On Monte Carlo Simulation for the HJM Model Based on Jump

Transcription

1 On Mone Carlo Simulaion for he HJM Model Based on Jump Kisoeb Park 1, Moonseong Kim 2, and Seki Kim 1, 1 Deparmen of Mahemaics, Sungkyunkwan Universiy , Suwon, Korea Tel.: , 734 {kisoeb, skim}@skku.edu 2 School of Informaion and Communicaion Engineering Sungkyunkwan Universiy , Suwon, Korea Tel.: moonseong@ece.skku.ac.kr Absrac. We derive a form of he HJM model based on jump. Heah, Jarrow, and Moron(HJM) model is widely acceped as he mos general mehodology for erm srucure of ineres rae models. We represen he HJM model wih jump and give he analyic proof for he HJM model wih jump. We perform he Mone Carlo simulaion wih several scenarios o achieve highly precise esimaes wih he brue force mehod in erms of mean sandard error which is one measure of he sharpness of he poin esimaes. We have shown ha bond prices in HJM jump-diffusion version models of he exended Vasicek and CIR models obained by Mone Carlo simulaion correspond wih he closed form values. 1 Inroducion Approaches o modeling he erm srucure of ineres raes in coninuous ime may be broadly described in erms of eiher he equilibrium approach or he no-arbirage approach even hough some early models include conceps from boh approaches. The no-arbirage approach sars wih assumpions abou he sochasic evoluion of one or more underlying facors, usually ineres rae. Bond prices are assumed o be funcions of he hese driving sochasic processes. Heah, Jarrow and Moron (HJM)[4] is widely acceped as he mos general mehodology for erm srucure of ineres rae models. The major conribuion of he HJM model [4], as i allows he model o be no-arbirage, a major improvemen over he Ho and Lee[5] and oher similar models. We will represen he HJM model wih jump. In pricing and hedging wih financial derivaives, jump models are paricularly imporan, since ignoring jumps in financial prices will cause pricing and hedging raes. Term srucure model soluions under HJM model wih jump is jusified because movemens in forward raes display boh coninuous and disconinuous behavior. These jumps are caused by several marke Corresponding auhor. V.N. Alexandrov e al. (Eds.): ICCS 26, Par I, LNCS 3991, pp , 26. c Springer-Verlag Berlin Heidelberg 26

2 On Mone Carlo Simulaion for he HJM Model Based on Jump 39 phenomena money marke inervenions by he Fed, news surprise, and shocks in he foreign exchange markes, and so on. The HJM model wih jump uses as he driving sochasic dynamic variable forward raes whose evoluion is dependen on a specified volailiy funcion. The mos models of forward raes evoluion in he HJM framework resul in non-markovian models of he shor erm ineres rae evoluion. This model depend on he enire hisory of forward raes. Therefore, his model is difficul of he acual proof analysis of he HJM model wih jump. In his sudy, we go achieved o make he acual proof analysis of he HJM model wih jump easy. The HJM model wih volailiy funcion was sudied by Hull and Whie, Carverhill, Richken and Sankarasubramanian (RS)[9], Inui and Kijima, and Bhar and Chiarella in heir aemp o obain Markovian ransformaion of he HJM model. We examines he one-facor HJM model wih jump which we use resricive condiion of RS. We invesigae he resricive condiion of RS. In addiion, we inroduce he Mone Carlo simulaion. One of he many uses of Mone Carlo simulaion by financial engineers is o place a value on financial derivaives. Ineres in use of Mone Carlo simulaion for bond pricing is increasing because of he flexibiliy of he mehods in handling complex financial insiuions. One measure of he sharpness of he poin esimae of he mean is Mean Sandard Error. Numerical mehods ha are known as Mone Carlo mehods can be loosely described as saisical simulaion mehods, where saisical simulaion is defined in quie general erms o be any mehod ha uilizes sequences of random numbers o perform he simulaion. The srucure of he remainder of his paper is as follows. In he secion 2, he HJM model wih jump are inroduced. In he secion 3, we calculae numerical soluions using Mone Carlo simulaion for he HJM model wih jump. In he secion 4, we invesigae he HJM model wih he jump version of he exended Vasicek and CIR models. This paper is finally concluded in secion 5. 2 Heah-Jarrow-Meron(HJM) Model wih Jump The HJM consider forward raes raher han bond prices as heir basic building blocks. Alhough heir model is no explicily derived in an equilibrium model, he HJM model is a model ha explains he whole erm srucure dynamics in a no-arbirage model in he spiri of Harrison and Kreps[6], and i is fully compaible wih an equilibrium model. If here is one jump during he period [, + d] hen dπ() = 1, and dπ() = represens no jump during ha period. We will ignore axes and ransacion coss. We denoe by V (r, r, T ) he price a ime of a discoun bond. I follows immediaely ha V (r, T, T )=1.We consider he muli-facor HJM model wih jump of erm srucure of ineres rae is he sochasic differenial equaion(sde) for forward rae df (, T )=μ f (, T )d + n σ fi (, T )dw i ()+ i=1 n J i dπ i () (1) i=1

