where lnp(, ) f(, ) = P(, ) = exp { f(, u)du} = exp{q(, )} Q(, ) = f(, u)du Heah, Jarrow, and Moron (1992) claimed ha under risk-neural measure, he dr

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1 HJM Model HJM model is no a ransiional model ha bridges popular LIBOR marke model wih once popular shor rae models, bu an imporan framework ha encompasses mos of he ineres rae models in he marke. As he firs muli-facor model, is funcionaliy needs o be appreciaed. In he beginning of his chaper, he model is inroduced. I is followed by he demonsraion of how o ge from HJM o shor rae models and LIBOR marke models. HJM Principal Componen analysis (PCA) is a hisorical esimaion/calibraion mehod o find HJM model parameers from hisorical daa, as opposed o from curren marke quoes. I sheds ligh on hisorical esimaion echnique for LIBOR marke models in he nex chaper. Conens HJM model From HJM o HW From HJM o G2++ From HJM o LMM HJM PCA HJM Model Appendix 1: PCA Appendix 2: aylor Expansion and Io Lemma Heah, Jarrow, and Moron (1992) discovered he no-arbirage condiion beween he insananeous sandard deviaion and drifs of forward raes under risk-neural measure Q, where he numeraire is he bank accoun B(). I is assumed ha, for a fixed mauriy, he insananeous forward rae f(, ) under Q follows: df(, ) = α(, )d + σ(, ) dw() f(, ) = f M (, ) where operaor is he inner produc of wo vecors; W = (W 1,, W N ) and σ(, ) = (σ 1 (, ),, σ N (, )). If in addiion σ(, ) does no depend on f(, ), i is known as he Gaussian HJM, where he insananeous forward raes are normally disribued. Noe ha for each fixed here is one f(, ), so in oal here are infinie many/coninuum equaions. σ(, ) is coninuous on boh firs and second variables. By definiion, we have leianquan.wordpr

2 where lnp(, ) f(, ) = P(, ) = exp { f(, u)du} = exp{q(, )} Q(, ) = f(, u)du Heah, Jarrow, and Moron (1992) claimed ha under risk-neural measure, he drif erm α(, ) can be arbirary bu deermined by he diffusion erm σ(, ). Le s show his relaionship. Firs, ake derivaive on Q(, ), Le So Q(, ) can be re-wrien as hen dq(, ) = f(, )d df(, u)du = r()d {α(, u)d + σ(, u) dw()}du = r()d ( α(, u)du) d ( σ(, u)du) dw() α (, ) = α(, u)du σ (, ) = σ(, u)du dq(, ) = r()d α (, )d σ (, ) dw() leianquan.wordpr dp(, ) = exp{q(, )} {dq + 1 d Q, Q } 2 = P(, ) {(r() α (, ) σ (, ) σ (, )) d σ (, ) dw()} (1) (2) Since he zero-coupon bond, as a radable asse, is discouned price P(, )/B() is a maringale under risk-neural measure, or in oher words, he drif erm of P(, ) should be r()p(, ), we have α (, ) = 1 2 σ (, ) σ (, )

3 ake derivaive w.r.., i becomes α(, ) = σ(, ) σ(, u)du = σ(, ) σ (, ) (3) In oher words, he drif erm is compleely deermined by he volailiy erm. his is he main resul of Heah, Jarrow, and Moron (1992). herefore under risk-neural world, he HJM model is df(, ) = {σ(, ) σ(, u)du} d + σ(, ) dw() (4) f(, ) = f M (, ) (5) where f M (, ), as exogenous inpus, ensuring ha hese models are auomaically consisen wih discoun bond prices a ime. In addiion, we have f(, ) = f(, ) + {σ(u, ) σ(u, s)ds} du + σ(s, ) dw(s) dp(, ) P(, ) u = r()d ( σ(, s)ds) dw() = r()d σ (, ) dw() (7) P(, ) = P(, ) + P(s, )r(s)ds P(s, )σ (s, )dw(s) r() = f(, ) = f(, ) + {σ(u, ) σ(u, s)ds} du + σ(s, ) dw(s) From (9), he shor rae process r() is no a Markov process in general. From HJM o HW u leianquan.wordpr (6) (8) (9) HW (Chaper 9) is a special case of one facor HJM (N = 1), and assume ime-homogeneous exponenial form df(, ) = {σ(, ) σ(, u)du} d + σ(, )dw() σ(, ) = σe a( ) (1)

