A Class of Jump-Diffusion Bond Pricing Models within the HJM Framework

Transcription

1 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 132 Sepember 24 A Class of Jump-Diffusion Bond Pricing Models wihin he HJM Framework Carl Chiarella and Chrisina Nikiopoulos Sklibosios ISSN

2 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS School of Finance and Economics Universiy of Technology, Sydney, PO Box 123, Broadway, NSW 27, Ausralia, Tel: , Fax: carl.chiarella@us.edu.au, Chrisina.Nikiopoulos@us.edu.au Absrac. This paper considers a class of erm srucure models ha is a parameerisaion of he Shirakawa 1991) exension of he Heah, Jarrow & Moron 1992) model o he case of jump-diffusions. We consider specific forward rae volailiy srucures ha incorporae sae dependen Wiener volailiy funcions and ime dependen Poisson volailiy funcions. Wihin his framework, we discuss he Markovianisaion issue, and obain he corresponding affine erm srucure of ineres raes. As a resul we are able o obain a broad racable class of jump-diffusion erm srucure models. We relae our approach o he exising class of jump-diffusion erm srucure models whose saring poin is a jump-diffusion process for he spo rae. In paricular we obain naural jump-diffusion versions of he Hull & Whie 199, 1994) one-facor and wo-facor models and he Richken & Sankarasubramanian 1995) model wihin he HJM framework. We also give some numerical simulaions o gauge he effec of he jump-componen on yield curves and he implicaions of various volailiy specificaions for he spo rae disribuion. Keywords: Markovian HJM model, jump-diffusions, sae dependen volailiy. 1. Inroducion This paper considers a muli-facor jump-diffusion model of he erm srucure of ineres raes under a specific volailiy srucure. The forward rae dynamics are driven by muli-dimensional Wiener and Poisson processes and he volailiy srucure is such ha he Wiener volailiy funcions are sae dependen whils he Poisson volailiy funcions are ime dependen. Working wihin he Heah, Jarrow & Moron 1992)hereafer HJM) framework we obain bond prices in an arbirage free environmen, even hough he spo rae dynamics are non-markovian. Imposing resricions on he volailiy srucure, a Markovian muli-facor model is obained. I urns ou ha he sae variables of To appear in Asia-Pacific Financial Markes; Received April 24; Acceped Augus 24. 1

3 2 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS his model, can be expressed as funcions of a finie number of benchmark forward raes or yields. The model ha we hereby develop provides a fairly broad racable class of jump-diffusion erm srucure models ha would be suiable for boh calibraion and economeric esimaion. The lieraure on he incorporaion of jump componens ino erm srucure models is no a very exensive one. For lieraure on jump-diffusion ineres rae models ha usually have as heir saring poin he spo rae dynamics, we would cie in paricular Das 22) and Chacko & Das 22). Wihin he HJM erm srucure modelling framework, where he focus is on he forward rae dynamics, Shirakawa 1991) was he firs aemp o incorporae disconinuous forward rae dynamics. Subsequenly a very general framework for erm srucure modelling under marked poin processes was developed by Björk, Kabanov & Runggaldier 1997). More recen work on jumpdiffusion versions of he HJM framework include Glasserman & Kou 23), who consider he marke model, and Das 2) who reas a discree ime version of he HJM model. Here we use he Shirakawa 1991) framework, which assumes only a finie number of possible jump sizes and ha here exiss a sufficien number of raded bonds o hedge away all of he jump risks, in his way guaraneeing marke compleeness. We derive a Markovian represenaion of he sochasic dynamic sysem driving bond prices by considering cerain specificaions of he volailiy funcions of he insananeous forward rae. Essenially we exend o he jump diffusion case he approach of he Markovianisaion of HJM models developed by a number of auhors. Early papers on he Markovianisaion of HJM models under Wiener diffusions include Cheyee 1992), Carverhill 1994), Richken & Sankarasubramanian 1995) and Bhar & Chiarella 1997), where he condiions on he volailiy srucure for he spo rae process o be Markovian are examined for he one facor HJM models. Inui & Kijima 1998) and de Jong & Sana- Clara 1999) exend hese condiions o muli facor HJM models. Duffie & Kan 1996) developed a square roo volailiy model. Furher, Björk & Landèn 22), Björk & Svensson 21) and Chiarella & Kwon 21b),23) generalise he above resuls in various direcions by assuming more general forward rae volailiy specificaions. We exend some of hese resuls o he Markovianisaion of he jump-diffusion version of he HJM class of models. Using ideas from sae space heory, Björk & Gombani 1999) allow forward raes o be driven by a muli dimensional Wiener process as well as by a marked poin process and give he necessary and sufficien condiions on a deerminisic volailiy srucure, for he exisence of finie dimensional realizaions. They also showed ha he sae variables consiue a minimal se of benchmark forward raes. Our model may be viewed as providing an exension o he framework of Björk & Gombani 1999) since we incorporae level dependen volailiy srucures wih he Wiener process noise.

4 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 3 This paper makes wo main conribuions: Under he generalised HJM jump-diffusion framework and a specific formulaion of level and ime dependen volailiy specificaions, Markovian spo rae and bond price dynamics are obained. In addiion, finie dimensional affine realisaions of he erm srucure in erms of forward raes and yields are obained. Wihin his framework we develop some paricular classes of jump-diffusion erm srucure models. In paricular we develop wha we believe is he naural exension of he Hull & Whie 199), 1994) class of models and he Richken & Sankarasubramanian 1995) class of models o he jump-diffusion case. The srucure of his paper is as follows. In Secion 2 we review he Shirakawa jumpdiffusion erm srucure framework focusing on an economic inerpreaion of he underlying hedging argumen. In Secion 3 we assume a specific volailiy srucure, and obain he corresponding Markovian represenaion of he spo rae and bond price dynamics in erms of a finie number of sae variables ha are driven by Markovian diffusion and jump processes. In Secion 4, we express hese sae variables as finie dimensional affine realisaions in erms of economic quaniies observed in he marke, such as forward raes and yields. In Secion 5 we consider he case in which he Poisson volailiies are also sae dependen and give some insigh ino he reason why a Markovian represenaion may no be possible in his case. We do however sugges a way in which an approximae Markovian represenaion may be developed, given he magniude of he jump volailiies suggesed by empirical sudies. In Secion 6 we develop some specific models, in paricular a class of muli facor Hull & Whie 199), 1994) and Richken & Sankarasubramanian 1995) jump-diffusion models and he equivalen forward rae curves. We also carry ou a number of numerical simulaions o gauge he implicaions of he various volailiy specificaions ha generae hese models. In Secion 7 we conclude and discuss fuure direcions for research. 2. The Model In his secion we review some fundamenal relaionships of he bond marke and he main feaures of he Shirakawa 1991) model. Our exposiion is in a less absrac seing han ha of Shirakawa, as we wish o emphasize more he economic inuiion of he underlying hedging argumen. Using f, T) o denoe he insananeous forward ineres rae a ime for insananeous borrowing a ime T ), we define as P, T), he price a ime of a defaul-free discoun zero-coupon bond wih mauriy T, i.e., ) P, T) = exp f, s)ds, 1) so ha PT, T) = 1.

