Numerical Methods for the Markov Functional Model

Transcription

1 Numerical Mehods for he Markov Funcional Model Simon Johnson Financial Engineering Commerzbank Corporaes and Markes 60 Gracechurch Sree, London EC3V 0HR Absrac: Some numerical mehods for efficien implemenaion of he 1- and 2-facor Markov Funcional models of ineres rae derivaives are proposed. These mehods allow a sufficienly rapid implemenaion of he sandard calibraion mehod, ha join calibraion o caples and swapions becomes possible wihin reasonable CPU ime. Prices for Bermudan swapions generaed wihin he Markov Funcional model are found o be very close o marke consensus prices. Bermudans are herefore a good example of a produc ideally suied o his model. 1 Inroducion The Libor Marke Model of Brace Gaarek and Musiela (BGM) (1997) is he marke sandard model for pricing and hedging exoic ineres rae derivaives. Is advanages include model parameers which are easy o inerpre in erms of financial variables, abiliy o define realisic correlaion dynamics, and he abiliy o price essenially any callable Libor exoics by means of Mone Carlo valuaion. There are wo main difficulies in pracical implemenaion of he BGM Model. Firsly he drif is srongly sae-dependen and canno be reduced o a low-dimensional Markovian form. Whils simulaions wih long ime-seps can sill be performed using a suiable differencing scheme (Huner e al., 2001; Joshi, 2005), his does mean ha Mone Carlo simulaion is he only pracical mehod for pricing. One mus work hard o achieve accepable convergence, paricularly when compuing hedge raios of callable Libor exoics Pierbarg (2004). Secondly, alhough calibraion o marke prices of caples is of course rivial, here is a grea deal of debae surrounding he bes way o perform global calibraion of a BGM model o marke prices of all a-he-money swapions. The choice of calibraion mehod becomes even richer when exensions o BGM, including displaced diffusion, local volailiy or sochasic volailiy are considered Rebonao (2004). However when he model is used in a producion environmen, i is by no means simple o ensure ha a small change in marke quoes give rise o a correspondingly small change in model parameers and hence sable hedges are obained. A second ype of marke model is he swap marke model Galluccio e al. (2004). The advanages and disadvanages of his model are closely relaed o hose of he BGM model. For insance, in he case of a co-erminal swap marke model, calibraion o a se of coerminal swapions is 68 Wilmo magazine

2 TECHNICAL ARTICLE 2 rivial, bu achieving good numerical convergence of greeks and sable global calibraion are again very challenging. A hird ype of Marke Model is he Markov Funcional model proposed by Hun, Kennedy and Pelsser (2000), Hun and Kennedy (2005). The derivaion of his model has he unusual saring poin of proposing a sae variable which follows a simple drifless Brownian moion wih ime-dependen volailiy, which is ypically 1- or 2-dimensional. dx i = σ i ()dw i (1) dw i dw j =ρ ij d (2) The calibraion of he model corresponds o he choice of he numeraire as some funcion N( x, ) of his sae variable. The sandard choice of numeraire, and he one which is used in his paper, is he zero-coupon bond wih mauriy T N. The model can herefore be used o price any (non-pah dependen) Libor exoics using a backwards finie difference solver. Of course srongly pah-dependen producs can also be priced using Mone Carlo mehods. Whils global calibraion of he 1-facor Markov Funcional model is no possible, local calibraion o he complee volailiy smile of one swap or libor rae per mauriy is possible. Because he model allows calibraion o he smile, i is paricularly suiable for highly smile-sensiive producs such as high srike payers Bermudan swapions. The purpose of his paper is o presen a range of numerical mehods which can be used in he calibraion of he Markov Funcional model. These improve he speed of he sandard calibraion mehod described by Hun, Kennedy and Pelsser, bu more imporanly hey make exensions of he calibraion, including join calibraion o caples and swapions, more pracical. 