Available online at Math. Finance Lett. 2014, 2014:1 ISSN

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1 Available online a hp://scik.org Mah. Finance Le : ISSN CLOSED-FORM SOLUION FOR GENERALIZED VASICEK DYNAMIC ERM SRUCURE MODEL WIH IME-VARYING PARAMEERS AND EXPONENIAL YIELD CURVES YAO ZHENG Hull College of Business Georgia Regens Universiy Augusa GA USA Copyrigh 04 Y. Zheng. his is an open access aricle disribued under he Creaive Commons Aribuion License which permis unresriced use disribuion and reproducion in any medium provided he original work is properly cied. Absrac. In his paper I sudy a generalized Vasicek dynamic erm srucure model wih ime-varying parameers where he shor rae r is unbounded and he ime o mauriy for he exponenial yield curve model is an exponenial funcion of he shor rae. Closed-form soluions are derived for wo cases by funcion analysis echnique wih he classical Vasicek equaion used as a special case. he mehods employed in his paper may have significance in he sudy of oher aspecs of finance. Keywords: generalized Vasicek equaion ime-varying parameer exponenial yield curve closed-form soluion. 00 AMS Subjec Classificaion: 9B8.. Inroducion As wih many opics in modern finance he origin of erm srucure modeling can be raced back o Meron s work. Meron (974) develops he firs so-called shor-rae model for he erm srucure of ineres rae. Many refinemens have followed mos noably Vasicek (977) i.e. he Vasicek model. he Vasicek model is a well known mahemaical model describing he evoluion of ineres raes. I specifically describes ineres rae movemens as driven by a Received January 9 04

2 YAO ZHENG single facor - marke risk. he Vasicek model is commonly used in he valuaion of ineres rae derivaives and has also been adaped for credi markes. Vasicek (977) which uilizes a Gaussian model generally assumes he marke prices of risk o be consan. he parial differenial equaion for he classic Vasicek model can be wrien as P = + σ P rr + ap r rp 0 0 < < < r < + () P( r) = < r < +. () where P is price r is he shor rae P is he firs derivaive of price wih respec o ime σ is he variance Prr is he second derivaive of price wih respec o he shor rae and P r is firs derivaive of price wih respec o he shor rae. he soluion o equaions ()-() follows he form: σ 3 P( r) = exp[ ( ) r a( ) + ( ) ]. (3) 6 Despie he uiliy of his model here are shorcomings. For example i is well known ha financial markes are affeced by numerous facors many of which relae o ime. Babbs and Nowman (999) presen srong evidence in favor of ime-varying marke prices of risk hus making consan marke prices of risk very difficul o esimae. In order o address his ime-relaed issue Gaussian Vasicek ype models can be modified as suggesed by Hull and Whie (990) or more recenly Dai and Singleon (00). Furher examples of Vasicek models wih ime-varying parameers include Mamon (004) which presens hree differen mehods for finding closed-form soluions o bond prices using a Vasicek model. Lemke (008) invesigaes an affine macro-finance erm srucure model for he euro area. Bernaschi acconi and Vergni (008) presen an analysis of he dynamics of he erm srucure of ineres raes based on he sudy of he ime evoluion of he parameers of a variaion of he Nelson Siegel model. Dai and Singleon (000) esimae several affine yield models using he simulaion-based efficien mehod of momens. he auhors show ha a leas in heory efficiency can be achieved if he number of momen condiions goes o infiniy wih he number of daa observaions. Ai-Sahalia and Kimmel (00) presen a way o esimae affine mulifacor erm srucure models using closed-form likelihood expansions. hey develop and

