QUANTITATIVE FINANCE RESEARCH CENTRE

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1 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 167 Sepember 25 A Conrol Variae Mehod for Mone Carlo Simulaions of Heah-Jarrow-Moron Models wih Jumps Carl Chiarella, Chrisina Nikiopoulos, and Erik Schlögl ISSN

2 A CONTROL VARIATE METHOD FOR MONTE CARLO SIMULATIONS OF HEATH-JARROW-MORTON MODELS WITH JUMPS CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL School of Finance and Economics Universiy of Technology, Sydney PO Box 123 Broadway, NSW 27 Ausralia Ph: Fax: carl.chiarella@us.edu.au, chrisina.nikiopoulos@us.edu.au, erik.schlogl@us.edu.au Absrac. This paper examines he pricing of ineres rae derivaives when he ineres rae dynamics experience infrequen jump shocks modelled as a Poissorocess and wihin he Markovian HJM framework developed in Chiarella & Nikiopoulos 23. Closed form soluions for he price of a bond opion under deerminisic volailiy specificaions are derived and a conrol variae numerical mehod is developed under a more general sae dependen volailiy srucure, a case in which closed form soluions are generally no possible. In doing so, we provide a novel perspecive on he conrol variae mehods by going ouside a given complex model o a simpler more racable seing o provide he conrol variaes. Keywords: HJM model, jump process, bond opiorices, conrol variae, Mone Carlo simulaions. JEL Classificaion: E43, G33, G INTRODUCTION Ineres rae derivaives are securiies, he payoffs of which depend in some way on he level of ineres raes. The value of an ineres-rae opion is subsanially affeced by he presence of skewness and kurosis in he ineres raes. The kurosis explains he smile effec 1 and resuls in fa-ailed disribuions. The skewness resuls in asymmeric ineres rae disribuions ha mach wih he empirically observed disribuional profile of he ineres raes. Jump-diffusion and sochasic volailiy models demonsrae an abiliy o accommodae hese feaures, providing a modelling seing which explicily incorporaes Dae: Curren Version Sepember 12, The shape of he implied volailiies exraced from raded opiorices by invering he Black & Scholes 1973 or Black 1976 opioricing formula, whichever is applicable for a range of differen srikes is called he smile. The smile implies ha a-he-money opions rade a lower volailiies while he opions away from he money rade a higher volailiies. 1

3 2 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL ail risk o more accuraely reflec realiy. However, hese classes of models come a he expense of an increasing complexiy ha makes i impossible in mos cases o derive closed form or compuaionally racable soluions for derivaive prices. Mos of he ineres rae models under jump-diffusions do no readily admi closed form soluions for derivaive prices even when he jump sizes are consan or drawn from well known disribuions such as normal and log-normal. Therefore, mos of he sudies in his area use numerical approximaion mehods o evaluae ineres rae insrumens, including hose of Ahn & Thompson 1988, Ahn 1988, Mercurio & Runggaldier 1993, Naik & Lee 1995, Baz & Das 1996 and Das Baes 1996, Duffie & Kan 1996 and Chacko & Das 22 have considered more advanced models of sochasic volailiy wih jumps. The work of Jamshidian 1989, Shirakawa 1991, Heson 1993 sochasic volailiy seing, Das & Foresi 1996 and Glasserman & Kou 23 provides closed form evaluaion formulas for bonds and bond opions in more specialised cases. This paper presens wo classes of erm srucure models ha incorporae jump behavior of ineres raes and more general volailiy specificaions bu also mainain racabiliy in he pricing of ineres rae derivaives. More specifically, we derive closed form soluions for bond opions under deerminisic volailiy specificaions and a numerical soluion under he more general sochasic volailiy case, which is, however, numerically racable and efficien due o he fac ha he erm srucure model developed admis finie dimensional Markovian represenaions. The Markovianisaion of he jump-diffusion version of he HJM model employed here, even under sae dependen volailiy specificaions, has been achieved by a suiable choice of volailiy funcions, as explained in Chiarella & Nikiopoulos 23. For he deerminisic volailiy se-up, we consider a parameerisaion of he Shirakawa 1991 model of he erm srucure of ineres raes under jump-diffusions. Under an appropriae equivalen probabiliy measure, we consider opioricing wihin his framework. We use Fourier ransform echniques o obain a represenaion of he soluion. A racable Black-Scholes ype pricing formula is derived under he assumpion of a consan jump volailiy funcion. An exension of he Shirakawa 1991 framework is also considered, in which he volailiy evolves sochasically, by means of a volailiy funcion dependen on he sae variables of he sysem. Again under an appropriae equivalen probabiliy measure, we sudy he pricing of bond opions. In his case, however, closed form valuaion formulas are no available. Taking he sae dependen volailiy specificaions of he ype discussed in Chiarella & Nikiopoulos 23, he ineres rae dynamics become Markovian in a finie dimensional sae variable and hus all he quaniies involved such as forward raes or bond prices can be expressed in erms of his sae variable. Taking advanage of hese Markovian represenaions, we employ hese paricular Markovian srucures o obain approximae bond opiorices by use of he Mone Carlo mehod. We furher improve he efficiency of he Mone Carlo mehod by using a conrol variae echnique, ha makes use of he closed form soluions obained in he deerminisic volailiy seing. This paper is srucured as follows. In Secion 2 we develop he deerminisic volailiy model. We solve he bond opioricing equaion using Fourier Transform echniques and we obain closed form soluions for European bond opions under consan jump volailiy specificaions. In Secion 3 he sae dependen volailiy model is considered

4 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 3 and he volailiy resricions ha lead o Markovian erm srucures are discussed. Secion 4 deals wih he numerical implemenaion of he wo models developed. We es he accuracy of he Mone Carlo resuls in he deerminisic volailiy model, since closed form soluions are available in his case. In addiion, we numerically evaluae bond opions under he sochasic volailiy model. Finally by combining boh models and closed form soluions, we develop a conrol variae mehod ha significanly reduces compuaional effor and improves accuracy. Secion 5 concludes and provides fuure direcions for research. 2. The Deerminisic Volailiy Model We denoe as f, T he insananeous forward ineres rae a ime for insananeous borrowing a ime T. Then he price a ime of a discoun zero-coupon bond wih mauriy T, denoed by P, T, is defined as P, T = exp T f, sds, 1 so ha PT, T = 1. On he filered probabiliy space Ω, F, P, 2 we assume ha he dynamics of he insananeous forward rae f, T are driven by boh Gaussian and Poisson risk erms and given by df, T = α, Td n w σ i, TdW i β i, T[dQ i λ i d], 2 where α : [, T] R is he drif funcion, W i are sandard Wiener processes i = 1, 2,...,n w, σ i : [, T] R are ime deerminisic volailiy funcions associaed wih he Wiener noise processes, Q i is a Poissorocess wih consan inensiy λ i i = 1, 2,...,, and β i : [, T] R are ime deerminisic volailiy funcions associaed wih he Poisson noise processes. The Poissorocess Q i is employed o model he arrival ime of he jump evens. Thus, he jump feaure is modelled by a mulivariae poin process, allowing for a finie number of jumps. 3 A number of empirical sudies, including Chiarella & Tô 23, sugges ha differen ypes of shocks impac differenly across he forward curve. The specificaion in equaion 2 generalises slighly he Shirakawa 1991 volailiy srucure by allowing he Gaussian noise and Poisson noise o have separae volailiy srucures. In sochasic inegral equaion form, equaion 2 may be wrien as f,t = f,t αs,tds n w σ i s,tdw i s β i s,t[dq i s λ i ds]. 3 Thus he sochasic inegral equaion for he insananeous spo rae r is expressed as n w n p r f, = f, αs, ds σ i s, dw i s β i s, [dq i s λ i ds], 2 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 Brownian moions W i i = 1, 2,..., n w and he Poisson processes Q i wih inensiy λ i i = 1, 2,...,, indexed on he ime inerval [, T]. 3 Runggaldier 22 provides a good survey of jump-diffusion models. 4

