Markov-Functional Interest Rate Models*

Transcription

1 Markov-Funcional Ineres Rae Models* Phil Hun 1, Joanne Kennedy 2, Anoon Pelsser 3 1 Global Derivaives and Fixed Income Markes, Wesdeusche Landesbank, 33/36 Gracechurch Sree, London EC3V 0AX, Unied Kingdom ( phil.hun@weslb.co.uk) 2 Deparmen of Saisics, Universiy of Warwick, Covenry CV4 7AL, Unied Kingdom ( j.e.kennedy@warwick.ac.uk) 3 Srucured Producs Group (AA4410), ABN-Amro Bank, P.O. Box 283, 1000 EA Amserdam, The Neherlands ( anoon.pelsser@nl.abnamro.com) and Deparmen of Finance, Erasmus Universiy Roerdam, P.O. Box 1738, 3000 DR Roerdam, The Neherlands. Absrac We inroduce a general class of ineres rae models in which he value of pure discoun bonds can be expressed as a funcional of some (low-dimensional) Markov process. A he absrac level his class includes all curren models of pracical imporance. By specifying hese models in Markov-funcional form, we obain a specificaion which is efficien o implemen. An addiional advanage of Markov-funcional models is he fac ha he specificaion of he model can be such ha he forward rae disribuion implied by marke opion prices can be fied exacly, which makes hese models paricularly suied for derivaives pricing. We give examples of Markov-funcional models ha are fied o marke prices of caps/floors and swapions. Key words: yield curve modelling, derivaives pricing, Markov processes JEL classificaion: G13, E43 Mahemaics Subjec Classificaion (1991): 60G44, 60J25, 90A09 * The auhors would like o hank an anonymous referee, Seve Shreve, Klaus Sandmann, paricipans a he fourh CAP Workshop on Mahemaical Finance a Columbia Universiy and paricipans a he Risk Global Summi meeing in London for heir commens and suggesions. Manuscrip received:...; final version received:... 1

2 1. Inroducion Amongs praciioners in he ineres rae derivaives marke a consensus is saring o emerge as o he desirable and mos imporan properies of an ineres rae pricing model. These properies sem from he way hese models are used in pracice. To deermine he prices of exoic derivaives, pricing models are used as exrapolaion ools. Firs he model parameers are chosen so ha he model values of relevan liquid insrumens agree wih marke prices, hen he model is used o price he exoic. From his perspecive, he properies of a good pricing model for derivaives can be summarised as follows. The model should a) be arbirage-free; b) be well-calibraed, correcly pricing as many relevan liquid insrumens as possible wihou overfiing; c) be realisic and ransparen in is properies; d) allow an efficien implemenaion. So far, models proposed in he lieraure have no been able o combine all four requiremens. The conribuion of his paper is o presen a modelling framework wihin which one can develop models possessing all four aribues. The condiio sine qua non for all models is, of course, freedom from arbirage. The general framework for arbirage-free ineres rae models was laid ou by Heah, Jarrow and Moron (1992), hjm hereafer. All models we will consider are arbirage-free ineres rae models and represenaives of he class of hjm models. The radiional approach for specifying an ineres rae model is o ake one or more mahemaically convenien underlying variables and o make cerain disribuional assumpions o reflec he fuure uncerainy of hese variables. One possible choice for he underlying variables are he insananeous forward raes. This is he seup for he model proposed by hjm. Unforunaely, for mos disribuional assumpions one can make for he forward raes (in he hjm framework his is known as specifying he volailiy funcion ), he dynamics of he forward raes become pah-dependen which makes numerical implemenaions of he model very cumbersome. An ineresing sub-class was proposed independenly by Cheyee (1992) and Richken and Sankarasubramanian (1995). They find resricions on he volailiy funcions such ha he pah-dependency can be refleced 2

3 by one addiional sae-variable, which makes reasonably efficien numerical implemenaion possible. Insead of explicily resricing he volailiy funcions one can also choose no o model he complee forward curve direcly, bu o focus aenion on a single rae, he insananeous shor rae. A few of he bes known examples of his approach are he models proposed by Hull-Whie (1994), Cox-Ingersoll-Ross (1985) or Black-Derman- Toy (1990) / Black-Karasinski (1991). These models assume ha he shor rae has a normal, chi-square or log-normal disribuion respecively. The aracive feaure of hese models is heir simple and efficien numerical implemenaion. A general problem wih models from he radiional approach is ha neiher he insananeous forward raes, nor he insananeous shor rae can be observed in he markes. To make maers worse, prices for insrumens which can be observed in he markes such as caps, floors and swapions ofen depend in complicaed non-linear ways on he underlying model parameers. Hence, o replicae marke prices one chooses a se of reference insrumens and solves for he correc parameer values by solving a nonlinear opimizaion problem such ha he pricing error of he model is minimised. These procedures can be numerically quie inensive and are known o be plagued by numerical insabiliies. A radical deparure from he radiional approach has emerged recenly in he lieraure. Insead of using some unobserved raes as he underlying variables, hese new models ake raes which are raded in he markes such as libor or swap raes as he underlying variables. Consequenly, hese new models have become known as Marke Models ; hey were inroduced independenly by Milersen, Sandmann and Sondermann (1997) and Brace, Gaarek and Musiela (1997), and have been exended by Jamshidian (1998). In he case when marke opion prices are given by some simple formula such as Black s formula i is easy o find specificaions of hese models such ha marke prices can be fied exacly and model paramerizaion is herefore rivial. However, he dynamics of forward raes are, jus like in he hjm models, pah-dependen which makes efficien numerical implemenaion very difficul, especially for American-syle producs. Furhermore, when marke prices are no given by Black s formula, as is he case in currencies such as Yen where ineres raes are very low, model paramerizaion is no easier han for he earlier models in he lieraure (see, for example, Andersen and Andreasen, 1998). 3

