TECHNICAL REPORT. CTR/54/06 November 2006

Transcription

1 The Centre for the Analyss of Rsk and Optmsaton Modellng Applcatons A mult-dscplnary research centre focussed on understandng, modellng, quantfcaton, management and control of RISK TECHNICAL REPORT CTR/54/06 November 2006 Parameter estmaton of an nterest rate model va a HMM flterng method n dscrete tme Chrstna Erlwen, Rogemar Mamon School of Informaton Systems, Computng and Mathematcs, Brunel Unversty, Uxbrdge, Mddlesex, UB8 3PH

2 Parameter estmaton of an nterest rate model va an HMM flterng method n dscrete tme Chrstna Erlwen Rogemar Mamon Abstract Ths paper consders the mplementaton of a mean-revertng nterest rate model wth Markov-modulated parameters. Hdden Markov model flterng technques n Ellott [8] and Ellott et al. [9] are employed to obtan optmal estmates of the model parameters va recursve flters of auxlary quanttes of the observaton process. Algorthms are developed and mplemented on a fnancal data set of 30-day T-bll yelds. We found that wthn the data set and perod studed, a model wth three regmes s suffcent to descrbe the nterest rate dynamcs on the bass of very small predctons errors. Keywords: term structure - regme-swtchng model - change of probablty measure technque - optmal parameter estmaton - hdden Markov model 1 Introducton The short term nterest rate s a key varable n fnancal modellng because of ts mportance n prcng and hedgng of fxed ncome securtes and other fnancal dervatves. Well-known short rate models nclude the no-arbtrage sngle-factor model of Vascek [27], where the nterest rate s mean-revertng and ts extenson developed by Hull and CARISMA School of Informaton Systems, Computng and Mathematcs, Brunel Unversty, Uxbrdge, Mddlesex, UB8 3PH, Unted Kngdom Department of Statstcal and Actuaral Scences, 2nd Floor Western Scence Centre, Unversty of Western Ontaro, London, ON Canada N6A 5B7 1

3 Whte [21]. The mean reverson level n the Hull-Whte model s tme varyng and the yeld curve can be ftted to today s term structure. Cox, Ingersoll and Ross [5] on the other hand proposed a sngle-factor model, where postve nterest rates are guaranteed. Mult-factor models by Duffe and Kan [7] and by Longstaff and Schwartz [23] are able to provde a better ft to the yeld curve, but the analyss and parameter estmaton for these models are more dffcult. Other popular nterest rate models are the lognormal short rate model of Black and Karasnsk [3] and the Heath-Jarrow-Morton methodology [20], where the entre forward-rate curve s modelled. A detaled dscusson of these models can be found n Brgo and Mercuro [4]. Most recent term structure models nclude the possblty of regme-swtchng for shortterm nterest rates. Garca and Perron [17] analysed the tme seres behavour of U.S. real nterest rates from 1961 to 1986 and found emprcal evdence for jumps caused by mportant structural events. Hamlton [19] ntroduced changes n regmes by modellng parameters of an autoregresson wth a dscrete Markov chan appled to a busness cycle. Gray [18] used a generalsed regme-swtchng model for short-term nterest rates, whch allows mean-reverson and captures condtonal heteroskedastcty. The model outperforms sngle-regme models n out-of-sample forecastng. Smlar evdence supportng good performance of regme-swtchng short-term nterest-rate models s descrbed by Bansal and Zhou [2]. Here the effcent method of moments s used for the estmaton of the model parameters. Amongst others, Nak and Lee [24] and Evans [15] nclude regme-swtchng n short-term nterest rate models. Drffll, Kenc and Sola [6] found emprcal evdence, that regme shfts add more realsm to nterest rate models. A study by Smth [26] supports Markov swtchng models over stochastc volatlty models, because the volatlty seems to depend on the level of the short rate. Landen [22] developed a hdden Markov model (HMM) for short-term nterest rates, where drft and dffuson parameters are modulated by an underlyng Markov process. Whlst the paper of Ellott, Fscher and Platen [11] addresses the HMM flterng of a mean-revertng model, the derved flters are n contnuous tme settng and only a smulaton s gven. In a model by Ellott, Hunter and Jameson [12] the short rate process r s a functon of a Markov chan. A closed form soluton for bond prces, where the underlyng short rate s modelled by a mean reverson level governed by a contnuous tme Markov chan s derved n related work of Ellott and Mamon [13], however the volatlty process s constant n ther model formulaton. A central concern for these models s parameter estmaton. In ths paper we provde recursve estmates for the parameters followng the approach of Ellott [8] n dscrete tme. We demonstrate the calculaton of exact adaptve flters for Markov chans observed n 2