5 4 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL and he corresponding sochasic differenial equaion is dr = ϑd where ϑ is defined as n w ϑ = f, α, n w σ i, dw i β i, [dq i λ i d], 5 αs, ds σ is, dw i s The corresponding dynamics for he bond price are 4 where β is, [dq i s λ i ds]. nw n dp, T P, T = [r H, T]d p ζ i, TdW i e ξi,t 1dQ i, 7 T ζ i, T ξ i, T T H, T T 6 σ i, udu, 8 β i, udu, 9 α, udu n w 1 2 ζ2 i, T λ i ξ i, T. 1 Consider a European call opion of mauriy T C wrien on a bond having mauriy T T > T C and denoe by C = Cr,, T C he value of his bond opion a ime. Taking ino accoun ha he dynamics of he spo rae are given by 5 and using he jump-diffusion version of Io s lemma we derive he sochasic differenial equaion for he bond opiorice as dc = C n w ϑ β i, λ i σ i, C r dw i C r 1 2 n w σi 2, 2 C r 2 d 11 [Cr β i,,, T C Cr,, T C ]dq i. In he nex secion, we develop he classical hedging porfolio argumen in he bond opion marke, in he spiri of he original Black-Scholes hedging approach, o derive he bond opioricing parial differenial-difference equaion Hedging Argumen in he Bond Opion Marke. In his bond opion marke, given n w sources of risk, n w due o he Gaussiarocesses W i and due o he Poissorocesses Q i, we consider a hedging porfolio conaining a bond wih mauriy T and n o = n w bond opions of mauriies T 1, T 2,, T no. All hese opions are wrien on he same bond having mauriy T. By aking an appropriae posiion in bonds and bond opions, i is possible o eliminae boh Gaussian and Poisson risks. The condiion ha he riskless hedged porfolio earns he risk-free rae of ineres r, implies 4 See Björk, Kabanov & Runggaldier 1997 for deails of he manipulaions.

6 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 5 ha here mus exis 5 a vecor Φ = φ 1,...,φ nw and a vecor Ψ = ψ 1,...,ψ np such ha for bond opions of any mauriy T C i mus be he case ha 6 C ϑ β i, λ i n w φ i σ i, C r 1 2 ψ i [Cr β i,, Cr, ] =. n w σ 2 i, 2 C r 2 rc Also as a resul of he no-riskless arbirage condiion, he following drif resricion holds α, T = n w 12 σ i, T φ i ζ i, T β i, Tψ i e ξi,t λ i. 13 Equaion 12 is he parial differenial-difference equaion for he bond opiorice ha is solved over T c and under boundary condiions appropriae o he ype of opion being evaluaed. The boundary condiions in he case of a call bond opion price wih exercise price E are and CrT C, T C, T C = PrT C, T C, T E, 14 C,, T C =, as a resul of he condiion on he bond price ha P,, T =. Noe ha he gis of he argumen is esablishing no-arbirage consisency among a se of insrumens sufficien o complee he marke. Thus we may derive he condiions 12 and 13 by using alernaive porfolios, for insance a porfolio consising of a bond opion and n o bonds. In deriving he maringale represenaion of he bond opiorice, he money marke accoun has been iniially used as he numeraire. By changing he numeraire, he bond opioricing equaion can be formulaed wihin a framework similar o ha used by Meron 1976 o evaluae sock opions involving Gaussian-Poisson risk. Furher he price of he zero-coupon bond wih mauriy T C will be employed as he numeraire for bond opioricing. For every fixed finie ime horizon T, we can obain a unique equivalen probabiliy measure P, under which he W i = φ isdsw i are sandard Wiener processes and he Q i are Poissorocesses wih inensiy ψ i. Thus imposing he drif resricion 13 on equaion 7, he dynamics for P, T C, he zero coupon bond mauring a bond opion mauriy, under P, are given by nw n dp, T C P, T C = rd p ζ i, T C d W i 1 e ξ i,t C [dq i ψ i d] Noe ha he underlying Gaussian and jump risks dwi, dq i driving he opiorice dynamics are he same as hose driving he bond price dynamics and he insananeous spo rae dynamics, hus he marke price of hese risks will be he same as hose in he bond hedging porfolio. 6 See Appendix 1 for deails on he developmen of he coninuous hedging argumen.

7 6 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL Using equaions 11 and 12, he dynamics for Cr,, T C under P are given by dc C =rd 1 C C r n w σ i, d W i 1 [Cr β i,,, T C Cr,, T C ][dq i ψ i d]. C Define he relaive opion and bond prices wih respec o he price of a zero-coupon bond ha has he same mauriy T C as he opion as and 16 Y = Cr,, T C Pr,, T C, 17 X = Pr,, T Pr,, T C, 18 respecively. An applicaion of he jump-diffusion version of he Io s lemma gives he dynamics for Y 7 as nw dy Y = ζ i, T C σ i C [d W i ζ i, T C d] C r Cr βi,, T C Cr,, T C e ξ i,t C 1 [dq i ψ i e ξ i,t C d], and he dynamics for X as nw dx X = ζ i, T C ζ i, T [d W i ζ i, T C d] e ξ i,t e ξ i,t C 1[dQ i ψ i e ξ i,t C d]. By an applicaion of Girsanov s heorem, a new measure P may be found under which he new processes specified here in incremen form are sandard Gaussiarocesses, and 19 2 dw i = d W i ζ i, T C d, 21 dq i = dq i ψ i e ξ i,t C d, 22 are Poissorocesses associaed wih he inensiy vecor Ψ = ψ 1 e ξ 1,T C, ψ 2 e ξ 2,T C,... ψ n e ξn,t C. I follows from 19 and 2 ha he relaive opiorice Y and he relaive bond price X are maringales under P and using he expecaion operaor E under his new measure, we may wrie Y = E [Y T C F ], 23 7 See Appendix 2 for deails.