3 4 K.Park,M.Kim,andS.Kim where, μ f (, T ) represens drif funcion; σ 2 f i (, T ) is volailiy coefficiens; J i is he magniude of a jump wih J i N(θ, δ 2 ); in his sochasic process n independen Wiener processes and Poisson processes deermine he sochasic flucuaion of he enire forward rae curve saring from a fixed iniial curve. The main conribuion of he HJM model is he parameers μ f (, T )and σ fi (, T ) canno be freely specified; drif of forward raes under he risk-neural probabiliy are enirely deermined by heir volailiy and by he marke price of risk. We inroduce he no-arbirage condiion as follows: n μ f (, T )= σ fi (, T )(λ i () σ fi (, s)ds) (2) i=1 where, λ i () represens he insananeous marke price of risk and ha is independen of he mauriy T. Furhermore, by an applicaion of Girsanov s heorem he dependence on he marke price of ineres rae risk can be absorbed ino an equivalen maringale measure. Thus, he Wiener processes is dw Q i () =dw i()+λ i ()ds We consider he one-facor HJM model wih jump of he erm srucure of ineres rae(ha is, n = 1). Subsiuing he above he equaion ino no-arbirage condiion(3), we represen he sochasic inegral equaion he following: f(, T ) f(,t)= + σ f (u, T ) σ f (, s)dsdu π() σ f (s, T )dw Q (s)+ J j (3) where, dw Q i is he Wiener process generaed by an equivalen maringale measure Q. Thesporaer() =f(, ) is obained by seing T = in he equaion (5), so ha π() r() =f(,)+ μ f (s, T )ds + σ f (s, T )dw Q (s)+ J j (4) where, μ f (, T ) = σ f (, T ) σ f (, s)ds, and dw Q () is a sandard Wiener process generaed by he risk-neural measure Q. Under he corresponding riskneural measure Q, he explici dependence on he marke price of risk can be suppressed, and we obain he differenial form of (3) is given by df (, T )=μ f (, T )d + σ f (, T )dw Q ()+Jdπ. (5) We know ha he zero coupon bond prices are conained in he forward rae informaions, as bond prices can be wrien down by inegraing over he forward rae beween and T in erms of he risk-neural process ( ) T V (, T )=exp f(, s)ds. (6) j=1 j=1

4 On Mone Carlo Simulaion for he HJM Model Based on Jump 41 From he equaion (3), we derive he zero coupon bond prices as follow: V (, T )=e R T f(,s)ds = V (,T) e (R V (,) R T R R P μ f (u,s)dsdu+ T σ f (u,s)dsdw Q π() R (u)+ T j=1 Jj ds) (7) where, we define as V (,)=e R f(,s)ds,v(,t)=e R T f(,s)ds,and μ f (, T )=σ f (, T ) σ f (, s)ds. The mos models of forward raes evoluion in he HJM framework resul in non-markovian models of he shor erm ineres rae evoluion. As above he equaion (7), hese inegral erms depend on he enire hisory of he process up o ime. Bu, numerical mehods for Markovian models are usually more efficien han hose necessary for non-markovian models. We examines he one-facor HJM model wih jump which we use resricive condiion of RS[9]. RS have exended Carverhill resuls showing ha if he volailiies of forward raes were differenial wih respec o mauriy dae, for any given iniial erm srucure, if and only if for he prices of all ineres rae coningen claims o be compleely deermined by a wo-sae Markov process is ha he volailiy of forward rae is of he form ( σ f (, T )=σ r ()exp ) a(s)ds where, σ r and a are deerminisic funcions. For he volailiy of forward rae is of he form (8), he following formula for he discoun bond price V (, T )was obained in resricive condiion of RS. Theorem 1. Le σ f (, T ) be as given in (8), hen discoun bond price V (, T ) is given by he formula where, φ() ξ() V (, T )= V (,T) V (,) exp ϕ(, T )= = exp σ f 2 (s, )ds = [f(,) r()] { 1 } 2 ϕ2 (, T )φ()+ϕ(, T )ξ()] ( u a(s)ds ) du As we menioned already, a given model in he HJM model wih jump will resul in a paricular behavior for he shor erm ineres rae. We inroduce relaion beween he shor rae process and he forward rae process as follows. In his sudy, we jump-diffusion version of Hull and Whie model o reflec his resricion condiion. We know he following model for he ineres rae r; dr() =a()[θ()/a() r()]d + σ r ()r() β dw Q ()+Jdπ(), (1) where, θ() is a ime-dependen drif; σ r () is he volailiy facor; a() ishe reversion rae. We will invesigae he β = case is an exension of Vasicek s jump diffusion model; he β =.5 case is an exension of CIR jump diffusion model. (8) (9)

