Markov Functional Market Model and Standard Market Model

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1 Markov Funcional Marke Model and Sandard Marke Model Tianang Sun S Hugh s College Universiy of Oxford A disseraion submied in parial fulfillmen for he degree of Maser of Science in Mahemaical and Compuaional Finance Triniy 28

2 This disseraion is dedicaed o My parens

3 Acknowledgemens Firs of all, I would like o hank Dr. Ben Hambly for aking me o do his projec. His supervision of he projec as well as his commens on he draf has been invaluable. Of course, i goes wihou saying ha his fixed income lecures during he maser course were also excellen. There are also many friends who I wan o hank for heir help hroughou he maser course a Oxford, paricularly my hanks goes o Wu Chen and Li Shanshan for heir coninuous encouragemen and suppor on boh my sudy and job seeking. I hank Jonahan Buckler for being a nice flamae and for his help on checking he grammar and spelling. Also, I would like o express my hanks o Mahew Clarke, who is a Mancheser alumnus of mine, for his commen on he draf of he paper and for, simply, being a grea friend a Oxford. Las, bu no leas, I would like o hank my parens for heir undersanding, rus and more imporanly for heir love.

4 Absrac The inroducion of so called Marke Models (BGM) in 199s has developed he world of ineres rae modelling ino a fresh period. The obvious advanages of he marke model have generaed a vas amoun of research on he marke model and recenly a new model, called Markov funcional marke model, has been developed and is becoming increasingly popular. To be clearer beween hem, he former is called sandard marke model in his paper. Boh sandard marke models and Markov funcional marke models are pracically popular and he aim here is o explain heoreically how each of hem works in pracice. Paricularly, implemenaion of he sandard marke model has o rely on advanced numerical echniques since Mone Carlo simulaion does no work well on pah-dependen derivaives. This is where he srengh of he Longsaff-Schwarz algorihm comes in. The successful applicaion of he Longsaff-Schwarz algorihm wih he sandard marke model, more or less, adds anoher weigh o he fac ha he Longsaff-Schwarz algorihm is exensively applied in pracice.

5 Conens 1 Inroducion 1 2 Ineres Rae Modelling Shor rae modelling HJM modelling framework Sandard marke model LIBOR marke model (LMM) Exisence of arbirage-free srong Markov marke model Change of Numeraire Valuaion in he sandard marke model Markov funcional marke model Definiion Implying he funcional form of he numeraire Swap Markov funcional model Longsaff-Schwarz algorihm Noaion Valuaion algorihm A numerical example Model implemenaion and numerical resul Bermudan swapion Implemenaion of LMM Simulaing he LIBOR rae Longsaff Schwarz algorihm Implemenaion of Markov Funcional model Polynomial fiing Inegraing agains Gaussian i

6 5.3.3 Expecaion calculaion Non-parameric implemenaion Numerical resuls Conclusion 35 Bibliography 37 ii

7 Chaper 1 Inroducion The rading volume in ineres rae derivaives, in boh he over-he-couner (OTC) and exchange-raded markes, has been growing rapidly since he 198s. Pricing ineres rae derivaives accuraely, however, is usually more difficul han valuing equiy and foreign exchange derivaives; one of he reasons is because an individual ineres rae has a more complicaed behaviour han ha of a sock price or an exchange rae. I is, hus, fundamenally imporan o model ineres rae, he non-raded underlying asse in he fixed income marke, effecively in he hope of correcly pricing is derivaives. Tradiionally, here are hree perspecives in modelling ineres raes, namely, shor rae models, insananeous forward rae modelling and marke models. So far, here has been a large number of classical shor rae models, such as Vasicek Model [37], Cox, Ingersoll & Ross (CIR) Model [9], Ho-Lee Model [17] and Hull-Whie-Vasicek Model [18], which have araced much aenion from boh he academic and praciioners due o heir racabiliy and ransparency (see Secion 2.1). As a resul, shor rae models have been widely used and i is sill being popular in he banking indusry. On he oher hand, mos shor rae models involve only one source of uncerainy, namely one-facor shor rae models, making all raes perfecly correlaed and hence hey are no accurae in modelling shifs in he yield curve ha are significanly differen a differen mauriies [38].This drawback of he shor rae model ends o be more obvious for complex producs which may well depend on he difference beween yields of differen mauriies; even hough exended wo-facor shor rae models can, o some exen, allow for a richer yield curve srucure, here is anoher more severe weakness of shor rae models. The volailiy srucure in he shor rae models, afer being made ime-dependen, is nonsaionary, which, consequenly, leads o model calibraion inconsisen; his apparenly is significan from a pracical poin of view [19]. A major breakhrough in arbirage-free modelling of ineres rae was he approach 1

8 o he erm srucure modelling proposed in [16] by Heah, Jarrow and Moron and i is now ofen referred o as he HJM modelling framework. The disinguishing feaure of he HJM framework is ha i covers a large number of previously proposed models and ha insead of modelling a shor-erm ineres rae, insananeous forward raes are modelled; hence he difficuly of calibraion ha shor rae models have is resolved naurally [39]. Though i is even possible o ake real daa for he random movemen of he forward raes and incorporae ino pricing derivaives, a vial weakness of he HJM modelling framework is ha i can be relaively inefficien o price derivaives especially for callable producs, as i requires a cerain degree of smoohness wih respec o he enor of he bond prices and heir volailiies [38](see Secion 2.2). An alernaive way of modelling ineres raes in an arbirage-free bond marke ha has been increasingly popular is o ake marke raded raes as he underlying variables in he model. The foundaion of his consrucion was buil in [35] where he focus was on he effecive annual ineres rae. The idea was furher developed in [6] by Brace, Gaarek and Musiela focusing direcly on modelling forward LIBORs, which is ofen considered as a milesone in so called he Marke Model. In he meanime, similar developmen was independenly done in [26] by Jamshidian bu more aenion insead was paid o modelling swap raes. Generally speaking, Marke Models, also known as BGM models, are essenially arbirage-free erm srucure models which are formulaed direcly in erms of marke observable raes, like LIBORs and swap raes, heir volailiies and correlaions. By enforcing he log-normaliy of he forward LIBOR (or swap) rae under he corresponding forward maringale measure, marke models are hen compaible wih he common pracice of pricing sandard fixed income producs, such as caps and swapions, hrough jusified Black s formula (see Secion 2.3.4). Whereas i is easy o specify he sandard Marke Models so as o have marke prices fied exacly and model calibraion is herefore rivial, Marke Models do have a raher undesirable characerisic. To accuraely implemen a Marke Model, i has o be done by Mone Carlo simulaion because of he high dimensionaliy of he model caused by each LIBOR (or swap) rae ypically having is own sochasic driver [23]. This is consequenly problemaic for pricing even non-callable, pah-dependen producs since i is compuaionally expensive o generae enough Mone Carlo pahs o ge a sufficienly accurae price so ha he Greeks (risk sensiiviies) will be accuraely usable in risk managemen. No surprisingly, his problem becomes more serious for callable producs because simulaion is usually poor in performing calculaions backwards in ime. Moreover, in he case of currencies, such as Yen, wih very 2