8 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 7 where he Wiener processes Wi and he Poissorocess Q i wih inensiy Ψ generae he P-augmenaion of he filraion F. By using he definiion of Y, 8 equaion 23 expands o Cr,, T C Pr,, T C = E [ ] Cr, TC, T C Pr, T C, T C F = E [ Pr, T C, T E ] F = E [ XT C E F ]. Therefore, he relaive opiorice Y can be expressed as a funcion of he relaive bond price X, i.e., 24 Y X, = E [ XT C E F ]. 25 The value of he adjused opion 9 Y X, is driven by he dynamics for X, which are given by equaion 2. Given he assumpion on he volailiy funcion, his process reduces o a form ha pus us essenially in he framework used by Meron o price sock opions under a geomeric jump-diffusiorocess, he only difference being ha he coefficiens of he sochasic differenial equaion are ime dependen. Applicaion of he Feynman-Kac Theorem for processes wih jumps o equaion 25 leads o he parial differenial-difference equaion Y n w e ξ i,t C e ξ i,t ψ i X Y X 1 ζ i, T C ζ i, T 2 X 2 2 Y 2 X 2 ψ i e ξ i,t C subjec o he boundary condiion Y X e ξ i,t e ξ i,t C Y X =, 26 lim Y X, = XT C E. 27 T C In he nex secion, a echnique o solve he parial differenial-difference equaion 26 is proposed, by employing Fourier Transform mehods, ha will lead o a pricing formula for he bond opion. The Fourier Transform provides a quie general framework for solving parial differenial equaions of financial economics, since i handles a variey of pricing frameworks such as he jump-diffusion seing or he American opioroblem Soluion o he Opion Pricing Equaion by Fourier Transform Techniques. By changing he variable X o he logarihmic variable Z = lnx and defining he new funcion ΥZ, = Y e Z, 8 Recall ha Cr,, TC is he value of a European opion wrien on a bond wih mauriy T hus Cr, T C, T C = Pr, T C, T E and Pr, T C, T C = 1. 9 The value of he opion under he new TC-forward measure is given by C, T C = P, T CE PTC, T E F. 1 See for example Carr & Madan 1999 and Chiarella, Kucera & Ziogas 1999.

9 8 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL he parial differenial-difference equaion 26 becomes Υ [ np n w ψ i e ξ i,t C e ξ i,t ζ i, T C ζ i, T 2 2 Υ 2 Z 2 [ ψ i e ξ i,t C Υ Z ln subjec o he boundary condiion n w e ξ i,t e ξ i,t C ζ i, T C ζ i, T 2 ] ] ΥZ =, Υ Z 28 lim ΥZ, = e ZTC E. 29 T C Define he Fourier ransform of he soluion Υ = ΥZ, o he parial differenialdifference equaion 28 by Υω, = ΥZ, e iωz dz, 3 where i = 1 is he imaginary number. By employing Fourier ransform echniques, as Appendix 3 shows, he funcion Υω, saisfies an ordinary differenial equaion wih complex coefficiens having soluion Υω, = Υω,T C exp where { T C c, T C = 1 T C υ, T C = 1 T C σ 2, T C = 1 T C ξω,, T C = 1 T C c,t C iω[υ,t C 12 σ2,t C ] ω2 2 σ2,t C ξω,,t C TC TC TC n w TC By he Fourier inversion heorem, we have ha ΥZ, = 1 2π 31 ψ i se ξ is,t C ds, 32 ψ i s e ξ is,t C e ξ is,t ds, 33 ζ i s, T C ζ i s, T 2 ds, 34 iω ψ i se ξ is,t C e ξ is,t e ξ ds. 35 is,t C Thus, by subsiuing 31 ino 36 we obain ΥZ, = e T C c,t C 2π Υω, e iωz dω. 36 }, Υω, T C e iω[υ,t C 1 2 σ2,t C ]T C Z ω2 2 σ2,t C T C ξω,,t C T C dω,

10 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 9 and by changing he variable Z back o he variable X recall ha Z = lnx, we obain Y X, = e c,t CT C where he kernel K is defined by KZ, X, = 1 2π Y e Z, T C KZ, X, dz, 37 e iω[υ,t C 1 2 σ2,t C ]T C ln X Z ω2 2 σ2,t C T C ξω,,t C T C dω. Thus, he value of a call bond opion can be expressed as Cr,, T C = e c,t CT C Pr,, T C ln E 38 e Z EKZ, X, dz. 39 Unlike he corresponding resul in Meron s jump-diffusion sock opion model, i does no seem possible o proceed furher wih 39 and obain a closed form soluion under he more general volailiy specificaions. This is apparenly due o he erm e ξ i s,t e ξ i s,t C iω in he expression for ξω,, TC in equaion 38 for he kernel K. Since in Meron s analysis he coefficiens of his inegro parial differenial equaion were no ime varying his erm reduces o 1 allow him o obain closed form soluions Consan Jump Volailiy Case. We will show in his secion ha resricing he model o consan jump volailiies will provide closed form soluions for he opion price. Assumpion 2.1. For i = 1,...,, he deerminisic Wiener volailiy srucure is of he form σ i s, = σ i se Ê s κ σiudu, 4 and for i = 1,...,n w, he Poisson volailiy funcions are of he form where β i are consan. β i s, = β i, 41 Under Assumpion 2.1 he quaniy ξω,, T C in 35 simplifies o ξω,, T C = 1 T C ψ i e β it T C iω β i 1 e β it C, 42 which simplifies furher he erm e ξω,,t CT C and allows us o complee he inversion of he Fourier Transform. Proposiion 2.1. Under he volailiy specificaions of Assumpion 2.1, he value of he bond opion is given by Cr,, T C = e c,t CT C ς p 1 1 ς p p p 1 = p 2 = p np = 1! p 2!... ς p np p np! [Pr,, Te υ,t CT C È ] np p iµ i Φd 1 p EPr,, T C Φd 2 p, 43 where d 1 p = ln X E p 1µ 1 p 2 µ 2... p np µ np [υt C, 1 2 σ2 T C, ]T C σt C,, 44 T C

11 1 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL d 2 p = d 1 p σt C, T C, 45 and he sandard normal cumulaive disribuion funcion Φz = 1 2π z e 2 2 d. Proof. By an applicaion of he Taylor expansion of e x = p= xp p! we wrie [ e ξω,,t CT C ψi 1 e β i T C ] = exp = = p= p 1 = p 2 = ς p i p! epµ iiω... p np = β i ς p 1 1 p 1! ς p 2 2 e β it T C iω p np p 2!... ς p np! ep 1µ 1 p 2 µ 2...p np µ np iω 46 where ς i = ψ i1 e β i T C β i and µ i = β i T T C. Subsiuing he expression 46 ino he kernel funcion 38 and simplifying, we obain KZ, X, = 1 2π... p 1 = p 2 = p np = ς p 1 1 p 1! ς p 2 2 p np p 2!... ς p np! e iω[υ,t C 1 2 σ2,t C ]T C ln X Zp 1 µ 1 p 2 µ 2...p np µ np ω2 2 σ2,t C T C dω. 47 By using he resul π e qω ω2 dω = e q 2 4, 48 we are able o evaluae he inegral expression in 47, o derive 1 ς p 1 KZ, X, = 1 ς p πTC σ, T C p p 1 = p 2 = p np = 1! p 2!... ς p np p np! { exp [υ, T } C 1 2 σ2, T C ]T C lnx Z p 1 µ 1 p 2 µ 2... p np µ np 2 2σ 2., T C T C 49 Thus equaion 37 reduces o Y X, = e c,t CT C ln E... p 1 = p 2 = p np = ς p 1 1 p 1! ς p 2 2 p np p 2!... ς p np! e Z E 2πTC σ, T C e [υ,t C 1 2 σ 2,T C ]T C ln X Zp 1 µ 1 p 2 µ 2...pnp µnp2 2σ 2,T C T C dz.

