Research Article Binary Tree Pricing to Convertible Bonds with Credit Risk under Stochastic Interest Rates

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1 Absrac and Applied Analysis Volume 03, Aricle ID 70467, 8 pages hp://dx.doi.org/0.55/03/70467 Research Aricle Binary Tree Pricing o Converible Bonds wih Credi Risk under Sochasic Ineres Raes Jianbo Huang, Jian Liu, and Yulei Rao School of Business, Cenral Souh Universiy, Changsha, Hunan 40083, China School of Economics & Managemen, Changsha Universiy of Science & Technology, Changsha 40004, China Correspondence should be addressed o Yulei Rao; yuleirao@sina.com Received 8 January 03; Acceped March 03 Academic Edior: Chuangxia Huang Copyrigh 03 Jianbo Huang e al. This 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. The converible bonds usually have muliple addiional provisions ha make heir pricing problem more difficul han sraigh bonds and opions. This paper uses he binary ree mehod o model he finance marke. As he underlying sock prices and he ineres raes are imporan o he converible bonds, we describe heir dynamic processes by differen binary ree. Moreover, we consider he influence of he credi risks on he converible bonds ha is described by he defaul rae and he recovery rae; hen he wo-facor binary ree model involving he credi risk is esablished. On he basis of he heoreical analysis, we make numerical simulaion and ge he pricing resuls when he sock prices are CRR model and he ineres raes follow he consan volailiy and he ime-varying volailiy, respecively. This model can be exended o oher financial derivaive insrumens.. Inroducion Converible bonds wih he characerisic of bonds and sock are a complex financial derivaive. They provide he righ o holders ha hey give up he fuure dividend o obain some sock wih specified quaniy. The pricing of converible bonds is more difficul han sraigh bonds and opions; hemainreasonishainoonlyhasvalueofbonds,bu also involves all kinds of embedded opion value brough by provisions of conversion, call, pu, and so on. Wha is more, he embedded opions in mos imes are American opions. So, generally speaking, he pricing of converible bonds canno ge closed-form soluion; in mos condiions, numerical mehod was adoped, for example, binary ree mehod, Mone Carlo mehod, finie difference mehod, and so on. As for Mone Carlo mehod, firsly i uses differen sochasic differenial equaions o describe he pricing facor models in he marke for simulaion, hen i makes pricing based on he characerisic of converible bonds, for example, he boundary condiions acquired by all kinds of provisions, o make pricing for conversion bonds (Ammann e al. [], Guzhva e al. [], Kimura and Shinohara [3], Yang e al. [4], and Siddiqi [5]). Bu because he supposed parameers of sochasic differenial equaions are exogenous, his mehod no necessarily makes a beer fiing for he exising marke condiions. The binary ree mehod can solve he above problems. The binary ree mehod is firsly pu forward by Cox e al. [6], Cox-Ross-Rubinsein (CRR) binomial opion pricing model. Afer ha, many researchers revised and popularized i. Cheung and Nelken [7] firsly apply binary ree o converible bonds pricing and obain he pricing soluion of a wofacor model which is based on sock prices and ineres raes. Carayannopoulos and Kalimipalli [8] apply he rigeminal ree pricing model o converible bonds pricing research wih single facor. Hung and Wang [9] also apply binary ree model o converible bonds pricing which embodies defaul risk and considers he influences of sock prices and ineres raes. Chambers and Lu [0] furher considered he correlaion of sock prices and ineres raes and expanded he model of Hung and Wang. Binary ree model has been widely used in he pricing of coningen claims such as sock opions, currency opions, sock index opions, and fuure opions. Xu [] proposes a rinomial laice model o price converible bonds and asse swaps wih marke risk and counerpary risk.

