A Hybrid Model for Pricing and Hedging of Long Dated Bonds

Transcription

1 A Hybrid Model for Pricing and Hedging of Long Daed Bonds This version: 24 April 25 Jan Baldeaux a, Man Chung Fung b, Kaja Ignaieva c, Eckhard Plaen d a Finance Discipline Group, Business School, Universiy of Technology Sydney, Ausralia (jan.baldeaux@us.edu.au) b School of Risk and Acuarial Sudies, Business School, Universiy of New Souh Wales Sydney, Ausralia (m.c.fung@suden.unsw.edu.au) c School of Risk and Acuarial Sudies, Business School, Universiy of New Souh Wales Sydney, Ausralia (k.ignaieva@unsw.edu.au) d Quaniaive Finance Research Cenre, Finance Discipline Group, Business School, Universiy of Technology Sydney Ausralia (eckhard.plaen@us.edu.au) Absrac Long daed fixed income securiies play an imporan role in asse-liabiliy managemen, in life insurance and in annuiy businesses. This paper applies he benchmark approach, where he growh opimal porfolio (GOP) is employed as numéraire ogeher wih he real world probabiliy measure for pricing and hedging of long daed bonds. I employs a ime dependen consan elasiciy of variance model for he discouned GOP and akes sochasic ineres rae risk ino accoun. This resuls in a hybrid framework ha models he sochasic dynamics of he GOP and he shor rae simulaneously. We esimae and compare a variey of coninuous-ime models for shor-erm ineres raes using non-parameric kernel-based esimaion. The hybrid models remain highly racable and fi reasonably well he observed dynamics of proxies of he GOP and ineres raes. Our resuls involve closed-form expressions for bond prices and hedge raios. Across all models under consideraion we find ha he hybrid model wih he 3/2 dynamics for he ineres rae provides he bes fi o he daa wih respec o lowes prices and leas expensive hedges. Key words: Long daed bond pricing, sochasic ineres rae, growh opimal porfolio, nonparameric kernel

2 Inroducion Asse-liabiliy managemen is a major componen of insurance businesses. For life insurance companies, pension funds and annuiy providers, he managemen of long erm asse-liabiliy depends crucially on he availabiliy of long daed fixed income securiies o cover fuure obligaions ha can span over -3 years. Among a variey of ineres rae relaed securiies and derivaives, long daed governmen bonds hold a special place in he porfolios of radiional annuiy firms. Under he risk-neural pricing framework, bond securiies are priced by modelling he underlying ineres rae dynamics, such as shor rae models, under he risk-neural measure. In his paper we propose o apply a more general framework, namely he benchmark approach (see Plaen and Heah (2)), o price and hedge long daed zero coupon bonds. Under he benchmark approach, he numéraire is he growh opimal porfolio (GOP), which can be approximaed by a well-diversified equiy index, and he pricing measure is he real-world probabiliy measure. One of he consequences is ha under realisic long erm modelling he price dynamics of a long daed zero coupon bond depends no only on ineres rae risk, bu also on he marke price of equiy risk. Shor rae models have been sudied inensely in he lieraure. Saring from he seminal work of Meron (973) in coninuous ime finance, hey have become fundamenal and imporan price deerminans when evaluaing ineres rae coningen claims and hedging ineres rae risk. Modelling sochasic ineres raes has also become increasingly imporan in solving imporan acuarial problems. For example, Peng e al. (22) price guaraneed minimum wihdrawal benefis in variable annuiies under sochasic ineres raes. In Nowak and Romaniuk (23), sochasic ineres raes are aken ino accoun when pricing and simulaing caasrophe bonds. Many coninuous ime models have been pu forward o reflec he behaviour of ineres raes. These include models inroduced in Brennan and Schwarz (977, 979, 98); Vasicek (977); Dohan (978); Cox e al. (985); Longsaff (989); Black e al. (99); Hull and Whie (992) and Longsaff and Schwarz (992). For more recen accouns in his area and overviews on exising models one can refer o Sanon (997); Rebonao (998) and Brigo and Mercurio (2). Despie he exisence of a large number of shor-erm ineres rae models, he model choice for applicaions in pricing specific fixed income securiies is ypically made on a case by case basis. Ofen differen producs moivae differen models. By choosing a model one aims for realisic behaviour, ease of implemenaion, and also analyical racabiliy. Each produc ypically requires a model which combines he above feaures, and in some way ouperforms he compeing models. A more advanced model should have he propery ha i works well for a wide range of producs and also over long ime periods. Noe ha Inernaional Financial Reporing Sandards (IFRS) and Inernaional Accouning Sandards (IAS) recognise he imporance of valuaion and hedging of long daed derivaives, e.g. pensions, invesmens, zero coupon bonds and oher fixed income securiies, which appear in he marke exposure of many firms. Long daed derivaives are no well covered by he currenly favoured risk neural pricing and hedging mehodology, as argued in Plaen and Heah (2). This paper aims o search for suiable models ha deal well wih pricing and hedging of long daed zero coupon bonds under sochasic ineres raes, by employing he benchmark approach. We place ourselves in a hybrid framework, o model simulaneously boh he dynamics of he shor rae and he growh opimal porfolio (GOP), used as numéraire porfolio or benchmark for pricing purposes and also for hedging. For modelling ineres rae risk we consider differen shor rae models. These include he Vasicek model, see Vasicek (977); he CIR model, see Cox e al. (985); he 3/2 model, inroduced in Ahn and Gao (999); he Dohan model, see Dohan (978) and he Craddock-Lennox shor rae model, see Craddock and Lennox (27). For he GOP modelling we build upon he ime dependen