12 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 11 Furher by evaluaing separaely he inegrals 11 in he las expression and using he sandard normal cumulaive disribuion funcion Φz = 1 z e 2 2 d, 2π we obain Y X, = e c,t CT C... p 1 = p 2 = p np = ς p 1 1 p 1! ς p 2 2 p np p 2!... ς p np! [Xe υ,t CT C p 1 µ 1 p 2 µ 2...p np µ np Φd 1 p EΦd 2 p], 5 where p = p 1, p 2,...,p np. By recalling he definiions 17 and 18 of Y and X, we derive he resul 43. The closed form bond opioricing formula derived is in he spiri of Shirakawa s 1991 closed form bond opion evaluaion resuls, in which he Poisson risk was assumed o be binomial, however here we provide a pricing formula allowing for muli-facor Poisson risk Markovian Spo Rae Dynamics under a Deerminisic Volailiy Srucure. Wihin he jump-diffusion framework and under paricular volailiy specificaions, he above erm srucure model admis a finie dimensional Markovian represenaion. These resuls are obained and presened in Nikiopoulos 25, however in his secion, we summarise he main resuls obained under deerminisic Wiener volailiies and consan jump volailiies, which are he volailiy specificaions ha lead o a closed form soluion for he bond opiorice as we have seen in he previous secion. Subsiuion of he condiion 13 ino 4, leads o he spo rae dynamics under he risk neural measure, which are of he form r = f, n w n w σ i s, ζ i s, ds σ i s, d W i s ψ i sβ i s, [1 e ξ is, ]ds β i s, [dq i s ψ i sds]. 51 Assume he case of deerminisic volailiy specificaions wih consan jump volailiies described in Assumpion 2.1. Using resuls from Nikiopoulos 25 and Chiarella & Nikiopoulos 23 Markovian spo rae dynamics can be obained under hese volailiy specificaions. For volailiy specificaions saisfying Assumpion 2.1, he dynamics for he spo rae 51 can be expressed in erms of a number of Markovian sochasic quaniies, See Proposiion of Nikiopoulos 25 for deails as [ n w n p ] dr = D ˆκ σi D σi κ σ1 D βi κ σ1 r d n w i=2 σ i d W i β i [dq i ψ i d], See Appendix 4 for deailed evaluaion of he inegrals.

13 12 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL where and D = κ σ1 f, f, E βi n w E σi, 53 ˆκ σi = κ σi κ σ1, 54 E σi = E βi = D σi = D βi = σ 2 i s, ds, 55 ψ i sβ 2 ie β i s ds, 56 σ i s, ζ i s, ds ψ i sβ i [1 e β i s ]ds σ i s, d W i s, 57 β i dq i s ψ i sds. 58 As he sochasic quaniies D σi and D βi display Markovian dynamics, he insananeous spo rae dynamics 52 are Markovian under he forward rae volailiy specificaions of Assumpion 2.1. The corresponding muli-facor bond price formula in erms of r and he sochasic quaniies D σi and D βi assumes he muli-facor exponenial affine represenaion See Proposiion of Nikiopoulos 25 for deails { P, T P, T = P, exp n w M, T N σ1, Tr N σi, T N σ1, TD σi where i=2 } T N σ1, TD βi, M, T = N σ1, Tf, 1 2 and T n w ψ i sβ i [1 e βiy s ]dyds T N σi, T 59 N 2 σi, TE σi 6 T ψ i sβ i [1 e β i s ]ds, e Ê y κσ i udu dy. 61 These Markovian represenaions of he jump-diffusion version of he Hull & Whie 199 ype model developed in his secion, will be used in Secion 4 where he simulaed bond opiorices are compared o he closed form soluion 43 for he bond opion price obained in Secion

14 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS The Sae Dependen Volailiy Model The sae dependen forward rae volailiy srucure is specified by allowing he Wiener volailiies o depend on f = r, f, T 1, f, T 2,...,f, T ns, a vecor of sae dependen variables including he insananeous spo rae and insananeous forward raes of differen fixed mauriies. This specific volailiy srucure is deermined by he following assumpion Assumpion 3.1. For i = 1,...,, he sae dependen Wiener volailiy srucure is of he form σ i s,, fs = σ i s, fse Ê s κ σiudu, 62 and for i = 1,...,n w, he ime dependen Poisson volailiy funcions coninue o be of he form, β i s, = β i se Ê s κ βiudu, where κ σi, κ βi and β i are deerminisic funcions of ime and σ i, f are ime and sae dependen funcions. I has been shown in Nikiopoulos 25 ha he above volailiy specificaions allow for Markovian represenaions of he erm srucure model. Consider a European call opion of mauriy T C wrien on a bond having mauriy T T > T C and denoe by C = C f,, T C he value of his bond opion a ime. Now however he underlying variables include no only r bu also a number of forward raes of fixed mauriies. For noaional convenience se f i = f, T i, wih i = 1, 2,..., n s. Thus using he jumpdiffusion version of Io s lemma o derive he sochasic differenial equaion for he bond opiorice, we have o ake ino accoun he dynamics of all he underlying facors, namely 2 for he f, T i, i = 1, 2,..., n s and 5 for he r. However he Wiener volailiies now depend on he vecor f. The muli-dimensional version of Io s lemma is applied o derive n C w n dc = KC p d D i CdW i J i CdQ i, 63 where KC ϑ β i, λ i 1 2 C 2 r n w n s n s k=1 j=1 ns C r σ 2 i,, f C f k f j n w j=1 n s j=1 α, T j 2 C r f j n w σ i, T k, fσ i, T j, f, β i, T j λ i C f j σ i,, fσ i, T j, f 64 D i C σ i,, ns C f r σ i, T j, j=1 f C f j, 65

15 14 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL for i = 1, 2,...,n w, and, J i C = Cr β i,, f 1 β i, T 1,...,f ns β i, T ns,, T C C f,, T C, 66 for i = 1, 2,...,. Developing he coninuous hedging argumen in a bond opion marke and imposing he condiion ha he riskless hedge porfolio earns he risk-free rae of ineres r, i follows ha here exis a vecor Φ = φ 1,...,φ nw and a vecor Ψ = ψ 1,...,ψ np such ha for bond opions of any mauriy T C i mus be he case ha nw n C KC p φ i D i C ψ i J i C rc =. 67 Furhermore in he curren conex, he no-arbirage forward rae drif resricion becomes α, T = n w σ i, T, f φ i ζ i, T, f β i, Tψ i e ξi,t λ i. 68 As in Secion 2, he parial differenial equaion 67 would need o be solved over T c and under boundary condiions appropriae o he ype of opion being evaluaed Markovian Spo Rae Dynamics under a Sae Dependen Volailiy Srucure. Similarly as in Secion 2.3, he spo rae dynamics under he risk neural measure, are of he form r = f, n w n w σ i s,, fsζ i s,,, fsds σ i s,, fsd W i s ψ i sβ i s, [1 e ξ is, ]ds β i s, [dq i s ψ i sds]. 69 Given he volailiy srucure consising of sae dependen Wiener volailiy funcions and deerminisic Poisson volailiy funcions as expressed in Assumpion 3.1, he spo rae dynamics 69 can be expressed in Markovian form, as saed in he following proposiion. Proposiion 3.1. Le σ i s,, fs and β i s,, saisfy Assumpion 3.1. Then he dynamics for he spo rae 69 can be expressed as [ n w n w n p ] dr = D E σi ˆκ σi D σi ˆκ βi D βi k σ1 r d n w i=2 σ i, fd W i β i [dq i ψ i d], 7

