Lognormal random eld approxmatons to LIBOR market models O. Kurbanmuradov K. Sabelfeld y J. Shoenmakers z Mathemats Subet Classaton: 60H10,65C05,90A09 Keywords: LIBOR nterest rate models, random eld smulaton, Monte Carlo smulaton of stohast derental equatons. S.Teh. Center Clmate Turkmenan Hydrometeorology Comm., Azad 81, 744000, Ashgabad, Turkmenstan, E-mal: sed@lmat.ashgabad.su y Insttute of Computatonal Mathemats and Mathematal Geophyss, Russan Aad. Senes, Akad. Lavrenteva, 6, 60090, Novosbrsk, Russa, E-mal: karl@osmf.ss.ru and Weerstra-Insttut fur Angewandte Analyss und Stohastk, Mohrenstrae 9, 10117, Berln, E-mal: sabelfel@was-berln.de z Weerstrass Insttute, Mohrenstrasse 9, D-10117 Berln, E-mal: shoenma@was-berln.de 0

Abstrat We study several approxmatons for the LIBOR market models presented n [1,, 5]. Speal attenton s payed to log-normal approxmatons and ther smulaton by usng dret smulaton methods for log-normal random elds. In ontrast to the onventonal numeral soluton of SDE's ths approah smulates the soluton dretly at the desred pont and s therefore muh more eent. We arry out a path-wse omparson of the approxmatons and gve applatons to the valuaton of the swapton and the trgger swap. 1 Introduton By far the most mportant lass of traded nterest rate dervatves s onsttuted by dervatves whh are speed n terms of LIBOR rates. The LIBOR 1 rate L s the annualzed eetve nterest rate over a forward perod [T 1 T ] and an be expressed n terms of two zero-oupon bonds B 1 and B wth fae value $1 maturng at T 1 and T respetvely, L(t T 1 T ):= B 1 (t) B (t) ; 1 T ; T 1 (1) where as usual T s the settlement date for the arual LIBOR perod. Brae, Gatarek and Musela [1], as well as Jamshdan [], onstruted an arbtrage free model for the LIBOR rate proess n order to pre LIBOR dervatves suh as aps, swaptons and more omplated types n a dret way. In [1] the dynams of the ontnuous famly of proesses fl(t T T + ) T 0 0 t T g s studed for a xed > 0 whereas Jamshdan [] onsdered for a dsrete set of tenors ft 1 ::: T n g the proesses fl (t) :=L(t T T +1 ) t T = 1 ::: n; 1g: In both papers [1, ] speal attenton s payed to so alled LIBOR market models whh are models where for every settlement date the LIBOR proess has determnst volatlty. In a market model, eah LIBOR s a log-normal martngale under the numerare measure gven by the bond whh termnates at the LIBOR's settlement date. In ths sequel we onentrate on a LIBOR market model for a dsrete set of tenors gven by a stohast derental equaton (SDE) n the termnal bond measure as developed n Jamshdan [], equpped wth a speal orrelaton struture proposed by Shoenmakers and Coey [5]. In ths model we wll test the valuaton of several LIBOR dervatves suh as the 'plan vanlla' swapton and the more 'exot' trgger swap. For a detaled analyss of the trgger swap and the valuaton of exot LIBOR dervatves n general we refer to Shoenmakers and Coey [5]. For the LIBOR proess, as beng a soluton of the SDE, we have onstruted derent path-wse approxmatons and n partular log-normal approxmatons and arred out mplementatons. The results are subeted to mutual omparson and a rankng between the derent approxmatons s thus obtaned. The man advantage of the log-normal approxmatons s that ther dstrbutons an be smulated very fast by a Gaussan random eld of log-libors wth a drft and orrelaton struture determned by the speapproxmaton. As the valuaton of a LIBOR dervatve generally omes down to the omputaton of the expeted value of some funtonal of the LIBOR proess, a large lass of dervatves an be valuated qute fast by random eld smulaton. Several approxmatons are derved n seton () where a mutual omparson s studed. In seton () we onstrut a log-normal random eld smulaton algorthm and n seton (4) we 1 LIBOR stands for London Inter Bank Oer Rate. 1

onsder the valuaton of swaptons and trggerswaps and ompare the results for derent smulaton algorthms and derent orrelaton strutures. In partular, n seton (4) t s observed that swapton pres depend on the nput orrelaton parameters of the LIBOR model under onsderaton n a numeral stable way, n ontrast to orrelaton parameters n low-fator LI- BOR models whh tend to unstable behavour under albraton to swapton pres. See, for a more detaled dsusson of ths ssue, Shoenmakers and Coey[5]. Derent approxmatons, log-normal approxmatons For a gven tenor struture 0 <T 1 <T <:::<T n we onsder a Jamshdan LIBOR market model [] for the forward LIBOR proesses L n the termnal bond numerare IP n dl = ; =+1 L L (1 + L ) dt + L dw (n) () where, for = 1 ::: n ; 1 the L are dened n the ntervals [ T ] = T +1 ; T and =( 1 ::: ) are gven determnst funtons, alled fator loadngs, dened n [ T ] respetvely. In (), (W (n) (t) t T ) s a standard n ; 1-dmensonal Wener proess under IP n. It s onvenent to deal wth the followng ntegral form of (): Z ln L (t) t L ( ) = ; =+1 L ds ; 1 1+ L ds + dw (n) () where = =. In prate, we may dene the vetors = through the matrx ( )by applyng a Cholesky deomposton. Note that only the rst term n the rght hand sde of () s generally non-gaussan. Let us onsder the ontrbuton of the non-gaussan term where we assume for smplty that the funtons are onstants. We ntrodue the notatons: = and =max Then, we may wrte () as P =+1, =max = max,. Let us denote by ~ L the maxmum value of the L,.e., ~ L = max ln L (t) L ( ) = " ; 1 (t ; )+ p t ; Z (t) sup L (t). tt where Z (t) s a standard normal dstrbuted random varable and " an be estmated by " (t ; ) ~ L : So, by negletng " weausenl only a small relatve error of order of " when " (t ; ) ~ L (t ; ) ~ L <<1: (4) Note that e.g. for typal values, = 0:5, = 0:4 L ~ = 0:07, t ; t0 = 5 ths relatve error s about 1:4 %: However, dependent on and the length of the tenor struture ths error an beome rather large n prate. The approxmaton by negletng the non-gaussan terms " n () wll be alled (0);approxmaton to () whh satses dl (0) = L (0) dw (n) (5)