4 Gaussan nose together wth the recursve flters for the jump process and occupaton tmes. These flters provde optmal estmates of the parameters of the proposed nterest rate model. All calculatons are made under an dealsed measure P, equvalently to the real world measure P. The paper s organsed as follows. Secton 2 descrbes the model framework and a dscusson on how to ncorporate a HMM n the sngle-factor Hull-Whte [21] model. In secton 3, related processes of the Markov chan are estmated usng the flterng technque. The recursve parameter estmatons are outlned n Secton 4. In Secton 5, one-step-ahead forecasts for short-term nterest rates are generated on 30-day Treasurybll rates. The last secton concludes wth some remarks on the results of the model mplementaton. 2 Model descrpton The short term nterest rate n the Hull-Whte model [21] follows the stochastc dfferental equaton (SDE) dr t [θ t a t r t ] dt + ξ t dw t. (1) The parameters a t, θ t and ξ t are determnstc functons and θ t s chosen so that the model matches the ntal term structure. In equaton (1), W {W t : 0 t T } s a Wener process ndependent of θ t. The volatlty of the short rate s descrbed by ξ t whlst a t s the mean-reverson rate. We can re-arrange the SDE n (1) to get dr t a t [β t r t ] dt + ξ t dw t (2) where β θ. Equaton (2) s a partcular case of the Ornsten-Uhlenbeck process wth a mean reverson level β. It has the soluton t r t r 0 e at + (1 e at )β + ξe at e au dw u. (3) Throughout the succeedng dscusson, all vectors wll be denoted by bold letters n lowercase whlst matrces wll be denoted by Englsh or Greek letters n uppercase. In ths exposton, t s supposed that the short rate process r can be proxed by the yeld rates of T-blls wth very short maturty observed n dscrete tme. A dscrete 0 3

5 Markov chan x k, whch represents the state of the economy, s assumed hdden n these observed values. Let (Ω, F, P ) be the underlyng probablty space of a homogeneous Markov chan x k wth fnte state n dscrete tme (k 0, 1,...). The dstrbuton of x 0 s known and the state space of x k s assocated wth the canoncal bass {e 1, e 2,..., e n } of R n. The th vector e s gven by e (0,..., 1,..., 0), where denotes the transpose. Let Fk 0 σ{x 0,..., x k } be the σ feld generated by x 0,..., x k, and F k be the complete fltraton generated by Fk 0. Furthermore let R k denote the complete fltraton generated by r, so that H k F k R k s the global fltraton generated by x and r. Under the real world probablty measure P, the Markov chan x has dynamcs x k+1 Πx k + v k+1 (4) where v k+1 s a martngale ncrement wth E[v k+1 F k ] 0, Π (π j ) s the transton probablty matrx and π j P (x k+1 e j x k e ). In our proposed model, the nterest rate r follows the stochastc process dr t a(x t )[β(x t ) r t ] dt + ξ(x t ) dw t (5) for r 0 0 wth a(x t ) a, x t, β(x t ) β, x t and ξ(x t ) ξ, x t, where, s the usual Eucldean scalar product. All three parameters are governed by a Markov chan, whch ensures, that the model s swtchng from one economc regme to another through tme. Consder the nterest rate process over the tme nterval [s, t]. Then, f t s s small and x s constant over ths nterval, the soluton of equaton (5), after nvokng (3), s r t e a(xs)(t s) r s + β(x s )(1 e a(xs)(t s) ) +ξ(x s )e a(x s)t t s e a(x s)u dw u (6) The stochastc ntegral e a(xs)t t s ea(xs)u dw u n (6) s normally dstrbuted wth mean zero and varance t s e 2a(xs)(u t) du (1 e 2a(x s)(t s) ). 2a(x s ) From equaton (6) we can derve the dscrete representaton of the nterest rate as r k+1 α(x k )r k + γ(x k ) + η(x k )z k+1 (7) 4