16 where and E σi = E βi = D σi = D βi = MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 15 D = κ σ1 f, f, E βi, 71 ˆκ σi = κ σi κ σ1, 72 ˆκ βi = κ βi κ σ1, 73 σ 2 i s,, fsds, 74 ψ i sβ 2 i s, e ξ is, ds, 75 σ i s,, fsζ i s,, fsds ψ i sβ i s, [1 e ξ is, ]ds σ i s,, fsd W i s, 76 β i s, dq i s ψ i sds. 77 Proof. See Proposiion of Nikiopoulos 25 for deails. The corresponding muli-facor bond price formula in erms of r and he sochasic quaniies E σi, D σi and D βi is given by { P, T P, T = P, exp M, T N σ1, Tr 1 n w Nσ 2 2 i, TE σi 78 n w n p } N σi, T N σ1, TD σi N βi, T N σ1, TD βi, where, and i=2 M, T =N σ1, Tf, N βi, T N x, T T T ψ i sβ i s, y[1 e ξ is,y ]dyds ψ i sβ i s, [1 e ξ is, ]ds, 79 e Ê y κxudu dy, x {σ i, β i }. 8 Using he exponenial affine erm srucure of ineres raes 78, we can express he insananeous forward rae in erms of r and he sochasic quaniies E σi, D σi and D βi, as f, T = f, T M, T T n w i=2 N σi, T T N σ 1, T T N σ 1, T T D σi n w N σi, T N σi, T E σi 81 T Nβi, T N σ 1, T D βi. T T

17 16 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL The relaionship 81 can be used o express he benchmark forward raes f, T j, wih j = 1, 2,..., n s of he sae dependen volailiy funcions in erms of he sochasic sae variables r, E σi, D σi and D βi. In addiion, by aking he number n s of he fixed forward rae mauriies used in he sae dependen volailiy srucure f = r, f, T 1, f, T 2,...,f, T ns, equal o he number of sochasic quaniies 2n w, we have a fully specified sysem wih he forward raes of any mauriy evaluaed by 81. See Secion 2.4 of Nikiopoulos 25 for more discussion on how finie dimensional affine realisaions in erms of forward raes may be obained. These Markovian represenaions of he jump-diffusion version of he sae dependen erm srucure model developed here, will be used in Secion 4 where he above Markovian erm srucure of ineres raes is simulaed in order o obain bond opiorices. In paricular, we use he Markovian represenaion in erms of he sochasic quaniies as in Proposiion 3.1, raher han in erms of a se of benchmark forward raes. 4. Mone Carlo Simulaions Mone Carlo simulaion for derivaive pricing, when he underlying asse follows a mulivariae sae dependen volailiy jump-diffusiorocess, is exremely inensive compuaionally, as he variance of he sampled variable is usually large and for N sample pahs he sandard errors of he Mone Carlo simulaions decreases only as 1/ N. To improve he Mone Carlo efficiency, one should employ some sor of variance reducion mehodology, namely aniheic variable, conrol variaes, sraified sampling and imporance sampling and/or use low discrepancy sequences. A conrol variae echnique was developed by Chiarella, Clewlow & Musi 23 for a sae dependen volailiy HJM model when he forward rae dynamics are driven by diffusiorocesses. We exend his o accommodae our jump-diffusion seing, also aking advanage of he Markovian represenaions ha have been obained under he paricular volailiy specificaions. For he one Wiener/wo Poisson case we examine wo classes of models. The firs one is he deerminisic volailiy DV model wih volailiies and σ, T = σ e κσt, 82 β i, T = β i, wih i = 1, This model yields closed form soluions of he form 43 for bond opiorices. 12 The second model is he sae dependen volailiy SV model, σ, T, f = σ, fe κσt, 12 For he volailiy specificaions 82 and 83, he quaniies 32, 33 and 34 simplify c, T C = υ, T C = σ 2, T C = 1 T C 1 T C 1 T C n p n p n w ψ i β i 1 e β it C, 84 ψ i β i 1 e β it C e β it e β it T C, 85 σ 2 2κ 3 i e κ it c e κ it 2 e 2κ it e 2κ i 2κ i T 2κ i T c. 86

18 where MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 17 σ, f = {.5 σ, L f <.5; σ [L f.5 γ.5], L f.5; wih L f = c r 3 h=1 c hfs, T h and γ = 1 2. Also we consider β i, T = β i e k βit and consan ψ i. Recall ha in he curren seup f = r, f, T 1, f, T 2, f, T 3. Also noe ha, since hese Markovian srucures may drive he forward rae o negaive values, he sae dependen volailiy funcions 87 have been seleced so as o be well-defined in such a case see Appendix Simulaion Scheme. Le be ime, T be mauriy, and T be he ime horizon where T T. The ime horizon, T is subdivided ino N inervals of lengh = T N so ha = n and T = m. This scheme requires he knowledge of he iniial forward curve f, T. The iniial forward rae curve considered here 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 gives a reasonable fi o he US zero yields on July 2, 21, wih mauriies up o 1 years and including he overnigh rae The iniial bond price. We recall ha he iniial bond price P, T is given by relaionship T P, T = exp f, sds. 88 The bond price P, T can be also expressed in erms of a risk neural expecaion as [ T ] P, T = Ẽ exp rsds F, 89 so ha he iniial bond price P, T may also be expressed as [ T ] P, T = Ẽ exp rsds F. 9 The bond price esimaed by performing simulaions over Π pahs, should coincide closely wih he inpu iniial bond price P, T 1 Π N exp r i j. 91 Π i= j= 87 σ.15 κ σ.18 β 1.2 β 2.3 ψ 1 1 ψ Table 1. Parameer Values. Equaion 91 can be used o provide a check on he accuracy of our simulaion schemes, iaricular giving an indicaion of he size of he discreisaion bias. Table 2 provides