3 consan elasiciy of variance (TCEV) model inroduced in Baldeaux e al. (24), which models he GOP and is exended in our analysis by aking ino accoun sochasic ineres rae risk. The TCEV model represens a model class which ness several known model specificaions. Among oher models i includes he consan elasiciy of variance (CEV) model, see Cox (996), and he minimal marke model (MMM), see Plaen and Heah (2). When using real world pricing, derivaive price processes, when denominaed in unis of he GOP, become maringales under he real world probabiliy measure. Thus, coningen claim prices can be calculaed via condiional expecaions using he real world probabiliy measure. Under classical assumpions he resuling prices coincide wih classical risk neural prices, however, real world pricing makes also perfec sense for models beyond he classical paradigm, see Du and Plaen (24). Ceci e al. (24) sugges anoher applicaion of he benchmark approach o develop locally risk-minimizing hedging sraegies in a semimaringale financial marke model under parial informaion. For more deails on he GOP and he numéraire porfolio we refer o Kelly (956), see also Laané (959), Breiman (96), Thorp (96), Markowiz (976), and Long (99). Furher sudies on valuaion porfolios in asse-liabiliy managemen in he acuarial conex can be found e.g. in Buchwalder e al. (27) and Lim and Wong (2). In he curren paper we consider he problem of pricing long daed coningen claims using real world pricing under sochasic ineres raes and a realisic model for he numéraire porfolio, he GOP, for which a risk neural probabiliy measure does no exis. The laer provides a more realisic model for he long erm dynamics of he financial marke han is obainable under he classical risk neural paradigm. We use a non-parameric kernel-based echnique o esimae he diffusion coefficien funcion, and poenially, he drif coefficien, of he shor rae model, and refer o Florens-Zmirou (993), Sanon (997), Jiang and Knigh (997), Soulier (998) and Jacod (2) for deails on he esimaion echnique. We aim o find a model ha is racable, fis well he hisorical daa, i.e., canno be easily falsified or rejeced as oher models based on he daa fi. We show ha he 3/2 model ends o capure beer he dynamics of he shor-erm ineres raes han he compeing models. We also apply he non-parameric echnique o esimae he diffusion coefficien funcion of he discouned GOP, wihou assuming any paricular scalar diffusion dynamics. I urns ou ha he diffusion coefficien funcion of he TCEV model fis surprisingly well he non-paramerically esimaed diffusion coefficien funcion. 2 To idenify he bes performing model for he purposes of long daed bond pricing we evaluae and compare hisorical marke bond prices wih heoreical prices derived by he benchmark approach. I urns ou ha he proposed hybrid model is highly racable. I allows us o derive closed-form soluions for bond prices and hedge raios. Furhermore, he hybrid model wih he 3/2 dynamics for he shor rae leads o he smalles prices when compared o oher alernaive models. Finally, his model performs successfully when empirically hedging long daed zero coupon bonds by employing shor-mauriy zero coupon bonds, he savings accoun and he GOP for he hedge. This paper is organised as follows. Secion 2 develops a hybrid modelling framework for sochasic shor-erm ineres raes ogeher wih he GOP. Secion 3 discusses he non-parameric kernel-based echnique, used o esimae he drif and he diffusion coefficien funcions in he shor rae and he GOP dynamics. Esimaion resuls are presened in Secion 4. Secion 5 summarises he real world pricing approach and shows how o calculae prices of zero coupon bonds under he hybrid framework wih differen shor rae specificaions. Pricing resuls and performance are evaluaed in Secion 6 by comparing model prices o marke prices. Finally, Secion 7 discusses hedging resuls obained for he bes performing model, and Secion 8 concludes he paper. 2 This holds for he GOP denominaed in differen currencies, whereas his sudy focuses on USD denominaion of he GOP. 2

4 2 Modelling Framework This secion describes a hybrid framework for modelling he dynamics of sochasic shor raes and he GOP simulaneously. For he dynamics of shor raes we consider a general economeric framework. I ness several well-known model specificaions, including he Vasicek model (see Vasicek (977)), he CIR model (see Cox e al. (985)), 3/2 model, inroduced in Ahn and Gao (999) and he Craddock- Lennox shor rae model, see Craddock and Lennox (27). For he GOP we use he ime dependen consan elasiciy of variance (TCEV) model inroduced in Baldeaux e al. (24), which assumes ha he drif of he discouned GOP is an exponenially growing funcion of ime. 2. Shor Rae Models We employ here a sandard economeric framework o compare he performance of a variey of wellknown ineres rae models in capuring he sochasic behaviour of he shor-erm ineres rae. To specify he mos general model for he shor rae dynamics sudied in his paper, we consider he following SDE: dr = g(r )d + σr α dw, (2.) where σ >, α [, ) and W = {W, } is a sandard Wiener process on a filered probabiliy space (Ω, A, A, P ) wih filraion A = (A) saisfying he usual condiions, see Karazas and Shreve (99). These Markovian dynamics imply ha he condiional mean and variance of changes in he shor rae depend only on he level r. This allows us o concenrae on reasonably racable shor rae models. Depending on he specificaion of he drif g(r ) and he diffusion coefficien funcion σr α, we can selec a paricular model. We concenrae on five models considered in he lieraure, wih specificaions lised in Table. Table Alernaive model specificaions for he shor-erm ineres rae dynamics. Model SDE α g(r ) Param. Consr.. Vasicek 2. CIR dr = (a br )d + σdw dr = κ(θ r )d + σ r dw /2 (a br ) κ(θ r ) a, b, σ > κ, θ, σ > 3. 3/2 dr = (pr + qr 2 )d + σr 3/2 dw 3/2 (pr + qr 2 ) q < σ2 2, σ > 4. Dohan dr = λr d + σr( dw ) λr σ >, λ R 5. C&L dr = 2r ξ co r d + ( ) 2ξr dw /2 2r ξ co r r > ξ ξ Model is he Vasicek (977) ineres rae model inroduced in Vasicek (977). I uses he Ornsein- Uhlenbeck process o provide an equilibrium model for shor raes. I yields a saionary Gaussian process, which assumes ha he shor rae r is linearly mean revering. For going o infiniy, he expeced shor rae ends o he long erm average rae a/b. In he lieraure i is argued ha he drawback of he Vasicek model is ha he shor rae r can become negaive. However, he analyical racabiliy, which is implied by is Gaussian ransiion densiy, is hardly achieved when assuming oher dynamics for he process r. The Vasicek model and is generalisaions have been used inensely o value bond opions, fuures, fuures opions and oher ypes of coningen claims; see e.g. Jamshidian (99) and Gibson and Schwarz (99). Model 2 is he CIR model, which uses he square roo process for he shor rae dynamics. I appears in Cox e al. (985) as a single-facor general equilibrium shor rae model. Here, we follow he presenaion in Brigo and Mercurio (2). This model has been a benchmark model for shor-rae 3