5 42 K.Park,M.Kim,andS.Kim Theorem 2. Le be he jump-diffusion process in shor rae r() is he equaion (1). Le be he volailiy form is ( wih η(, T )=exp a(s)ds σ f (, T )=σ r ()( r()) β η(, T ) (11) ) is deerminisic funcions. We know he jumpdiffusion process in shor rae model and he corresponding compaible HJM model wih jump df (, T )=μ f (, T )d + σ f (, T )dw Q ()+Jdπ() (12) where μ f (, T )=σ f (, T ) σ f (, s)ds. Then we obain he equivalen model is f(,t)=r()η(,t)+ θ()η(s, T )ds σ 2 r(s)(r(s) 2 ) β η(s, T ) s (η(s, u)du)ds (13) ha is, all forward raes are normally disribued. Noe ha we know ha β = case is an exension of Vasicek s jump diffusion model; he β =.5 case is an exension of CIR jump diffusion model. Noe ha he forward raes are normally disribued, which means ha he bond prices are log-normally disribued. Boh he shor erm rae and he forward raes can become negaive. As above, we obain he bond price from he heorem 1. By he heorem 2, we drive he relaion beween he shor rae and forward rae. Corollary 1. Le be he HJM model wih jump of he erm srucure of ineres rae is he sochasic differenial equaion for forward rae f(, T ) is given by df (, T )=σ f (, T ) σ f (, s)dsd + σ f (, T )dw Q ()+Jdπ() (14) where, dw Q i is he Wiener process generaed by an equivalen maringale measure Q and σ f (, T )=σ r ()( ( r()) β exp ) T a(s)ds. Then he discoun bond price V (, T ) for he forward rae is given by he formula V (, T )= V (,T) V (,) exp{ 1 2 wih he equaion (13). ( σ f (, s)ds σ f (, T ) σ f (, s)ds [f(,) r()]} σ f (, T ) ) 2 σf 2 (s, )ds Noe ha we know ha β = case is an exension of Vasicek s jump diffusion model; he β =.5 case is an exension of CIR jump diffusion model.

6 On Mone Carlo Simulaion for he HJM Model Based on Jump 43 3 Mone Carlo Simulaion of he HJM Model wih Jump Recen mehods of bond pricing do no necessarily exploi parial differenial equaions(pdes) implied by risk-neural porfolios. They res on convering prices of such asses ino maringales. This is done hrough ransforming he underlying probabiliy disribuion using he ools provided by he Girsanov s heorem. A risk-neural measure is any probabiliy measure, equivalen o he marke measure P, which makes all discouned bond prices maringales. We now move on o discuss Mone Carlo simulaion. A Mone Carlo simulaion of a sochasic process is a procedure for sampling random oucomes for he process. This uses he risk-neural valuaion resul. The bond price can be expressed as: V (, T )=E Q [e R T f(,s)ds] (15) where, E Q is he expecaions operaor wih respec o he equivalen riskneural measure. Under he equivalen risk-neural measure, he local expecaion hypohesis holds(ha is, E Q [ dv ] V ). According o he local expecaion hypohesis, he erm srucure is driven by he invesor s expecaions on fuure shor raes. To execue he Mone Carlo simulaion, we discreized he equaion (15). We divide he ime inerval [, T ]inom equal ime seps of lengh Δ each(ha is, Δ = T m ). For small ime seps, we are eniled o use he discreized version of he risk-adjused sochasic differenial equaion (14): [ ] T f j = f j 1 + σ f (, T ) σ f (, s)dsd Δ + σ f (, T )ε j Δ + Jj N Δ (16) where, σ f (, T )=σ r ()( ( r()) β exp ) T a(s)ds, j =1, 2,,m,ε j is sandard normal variable wih ε j N(, 1), and N Δ is a Poisson random variable wih parameer hδ. Noe ha we know ha β = case is an exension of Vasicek s jump diffusion model; he β =.5 case is an exension of CIR jump diffusion model. We can invesigae he value of he bond by sampling n spo rae pahs under he discree process approximaion of he risk-adjused processes of he equaion (16). The bond price esimae is given by: V (, T )= 1 n n exp i=1 m 1 j= f ij Δ, (17) where f ij is he value of he forward rae under he discree risk-adjused process wihin sample pah i a ime +Δ. Numerical mehods ha are known as Mone Carlo mehods can be loosely described as saisical simulaion mehods, where saisical simulaion is defined in quie general erms o be any mehod ha uilizes sequences of random numbers o perform he simulaion. The Mone Carlo simulaion is clearly less efficien compuaionally han he numerical mehod.