19 18 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL he simulaed iniial bond prices for he deerminisic volailiy models when he parameer values are se as in Table 1. The discreised spo rae dynamics used in he simulaion scheme are he Markovian dynamics presened in Secion 2.3, recall equaion 52. P, 1 = exp 1 f, sds = N Π P,1 S. Dev. S. Err. 2 5, , , , , , , , , Table 2. Iniial Bond Prices - DV models. Table 3 presens he simulaed iniial bond prices, of a bond mauring in 1 year, for he sae dependen volailiy models, and when he parameer values are se as in Table 1. In addiion, we se κ β1 =.31, κ β2 =.17, c = 1, c 1 = 2, c 2 = 1, c 3 = 2. The discreised spo rae dynamics are he Markovian dynamics described in Secion 3.1, see iaricular equaion 7, wih he benchmark forward raes expressed in erms of he sochasic facors of he sysem, by using equaion 81. P, 1 = exp 1 f, sds = N Π P,1 S. Dev. S. Err. 2 5, , , , , , , , , Table 3. Iniial Bond Prices - SV models. We consider discreisaion error 13 o be eviden when he disance beween he rue and simulaed bond prices exceeds wo sandard deviaions, i.e. when he rue price lies ouside he 95% confidence inerval around he Mone Carlo esimae. In Table 2, his is he case for 5, pahs or more, wih he error appearing o be somewha reduced when is reduced as would be expeced. Iaricular, discreisaion error is no longer 13 Sandard error is defined as he difference beween exac price and simulaed price divided by he sandard deviaion of he simulaed prices.

20 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 19 eviden a he curren number of pahs when he discreisaion level is increased o 8. I is imporan o keep he magniude of he error from his source in mind when inerpreing he resuls from he simulaions in he subsequen secions. The iniial bond price resuls obained by he simulaions for boh models deerminisic volailiy DV and sochasic volailiy SV model are consisen, o four decimal place accuracy, especially when we reduce he discerisaion bias by seing he discreisaion level o N = 8 wih he value obained from he analyical bond price 88, providing evidence of he effeciveness of his numerical scheme Bond Opion Price Evaluaion. Denoe wih C, T c, T he ime -value of a European call opion mauring a T c on he zero-coupon bond wih mauriy T, where T c T. The curren value of a European call opion C, T c, T can be evaluaed, under he risk neural measure as he expeced discouned payoff of he opion a he opion s mauriy [ { Tc } ] C, T c, T = Ẽ exp rsds PT c, T E F, 92 or, alernaively, under he T c -forward measure, as C, T c, T = P, TE [ PT c, T E F ]. 93 For simulaion based approaches o bond opioricing, we have found ha he use of one or he oher probabiliy measure does no seem o provide any significan advanage. Here we repor he simulaions under he risk neural probabiliy measure. Given he Markovian spo rae dynamics under he risk neural measure, which are equaion 52 for he deerminisic volailiy models and equaion 7 for he sae dependen volailiy models, he bond opiorice is evaluaed using formula 92, by using he Euler-Maruyama scheme for he inegraion, as { } C, T c, T = 1 K N exp r k i P k T c, T E. 94 K k=1 The bond price P k T c, T is compued by he exponenial affine erm srucure 59 for he deerminisic volailiy model and 78 for he sae dependen volailiy model. Table 4 shows he simulaed bond opiorices for he deerminisic volailiy model under he parameer values given in Table 1. The exercise price is se E =.95 and he exac value for he bond opiorice is evaluaed from 43. Exac Opion Price N Π CMC DV C DV exac,.5, 1 = ,.5, 1 S. Dev. S. Err. 2 5, , , , , , Table 4. Call Bond Opion Prices - DV models.

21 2 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL The bond opiorices obained by he Mone Carlo simulaions for he deerminisic volailiy model are consisen wih he value obained from he analyical bond opion price 43 wih accuracy reaching hree significan figures for he 5, simulaed pahs and over. This provides evidence ha he numerical scheme employed here is effecive. Table 5 shows he simulaed bond opiorices for he sochasic volailiy ype of models under he parameer values shown in Table 1, and for κ β1 =.31 and κ β2 =.17. Recall ha he Wiener sae dependen volailiies have he funcional form 87, wih c = 1, c 1 = 2, c 2 = 1 and c 3 = 2. The exercise price is se o be E =.95. N Π CMC SV,.5, 1 S. Dev. 2 5, , , , , , Table 5. Call Bond Opion Prices - SV models. The bond opiorices obained by he Mone Carlo simulaions for he sochasic volailiy model are consisen o a leas wo significan figures, however in he nex secion we will aemp o improve convergence of he sochasic volailiy numerical scheme by an applicaion of a conrol variae mehod Conrol Variae Mehod. The applicaion of Mone Carlo simulaions o evaluae bond opiorices under he HJM framework comes a he expense of significan compuaional effor. To improve convergence we propose o use a conrol variae mehod. The DV model under deerminisic Wiener volailiies and consan jump sizes accommodaes a closed form opioricing formula and hus we can compue he opion price Cexac. DV The SV model sae dependen Wiener volailiies and deerminisic jump volailiies can only be evaluaed numerically. Running simulaions of hese wo models, he opiorices CMC DV under he deerminisic volailiy model and he opiorices CMC SV under he sochasic volailiy model are esimaed. The conrol variae adjusmen proposes ha he approximaed opion value of he sochasic volailiy model is evaluaed by C SV = C SV MC C DV MC C DV exac. 95 The raionale of he conrol variae mehod is ha he known error imposed by he Mone Carlo simulaions in he case of he deerminisic volailiy model CMC DV Cexac, DV is assumed o be close o he error of he Mone Carlo esimaion for he case of sochasic volailiy model, namely CMC SV C SV. Evaluaing 95 can be ime consuming since i requires he resuls of wo simulaions. However, use of he Markovian represenaions of he models considered have considerably simplified and sped up he calculaion. As Table 6 shows he sandard errors of he

22 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 21 opion values esimaed by he conrol variae mehod are of he order of approximaely one sevenh wih respec o he values obained by he sandard Mone Carlo simulaion of he Markovian sochasic volailiy erm srucure model. 14 This reducion is uniform across he order of discreisaion and he number of simulaed pahs. The accuracy on he bond opiorice has increased o hree significan figures by he applicaion of he conrol variae scheme compared o he wo significan figures accuracy obained by he applicaion of he sandard Mone Carlo simulaion. N Π C SV,.5, 1 S. Dev. C CV,.5, 1 S. Dev. 2 5, , , , , , Table 6. Call Bond Opion Prices - SV models; Conrol Variae Mehod. To jusify he efficiency of he conrol variae mehod, we ensure firsly ha E[C DV MC C DV exac] =. 96 From Table 4, we conclude ha condiion 96 holds since insignifican discreisaion error exiss iaricular when we increase he order of he discreisaion o 4. The conrol variae mehod is employed here o price he same produc - bond opions - under wo differen models. This is a somewha unorhodox and - o our knowledge - new perspecive on conrol variae mehods iricing derivaives. Typically a conrol variae mehod is applied o anoher closely relaed insrumen priced in he same model, whereas here he conrol variae is he same insrumen priced in a closely relaed model. Under hese model specificaions he sae variables evolve differenly. However, he sae variables can be seen as simply wo differen ses of funcions of he driving Wiener and Poissorocesses, which are he same in boh models. Therefore, he conrol variae mehod will be correcly used if he sae variables of he wo models considered are highly correlaed. Table 7 presens he correlaion coefficiens of he sae variables, which are common in hese wo classes of models, and hese are he r, D β1 and D β2. The sae variables are clearly highly posiively correlaed, hus one can reasonably expec he conrol variae mehod o be effecive in his conex. Thus, he combinaion of hese Markovian srucures wih a conrol variae mehod provides an efficien numerical scheme ha may yield good resuls in Mone Carlo simulaion even wih a relaively small number of simulaed pahs. Given ha he conrol variae mehod improves he sandard error by seven imes relaive o he sandard error obained from he sandard Mone Carlo simulaion on he sochasic volailiy model, one may obain he same order of accuracy wih fory-nine imes less he number of simulaed pahs. An imporan conribuion o he efficiency of his numerical scheme mus 14 Thus a back-of-an-envelope calculaion indicaes ha compuaional efficiency can be gained by using he mehod proposed here: Conrol variaes a mos double he compuaional effor compared o he same number of simulaions wihou conrol variaes, whereas he number of simulaions wihou conrol variaes would have o be increased by a facor of 7 2 = 49 in order o achieve he same accuracy as a given number of simulaions wih conrol variaes.