5 4 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS Generalising he basic assumpion of Shirakawa 1991), on he filered probabiliy space Ω, F, P), 1 he sochasic differenial equaion for he insananeous forward rae f, T) driven by boh Wiener and Poisson risk is given by m i df, T) = α, T)d σ i, T)dW i ) β ij, T)[dQ ij ) λ ij d], 2) where α : [, T] R is he drif funcion, W i ) are sandard Wiener processes i = 1, 2,...,n), Q ij ) is a Poisson process wih consan inensiy λ ij j = 1, 2,...,m i ). The Poisson process Q ij models he arrival ime of he jump evens. Recall ha, by definiion 1, if a jump occurs in he ime inerval, d) wih probabiliy λ ij d), dq ij ) =, oherwise wih probabiliy 1 λ ij d), and E[dQ ij ) F ] = λ ij d, E[dQ 2 ij ) F ] = λ ij d. A he Poisson jump imes, he jump size is equal o β ij, T). Under hese assumpions, he jump feaure is modelled by a mulivariae poin process, allowing for a finie number of jumps. 2 The volailiy specificaions allow for σ i : [, T] R, he volailiy funcions associaed wih he Wiener noise processes, which are posiive valued, o be sae dependen. Here we consider a specificaion of he general form σ i, T) = σ i, T, f)), for i = 1,...,n, 3) where σ i are well-defined funcions ha depend on ime, mauriy and f) is a vecor of pah dependen variables such as he insananeous spo rae and/or insananeous forward raes of differen fixed mauriies. By omiing his level dependence, we would obain he special case of ime deerminisic Wiener volailiy funcions. The β ij : [, T] R, he volailiy funcions associaed wih he Poisson noise processes are assumed o be only ime and mauriy dependen. These volailiy specificaions generalise he Shirakawa framework by allowing he Wiener noise and Poisson noise o have separae volailiy srucures. Such a framework is appropriae if one believes ha hese differen ypes of shocks impac differenly across he forward curve. The empirical sudy of jump-diffusion ineres rae models by Chiarella & Tô 23), suggess ha his may in fac be he case in some markes. In sochasic inegral equaion form, equaion 2) may be wrien f,t) = f,t) αs,t)ds σ i s,t, fs))dw m i i s) β ij s,t)[dq ij s) λ ij ds]. 1 In more formal noaion we assume ha Ω, F, F) T, P) is he probabiliy space equipped wih he naural filraion of a vecor of sandard Wiener processes W i) i = 1, 2,..., n) and he Poisson processes Q ij) wih inensiy λ ij j = 1, 2,..., m i), indexed on he ime inerval [, T]. 2 See Runggaldier 23) for a good survey of jump-diffusion models. 4)

6 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 5 Seing T = in equaion 4), he sochasic inegral equaion for he insananeous spo rae is given by r) f,) = f,) αs,)ds σ i s,, fs))dw m i i s) β ij s,)[dq ij s) λ ij ds]. Wih applicaion of he jump-diffusion version of Io s lemma, he dynamics for he bond price driven by Wiener and Poisson risk, may be expressed as 3 dp, T) P, T) = [r) H, T, f))]d ζ i, T, f))dw m i i ) 1 e ξij,t) )dq ij ), where ζ i, T, f)) = ξ ij, T) = H, T, f)) = α, u)du σ i, u, f))du, 7) β ij, u)du, 8) 1 m 2 ζ2 i, T, f)) i λ ij ξ ij, T). 9) In his economy we have n m 1 m 2... m n sources of risk, n due o he Wiener processes W i ) i = 1,, n), and m 1 m 2... m n due o he Poisson processes Q ij j = 1,, m i ). Using he classical hedging porfolio argumen of Vasicek 1977) ha carries over o ineres rae models he original Black-Scholes hedging approach, we hus place bonds of n n m i 1 mauriies in he hedging porfolio. 4 By aking an appropriae posiion in he n n m i 1 bonds i is possible o eliminae boh Wiener and Poisson risks and afer some manipulaions 5 o derive he forward rae drif resricion ha exends he HJM forward rae drif resricion o now incorporae he 5) 6) 3 See Proposiion 2.2 of Björk e al. 1997). 4 The suble issue in he hedging argumen concerns wheher or no he se of bonds in he hedging porfolio remains fixed over ime. The Shirakawa analysis only esablished he exisence of a se of bonds ha would possibly change over ime. Björk e al. 1997) esablished ha he se of hedging bonds can in fac remain fixed over ime. 5 See Appendix 1 for full deails of he hedging porfolio argumen in he curren conex. The reader may refer o Björk e al. 1997), for he mos general approach o deriving he arbirage free dynamics for ineres rae models under marked poin processes.

7 6 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS jump feaure, namely, α, T) = σ i, T, f)) φ i ) ζ i, T, f))) m i β ij, T)ψ ij )e ξij,t) λ ij ). In equaion 1) he φ i are he marke prices of diffusion risk associaed wih he Wiener process sources of uncerainy W i, whils he ψ ij are he marke prices of jump risk associaed wih he Poisson process sources of uncerainy Q ij. 1) 2.1. The Risk Neural Dynamics under a General Volailiy Specificaion. By an applicaion of Girsanov s heorem Bremaud 1981)), for every fixed finie ime horizon T, we can obain a unique equivalen probabiliy measure P 6, under which he W i ) = φ is)ds W i ) are sandard Wiener processes for i = 1,...,n) and he Q ij are Poisson processes for i = 1,...,n and j = 1,...,m i ) associaed wih inensiy ψ ij ) such ha W i and Q ij are muually independen. Subsiuion of 1) ino 9) reduces he sochasic differenial equaion for he bond price in he now arbirage free economy o dp, T) P, T) = r)d ζ i, T, f))d W m i i ) 1 e ξij,t) )[dq ij ) ψ ij )d]. In addiion, by obaining he dynamics of he bond price measured in unis of he money marke accoun, he bond price can be expressed as [ ] [ B) T ) ] P, T) = Ẽ BT) F = Ẽ exp rs)ds F, 12) where Ẽ is expecaion given informaion a ime ) wih respec o he equivalen probabiliy risk neural) measure P and B) is he accumulaed money accoun ) B) = exp rs)ds. Consequenly, by subsiuion of he drif resricion 1) for αs, ) ino he equaion 5), we obain he dynamics of he spo ineres rae r) under he risk neural measure 11) 6 The Wiener processes Wi) i = 1,, n) and he Poisson processes Q ij) j = 1,, m i) wih inensiy Ψ i generae he P -augmenaion of he filraion F.