5 dynamics because of is analyical racabiliy and he fac ha, conrary o he Vasicek model, he shor rae is always nonnegaive. I has been applied in consrucing racable valuaion models for ineres-rae-sensiive coningen claims, including discoun bond opions, fuures and fuures opions and swaps, see e.g. Cox e al. (985), Ramaswamy and Sundaresan (986), and Longsaff (99). Model 3 is he 3/2 model inroduced in Ahn and Gao (999), see also Plaen (999), for modelling he shor rae dynamics. Here, we follow he represenaion in Carr and Sun (27), who employed his ype of model o describe he insananeous variance in equiy models. I differs from he oher popular shor rae models in he power α = 3 > of he diffusion coefficien. Such a high power 2 in he diffusion coefficien funcion has been suppored by an earlier empirical resul in Chan e al. (992), who show via parameric mehods ha models wih power α capure he dynamics of he shor-erm ineres rae beer han hose which assume α <. Model 4 is he Dohan model, inroduced in Dohan (978), where we follow he presenaion in Pinoux and Privaul (2). Here, he shor-erm ineres rae process follows a geomeric Brownian moion and, herefore, uses α =. This model leads o he ineres rae being condiionally log-normally disribued, which implies ha he shor rae canno become negaive and, herefore, overcomes a main criicism of he Vasicek model. However, he process is only mean revering for λ < wih mean-reversion level equal o zero. The resuling dynamics do no seem o be realisic for long ime periods, since he variance of he log-shor rae is growing proporionally o ime. Neverheless, he Dohan model appears o be analyically racable. In fac, i seems o be he only log-normal shor rae model in he lieraure wih analyical formulas for pure discoun bonds, see Brigo and Mercurio (2) and Pinoux and Privaul (2). Model 5, which in he following will be referred o as he C&L model, is he Craddock-Lennox shor rae model. I was inroduced in Craddock and Lennox (27), who consruced shor rae models using Lie symmery group mehods and showed ha bond prices can be derived in closed form for his model. To our knowledge, he C&L model has no been empirically esed so far agains differen alernaive model specificaions. 2.2 Modelling he GOP Firs, le us recall he general sochasic differenial equaion (SDE) for he dynamics of he GOP in a coninuous financial marke. We denoe by S he value of he discouned GOP a ime. Following Plaen and Heah (2), he discouned GOP S saisfies in a coninuous financial marke he SDE d S = S θ ( θ d + d W ) for. Here, S > is he iniial value and θ denoes he volailiy of he GOP or marke price of risk, where W = { W, } denoes a sandard Wiener process. Following Baldeaux e al. (24), we inroduce he so-called ime dependen consan elasiciy of variance (TCEV) model, which assumes ha he GOP volailiy, or marke price of risk, is of he form (2.2) ( ) S θ = g. (2.3) α Here g( ) is a given funcion, and α is a deerminisic funcion of ime ha is defined below. Using 4

6 his noaion, we can wrie he following SDE for he discouned GOP: ( d S = S S g α ) 2 ( d + S S g α ) d W. (2.4) For our model we assume ha α is a funcion of ime growing exponenially like he average of he discouned GOP, ha is, α = α exp {η} (2.5) wih η >. We remark, assuming ha ( ) ( ) a S g S = c, (2.6) α α where a (, ) and c >, leads o he TCEV model and by (2.4) o he SDE d S = c 2 α 2 2a ( ) 2a ( ) S d + cα a a S d W, (2.7) for [, ) and S >. For he purpose of our analysis we also inroduce he normalized GOP process Y = { Y = S α, }, which follows he SDE dy = ( c 2 Y 2a ηy ) d + cy a d W (2.8) for and Y = y >, and exponen a (, ). Noe ha Y = {Y, } is ime homogenous. Using he SDE (2.8) and he Iˆo formula, one can easily verify ha S can be represened as he produc S = α Y. (2.9) Noe ha for a = and c = he TCEV model recovers he minimal marke model in Plaen and 2 Heah (2). 3 Nonparameric Esimaion of Drif and Diffusion Coefficien Funcions Non-parameric kernel-based esimaion of diffusion coefficien funcions has been inroduced in Florens- Zmirou (993) and is discussed in several sudies; see e.g. Soulier (998) and Jacod (2). Following he mehodology of Sanon (997) and Jiang and Knigh (997), we apply he nonparameric kernelbased echnique o he esimaion of he diffusion coefficiens in he dynamics of he shor rae, as well as he equiy index, as discussed in Ignaieva and Plaen (22) and Baldeaux e al. (24). Given n sample poins (x,..., x n ), he kernel esimaion densiy ˆf(x, h n ) is defined as ˆf(x, h n ) = nh n n ( x xi K i= h n ), (3.) where K( ) is a kernel funcion and h n is a bandwidh parameer conrolling he degree of smoohness of he esimaor. In his paper, we employ a Gaussian kernel funcion, i.e. we se K(z) = (2π) 2 exp { 2 z2}. I has been shown in he lieraure, ha he specific choice of he kernel K( ) 5

7 does no affec much he performance of he resuling esimaor. In fac, as argued in Epanechnikov (969), any reasonable kernel gives almos opimal resuls. For he bandwidh h n, one ypically chooses he bandwidh which minimizes he asympoic mean inegraed squared error (MISE), see Wand and Jones (995) and Boev e al. (2). For he purpose of our analysis we use a Gaussian kernel wih cross-validaed bandwidh seleced using Silverman (986) s rule of humb, ha is, h n = h cˆσn 5, where ˆσ is he dispersion of observaions, n is he number of sample poins and h c =.6. We can calculae he esimaor of he drif and he squared diffusion coefficien funcions, as suggesed by Sanon (997), in he following way: µ (x i ) = E((x i x i ) x i = x) + O( i i ) i i σ(x 2 i ) = E((x i x i ) 2 x i = x) + O( i i ), (3.2) i i using he approximaions E((x i x i ) x i = x) ni= (x i x i )K ( ) x x i ni= K ( x x i h h ) (3.3) and ni= (x E((x i x i ) 2 i x i ) 2 K ( ) x x i h x i = x) ni= K ( ) x x i, (3.4) h respecively, where i i denoes he ime sep size beween successive observaions. In addiion o he above firs order approximaions, Sanon (997) developed higher order approximaions based on Taylor expansions. However, he showed ha using approximaions of higher order does no affec he order of convergence bu may improve he approximaions for a chosen ime sep i i. For he purpose of our analysis we do no require higher order approximaions o fi a parameric form o he esimaed funcions for he drif and diffusion coefficiens and, hus, he firs order approximaion (3.2) will be sufficien. 4 Esimaion Resuls This secion discusses he daa used in our empirical analysis and repors esimaion resuls for he hybrid model. We sar wih he esimaion of he shor rae models followed by he esimaion of he GOP. Boh mehodologies will apply he nonparameric esimaion procedure described in Secion Shor Rae Daa In our empirical analysis we will use daily (3-monh) T-Bill raes obained from Daasream Thomson Financial. The observaion period covers he ime period from January 973 o July 2. Figure shows he ineres rae level (op lef panel) and ineres rae changes (op righ panel), defined as r + r wih corresponding o one day. One observes ha during he ime period from 978 o 985, 6