2 Absrac and Applied Analysis Ineres rae is a very imporan facor in he financial marke;allsecuriypricesandyieldsarerelaedoi.the ineres raes model has equilibrium model, no-arbirage model,andsoon.inheequilibriummodel,ineresraes are generally described by some sochasic process, which is mean-reversion, o make ineres raes show he rend of convergence o a long-erm average wih he passing of ime, including Vasicek model, Rendleman-Barer model, and CIR model. The parameers of hese models should be esimaed wih hisory daa, by selecing parameers purposely, bu generally his fiing is no accurae and even reasonable fiing formula canno be found. No-arbirage models make iniial erm srucure as model inpu and consruc a binary ree such as CRR model for ineres rae process, so ha he ermsrucurecanfiherealiybeerandmoreconcise.the relaively wide applicaion no-arbirage models include Holee model, Hull-Whie model, Black-Derman-Toy model, and Heah-Jarrow-Moron model. Differen from he ineres rae model of Hung and Wang and Chambers and Lu, his paper uses consan volailiy and ime-varying volailiy binary ree modelodescribeshorineresraeswhicharemoreinuiive and convenien. Thispapermakespricingresearchofconveriblebonds wih he call and pu provisions and uses binary ree mehod for he modeling of sae variables in he financial marke. Asheduraionofconveriblebondsisrelaivelylonger han sraigh bonds, heir prices are subjec o he impac of ineres raes. Moreover, as one of he corporae bonds, converible bonds may have credi risk. So his paper uses differen binary rees o model he process of sock prices and ineres raes, and considering he impac of sock dividends and credi risk o converible bonds, i adops defaul rae and recovery rae o describe credi risk and ge wo-facor binary ree model involving credi risk; on his basis, wih example simulaion o ge he converible bonds, pricing resuls under he condiion of sock price obey CRR model, consan volailiy ineres rae binary ree model, and imevarying volailiy ineres rae binary ree model.. Marke Model.. Ineres Raes Binary Tree... Consan Volailiy Binary Tree Model. Richken [] deduces he coninuous form of shor-erm r() in Ho- Lee model [3] meeing he following sochasic differenial equaions: dr () =μ() d + σ () dz (), > 0, () where μ() is he drif, σ() is he insananeous volailiy, boh can be he funcionofime,and z() is Brownian moion. Gran and Vora [4] ge he discree form of () as follows: Δr () =μ() Δ + σ () Δz (), 0. () Make f(j) o be he forward ineres rae in he inerval [j, j+ ].Thenge μ (0) Δ = f () r(0) + Δ σ (r ()), μ () Δ = f () f() + Δ σ ( r (j)) Δ σ (r ()), μ ( ) Δ = f () f( ) + Δ σ ( r (j)) n=0 where σ ( Δ σ ( r (j)) + Δ σ ( n= n r (j)), 3, μ (n) Δ = f (+) r(0) + δ (n),, n=0 n= δ (n) = Δ + σ ( r (j)) Δ σ ( r (j)), r (j)) = σ ( = ( j+)σ j Δz j ) ( j+) σ j Δ., Suppose ha volailiy is consan; ha is, σ() = σ c, > 0,andhen σ ( r (j)) = σ ( =σ c ( j+)σ c Δz j ) ( j+) Δ. And ge he consan volailiy ineres raes binary ree as shown in Figure.... Time Varying Volailiy Binary Tree Model. Jarrow and Turnbull [5] supposed ha he volailiy of shor-erm is changeable in differen inervals, bu is consan in he same ime inerval. Le Δ =, and hen he discree form of ineres raes can mee r () =r(0) + μ (j) + σ ( ) (3) (4) (5) Δz (j). (6)