8 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 7 P, in he form r) = f, ) σ i s,, fs))ζ i s,, fs))ds m i σ i s,, fs))d W m i i s) ψ ij s)β ij s, )[1 e ξijs,) ]ds β ij s, )[dq ij s) ψ ij s)ds]. 13) Under a general specificaion for σ i s,, fs)) and β ij s, ) he dynamics for r) implied by 13) are non-markovian due o he pah dependency of some or all of he inegral erms on he righ-hand side of 13). 3. A Specific Volailiy Srucure In order o generae specific erm srucure models and o be able o obain Markovian represenaions of he spo rae dynamics 13), we shall consider more specific volailiy srucures. To make he discussion explici, we shall also assume ha f) = r), f, T 1 ), f, T 2 ),...,f, T k )) where T 1, T 2,..., T k are a se of fixed mauriies. Assumpion 3.1. For i = 1,..., n, he sae dependen Wiener volailiy srucure 3) is of he form σ i s,, fs)) = σ i s, fs))e s κ σiu)du, 14) and for i = 1,...,n and j = 1,...,m i, he ime dependen Poisson volailiy funcions are of he form β ij s, ) = β ij s)e s κ βiju)du, 15) where κ σi ), κ βij ) and β ij ) are ime deerminisic funcions and σ i, f)) are ime and sae dependen funcions. We recall ha in he no jump siuaion, he funcional form 14) for he forward rae volailiy derives, wihin he HJM framework, he exended Vasicek model of Hull-Whie one-facor model) see Baxer & Rennie 1996), Chiarella & El-Hassan 1996)) and he Hull-Whie wo-facor and muli-facor models see Chiarella & Kwon 21a)). We shall now show ha his case gives a Markovian represenaion of 13) ha may be viewed as a generalisaion of he Markovian muli-facor models o he jump-diffusion case. The crucial propery of he volailiy funcions 14) and 15) is ha heir derivaives wih respec o he second argumen mauriy) are given by σ is,, fs)) = κ σi )σ i s,, fs)), 16)

9 8 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS for i = 1,...,n, and β ijs, ) = κ βij )β ij s, ), 17) for i = 1,...,n, and j = 1,...,m i. This is a naural consequence of he funcional forms 14) and 15), ha allows he separaion of he ime dependen componen from he mauriy dependen componen. As poined ou by Chiarella & Kwon 23), his is in fac he key propery of he volailiy funcions ha leads o he Markovianisaion of he model. In many of he common models, he insananeous spo rae iself is included in he se of sae variables. Thus, in he following proposiion, we derive he spo rae dynamics in boh inegral and differenial form in erms of a number of sochasic facors and he spo rae. Proposiion 3.1. Le σ i s,, fs)) and β ij s, ), for i = 1, 2,...,n and j = 1, 2,...,m i, saisfy Assumpion 3.1. Then he dynamics for he spo rae can be expressed as m i r) = f, ) D σi ) D βij ), 18) in sochasic inegral equaion form, or, m i dr) = D) E σi ) ˆκ σi )D σi ) ˆκ βij )D βij )) k σ1 )r) d i=2 σ i, f))d W m i i ) β ij )[dq ij ) ψ ij )d], 19) in sochasic differenial equaion form, where D) = κ σ1 )f, ) m i f, ) E βij ), 2) and E σi ) = E βij ) = D σi ) = D βij ) = ˆκ σi ) = κ σi ) κ σ1 ), 21) ˆκ βij ) = κ βij ) κ σ1 ), 22) σ 2 i s,, fs))ds, 23) ψ ij s)β 2 ijs, )e ξ ijs,) ds, 24) σ i s,, fs))ζ i s,, fs))ds ψ ij s)β ij s, )[1 e ξ ijs,) ]ds σ i s,, fs))d W i s), 25) β ij s, )dq ij s) ψ ij s)ds). 26)

10 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 9 Proof. Subsiuion of he sochasic quaniies 25) and 26) ino 13) derives 18). For he sochasic differenial represenaion, ake he sochasic differenial of 18) and make use of properies 16) and 17), o obain he sochasic differenial equaion for he insananeous spo rae under he risk neural measure, as, dr) = n f,) σ i s,, fs))ζ i s,, fs))ds m i ψ ij s)β ij s,)[1 e ξijs,) ]ds) κ σi ) σ i s,, fs))d W m i i s) κ βij ) β ij s,)dq ij s) ψ ij s)ds) d σ i, f))d W m i i ) β ij )[dq ij ) ψ ij )d], which, by using he resuls of Appendix 2, may be expressed as [ dr) = f, ) σi 2 s,, fs))ds κ σi ) σ i s,, fs))ζ i s,, fs))ds m i ) ψ ij s)βijs, 2 )e ξijs,) ds κ βij ) ψ ij s)β ij s, )[1 e ξijs,) ]ds κ σi ) σ i, f))d W i ) σ i s,, fs))d W i s) m i κ βij ) 27) β ij s, )[dq ij s) ψ ij s)ds] d m i β ij )[dq ij ) ψ ij )d]. 28) Relaion 18) allows one of he sochasic facors o be expressed in erms of he spo rae r) and he remaining sochasic facors and here we ake m i D σ1 ) = r) f, ) D σi ) D βij ). 29) i=2 Use of expressions 23), 24), 25) and 26), and subsiuion of 29) ino he sochasic differenial equaion 28) leads o he dynamics 19). The E βij ) are deerminisic ime funcions, whereas he E σi ), D σi ) and D βij ) are sochasic quaniies depending on he pah hisory up o ime. These sochasic quaniies saisfy sochasic differenial equaions wih drifs and diffusion erms ha depend on hemselves and he sae variables f), as he nex Proposiion shows. Proposiion 3.2. Given he forward rae volailiy specificaions of Assumpion 3.1 and assuming ha he marke prices of jump risk are non-sochasic, he sochasic quaniies E σi ), D σi ) and D βij ) saisfy he sochasic differenial equaions, for i = 1,, n,