4 hen r() = f(, ) + {σ(u, ) σ(u, s)ds} du + σ(s, )dw(s) u = f(, ) + {σe a( u) σe a(s u) ds} du + σe a( s) dw(s) = f(, ) + σ2 2a 2 (1 e a ) 2 + σ e a( u) dw(s) u = r()e a + f(, ) + σ2 2a 2 (1 e a ) 2 f(,)e a + σ e a( u) dw(s) Compare i wih he HW model (Chaper 9) and noice ha f(,) = r(), we have From HJM o G2++ dr() = [θ() ar()]d + σdw() f(, ) θ() = + af(, ) + σ2 2a (1 e 2a ) G2++ is also a special case of HJM where here are wo facors (N=2). I becomes df(, ) = α(, )d + σ 1 (, )dw 1 () + σ 2 (, )dw 2 () dw 1 ()dw 2 () = ρd In realiy, he wo facors can be deermined by principal componens analysis (PCA): W 1 () sands for he change of slope, while W 2 () sands for he change of curvaure. Assume ime-homogeneous exponenial form Follow he same logic as in he previous secion Le σ(, ) = σe a( ) + ηe b( ) (11) leianquan.wordpr r() = f(, ) + r()e a + { σ2 2a 2 (1 e a ) 2 f(,)e a } + σ e a( u) dw 1 (s) +r()e b + { σ2 2b 2 (1 e b ) 2 f(,)e b } + η e b( u) dw 2 (s)

5 hen φ() = f(, ) + r()e a + { σ2 2a 2 (1 e a ) 2 f(,)e a } +r()e b + { σ2 2b 2 (1 e b ) 2 f(,)e b } x() = σ e a( u) dw 1 (s) y() = η e b( u) dw 2 (s) dx() = ax()d + σdw 1 (), x() = dy() = bx()d + ηdw 2 (), y() = dw 1 ()dw 2 () = ρd and i arrives he G2++ model under he risk-neural measure From HJM o LMM r() = x() + y() + φ() In his secion we discuss he forward measure, caple pricing, and LIBOR marke model. Consider he forward rae Apply Io lemma, F(; S, ) = 1 S) (P(, τ(s, ) P(, ) 1) leianquan.wordpr df(; S, ) = 1 τ(s, ) { 1 P(, ) dp(, S) + P(, S)d ( 1 P(, ) ) + dp(, S)d ( 1 )} (12) P(, ) Noice ha from equaion (7), dp(, S) = P(, S)[r()d σ (, S) dw()] 1 d ( P(, ) ) = 1 P(, ) 2 dp(, ) + 1 d P(, ), dp(, ) P(, ) 3 = 1 P(, ) {( r() + σ (, ) 2 )d + σ (, ) dw()}

6 so ha equaion (12) can be re-wrien as ha is, df(; S, ) = 1 τ(s, ) { 1 P(, ) dp(, S) + P(, S)d ( 1 P(, ) ) + dp(, S)d ( 1 P(, ) )} = 1 P(, S) τ(s, ) P(, ) {( σ (, ) 2 σ (, S) σ (, ))d + (σ (, ) σ (, S)) dw()} = [F(; S, ) + 1 τ(s, ) ] [σ (, ) σ (, S)] [σ (, )d + dw()] df(; S, ) = [F(; S, ) + 1 τ(s, ) ] [σ (, ) σ (, S)] [σ (, )d + dw()] (12) Now define he shifed LIBOR forward rae as hen i follows lognormal process F (; S, ) = F(; S, ) + 1 τ(s, ) df (; S, ) = F (; S, ) [ σ(, u)du] [σ (, )d + dw()] (14) S We know under he -forward measure, he (shifed) LIBOR forward rae is a maringale. o change he measure from risk-neural o -forward measure, use Girsanov heorem (Wikipedia: Girsanov heorem) X() = ( σ(s, u)du) dw(s) = σ (s, )dw(s) s dx() = ( σ(, u)du) dw() = σ (, )dw() leianquan.wordpr Ԑ(X()) = exp {X() 1 2 [X()]} W () = W() X() (13) dw () = dw() [dw(), dx()] = dw() + σ (, )d (15) So ha under -forward measure df (; S, ) = F (; S, ) [ σ(, u)du] W () (16) S

7 In addiion, from equaion (4), under -forward measure, he insananeous forward rae is a maringale, or his can be seen from df(, ) = σ(, ) dw () (17) df(, ) = σ(, ) σ (, )d + σ(, ) dw() = σ(, ) σ (, )d + σ(, ) [dw () σ (, )d] = σ(, ) dw () In oher words, equaion (16) shows ha he expecaion hypohesis holds under he forward measure, or E Q [r() F ] = f(, ) Caple/Floorle on rae F(S; S, ) wih srike K can be re-defined on rae F (S; S, ) wih srike K + 1 τ(s,). he Black volailiy is σ(, u)du S Anderson and Pierbarg (21) poined ou ha Log-Normal (proporional volailiy) HJM S σ(,, f(, )) = σ(, )f(, ) (19) Leads o log-normally disribued insananeous forward rae under -forward measure 2 d df(, ) = σ(, ) f(, )dw () bu he forward raes will explode in finie ime o infiniy wih non-zero probabiliy. In order o fix his drawback, i comes he LIBOR marke model (Chaper 11). Acually, LIBOR marke model can be viewed as a special case of HJM model. Suppose here exiss an deerminisic funcion σ (, ) such ha hen from equaion (12), leianquan.wordpr σ(, u)du S = F(; S, ) F(; S, ) + 1 σ (, ) τ(s, ) df(; S, ) = σ (, )F(; S, )dw () (18)