4 In his paper we propose models ha can fi he observed prices of liquid insrumens in a similar fashion o he Marke Models, bu which also have he advanage ha prices can be calculaed jus as efficienly as in he shor-rae models of he radiional approach. To achieve his we consider he general class of Markov-funcional ineres rae models, he defining characerisic of which is ha pure discoun bond prices are a any ime a funcion of some low-dimensional process which is Markovian in some maringale measure. This ensures ha implemenaion is efficien since i is only necessary o rack he driving Markov process. Marke Models do no possess his propery (for a low-dimensional Markov process) and his is he impedimen o heir efficien implemenaion. The freedom o choose he funcional form is wha permis accurae calibraion of Markov-funcional models o relevan marke prices, a propery no possessed by shor-rae models. The remaining freedom o specify he law of he driving Markov process is wha allows us o make he model realisic. The idea of describing ineres rae models in erms of funcional forms of an underlying Markov process has also been proposed in he papers by Hagan and Woodward (1997) and Schmid (1997). The focus of hese papers is primarily on he form of he Markov process. By conras, our focus is on he funcional forms and showing how hey can be chosen such ha he Markov-funcional model has cerain pre-specified properies. In paricular, we show how o generae Markov-funcional models ha replicae marke prices for libor- and swap-based insrumens. In a forhcoming paper Balland and Hughson (1999) presen a model for libor raes which was derived independenly from our work and employs similar ideas. This paper is organised as follows. We begin in Secion 2 wih some noaion and a brief summary of he derivaive pricing resuls ha we shall need. Secion 3 conains a descripion of a general Markov-funcional ineres rae model and examples wihin his class. I concludes by examining he relaionship of his class o exising Marke Models. In Secion 4, we discuss he properies of he driving Markov process which will ensure he resuling Markov-funcional model is realisic and suiable for he pricing problem a hand and in Secion 5 we presen some numerical resuls o compare his model wih some oher sandard models currenly in use. Finally, we conclude in Secion 6. 4

5 2. Preliminaries The resuls and ideas presened in his paper rely on a knowledge of he general L 1 heory of opion pricing. To presen his heory in any deail here would be onerous. Insead we will very briefly summarise he key ideas ha we need, boh wihou proof and wihou including all he echnical regulariy condiions on he economy required o make hese resuls hold rue. A full and rigorous reamen of he background heory can be found in Hun and Kennedy (2000). Throughou we will be working in a single currency economy E in which he underlying asses are a se of pure discoun bonds. We denoe by D T he value a ime of he bond which maures and pays uniy a T, and hus in paricular D TT = 1. For he presenaion here i will be enough o consider an economy comprising only a finie number of hese bonds, T T where T = {,i =1,2,...,n}, bu he resuls can be exended o a coninuum T = R +.WedenoebyF he informaion available a ime from observing he prices of hese asses, F = σ(d ut : u, T T). We will allow rading in his economy, buying and selling of he asses hroughou ime, bu will proclude he injecion of exernal funds ino he economy all rading sraegies mus be self-financing. The value of a porfolio generaed in his way by rading in he asses of he economy will be called a price process and any price process ha is almos surely posiive is a numeraire. We shall be ineresed in he problem of finding he value of a derivaive wihin his economy. We suppose ha here is some ime on which he value of he derivaive, V, will have been deermined from he evoluion of he asse prices, and hus we need only consider he prior evoluion of he economy up unil he ime. We assume ha he derivaive in quesion can be replicaed and ha he economy admis a numeraire pair (N,N), meaning a numeraire N and a measure N equivalen o he original measure P (wih respec o which he economy is defined) such ha he vecor process (D T /N,T T)isan{F } maringale. The measure N is called a maringale measure. I hen follows (see for example Hun and Kennedy (2000), Musiela and Rukowski (1997)) ha E is arbirage-free and he value of he derivaive a any ime prior o is given by for any T. V = N E N [V N 1 F = N E N [V T N 1 T F (1) 5