9 low ineres raes, marke opion prices canno be simply given by Black s formula, consequenly, calibraing Marke Models is almos as cumbersome as he case of shor rae models (see, for example, [1]). By now, here have been several approaches proposed o overcome he pracical difficulies sandard Marke Models face (see, for example, [33] [13] [27]). This paper parially focuses on anoher recen developed model which canno only fi he observed prices of liquid insrumens similarly as in he sandard Marke Models bu which also enjoys a low-dimensional propery in pricing derivaives. This approach, primarily proposed in [24] [21] [2], is ermed he Markov-Funcional Marke Model since is defining characerisic is ha zero-coupon bond (ZCB), also called pure discoun bond, prices are a any ime a funcion of some low-dimensional Markovian process in some maringale measure. This hen implies an efficien implemenaion as i need only rack he driving Markov process. The second main goal of his paper lies on explaining he usage of he Longsaff-Schwarz algorihm [29], one of he mos successful algorihms developed o price American syle producs, ogeher wih he sandard Marke Model o price high dimensional fixed income insrumens. The organisaion of his paper is as follows. While he firs wo secions of Chaper 2 are devoed o concisely describing shor rae modelling and he HJM approach, secion 3 examines he sandard Marke Model wih more deails. Though i is possible o use he Markov-Funcional Marke Model o price European syle producs, he focus of whole Chaper 3 is predominaingly on pricing muli-emporal (such as American syle) producs. Chaper 4 succincly describes he famous Longsaff-Schwarz algorihm which is a powerful ool in pricing muli-emporal producs, indicaing ha he implemenaion of he sandard marke model pricing complex ineres rae derivaives is indeed possible. In Chaper 5, implemenaion of he sandard marke model and he Markov-Funcional Marke Model will be presened wih some numerical resuls, pricing a Bermudan swapion as an example, in order o make he heoreical comparison ino numerical. Finally, he conclusion is in Chaper 6. 3

10 Chaper 2 Ineres Rae Modelling 2.1 Shor rae modelling The class of shor rae models is, in fac, a special case of arbirage-free models of he erm srucure for which he shor rae (r ) is, in he risk neural measure Q, a (ime-inhomogeneous) Markov process [22]. Normally, hough no necessarily, shor rae models are driven by a univariae Brownian moion and his class of models, due o heir convenien numerical implemenaion propery, has been significanly imporan hisorically. Almos all he shor rae models are specified hrough a sochasic differenial equaion (SDE) dr = µ(,r)d + σ(,r)dw, where W is a Brownian moion in he risk neural measure Q and funcions µ and σ are carefully chosen o make he model paricularly racable, in a sense ha he soluion process is, mos ofen, a Gaussian process and herefore he model can be analyically developed furher, and arbirage-free. By he risk neural pricing formula, he price of a zero-coupon bond (ZCB) P(,T), mauring a T, a ime ( < T) is given by P(,T) = E Q [ exp ( T r s ds ) F ], (2.1) where F is he augmened naural filraion generaed by he Brownian moion W. The Markov propery of r ensures ha (2.1) is a funcion of he riple (r,,t) for all pairs (,T). In oher words, he sae of he marke a is compleely deermined by he pair (r,). I is his propery ha allows one o price derivaives by mos sandard numerical mehods such as simulaion and finie-difference algorihms (see 4

11 [1] [11] [15] in his regard). A simplified version of he Hull-Whie-Vasicek (HWV) model akes he form dr = ( ) θ() kr d + σdw, (2.2) where k, σ are consans bu θ, he mean reversion level, is a deerminisic funcion of ime. This simplified form is a suiable candidae for a quick example showing he imperfec calibraion propery of he shor rae models as well as he racabiliy of shor rae models in erms of he exsience of close-form bond prices. Theorem 2.1. The bond prices in shor rae model wih µ(,r) = α()r + β(), σ 2 (,r) = γ()r + δ(), are of he form ) P(,T) = exp (A(,T) B(,T)r, (2.3) where equaions A(,T) and B(,T), respecively, saisfy A βb δ()b2 =, and B + α()b 1 2 γ()b2 + 1 =. A,B denoe he firs derivaive of A,B wih respec o, wih he boundary condions A(T,T) =, B(T,T) =. By Theorem 2.1 1, in he simplified HWV model, i immediaely follows ha α() = k, β() = θ(). γ() =, δ() = σ 2. Whence for equaions A and B hey become A = θ()b(,t) 1 2 σ2 B(,T) 2, (2.4) B = kb(,t) 1. (2.5) 1 For he proof of his sandard resul see, for example, Chaper 21 of [5] or Chaper 3 of [7] 5

12 Solving (2.5) wih B(T,T) = gives Subsiuing (2.6) ino (2.4) gives A(T,T) A(,T) = B(,T) = T Solving (2.7) wih A(T,T) = leads o A(,T) = T ) 1 e k(t. (2.6) k [θ(u)b(u,t) σ2 B(u,T) 2 [ θ(u)b(u,t) + σ2 B(u,T) 2 Fiing he observed iniial forward price f (,T) resuls 2 2 ] du. (2.7) ] du. (2.8) f (,T) = T log P (,T) = A T (,T) + B T (,T)r, (2.9) where A T,B T denoe he firs derivaive of A,B wih respec o T. From (2.6) and (2.8) respecively, i is easy o, differeniaing wih respec o T, ge and A T = T B T = e k(t ), ( T [ θ(u)b(u,t) + σ2 B(u,T) 2 ] ) du 2 = θ(t)b(t,t) + σ2 B(T,T) 2 T [ + θ(u)bt (u,t) + σ 2 B T (u,t)b(u,t) ] du 2 T [ = θ(u)bt (u,t) + σ 2 B T (u,t)b(u,t) ] du. Hence (2.9) becomes f (,T) = r B T (,T) A T (,T) = r e kt σ2 k T = r e kt σ2 2k 2(1 e kt ) 2 + = r e kt σ2 2 B(,T)2 + e k(t u) (1 e k(t u) )du + T T T θ(u)e k(t u) du θ(u)e k(t u) du θ(u)e k(t u) du Seing x(t) =: r e kt + T θ(u)e k(t u) du, 6