11 1 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS and j = 1, 2,, m i, de σi ) = [σ 2 i, f)) 2κ σi )E σi )]d, 3) and dd σi ) = [E σi ) κ σi )D σi )]d σ i, f))d W i ), 31) dd βij ) = [E βij ) κ βij )D βij )]d β ij )[dq ij ) ψ ij )d]. 32) Proof. Taking he differenial of he quaniies 23), 24) and 25), he saed resuls follow. Secion 4 shows how he f) can be expressed in erms of he sochasic quaniies E σi ), D σi ), D βij ) or vice versa). Thus, he insananeous spo rae dynamics 19) are Markovian under he forward rae volailiy specificaions 14) and 15), since he sochasic quaniies E σi ), D σi ), D βij ) display Markovian dynamics. 7 We noe ha he drif erm in 19) is a linear combinaion of 2n m 1... m n 1 sochasic variables, deermined by 3), 31) and 32) and he spo rae. In he following secion, an exponenially affine erm srucure of ineres raes in erms of he hese sochasic variables is obained Affine Term Srucure of Ineres Raes. We obain he muli-facor bond price formula in erms of he sochasic variables E σi ), D σi ), and D βij ), by using he Inui & Kijima 1998) approach. This consiss of a direc subsiuion of he risk neural forward rae dynamics and he volailiy specificaions 14) and 15) ino he fundamenal relaionship beween bond prices and forward raes in equaion 1) and manipulaing he resuling inegrals. Proposiion 3.3. Under Proposiion 3.1 he bond price assumes he muli-facor exponenial affine form given by { P, T) P, T) = P, ) exp M, T) 1 Nσ 2 2 i, T)E σi ) N σi, T) N σ1, T))D σi ) i=2 m i N βij, T) N σ1, T))D βij ) N σ1, T)r), 33) 7 As saed in Proposiion 3.2, he Markovianisaion obained depends on he assumpion ha he marke prices of jump risk are non-sochasic. If one in fac wished o allow hese o be sochasic say for empirical sudies) hen one could sill obain a Markovian represenaion if he ψ ij were assumed o follow some Markovian sysem of sochasic differenial equaions.

12 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 11 where, and M, T) =N σ1, T)f, ) m i N βij, T) N x, T) Proof. See Appendix 3 for deails. m i ψ ij s)β ij s, y)[1 e ξ ijs,y) ]dyds ψ ij s)β ij s, )[1 e ξ ijs,) ]ds, 34) e y κxu)du dy, x {σ i, β ij }. 35) The bond price formula 33) displays a finie dimensional affine srucure in erms of a number of sae variables n s = 2n m 1... m n in our case) ha are driven by diffusion processes and jump processes. In paricular, he sae variables E σi ) are driven by jump-diffusion processes due o he dependency on he f), he sae variables D σi ) are driven by pure diffusion processes, whereas he sae variables D βij ) are driven by pure jump processes. These sochasic facors namely E σi ), D σi ) and D βij )) have no easy economic inerpreaion. I would be very convenien and more inuiive for applicaions if we could express hese sochasic facors in erms of economic quaniies observed in he marke, like forward raes, whose dynamics would be driven by combined jump-diffusion processes. In he nex secion, we will show ha hese sochasic facors can indeed be expressed in erms of benchmark forward raes wih dynamics driven by jump-diffusion processes. 4. Finie Dimensional Affine Realisaions in Terms of Forward Raes We employ he basic ideas from Chiarella & Kwon 23) and Björk & Svensson 21) who show ha, in a Markovian HJM framework wih dynamics driven by diffusion processes, he sae variables can be expressed as affine funcions of a finie number of forward raes and yields. We inroduce he jump componen ino heir modelling framework and we assume sae dependen Wiener volailiy funcions and ime deerminisic Poisson volailiy funcions. I seems ha he inclusion of jumps makes i very hard or probably impossible o derive Markovianisaion resuls under more general volailiy specificaions ha allow he jump volailiy funcions o be sochasic. However in Secion 5 we indicae how an approximae Markovianisaion may be found in his case. We use he exponenial affine erm srucure of ineres raes 33), where he bond price is a funcion of he insananeous spo rae r), and he sochasic quaniies E σi ), D βij ), and D σi ). We can hen express he insananeous forward rae as

13 12 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS from equaion 1)) f, T) f, T) Nσi, T) i=2 M, T) N σ 1, T) N σ 1, T) r) = ) m i Nβij, T) D σi ) where he N x, T) x {σ i, β ij }) are defined in equaion 35). N σi, T) N σi, T) E σi ) 36) N ) σ 1, T) D βij ), We now ake a number of fixed forward rae mauriies equal o he number of sae variables remaining afer excluding he insananeous spo rae r). We express hese sae variables in erms of forward raes wih differen fixed mauriies. Thus, n s = n s 1) forward raes of differen fixed mauriies T h are required. Proposiion 4.1. The forward rae of any mauriy can be expressed in erms of he n s benchmark forward raes and he insananeous spo rae r) 8 as f, T) = f, T) Q, n s T) R h, T)f, T h ) S, T)r), 37) where, for l = n q 1 and k = 2n i 1, and Q,T) = M,T) q=2 ϖ lh Nσq,T) R h, T) = S, T) = N σ 1, T) q=2 n s M,Th ) h f,t h ) N σ 1,T) ) m i N σi, T) ϖ ih N σi, T) m i n s ϖ lh Nσq, T) ϖ kh Nβij, T) N σ1, T h ) h N σ 1, T) q=2 ) [ n ϖ kh Nβij,T) N σi,t) ϖ ih N σi,t) 38) ϖ lh Nσq, T) N σ 1, T) N ) σ 1,T), N ) σ 1, T) ), 39) N σi, T) ϖ ih N σi, T) 4) ) m i ϖ kh Nβij, T) N ) σ 1, T), 8 Only up o ime = min Th h. By reparameerising in erms of fixed ime-o-mauriy forward raes f, T h ), we may allow for any R, a represenaion which would acually be more amenable o empirical esimaion.

14 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 13 and ϖ l denoes he l h elemen of he marix Ō 1, he inverse of he square marix Ō), defined such ha for i = 1, 2,...,n, q = 2,...,n, and j = 1, 2,...,m i, [ ] Ō) = ϕ 1 ) ϕ 2 ) ϕ 3 ), [ ] where, ϕ 1 ) = Nσi,T h ) h N σi, T h ) is a n s n marix, ϕ 2 ) = ϕ 3 ) = [ ] Nσq,T h ) h Nσ 1,T h) h, is a n s n 1) marix, and [ ] Nβij,T h ) h Nσ 1,T h) h, is a n s m 1... m n ) marix. Assume ha Ō) is inverible for all {; = min T h}. h Proof. Considering equaion 36) for he mauriies T 1, T 2,, T ns we obain he sysem E σ1 ). f, T 1 ) f, T 1 ) M,T 1) 1 Nσ 1,T 1) E 1 r) σn ) f, T 2 ) f, T 2 ) M,T 2) 2 Nσ 1,T 2) 2 r) D σ2 ) =. Ō).. f, T ns ) f, T ns ) M,T ns) Nσ 1,T ns) D σn ) r) ns ns D β11 ). D βnmn ) By invering he marix Ō), he sae variables E σi), D σi ) and D βij ) are expressed in erms of forward raes of n s disinc mauriies as E σ1 ). E σn ) f, T 1 ) f, T 1 ) M,T 1) D σ2 ) 1 Nσ 1,T 1) 1 r) f, T 2 ) f, T 2 ) M,T 2). = Ō 1 ) 2 Nσ 2,T 1) 2 r). 41) D σn ). f, T ns ) f, T ns ) M,T ns) Nσ 1,T ns) r) D β11 ) ns ns. D βnmn ) By subsiuion of expressions 41) for he sae variables ino he forward rae formula 36), one obains 37) which expresses he forward rae of any mauriy in erms of he n s forward raes and he insananeous spo rae r). The following proposiion displays he corresponding bond price formula.