3 CLOSED-FORM SOLUION FOR GENERALIZED VASICEK DYNAMIC ERM SRUCURE 3 implemen a echnique for closed-form maximum likelihood esimaion (MLE) of mulifacor affine yield models and derive closed-form approximaions o likelihoods for nine Dai and Singleon (000) affine models. Curren models uilizing an equilibrium erm srucure for ineres raes are ofen based on a represenaive agen framework wih specific parameric assumpions abou he preferences of he represenaive agen. I is imporan ha when dealing wih a Vasicek bond pricing problem wih ime relaionships ha he shor rae r be unbounded from wha essenially remains an open quesion because he ime o mauriy is a funcion of he shor rae. Some approaches o his may be seen in Sampfli and Goodman (00) Chrisiansen (005) Lemke (008) Realdon (009) Jiang and Yan (009) Laurini and Moura (00) Jang and Yoon (00) Yu and Zivo(0) Wang (996) Honda amaki and Shioham (00) Chen Liu and Cheng (00) Mahdavi (008) and Fouque Papanicolaou and Sircar (000). he purpose of his paper is o presen a heoreical research for generalized Vasicek dynamic erm srucure models wih ime-varying parameers where he shor rae r is unbounded and he ime o mauriy for he exponenial yield curve model is an exponenial funcion of he shor rae. Closed-form soluions are derived for wo cases by funcion analysis echnique wih he classical Vasicek equaion used as he special case.. Soluion for generalized non-homogeneous Vasicek equaion he original Vasicek equaion (Equaion ()) uilizes a aylor expansion echnique. herefore cerain iems are lef ou of he complee equaion/series. I nex assume hese iems as a funcion wih ime-varying parameers r and funcion f (r). From his I obain he generalized non-homogeneous Vasicek equaion wih ime-varying parameers which can be wrien as Equaion (4). P + σ () Prr + ap () r rp= f() r 0 < < < r < + (4) Pr ( ) = < r< +. (5) An emphasis is given in his secion o obain he closed-form exac analyical soluion for Equaions (4)-(5) and for he special case of ( ) r f() r = e g () by using he variable funcion analysis echnique o derive he closed-form exac analyical soluion.

4 4 YAO ZHENG. Closed-form soluion Consider he characerisic and form of a Vasicek equaion wih ime-varying parameers (Equaions (4)-(5)). I assume ha he soluion of Equaions (4)-(5) has he following form: Pr ( ) = b ()exp[ Ar () + B ()] (6) where b () A () B () are funcions which need o be deermined. I is from Equaion (6) I obain P = b ()exp[ Ar () + B ()] + b ()[ Ar () + B ()]exp[ Ar () + B ()] = b ()exp[ Ar () + B ()] + [ Ar () + B ()] P P = b () A ()exp[ Ar () + B ()] = A () p r P r = () A P aking he above expressions ino Equaion (4) I obain b ()exp[ Ar () + B ()] + [ Ar () + B ()] p++ σ A() p + a() A() p rp = f () r i.e. + σ + + =. [ A() r B() + () A() + a () A () r] p b()exp[ Ar () B ()] f() r (7) i.e. If A () = I obain A ( ) = + C. By choosingc = his yields Furhermore le Inegraing Equaion (8) from o I obain If B ( ) = 0 his yields A( ) =. (8) B () + σ () A () + a() A() = 0 (9) B ( ) = a( ) A( ) σ ( ) A ( ). (0) B ( ) = B ( ) + a ( )( ) d + σ ( )( ) d.

5 CLOSED-FORM SOLUION FOR GENERALIZED VASICEK DYNAMIC ERM SRUCURE 5 and σ B ( ) = a ( )( ) d + ( )( ) d () b ()exp[ Ar () + B ()] = f() r () For ( ) r f() r = e g () he special case le B () b () = e g () B( ) () ( ) ( ) b b e g d =. Subsiuing b () A () and B () ino Equaion (6) yields b ( ) = i.e. [ a( s)( s ) + s ( s)( s ) ] ds b () = e g( ) d. (3) I follows from (6) ha he soluion of Equaions (4)-(5) can be wrien as [ a( s)( s ) + s ( s)( s ) ] ds Pr ( ) = [ e g( ) d] + + exp[( r ) a( )( d ) s ( )( ) d].. Specific examples Uilizing he closed-form soluion derived in secion. I give he soluion for wo specific cases boh of which indicae he advanage of expressing he soluion as Equaion (3) of his paper. Example. Le a ( ) = a (consan) σ () = σ (consan) and g () = 0. By doing so I immediaely obain he soluion o he classical Vasicek Equaions ()-() σ 3 P( r ) = exp[ ( ) r a( ) + ( ) ]. 6 wih he soluion being idenical o Equaion (3). Example. Le a () = σ () = and () g =. From his I obain i.e. 4 [ s( s ) s ( s ) ] ds Pr ( ) = [ e d ] 4 exp[( r ) ( d ) ( ) d] + +