2 The Markov Funcional Model The sandard calibraion of he 1-facor Markov Funcional model, as described by Hun, Kennedy and Pelsser, relies on a Jamshidian-ype rick. To be more specific, using he marke prices of European swapions we can find he marke price of a digial swapion which pays an annuiy if he swap rae is greaer han some srike: [ A (SR K) ] F D marke (K) = N 0 E where (x) represens he Heaviside (sep) funcion. In his calibraion mehod, caples can be considered o be a single period swapion. Moreover, having calibraed he numeraire a canonical daes N, N 1,..., i+1 we can price in he model a produc which pays an annuiy if he sae variable x is greaer han some criical value x [ A (x ) (x x ) ] D model (x ) = N 0 E F N(x, ) N (3) (4) We make he ansaz ha he swap rae is a monoonic funcion of he sae variable and solve for he swap rae as a funcion of sae variable SR(x) = D 1 marke (D model (x)) (5) So given a coninuum of European swapion prices, we can exrac he swap rae, and hence he numeraire, as a funcion of he sae variable. A similar backwards-rolling calibraion mehod can be used in he 2-facor Markov Funcional model. As suggesed in Hun and Kennedy (2005) we make he ansaz ha he swap rae of ineres is a monoonic funcion of a 1-d projecion of he 2-d sae variables z(x, y). Hun and Kennedy choose he projecion funcion using a low-dimensional, Markovian approximaion o a BGM model which hey call he pre-model. We propose anoher choice, which is moivaed by an approximaion o a Hull-Whie model and which addiionally allows he efficien numerical inegraion mehods in he calibraion described in secion 5. Consider he wo-facor Hull-Whie shor rae model: df 1 = λ 1 f 1 + σ 1 dw 1 df 2 = λ 2 f 2 + σ 2 dw 2 r = f 1 + f 2 + φ() where φ() is a deerminisic erm. By making he subsiuion x = exp(λ 1 )f 1, y = exp(λ 2 )f 2 we obain driving facors in he form of (1). dx = σ 1 exp(λ 1 )dw 1 dy = σ 2 exp(λ 2 )dw 2 r = exp( λ 1 )x + exp( λ 2 )y + φ() In he wo-facor Hull-Whie model, hen, he shor rae is a monoonic funcion of he bilinear projecion funcion or more generally z(x, y ) = exp( λ 1 )x + exp( λ 2 )y (8) z(x, y ) = x / var(x ) + y / var(y ) (9) In he case of a 2-facor Markov Funcional model, we herefore expec realisic behaviour by assuming ha he Libor rae or swap rae is a monoonic funcion of (9). Moreover, as we shall find in secion 5, his simple bilinear form allows very efficien performance of he inegrals used in calibraion. 3 Exrapolaion of Marke Swapion Prices The calibraion mehod described in he previous secion is simple and numerically well-behaved. However, marke daa for European swapions (6) (7) ^ Wilmo magazine 69

3 wih exremely high or exremely low srikes migh no exis or may be arbirageable for several reasons: Marke daa for a-he-money swapions and for swapions of exreme srikes may no come from he same source or may no have been updaed a he same ime, and hence may be inconsisen. For example quoes for a-he-money swapions are very liquid, and will be ypically updaed on a coninuous basis by marke daa providers. However swapions wih exreme srikes will be much less liquid and may be updaed only occasionally. In many banks, smile surfaces are paramerised by means of a sochasic volailiy model such as he SABR model Hagan e al. (2002). Hagan gives an asympoic expansion for he price of a European opion in his model which is exremely accurae for srikes close o he forward. However in he wings of he disribuion, he approximaion derived by Hagan for he implied volailiy of a European opion can give negaive probabiliy densiies and hence arbirageable marke prices. Of course he problem here is no he use of he SABR model