6 3. Markov-Funcional Ineres Rae Models The class of models wih which we shall work we refer o as Markov-funcional Ineres Rae Models (M-F models). The assumpions we make here are moivaed by wo key issues: firs, he need for a model o be well-calibraed o marke prices of relevan sandard marke insrumens and, secondly, he requiremen ha he model can be efficienly implemened. Cenral o he approach is he assumpion ha he uncerainy in he economy can be capured via some low dimensional (ime-inhomogeneous) Markov process {x :0 }in ha, for any, he sae of he economy a is summarised via x. This is clearly he defining propery of a model ha can be implemened in pracice. I is rue for all classical spo rae models, in which case x = r. The examples of Consaninides (1992), generaed by direcly modelling he sae price densiy (for more on his in a general non-markovian seing, see Jin and Glasserman (1997)), also share his propery, as does he Raional Log-normal Model of Flesaker and Hughson (1996). Models no possessing his propery are Marke Models, where he dimension of he Markov process x is much higher, and, of course, general HJM models. Wih he excepion of Marke Models, which suffer from high dimensionaliy, exising models fail o calibrae well o he disribuion of relevan marke raes. The key o achieving his, wihou he exra dimensionaliy, is firs o define a process x of low dimension and hen o define is relaionship o he asses in he economy in a way which yields he desired disribuions. In applicaions here will usually be an obvious and naural choice for he process x, and ypically i will be Gaussian which is paricularly easy o implemen. In pracice x will be one- or a mos wo-dimensional, he examples in his paper all being one-dimensional Definiion We now describe a general Markov-funcional ineres rae model. Le (N,N) denoea numeraire pair for he economy E. We make he following assumpions (i) he process x is a (ime-inhomogeneous) Markov process under he measure N, (ii) he pure discoun bond prices are of he form D S = D S (x ), 0 S S, 6

7 for some boundary curve S :[0, [0,, (iii) he numeraire N, iself a price process, is of he form N = N (x ) 0. The boundary curve S :[0, [0, will be chosen o be appropriae for he paricular pricing problem under consideraion. In he examples we discuss laer he producs of ineres share some common erminal ime T and he curve S is aken o be { S, if S T, S = T, if S>T. (2) (see Figure 1). For all pracical applicaions, M-F models will have a boundary curve given by (2). Inser Figure 1 here Figure 1 Boundary Curve Noe ha if S = S hen Assumpion (ii) follows from Assumpion (iii) via he valuaion formula (1). Conversely, Assumpion (iii) regarding he numeraire is no addiional assumpion over (ii) if he numeraire is one of he pure discoun bonds, or a linear combinaion hereof. However, his assumpion does rule ou working in he risk-neural measure wih he cash accoun as numeraire. The Markovian assumpion on x is a naural one. For suppose we consider a produc which makes paymen a some ime T of an amoun which is deermined by he asse 7

8 prices over he ime inerval [, T. If we insis for all such producs ha he value a is of he form V (x ) (i.e. a funcion of (x,)), hen i follows ha he process x mus be Markovian in any maringale measure M corresponding o a numeraire M of he form M (x ). Wih hese assumpions, o compleely specify he model i is sufficien o know (P.i) he law of he process x under N, (P.ii) he funcional form of he discoun facors on he boundary S, i.e. D S S(x S ) for S [0,, (P.iii) he funcional form of he numeraire N (x ) for 0. From his we can recover he discoun facors on he inerior of he region bounded by S via he maringale propery for numeraire-rebased asses under N, [ D S S(x S ) D S (x )=N (x )E N F. (3) N S (x S ) The M-F models we will develop in his paper will be suiable for pricing muliemporal (such as American-syle) producs. I is also possible o consruc M-F models which are suied for pricing European-syle producs and he ineresed reader is referred o Hun-Kennedy-Sco (1996). For pricing muli-emporal producs, we need o fi he join disribuion of a se of swap-raes a differen poins in ime. As we shall see, given a one-dimensional Markov process, i is possible o fi he marginal disribuions. However, for successfully pricing he produc i is imporan o capure he join disribuion of he swap-raes under consideraion. We can affec his join disribuion hrough he choice of he underlying Markov process. We will discuss his imporan poin furher in Secion Implying he Funcional Form of he Numeraire from Swapion Prices Consider an exoic produc which depends on a se of forward swap raes or forward libor s each observed a a disinc ime. Denoe hese raes by {y (i),i =1,...,n} and le,i =1,...,n, denoe he seing daes for hese par swap raes. We make he assumpion ha for each of he paymen daes S (i) j,j =1,2,...,n i,eihers (i) j >T n or S (i) j = T k,somek>i. If his assumpion is no valid i can be made o hold by inroducing auxillary swap raes y (k) as necessary. Our aim in his subsecion is o consruc a one-dimensional M-F model which correcly prices opions on he swaps associaed wih hese forward raes. 8

9 We will assume ha (i) we are given he law of he process x under N, and (ii) he funcional forms D TnS(x Tn ) are known for S T n. From he discussion above i is clear ha o compleely specify our model i remains o specify he funcional form of he numeraire. We will make wo furher assumpions: (iii) our choice of numeraire is such ha N Tn (x Tn ) can be inferred from he funcional forms of he discoun facors a he boundary, i.e. hose in (ii) above, and (iv) he ih forward rae a ime, y (i), is a monoonic increasing funcion of he variable x Ti. In he remainder of his subsecion we show how marke prices of he calibraing vanilla swapions can be used o imply, numerically a leas, he funcional forms N Ti (x Ti ) for i =1,...,n 1. Equivalen o calibraing he model o he vanilla swapions is o calibrae i o he inferred marke prices of digial swapions, see e.g. Dupire (1994). The digial swapion corresponding o y (i) having srike K has payoff a ime of Ṽ (i) (K) =P (i) 1(y (i) >K), where P (i) denoes he accrual facor for he payoff. Applying (1), he value of his opion a ime zero is given by Ṽ (i) [ˆP(i) 0 (K) =N 0 (x 0 )E N (x Ti )1(y (i) (x Ti )>K) (4) where ˆP (i) (x Ti )= P(i) (x Ti ) N Ti (x Ti ). To deermine he funcional forms of he numeraire N Ti (x Ti ) we work back ieraively from he erminal ime T n. Consider he ih sep in his procedure. Assume ha N Tk (x Tk ),k =i+1,...,n, have already been deermined. We can also assume ˆD Ti S(x Ti )= D S(x Ti ) N Ti (x Ti ) (i) for relevan S> are known (and hence so is ˆP ) having been deermined using (3) and he known (condiional) disribuions of x Tk,k=i,...,n. 9