15 14 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS Proposiion 4.2. The zero-coupon bond prices in erms of he n s benchmark forward raes and he insananeous spo rae r) is given by he exponenial affine form { } P, T) P, T) = P, ) exp n s Q P, T) R P h, T)f, T h) S P, T)r), 42) where Q P, T) = Q, s)ds, RP h, T) = R h, s)ds, and S P, T) = Proof. By subsiuion of 37) ino he fundamenal relaionship 1). The risk neural dynamics for each benchmark forward rae f, T h ) are given by m df, T h ) = σ i, T h, f))ζ i, T h, f)) i ψ ij )β ij, T h )e ξ ij,t h ) d S, s)ds. σ i, T h, f))d W m i i ) β ij, T h )dq ij ), 43) which are driven by boh Wiener and Poisson processes. By using he sysem 41), he dynamics 19) of he spo rae r) can be expressed in erms of he sae vecor se k = n s ) f) = r), f, T 1 ), f, T 2 ),..., f, T ns )), as [ ] n s dr) = D f ) R f h )f, T h) S f )r) d σ i, f))d W m i i ) β ij )[dq ij ) ψ ij )d], 44) where, for l = n q 1 and k = 2n i 1, R f h ) = ϖ ih ˆκ σq )ϖ lh q=2 m i ˆκ βij )ϖ kh, 45) and n s D f ) = D) n s S f ) = k σ1 ) R f h ) f, T h ) M, T ) h), 46) h R f h ) N σ h, T 1 ) h. 47) Thus a closed Markovian sysem for all he elemens of he sae vecor has been obained.

16 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 15 The advanage in obaining he bond pricing formula 42) and he forward rae formula 37) is ha hey allow us o ransfer he marke informaion of a cerain se of disinc forward rae curves and he insananeous spo rae in addiion o he iniial forward curves included in he erms M,T h) h ) ino he bond price and he forward rae curve respecively. Moreover, he yield o mauriy which is defined as R, T) lnp, T)/T ), becomes in erms of he forward rae R, T) = f, u)du, 48) T and using expression 37) we could express he yield o mauriy in erms of he same se of forward raes menioned above. Applying similar inveribiliy argumens 9 we may express he forward curve in erms of a se of bonds or yields o mauriy. Given ha yields of differen mauriies are observed in he marke, his model seup would prove o be very suiable for parameer esimaion and model calibraion. Remark 4.1. Wheher or no one includes he spo rae in he se of he sae variables depends on he paricular applicaion. By doing so in he presen paper allows us o relae he class of models developed here o he radiional models e.g. Hull-Whie, Richken-Sankarasubramanian) ha do ake he insananeous spo rae as he underlying sae variable, as we shall show in Secion 6. However he general framework developed does no ie us o such a choice, we may use any convenien se of ineres raes as he sae variables. Appendix 5 ses ou he resuls of his secion for he case when r) is no one of he sae variables. 5. Sae Dependen Poisson Volailiy Srucure In previous secions, we considered he case in which only he Wiener volailiy funcions depend on a number of sae variables. The case where boh Wiener and Poisson volailiies are sae dependen, poses some problems. We now indicae why, in he case ha boh Wiener and Poisson volailiies are sae dependen, i seems impossible o obain Markovian represenaion of he spo rae dynamics 13) and so we propose an approximae soluion o he problem. Assume ha he Wiener volailiies follow he srucure 14) for i = 1,...,n, and ha for j = 1,...,m i he Poisson volailiies are expressed as β ij s,, fs)) = β i s, fs))e s κ βiu)du. 49) The derivaive of he volailiy funcions 49) wih respec o he second argumen mauriy) sill saisfies 17), so he dynamics of he spo rae 19) sill follow. Given he sae dependen volailiy specificaions 14) and 49) assume ha he marke prices 9 See Corollary 2 of Chiarella & Kwon 23).

17 16 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS of jump risk are non-sochasic), all he quaniies E σi ), E βij ), D σi ) and D βij ) are now sochasic. The problem wih his case arises from he process E βij ). Recall ha E βij ) = and inroduce for n = 2, 3,..., he quaniies E n) βij, f) = ψ ij s)β 2 ijs, )e ξ ijs,) ds, 5) ψ ij s)β n ijs,, fs))e ξ ijs,, fs)) ds. We seek o obain he sochasic differenial equaion for E βij ), which from 5) urns ou o be de βij = ψ ij )β ij, f)) κ βij )E βij, f)) E 2) β ij, f)))d. The process E 2) βij, f) in urn saisfies he sochasic differenial equaion de 2) βij, f)) = ψ ij )βij, 2 f)) 2κ βij )E 2) βij, f)) ) E 3) βij, f)) d. Thus, we are dealing wih an infinie sequence of processes E n) βij, f), since for n = 2, 3,... we find ha de n) βij, f)) = ψ ij )β n ij, f)) nκ βij )E n) βij, f)) E n1) βij, f)) ) d. Therefore, when boh Wiener and Poisson volailiies are sae dependen, i seems ha we canno obain a Markovian represenaion a leas no by an approach similar o he one ha led o he spo rae dynamics equaion 19). To close his sequence will require some approximaion. In pracice, if would be he case ha β n ), for sufficienly large n see he magniudes of he jump componen obained by Chiarella & Tô 23)) so in his way i is possible o achieve an approximae Markovianisaion and have an approximae affine erm srucure. 6. Model Applicaions In his secion, we illusrae examples of jump-diffusion models ha can be generaed wih he framework of his paper. We also relae our general model o known models and exend hese models o incorporae jumps componens. In paricular, we consider examples of he Hull-Whie ype models hereafer HW) and he Richken & Sankarasubramanian 1995) class of models hereafer RS) exended o he muli-facor jumpdiffusion case and invesigae heir disribuional profiles.