13 hen Observe ha x(t) = f (,T) + σ2 2 B(,T)2. (2.1) i.e. dx dt = kr e kt + θ(t) k So using (2.1) gives θ(t) = T θ(u)e k(t u) du = kx(t) + θ(t), θ(t) = dx dt + kx(t). ( f (,T) + σ2 T 2 B(,T)2) + k [ f (,T) + σ2 2 B(,T)2] = f T(,T) + σ 2 B T (,T)B(,T) + k [ f (,T) + σ2 2 B(,T)2]. Thus even in his simplified case, {θ()} T could be found bu no so sraighforwardly from observed forward rae curve and hence bond prices will mach he observed marke prices a anyime, = in his case, before he mauriy. In pracice, o beer calibrae he model, k, σ will be allowed o be ime-dependen as well; consequenly calibraion would have o employ numerical echinques which are ofen compuaionally inensive and unsable. Neverheless, subsiuing he expression for θ ino (2.8), simplifying algebraically, gives ( P (,T ) A(,T) = log + B(,T)f (,) σ2 P (,) 4k B(,T)2 (1 e 2k ). (2.11) Whence he close-form of ZCB price, in his special case, follows naurally by subsiuing (2.11) ino (2.3) P(,T) = P (,T) P (,) exp [ B(,T)f (,) σ2 4k B(,T)2 (1 e 2k ) r B(,T) ], where B(,T) can be found from (2.6) and P (,T),P (,) are usually observable from he marke. 2.2 HJM modelling framework The HJM modelling framework relies on exogenously specifying he dynamics of insananeous coninuously compounded forward raes f(,t). For any fixed mauriy T < T, he dynamics of he forward rae f(,t) are df(,t) = α(,t)d + σ(,t)dw, 7

14 where α(,t) R, σ(,t) R d are adaped sochasic processes and W is a d- dimensional sandard Brownian moion wih respec o he underlying real probabiliy measure P. I is also assumed ha T and T α(,t) d <, σ 2 i (,T) d < 1 i d. Hence, i is equivalen, for every fixed T < T where T > is he horizon dae, o have f(,t) = f(,t) + α(s,t) ds + σ(s,t) dw s, (2.12) for some Borel-measurable funcion f(, ) : [,T ] R and sochasic processess α(,t) and σ(,t). I is worhwhile noicing ha in he HJM seing, for any fixed mauriy T < T, he iniial condiion f(,t) is deermined by he curren yield curve, which can be esimaed using observed marke prices of bonds and/or oher relevan insrumens; his is exacly why calibraion in his seing becomes rivial. As in Secion (2.1), P(,T) denoes he price a ime < T of a uni ZCB mauring a ime T < T. By he definiion of he forward rae, P(,T) can be recovered from he formula ( T ) P(,T) = exp f(,u) du. (2.13) Theorem 2.2 (HJM Drif Condiion Theorem). In he HJM forward rae modelling framework, he bond marke is arbirage free under he risk neural measure Q if α(,t) = σ(,t) T A more general form of his resul is d ( α(,t) = σ i (,T) i=1 T where d is he number of sochasic drivers. σ(,s) ds T. (2.14) ) σ i (,s) ds T, (2.15) Proof. The proof 2 of (2.14) begins wih deriving he bond prices dynamics in he real measure P. I is easy o see ha (2.13) can also be wrien in he following wo forms log P(,T) = T 2 Proving (2.15) is relaively sraighforward based on proof of (2.14) f(,u) du, (2.16) 8

15 f(,u) du = log P(,T) + Subsiuing (2.12) ino (2.16) gives log P(,T) = = = T T T f(,u) du ( f(,u) + f(,u) du α(s,u) ds + T T α(s,u) ds du Then subsiuing (2.17) ino he above expression gives log P(,T) = = T By Fubine s Theorem T Hence log P(,T) = f(,u) du f(,u) du + log P(,T) T T T T α(s,u) ds du = σ(s,u) dw s du = u α(s,u) ds du α(s,u) ds du + σ(s,u) dw s du + T s T s f(,u) du. (2.17) ) σ(s,u) dw s du T T u u α(s,u) du ds, σ(s,u) du dw s. u σ(s,u) dw s du. σ(s,u) dw s du α(s,u) ds du σ(s,u) dw s du. f(,u) du + α(s,u) ds du + σ(s,u) dw s du }{{} + log P(,T) Noe ha = f(u,u) = r u herefore log P(,T) = log P(,T) + := log P(,T) + T s r u du r u du + α(s,u) du ds T s T s α(s,u) du ds A(s,T) ds + Now wrie P(,T) = e log P(,T) = e X and applying Iô formula gives σ(s,u) du dw s T s S(s,T) dw s σ(s,u) du dw s dp(,t) = e X dx ex d < X > ) = P(,T) (r + A(,T)d + S(,T)dW P(,T) S(,T) 2 d 9

16 Hence in he HJM seing, he bond prices dynamics follow dp(,t) = P(,T) (r + A(,T) + 1 ) 2 S(,T) 2 d + P(,T)S(,T)dW, (2.18) where A(,T) = T α(,s) ds and S(,T) = T σ(,s) ds. I is imporan o noe ha he ruh of (2.18) is independen of he measure used, hen under he risk neural measure Q, where he discouned bond prices are maringales, he bond prices have he shor rae r as he drif. Namely, under Q (2.18) is reduced o dp(,t) = P(,T) (r()d + S(,T) d W ), (2.19) where W is a Q-maringale. Meanwhile, i is also rue, under Q, o have A(,T) S(,T) 2 =. Equivalenly T α(,s) ds + 1 T T σ(,s) ds σ(,s) ds =. 2 Differeniaing w.r. T gives he resul α(,t) = σ(,t) T σ(,s) ds I is, hus, obvious o see ha in an arbirage-free marke he drif of he forward rae is compleely deermined by he volailiy. This, however, causes some pahs of forward rae, excep some special cases where he coefficien σ follows a deerminisic funcion, o explode if log-normaliy is embedded in forward raes [36]. Tha is o say he HJM framework can easily lead o non-markovian forward rae models, which srongly limis is pracical applicaion in pricing ineres rae derivaives. 2.3 Sandard marke model The inroducion of marke models presened an exraordinarily fresh way of hinking, one ha direcly models he marke ineres raes. As a resul, when he opion price is given by Black s formula (see Secion 2.3.3), he link beween he SDE governing he evoluion of he appropriae marke ineres raes and he erminal disribuions of hese raes is clear; his deduces an easy specificaion of marke models such ha hey can exacly mach marke prices [22]. Describing LIBOR marke model serves as a good example of explaining sandard marke models since ha was wha he firs marke models did. 1