6 where we set α(x k ) e a(xk), γ(x k ) β(x k )(1 e a(xk) ) and η(x k ) 1 e ξ(x k ) 2a(x k ) 2a(x k. Here, {x ) k } s a dscrete-tme Markov chan and {z k } s a sequence of IID standard normal random varables. Now we assume that we have a seres of yeld observatons {y k : k N} of the form y k+1 α(x k )y k + γ(x k ) + η(x k )z k+1. (8) The fltratons generated by the processes are defned by F y σ(y 1, y 2,...), F x σ(x 1, x 2,...) and G F y F x. We start wth a reference probablty measure P under whch we can fnd optmal estmates for the unobservable process x and related quanttes. To back out the real world measure P from P we follow Ellott, Aggoun and Moore [9], Chapter 8. We defne the measure P by dp d P G t Λ t wth λ l exp [ α, x l 1 y l 1 + γ, x l 1 η, x l 1 Λ l l k1 ( α, x l 1 y l 1 + γ, x l 1 ) 2 2 η, x l 1 2 ] y l η, x l 1 (9) λ k (10) wth Λ 0 1, { λ l : λ N + } and { Λ l : l N}. The process { Λ l } s a G-martngale under P. 3 Calculaton of flters We wsh to determne the expectaton of any G adapted stochastc process H gven the fltraton F y k. Usng Bayes-theorem, a flter for any adapted process H s gven by ] E [H k F y k ] E[ H k Λk F y k E [ Λk ] F y. k We defne σ(h k ) : E [ ] [ ] H k Λk F y k, so that E Hk F y k σ(h k ). Clearly, σ(h σ(1) 0) E[H 0 ]. We shall fnd a recursve formula for σ(h k 1 x k 1 ). Snce H s a scalar, σ(h k 1 x k 1 ) s a vector. To relate σ(h k ) and σ(h k x k ) we note that 1, x k 1. Hence, 1, σ(h k x k ) σ(h k 1, x k ) σ(h k ). (11) 5

7 Therefore E [ ] H k F y k 1,σ(H k x k ) 1,σ(x k. Suppose H ) l s a scalar G adapted process, H 0 s F0 x measurable and H l H l 1 + a l + b l, v l + g l f(y l ) (12) where a, b and g are G-predctable, f s a scalar-valued functon and v l x l + Πx l 1. From theorem 5.3 of Ellott [8], a recursve relaton for σ k (H k x k ) s gven by n σ k (H k x k ) Γ (y k ) [ e, σ k 1 (H k 1 x k 1 ) Πe wth + e, σ k 1 (a k x k 1 ) Πe + (dag(πe ) Πe Πe )σ k 1 (b k e, x k 1 ) + σ k 1 (g k e, x k 1 )f(y k )Πe ] [ Γ (y l ) exp α y l 1 + γ yl (α ] y l 1 + γ ) 2 η η 2η 2 where denotes the tensor product of vectors n R n and dag(b) s a dagonal matrx B wth (b 1, b 2,..., b N ) n the dagonal. (13) (14) 4 Recursve estmaton of processes Now, for the estmaton of the unknown parameters we need an estmator for the state of the Markov chan as well as for the three related processes. These processes are specal cases of the general form H l H l 1 +a l + b l, v l +g l f(y l ), where H 0 s F x 0 measurable. The estmator σ(x k ) can be derved from σ(h k x k ) by settng H k 1, a k 0, b k 0 and g k 0. From (13), ths mples Let J sr k σ(x k ) N Γ (y k ) e, σ k 1 (x k 1 ) Πe. (15) represent the number of jumps of x k from state e r to state e s n tme k. So, J sr k k x l 1, e r x l, e s J sr k 1 + x k 1, e r π sr + x k 1, e r v k, e s. (16) 6