18 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK Hull & Whie Type models. One of he characerisic feaures of HW ype models is ha he underlying dynamics involve a mean revering process for he insananeous spo rae of ineres as his is he underlying sae variable 1 in his class of models and he volailiy funcion is only ime deerminisic. So, o obain HW ype models under jump diffusions, we resric our general model of sae dependen Wiener volailiies o ime deerminisic volailiies. Thus he volailiy specificaions of Assumpion 3.1 are simplified o Assumpion 6.1. For i = 1,..., n, he ime dependen Wiener volailiy srucure 3) is of he form σ i s, ) = σ i s)e s κ σiu)du, 51) and for i = 1,...,n and j = 1,...,m i, he ime dependen Poisson volailiy funcions are of he form 15) where κ σi ) and σ i ) are ime deerminisic funcions. This assumpion will reduce he number of he sochasic sae variables. To see his noe ha he sae variables E σi ) deermined by 23) are now non sochasic as hey assume he ime deerminisic form E σi ) = σ 2 i s, )ds. 52) In addiion, he sochasic sae variables D σi ) defined by 25) are of he form D σi ) = σ i s, )ζ i s, )ds σ i s, )d W i s). 53) The dynamics 19), under Assumpion 6.1, herefore become m i dr) = [Λ) S) k σ1 )r)] d σ i )d W i ) β ij )dq ij ) ψ ij )d), where we se Λ) = f, ) κ σ1)f, ) he deerminisic par of he drif erm, and S) = [κ σi ) κ σ1 )]D σi ) i=2 E σi ) 54) m i E βij ), 55) m i [κ βij ) κ σ1 )]D βij ), 56) he sochasic par of he drif erm. The spo rae dynamics 54) generalise he srucure of he Hull & Whie 1994) wo-facor model where he spo rae was driven only by one Wiener process. We recall ha he basic idea of Hull & Whie 1994) was o 1 Of course, here is no reason why one could no define a class of HW or RS models where say some oher rae e.g. he 6-monh LIBOR rae serves as he underlying sae variable. However i has become radiional for a wide class of models o use he insananeous spo rae as he underlying sae variable.

19 18 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS add o he drif erm a sochasic facor driven by anoher Wiener process. In 54) we see ha he drif conains he deerminisic erm Λ), he sochasic erm S) consising of n m 1... m n 1 sochasic facors and he mean revering erm for he insananeous spo rae r). I is also worh poining ou ha as i is defined wihin he HJM framework, he spo rae process 54) is auomaically calibraed o he currenly observed yield curve hrough he Λ) erm. For hese reasons we sugges ha his represenaion is he naural exension of he HW model o he muli-facor jump-diffusion siuaion. In Appendix 4, 11 by following similar argumens as in Secion 4, he corresponding finie dimensional affine realisaions, for he HW models, in erms of forward raes of ˆn s = n m 1... m n 1 differen fixed mauriies are derived. In he following examples, he iniial forward rae curve considered has he funcional form f, ) = a a 1 a 2 2 )e v wih parameers being esimaed as a =.33287, a 1 =.14488, a 2 =.117, and v =.925, which resul in an upward sloping forward curve. The daa used for inerpolaion are he US zero yields from he US zero curves up o 1 years including he spo US zero curve on July 2, 21. We will now consider he case of he one Wiener-wo Poisson HW ype of model. 6.4% 6.2% 6.% 5.8% f,5) = 5.8% f,5) = 6% f,5) = 6.2% 5.6% Time o Mauriy Figure 1. Forward rae curves a = 6 monhs, for he one Wiener and wo Poisson HW ype model when σ = 3.2%, κ σ =.18, β 1 =.6%, κ β1 =.31, ψ 1 = 1, β 2 = 1.28%, κ β2 =.17 and ψ 2 = 1.5. The corresponding curves represen he f, T) when f, 1) = 6.2% and f, 5) akes he values of 5.8%, 6% and 6.2%. 11 The curren case when he Wiener volailiy funcion is independen of he sae variables yields slighly differen looking represenaions because he E σi) quaniies became deerminisic. This means ha he elemens ϕ 1) of he marix Ō) in Proposiion 4.1 does no appear.

20 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 19 Thus, for n = 1 and m 1 = 2, consider he volailiy funcions σ, T) = σ )e κ σu)du, 57) and β i, T) = β i )e κ βi u)du, wih i = 1, 2. 58) The number of he sochasic sae variables, in his case, is 3= n m i ) including he r), and using he resuls of Appendix 4 we may express hese sae variables in erms of 2 benchmark forward raes and he spo rae. In urn, he forward rae f, T) and he bond prices P, T) can be expressed in erms of he spo rae r) and hese benchmark forward raes. The sae variables used here are he spo rae, he 5-year forward rae f, 5) and he 1-year forward rae f, 1). Furher he parameer values of he Wiener volailiy are σ = 3.2%, κ σ =.18. The parameer specificaions for he jump volailiy erms are β 1 =.6%, κ β1 =.31, β 2 = 1.28%, κ β2 =.17. The jump inensiies are ψ 1 = 1 and ψ 2 = 1.5 respecively. The forward rae curves shown in Figure 1 are a 6 monhs ime when i is assumed ha r = 6%, he 1-year forward rae f, 1) is 6.2% and he 5-year forward rae akes he values of 5.8%, 6% and 6.2%. For hese volailiy specificaions, he spo rae volailiy is 3.5% and he 1-year forward rae volailiy is 16.5% of he spo rae volailiy Richken & Sankarasubramanian Type Models. The RS class of models 12 considered in his example is characerised by sae dependen Wiener volailiy funcions, so ha σ, T) = σ 1 T )σ 2 f))e κ σu)du. 59) We consider he case ha n = 1 and m 1 = 2. The number of he sae variable, in his case, is 4= 2n m i ) including r). Using he resuls from Proposiion 4.1 and Proposiion 4.2 we may express hese sae variables in erms of 3 benchmark forward raes and he spo rae. Thus we may se f) = r), f, T 1 ), f, T 2 ), f, T 3 )). In urn, he forward rae f, T) and he bond prices P, T) can be expressed in erms of he spo rae r) and hese benchmark forward raes. The sae variables used now are he spo rae, he 2.5-year forward rae, he 5-year forward rae and he 1-year forward rae. The iniial forward rae curve and he volailiy specificaions considered here are he same as in Secion 6.1. The forward rae curves shown in Figure 2 are in 6 monhs ime 12 In he original Richken & Sankarasubramanian 1995) paper, he forward rae volailiy funcions considered are of he form σr)e k σu)du. Subsequenly, Richken & Chuang 1999) consider he forward rae volailiy funcions a a 1T ))e kt ). The form 59) generalises his ype of volailiy srucure.