3 Absrac and Applied Analysis 3 r 0 + μ j Δ + 3σ c Δ r 0 r 0 +μ 0 Δ + σ c Δ r 0 +μ 0 Δ σ c Δ r 0 + μ j Δ + σ c Δ r 0 + μ j Δ r 0 + μ j Δ σ c Δ r 0 + μ j Δ + σ c Δ r 0 + μ j Δ σ c Δ r 0 + μ j Δ 3σ c Δ Figure : 4-period consan volailiy ineres raes binary ree. The variance formula of he sum of shor-erm ineres rae is σ ( r j ) = (j+) σ j + k=j+ (j+) σ j σ k. (7) And hen ge ime-varying ineres raes ree as shown in Figure... Sock Price Binary Tree. Suppose ha he curren momen is 0 and he expiraion dae of converible bonds is T. According o inerval Δ, wedivideheperiod[0, T] o L subinervals: [ i, i+ ], 0 i L, 0 =0, L =T, T=LΔ. In each inerval [ i, i+ ], here are wo possible saes in he marke, up or down. The change of every marke sae is independen. U means he up sae and D means he down sae. The sock prices will have wo saes; p means he probabiliyofmarkeup,andhenheprobabiliyofmarke down is p.ifhecurrensockpriceiss, hen he sock priceoflaerperiodmayhavewopossibiliies:s u, S d,and S u =S u, S d =S d; u, d separaely mean he magniude ofupanddown.ifheiniialpriceofsockisknown,hen hesockpricereecanbedeerminedbyhegivenmodel parameers p, u,andd. Model parameers p, u, andd will direcly impac he resuls of binary ree; he selecion of hem should follow no-arbirage principle. Generally speaking, here are wo selecions: CRR model [4], equal-probabiliy binomial model (Roman [6], Hull [7]). This paper adops CRR model o describe pricing process of sock. CRR model selecs parameers as follows: u=e σ s Δ, d = u =e σ s Δ, p= [ + μ s σ s Δ]. Especially, if he acual financial marke is changed o risk neural marke. Then he expeced profi μ s of sock will (8) change o risk-free ineres rae r, buhevolailiyσ S is he same.theprobabiliyofpriceupinhismodelisp=(e rδ d)/(u d); among hem,r is risk-free ineres rae. Under his condiion, he pricing resul is no arbirage..3. Credi Risk. Consider converible bonds wih credi risk. We adop he mehod of Jarrow and Turnbull [8] o model he credi risk of converible bonds. Suppose ha he probabiliy of defaul risk in ime inerval [ i, i ] is λ i and he rae of recovery is ξ i when defaul. If here are serials differen deadline risk-free zero-coupon bonds in he financial marke and he prices are {P(), P(), P(3),..., P(n)}, heserialsof differen deadline risk company zero-coupon bonds and he prices are {D(), D(), D(3),..., D(n)}. Wecangeheriskfree ineres erm srucure and risk ineres erm srucure from hem. If he recovery rae ξ i is already known, hen he rae of risk λ i,innumberi period of bonds, can be acquired. The deail analysis process is as follows. If he risk-free ineres rae of one-year period is r 0 and risk ineres rae is r,hen e r = [ ( λ ) +ξ λ ] e r 0, and ge λ = er 0 r. ξ (9) If he risk ineres rae of wo-year period is r,hen e r = {[( λ )+ξ λ ]e r u ( λ ) +[( λ )+ξ λ ]e r d ( ) ( λ ) +ξ λ }e r 0. (0) When λ, λ can be go by he above formula, wih he same mehod, we can ge he risk ineres rae {λ i,i } of each period.

4 4 Absrac and Applied Analysis r 0 r 0 +μ 0 +σ 0 r 0 +μ 0 σ 0 r 0 + μ j +σ r 0 + μ j r 0 + μ j +3σ r 0 + μ j +σ r 0 + μ j σ r 0 + μ j σ r 0 + μ j 3σ Figure : 4-period ime-varying ineres rae ree. 3. Sock and Ineres Rae Binary Tree Model wih Credi Risk For converible bonds wih credi risk, suppose ha he underlying sock price and risk-free ineres rae process are random, and he underlying sock price process is described bycrrmodel,wherehesockmagniudeofupanddown is u=e σ S Δ, d=e σ S Δ respecively. Suppose ha he sock price is 0 when defaul, and hen he possible price of sock is 0, S u, S d. In risk-neural world, he expeced yield rae is risk-free ineres rae r, and he sock coninuous dividends yield is q, hen he expeced yield rae is r q;soomeeheno-arbirage condiion, here is Se (r q)δ =p( λ) Su + ( p) ( λ) Sd+0 λ. () Then ge p = (e (r q)δ /(( λ) d))/(u d). p is he up probabiliy of sock wih credi risk. As he risk-free ineres rae of all periods is random, suppose ha he risk-free ineres rae of number i period is r i,hevolailiyofsock is consan σ S, and dividends rae is q i, so he parameers of sock price in all periods can be