iself, bu in pushing an asympoic expansion beyond is region of applicabiliy. Oher inerpolaion mehods have been suggesed, such as Gaheral s SVI (sochasic volailiy inspired) parameerisaion Gaheral (2004), which could also be used o miigae his problem. Typically he degree of arbirage from hese causes will be much oo small o exploi once ransacion coss are aken ino accoun. However i prevens he funcional inversion in (5), and as a resul, calibraion o digial swapions can only be performed for a finie range of srikes. Ouside of his range, some exrapolaion mehod should be used, however i is vially imporan ha he exrapolaion mehod chosen for digial opions mus preserve he price of a european opion. In oher words, if he marke price of an a-he-money European swapion (payers or receivers) is E marke hen he exrapolaion mus give he correc value for a call opion and a pu opion E marke = F E marke = (df D marke (K))dK (10) F D marke (K)dK (11) where df is he discoun facor on he paymen dae. If we only have rusworhy, non-arbirageable prices for European swapions wih srikes in he range k min o k max, we mus ensure ha our exrapolaed marke prices saisfy he following: kmin (df D marke (K)) = E marke kmax F kmin (df D marke (K))dK kmax D marke (K)dK = E marke D marke (K)dK F Oherwise, we may have a Markov Funcional model which reprices digial swapions perfecly wihin he range of ineres, bu which fails (12) significanly for a-he-money European swapions. One way o achieve his is by consrucing an inerpolaing objec and performing he inegals on he RHS of (12) using a sandard mehod such as adapive Gauss- Lobao inegraion. We can choose any exrapolaion mehod saisfying (12), bu if we choose an exrapolaion ype such as exponenial, hen he inegrals on he LHS can be performed analyically and he ask is paricularly easy. 4 Inerpolaion of he Numeraire Funcion The main requiremens on he numeraire are ha I mus be posiive, for all choices of marke daa, sae variable x and ime. If he numeraire is a zero-coupon bond wih mauriy T N, he expecaion of 1/N(x, ) mus equal he discoun facor raio DF()/DF(T N ) for all so ha he model will mach marke prices for zero-coupon bonds sripped from a yield curve During he model calibraion, he numeraire funcion is only consruced on a se of canonical daes. Typically hese canonical daes may be quarerly, semiannual, or annual, wih he choice made according o he period of Libor underlying he caples, or else he fixed period of he swapions used for calibraion. In general we will need o price producs wih paymen daes differen from he canonical daes, so some form of inerpolaion of he numeraire funcion is necessary. For a 1-facor Markov Funcional model, N(x, ) is sored as a 2-d inerpolaing objec. The firs requiremen is generally achieved by choosing an inerpolaion mehod which preserves posiiviy. The second requiremen is achieved by defining a normalised numeraire funcion where f i (x/ v()) = DF(T N) DF( i ) v() = 0 1 N(x, i ) (13) σ 2 (s)ds (14) is he variance of he underlying Wiener process. The funcions f i (x) can use any x-inerpolaion mehod, alhough we have found ha log-linear inerpolaion and fla exrapolaion gives paricularly good resuls. For - inerpolaion, we can inerpolae beween he differen ime-slices f i (x) linearly in variance v(). This inerpolaion mehod is chosen because i preserves normalisaion beween he inerpolaion daes. For example, if he numeraire funcion is correcly normalised a canonical daes: DF(T N ) DF( 1 ) DF(T N ) DF( 2 ) dx exp( x 2 /2/v 1 ) = f 1 ( x) exp( x 2 /2) d x = 1 v1 N(x, 1 ) dx exp( x 2 /2/v 2 ) = f 2 ( x) exp( x 2 /2) d x = 1 v2 N(x, 2 ) (15) 70 Wilmo magazine