6 6 YAO ZHENG [ a( s)( s ) + s ( s)( s ) ] ds Pr ( ) = [ e g( ) d] + + exp[( r ) a( )( d ) s ( )( ) d] = r [ exp( ) ( ( 7 ) 7 ( 5 )] exp[( ) ( )]. 3. Soluion for generalized Vasicek equaion wih exponenial mauriy value model In his secion I invesigae he generalized Vasicek equaion wih he exponenial mauriy value model. he exponenial mauriy value can be expressed as a funcion of shor rae r. he generalized Vasicek equaion can be wrien as: 3. Closed-form soluion P + σ () Prr + ap () r rp= 0 0 < < < r < + (4) λr Pr ( ) = e < r< +. (5) Consider he characerisic and form of he Vasicek equaion wih ime-varying parameers (4)-(5) I assume ha he soluion o Equaions (4)-(5) has he following form: Pr ( ) = crb ( ) ()exp[ Ar () + B ()] (6) where cr ( ) b ( ) A ( ) B ( ) are funcions which need o be deermined. Following from Equaion (6) I obain P = crb ( ) ()exp[ Ar () + B ()] + crb ( ) ()[ Ar () + B ()]exp[ Ar () + B ()] = crb ( ) ()exp[ Ar () + B ()] + [ Ar () + B ()] P P = c ( rb ) ()exp[ Ar () + B ()] + crb ( ) () A ()exp[ Ar () + B ()] r = c ( rb ) ()exp[ Ar () + B ()] + A () p P = ( ) ()exp[ () ()] ( ) () ()exp[ () ()] c rb Ar + B + c rb A Ar + B r + A (){ c ( rb ) ()exp[ Ar () + B ()] + A () P} = c rb Ar+ B + c r Ab Ar+ B + A P ( ) ()exp[ () ()] ( ) () ()exp[ () ()] ().

7 CLOSED-FORM SOLUION FOR GENERALIZED VASICEK DYNAMIC ERM SRUCURE 7 aking he expressions above ino Equaion (4) I obain i.e. crb Ar+ B + Ar + B p+ σ c rb Ar+ B + σ () c ( rb ) () A ()exp[ Ar () + B ()] + σ () A() p + ac () ( rb ) ()exp[ Ar () + B ()] + a () A () p rp= 0 ( ) ()exp[ () ()] [ () ()] () ( ) ()exp[ () ()] A r+ B + σ A + a A r p + crb + σ c r + σ c ra + ac r b Ar+ B = [ () () () () () () ] {( ) () [ () ( ) () ( ) () () ( )]()}exp[ () ()] 0. If A () = I obain A ( ) = + C. By choosing C = his yields A( ) =. Furhermore le Since exp[ Ar ( ) + B ( )] 0 his yields In erms of Equaion (8) (7) B () + σ () A () + a() A() = 0 (8) crb ( ) () + [ σ () c ( r) + σ () c ( ra ) () + ac () ( r)]() b = 0. (9) Inegraing Equaion (8) from o I obain B ( ) = a( ) A( ) σ ( ) A ( ). (0) B ( ) = B ( ) + a ( )( ) d + σ ( )( ) d. Subsiuing he expressions of A( ) B( ) ino Equaion (6) I obain P ( ) r ()( ) B ( ) = crb e () Le cr () = e λr and B ( ) = 0. In view of condiion (5) i yields b ( ) = and σ B ( ) = a ( )( ) d + ( )( ) d. () Nex I deermine he funcion of b (). I follows from Equaion (9) ha

8 8 YAO ZHENG σ () c ( r) [ σ () A () a ()] c ( r) b + + () =. b () cr ( ) One sees immediaely ha for cr () = e λr b () = { σ () λ [ σ () A () + a ()] λ}. b () I follows hen ha i.e. ln b ( ) = { σ ( ) l [ σ ( ) A( ) + a( )] l} d+ ln c b A a d c ln ( ) = { σ ( ) l [ σ ( ) ( ) + ( )] l} + ln { σ ( ) λ [ σ ( ) A( ) + a( )] λ} d b () = ce. In view of c = b ( ) = his yields { σ ( ) λ [ σ ( ) A( ) + a( )] λ} d b () = e. herefore I obain he closed-form soluion o a generalized Vasicek dynamic erm srucure models wih ime-varying parameers where he ime o mauriy is a funcion of shor rae r. { σ ( ) λ [ σ ( )( ) + a( )] λ} d λr ( ) = exp[( ) + ( )( ) + σ ( )( ) ] i.e Pr e e r a d d Pr r a d ( ) = exp ( λ+ ) { [ λ ( ) λ ( ) ] σ ( ) + [ λ ( )] ( )}. (3) 3. Specific examples he resuls in secion 3 are nex used o give he soluion for wo specific examples which again indicaes he advanage of expressing he soluion as Equaion (3) of his paper. Example. Le a ( ) = a (consan) σ () = σ (consan) and λ = 0. From his I immediaely obain he soluion o he classical Vasicek Equaions ()-() σ 3 P( r ) = exp[ ( ) r a( ) + ( ) ]. 6