8 Settng H k J (sr) k,h 0 0, a k x k 1, e r π sr, b k x k 1, e r e s and g k 0 n equaton (13) we get σ k (J sr k x k ) N Γ (y k )E [ σ k 1 x k 1, e {J sr k 1Πe + x k 1, e r π sr Πe + x k 1, e r e s(dagπe Πe Πe )} ] N Γ(y k ) σ k 1 (Jk 1x sr k 1 ), e Πe +Γ r (y k )σ k 1 ( x k 1, e r )π sr e s. (17) Let Ok r denote the occupaton tme of the Markov process x, that s the length of tme x spent n state r up to tme k. Then, O r k k x l 1, e r Ok 1 r + x k 1, e r (18) Here we set H k O r k, H 0 0, a k x k 1, e r, b k 0 and g k 0 n equaton (13) to obtan σ k (O r kx k ) N Γ (y k ){ σ k 1 (Ok 1x r k 1 ), e Πe +σ k 1 ( x k 1, e r x k 1, e )Πe } N Γ (y k ) σ k 1 (Ok 1x r k 1, e Πe +Γ r (y k ) σ k 1 (x k 1 ), e r Πe r. (19) Fnally, defne the process Tk r (f) as T r k (f) : k x l 1, e r f(y l ) T r k 1(f) + x k 1, e r f(y k ) (20) where f s a functon of the form f(y) y, f(y) y 2 or f(y) y l+1 y l, 1 l k. We apply formula (13) wth substtuton H k T r k (g), H 0 0, a k 0, b k 0 and 7

9 g k x k 1, e r and get σ k (T r k (f)x k ) N Γ (y k ){ σ k 1 (Tk 1(f)x r k 1 ), e Πe +σ k 1 ( x k 1, e r x k 1, e )f(y k )Πe } N Γ (y k ){ σ k 1 (Tk 1(f)x r k 1 ), e Πe +Γ r (y k ) σ k 1 (x k 1 ), e r f(y k )Πe r (21) The recursve optmal estmates of J, O and T can be calculated usng equaton (11). 5 Parameter estmaton In ths secton we dscuss the estmaton of the model parameters for the observaton process y k+1 α(x k )y k + γ(x k ) + η(x k )z k+1 (22) and the transton probablty matrx Π of the Markov chan. The set of parameters ρ, whch determnes our model s ρ {π j, α, γ, η, 1, j n}. (23) Intal values of these parameters are assumed to be gven. Wth the EM algorthm (see [10] for detals) we wsh to fnd a new set of parameters ˆρ, whch maxmses the condtonal expectaton of the log-lkelhoods. Wrte Ĥl E[H l Y k ]. To fnd an estmate for the transton probablty matrx Π π j, where n π j 1 we consder the Radon- Nkodym dervatve d ˆP dp Λ k Yk wth Λ 0 1 The log-lkelhood s therefore ( k n r,s1 ( ˆπsr π sr ) xl,e s x l 1,e r ) log Λ k n s,r J sr k log ˆπ sr + Remander 8