23 22 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL N Π r D β1 D β2 2 5, , , , , , Table 7. Correlaion Coefficiens. be aribued o he fac ha he models developed here possess Markovian dynamics. All he parameers used in he simulaions such as bond prices, benchmark forward raes used in he volailiy srucure could be expressed in erms of he sae variables of he Markovian sysem. Discreisaion has only been applied o he dynamics of he sae variables of he sysem, herefore a lo of numerical evaluaions have been avoided. 5. Conclusions This paper develops wo models o price bond opions when ineres raes are subjec o jumps. In he firs model, boh Wiener and Poisson volailiies are ime dependen, and working wihin he Shirakawa general HJM model, we have derived he parial differenial-difference equaion for he pricing of bond opions. In addiion, by employing Fourier ransform echniques, bond opiorices have been evaluaed and an easily racable Black-Scholes ype bond opioricing formula under he assumpion of consan jump volailiy has been derived. In he second model, he volailiy srucure is more general, by allowing for sae dependen Wiener volailiies and ime dependen Poisson volailiies. In his second model, i is difficul o explicily solve he bond opion pricing problem, herefore Mone Carlo simulaion echniques are used o evaluae bond opions. However, under appropriae volailiy funcions, he erm srucures obained for boh models display Markovian dynamics. These Markovian represenaions conribue o increase he efficiency and accuracy of he applicaion of he Mone Carlo simulaions. Addiionally, aking advanage of he closed form soluions obained under he deerminisic volailiy seing, we employ a conrol variae mehod ha significanly improves he efficiency of he numerical procedure. The imporan characerisic of he soluions for bond opiorices proposed in his paper is ha hey incorporae he complexiy of a sochasic volailiy and/or jumpdiffusion model alhough hey enjoy compuaional racabiliy due o he Markovian srucures used. A worhwhile exension of his work would be o fi o empirical informaion o calibrae he model parameers as well as he volailiy smile. Appendix 1. The No-Arbirage Condiion in he Bond Opion Marke Recall he sochasic differenial equaion for he spo rae dr = ϑd n w σ i, dw i β i, [dq i λ i d], 97

24 where ϑ is defined as MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 23 ϑ = f, α, n w αs, ds σ is, dw i s β is, [dq i s λ i ds]. 98 Using he jump-diffusion version of Io s lemma we derive he sochasic differenial equaion for he bond opiorice dc = C ϑ n w n p β i, λ i σ i, C r dw i C r 1 2 n w σi 2, 2 C r 2 d [Cr β i,,, T C Cr,, T C ]dq i, 99 We consider a hedging porfolio conaining a bond wih mauriy T and n o = n w bond opions of mauriies T 1, T 2,, T no iroporions w 1, w 2,, w no wih w 1 w 2 w no1 = 1, where w no1 is he proporion corresponding o he bond. All hese opions are wrien on he bond having mauriy T. If we denoe wih C i = C, T i i = 1, 2,...,n o he value of he i h bond opion, we may wrie he sochasic differenial equaion for C i in he general form dc i C i = µ C i d n w j=1 ν Ci,j dw χ Ci,j dq j, j=1 where µ Ci = 1 C i ν Ci,j = 1 C i C i r C i n w j=1 ϑ β i, λ i σ ij,, χ Ci,j = 1 n p [C i r β ij,, C i r, ]. C i j=1 C i r 1 2 C i 2 r 2 n w j=1 σij, 2, Also recall he sochasic differenial equaion for he bond price P where dp P = µ Pd n w ν Pi dw i χ Pi dq i, µ P = r H, T, ν Pi = ζ i, T, andχ Pj = η j, T 1.

25 24 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL Le V be he value of he hedging porfolio hen he reurn on he porfolio is given by dv V = w dc 1 dc 2 dc no 1 w 2 w no C 1 C 2 C no n o n o = w i µ Ci d w no1µ P d n o w i dp w no1 P n w w i ν Cij dw j w no1 j=1 χ Ci,j dq j w no1 χ Pj dq j. j=1 j=1 n w j=1 ν Pj dw j In order o eliminae boh Gausian and Poisson risks we need o choose w 1, w 2,, w no1 so ha n o n o w i ν Ci,j w no1ν Pj =, when j = 1, 2,...,n w 1 w i χ Ci,j w no1χ Pj =, when j = 1, 2,...,. 11 The hedging porfolio hen becomes riskless, hus, i should earn he risk-free rae of ineres r, of he Gaussian bond marke, i.e., which can be simplified o no dv V = w i µ Ci d w no1µ P d = rd, n o w i µ Ci r w no1µ P r =, 12 using also he fac ha w 1 w 2 w no1 = 1. Equaions 1, 11 and 12 form a sysem of n o 1 equaions wih n o 1 unknowns w 1, w 2,, w no1. This sysem can only have a non-zero soluion if he deerminan ν C1,1 ν C2,1 ν Cno,1 ν P1.... ν C1,nw ν C2,nw ν Cno,nw ν Pnw χ C1,1 χ C2,1 χ Cno,1 χ P1.... χ C1,np χ C2,np χ Cno,np χ Pnp µ C1 r µ C2 r µ Cno r µ P r is equal o zero. The above equaion implies ha for h = 1, 2,...,n o here exis φ 1, φ 2,..., ψ nw and ψ 1, ψ 2,..., ψ np such ha and µ Ci r = µ P r = n w j=1 n w φ j ν Ci,j ψ j χ Ci,j, 13 j=1 φ i ν Pi ψ i χ Pi. 14