8 IR level.5. IR changes Ineres rae level Ineres rae changes ime ime 6 Nonparameric Densiy Esimae 4 2 Densiy Ineres rae level Fig.. Ineres rae level (op lef panel) and ineres rae changes (op righ panel); nonparameric densiy esimae (boom panel). shor raes are paricularly volaile. During his ime he inflaion drove US yields o heir all-ime high in 98. While he volailiy decreases gradually owards he end of he observaion period, i is ineresing o noe ha he shor raes are rending downwards as well. Table 2 Summary saisics for he shor rae levels and shor rae changes for he US. Mean Sd.Dev. Skewness Kurosis Firs Auocorr. r r + r Table 2 provides summary saisics for he shor rae levels and shor rae changes. I shows ha he kurosis of daily 3-monh T-bill rae incremens exceeds 27, while he skewness is posiive indicaing a high level of non-normaliy. Furhermore, as an illusraion of he nonparameric kernel mehodology, and o give some idea of he disribuion of ineres raes, he probabiliy densiy esimae for he daily changes is shown in he boom panel of Figure. One observes a long righ ail in he disribuion of shor rae incremens, which ogeher wih momen saisics indicaes ha he disribuion is lepokuric. 4.2 Esimaing Shor Rae Models To esimae he parameers of he drif and diffusion coefficien funcions of he shor rae models we apply he same mehodology as for he discouned equiy indices. Thereby, we esimae no only he diffusion bu also he drif coefficien funcion since drif parameers are required for he pricing of derivaive producs, as will be discussed below. 7

9 Esimaed Shor Rae Drif Coefficien (Vacisek) Nonparam. kernel esimaor Fied drif a=.22, b= % Confidence band (nonpar.).7.6 Esimaed Shor Rae Diffusion Coefficien (Vacisek) Nonparam. kernel esimaor Fied diffusion alpha=, sigma= % Confidence band (nonpar.) Drif.2 Diffusion Ineres rae level Ineres rae level.8.6 Esimaed Shor Rae Drif Coefficien (CIR) Nonparam. kernel esimaor Fied drif kappa=.2297, hea= % Confidence band (nonpar.).7.6 Esimaed Shor Rae Diffusion Coefficien (CIR) Nonparam. kernel esimaor Fied diffusion alpha=.5, sigma= % Confidence band (nonpar.) Drif.2 Diffusion Ineres rae level Ineres rae level.8.6 Esimaed Shor Rae Drif Coefficien (3/2 Model) Nonparam. kernel esimaor Fied drif p=.343, q= % Confidence band (nonpar.).7.6 Esimaed Shor Rae Diffusion Coefficien (3/2 Model) Nonparam. kernel esimaor Fied diffusion alpha=.5, sigma= % Confidence band (nonpar.) Drif.2 Diffusion Ineres rae level Ineres rae level.6.4 Esimaed Shor Rae Drif Coefficien (Dohan Model) Nonparam. kernel esimaor Fied drif lambda= % Confidence band (nonpar.).7.6 Esimaed Shor Rae Diffusion Coefficien (Dohan Model) Nonparam. kernel esimaor Fied diffusion alpha=, sigma= % Confidence band (nonpar.).2.5 Drif.2 Diffusion Ineres rae level Ineres rae level.8.6 Esimaed Shor Rae Drif Coefficien (Craddock Lennox) Nonparam. kernel esimaor Fied drif ksi= % Confidence band (nonpar.).7.6 Esimaed Shor Rae Diffusion Coefficien (Craddock Lennox) Nonparam. kernel esimaor Fied diffusion ksi= % Confidence band (nonpar.) Drif.2 Diffusion Ineres rae level Ineres rae level Fig. 2. Nonparameric esimaes of he drif coefficien funcion (lef panels) and he diffusion coefficien funcion (righ panels) for he models: Vasicek, CIR, 3/2, Dohan, Craddock-Lennox (from op o boom). 8

10 In he firs sep we esimae he parameers of he diffusion coefficien funcion σr α in he shor rae dynamics (2.) in order o decide on he parameric model which fis he daa bes. The esimaed coefficien for he consan σ and he diffusion coefficien power α are σ =.63 and α =.3694, respecively. Thus, by jus looking a he esimaed power, i urns ou ha among all models under consideraion he diffusion coefficien of he shor rae can be capured mos appropriaely using he 3/2 model, as he power of is diffusion coefficien is closes o he esimaed unresriced diffusion coefficien power. Table 3 Nonparameric esimaes for he diffusion consan and drif parameers in he ineres rae dynamics. Diffusion Param. Drif Parameers σ α a b κ θ λ p q ξ = σ 2 /2 Vasicek CIR.938 / / / Dohan C&L.938 / Now we assume a cerain model specificaion. Tha is, we fix he diffusion coefficien power α o, /2,, 3/2 and /2 for he Vasicek, CIR, 3/2, Dohan, and C&L model, respecively, and esimae only he diffusion consan and he drif (resriced esimaion). The resuls of he fied parameer esimaes obained using non-linear leas squares are summarised in Table 3. The esimaed drif and diffusion coefficien funcions for he Vasicek, CIR, 3/2, Dohan, and Craddock-Lennox models (from op o boom) are ploed in he lef, respecively, he righ panel of Figure 2 using a solid line. 3 The fied diffusion coefficien funcions for he respecive model obained, using he parameer esimaes displayed in Table 3, are shown in hese figures by using dashed-doed lines. The figure also shows a poinwise 95% confidence band (doed line), calculaed using ieraions of he block boosrap algorihm, see Künsch (989). In order o preserve serial dependence in he daa we apply he moving block boosrap mehod which uses blocks of observaions raher han single observaions in he boosrap algorihm. The algorihm resamples he observed ime series using approximaely independen (non-overlapping) moving blocks of lengh l. For n observaions, we consider k blocks of lengh l (n = lk), consruced in he following way: Block one comprises observaions from o l, block wo comprises observaions from l + o 2l, ec. The las block k comprises observaions from n l + o n. The esimaion algorihms draws k blocks wih replacemen from a se of blocks. Aligning hese blocks in he order hey were picked provides us wih he boosrap observaions, which can hen be used o compue sandard errors and confidence bands. Comparing visually he resuls for he parameer esimaes wih he diffusion coefficien funcion plos, we observe ha he fied diffusion coefficien funcion line for he 3/2 model falls almos enirely ino he esimaed 95% confidence band and hereby, seems o ouperform is compeiors. I indicaes ha he 3/2 model is likely o be a good candidae for describing he shor rae dynamics when aiming for a simple scalar diffusion shor rae model. This makes he shor rae model proposed in Ahn and Gao (999), as well as Plaen (999), reasonably realisic. 3 From he figure we observe ha he esimaed diffusion coefficien is greaer han zero for r =, which suggess ha no consrain (ha he diffusion coefficien is zero for r = ) is required for he esimaion of he diffusion coefficien funcion. However, in order o preven ineres raes from becoming negaive, we impose his consrain, as suggesed in Sanon (997) for he esimaion of he drif coefficien funcion. 9