10 Now consider y (i) which can be wrien as Rearranging equaion (5) we have ha N 1 y (i) D Ti N 1 S (i) T n i = i. (5) P (i) N 1 N Ti (x Ti )= 1 ˆP (i) (x Ti )y (i) (x Ti )+ ˆD Ti (x S (i) Ti ). (6) n i Thus o deermine N Ti (x Ti ) i is sufficien o find he funcional form y (i) (x Ti ). By assumpion (iv) here exiss a unique value of K, sayk (i) (x ), such ha he se ideniy {x Ti >x }={y (i) >K (i) (x )} (7) holds almos surely. Now define J (i) [ˆP(i) 0 (x )=N 0 (x 0 )E N (x Ti )1(x Ti >x ). (8) For any given x we can calculae he value of J (i) 0 (x ) using he known disribuion of under N. Furher, using marke prices we can hen find he value of K such ha x Ti J (i) 0 (x )=Ṽ(i) 0 (K). (9) Comparing (4) and (8) we see ha he value of K saisfying (9) is precisely K (i) (x ). Clearly, from (7), knowing K (i) (x ) for any x is equivalen o knowing he funcional form y (i) (x Ti ) and we are done. I is common marke pracice o use Black s formula o deermine he swapion prices V (i) 0 (K). Observe, however, ha he echniques here apply more generally. In paricular, if he currency concerned is one for which here is a large volailiy skew, meaning he volailiy used as inpu o Black s formula is highly dependen on he level of he srike K, hese echniques can sill be applied. This is paricularly imporan in currencies such as Yen in which i is no reasonable o model raes via a log-normal process. The abiliy of M-F models o adap o all differen markes is one of heir major advanages over oher models. 10

11 3.3. LIBOR Model In his subsecion and he nex we inroduce wo example models which can be used o price libor and swap based ineres rae derivaives. The se of marke raes ha concern us here are he forward libor s L (i) for i = 1, 2,...,n. We assume ha he swapion measure S (n) exiss corresponding o he numeraire D Tn+1, a measure under which he D Tn+1 -rebased asses D S /D Tn+1 are maringales. Noe, ha in he case of libor raes he measure S (n) is ofen called a forward measure. As described in Secion 3.1, an M-F model is specified by Properies (P.i) (P.iii). We will be consisen wih Black s formula for caples on L (n) log-normal maringale under S (n), i.e. dl (n) if we assume ha L (n) is a = σ (n) L (n) dw (10) where W is a sandard Brownian moion under S (n). In applicaions we will ofen ake σ (n) = σe a, some σ and some mean reversion parameer a, for reasons explained in Secion 4. I follows from (10) ha we may wrie L (n) ( = L (n) 0 exp (σ (n) u ) 2 du + x ), where x, a deerminisic ime-change of a Brownian moion, saisfies dx = σ (n) dw. (11) We ake x as he driving Markov process for our model, and his complees he specificaion of (P.i). The boundary curve S for his problem is exacly ha of Equaion (2). For his applicaion he only funcional forms needed on he boundary are D Ti (x Ti ) for i = 1, 2,...,n, rivially he uni map, and D TnTn+1 (x Tn ). This laer form follows from he relaionship 1 D TnTn+1 =, 1+α n L (n) T n which yields 1 D TnTn+1 = 1+α n L (n) 0 exp( 1 Tn (σ (n) 2 0 u ) 2 du + x Tn ). 11