generally presened as u i =e σ S Δ, d i =e σ S Δ, p i = e(r i q )Δ i /( λ i ) d. u d () Suppose risk-free ineres raes are sochasic and described by binary ree model, hen he sock ree and ineres rae ree involving credi risk are combined as shown in Figure 3. In his paper, we suppose ha he correlaion coefficien of ineres rae and sock price is 0. Afer obaining he process of sock prices and risk-free ineres raes, he value of converible bonds can be go by backward inducion. We divide he value of converible bonds ino wo pars; one is he value of equiy go by convering o sockorexerciseembeddedopions;heoherisbondsvalue λ r, S ξ p ( λ ) ( )( p )( λ ) r u,s u r d,s u ( )p ( λ ) ( p )( λ ) r u,s d r d,s d Figure 3: 4-period wo-facor binary ree wih credi risk added. ha is he presen value of bonds when repaying capial and ineres and he presen value of he residual value. Suppose ha he defaul probabiliy of converible bonds in he inerval [ k, k ] is λ k,recoveryvalueisξ k,andhen he holding value a k ime is EV k,givenby EV k = (he expeced equiy value a k+ -ime + he expeced bonds value a k+ -ime) e r k Δ. So he value of converible value a ime- k is V k = max [min (heholdingvaluea k -ime, call value), conversion value, pu value] = max [min (EV k,v call k ),V con k,v pu k ]. (3) (4)

5 Absrac and Applied Analysis 5 4. Numerical Examples We ake a four-period binary ree model as an example o expound he converible bonds pricing process wih call provision and pu provision in he above models and compare he resuls under he consan volailiy ineres rae model and he ime-varying volailiy ineres rae model. 4.. Process of Ineres Rae and Sock. The iniial parameers ofheconveriblebondareallhesameoheconsan volailiy ineres rae model and he ime-varying volailiy ineres rae model. Suppose ha ime inerval Δ =, he up probabiliy of ineres raes is = /,andhe 4- year period yields of risk-free zero-coupon bonds are 6.45%, 6.366%, 6.837%, and 6.953%, respecively; he volailiy of shor-erm ineres raes is.5%. So he oher binary ree parameers of consan volailiy ineres rae can be go as shown in Table. In he same way, he oher binary ree parameers of ime-varying volailiy ineres rae can be go as shown in Table. Then we ge he wo ineres rae binary rees. The process of underlying sock prices uses CRR model o describe. The seleced parameers 8 are as follows: S 0 =5, σ S = 0.85, Δ =, T=4,andq = So all he parameers under consan volailiy ineres rae are u =.03, d = 0.83, p r0 = , p r()u = 0.684, p r()d = 0.593, p r()uu = 0.757, p r()ud = , p r()dd = , p r(3)uuu = 0.96, p r(3)uud = 0.870, p r(3)udd = 0.709, p r(3)ddd = (5) In he same way, he parameers under ime-varying volailiy ineres rae model are u =.03, d = 0.83, p r0 = , p r()u = 0.686, p r()d = , p r()uu = , p r()ud = , p r()dd = , p r(3)uuu = , p r(3)uud = 0.807, p r(3)udd = , p r(3)ddd = Thenwegehefour-periodsockpricesbinaryree. (6) 4.. Defaul Raes. We ake he corporae bonds as reference risk bonds; suppose ha he 4-year period yields of corporae zero-coupon bonds are 7.645%, 8.55%, 8.557%, and9.8%,respecively,andherecoveryraeofconverible bondsisconsanξ = 45%, and one-year risk-free ineres rae r 0 = 6.45%. Ineres rae binary ree indicaes ha he branch poin of number n period is n. If he ineres rae of number i branch poin in number n period is r(n ) ω, ω is he ineres raes sae from sar o curren, and hen he derived ineres rae branchpoinofnumbern+period is r(n) ωu, r(n) ωd.as= / =, o mee he no-arbirage principle, parameers {λ,λ,λ 3,λ 4 } mee he following four equaions: e r +r 0 =( λ )+ξ λ, e r +r 0 =( λ )( λ +ξλ ) (e r() u +e r() d )+ξλ, e 3r 3 +r 0 = ( λ )( λ )( λ 3 +ξλ 3 ) [e r() u (e r() uu +e r() ud ) +e r() d (e r() ud +e r() dd )] +( λ )ξλ (e r() u +e r() d )+ξλ, e 4r 4 +r 0 = 3 ( λ )( λ )( λ 3 )( λ 4 +ξλ 4 ) {e r() u [e r() uu (e r(3) uuu +e r(3) uud ) +e r() ud (e r(3) uud +e r(3) udd )] +e r() d [e r() ud (e r(3) uud +e r(3) udd ) + ( λ )( λ )ξλ 3 +e r() dd (e r(3) udd +e r(3) ddd )]} [e r() u (e r() uu +e r() ud ) +e r() d (e r() ud +e r() dd )] +( λ )ξλ (e r() u +e r() d )+ξλ. (7) By he above equaions and he consan volailiy ineres rae binary ree, he defaul raes of bonds in every period are shown in Table 3. Similarly, he defaul raes of corporae bonds in every period under ime-varying volailiy ineres rae binary ree are shown in Table PriceProcessofConveribleBonds. The converible bond conains call and pu provisions, he duraion is T=4, he face value go in mauriy dae is 00, conversion rae is 3, callable price is V call = 06, and puable price is V pu =80. We suppose ha he invesors can exercise he puable righ afer one year. Now we ake he converible bond under ime-varying volailiy ineres rae binary ree o explain is pricing process. Take four poins A, B, C, and D in pricing ree of Figure 4 ino consideraion; among hem, C, D are a he end of period, 4, B are a he end of period 3, and A is a

6 6 Absrac and Applied Analysis Deadline year(s) Price of bonds (yuan) P() Volailiy of shor-erm ineres rae σ() Table : Consan volailiy ineres rae parameers. Annual profi rae of bonds y() -year long-erm ineres rae f() Variance σ ( r(j)) Sum of Dela δ(j) Dela δ() Drif iem μ() Sum of drif iems μ(j) Expecaion E Q 0 [r()] Deadline year(s) Price of bonds (yuan) P() Volailiy of shor-erm ineres rae σ() Table : Time-varying volailiy ineres rae parameers. Annual profi rae of bonds y() -year long-erm ineres rae f() Variance σ ( r(j)) Sum of Dela δ(j) Dela δ() Drif iem μ() Sum of drif iems μ(j) Expecaion E Q 0 [r()] Table 3: Defaul rae of corporae bonds under consan volailiy ineres rae model. Time period 0- (λ ) - (λ ) -3 (λ 3 ) 3-4 (λ 4 ) Defaul rae In he same way, we can calculae he value of oher hree branch poins E, F, and G a he end of period 3 ha are [5.5,.96], [5.5, 3.03], and[79.00, 3.].Soheequiy value a A poin is Table 4: Defaul rae of corporae bonds under ime-varying volailiy ineres rae model. Time period 0- (λ ) - (λ ) -3 (λ 3 ) 3-4 (λ 4 ) Defaul rae ( ) e = The bonds value a A poin is (9) he end of period. A poin C, as he converible value is ha is larger han he face value 00, he value of converiblebondis08.57,whilehebondsvalueis0,so wrie i as [08.57, 0]; he firs number means he equiy value and he second number means he bonds value. In he same way, we can ge he converible value of D poin ha can be wrien as [0, 00]. And he up probabiliy of B poin is p r(3)uud ( λ 4 ) = ( ) = Ahesame way, he down probabiliy is Then he equiy value of B poin is e = 74..Thebondsvalue of B poin is ( )e = 9.8. Then he holding value EV B is And he converible valueabpoinis90.4,sohevalueofconveriblebonds in B poin is V B = max [min (EV B,V call ),V con B,V pu] = max [min (93.93, 08), 90.4, 80] = 93.93, wrien as [74., 9.8]. (8) ( ) e = (0) Then he holding value EV A is And he converible valueaapoinis08.57,sohevalueofconveriblebondsin Apoinis V A = max [min (EV A,V call ),V con A,V pu] = max [min (04.96, 08), 08.57, 80] = 08.57, () wrien as [08.57, 0]. The oher branch poin in he pricing binary ree of converible bonds can be go in he same way. A las, under ime-varying volailiy ineres rae binary ree model, we can ge he price of converible bond which conains credi risk ha is 79.3 a he ime of =0.Similarly, under consan volailiy ineres rae binary ree model, we can ge he price of converible bond which conains credi risk ha is 78.5 a he ime of =0;hisislesshanheformer.