11 7 discouned EWI4 9 log(s ) 6 8 log( ) 5 7 EWI log( S) ime ime 2.2 Normalized discouned EWI4 Y Y ime Fig. 3. Upper lef panel: Discouned EWI4 S as funcion of ime. Upper righ panel: Logarihm of he discouned EWI4, ln( S ), which is on average a linearly growing funcion of ime. Lower panel: Normalised discouned EWI4 Y = S /α Nonpar. densiy esimae.4.3 Esimaed Diffusion Coefficien Nonparam. kernel esimaor Fied diffusion alpha=.28678, sigma=.99 95% Confidence band (nonpar.).5.2 f(y ) Diffusion Y Y Fig. 4. Nonparameric densiy esimae (lef panel) and diffusion coefficien esimae (righ panel) for he normalised discouned EWI4 denominaed in USD. 4.3 Esimaing he GOP As equiy index and approximaion of he GOP, we consider he well diversified equi-weighed index, he EWI4, which is available on a daily basis for he ime period from January 973 o July 2. In is consrucion in Plaen and Rendek (22) he EWI4 uses indusry sub-secor indices as consiuens provided by Daasream Thomson Financial. Here we discuss esimaion resuls for he GOP.

12 ..8 Covariaion Covariaion ime Fig. 5. Covariaion beween he logarihms of he EWI4 and he US shor rae. The discouned EWI4, denominaed in USD, is ploed in he upper lef panel of Figure 3. 4 In order o obain an exponenially increasing drif funcion α, which is given by (2.5), we have o esimae he scaling parameer α and he ne growh rae η. For his purpose we apply a leas squares esimaion o fi he logarihm ln(α ) o he logarihm ln( S ), as shown in he upper righ panel of Figure 3. The resuling parameer values are α = and η =.239. Given he funcion α, he normalised discouned GOP can be compued as Y = S /α. We plo his normalised process in he lower panel of Figure 3. Y can now be used o esimae he diffusion coefficien funcion in a non-parameric way, as described in Secion 3. The kernel densiy is esimaed using a Gaussian kernel wih a rule-of-humb bandwidh and is ploed in he lef panel of Figure 4. The resuling esimaed approximaion of he diffusion coefficien funcions is ploed in he righ panel of Figure 4. Fiing he esimaed diffusion coefficien funcion o he funcional form assumed for he diffusion coefficien funcion (cy a ) of he process Y, we obain for he parameer esimaes of c and a he values. and.2868, respecively. The fied diffusion coefficien funcion is represened by he dashed-doed line in he righ panel of Figure 4. Again, he doed lines represen a poinwise 95% confidence band, calculaed using ieraions of he block boosrap algorihm. Finally, comparing he resuls for he parameer esimaes wih he diffusion coefficien funcion plo, we observe ha he fied diffusion line (dashed-doed line), obained by using parameer esimaes of c =. and a =.2868, falls ino he esimaed 95% confidence band for he TCEV diffusion process, indicaing ha he TCEV model is likely o be a good candidae model for describing he normalised EWI4 dynamics. 5 Zero Coupon Bond Pricing We use B o denoe he value of he savings accoun a ime, assuming db = r B d, and B =. The adaped shor rae process is denoed by r = {r, }. In Figure 5 we plo he covariaion beween he logarihm of he discouned EWI4 and he US shor rae. Noice ha he 4 Noe, in he following we apply he esimaion procedure described above o he EWI4 denominaed in USD. When considering denominaions of he EWI4 in oher currencies, as suggesed in e.g. Ignaieva and Plaen (22), he resuls appear o be qualiaively similar.

13 absolue values aken by he covariaion process are exremely small wihou exhibiing any clear rend. I does no seem ha he noise processes driving he shor rae and he discouned equiy index are significanly correlaed. Therefore, for simpliciy, we assume ha he driving noise source of he EWI4 and ha of he shor rae are independen, which leads o he following assumpion: Assumpion 5. We assume ha he Wiener processes W and W are independen. This makes he marke price for shor rae risk zero. Of course, he case when he marke price of shor rae risk is no zero can be heoreically and numerically handled as well. 5. Real World Pricing To perform pricing under he above discussed model, we are no covered by he classical risk neural paradigm since an equivalen risk neural probabiliy measure does no exis for our specificaion of he TCEV model. More precisely, he Radon-Nikodym derivaive of he puaive risk neural measure Λ = S can be shown o be a sric local maringale, and is, hus, no a maringale, see Plaen and S Heah (2). Therefore, we briefly recall he basics of he benchmark approach; see Plaen and Heah (2), which allows pricing and hedging also in cases when he risk neural approach fails. Here, S = B S denoes he GOP, approximaed by he EWI4, a ime, when denominaed in unis of he domesic currency (here USD). Definiion 5. A price process U = {U, } wih E( U ) < for, is called fair, if he S corresponding benchmarked price process Û = { Û = U, } forms an (A, P )-maringale. S Definiion 5.2 We define a coningen claim H T ha maures a a sopping ime T as an A T measurable, non-negaive payoff wih E( H T A ST ) < for all [, T ]. Following he benchmark approach, see Plaen and Heah (2), he minimal price process U H T { = U H T, [, T ] } ha possibly replicaes a hedgeable coningen claim H T, is given by he condiional expecaion ( ) S = E H T A U H T for [, T ], which represens he real world pricing formula. The process Û H T = { Û H T S T = U H T S, [, T ] is an (A, P )-maringale and, hus, by Definiion 5. U H T consiues a fair price process. In he case of no fully replicable claims Du and Plaen (24) obain also he real world pricing formula employing he concep of benchmarked risk minimizaion, where benchmarked profi and losses are orhogonal o benchmarked raded wealh. For calculaing prices of derivaives, recall ha for he TCEV model we have he SDE (2.7) for he discouned GOP and S = Z 2 2a ϕ(), } 2