17 2.3.1 LIBOR marke model (LMM) Le T < T 1 <... < T n be a sequence of fixed daes for i = 1,...,n and δ i = T i T i 1, he corresponding forward LIBORs are defined as L i () = P(,T i 1) P(,T i ). (2.2) δp(,t i ) To be able o use P(,T n ) as a numeraire laer, define and P i () := P(,T i) P(,T n ) π i () := i =, 1,...,n (2.21) i (1 + δ j L j ()). (2.22) j=1 In (2.22), since he produc over he empy se is uniy so π = 1. For convenience, wihou loss of generaliy, also define P n+1 1 and L n+1. Then from (2.2) and (2.21) i immediaely follows ha ( P i () = 1 + δ i+1 L i ()) Pi+1 (). (2.23) Furhermore by using (2.22) P i () = n j=i+1 ( 1 + δj L j () ) = π n() π i (). Le {W i }, i = 1,...,n, be a se of n correlaed Brownian moions, wih dwdw i j = ρ ij d, under he forward measures F where, if choose P(,T n ) as a numeraire, all he radable discouned by P(,T n ) are maringales, namely, all Pi () are maringales. Then under F,he forwrad LIBORs L i () mus saisfy dl i () = µ i (,L)L i ()d + σ i ()L i ()dw i. (2.24) Since P i () are maringales under F, applying Iô o (2.23) gives d P i () = [1 + δ i+1 L i+1 ()]d P i+1 () + δ i+1 Pi+1 ()dl i+1 () + δ i+1 d P i+1 ()dl i+1 ()(2.25) Then subsiuing (2.24) ino (2.25) yields d P i () = [1 + δ i+1 L i+1 ()]d P ( ) i+1 () + δ i+1 Pi+1 () µ i+1 (,L)L i+1 ()d + σ i+1 ()L i+1 ()dw i+1 + δ i+1 σ i ()L i+1 ()dw i+1 d P i+1 = [1 + δ i+1 L i+1 ()]d P i+1 () + δ i+1 Pi+1 ()σ i+1 ()L i+1 ()dw i+1 + δ i+1 Pi+1 ()µ i+1 (,L)L i+1 ()d + δ i+1 σ i+1 ()L i+1 ()dw i+1 d P i+1 11

18 Thus for d P i () o be a maringale, i requires, i =,,n 1, d P i () = ( 1 + δ i+1 L i () ) d P i+1 () + δ i+1 Pi+1 ()σ i+1 ()L i+1 ()dw i+1 (2.26) and i.e. δ i+1 Pi+1 ()µ i+1 (,L)L i+1 ()d + δ i+1 σ i+1 ()L i+1 ()dw i+1 d P i+1 =, µ i+1 (,L) P i+1 ()d = σ i+1 ()dw i+1 d P i+1. (2.27) Now muliplying (2.26) by π i (), by backward inducion, gives π i ()d P i () = π i () ( 1 + δ i+1 L i () ) d P i+1 () + π i ()δ i+1 Pi+1 ()σ i+1 ()L i+1 ()dw i+1 = π i+1 ()d P ( π i+1 () ) i+1 () + δ i+1 L i+1 () 1 + δ i+1 L i+1 () P i+1 ()σ i+1 ()dw i+1 n = π j () P ( δj L j () ) j () σ j ()dw j 1 + δ j L j () j=i+1 hus i.e. d P i () = P i () n j=i+1 d P i () = P i () π j () P j () π i () P i () n j=i+1 ( δj L j () ) σ j ()dw j, 1 + δ j L j () ( δj L j () ) σ j ()dw j. (2.28) 1 + δ j L j () Now subsiuing (2.28) ino (2.27), by backward inducion again, yields µ i (,L) P i ()d = σ i ()dw i P i () j=i+1 n j=i+1 δ j L j () 1 + δ j L j () σ j()dw j. Then he drif condiion in LMM is n ( δj L j () ) µ i (,L) = σ i ()σ j ()ρ ij. 1 + δ j L j () Finally, he original SDE (2.24) becomes [ dl i () = n j=i+1 ( δj L j () ] )σ i ()σ j ()ρ ij L i ()d + σ i ()L i ()dw i 1 + δ j L j (). (2.29) The procedure for deriving he swap-rae marke models (SMM) is idenical o ha for LIBOR marke models excep ha he algebra is slighly more complicaed. The 12

19 resul is saed below (for a deailed derivaion see Chaper 18 of [22]). For each i = 1,,n, he forward par swap raes y i, in he forward measure F, saisfy he SDE of he form ( n dy i = j=i+1 Γ j 1 Γ i 1 P j P i ( δ j 1 y j 1 + δ j 1 y j where as always P(,T) denoes ZCBs, dwdw i i = ρ ij d and n P i P(,T j ) := δ j P(,T n ), and, for 1 i n j=i y i := P(,T i 1) P(,T n ) n j=i δ, jp(,t j ) Γ i := Also, P n+1 y n+1 : and Γ 1. i j=1 ) σ i σ j ρ ij )y i d + σ i y i dw i, (2.3) (1 + δ j y j+1 ) Exisence of arbirage-free srong Markov marke model To show ha he marke model is srong Markov and consisen wih a full arbiragefree erm srucure model, a few general SDE heories are saed below. These resuls are so classical ha almos any Sochasic Calculus ex conains heir proof, see, for example, [22] [28] Definiion 2.3. On a filered probabiliy space (Ω, {F }, F, P), R n is adaped o {F }. X is a srong Markov process if, given any almos surely (a.s.) finie {F } sopping ime τ, any Γ B(R n ), and any, P(X τ+ Γ F τ ) = P(X τ+ Γ X τ ) a.s. (2.31) An equivalen formulaion o (2.31) is he following sandard resul which is appealing when verifying he srong Markov propery. Theorem 2.4. The process X is srong Markov if and only if, for a.s. finie {F } sopping imes τ and all >, for all bounded coninuous funcions f. E[f(X τ+ ) F τ ] = E[f(X τ+ ) X τ ] (2.32) 13

20 In fac, he srong Markov propery of he soluion process for a locally Lipschiz SDE (σ, b) is inheried from he srong Markov propery of he driving Brownian moion. The following heorem confirms his connecion. Theorem 2.5. If he SDE (σ,b) is locally Lipschiz and le (Ω, {F }, F, P,W,X) be some soluion. Then he soluion process X is srong Markov, i.e. (2.32) holds, for all bounded coninuous funcions f and all a.s. finie F sopping ime τ. The following heorem shows ha here are indeed processes L and y saisfying he SDE (2.29) and (2.3) and hence i is a necessary condiion for he model o be arbirage-free. Theorem 2.6. Suppose ha, for i = 1,,n, he funcions σ i : R n R R are bounded on any ime inerval [, ]. Then srong exisence and pahwise uniqueness hold for he SDE (2.29) and (2.3). Furhermore, he soluion processes L and y are srong Markov processes. The sufficien condiion is easily verified by noing ha from (2.28) P i can be wrien as a Dolean exponenial, P i () = P ( i () exp and similarly for SMM P i = P ( i exp n j=i+1 n j=i+1 Γs j 1 Γ i 1 s ( δ j L j (s) ) ) σj (s)dws j 1 + δ j L j (s) P j s P i s ( δ j 1 y j s 1 + δ j 1 y j s ) σ j s dw j s ) and observe ha he exponenial erm has bounded quadraic variaion over any ime inerval [,], hence, by Novikov s condiion, P i () and P i are indeed rue maringales Change of Numeraire Changing numeraire is a very imporan echnique in mahemaical finance and i, mos ofen, can dramaically simplify he calculaion especially when pricing complex producs. This secion gives a raher brief examinaion of changing numeraire which will be used frequenly hroughou he res of he paper (for deails of change of numeraire, see Chaper 9 in [36]). A numerarie is he uni of accoun in which oher asses are denominaed [36]. In principle, any posiively priced asse can be aken as a numeriare and hence all oher asses are denominaed by he chosen numeraire. In a fixed income marke, a convenien choice of numeraire is a ZCB mauring a ime T and he associaed risk neural measure is ofen called he T-forward measure. 14