10 where the remander does not nvolve ˆπ sr. Subject to the constrant n ˆπ j 1 we maxmse the log-lkelhood. Consequently, the optmal estmates for the parameters ˆπ, ˆα, ˆγ, ˆη are ˆπ j Ĵ j l Ôl ˆα ˆT l (y l+1, y l ) ˆT l (y)γ ˆT l (y2 ) ˆγ ˆT l+1 (y) ˆT l (y)ˆα Ô l (24) (25) (26) ˆη ˆT l+1 (y2 ) + ˆα 2 ˆT l (y2 ) + ˆγ 2Ô l 2ˆα ˆT l (y l+1, y l ) Ôl 2ˆγ ˆT l+1 (y) + 2ˆα ˆγ ˆT l (y). (27) Ôl The sketch of the proofs of equatons (25) (27) are provded n the Appendx. As n Ellott, Sck and Sten [14] the frst parameter α s updated usng the parameter estmate γ from the prevous calculated optmal parameter set. However, once α s updated, ths new optmal parameter estmate s used for updatng the remanng parameters. Evdently, the above parameter estmates rely on the processes σ(j j k ), σ(o k ) and σ(t k (f)), whch were explctly specfed by recursve equatons (17), (19) and (21), respectvely. Snce these optmal estmates are updated upon the arrval of new nformaton throughout the mplementaton the flterng procedure produces a self-calbratng model. 6 Implementaton We use 30-day Treasury-bll rates as a proxy for the short-term nterest rates. The data set s compled by the Bank of Canada and conssts of daly 30-day Treasury-bll yelds between 1996 and 2005 and there are 2500 data ponts n total. The data s processed n batches of 20 data ponts, therefore the parameters are roughly updated monthly. For ths data set, the parameters were updated 125 tmes. A prelmnary analyss of the actual data reveals that evoluton of the T-bll rates undergoes several dstnct regmes charactersed by states wth hgh and low means as well as hgh and low standard devatons. The regme-swtchng model s developed to capture ths partcular behavour. Tables 1 and 2 present the segregaton of actual data nto 9

11 ether two or three states. Table 1: Segregaton of the perod of actual data nto 2 states Table 2: Segregaton of the perod of actual data nto 3 states The calculated means and standard devatons are used as rough gudes for the ntal values n estmatng the parameters α, γ, η and the transton probablty matrx Π. The values of the parameters after 125 passes are also ncluded n Tables 3 and 4. The one-step ahead predcted yelds of the T-bll rates are calculated by E [ y k+1 G k ] E [ α(xk )y k + γ(x k ) + η(x k )z k+1 G k ] α, Πˆx k y k + γ, Πˆx k (28) 10

12 Table 3: Results of parameter estmaton for a 2-state HMM-based nterest rate model Table 4: Results of parameter estmaton for a 3-state HMM-based nterest rate model where ˆx k E [ x k G k ]. Fgure 1 shows the actual tme seres and the resultng one-step ahead forecasts for a 3-state HMM-based nterest rate model between 1996 and Apparently, we mpose the number of states n the mplementaton. Of course, n dong ths, we are guded by the realstc features of the actual data. We put them nto dfferent categores accordng to mean and standard devaton. Followng the date segregaton dsplayed n tables 1 and 2 we generate a 1-step ahead forecast wth a 2- and 3-state HMM. It s evdent that all forecasts and the actual data are very close to each other. We dd generate forecasts n a 4-state HMM-based nterest rate model to see f any further mprovement can be ganed. However, we dd not fnd evdence of ths. The evoluton of the parameters after 125 passes for the 3-state HMM-based nterest rate model can be seen n fgure 2. We adopt the crteron of Armstrong and Collopy [1] n assessng the goodness of ft of the one-step ahead forecasts. We evaluate the Medan Relatve Absolute Error (MdRAE) and the Medan Absolute Percentage Error (MdAPE) for the 2- and 3-state HMM respectvely. Furthermore we calculate the Mean Square Error (MSE), whch can be used for a comparson of the models. The results of ths error analyss are gven n Table 5. 11