4 TECHNICAL ARTICLE 2 hen he inerpolaed numeraire funcion will also be correcly normalised DF(T N ) DF() dx exp( x 2 /2/v) = v N(x, ) = f ( x) exp( x 2 /2) d x ( v v1 f 1 ( x) + v ) 2 v f 2 ( x) v 2 v 1 v 2 v 1 exp( x 2 /2) d x = v v 1 + v 2 v = 1 v 2 v 1 v 2 v 1 and all zero-coupon bonds will be exacly repriced. 5 Numerical Mehods for Expecaion Inegrals The calibraion of he 1-facor model requires calculaion of wo inegrals, he firs of which is a convoluion: I 1 (x, ) = E[ f (x T, T) F ] = (16) f (x T, T)dx exp ( (x ) T x ) 2 (17) (v(t) v()) 2(v(T) v()) Alhough he convoluion form suggess he use of Fourier mehods, our experience is ha i can be difficul o preven edge-effecs from diffusing ino he soluion domain. Insead we perform hese inegrals using sraighforward Gauss-Hermie inegraion Press e al. (1992). The second inegral used in he 1-facor model is he condiional expecaion: I 2 = E[ f (x T, T) (x T x ) F 0 ] ( ) f (x T, T)dx (18) = exp x2 T v(t) 2v(T) x where again he funcion f (x T, T) is smooh. In he 1-facor case, he cos of hese inegrals is no oo grea, so ha he choice of numerical mehods used is no oo criical. However some improvemen in performance is achieved by calculaing he condiional expecaion inegral by a change of variable y = N(x/ v()) I 2 = 1 y f (x(y), T)dy where N(x) is he cumulaive normal disribuion. The inegral can now easily be performed using Gauss-Legendre inegraion. Noe ha he mehod of an inverse cumulaive normal ransform followed by Gauss- Legendre quadaraure only gives good resuls if he inegrand mees some regulariy condiions. In paricular, for some funcions f (x), he (19) polynomial fi implici in he Gauss-Legendre mehod is very far from he original funcion. However in his case, paricularly when using fla exrapolaion of he numeraire funcion, he algorihm works well. In he 2-facor Markov Funcional model, he choice of inegraion mehod becomes much more imporan. The convoluion inegral: I 3 ( x, ) = E[f ( x T, T) F ] = i dx i exp ( 1 ) f 2 ( x T x ).v 1 (xt, T) (20).( x T x ) de v is an inegral of a smooh funcion over a Gaussian kernel. One possible approach o inegrals of his ype is o diagonalise he marix v, and o use 1-d Gauss-Hermie inegraion for each eigenvecor direcion. This is known as repeaed quadraure. Wih Gauss-Hermie quadraure of order n for each direcion, his will require n 2 evaluaions of he inegrand. Anoher approach, which we have found gives improved speed wih no loss of accuracy, is o use a cubaure formula. This echnique is no widely known in Quaniaive Finance, and he reader is referred o Cools (1997) for an inroducion o he subjec. Briefly, insead of he 1-d se of orhogonal polynomials which are used o consruc Gauss-Hermie poins and weighs Press e al. (1992), cubaure echniques sar wih a basis se of 2-d orhogonal polynomials. Again he aim is o find a se of weighs and poins such ha he inegral is approximaed by K( x)f ( x)d x N w i f ( x i ) (21) Finding he opimal se of poins x i a which he inegrand should be evaluaed is a grea deal more difficul han in he 1-dimensional case and relies on finding poins which are simulaneously zeros of as many as possible of he basis funcions. This uses advanced group-heoreic echniques and efficien formulae are only known for a few values of N. In he case of a 2-dimensional inegraion wih a Gaussian kernel, for example, a number of efficien formulae of degree 5 are known (i.e. inegraing exacly all bivariae polynomials of order 5 and less) [3] for values of N beween 7 and 12. A number of formulae of degree 9 are known, wih values of N beween 18 and 25. A number of higher order schemes are known, bu hose described give remarkably quick and robus resuls for he cos of only a small number of funcion evaluaions. The inegral I 4 is defined by i=1 I 4 = E[ f ( x T, T) (z(x, y) z ) F 0 ] ( dx i = (z(x, y) z ) exp 1 ) f ( xt, T) (22) 2 xt T.v 1. x T de v i and since we chose a bilinear projecion funcion (9) z(x, y) = c x x + c y y = r(x cos θ + y sin θ) (23) we can roae he coordinaes of he inegral so ha he digial condiion affecs only one direcion. ^ Wilmo magazine 71