9 CLOSED-FORM SOLUION FOR GENERALIZED VASICEK DYNAMIC ERM SRUCURE 9 wih he soluion being idenical o Equaion (3). Example. Le a () = σ = and () λ = ( Pr ( ) = e r ) from which I obain 4 { { } Pr ( ) = exp ( + ) r [ ( ) ( ) ] + [ ( )] } d = + r exp[( ) ( ( ) 50 ( ) ( ) ( ))]. 4. Conclusion In his paper I invesigae generalized Vasicek dynamic erm srucure models wih ime-varying parameers where he shor rae r is unbounded and he ime o mauriy for he exponenial yield curve model is an exponenial funcion of he shor rae. Closed form soluions are derived for wo cases by funcion analysis echnique wih he classical Vasicek equaions as a special case. he mahemaical echnique employed in his paper may have significance in sudying oher problems relaed o financial engineering. Conflic of Ineress he auhor declares ha here is no conflic of ineress. REFERENCES [] M. Mahdavi A comparison of inernaional shor-erm raes under no arbirage condiion Global Finance Journal 8(3) (008) [] M. Bernaschi E. acconi and D. Vergni A parameric sudy of he erm srucure dynamics Physica A: Saisical Mechanics and is Applicaions 387(5) (008) [3] B. G. Jang and J. H. Yoon Analyic valuaion formulas for range noes and an affine erm srucure model wih jump risks Journal of Banking & Finance 34(9) (00) [4] W. Lemke An affine macro-finance erm srucure model for he euro area he Norh American Journal of Economics and Finance 9() (008) [5] O. Vasicek An equilibrium characerizaion of he erm srucure Journal of Financial Economics 5() (977)

10 0 YAO ZHENG [6] M. Polei Laurini and M. Moura Consrained smoohing B-splines for he erm srucure of ineres raes Insurance: Mahemaics and Economics 46() (00) [7] Y. Aï-Sahalia and R. L. Kimmel Esimaing affine mulifacor erm srucure models using closed-form likelihood expansions Journal of Financial Economics 98() (00) [8] Q. Dai and K. J. Singleon Expecaion puzzles ime-varying risk premia and dynamic models of he erm srucure (No. w867) (00) Naional Bureau of Economic Research. [9] M. Realdon Exended Black erm srucure models Inernaional Review of Financial Analysis 8(5) (009) [0] W. C. Yu and E. Zivo Forecasing he erm srucures of reasury and corporae yields using dynamic Nelson-Siegel models Inernaional Journal of Forecasing 7() (0) []. Honda K. amaki and. Shiohama Higher order asympoic bond price valuaion for ineres raes wih non-gaussian dependen innovaions Finance Research Leers 7() (00) [] G. De Rossi Kalman filering of consisen forward rae curves: a ool o esimae and model dynamically he erm srucure Journal of Empirical Finance () (004) [3] S. H. Babbs and K. Ben Nowman Kalman filering of generalized Vasicek erm srucure models Journal of Financial and Quaniaive Analysis 34() (999). [4] G. Jiang and S. Yan Linear-quadraic erm srucure models oward he undersanding of jumps in ineres raes Journal of Banking & Finance 33(3) (009) [5] C. Chrisiansen Mulivariae erm srucure models wih level and heeroskedasiciy effecs Journal of Banking & Finance 9(5) (005) [6] R. C. Meron On he pricing of corporae deb: he risk srucure of ineres raes he Journal of Finance 9() (974) [7] J. Hull and A. Whie Pricing ineres-rae-derivaive securiies Review of financial sudies 3(4) (990) [8] R. R. Chen B. Liu and X. Cheng Pricing he erm srucure of inflaion risk premia: heory and evidence from IPS Journal of Empirical Finance 7(4) (00) [9] Q. Dai and K. J. Singleon Specificaion analysis of affine erm srucure models he Journal of Finance 55(5) (000) [0] J. P. Fouque G. Papanicolaou and K. R. Sircar Sochasic Volailiy Correcion o Black-Scholes Risk 3() (000) 89-9.

11 CLOSED-FORM SOLUION FOR GENERALIZED VASICEK DYNAMIC ERM SRUCURE [] V. Goodman and J. G. Sampfli he mahemaics of finance: modeling and hedging (Vol. 7) American Mahemaical Sociey (00). [] J. Wang he erm srucure of ineres raes in a pure exchange economy wih heerogeneous invesors Journal of Financial Economics 4() (996) [3] R. S. Mamon hree ways o solve for bond prices in he Vasicek model Advances in Decision Sciences 8() (004) -4.