14 where Z = {Z ϕ, ϕ } is a squared Bessel process of dimension ν = 3 2a and he funcion ϕ() = ( a)α 2 2a c 2 2η (exp {2( a)η} ). Furhermore, by seing ϕ() = ϕ(t ) ϕ(), we define a λ(, S ) = ( S ) 2( a) ϕ() ( K and x() = BT ) 2( a) ϕ(), and noe ha in disribuion we have Z ϕ() ϕ() d = χ 2 ν( x ϕ() ), where χ 2 ν(µ) denoes a non-cenral chi-square disribued random variable wih ν degrees of freedom and non-cenraliy parameer µ. We now presen generic formulas for prices of zero coupon bonds under he shor rae models discussed in his paper. 5.2 Zero Coupon Bond Prices By Assumpion 5., he price of a fair zero coupon bond is given by where see Miller and Plaen (25), and ( ) ( S P (, T ) = E S A ST = E S T S T ) B A = M(, B S, T )G(, r, T ), (5.) T ( ) M(, S S, T ) = E A = Ψ(λ(, S ),, ), (5.2) a G(, r, T ) = E ( ) B A. (5.3) B T Recall ha Ψ(, ν, µ) denoes he cumulaive disribuion funcion of a non-cenral chi-square disribued random variable wih ν degrees of freedom and non-cenraliy parameer µ. In he following, G(, r, T ) will be referred o as he shor rae conribuion (SRC), whereas M(, S, T ) defines he marke price of risk conribuion (MPRC). 5.3 Tracable Shor Rae Models In his secion we obain zero coupon bond pricing formulas for he Vasicek, CIR, 3/2, Dohan and Craddock-Lennox shor rae models. 3

15 5.3. Vasicek Model In he Vasicek model, he shor rae is given by he SDE dr = (a br )d + σdw, where a, b and σ are assumed o be posiive. The following se of formulas is derived in Musiela and Rukowski (25). These employ he Laplace ransform of T r s ds, which gives he formula for he zero coupon bond. Lemma 5. Condiional on A, he inegral T r s ds is a Gaussian random variable wih mean µ(, T ) = a b (T ) + (r a ( ) exp{ b(t )} b ), b and variance σ 2 (, T ) = σ2 2b + σ2 b 2 ( exp{ b(t )})2 3 ( T exp{ b(t )} b ), i.e. we have E ( exp{ u T r s ds} A ) = exp{ uµ(, T ) + u2 2 σ2 (, T )}. Seing u =, gives he respecive formula for he SRC G(, r, T ) CIR Model The CIR model was inroduced in Cox e al. (985), and we use here a bond price derivaion given in Brigo and Mercurio (2). Under he real world probabiliy measure P one has he SDE where r > and κ, θ, σ are posiive parameers. dr = κ(θ r )d + σ r dw, (5.4) The Laplace ransform of T r s ds, condiional on A, is given by he following formula, see Jeanblanc e al. (29), Corollary Lemma 5.2 The Laplace ransform of T { { } } T E exp u r s ds A r s ds saisfies he formula = exp { ( κ 2 θτ/σ 2} ( ) γτ cosh + κ ( ) ) 2κθ/σ γτ 2 2 γ sinh 2 4

16 ( ) 2ur exp κ + γ coh( γτ ), 2 where τ = T and γ 2 = κ 2 + 2uσ 2. To obain he corresponding SRC G(, r, T ), we simply se u = /2 Model The 3/2 model was inroduced in Ahn and Gao (999) and Plaen (999). We se dr = (pr + qr 2 )d + σr 3/2 dw, where q < σ2 2 and σ > o avoid explosions in r : Seing R = r, we obain he SDE dr = ( σ 2 q pr ) d σ R dw, which shows ha he 3/2 model is he inverse of a square-roo process. We now use Theorem 3 in Carr and Sun (27) o obain he Laplace ransform of he inegraed shor rae process. Lemma 5.3 The Laplace ransform of T { } T E exp{ u r s d s } A = where r s ds has he form Γ(γ α) Γ(γ) y(, p) = T ( 2 ) α M(α, γ, σ 2 ry(, p) e ( )p d, 2 σ 2 ry(, p) ), M(α, γ, z) is he confluen hypergeomeric funcion, see Abramowiz and Segun (972), wih and α = ( 2 q σ 2 ) + Seing above u = yields he SRC G(, r, T ). γ = 2(α + q σ 2 ). ( 2 q σ 2 )2 + 2u σ Dohan Model The Dohan model was inroduced in Dohan (978), and we follow in his paper he presenaion in Pinoux and Privaul (2). The shor rae is modelled under he real world probabiliy measure P as a geomeric Brownian moion via he SDE dr = λr d + σr dw, where σ > and λ R. Zero coupon bond prices were derived in Pinoux and Privaul (2). We follow heir probabilisic approach and compue he Laplace ransform of he random variable T r s ds. 5

17 Corollary 5. The Laplace ransform can be given for all p R, where u >, by he formula ( { } ) T E exp u r s ds A = 8 r u σ 2 π 3 τ exp( σ2 p 2 τ/8 + 2π 2 /(σ 2 τ)) e 2ξ2 /(σ 2 τ) sinh(ξ) sin(4πξ/(σ2 τ)) K ( 8r u (z + ξ)(z + ξ )/σ)dξ dz (z + ξ)(z + ξ ) z p where τ = T. Proof. The proof can be given in he same way as Corollary 2.2 in Pinoux and Privaul (2). The following resul, which involves only a single inegraion, only holds for p < : Corollary 5.2 The Laplace ransform is given for all p <, where u >, by he expression ( { } ) T E exp u r s ds A = 2e σ2 p 2 τ/8 (v 2 + 8ur /σ 2 ) p/2 θ(v, σ2 τ 4 )K p( v 2 + 8ur /σ 2 ) dv v. p+ Proof. The proof can be given in he same way as he one for Corollary 2.3 in Pinoux and Privaul (2). Also here we obain he respecive SRC by seing u = Craddock-Lennox Model In Craddock and Lennox (27), he following shor rae model was analysed using Lie symmery groups mehods. The respecive shor rae saisfies he SDE dr = 2r ξ co ( ) r d + 2ξr dw ξ, where r. The following funcion, he fundamenal soluion o he PDE for he ransiion densiy can be used in bond pricing: p µ (, x, y) = exp { (x + y) ξ } ( ( ) x 2 xy I + δ(y) yξ ξ ( ) sin ( sin y ) µ ξ x ). µ ξ Here, I ( ) denoes he modified Bessel funcion of he hird kind of order and δ( ) he Dirac dela measure. For he required Laplace ransform Craddock and Lennox (27) provide he following resul: Lemma 5.4 The Laplace ransform of T r s ds saisfies for u > he formula 6