7 Absrac and Applied Analysis 7 λ 4 E p r(3)uuu ( λ 4 ) r(3) uuu = S(3) uuu = ( p r(3)uuu )( λ 4 ) S(4) uuuu = 5.39 S(4) uuud = 36.9 λ 3 A r() uu = S() uu = 36.9 λ 4 F p r(3)uud ( λ 4 ) r(3) uud = S(3) uuu = ( p r(3)uud )( λ 4 ) λ 4 G p r(3)uuu ( λ 4 ) r(3) uuu = S(3) uud = ( p r(3)uuu )( λ 4 ) S(4) uuuu = 5.39 S(4) uuud = 36.9 S(4) uuud = 36.9 S(4) uudd =5 λ 4 B C p r(3)uud ( λ 4 ) S(4) uuud = 36.9 r(3) uud = S(3) D uud = ( p r(3)uud )( λ 4 ) S(4) uudd =5 Figure 4: Consrucing pricing ree under ime-varying volailiy ineres rae model. 5. Conclusions Binary ree mehod is a classical pricing mehod, by consrucing he binary ree of sae variable o describe he possible pahs of sae variable in he duraion of coningen claims and hen o make pricing research. Binary ree mehod can effecively solve he pah-dependen opions pricing, inuiive and easy o operae. As he embedded opions in he converible bonds are all American opions, binary ree mehod becomes one of he main pricing mehods of converible bonds. Ineres rae is he main facor which impacs he price of converible bonds; he descripion of is binary ree model is he main problem of converible bonds pricing. This paper adops consan volailiy and imevarying volailiy binary ree model o describe ineres raes and furher consider he impac of sock dividends and credi risk o he price of converible bonds, adop defaul rae and recovery rae o describe he credi risk, and ge he wofacor binary ree model wih credi risk added. Based on his, we make a numerical example and ge he converible bonds pricing resul under he sock prices obeying CRR model and he consan and ime-varying volailiy ineres rae binary ree model. The model can be popularized o he pricing of converible bonds wih more complex provisions and oher financial derivaives such as bond opions, caasrophe bonds, and morgage-backed securiy. Acknowledgmens This work was suppored in par by he Naural Science Foundaion of China (no , no. 7003, and no ) and he Minisry of Educaion of Humaniies and Social Science Projec of China (no. YJC6308). References [] M. Ammann, A. Kind, and C. Wilde, Simulaion-based pricing of converible bonds, Empirical Finance, vol. 5, no.,pp.30 33,008. [] V.S.Guzhva,K.Belsova,andV.V.Golubev, Markeundervaluaion of risky converible offerings: evidence from he airline indusry, Economics and Finance, vol. 34, no., pp , 00. [3] T. Kimura and T. Shinohara, Mone Carlo analysis of converible bonds wih rese clauses, European Operaional Research,vol.68,no.,pp.30 30,006. [4] J. Yang, Y. Choi, S. Li, and J. Yu, A noe on Mone Carlo analysis of converible bonds wih rese clause, European Operaional Research,vol.00,no.3,pp.94 95,00. [5] M. A. Siddiqi, Invesigaing he effeciveness of converible bonds in reducing agency coss: a Mone-Carlo approach, Quarerly Review of Economics and Finance, vol.49,no.4,pp , 009.

8 8 Absrac and Applied Analysis [6]J.C.Cox,S.A.Ross,andM.Rubinsein, Opionpricing:a simplified approach, Financial Economics,vol.7,no. 3, pp. 9 63, 979. [7] W. Cheung and L. Nelken, Cosing he convers, RISK, vol.7, pp.47 49,994. [8] P. Carayannopoulos and M. Kalimipalli, Converible bonds prices and inheren biases, Working Paper, Wilfrid Laurier Universiy, 003. [9] M. W. Hung and J. Y. Wang, Pricing converible bonds subjec o defaul risk, The Derivaives, vol.0,pp.75 87, 00. [0] D. R. Chambers and Q. Lu, A ree model for pricing converible bonds wih equiy, ineres rae, and defaul risk, The Journal of Derivaives,vol.4,pp.5 46,007. [] R. Xu, A laice approach for pricing converible bond asse swaps wih marke risk and counerpary risk, Economic Modelling,vol.8,no.5,pp.43 53,0. [] P. Richken, Derivaive Markes, HarperCollinsCollege,New York, NY, USA, 996. [3] T. S. Y. Ho and S. B. Lee, Term srucure movemens and pricing ineres rae coningen claims, Finance,vol. 4, pp. 0 09, 986. [4] D. Gran and G. Vora, Analyical implemenaion of he Ho and Lee model for he shor ineres rae, Global Finance Journal, vol.4,no.,pp.9 47,003. [5] R. Jarrow and S. Turnbull, Derivaive Securiies,Souh-Wesern College, Cincinnai, Ohio, USA, 996. [6] S. Roman, Inroducion o he Mahemaics of Finance, Undergraduae Texs in Mahemaics, Springer, New York, NY, USA, nd ediion, 0. [7] J. Hull, Opions, Fuures, and Oher Derivaives, Tsinghua Universiy Press, 6h ediion, 009. [8] R. A. Jarrow and S. M. Turnbull, Pricing derivaives on financial securiies subjec o credi risk, Finance,vol. 50, pp , 995.

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