26 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 25 Thus using equaions 13 for bond opions of any mauriy T C we mus have ha µ C r = n w φ i ν Ci ψ i χ Ci, and subsiuing he expressions for µ C, ν Ci and χ Ci, we have ha 1 C C C ϑ β i, λ i r 1 n w σi 2, 2 C 2 r 2 r = 1 C C r n w or afer furher manipulaions φ i σ i, ψ i 1 C [Cr β i,, Cr, ], 15 C ϑ β i, λ i n w φ i σ i, C r 1 2 ψ i [Cr β i,, Cr, ] =. n w σ 2 i, 2 C r 2 rc 16 By subsiuing he expressions for µ P, ν Pi and χ Pi in equaion 14, we derive he drif resricion α, T = n w σ i, T φ i ζ i, T β i, Tψ i e ξi,t λ i. 17 Appendix 2. Applicaion of Io s lemma on Y The dynamics for P, T C are given by dp, T C = rp, T C d and he dynamics for Cr,, T are dc =rcd C r n w ζ i, T C P, T C d W i P, T C e ξ i,t C 1[dQ i ψ i d], n w σ i d W i [Cr β i,, T Cr,, T][dQ i ψ i d] Define he new quaniy Y C, P = Cr,, T Pr,, T C 11

27 26 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL hen Y C = 1 P, 2 Y C 2 =, Y P = C P 2 2 Y P 2 = 2C P 3, 2 Y P C = 1 P 2. Applicaion of he muli-dimensional jump-diffusion version of Io s Lemma leads o [ Y dy = rp e ξ i,t C 1ψ i P Y P rc [Cr β i Cr]ψ i Y C 1 2 n w n w ζ 2 i, T C P 2 2 Y P 2 2ζ C i, T C Pσ i r d W i ζ i, T C P Y P σ C Y i r C 2 Y P C σ2 i C r 2 2 Y C 2 ] d 111 [{Y Cr Cr β i Cr, P Pe ξ i,t C 1 Y Cr, P}]dQ i, 112 and afer subsiuion of he parial derivaives i is simplified o [ dy = ry ry 1 n w 2ζi 2, T C Y 2ζ i, T C σ ] i C 2 C r Y d and furher o n w Y dy Y = nw e ξ i,t C Cr β i C ζ i, T C Y σ i C Cr βi e ξ i,t C Cr 1 ζ i, T C σ i C C r Y ψ i Y d d W i dq i, n Cr βi e ξ i,t C 1 Cr C [d W i ζ i, T C d] r [dq i ψ i e ξ i,t C d] Appendix 3. Fourier Transform Technique Define he Fourier ransform of he soluion Υ = ΥZ, o he parial differenial equaion 28 by Υω, = ΥZ, e iωz dz, 115

28 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 27 where i = 1 is he imaginary uni. Then while 15 and Also noe ha Υ e iωz dz = Υω,, 116 Υ Z e iωz dz = Υe iωz iω Υe iωz dz = iωυω,, Υ Z 2 e iωz dz = Υ Z e iωz Υ iω Z e iωz dz = Υe iωz ω 2 Υe iωx dz ΥZ ln e ξ i,t e ξ i,t C e iωz dz = e e ξ iω ln i,t e ξ i,t C = ω 2 Υω,. 118 ΥZ ln e ξ i,t e ξ i,t C e iωzln e ξ i,t e ξ i,t C dz iω e ξ i,t = e ξ Υω,, 119 i,t C Using he resuls , he parial differenial equaion 28 for ΥZ, becomes an ordinary differenial equaion wih complex coefficiens for Υω,, i.e., Υω, = iω ω2 2 [ np n w ψ i e ξ i,t C e ξ i,t 1 2 ζ i, T C ζ i, T 2 n w ζ i, T C ζ i, T 2 ] n p iω ψ i e ξ i,t C ψ i e ξ i,t C e ξ i,t e ξ Υω,. i,t C Noe ha we assume Υe iωz = and Υ Z e iωz =. I is necessary laer o verify ha he soluion obained based on hese assumpions saisfies he parial differenial equaion 28. The assumpion is hen jusified on he basis of uniqueness of he soluion.

29 28 CARL CHIARELLA, CHRISTINA NIKITOPOULOS SKLIBOSIOS AND ERIK SCHLÖGL Equaion 12 may be expressed as [ { n p Υω, exp ψ i se ξ is,t C ds [ np iω ψ i s e ξ is,t C e ξ is,t ds 1 2 ω2 2 n w Inegraing from o T C { n p Υω, =Υω, T C exp Le iω [ np ω2 2 TC ζ i s, T C ζ i s, T 2 ds n w ζ i s, T C ζ i s, T 2 ds iω ψ i se ξ is,t C e ξ is,t ds e ξ =. 121 is,t C TC n w TC TC ψ i s ψ i se ξ is,t C ds e ξ is,t C e ξ is,t ds 1 TC 2 ζ i s, T C ζ i s, T 2 ds ψ i se ξ is,t C c, T C = 1 T C υ, T C = 1 T C σ 2, T C = 1 T C ξω,, T C = 1 T C TC n w ζ i s, T C ζ i s, T 2 ds iω e ξ is,t ds e ξ is,t C. 122 TC TC n w TC hen equaion 122 is simplified o Υω, = Υω, T C e T C By he Fourier inversion heorem, we have ha ψ i se ξ is,t C ds, 123 ψ i s e ξ is,t C e ξ is,t ds, 124 ζ i s, T C ζ i s, T 2 ds, 125 iω ψ i se ξ is,t C e ξ is,t e ξ ds, 126 is,t C c,t C iω[υ,t C 1 2 σ2,t C ] ω2 2 σ2,t C ξω,,t C ΥZ, = 1 2π. 127 Υω, e iωz dω. 128 ] ]

30 MONTE CARLO SIMULATIONS OF HJM MODELS WITH JUMPS 29 Thus, by subsiuing 31 ino 128 we obain ΥZ, = 1 Υω, T C 2π { exp T C c, T C iω[υ, T C 12 σ2, T C ] ω2 2 σ2, T C ξω,, T C = e T C c,t C 2π } iωz dω Υω, T C e iω[υ,t C 1 2 σ2,t C ]T C Z ω2 2 σ2,t C T C ξω,,t C T C dω. Noe ha by changing he variable Z back o he variable X recall ha Z = lnx, we obain and so Y X, = e c,t CT C 2π Y e Z, T C e iωz dz = e c,t CT C 2π Y e Z, T C = e c,t CT C Υω, T C = where he kernel K is defined by KZ, X, = 1 2π = ΥZ, T C e iωz dz Y e Z, T C e iωz dz, 129 e iω[υ,t C 1 2 σ2,t C ]T C ln X ω2 2 σ2,t C T C ξω,t C,T C dω e iω[υ,t C 1 2 σ2,t C ]T C ln X Z ω2 2 σ2,t C T C ξω,,t C T C dω dz Y e Z, T C KZ, X, dz, 13 e iω[υ,t C 1 2 σ2,t C ]T C ln X Z ω2 2 σ2,t C T C ξω,,t C T C dω. 131 Appendix 4. Derivaion of Black-Scholes ype Inegral We se as I he inegral I = ln E hen by performing furher manipulaions I = C 1 e Z Ee [υ,t 2 σ 2,T C ]T C ln X Z Ènp p i µ i 2 2σ 2,T C T C dz, 132 C 1 e Z e [υ,t 2 σ 2,T C ]T C ln X Z Ènp p i µ i 2 2σ 2,T C T C dz 133 ln E C 1 e [υ,t 2 E ln E σ 2,T C ]T C ln X Z Ènp p i µ i 2 2σ 2,T C T C dz.