12 This complees (P.ii). I remains o find he funcional form N Ti (x Ti ), in our case D Ti T n+1 (x Ti ), and for his we apply he echniques of Secion 3.2. Take as calibraing insrumens he caples on he forward libor s L (i). The value of he ih corresponding digial opion of srike K is given by [ Ṽ (i) 0 (K) =D DTi T 0T n+1 (x 0 )E i+1 (x Ti ) S (n) D Ti T n+1 (x Ti ) 1(L(i) (x Ti ) >K). If we assume he marke value is given by Black s formula wih volailiy σ (i), he price a ime zero for his digial is where Ṽ (i) 0 (K) =D 0Ti+1 (x 0 )Φ(d (i) 2 ) (12) d (i) 2 = log(l(i) 0 /K) σ (i) 1 T σ(i) T 2 i, i and Φ denoes he cumulaive normal disribuion funcion. (Observe ha Black s formula implies ha he marginal disribuions of he L (i) are log-normal in heir respecive swapion measures.) To deermine he funcional form D Ti T n+1 (x Ti ) for i<nwe proceed as in Secion 3.2. Suppose we choose some x R. Evaluae by numerical inegraion [ J (i) 0 (x DTi T )=D 0Tn+1 (x 0 )E i+1 (x Ti ) S (n) D Ti T n+1 (x Ti ) 1(x >x ) [ [ DTi+1 T =D 0Tn+1 (x 0 )E S (n) E i+1 (x Ti+1 ) S (n) F Ti 1(x Ti >x ) (13) D Ti+1 T n+1 (x Ti+1 ) [ 1 =D 0Tn+1 (x 0 ) D Ti+1 T n+1 (u) φ x Ti+1 x Ti (u) du φ xti (v) dv x where φ xti denoes he ransiion densiy funcion of x Ti and φ xti+1 x Ti he densiy of x Ti+1 given x Ti. Noe from (11) ha φ xti+1 x Ti is a normal densiy funcion wih mean x Ti and variance +1 (σ u (n) ) 2 du. Finally noe he inegrand in (13) only depends on D Ti+1 T n+1 (x Ti+1 ) which has already been deermined in he previous ieraion a +1. The value of D Ti T n+1 (x ) can now be deermined as follows. Recall from Equaion (6) ha o deermine D Ti T n+1 (x ) i is sufficien o find he funcional form L (i) (x ). From (7) and (9) L (i) (x )=K (i) (x ) 12

13 where K (i) (x )solves J (i) 0 (x )=Ṽ(i) 0 (K(i) (x )). (14) We have jus evaluaed he LHS of (14) numerically and K (i) (x ) can hus be found from (12) using some simple algorihm. Formally, [ L (i) (x )=L (i) 0 exp 1 2 ( σ(i) ) 2 σ (i) ( (i) J Φ 1 0 (x ) D 0Ti+1 (x ) Finally, o obain he value of D Ti T n+1 (x ) we use (6). An example of he ype of produc we migh wish o model wih a libor M-F model are flexible caps. A flexible cap is defined by a finie sequence of daes, i =1,2,...,n+1, a se of srikes K i, i =1,2,...,n, and a limi number m. The holder of a flexible cap has he righ o choose when o exercise up o a maximum of m caples. To deermine he opimal exercise daes for he m caples, he holder has o solve simulaneously m opimizaion problems. For a more deailed analysis of flexible caps we refer o Pedersen and Sidenius (1997) Swap Model For he consrucion of a swap M-F model we consider he special case of a cancellable swap for which he ih forward par swap rae y (i), which ses on dae, has coupons precisely a daes +1,...,T n+1. For his case he las par swap rae y (n) is jus he forward libor, L (n), for he period [T n,t n+1. As in he above example, we ake D Tn+1 as our numeraire and assume ha he swapion measure S (n) exiss. Furher, we specify properies (P.i) and (P.ii) exacly as for he libor model. However (P.iii), he funcional form for he numeraire D Tn+1 a imes,i=1,...,n 1, will need o be deermined. For his new model he value of he digial swapion having srike K and corresponding o y (i) is given by ). [ (i) P Ṽ (i) 0 (K) =D T 0T n+1 (x 0 )E i (x Ti ) S (n) D Ti T n+1 (x Ti ) 1(y(i) (x Ti ) >K), where P (i) (x )= n α j D Tj+1 (x ). j=i 13

14 If we assume ha he marke value is given by Black s formula hen he price a ime zero of his digial swapion has he form where Ṽ (i) 0 (K) =P (i) 0 (x 0 )Φ(d (i) 2 ) (15) d (i) 2 = log(y(i) 0 /K) σ (i) 1 T σ(i) T 2 i. i Here Black s formula implies ha he marginal disribuion of he y (i) are log-normally disribued in heir respecive swapion measures. Nex suppose we choose some x Rand evaluae by numerical inegraion [ (i) P J (i) 0 (x T )=D 0Tn+1 (x 0 )E i (x Ti ) S (n) D Ti T n+1 (x Ti ) 1(x >x ) [ E S (n) =D 0Tn+1 (x 0 )E S (n) [ =D 0Tn+1 (x 0 ) x [ (i) P +1 (x Ti+1 ) D Ti+1 T n+1 (x Ti+1 ) P (i) +1 (u) D Ti+1 T n+1 (u) φ x Ti+1 x Ti (u) du F Ti 1(x Ti >x ) φ xti (v) dv Noe o calculae a value for J (i) 0 (x ) we need o know D Ti+1 T j (x Ti+1 ),j>i. These will have already been deermined in he previous ieraion. Now, as in Secion 3.3, y (i) (x )=K (i) (x ). where K (i) (x )solves J (i) 0 (x )=Ṽ(i) 0 (K (i) (x )). (16) Having evaluaed he LHS of (16) numerically, K (i) (x ) can be recovered from (15). Formally we have [ y (i) (x )=y (i) 0 exp 1 2 ( σ(i) ) 2 σ (i) ( (i) J Φ 1 0 (x ) P (i) 0 (x ) The value of D Ti T n+1 (x ) can now be calculaed using (6). An example of a produc we migh wish o model wih a swap M-F model is a Bermudan swapion. Once again his produc is specified via a finie number of daes,,s (i) j,i=1,2,...,n, j =1,2,...,n i,and a se of srikes K i, i =1,2,...,n. The holder ). 14