7 44 K.Park,M.Kim,andS.Kim The precision of he mean as a poin esimae is ofen defined as he half-widh of a 95% confidence inerval, which is calculaed as P recision =1.96 MSE. (18) where, MSE = ν/ n and ν 2 is he esimae of he variance of bond prices as obained from n sample pahs of he shor rae: [ ( n ν 2 i=1 exp ) ] m 1 j= f ijδ ν =. (19) n 1 Lower values of Precision in Equaion(18) correspond o sharper esimaes. Increasing he number of n is a brue force mehod of obaining sharper esimaes. This reduces he MSE by increasing he value of n. However, highly precise esimaes wih he brue force mehod can ake a long ime o achieve. For he purpose of simulaion, we conduc hree runs of 1, rials each and divide he year ino 365 ime seps. 4 Experimens In his secion, we invesigae he HJM model wih he jump version of he exended Vasicek and CIR models. In experimen 1, he parameer values are assumed o be r =.5, a =.5, θ =.25, σ r =.8, λ =.5, =.5, β =,andt = 2. Fig. 1. Experimen 1: The relaive error beween he HJM model wih he jump version of he exended Vasicek and CIR models Experimen 2, conrass bond prices by Mone Carlo simulaion. In experimen 2, he parameer values are assumed o be r[] =.5, f[,]= , a =.5, θ =.25, σ r =.8, λ =.5, β =,Δ =(T )/m, m = 365, n = 1, =.5, and T = 2.

8 On Mone Carlo Simulaion for he HJM Model Based on Jump 45 Table 1. Experimen 2: Bond price esimaed by he Mone Carlo simulaion for he HJM model wih he exended Vasicek model, CIR model, he jump diffusion version of he exended Vasicek model and CIR model. HJME V HJME CIR Jump HJME V Jump HJME CIR CFS MCS CFS MCS E E Variance E Precision Conclusion In his paper, we derive and perform he evaluaion of he bond prices of he HJM-Exended Vasicek and he HJM-CIR models wih forward ineres raes insead of shor raes using numerical mehods. The resuls show ha he values obained are very similar. Even hough i is hard o achieve he value of bond prices o erm srucure models when forward raes follow jump diffusions, we have shown ha bond prices in HJM jump-diffusion version models of he exended Vasicek and CIR models obained by Mone Carlo simulaion correspond wih he closed form soluion. Lower values of precision in he HJM model wih jump of he exended Vasicek model correspond o sharper esimaes. References 1. C. Ahn and H. Thompson, Jump-Diffusion Processes and he Term Srucure of Ineres Raes, Journal of Finance, vol. 43, pp , J. Baz and S. R. Das, Analyical Approximaions of he Term Srucure for Jump- Diffusion Processes: A Numerical Analysis, Journal of Fixed Income, vol. 6(1), pp , J. C. Cox, J. Ingersoll, and S. Ross, A Theory of he Term Srucure of Ineres Rae, Economerica, vol. 53, pp , D. Healh, R. Jarrow, and A. Moron, Bond Pricing and he Term Srucure of Ineres Raes, Economerica, vol. 6, no.1, pp , T. S. Ho and S. Lee, Term Srucure Movemens and Pricing Ineres Rae Coningen Claims, Journal of Finance, vol. 41, pp , M. J. Harrison and D. M. Kreps, Maringales and arbirage in muliperiod securiies markes, Journal of Economic Theory, vol. 2. pp , J. Hull and A. Whie, Pricing Ineres Rae Derivaive Securiies, Review of Financial Sudies, vol. 3, pp , M. J. Brennan and E. S. Schwarz, A Coninuous Time Approach o he Pricing of Bonds, Journal of Banking and Finance, vol. 3, pp , P. Richken and L. Sankarasubramanian, Volailiy Srucures of Forward Raes and he Dynamics of he Term Srucure, Mahemaical Finance, vol. 5, pp , O. A. Vasicek, An Equilibrium Characerizaion of he Term Srucure, Journal of Financial Economics, vol. 5, pp , 1977.