21 Theorem 2.7 (Change of numeraire). Le N() be a numeraire and Q N be he associaed measure equivalen o he real world measure P such ha he asse prices S N are Q N maringales. Then for an arbirary numeraire U, here exiss an equivalen measure Q U such ha any coningen claim X T has price and moreover and (S/U) are maringales under Q U. V (,S ) = U E QU [X T U T F ] dq U dq N F = U TN N T U, Theorem 2.8 (Change of risk neural measure). Le M() and N() be he prices of wo asses denominaed in a common currency and le σ() = (σ 1 (),,σ d ()) and ν() = (ν 1 (),,ν d ()) denoe heir respecive volailiy vecor process: d(d()m()) = D()M()σ() dw(), d(d()n()) = D()N()ν() dw(), where D() := exp( r(s) ds) is called he discoun process. If aking N() as he numeraire hen ds N () = S N ()[σ() ν()] dw N () Valuaion in he sandard marke model Based on he resuls in he las wo secions, he common marke pracice of pricing vanilla syle (pah independen) producs is o assume [,T i ],δ = T i T i 1 dl i (,T i ) = L i (,T i )σ i ()d W i, dy k n() = y k n()σ n,k ()d W k n, where W is a 1-dimensional Q i Brownian moion, and σ i (),σ n,k () is some deerminisic funcion. Then a caple, one leg of a cap, a [,T i ], wih srike K, is priced by Proposiion 2.9 (Black s formula 3 ). ( ) Capl i () = δp(,t i ) L i (,T i )N(d 1 ) KN(d 2 ), (2.33) 3 proof, based on changing numeraire, is sandard, see, for example,[36] 15

22 where and d 1,2 = log L i(,t i ) K ± 1 2 Σ2 i(,t i ) Σ i (,T i ) Σ 2 i(,t i ) = Ti σ 2 i (s) ds wih N being he sandard normal cumulaive disribuion funcion N(x) = 1 2π x e z2 /2 dz, x R. Whence a cap seled in arreas a imes T i,i =,,n where T i T i 1 = δ i,t = T is priced, by definiion, Cap() = n Capl i () = i=1 where for every i =,,n 1 and n i=1 ( ) δ i P(,T i ) L i (,T i 1 )N(d 3 ) KN(d 4 ), (2.34) d 3,4 = log L i(,t i 1) K ± 1 2 Σ2 i(,t i 1 ) Σ i (,T i 1 ) Σ 2 i(,t i 1 ) = Ti 1 σ 2 i (s) ds. Then by cap-floor pariy, which is an immediae consequence of he no-arbirage propery, Cap() Floor() = n ( P(,Ti 1 ) (1 + kδ i )P(,T i ) ), i=1 he price of he floor is easily calculaed. In an almos idenical fashion, Black s formula for a payer s swapion V for he period beween [k,n], sruck a K wih swap raes y k n(), is where and V k n () = S k n()(y k n()n(d + ) KN(d )), d ± = Σ 2 k,n = yk n log () ± 1 K 2 Σ2 k,n Σ k,n Tk σ 2 k,n(s) ds. Obviously, pricing non-callable fixed income producs by Black s formula is jus like using he Black-Scholes formula for pricing vanilla producs in he equiy marke. In 16

23 boh cases, a raher simple srucure of volailiy of he underlying variable is a major assumpion, wihou which he valuaion has o swich o numerical mehods. While generally his swich works relaively well in he equiy marke for mos common exoic producs, i does no work so well in he fixed income marke. This is because he full marke model (i.e. SDE (2.29) or (2.3)) has o be used if he exoic produc depends on muliple marke ineres raes (LIBOR or swap rae) in a non-linear way, which hen leads o a very high dimensional problem. For example, pricing a 1 year Bermudan swapion exercisable quarerly involves solving a 39-dimensional problem. Thereby, Mone Carlo simulaion, he only feasible numerical mehod lef in his high dimensional case, could be inefficien wihou specific numerical echnique, especially when pricing and hedging srong pah dependen producs. This pracical drawback of he sandard marke model has generaed considerable research ineres and so far i has been solved relaively well. In Secion 5.2.2, i will show how o use he Longsaff-Schwarz algorihm o make he sandard marke model implemenaion pracically possible. By no means is he Longsaff-Schwarz algorihm he only way of doing his, i jus, o a cerain degree, ends o be more popular han oher mehods proposed in [3], [2] and [8]. 17

24 Chaper 3 Markov funcional marke model Generally speaking, a good pricing model for derivaives should, a leas from a pracical perspecive, have he following properies: 1. arbirage-free; 2. well-calibraed, accuraely pricing as many relevan liquid insrumens as possible wihou overfiing; 3. be realisic and ransparen in is properies; 4. allows an efficien implemenaion [2]. As can be seen from Chaper 2, shor rae modelling, he forward rae modelling in he HJM framework and sandard marke model have no been able o mee all hese four crierion. Moivaed by his observaion, a general class of Markov-funcional ineres rae models has been inroduced and received growing aenion paricularly from praciioners. I is because he Markov-Funcional Marke Model complemens shor rae models and sandard marke models in a way ha i allows an efficien implemenaion and permis accurae calibraion of he model hrough more freedom in choosing he funcional form. In addiion, he remaining freedom o specify he law of he driving Markov process enables he model o be realisic. The vial assumpion in he Markov-Funcional Marke Model is ha he uncerainiy can be capured by some low dimensional (ime-inhomogeneous) Markov process {m : α }, in ha, for any, he sae of he economy a is summarised via m and clearly his is he defining feaure of any pracically implemenable model [2]. α is some ime on which he value of he derivaive, V α, will have been deermined from he evoluion of he asse prices hence only prior evoluion of he economy up o α need be considered. 18

25 3.1 Definiion Le (N,M) be a numeraire pair for he economy E where he numeraire N, iself a price process, is of he form N = N (m ) α and he measure M, ofen called he maringale measure, is equivalen o he real world measure P and such ha (P T /N ) is maringale. Assume ha he process m is a Markov process under he measure M and ha ZCBs are of he form P,S = P,S (m ), α S S for some boundary curve α S : [,α ] [,α ]. Mauriy T α S =T T T Time Figure 3.1: Boundary curve For almos all pracical applicaions, he boundary curve (see Figure 3.1) is appropriaely chosen o be of he form α S = { S, if S T, T, if S > T, (3.1) for some consan T so ha he model need no be defined over he whole ime domain S < [2]. 19