15 of his opion is free o exercise on any of he daes anduponexerciseenersaswap wih corresponding par swap rae y (i) a and which on dae has value P (i) (y (i) K i ), where P (i) = n i j=1 α (i) j D, S (i) j he α (i) j being daycoun fracions Relaionship o Marke Models Recenly several auhors including Milersen, Sandmann and Sonderman (1997) and Brace, Gaarek and Musiela (1997) have sudied a class of ineres raes models paramerized by forward libor s. Jamshidian (1998) has exended his o models paramerized by general swap raes and hey are collecively known as Marke Models. When hese marke raes are aken o be log-normal maringales in heir respecive swapion measures (dy (i) = σ (i) y (i) dw (i) where W (i) is a sandard Brownian moion under S (i) and σ (i) is a deerminisic funcion of ime) he models yield marke prices for sandard swapion producs. The obvious quesion is how does our approach relae o ha of bgm and Jamshidian. Marke Models presen a general framework for modelling ineres rae derivaives and as such M-F models are a subse hereof, jus as Marke Models fi wihin he hjm framework. A his level Marke Models are jus an alernaive model paramerizaion. However, when Marke Models make he addiional assumpion ha forward par swap raes be log-normal maringales in heir associaed swapion measures, his assumpion is much sronger and more resricive han ours we only assume he maringale propery and a log-normal disribuion for swap raes on heir respecive fixing daes. This addiional resricion in he Marke Models is precisely wha makes hose models difficul o use for American-syle producs in pracice because hey canno be characerised by a low-dimensional Markov process. The following resul, which can be exended o include he more general framework of Jamshidian, makes his saemen precise for he libor Marke Models. 15

16 Theorem 1 Le L =(L (1),...,L (n) ), n > 1, be a non-rivial log-normal libor Marke Model, where L (i) denoes a forward libor rae. Then here exiss no one-dimensional process x such ha (i) L (i) = L (i) (x ) C 2,1 (R, R + ) for i =1,2,...,n, (ii) each L (i) (x ) is sricly monoone in x. Tha is, L is no a one-dimensional M-F model. Remark: We believe his resul exends o hold for any process x of dimension less han n and any funcional forms L (i) (x ). Proof: Suppose such a process x exiss. Then i follows from he inveribiliy of he map x L (n) (x )hawecanwrie L (i) =L (i) (L (n) ), i =1,2,...,n. (17) Since L is a log-normal libor Marke Model i follows (Brace, Gaarek and Musiela (1997)) ha i saisfies an SDE under S (n) of he form dl (i) = µ (i) d + σ (i) L (i) dw (i) (18) for some n-dimensional Brownian moion W =(W (1),...,W (n) ) and some process µ = (µ (1),...,µ (n 1), 0). On he oher hand, if we apply Iô s formula o (17), we obain [ L (i) L (n) ) 2 2 L (i) d + L(i) dl (n) (L (n) ) 2 L (n) [ = L (i) L (n) ) 2 2 L (i) d + L(i) σ (n) L (n) dw (n). (19) dl (i) = (L (n) ) 2 L (n) Noe, ha µ (n) =0sinceL (n) is a maringale under S (n). Equaing he local maringale erms in (18) and (19) yields W (i) W (n),alli,and L (i) L (n) = σ(i) L (i) σ (n) L (n). (20) Solving (20) we find L (i) = c i ()(L (n) ) β i() (21) where c i () is some funcion of and β i () =σ (i) /σ (n). 16

17 Having concluded ha W (i) = W (n),alli, Equaion (18) reduces (Jamshidian, 1998) o he form dl (i) ( n = j=i+1 α j σ (j) L (j) ) 1+α j L (j) σ (i) L (i) d + σ (i) L (i) dw. (22) Subsiuing (21) back ino (19) and equaing finie variaion erms in (19) and (22) now gives (afer some rearrangemen) 1 σ (i) ( c i () ) c i () + β i ()logl(n) σ(n) In paricular, aking i = n 1, yields σ (n) (β i () 1) = n j=i+1 α j σ (j) L (j). 1+α j L (j) = 0 and hus he model is degenerae. 4. Mean reversion, forward volailiies and correlaion 4.1. Mean reversion and correlaion Mean reversion of ineres raes is considered a desirable propery of a model because i is perceived ha ineres raes end o rade wihin a fairly ighly defined range. This is indeed rue, bu when pricing exoic derivaives i is he effec of mean reversion on he correlaion of ineres raes a differen imes ha is more imporan. We illusrae his for he Hull-Whie (1994) (hw) model because i is paricularly racable. In he sandard hw model he shor-rae process r solves he SDE dr =(θ a r )d + σ dw, (23) where a,θ and σ are deerminisic funcions of and W is a sandard Brownian moion. For he purposes of his discussion we ake a a, some consan. Suppose we fi he hw model o marke prices. The bes we can do is o fi he model so ha i correcly values one caple for each, he one having srike K i say. I urns ou ha, given he iniial discoun curve {D 0T,T >0}and a, he marke cap prices deermine v i =Var(r Ti ), each i. The general correlaion srucure ρ(r Ti,r Tj ) depends on a, via he relaionship ρ(r Ti,r Tj )=e a(t j ) vi. v j Noe ha, given he marke cap prices, he raio v i /v j is independen of a. Thus, increasing a has he effec of reducing he correlaion beween he shor rae a differen imes, 17