21 2 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS 6.4% 6.2% 6.% 5.8% f,5) = 5.8% f,5) = 6% f,5) = 6.2% 5.6% Time o Mauriy Figure 2. Forward rae curves a = 6 monhs, for he One Wiener and Two Poisson RS ype models when σ = 3.2%, κ σ =.18, β 1 =.6%, κ β1 =.31, ψ 1 = 1, β 2 = 1.28%, κ β2 =.17 and ψ 2 = 1.5. The corresponding curves represen he f, T) when f, 2.5) = f, 1) = 6.2% and f, 5) akes he values of 5.8%, 6% and 6.2%. when r = 6%, he 2.5-year forward rae and he 1-year forward rae is 6.2% and he 5-year forward rae akes he values of 5.8%, 6% and 6.2%. In order o compare he differen class of models examined, we selec he model parameers so as o mainain, for all models, he spo rae volailiy a 3.5% and he 1-year forward rae volailiy a 16.5% of he spo rae volailiy. To obain hese volailiy levels, he se of he Wiener and Poisson volailiy parameer values is he one used in each of he above examples. Comparing Figure 1 and Figure 2 we see ha he sae dependen volailiy models display forward rae curves wih sharper curvaure changes han he equivalen Hull Whie ype models of Secion 6.1. This is expeced since he sae dependen volailiy models incorporae a larger number of sae variables, which makes he model more flexible and able o capure more realisic forward rae behavior Simulaed Disribuions. In his secion we perform simulaions of he sochasic differenial equaion sysem under he risk neural measure ha resuls from he Markovianisaion procedure. We examine and compare he simulaed normalised disribuions of r) for he HW class of models and he RS class of models and in paricular when one Wiener and wo Poisson noise erms drive he forward rae dynamics. For all he simulaion examples performed in his secion, an Euler-Maruyama approximaion is employed and we discreize he ime inerval [, 1] ino N = 4 equal subinervals of lengh = 1/N, and generae 1, pahs for r). Furhermore, in order

22 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 21 o compare he lepokurosis levels of he wo classes of models, he volailiy parameers Wiener and Poisson) have been seleced as o provide he same variance of he simulaed disribuions, wih variance being.17 in all cases. For he One Wiener and Two Poisson HW ype of models, he volailiy specificaions considered, are σs, ) = σ e kσ s) and β i s, ) = β i e k βi s) and consan ψ i. We consider he discreised sysem of he insananeous spo rae dynamics 54) wih he wo sae variables D βi ) expressed in erms of he wo benchmark forward raes f, 5) and f, 1), by making use of he sysem 86) in Appendix 4. Simulaed Normalised Densiy - Hull Whie Models.4 NO Jump LowJump.3 High Jump Figure 3. Simulaed Normalised Densiy of he Insananeous Spo Rae for he HW ype of models a = 1. The volailiy magniudes are; for he high jump volailiy case σ =.9%, β 1 = 4%, β 2 = 2%; for he low jump volailiy case σ = 3.8%, β 1 = 2%, β 2 = 1.2%; and for he no jump volailiy case σ = 4.5%. Figure 3 shows he simulaed normalised disribuion of r) for he HW ype of models a = 1. The volailiy parameer values used are κ σ =.18, κ β1 =.31, κ β2 =.17, ψ 1 = 1 and ψ 2 = 1.5. We consider hree ses of volailiy magniude parameers, one wih high jump volailiy, one wih low jump volailiy and one wih no jump volailiy which are respecively; a) σ =.9%, β 1 = 4%, β 2 = 2%, b) σ = 3.8%, β 1 = 2%, β 2 = 1.2% and c) σ = 4.5%. We consider he no-jump volailiy case in order o compare he disribuional oucome wih he Gaussian case. In fac, in he absence of jumps, he model reduces o he Gaussian case. Figure 3 shows ha, compared o he normal disribuion, wih increasing jump magniude, he disribuion becomes asymmeric wih long ail o he righ. However he jump magniude needs o be of a reasonable size for his effec o become pronounced. For he RS ype models, he Wiener volailiies are sae dependen having he funcional form 59). In paricular, for he One Wiener and Two Poisson RS ype models, we need four sae variables o Markovianise he sysem and by considering he insananeous

23 22 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS spo rae r) as one of he sae variables, hen f) = r), f, T 1 ), f, T 2 ), f, T 3 )). We furher assume ha σ 1 T ) = σ consan, and [ γ σ 2 f)) 3 = r) c h f, T h )], wih γ =.5, so we considering a square roo process for he Wiener volailiies. 13 For he Poisson volailiy specificaions, we consider β i s, ) = β i e k βi s) and consan ψ i. Simulaed Normalised Densiy - Richken-Sankarasubramanian Models.4 NO JUMP LOW JUMP VOL.3 HIGH JUMP VOL Figure 4. Simulaed Normalised Densiy of he Insananeous Spo Rae for he RS ype models a = 1. The volailiy magniude is se as σ = 1.2%, β 1 = 4%, β 2 = 2% for he high jump volailiy case; σ = 5.2%, β 1 = 2.4%, β 2 = 1.5% for he low jump volailiy case; and σ = 6.8% for he no-jump volailiy case. We now consider he discreised sysem of he spo rae dynamics 44) which is he dynamics 19) wih he sae variables E σ ) and D βi ) expressed in erms of he hree benchmark forward raes f, 2.5), f, 5), f, 1) and he spo rae by using he sysem 41)). The simulaed normalised disribuion of r) a = 1 for he RS ype of models is shown in Figure 4. The volailiy parameer values used are κ σ =.18, κ β1 =.31, κ β2 =.17, ψ 1 = 1 and ψ 2 = 1.5. We also se c 1 = 2, c 2 = 1, c 3 = 2. For he hree cases of volailiy magniude, we consider σ = 1.2%, β 1 = 4%, β 2 = 2% for he high jump volailiy case, σ = 5.2%, β 1 = 2.4%, β 2 = 1.5% for he low jump volailiy case, and σ = 6.8% for he no-jump volailiy case. Considering he no-jump volailiy case, in oher words by relying on sae dependen volailiies only, he skewness obained is relaively large. Adding jumps does no change he order of he magniude of he skewness. However, in he HW models he jump magniude significanly change he order of magniude of he skewness see Table 1 and Table 2). 13 There is a posiive probabiliy ha his ype of dynamics may drive ineres raes o negaive values. Thus more general sae dependen volailiy funcions may be employed, ha are well defined for negaive values as i has been shown in Nikiopoulos 24).

24 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 23 Simulaed Normalised Densiy Simulaed Normalised Densiy.4 HW Model.4 HW Model RS Model.3 RS Model Figure 5. Comparison of Simulaed Normalised Densiy of he Insananeous Spo Rae for he HW and RS ype of models a = 1 when a) no jump and b) large jump volailiy is considered. Figure 5 compares he simulaed normalised disribuion of r) for he HW and RS ype of models a = 1 for he cases considered earlier. The no jump cases are o he lef and he large jump volailiy cases o he righ. In he large volailiy cases similar disribuions are obained, however he wo models show differences when we compare he saisical properies of he spo rae changes as Table 2 illusraes. Furher, in order o gauge he effec of he jump parameers and he sae dependen volailiy on he simulaed normalised disribuions, we compare in Table 1 and Table 2 he saisical properies of he simulaed disribuions of he spo rae and he spo rae changes recall ha variance of he spo rae is.17% in all cases and expressed in percenage erms). Saisical Informaion on r) no-jump low jump high jump HW RS HW RS HW RS Mean Variance Skewness Kurosis Table 1. The saisical measures of he spo rae from simulaed disribuions for differen jump magniudes under he HW and RS models. We observe ha when he jump volailiies are low he HW model is very close o a Gaussian one, alhough he RS model exhibis a variaion from he Gaussian model wih high skewness and kurosis. Also, he sae dependen models RS) wih or wihou jumps cerainly display higher kurosis and higher skewness of he spo rae compared o he equivalen wih respec o he jump size) deerminisic volailiy models HW). This indicaes ha sae dependen volailiies may capure more efficienly he asymmeric feaure of he empirical spo rae disribuion.