5 x = U x (24) ( ) cos θ sin θ U = sin θ cos θ so ha ( I 4 = dx dy exp 1 ) f (U x x 2 T T.U.v 1.U T T x. x T, T) T (25) de v We can hen perform he y inegral using Gauss-Hermie, ransform he x coordinae using (19) and hen perform he x inegral using Gauss-Legendre. This mehod gives excellen accuracy because i deals explicily wih he disconinuiy in he inegrand. 6 Calibraion of he Volailiy Funcion and Typical Resuls When implemened in C++, he numerical mehods described above allow an efficien implemenaion of he sandard calibraion of he Markov Funcional model. For a simple example wih 20 canonical daes (10 yrs, semiannual), calibraion of he 1-facor model on a grid of size 50 akes around 0.06 seconds on a 2.8GHz Inel Penium IV Xeon. For he 2- facor model, calibraion of he numeraire on a 30*30 grid akes abou 0.28 seconds. In each case, he calibraion is sufficienly fas ha we can perform some or all of his calibraion sweep ieraively, whils solving for he volailiy σ() of he Markovian process (1). As described in Hun e al. (2000), he erm-srucure of volailiy effecively conrols he inegraed correlaion of differen forward raes, whils preserving a link o a coninuous hedging argumen. To give a simple example, we have used a Levenberg-Marquard algorihm Press e al. (1992), Nielsen (1999) o calibrae he piecewise consan volailiy of he Markovian process, enabling calibraion o he whole (smile-consisen) disribuion of caples, whils addiionally calibraing o a se of a-he-money coerminal swapions. Each call o he error funcion of he nonlinear solver does he following seps: Typical resuls from he calibraion were gahered using marke daa for he Euro ineres raes marke, observed on 9h Augus For he 1-facor Markov Funcional model, he numeraire was sored on a 80 poin grid wih 20 canonical daes (10 yrs semiannual). Calibraion of he numeraire funcion and of σ() ook 35 seconds using he algorihm described above, and 3 ouer ieraions of he Levenberg-Marquard algorihm were used. As expeced, he complee smile for he caples was reproduced perfecly (see figure 1). The ATM coerminal swapions were also repriced, alhough again as expeced, ou-of-he-money he smile was no mached perfecly (figure 2). Figure 3 shows repricing errors for ATM opions across he swapion marix. The calibraed erm-srucure of σ() is shown in figure 4. The remaining calibraion error had wo sources. Firsly, numerical convergence error which could be reduced by increasing he number of poins on he numeraire grid and he number of Levenberg-Marquard ieraions. Secondly, a small error was inroduced by calibraing o canonical caples (wih no fixing lag) and hen repricing real caples wih a 2 business day fixing lag. Similar resuls were obained using a 2-facor model. For example, wih a 30*30 grid and 2 ouer ieraions of he Levenberg-Marquard solver, global calibraion o caples and coerminal swapions ook 99 seconds. Because of he somewha coarser numerical grid, he maximum calibraion error for he ATM caples or coerminal swapions in he calibraion se increased from 0.1% illusraed in figure 3, o 0.21%. Of course he use of a 2-facor se he piecewise consan volailiy of he Markovian process o he chosen value; perform he Markov-funcional calibraion sweep; using he newly calibraed model, compue he prices of he se of ahe-money coerminal swapions; he error funcion reurned is he difference beween model and marke prices for hese coerminal swapions; Such a calibraion ype migh be suiable for a produc such as a callable range accrual on Libor. Digial caples wih any srike are correcly repriced, so we can be confiden ha he underlying range accrual, which can be decomposed ino a sum of digial caples, is correcly priced. The opionaliy will be priced correcly, a leas in he limi ha he barrier levels are widely spaced, because we can reprice all of he underlying European opions ino which we migh exercise. Figure 1: Calibraion resuls for caples wih a sar dae of 4y. As hese are being used for he calibraion of he numeraire funcion, hey are all exacly repriced. 72 Wilmo magazine