13 Fgure 1: Plot of actual and one-step ahead forecasts generated by a 3-state HMMbased nterest rate model A comparson of the MdRAE, MdAPE and MSE gves the 3-state HMM-based nterest rate model a better ft than the 2-state HMM-based model, although the error dfferences between the 2-state and 3-state models s not that sgnfcant. Table 5: An error analyss of the 2-state and 3-state HMM-based nterest rate model. 12

14 Fgure 2: Evoluton of estmates for the parameters α, γ, η and the transton probabltes π j for a 3-state HMM-based nterest rate model 13

15 7 Concludng remarks By utlsng HMM flterng technques, we mplemented successfully an nterest rate model n whch the volatlty, and both the level and speed of mean-reverson are governed by a Markov chan n dscrete tme. In partcular, the proposed model s tested on a fnancal tme seres of T-bll yelds wth very short maturty. The model possesses self-tunng characterstcs snce the parameters can be updated every tme a new set of nformaton arrves. Ths s made possble through the use of recursve flters developed for the optmal estmate of the state of the Markov chan and other related quanttes of the observaton process. Our emprcal results show that an nterest rate model wth a Hull-Whte specfcaton and whch allows swtchng between two regmes s capable of capturng the dynamcs of the short-term nterest rate process. However, wthn the algorthms and procedures outlned n ths paper the systematc choce of ntal values and an approach to selectng the optmal number of states stll largely reman unexplored areas whch warrant further analyss and research. 8 Acknowledgement Both authors would lke to thank the fnancal support provded by a Mare Cure Fellowshp for Early Stage Researchers Tranng. Appendces A Optmal estmate for α To derve an optmal estmate for α we consder a new measure ˆP, whch s defned by d ˆP dp Λ α k Yk k λ α l 14

16 where λ α l ( ) exp 1 2η 2 (x l ) (y2 l+1 + (ˆα(x l)y l ) 2 + γ(x l ) 2 2y l+1 ˆα(x l )y l 2y l+1 γ(x l ) + 2ˆα(x l )y l γ(x l )) ( ) exp 1 2η 2 (x l ) (y2 l+1 + (α(x l)y l ) 2 + γ(x l ) 2 2y l+1 α(x l )y l 2y l+1 γ(x l ) + 2α(x l )y l γ(x l )) exp ( 1 2η 2 (x l ) (α(x l)y l ) 2 (ˆα(x l )y l ) 2 2y l+1 α(x l )y l Ths means that + 2y l+1 ˆα(x l )y l + 2α(x l )y l γ(x l ) 2ˆα(x l )y l γ(x l ) ). log Λ α k k [ (α(xl )y l ) 2 (ˆα(x l )y l ) 2 2y l+1 α(x l )y l + 2y l+1 ˆα(x l )y l + 2α(x l )y l γ(x l ) 2ˆα(x l )y l γ(x l ) ] /2η 2 (x l ) k ( n x l, e ( ) ) α 2 yl 2 ˆα 2 yl 2 2y l+1 y l α + 2y l+1 y l ˆα + 2y l α γ 2ˆα y l γ /2η 2 k ( n x l, e ( ) ) ˆα 2 yl 2 + 2y l+1 y l ˆα 2ˆα y l γ /2η 2 + R(α ) n ( T l (y 2 )ˆα 2 + 2T l (y l+1, y l )ˆα 2T l (y)γ ˆα ) /2η 2 + R(α ) where R(α ) s a remander and does not contan ˆα. The expectaton of the log-lkelhood condtonal on Y k s ] L(ˆα) E [log Λ αk Y k n ( ˆT l (y 2 )ˆα ˆT l (y l+1, y l )ˆα 2 ˆT l (y)γ ˆα ) /2η 2 + R(α ) where Ĥl E[H l Y k ]. We dfferentate L(ˆα) n ˆα and equate the result to 0. Ths gves us the optmal choce of the parameter ˆα. In partcular, ( 2ˆα ˆT l (y 2 ) + 2 ˆT l (y l+1, y l ) 2 ˆT l (y)γ )/2η 2 0, and solvng for ˆα, we get ˆα ˆT l (y l+1, y l ) ˆT l (y)γ. ˆT l (y2 ) 15