18 hence also he covariance of spo libor a differen imes. This is imporan for pricing pah dependen and American opions whose values depend on he join disribuion (r Ti,i=1,2,...,n). For oher models, including M-F models, he analyic formulæ above will no hold, of course, bu he general principle does. Given he marginal disribuions for a se of spo ineres raes {y (i),i=1,2,...,n}, a higher mean reversion for spo ineres raes leads o a lower correlaion beween he {y (i),i=1,2,...,n} Mean reversion and forward volailiies In he models of Secion 3 we have no explicily presened an SDE for spo libor or spo par swap raes. We herefore need o work a lile harder o undersand how o inroduce mean reversion wihin hese models. To do his we consider he hw model once again. We have a Equaion (10) paramerized our M-F example in erms of a forward libor process L (n). If we can undersand he effec of mean reversion wihin a model such as hw on L (n) we can apply he same principles o a more general M-F model. An analysis of he hw model defined via (23) shows ha L (n) = X α 1 n where ( e at n e at n+1 ) dx = σe a X d W, (24) a W is a sandard Brownian moion under he measure S (n). The forward libor is a lognormal maringale minus a consan. Noice he dependence on ime of he diffusion coefficien of he maringale erm: he volailiy is of he form consan X e a. This is he moivaion for he volailiy srucure we chose for L (n) in he M-F model definiion a Equaion (10). Wha is imporan is no so much he exac funcional form bu he fac ha he volailiy increases hrough ime. The faser he increase, he lower he correlaion beween spo ineres raes which se a differen imes Mean reversion wihin he M-F LIBOR model To conclude his discussion of mean reversion we reurn o he M-F libor model of Secion 3.3 and show how aking σ (n) = σe a in (10) leads o mean reversion of spo 18

19 libor. Suppose he marke cap prices are such ha he implied volailiies for L (1) and L (n) are he same, ˆσ say, and all iniial forward values are he same, L (i) 0 = L 0,alli. Inser Figure 2 here Figure 2 Mean Reversion Figure 2 shows he evoluion of L (1) and L (n) in he siuaion when he driving Markov process x has increased (significanly), x T1 >x 0.BohL (1) and L (n) have increased over he inerval [0,T 1 bu L (1) has increased by more. The reason for his is as follows. Over [0,T 1, L (1) has (roo mean square) volailiy ˆσ. By comparison, L (n) has (roo mean square) volailiy ˆσ over he whole inerval [0,T n, bu is volailiy is increasing exponenially and hus is (roo mean square) volailiy over [0,T 1 islesshanˆσ.since L (n) is a maringale under S (n), i follows ha [ (n) (n) E S (n) L T n F T1 = L T 1 <L (1) T 1. Tha is, in Figure 2 when spo libor has moved up, from is iniial value L 0 a ime zerooisvalueat 1,L (1) T 1, he expeced value of spo libor a T n is less han L (1) T 1. Conversely, when spo libor moves down by ime T 1 (x T1 <x 0 ) he expeced value of spo libor a T n is greaer han is value a T 1. This is mean reversion. 19

20 5. Numerical Resuls For a 30 year DEM Bermudan, which is exercisable every five years we have compared in Table 1 below, for differen levels of mean-reversion, he prices calculaed by hree differen models: Black and Karasinski (1991) (BK), MF and he Hull-Whie (1994) (HW) model. Noe ha our implemenaion of he Hull-Whie model prohibis negaive mean-reversions so we have no been able o include hese resuls. For every level of mean-reversion, we have given he prices of he embedded European swapions (5 25, 10 20, 15 15, 20 10, 25 5). Since all hree models are calibraed o hese prices, all models should agree exacly on hese prices. The differences repored in he able are due o numerical errors. We adoped he following approach. Firs we chose a level for he BK mean reversion parameer and reasonable levels for he BK volailiies. We hen used hese parameers o generae he prices, using he BK model, of he underlying European swapions. We hen calibraed he oher, HW and MF, models o hese prices. The resulan implied volailiies used for each case are repored in he second column. A he boom of each block in he able, we repor he value of he Bermudanswapionascalculaedbyeachofhehreemodels. From he able we see ha he mean-reversion parameer has a significan impac on he price of he Bermudan swapion. I is inuiively clear why his is he case. The reason a Bermudan opion has more value han he maximum of he embedded European opion prices is he freedom i offers o delay or advance he exercise decision of he underlying swap during he life of he conrac o a dae when i is mos profiable. The relaive value beween exercising now or laer depends very much on he correlaion of he underlying swap-raes beween differen ime poins. This correlaion srucure is exacly wha is being conrolled by he mean-reversion parameer. By conras, he effec of changing he model is considerably less, and wha difference here is will be due in par o he fac ha he mean-reversion paramer has a slighly differen meaning for each model. We conclude ha he precise (marginal) disribuional assumpions made have a secondary role in deermining prices for exoic opions relaive o he join disribuions as capured by he mean-reversion parameer. 20