6 TECHNICAL ARTICLE 2 Figure 4: Calibraion resuls for he Markovian process volailiy σ(). Figure 2: Calibraion resuls for 5y/5y swapions. As hese are being used for he calibraion of he volailiy σ(), he ATM coerminal swapions are exacly repriced. Bermudan prices (source: Marki Group). In an illusraion of he phenomenon described a he sar of secion 3, he daa supplier of he swapion and caple prices used as calibraion inpus was differen o he source of he Bermudan prices used o compare he model oupus. The daa supplier provided prices for caps and forward-saring caps which were hen subjeced o a proprieary caple sripping mehod. Resuls are presened in figure 6 for 10y no-call 1y EUR Bermudan swapions. As expeced, he prices of low srike payers and high srike receivers opions are dominaed by heir inrinsic value, and hence are no sensiive o deails of he model. The Hull-Whie model gives accepable accuracy for Figure 3: Calibraion resuls for he ATM swapion marix. The able shows he difference beween marke implied volailiies and implied volailiies from he calibraed model. The opions which were par of he calibraion se are shown in boldface. model would now enable more realisic modelling of producs which depend on CMS spread. As a simple example, figure 5 shows a scaer plo of Mone Carlo pahs, illusraing he decorrelaion beween he 5yr swap rae and he 1yr swap rae, observed in 5 years. As a final illusraion of he use of he Markov Funcional model, he 1- facor model was used o price Bermudan swapions. The prices of co-erminal swapions were used o calibrae he numeraire funcion, and he prices of ATM caples were used o calibrae he volailiy funcion. The resuls obained were compared wih prices from a Hull-Whie model calibraed o a-he-money coerminal swapions, and also wih marke consensus Figure 5: Decorrelaion beween he 5yr swap rae and he 1yr swap rae observed in 5yrs, using he 2-facor Markov Funcional model Mone Carlo samples are ploed. Inegraed lognormal correlaion beween hese wo swap raes is 90.8%, comparable wih ha observed in he marke. ^ Wilmo magazine 73

7 opions wihin a single Bermudan book. Second, i is no clear how o exend hese mehods o oher producs such as callable range accruals or snowballs, whils guaraneeing sable hedge parameers. I is imporan o ensure ha he exra degrees of freedom added o he Markov-Funcional model did no resul in over-fiing marke daa. The calibraion mehod described above decouples he roles of he differen model parameers: he numeraire funcion N(x, ) is calibraed o caples, whils he volailiy funcion σ() is calibraed o swapions. Hun, Kennedy and Pelsser give a simple financial inerpreaion of he erm-srucure of σ() in erms of a mean-reversion parameer, conrolling he erminal decorrelaion. And experimenally we find ha a small change in marke daa gives rise o a small change in model parameers, so ha we believe ha he sysem has a single global opimum. Figure 6: Comparison of marke prices for Bermudan payers and receivers swapions (Source: Marki Group) agains resuls from he Hull-Whie and Markov Funcional models. Prices are shown in basis poins; model pricing error is defined as 100%*((model price)/(marke price)-1). The Hull-Whie model was calibraed o ATM coerminal swapions see secion 6 for discussion of his. The Markov Funcional model was calibraed o he smile of all coerminal swapions and o ATM caples. Very in-he-money opions are omied for clariy as he values of hese are dominaed by heir inrinsic value. Noe ha he Markov Funcional model gives beer resuls han Hull- Whie for he case of higher srike payers swapions, mos clearly hose wih srikes 3.0%, 3.5%, and 5.5%. The Hull-Whie model performs saisfacorily for payers swapion wih srike 4.5% because his is sufficienly close o