18 E { exp ( u T ( ) } { r s ds A = exp r (T )u } ) sin ur ξ+ξ 3/2 (T ) 2 u + ξ(t ) 2 ( ). u sin r u ξ By seing u =,we obain he formula for he SRC. 6 Pricing Resuls This secion demonsraes how zero coupon bonds can be priced using he differen models under consideraion. Thereby, we evaluae model performance by comparing model prices and yields for bonds of differen mauriies o hose observed in he marke. In line wih he concep of benchmarked risk minimizaion of Du and Plaen (24), we will idenify he bes performing model as he model ha leads o he lowes prices compared o he marke and shows benchmarked hedge errors ha are orhogonal o benchmarked primary securiy accouns. The laer means ha he benchmarked profi and loss and heir producs wih benchmarked savings accouns are local maringales and do no exhibi sysemaic rends. We illusrae he pricing of zero coupon bonds of differen mauriies corresponding o he enors of 6 monh, year, 2 years, 5 years, years, 2 years and 3 years. Thereby, prices are compued using he differen model specificaions, as described in Secion 5.3. As an illusraion, we compare model prices o marke prices on four differen daes ha all fall ino our esimaion period. Marke prices are provided on a monhly basis by Bloomberg for he ime period from May 99 unil July 2. To obain a visual impression on how model prices evolve, compared o marke prices, we firs consider bond prices and yields 5 on several daes, followed by presening he summary saisics for he enire sample. To visualise pricing performance, we consider bond prices and bond yields compued for five randomly seleced daes in our sample period in he lef, respecively, he righ panel of Figure 6. 6 Marke prices and yields are presened using he mos solid line. Visually, one observes ha all models fi reasonably well he marke daa. However, he 3/2 model ends o ouperform for many periods he compeing models by leading o he smalles price and highes yields. The Dohan model seems o perform wors for mos of he sample periods. In addiion, Figure 7 shows in he op lef panel he marke price of risk conribuion (MPRC) M(, S ), which eners he pricing formula (5.) as a funcion of ime o mauriy and power a in he dynamics of he GOP. The op righ panel of Figure 7 shows M(, S ) as a funcion of ime o mauriy for he esimaed diffusion coefficien power a =.2868, which we obained in Secion 4 when esimaing parameers for he GOP dynamics. The boom lef panel shows he MPRC M(, S ) as a funcion of he diffusion power a for a fixed ime o mauriy of T = 5 years. We observe ha for he esimaed parameer se he MPRC remains nearly consan for mauriies T up o 2 years (op righ panel). Beyond his period, he marke price of risk conribuion sars o decrease markedly in value. Therefore, he MPRC does no impac he zero coupon bond price unil medium mauriies of he yield curve. This propery of he fair zero coupon bond price resuls from he fac ha he discouned GOP is a sric local maringale; see Plaen and Heah (2) and Plaen and Brui-Liberai (2). 5 log P (,T ) The yield o mauriy R(, T ) is defined as R(, T ) = T, where P (, T ) denoes he price a ime of a zero coupon bond mauring a ime T. 6 Figures covering oher monhs from he sample period are available upon reques from he auhors. 7

19 Bond Price (June 99) Marke Price Vasicek CIR Dohan 3/2 C&L Marke Price Vasicek CIR Dohan 3/2 C&L Bond Yield (June 99) Bond price Bond yield Time o mauriy Time o mauriy Bond Price (Oc 993) Marke Price Vasicek CIR Dohan 3/2 C&L Marke Price Vasicek CIR Dohan 3/2 C&L Bond Yield (Oc 993) Bond price Bond yield Time o mauriy Time o mauriy.2.8 Bond Price (March 22) Marke Price Vasicek CIR Dohan 3/2 C&L Bond Yield (March 22) Marke Price Vasicek CIR Dohan 3/2 C&L Bond price.6 Bond yield Time o mauriy Time o mauriy Bond Price (Sep 24) Marke Price Vasicek CIR Dohan 3/2 C&L Marke Price Vasicek CIR Dohan 3/2 C&L Bond Yield (Sep 24) Bond price Bond yield Time o mauriy Time o mauriy Bond Price (May 26) Marke Price Vasicek CIR Dohan 3/2 C&L..9.8 Marke Price Vasicek CIR Dohan 3/2 C&L Bond Yield (May 26) Bond price Bond yield Time o mauriy Time o mauriy Fig. 6. Zero coupon bond prices (lef panel) and yields (righ panel) for bonds of differen mauriies and five differen daes. Prices are compued under differen model specificaions (Vasicek, CIR, 3/2, Dohan, C&L). Marke prices are presened using solid lines. 8

20 Marke Price of Risk Conribuion (a,t) Marke Price of Risk Conribuion (for fixed a=.2868) MPR conribuion T 5.5 a T Marke Price of Risk Conribuion (for fixed T=5).9 Shor Rae Conribuion and Zero Bond Price Shor Rae Conribuion Zero Bond Price.98.8 MPR conribuion Bond price a T Fig. 7. Upper lef panel: marke price of risk conribuion o he zero coupon bond price as a funcion of ime o mauriy T and power a in he GOP dynamics. Upper righ panel: marke price of risk conribuion o he zero coupon bond price for esimaed a =.2868 and differen imes o mauriy. Boom lef panel: marke price of risk conribuion o he zero coupon bond price for consan ime o mauriy T = 5 years and differen powers a. Boom righ panel: bond price and shor rae conribuion versus ime o mauriy. For he fixed mauriy of T = 5 years he MPRC reaches a value of abou one for a greaer han.4, which again indicaes ha he MPRC does no impac he zero coupon bond price as soon as a exceeds a paricular hreshold. By using he esimaed parameers and observed shor rae and index value one can for all observaion imes (daily) calculae he respecive model bond prices. To compare he overall performance of he compeing models, we compue he relaive pricing error δ τ for all models under consideraion. I is defined as he difference beween he model price and he marke price, normalised by he marke price, ha is, δ τ = P (, T )i P (, T ) m P (, T ) m. (6.) Here P (, T ) i defines he model price i, i {Vasicek,CIR,3/2,Dohan,C&L}. P (, T ) m is he marke price and τ is he ime o mauriy, which we se equal o 6 monh, year, 5 years, years, 2 years, 3 years for zero coupon bonds. Noe ha we are no comparing he absolue value δ τ since we are ineresed in he direcion of he deviaion and, hereby, aim o idenify he model wih he leas expensive bonds compared o oher models and he marke. Since we use daily daa we have a reasonably large daa se. To compare he performance of he models wih respec o he direcion of he deviaion from he marke price, we consider he summary saisics (mean, median, sandard deviaion, 5% and 95% quaniles) for he pricing error δ τ defined in equaion (6.). The resuls are summarised in Table 4, 9