21 6. Conclusion We have inroduced a new class of models, moivaed very much by he pracicaliies ha a model boh be an accurae reflecion of marke prices and be one which can be implemened efficienly. The examples of Secion 3, which can easily be generalised o model markes for which opion prices exhibi significan volailiy skew, illusrae he ease wih which he models can be implemened and how well ailored hey are o pracical derivaive pricing. The ideas in his paper require furher sudy. The nex sep is o develop a pracical exension o muli-dimensional processes (wo dimensions being he mos imporan case), and o gain a deeper undersanding of how differen examples of hese models behave. In erms of fiing marke prices, he generaliy sacrificed by resricing o Markovian models is more han redressed by he exra flexibiliy offered by only fiing disribuions a erminal daes. This offers he abiliy, as demonsraed in he one-dimensional case in Hun and Kennedy (1998), o creae models which fi marke swapion and cap prices simulaneously, in direc conras o he exising Marke Models. 21

22 Table 1: Value of 30 year Bermudan, exercisable every 5 years Srike: , Currency: DEM, Valuaion Dae: 11-feb-98 Mean Reversion = European Receivers Payers Ma ImVol BK MF HW BK MF HW Bermudan Mean Reversion = 0.06 European Receivers Payers Ma ImVol BK MF HW BK MF HW Bermudan Mean Reversion = 0.20 European Receivers Payers Ma ImVol BK MF HW BK MF HW Bermudan Ma denoes mauriy and enor of embedded European opion. ImVol denoes implied volailiy of embedded European opion. BK are prices calculaed wih Black-Karasinski model. MF are prices calculaed wih Markov-Funcional model. HW are prices calculaed wih Hull-Whie model. 22

23 M-Func Figures Mauriy T S =T T T Figure 1 Time Libor () 1 L T1 L 0 ( n) L T 1 E S ( n) ( n [ L ) T F n T 1 T1 Figure 2 Tn 23

24 References 1. Andersen L. and Andreasen J.: Volailiy Skews and Exensions of he Libor Marke Model, Working Paper, General Re Financial Producs (1998) 2. Balland, P. and Hughson, L.: Markov Marke Model Consisen wih Caple Smile, forhcoming, Inernaional Journal of Theoreical and Applied Finance (1999) 3. Black F., Derman E. and Toy W.: A One-Facor Model of Ineres Raes and is Applicaion o Treasury Bond Opions. Financial Analyss Journal. Jan-Feb, (1990) 4. Black F., Karasinski P.: Bond and Opion Pricing when Shor Raes are Lognormal. Financial Analyss Journal. Jul-Aug, (1991) 5. Brace A., Gaarek D. and Musiela M.: The Marke Model of Ineres Rae Dynamics. Mahemaical Finance. Vol. 7, (1997) 6. Consaninides G.M.: A Theory of he Nominal Term Srucure of Ineres Raes. Review of Financial Sudies. Vol 5, (1992) 7. Cox J.C., Ingersoll J.E. and Ross, S.A.: A Theory of he Term Srucure of Ineres Raes. Economerica. Vol 53, (1985) 8. Cheyee O.: Term Srucure Dynamics and Morgage Valuaion. Journal of Fixed Income. Vol 1, (1992) 9. Dupire B.: Pricing wih a Smile. Risk. Vol.9(3), (1994) 10. Flesaker B. and Hughson L.P.: Posiive Ineres. Risk. Vol 1, (1996) 11. Hagan P. and Woodward D.: Markov Ineres Rae Models. Pre-prin, Banque Paribas and The Bank of Tokyo Misubishi (1997) 12. Heah D., Jarrow R. and Moron A.: Bond Pricing and he Term Srucure of Ineres Raes: A New Mehodology for Coningen Claims Valuaion. Economerica. Vol 61(1), (1992) 24

25 13. Hull J. and Whie A.: Numerical Procedures for Implemening Term Srucure Models I: Single-Facor Models. Journal of Derivaives. Fall 1994, 7 16 (1994) 14. Hun P.J. and Kennedy J.E.: Implied Ineres Rae Pricing Models. Finance and Sochasics. Vol. 2(3) (1998) 15. Hun P.J. and Kennedy J.E.: Financial Derivaives in Theory and Pracice. John Wiley & Sons: Chicheser (2000) 16. Hun P.J., Kennedy J.E. and Sco, E.M.: Terminal Swap-Rae Models. Pre-prin, Universiy of Warwick (1996). 17. Jamshidian F.: Libor and Swap Marke Models and Measures. Finance and Sochasics. Vol. 1(4), (1998) 18. Jin, Y. and Glasserman, P.: Equilibrium Posiive Ineres: A Unified View. Pre-prin, Columbia Universiy (1997) 19. Milersen K., Sandmann K. and Sondermann D.: Closed Form Soluions for Term Srucure Derivaives wih Log-normal Ineres Raes. Journal of Finance. Vol. 52, (1997) 20. Musiela M. and Rukowski M.: Maringale Mehods in Financial Modelling. Springer Verlag: Berlin (1997) 21. Pedersen M. and Sidenius J.: Valuaion of Flexible Caps. Working Paper, Skandinavia Enskilda Banken, Copenhagen (1997) 22. Richken P. and Sankarasubramanian L.: Volailiy Srucures of Forward Raes and he Dynamics of he Term Srucure. Mahemaical Finance. Vol 5, (1995) 23. Schmid J.: On a General Class of One-facor Models for he Term Srucure of Ineres Raes. Finance and Sochasics. Vol. 1(1),3 24 (1997) 25