26 Then by he fundamenal asse pricing formula he value of a derivaive, wih payoff V T a T, a any ime prior o α is given by for any T α V = N E[N 1 T V T F ] (3.2) Under hese assumpions, i is sufficien o compleely specify he Markov-Funcional Marke Model wih he knowledge of he law of he process m under M, P αs S(m αs ), for S [,α ], he funcional form of he discoun facors on he boundary α S he funcional form of he numeraire N (m ) for α. Tha is o say i is no necessary o explicily specify he funcional form of discoun facors on he inerior of he region bounded by α S. Thus, via he maringale propery for numeraire-rebased asses under M, discoun facors on he inerior of he region bounded by α S can be recovered by P S (m ) = N (m )E M [ PαS S(m αs ) N αs (m αs ) F ]. (3.3) 3.2 Implying he funcional form of he numeraire Defining he paymen daes for he swap associaed wih he rae y i by S i j,j = 1, 2,,m i ; hough no sricly necessary, for convenience i is assumed ha, for all i,j, eiher S i j > T n or S i j = T k, for some k > i. This assumpion generally holds for many common pracical producs and in he case where i does no hold one can always inroduce auxiliary swap raes y k o make i hold. To consruc a one-dimensional Markov-Funcional Marke Model which correcly prices opions on he swaps associaed wih hese forward raes, we need also o assume ha he ih forward rae a T i, y i T i, is a monoonic increasing funcion of he variable m Ti. To simplify calculaion, PVBP-digial swapions, which have a simple payoff srucure, are used because calibraing he model o vanilla swapions is equivalen o calibraing i o he inferred marke prices of digial swapions [12]. The PVBP-digial swapion corresponding o y i, wih srike K, has payoff a T i of Ṽ i T i (K) = B i T i I {y i Ti >K} 2

27 where B i := n δ j P Sj j=i is called he presen value of a basis poin (PVBP) of he swap corresponding o he swap rae y i and i represens he value of fixed leg of he swap if he fixed leg were uniy [22]. Applying (3.2), is value a ime zero is given by [ ] Ṽ(K) i = N (m )E M ˆBi Ti (m Ti )I {y i Ti (m Ti )>K}, (3.4) where ˆB i T i (m Ti ) = Bi T i (m Ti ) N Ti (m Ti ). Then o deermine he funcional form of N Ti (m Ti ), i involves working backward ieraively from he erminal ime T n. I is naural o assume ha N Tk (m Tk ), k = i + 1,,n, have been already deermined and o assume ˆP Ti S(m Ti ) = P T i S(m Ti ) N Ti (m Ti ) for relevan S > T i, is known, having been deermined by (3.3) and known (condiional) disribuions of m Tk, k = i,,n. This hen implies ha ˆB i T i is also known. Now consider y i T i which can be wrien as yt i i = N 1 T i Simplifying (3.5) algebraically gives P Ti S i n i N 1 T i P i T i N 1 T i. (3.5) N Ti (m Ti ) = 1 ˆB i T i (m Ti )y i T i (m Ti ) + ˆP Ti S i n i (m Ti ). (3.6) Hence, finding he funcional form yt i i (m Ti ) will be sufficien o deermine N Ti (m Ti ). Since yt i i is assumed o have monooniciy wih respec o m Ti, here exiss a unique value of K, say K i (m ), such ha he following holds {m Ti > m } = {yt i i > K i (m )}. (3.7) Now define [ J(m i ) = N (m )E M ˆBi Ti (m Ti )I {mti >m }]. (3.8) 21

28 Then, for any given m, he value of J i (m ) can be calculaed using he known disribuion of m Ti under M. Moreover, he value of K can be found using marke prices such ha J i (m ) = Ṽ i (K). (3.9) I is no hard o see ha he value of K saisfying (3.9) is precisely K i (m ) by comparing (3.4) and (3.8). Finally, he funcional form of y i T i (m Ti ) can be obained by noicing ha i is equivalen o knowing K i (m ) for any m from (3.7). Sandard marke pracice is o use Black s formula (Proposiion 2.9) o find swapion prices V i (K). In fac, he echniques above can be applied more generally, especially for currencies wih a large volailiy skew, meaning volailiy is highly dependen on he srike K, hese echniques are sill applicable. This is one of he major srenghs of he Markov funcional marke model, working well for currencies such as yen in which i is no suiable o model raes hrough a log-normal process. 3.3 Swap Markov funcional model This secion akes he swap Markov funcional model, suiable for pricing swap based producs, as an example o show generally how o consruc he Markov-Funcional Marke Model. To keep he noaion simple, a special case of a cancellable swap is considered for which he ih forward swap rae y i sars on dae T 1 and has coupons precisely a daes S 1,,S n wih exercise imes a T 1,,T n. As before, denoe by δ i he accrual facor for he period [T i,s i ]. Then i follows ha y i = P T i P Sn, B i where B i is, as before, he presen value of a basis poin (PVBP) of he swap. I is worhwhile o noe ha in his case he las par swap rae y n is jus he forward LIBOR, L n, for he period [T n,s n ]. To be consisen wih Black s formula, assume ha y n is a log-normal maringale under he swapion measure S n, i.e. dy n = σ n y n dw, (3.1) where W is a sandard Brownian moion under S n and σ n is some deerminisic funcion. From (3.1), i is equivalen o have ( y n = y n (σ n u) 2 du + m ),

29 where m, a deerminisic ime-change of a Brownian moion, saisfies dm = σ n dw. (3.11) Tha is o say m is aken as he driving Markov process of he model, which is he firs sage o compleely specify he model. As previously indicaed, he boundary curve α S, for his case, is exacly of he form in (3.1) and we only need he funcional form of P Ti T i (m Ti ) for i = 1, 2,,n, namely he uni map, and P TnS n (m Tn ) on he boundary. In his case, by definiion, i follows ha and his immediaely yields P TnS n (m Tn ) = δ n y n T n, P TnS n = 1 + δ n y n ( ), (σn u) 2 du + m which hen complees he second sage of specifying he swap Markov funcional marke model. To find he funcional form of he numeraire P Sn a imes T i,i = 1,,n 1, we need only follow he procedures in Secion 3.2. For his new model, he value of a PVBP-digial swapion wih srike K and corresponding o y i is given by [ B i Ṽ(K) i = P Sn (m )E Ti (m Ti ) ] S n I P {y i Ti (m Ti )>K}. Ti S n(m Ti ) Assuming he marke price obained from he Black s formua yields where Proceeding as in Secion 3.2, le m inegraion Ṽ i (K) = B i (m )N(d 2 ), (3.12) d 2 = log(yi /K) ˆσ i T i 1 2ˆσi T i. R and for i < n, evaluae by numerical [ B i J(m i ) = P Sn (x )E Ti (m Ti ) ] S n P Ti S n (m Ti ) I {m Ti >m } [ [ BT i = P Sn (x )E S n E i+1 (m Ti+1 ) ] ] S n F Ti I{mTi >m P Ti+1 S n (m Ti+1 ) } [ BT i = P Sn (x ) i+1 (u) ] P Ti+1 S n (u) φ m Ti+1 m Ti (u) du φ mti (v) dv m 23