17 B Optmal estmate for γ To calculate the optmal estmate ˆγ we consder agan the followng Radon-Nkodym dervatve d ˆP k ( Λ γ k dp 1 ( exp γ(xl ) 2 ˆγ(x 2η 2 l ) 2 2y l+1 γ(x l ) + 2y k+1ˆγ(x l ) (x l ) Yk Now + 2α(x l )y l γ(x l ) 2α(x l )y lˆγ(x l ) )). k log(λ γ k ) ( γ(xl ) 2 ˆγ(x l ) 2 2y l+1 γ(x l ) + 2y l+1ˆγ(x l ) + 2α(x l )y l γ(x l ) 2α(x l )y lˆγ(x l ) ) /2η 2 (x l ) k ( n x l, e ( ) ) ˆγ 2 + 2y k+1ˆγ 2y k α ˆγ /2η + R(γ ) where R(γ ) s ndependent of ˆγ. Thus L(ˆγ ) n ( Ô l ˆγ ˆT l+1(y)ˆγ 2 ˆT l (y)α ˆγ ) /2η + R(γ ). We dfferentate L(ˆγ ) and set the dervatve to 0. The optmal choce of ˆγ s gven by ˆγ ˆT l+1 (y) ˆT l (y)α Ôl. C Optmal estmate for η To fnd the optmal estmate ˆη, we start wth the Radon-Nkodym dervatve d ˆP k Λ η η(x l ) ( k dp ˆη(x l ) exp 1 ( yl+1 α(x 2η 2 l )y l γ(x l ) ) 2 (x l ) Yk 1 ( yl+1 α(x 2ˆη 2 l )y l γ(x l ) ) ) 2. (x l ) 16

18 Therefore the log-lkelhood s log Λ η k k ( 1 log(ˆη(xk )) 2 ˆη 2 (x l ) (y l+1 α(x l )y l γ(x l )) 2) + R(η) where R(η) s the remander ndependent of ˆη. Hence, [ k n ( L(ˆη) E x l, e log ˆη x l, e 2ˆη 2 ( ) ) ] yl (α y l ) 2 + γ 2 2y l+1 α y l 2y l+1 γ 2α y l γ Y k n ( log ˆη ˆ(O) l 1 2ˆη 2 + α ˆη 2 ˆT l (y l+1, y l ) + γ ˆη 2 ˆT l+1(y 2 ) α2 2ˆη 2 ˆT l (y 2 ) γ2 2ˆη 2 ˆT l+1(y) + α γ ˆT ˆη 2 l (y) Ô l ) + R(η). After dfferentatng L(ˆη) wth respect to ˆη and maxmsng L(ˆη) the optmal estmate for η may be shown as ˆη ˆT l+1 (y2 ) + α 2 ˆT l (y2 ) + γ 2 Ô l 2α ˆT l (y l+1, y l ) 2γ ˆT l+1 (y) 2α γ ˆT l (y) Ô l. References [1] Armstrong J and Collopy F (1992) Error measures for generalzng about forecastng methods: emprcal comparsons. Internatonal Journal of Forecastng 8: [2] Bansal R and Zhou H (2002) Term structure of nterest rate wth regme shfts. Journal of Fnance 57: [3] Black F and Karasnsk P (1991) Bond and Opton Prcng when Short Rates are Log Normal. Fnancal Analysts Journal 47: 52-9 [4] Brgo D and Mercuro F (2001) Interest Rate Models: Theory and Practce. Sprnger- Verlag, Hedelberg 17

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