21 and Figure 9 shows he corresponding box-plos. Here, he negaive (respecively posiive) sign of he pricing error indicaes ha he model undercus (respecively overshoos) he marke price. For shor mauriies of 6 monh, year and 5 years one observes ha he average pricing errors are close o zero for all considered models. However, he 3/2 and he C&L models lead o negaive pricing errors indicaing ha hese models undercu marke prices as well as produce lower prices han he oher compeing models. When increasing he ime o mauriy o years one observes ha he 3/2 model ouperforms he C&L model based on boh, mean and median saisic, and again, boh models lead o prices lower han he marke price and lower han hose of he oher compeing models. For he longes ime o mauriy of 3 years all models, excep he Dohan model, exhibi negaive pricing errors indicaing ha he marke is likely o be overpriced. One noes in Table 4 ha he sandard deviaions of he pricing errors are for he enors up o 5 years for all models raher similar. Remarkably, for he exreme mauriies he 3/2 model has a significanly smaller sandard deviaion. The 3/2 model leads o he lowes bond price (followed by he C&L model), and hus, ouperforms all compeing models for long and exreme mauriies. 7 Hedging he Zero Coupon Bond In his secion, we discuss he hedging of he zero coupon bonds. We firsly presen maringale represenaions of he benchmarked zero coupon bond price. Using such represenaion, we discuss a hedging sraegy involving a shor mauriy zero coupon bond, he savings accoun and he GOP. Finally, since we have idenified he 3/2 model as he bes performing model in he sense ha i leads o he lowes bond prices, we presen hedging resuls obained by using he respecive hybrid model wih he 3/2 dynamics for he ineres raes and he TCEV dynamics for he discouned index. 7. Hedge Raios for he 3/2 Model In his secion we demonsrae how a zero coupon bond can be hedged by means of dynamic rading in he shor mauriy bond, he savings accoun and he GOP. We resric ourselves o he hybrid model wih 3/2 dynamics for ineres raes. To derive hedge raios, we use he following noaion: ( { } ) T V (r,, T ) := E exp r s ds A ( ) α Γ(γ α) 2 2 = M(α, γ, Γ(γ) σ 2 ry(, p) σ 2 ry(, p) ), see Lemma 5.3, ( ) u(, ˆB, T ) = E A S T = E ( ) ˆBT A = ( S ) Ψ(λ(, S ),, ), (7.) a see (5.2), where ˆB = B. We remark ha explici formulae for V and u were derived in Ahn and S Gao (999) and Baldeaux e al. (24), respecively. The following lemma produces a maringale 2

22 represenaion for he benchmarked zero coupon bond ˆP (, T ) = P (, T ) S = E Corollary 7. Le r = {r, } be given by and hen ( ) A = E ( ) ( ) ˆBT A E A = u(, ˆB, T ) V (r,, T ). B T S T B T B d S = c 2 α 2 2a ( ( ) d ˆP (, T ) = V (r,, T ) σr 3/2 B r dr = (pr + qr 2 )d + σr 3/2 dw, ( ) 2a ( ) S d + cα a a S d W, u(, ˆB, T )dw + V (r,, T ) u(, ˆB, T ) ˆB ( cα a ( ) a S ˆB d W ) Proof. I follows ha V (r,, T ) B and u(, ˆB, T ) are maringales under P. Furhermore, W and W are independen, so ha V (r,,t ) B and u(, ˆB, T ) are independen. Hence we have by he Iˆo formula d ( ˆP (, T ) ) = u(, ˆB, T )d ( ) V (r,, T ) Again by he Iô-formula and he fac ha V (r,,t ) B d B ( ) V (r,, T ) B + V (r,, T ) B du(, ˆB, T ). is a maringale i follows ha = B V (r,, T ) r σr 3/2 dw. We now again use he Iô-formula and he fac ha u(, ˆB, T ) is a maringale d ( u(, ˆB, T ) ) = u(, ˆB, T ) ˆB d ˆB. ). Bu which complees he proof. d ˆB = d( S ( ) ) = cα a a S ˆB d W, We now discuss he hedging of a given zero coupon bond ( ) P (, T ) := S E A ST. From Corollary 7., we have d ( ) ˆP (, T ) = a(, T )dw + b(, T )d W, 2

23 where a(, ) and b(, ) are given in Corollary 7.. We propose o hedge P (, T ) wih longer mauriy using a zero coupon bond wih shorer mauriy ( ) P (, T H ) := S E A ST, H where T H << T. From Corollary 7., i follows ha d ( ˆP (, TH ) ) = a(, T H )dw + b(, T H )d W. We inves δ H unis a ime in P (, T H ), where δ H a(, T H ) = a(, T ), i.e. we use P (, T H ) o eliminae he effec of W. Furhermore, we inves δ B a ime in he savings accoun, where δ B ( θ ˆB ) + δ H b(, T H ) = b(, T ), so ha δ B eliminaes he effec of W. Finally, we use he GOP o make he porfolio self-financing, i.e. dp (, T ) = δ H dp (, T H ) + δ B db + δ S ds. The quaniies δ H and δ B can be shown o saisfy he following equaions and For compuing δ H δ H = a(, T ) a(, T H ) = V (r,, T ) r ( ) V (r,, T H ) u(, ˆB, T ) r u(, ˆB, T H ) δ B = a(, T ) V (r,, T H ) u(, ˆB, T ) a(, T H ) B ˆB + V (r,, T ) u(, ˆB, T ) B ˆB. and δ B, we find i useful o noe ha from equaion (7.) we ge u(, ˆB, T ) ˆB = Ψ(λ(, S ), a, ) ( ˆB 2a 2 2( a) ) ϕ() s(λ(, S ), a, ), where s(, ν, ) is a probabiliy densiy funcion of a χ 2 ν(λ) non-cenral chi-squared disribued random variable wih ν degrees of freedom and non-cenraliy parameer λ, see Baldeaux e al. (24). We also have V (r,, T ) r ( ) α Γ(γ α) α 2 = Γ(γ) r σ 2 r y(, p) ( ) 2 σ 2 γr y(, p) M(α +, γ +, 2 σ 2 r y(, p) ) M(α, γ, 2 σ 2 r y(, p) ). where he fac ha dm (a, b, z) = a M(a +, b +, z) is used (Abramowiz and Segun (972)). dz b 22