.phys.msu.ru Manuscrip Due December 1, 28 Firs Round of Reviews March 1, 29 Publicaion Dae June 1, 29 Hindawi Publishing Corporaion hp://
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