30 where φ mti denoes he ransiion densiy funcion of m Ti and according o (3.11), φ mti+1 m Ti denoes he normal condiional densiy funcion of m Ti+1 given m Ti wih mean m Ti and variance T i+1 T i (σ n u) 2 du. Then where K i (m ) solves y i T i (m ) = K i (m ), J i (m ) = Ṽ i (K i (m )). (3.13) Whence, having found J i (m ) numerically, K i (m ) can be recovered from (3.12) [ yt i i (m ) = yexp i 1 2 ( σi ) 2 T i σ i ( J T i N 1 i (m ) )]. B(m i ) Finally, he value of P Ti S n (m ) can now be calculaed by using (3.6). Here, he focus is on he case of one-dimensional Markov process m, which is sufficien for mos imporan ineres rae derivaives. The generalisaion o he mulidimension case is no difficul and necessary for some paricular producs, for example, Bermudan callable spread opion; however working in he muli-dimensional case is sill a a relaive early sage and some deails can be found in [23] [22]. Since he Markov funcional marke model has successfully ransformed he high dimensional sandard marke model ino a low dimensional (1-dim here) problem, he numerical mehods do no have o rely on Mone Carlo simulaion only. Numerical resuls of, employing one-dimensional Markov funcional marke model, pricing Bermudan swapions will be presened in Secion

31 Chaper 4 Longsaff-Schwarz algorihm The Longsaff-Schwarz algorihm is, so far, arguably he mos widely adoped mehod for pricing muli-dimensional American-syle financial insrumens in boh equiy and fixed income marke. This is mainly because on one hand i can be applied o a large number of common exoic derivaives; on he oher hand i has been demonsraed ha i is raher effecive in numerical implemenaion (see, for insance, [25] [32]). I is he use of leas squares o esimae he condiional expeced payoff o he opion-holders from coninuaion ha makes he approach a worhwhile subsiue for radiional finie difference mehods when pricing high-dimensional producs. Discussion here focuses on describing he general valuaion framework using he Longsaff-Schwarz algorihm bu he argumen is equally well applicable o any specific produc wih some minor modificaion. 4.1 Noaion As before, he framework is based on an underlying complee probabiliy space (Ω, F,P) wih finie ime horizon [,T]. To be consisen wih he no arbirage paradigm, i is assumed ha here exiss an equivalen maringale measure Q for he economy; also define F = {F : [,]} o be he augmened filraion generaed by he he relevan price processes for he securiies and assume F = F T. Le K be he srike price wih discree exercisable imes < 1 2 k = T; in case of coninuously exercisable producs he mehod can also be used by aking sufficienly large K. In addiion, le I(w,s;,T) denoe he pah of cash flows generaed by he securiy, condiional on he produc no being exercised a or prior o ime and on he holder of he securiy pursuing he opimal sopping sraegy for all s, < s T. 25

32 4.2 Valuaion algorihm A each exercisable ime k, invesors are able o know he cash flow from immediae exercise and he value of immediae exercise simply equals his cash flow. Of course, he coninuaion cash flows are no known a k, however, he fundamenal asse pricing formula implies ha he value of coninuaion can be obained by aking he expecaion, in he risk neural measure Q, of he remaining discouned cash flows I(w,s; k,t). More specifically, he value of coninuaion C(w; k ) a ime k is simply [ K C(w; k ) = E Q exp ( j j=k+1 k r(w,s) ds ) I(w, j ; k,t) F k ], (4.1) where r(w, ) is he riskless ineres rae, possibly in a sochasic form. Hence, he problem of opimal exercise is reduced o comparing he immediae exercise value I(w,s;,T) and he coninuaion value C(w; k ) in he sense ha exercise occurs as soon as I C >. As menioned earlier, in he Longsaff-Schwarz algorihm leas squares are used, working backwards, o approximae C(w; k ) a K 1, K 2,, 1. To be more specific, i is assumed 1 ha he unknown funcional form of C(w; K 1 ) in (4.1) can be expressed as a linear combinaion of a counable se of F K 1 measurable basis funcions. 4.3 A numerical example To quickly show an example of how he Longsaff-Schwarz algorihm works, resuls of pricing an American pu opion using he Longsaff-Schwarz algorihm are compared wih ha of using an implici finie difference echnique, a popular mehod of grea accuracy in pricing low-dimensional pah dependen producs. In finie difference, 6, ime seps and 1 sock price seps are used o discreize he Black-Scholes PDE. The L-S simulaion is based on 1, pahs and 5 exercise poins. As shown in he able, he difference beween he wo mehods is quie small and i is believed ha he resuls will be even closer if more simulaion pahs are used. I is worhwhile noing ha he differences in early exercise value could be eiher posiive or negaive, which indicaes ha Longsaff-Schwarz algorihm is capable of replacing he finie-difference o price pah-dependen producs. This is probably why L-S algorihm is being used inensively in pracice when pricing high-dimensional pah-dependen derivaives. In Chaper 5, i will become clearer 1 This assumpion can be formally jusified, for deails, see he original work [29] 26

33 ha he Longsaff-Schwarz algorihm is powerful ye simple enough o price mulidimensional pah-dependence ineres rae producs such as Bermudan swapion. S σ T FD American LS American Analyical European Difference Table 4.1: Comparison of Finie Difference and Longsaff-Schwarz algorihm As always, S denoes he spo price, T denoes he mauriy and σ denoes he volailiy. Oher parameers in his comparison are ineres rae r =.5, srike price K = 2. The Difference column refers o he difference in early exercise value beween wo mehods and early exercise value is he difference beween American opion value and analyical European opion value. The benefi of employing he Longsaff-Schwarz algorihm here may no be so obvious, indeed, he major srengh of Longsaff-Schwarz algorihm is o price muli-dimensional pah dependen producs; a deailed example of his case is in Secion