he ypical srikes of he ATM swapions used in calibraion. The performance of he Markov Funcional model is acually worse han he Hull-Whie model for he case of he 2.5% payers swapion. This is because arbirage in he inpu daa sars o become apparen for long-daed swapions wih srikes around 2.2%, and we have herefore chosen k min = 2.5% as he lower limi in he range of calibraion srikes (see secion 3). low srike receivers opions because he volailiy skew enforced by Hull- Whie is reasonably close o ha observed in he marke. However for highsrike payers swapions, we find ha he Markov Funcional model s abiliy o mach he smile exacly gives a significan improvemen in he accuracy of pricing. The Hull-Whie resuls were obained using calibraion o a-he-money opions, o highligh he smile-sensiiviy of Bermudan swapions. In his case here is some debae regarding wheher i is preferable o calibrae o a-he-money, a-he-srike, or a-he-exercise-boundary opions. The second and hird of hese choices will give a beer mach o marke prices han ahe-money calibraion. However here are srong objecions o he use of hese mehods in producion. Firsly, here is a risk of self-arbirage, inconsisency beween he various locally calibraed models used o price 7 Conclusion The following numerical mehods are proposed for use wih he Markov- Funcional model: a safe mehod o exrapolae digial opion prices beyond hose observed in he marke; sorage of a rescaled numeraire funcion, o enforce correc normalisaion a all even daes; in he case of he 2-facor model, use of a simple linear projecion funcion. This allows he inegrals in calibraion o be performed very efficienly; cubaure echniques which can give significanly higher speed han repeaed Gauss-Hermie quadraure. When hese mehods are used, sandard calibraion as described by Hun, Kennedy and Pelsser is exremely rapid. In fac, i can be fas enough ha i is possible o calibrae he Markovian volailiy funcions, hereby achieving join calibraion o swapions and caples. REFERENCES A. Brace, D. Gaarek and M. Musiela, The Marke Model of Ineres Rae Dynamics, Mahemaical Finance, 1997, 7, R. Cools, Consrucing Cubaure Formulae: he science behind he ar. Aca Numerica (1997) Encyclopedia of Cubaure Formulas, hp:// ecf/ecf.hml S. Galluccio, Z. Huang, J-M Ly, O. Scaille, Theory and Calibraion of Swap Marke Models, 2004, BNP Paribas, hp:// J. Gaheral, A parsimonious arbirage-free implied volailiy parameerizaion wih applicaion o he valuaion of volailiy derivaives, Merrill Lynch, (2004) hp://www. mah.nyu.edu/fellows_fin_mah/gaheral/madrid2004.pdf P. Hagan, D. Kumar, A.S. Lesniewski and D.E. Woodward, Managing Smile Risk, Wilmo Magazine, Sepember 2002, P.J. Hun and J.E. Kennedy and A. Pelsser, Markov-Funcional Ineres Rae Models, 2000, Finance and Sochasics, 4, Wilmo magazine

8 TECHNICAL ARTICLE 2 P.J. Hun and J.E. Kennedy, Longsaff-Schwarz, Effecive Model Dimensionaliy and Reducible Markov-Funcional models, 2005, hp://ssrn.com/absrac= C. Huner, P. Jäckel and M. Joshi, Geing he Drif, RISK Magazine, July M. Joshi, Rapid Compuaion of Drifs in a Reduced Facor Libor Marke Model, Wilmo Magazine, June H.B. Nielsen, Damping Parameer in Marquard s Mehod, Technical Repor, Technical Universiy of Denmark, (1999). V.V. Pierbarg, A Praciioner s Guide o Pricing and Hedging Callable Libor Exoics in Forward Libor Models, 2004, Bank of America Quaniaive Research, hp://ssrn.com/ absrac= W. H. Press, S. A. Teukolsky, W. T. Veerling and B. P. Flannery, Numerical Recipes in C, Cambridge Universiy Press, R. Rebonao, Volailiy and Correlaion, Wiley (2004). ACKNOWLEDGEMENTS Thanks for useful discussions o Denis Desbiez, Sergio Dura, Marin Forde, Peer Jäckel, Rhodri Wynne, and paricularly Ralph Sebasian. Thanks also o an anonymous referee for valuable commens. The views expressed in his aricle are personal and do no represen he views of Commerzbank. W Wilmo magazine 75