25 24 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS Saisical Informaion on dr) no-jump low jump high jump HW RS HW RS HW RS Mean Variance Skewness Kurosis Table 2. The saisical measures of he spo rae changes from simulaed disribuions for differen jump magniudes under he HW and RS models. By increasing he jump volailiies, boh models exhibi asymmeric normalised disribuions wih a long ail o he righ as we made he choice of he posiive jump size o dominae he negaive jump size. However, he sae dependen RS model wihou jump, has high kurosis of he spo rae changes bu no paricularly high negaive skewness. When he sae dependen model is combined wih he jump diffusion model, such as RS wih jumps, hen boh high kurosis and sufficienly negaive skewness of he spo rae changes are obained. Thus, jumps on one hand and sae dependen volailiy on he oher hand, yield models ha capure beer he sylised empirical facs of ineres rae movemens. However, he combinaion of boh sae dependen volailiies and jumps succeeds in accommodaing mos of he empirical disribuional behavior of he spo rae and he spo rae changes. This feaure should help o produce derivaive securiy pricing models wih improved valuaion accuracy. The feaure ha has made i racable and possible o quanify hese characerisics is he abiliy o obain Markovian srucures for he ineres rae dynamics. This Markovian class of models ha incorporaes he more realisic jump-diffusion processes combined wih sochasic volailiies may be employed for more accurae derivaive pricing and hedging and also in empirical sudies of ineres rae markes see Chiarella & Tô 24)). 7. Conclusion In his paper we have developed a class of jump diffusion erm srucure models wihin he framework of Shirakawa 1991). By an appropriae choice of a sae dependen and ime dependen forward rae volailiy funcions, we obained Markovian represenaions of he spo rae dynamics and derived he corresponding exponenial affine bond pricing formulas. Furhermore, he sae variables of he model have been expressed in erms of a se of benchmark forward raes and yields, a fac ha makes he model suiable for boh calibraion and parameer esimaion. Thus for sae dependen Wiener volailiies and ime deerminisic Poisson volailiies, we have been able o exend he resuls concerning finie dimensional affine realisaions of HJM models in erms of forward raes

26 A CLASS OF JUMP-DIFFUSION BOND PRICING MODELS WITHIN THE HJM FRAMEWORK 25 discussed in Chiarella & Kwon 23) o he jump diffusion case. We have provided some numerical examples o demonsrae he naure of he Hull-Whie and Richken & Sankarasubramanian class of models when hey are exended o incorporae jumps. Summarising hese resuls, he combinaion of sae dependen volailiies and jumps succeeds in accommodaing mos of he empirical disribuional behavior of he spo rae and he spo rae changes which will in urn provide more accurae derivaive securiy pricing models. In he case of sae dependen Poisson volailiy specificaions, we have given some insigh ino why i becomes difficul o obain a Markovian represenaion of he sysem, and we have proposed an approximae Markovian srucure. Furher developmens of his work would include esimaion and model calibraion. Incorporaion of he jump processes ino he HJM model as well as sae dependen volailiy srucures allows a more efficien fi o marke informaion. Addiionally and more imporanly, he racabiliy of he Markovian srucures obained provides an efficien and more accurae basis for Mone-Carlo simulaions, ha may be employed for derivaive pricing and hedging purposes. Furhermore he framework developed here may be exended o credi risk models, as shown in Chiarella, Schlögl & Nikiopoulos 23). This seems a naural exension as he fundamenal processes used in credi risk models are jump-diffusion processes. Appendix 1. The No-Arbirage Condiion in he Bond Marke Seing n H = n n m i, consider a hedging porfolio conaining bonds of mauriies T 1, T 2,, T nh 1 in proporions w 1, w 2,, w nh 1 wih w 1 w 2 w nh 1 = 1. We denoe wih P h ) = P, T h ) h = 1, 2,...,n H 1)) he value of hese n H 1 zero-coupon bonds, and for simpliciy of noaion we wrie he sochasic differenial equaion for P in he general form dp h ) P h ) = µ m i P h )d ν Ph,i )dw i ) χ Ph,ij )dq ij ), where µ Ph ) = r) H, T h, f)) ν Ph,i ) = ζ i, T h, f)), and χ Ph,ij ) = e ξ ij,t h ) 1.

27 26 CARL CHIARELLA AND CHRISTINA NIKITOPOULOS SKLIBOSIOS Le V be he value of he hedging porfolio, hen he reurn on he porfolio is given by dv V = w dp 1 dp 2 1 w 2 P 1 = H 1 dp nh 1 w nh 1 P 2 P nh 1 m i w h ν Ph,i dw i ) χ Ph,ij dq ij ). n H 1 w h µ Ph d In order o eliminae boh Wiener and Poisson risks we need o choose w 1, w 2,, w nh 1 so ha for every i = 1, 2,...,n H 1 and for i = 1, 2,...,n, and j = 1, 2,...,m i, H 1 w h ν Ph,i =, 6) w h χ Ph,ij =. 61) The hedging porfolio hen becomes riskless, hus, i should earn he risk-free rae of ineres r), i.e., nh1 dv V = w h µ Ph d = r)d. From he las equaliy and he fac ha w 1 w 2 w nh 1 = 1, we have H 1 w h µ Ph r)) =. 62) Equaions 6), 61) and 62) form a sysem of n H 1 equaions wih n H 1 unknowns w 1, w 2,, w nh 1. This sysem can only have a non-zero soluion if ν P1,1 ) ν P2,1 ) ν ) PnH 1,1.... ν P1,n ) ν P2,n ) ν ) PnH 1,n χ P1,11 ) χ P2,11 ) χ PnH 1,11 ) =..... χ P1,nmn ) χ P2,nmn ) χ ) PnH 1,nmn µ P1 r) µ P2 r) µ r)) PnH 1 This implies ha for h = 1, 2,...,n H 1) here exis φ 1 ), φ 2 ),...,φ n ) and ψ 11 ),..., ψ 1m1 )), ψ 21 ),..., ψ 2m2 )),..., ψ n1 ),..., ψ nmn )), such ha m i µ Ph r) = φ i )ν Ph,i ) ψ ij )χ Ph,ij ).