34 Chaper 5 Model implemenaion and numerical resul Having focused on he heoreical developmen of he sandard Marke Model and he Markov Funcional Marke Model, we are ready o carry ou model implemenaions and presen some numerical resuls, based on pricing an imporan fixed income derivaive Bermudan swapion. The aim is o show ha wih he help of he Longsaff- Schwarz algorihm, implemening he sandard Marke Model (LMM/BGM) o price Bermudan swapion, a high-dimensional problem, is indeed possible. Meanwhile, he Markov funcional marke model, as will be seen, reaches a srong agreemen wih he BGM model on valuaion resuls. 5.1 Bermudan swapion A financial insrumen is called Bermudan if i has muliple exercise daes, namely, here are imes T i a which he holder of a Bermudan may choose beween differen paymens or underlying producs. A Bermudan swapion is a swapion ha has a mauriy dae equal o he las rese dae of he underlying swap and ha has an iniial lockou period in which exercise is prohibied. Effecively speaking, a Bermudan swapion is equivalen o a Bermudan opion on a coupon bond wih srike equal o he par value of he bond and, as an opion on a coupon bond, a Bermudan swapion clearly has posiive probabiliy of early exercise. Le = T < T 1 < < T n = T denoe a given enor srucure and V (T 1,,T n ;T 1 ) denoe he price of a Bermudan swapion iniiaed a T 1. Then by definiion V (T i,,t n ;T i ) := max ( V (T i,,t n ;T i ), ˆV (T i,,t n ;T i ) ) i =,,n 28

35 where ˆV (T i,,t n ;T i ) denoes he value of a swap wih fixing daes T i,,t n 1 and paymen daes T i+1,,t n, observed a T i ; and V (T n ;T n ) :=. Moreover, wih a given numeraire N and a corresponding equivalen maringale measure Q N V (T i+1,,t n ;T i ) = N(T i )E QN( V (T i+1,,t n ;T i+1 ) F Ti ). N(T i+1 ) 5.2 Implemenaion of LMM This secion is o show, sep by sep, how o price a Bermudan swapion in LMM using Mone Carlo simulaion wih he applicaion of he Longsaff-Schwarz algorihm. The volailiy srucure can simply be fla bu more complex volailiy erm srucure can be obained from principal componen analysis (PCA) of correlaion marix and adjusing o calibraed volailiies (see [34] on his opic) Simulaing he LIBOR rae Recall he SDE (2.29) ha LIBOR rae follows under forward measure F [ dl i () = n j=i+1 ( δj L j () ] )σ i ()σ j ()ρ ij L i ()d + σ i ()L i ()dw i 1 + δ j L j (). Since Bermudan swapions are pah-dependen and SDE (2.29) canno be inegraed exacly, he Euler-Maruyama mehod (Euler scheme) needs o be applied here o simulae he LIBOR rae pah [15]. For a beer discreizaion, i is necessary o apply he Euler scheme o log L(); applying Iô s lemma o he above SDE (2.29) gives [ d log L i () = n j=i+1 ( δj L j () ) ] σ i ()σ j ()ρ ij σ2 i d + σ i ()dw i 1 + δ j L j () 2 which is hen suiable o be discreized, using he Euler scheme, as [ L i+1 () = L i () exp n j=i+1 ( δj L j () ) 1 + δ j L j () σ j()ρ ij σ i ()h σ2 i 2 + σ i() ] hz i where, following he same noaion as in Secion 2.3, i =, 1,,n, and Z 1,Z 2, Z n are independen n-dimensional sandard normal random vecors; h is fixed ime sep [14]. 29

36 5.2.2 Longsaff Schwarz algorihm When applying he Longsaff-Schwarz algorihm in he process of pricing a Bermudan swapion, he procedures are divided ino he following seps. 1. Simulaing a large number of pahs (D) of he underlying LIBOR raes so ha values of regression coefficiens are smooh. To conrol he discreizaion bias and o approximae coninuous exercise, he number of simulaion seps (N) is chosen equal o he number of exercise daes. Le disc() be he discree discouning facor a ime. Consider a payer Bermudan swapion V s,n (T i ) wih lockou dae T s exercise daes T i, i = s,,n 1,T n = T and δ = T i+1 T i hen, by definiion, n 1 V s,n (T i,t) = P(T i,t k+1 )δ[l k (T i ) K], k=i where L k (T i ) is he forward LIBOR rae observed a T i for period (T k,t k+1 ), K is he srike price and P(T k,t k+1 ) is he ZCB price a T i for period (T k,t k+1 ). 2. To find he Bermudan swapion price, i is necessary o carry ou dynamic programming backward from he final exercise ime T n 1 as a Bermudan swapion is srongly pah-dependen. Le I(T n 1 ), a ime T n 1, be he maximum of he value of exercising he opion and zero, i.e. I(T n 1 ) = max(v n 1,n (T n 1 ), ). Furhermore, define sop rule (sr) as he opimal sopping ime along a given pah (d) of he LIBOR rae process and he sop rule firsly is se equal o he final exercise ime, sr = T n Working backwards, a ime T n 1, make a regression of he basis funcions of sae variables a ha ime on Y (d) where Y (T n 1 ) = I(sr) disc(t i). disc(sr) Again, sop rule (sr) is he nex sopping ime along a given pah. Basis funcions, denoed by X j (T i ) j = 1 J, are chosen o be quadraic funcions of he curren value of he underlying swap V i,n (T i ) and discouning facor disc(t i ). Regression coefficiens, calculaed from ordinary leas square regression, of X j (T i ) are called β j (T i ). 3

37 4. Then we are ready o compare he coninuaion value C(T i ) = β j (T i )X j (T i ), corresponding o he esimaed condiional expecaion of he he payoff, wih he immediae exercise value I(T i ) = max(v i,n (T i ), ). If I(T i ) > C(T i ), hen presen ime is an opimal sopping ime and I is se o V i,n (T i ) and sop rule is se o T n 1 i.e. I(sr) = V i,n (T i ), sr(d) = T n Seps 3 and 4 are repeaed for all (T n 1 1) T i T s, T s is he lockou dae, unil one reaches he firs exercise ime of he swapion and all coefficiens β j (T i ) have been calculaed. 6. Finally, he value of he swapion can be calculaed by discouning he value a he opimal sopping ime back o presen ime and i is calculaed as 1 D D W(sr)/disc(sr). d=1 Once again, (sr) is he variable ha keeps racking opimal sopping ime along a given pah. 5.3 Implemenaion of Markov Funcional model The implemenaion of he Markov Funcional model replies heavily on numerical inegraion, as seen from Secion 3.3, when calculaing expecaions. I is, however, no advisable o apply simple numerical inegraion schemes such as rapezoid rule or Simpson s rule on a grid of fixed spacing for he Markov process m because, hough yielding reasonably accurae prices, Greeks would become very unsable [3]. Moreover, such simple numerical inegraion scheme on a fixed grid would lead o spiking inegrands, in which case he numerical inegraion is inaccurae when he calculaion dae is approaching a fixing dae. To overcome hese problems, Hun and Kennedy inroduced an idea by firsly fiing a polynomial o he payoff funcion defined on he grid and hen calculae analyically he inegral of he polynomial agains he Gaussian disribuion [3]. This is beer as he only error in he inegraion comes from he polynomial fi and he fiing error can be conrolled, by choosing a sufficienly high order of polynomial, as he inegraion of he polynomial is done analyically [3]. 31