8 HJM PCA In his secion i firs inroduces he hisorical esimaion /calibraion for HJM. hen i discusses he Principal Componen Analysis (PCA) for HJM. Firsly, one imporan assumpion abou hisorical esimaion is ime homogeneiy. Consider Gaussian HJM model in equaion (4) and (5), ime homogeneiy means ha he volailiy parameer σ(, ) depends only on he remaining ime o mauriy, or so ha equaion (4) can be rewrien as σ(, ) = σ( ) f(, ) = μ( ) + σ( ) W() I enables us o esimae σ (hence drif) from hisorical daa. For example, if we wan o esimae 3- monh volailiy σ(, = + 3M) = σ( ) = σ(3m) On 211-Jun-1, we record he change of insananeous forward rae ha ends on 211-Augus-31; on 211-Jun-2, we move forward and record he change of insananeous forward rae ha ends on 211- Sep-1; on 211-Jun-3, we record he hisorical move of insananeous forward rae ha ends on 211- Sep-2, and so on. echnically, we move from one forward rae equaion o anoher forward rae equaion (remember here are infinie many of hem). Since he ime o mauriy is kep a 3 monh, in his way we obain a series of hisorical observaion on he very same σ(3m). his is why we need ime homogeneiy. he hisorical esimaion can be done on insananeous forward rae, or on more observable coninuous zero rae. Recall ha coninuously-compounded spo ineres rae is defined by hen apply Io lemma on equaion (7), ln P(, ) R(, ) = τ(, ) 1 1 dr(, ) = d ( ) ln P(, ) τ(, ) τ(, ) { 1 P(, ) dp(, ) 1 d P(, ), P(, ) } 2P(, ) 2 ln P(, ) = ( ) 2 d 1 τ(, ) {(r() 1 2 σ (, ) 2 ) d σ (, ) dw()} = 1 τ(, ) {(R(, ) r() σ (, ) 2 ) d + σ (, ) dw()} which yields leianquan.wordpr dr(, ) == 1 τ(, ) {(R(, ) r() σ (, ) 2 ) d + σ (, ) dw()} (2)

9 In he res of his secion, we consider general processes N dx i () = μ i d + σ ij dw j (), i = 1,, m (21) j=1 where X can be insananeous forward rae (hen i becomes equaion (4)), or coninuouslycompounded spo rae (hen i becomes equaion (2)). In equaion (21), ime (or -) is replaced by subscrip i, o indicae explicily he discree enors (e.g. i=1 sandards for 3-monh ime o mauriy, i=2 sandards for 6-monh ime o mauriy, ec.). ha is, σ ij = σ j (, + τ i ) Secondly, in order o undersand beer he mechanism of PCA and laer he LIBOR marke model, i s helpful o re-wrie equaions (21) in he following forma. Define and define processes where Z i () = N d σ i = σ ij 2 j=1 σ ij j=1 W σ j (), i = 1,, m i N dz i ()dz i () = σ ij 2 d = d dz i ()dz k () = 2 j=1 σ i N σ ijσ kj j=1 ρ ik = 1 σ i σ k N σ i σ k j=1 σ ij σ kj d = ρ ik d hen he equaions (21) can be re-wrien in erms of he Brownian moion Z i () as leianquan.wordpr dx i () = μ i d + σ i dz i (), i = 1,, m (22) dz i ()dz k () = ρ ik d (23) As explained before, he volailiy σ i is consan here because of ime-homogeneiy. Also he drif erm μ i is compleely deermined by σ i from no-arbirage argumen. herefore one can esimae he variance-covariance marix and obain σ i, ρ i,k from he hisorical movemens of X i (). he esimaion process is sandard (sample variance and covariance) hence is omied here. If one s ask is o esimae/calibrae he HJM model, he ask is achieved by now.

10 Ofen in pracice, we go one sep furher and simulae he esimaed/calibraed HJM models. he Mone Carlo simulaion of HJM is commonly seen in pricing exoic producs (such as swapions), or in counerpary credi risk calculaions (such as PFE calculaion). I is suffered by curse of dimensionaliy, in his case dimension is m. In he nex sep we reduce he dimension via PCA. Usually firs hree PCAs are picked, sanding for hree main yield curve movemens: parallel shifs, wiss (seepen/flaen), and curvaure (buerfly). he logic is simple: we have m variables ha are correlaed in equaion (22), and we wan o find m uncorrelaed variables in equaion (21). Use he jargons of PCA, we wan o {dz 1,, dz m } PCA {dw 1,, dw m } and hen pick he larges 3 PCAs by seing N = 3 in equaion (21). PCA is also a sandard saisics pracice hence is negleced here. he mehod is briefly inroduced in he appendix. In he nex chaper LMM model will be discussed. In erms of calibraion, yield correlaion marix can be obained from eiher hisorical daa or from he prices of caps and swapions. Readers may have already figured ou how o calibrae LMM hrough hisorical esimaion, yes, i is quie similar o wha we have seen in his secion. How o calibrae o caps and swapions is he main opic of he nex chaper. Appendix 1 Principal Componen Analysis Denoe X R n he marix of daa ime series, where is he number of daa poins and n is he number of variables. X = [x 1, x 2,, x n ] R n We assume each column of X has zero mean. If in addiion hey are normalized, hen he correlaion marix C = 1 X X R n n leianquan.wordpr Denoe is eigenvalues by λ 1 λ 2 λ n and heir corresponding eigenvecors w 1, w 2,, w n, and Cw i = λ i w i, C = WΛW 1 λ 1 Λ = [ ] R n n λ n W = [w 1, w 2,, w n ] R n n so ha Λ is he diagonal marix of eigenvalues and W is he orhogonal marix of eigenvecors (W = W 1 ) because C is symmeric.

11 he marix of principal componens of C is a n marix P defined by P = [p 1, p 2,, p n ] = XW he mh principal componen of C is defined as he mh column of P, p m = w m1 x 1 + w m2 x w mn x n where w m = [w m1, w m2,, w mn ] is he eigenvecor corresponding o λ m. hus p 1 belongs o he firs and he larges eigenvalue λ 1, p 2 belongs o he firs and he larges eigenvalue λ 2, and so on. Each principal componen is a ime series, i.e., a 1 vecor. he covariance marix of he principal componens is Λ, or 1 P P = 1 W X XW = W CW = W 1 WΛ = Λ So ha he principal componens are uncorrelaed, he variance of he mh principal componen is λ m. Now X = PW 1 = PW So ha we can wrie each of he original variables as a linear combinaion of he principal componens, as x i = w 1i p 1 + w 2i p w ni p n w 1i p 1 + w 2i p 2 + w 3i p 3 Appendix 2 aylor Expansion and Io Lemma 1-D aylor expansion 2-D aylor expansion f(x) = f(x ) + f x (x x ) f 2 x 2 (x x ) 2 + leianquan.wordpr f(x, y) = f(x, y ) + [(x x ) x + (y y ) y ] f [(x x ) x + (y y ) y ] 2 f + 1-D Io lemma df = f x dx f d x, x 2 x2 2-D Io lemma

12 In paricular, Reference df = f f dx + x y dy + 1 f 2 { 2 x 2 d x, x f x y d x, y + 2 f d y, y } y2 d(xy) = ydx + xdy + d x, y [1] Anderson, L. and Pierbarg, V. (21). Ineres Rae Modeling. Volume 1: Foundaions and Vanilla Models. Alanic Financial Press. [2] Anderson, L. and Pierbarg, V. (21). Ineres Rae Modeling. Volume 2: erm Srucure Models. Alanic Financial Press. [3] Brace, A. and M. Musiela (1994). A muli-facor Gauss Markov implemenaion of Heah, Jarrow and Moron. Mahemaical Finance, 4, pp [4] Brace, A. and M. Musiela (1997). Swap derivaives in a Gaussian HJM framework. In M. Dempser and S. Pliska (Eds.). Mahemaics of Derivaives Securiies, pp Cambridge Universiy Press. [5] Brigo, D. and Mercurio, F (26). Ineres rae models: heory and pracice: wih smile, inflaion, and credi. Springer Verlag. [6] Duffie, D. and Kan, R. (1996). A yield-facor model of ineres raes. Mahemaical finance, 6(4). [7] Filipovic, D. (29). erm-srucure models: A graduae course, Springer Verlag. [8] Heah, D. and Jarrow, R. and Moron, A.(1992). Bond pricing and he erm srucure of ineres raes: A new mehodology for coningen claims valuaion. Economerica, 6(1), pp [9] Jamshidian, F. and Zhu, Y. (1997). Scenario Simulaion: heory and mehodology. Finance and Sochasics. 1, pp leianquan.wordpr [1] Shreve, S.E. (24). Sochasic calculus for finance. Springer