and s gven by the explt soluton L (0) 8 < (t) =L ( )exp : ;1 (s)ds + 9 = (s) dw (n) (s) : (6) Below we show for llustraton (see Fgs. 1,) some typal samples of L (t) and L (0) (t) where we hose n =1 1 = ::: = =0:4, and = b ^ b b _ b b =expf g: The orrelatons are thus dened va two parameters, and see also [5]. In our smulatons, presented n the gures below, we took = 0:8 = 0:1 and = 0:8 = 0: respetvely. Further we hose = 0 and a unform tenor struture T = wth =0:5, =1 ::: 1. The ntal L values were taken to be L (0) = 0:061. L 10 0.1 0.11 0.1 0.09 0.08 0.07 0.06 L (0) 10 (t) ;; L 10 (t) ;; 0.05 0.5 1 1.5.5 tme Fg.1 A sample of L 10 (t) andl (0) 10 (t), for =0:8 and =0:1:

L 10 0.11 0.1 0.09 0.08 0.07 0.06 0.05 L (0) 10 (t) ;; L 10 (t) ;; 0.04 0.5 1 1.5.5 tme Fg. A sample of L 10 (t) andl (0) 10 (t), for =0:8 and =0:: From the traetores presented n Fgs.1- t s seen that on the ntal tme nterval, the funton L (0) 10 approxmates the funton L 10 very good. For nreasng tme, however, the dsrepany nreases. Note that the larger the lesser the orrelaton tme and by (4) the lesser the dsrepany. Ths s onrmed by our observaton presented above. From the ptures n Fgs.1- we see that the (0);approxmaton s good for small tmes, whereas from (4) we see that for large the (0)-approxmaton s also good beause dereases wth (e.g., vanshes). More detals about the (0) and other approxmatons are presented n Tables 1-5. In Fg. we show a sample for the Bond pre B 1 (T ) and ts (0)-approxmaton B (0) (T 1 ) =0 ::: 1. In ontrast to the results presented n Fgs.1-, the maxmum desrepany happens around the mddle of the tme nterval (0 T 1 ). The reason s that the Q Bond pre B 1 (T ) 0 nvolves the produt of all lbor rates L = : : : 0 by B 1 (T ) = = (1 + L (T )) ;1 : Indeed, ether when s lose to zero or when s lose to 0 where the drft terms beome small, the approxmatons L (0) (T ), = : : : 0 are lose to L (T ) and so B (0) (T 1 )slosetob 1 (T ): 4

B 1 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 B (0) 1 (t) ;; B 1 (t) ;; 1 4 5 6 7 tme Fg. A sample of Bond pres B 1 (t) and ts (0)-approxmaton, for = 0:8, = 0:1, n =1. It s of nterest to onsder more rened approxmatons to L and n partular to look for lognormal approxmatons mprovng L (0) : By replang L n the rght-hand sde of () wth L (0) we ome to what we all the (1)-approxmaton: dl (1) = ; =+1 The soluton to (7) s gven expltly by: L (1) L (0) 1+ L (0) dt + L (1) dw (n) (7) ln L(1) (t) L ( ) = ; =+1 L (0) (s) (s) ds ; 1 1+ L (0) (s) (s)ds + (s) dw (n) (s): (8) It turns out that ths approah mproves very muh the (0)-approxmaton ndeed, and the results presented n Tables 1-5 below onrm ths onluson. It should be noted, however, that the (1)-approxmaton s unfortunately non-lognormal, n ontrast to the (0)-approxmaton. Therefore, for eah we approxmate the proess Z (t) := L (0) 1+ L (0) 5

wth a Gaussan proess n (8) as follows. Let the funton f be dened as f(x) :=x=(1 + x) so f (;1) (x) =x=(;x +1) andz = f( L (0) ): Hene Z satses the SDE dz = f 0 ( L (0) ) L (0) dw (n) + 1 f 00 ( L (0) )[ L (0) ] dt = f 0 f (;1) (Z ) f (;1) (Z ) dw (n) + 1 f 00 f (;1) (Z )[f (;1) (Z ) ] dt = : a(z t)dt + b(z t) dw (n) wth ntal ondton Z ( )=f( L ( )): The Pard;0 and Pard;1 teraton for the soluton of ths SDE are respetvely Z (0) (t) : Z ( )= L ( ) 1+ L ( ) and R Z (1) t (t) =Z ( )+ [a(z ( ) s)ds + b(z ( ) s) dw (n) (s)] = f( L ( )) + 1 f 00 ( L ( )) L () R t ds + f 0 ( L ( )) L ( ) R t dw (n) (s) whh are learly both Gaussan. The next Pard teraton, however, wll be non-gaussan n general. By usng Z (0) n (8) we nd a lognormal approxmaton whh we all the (g); approxmaton, Z ln L(g) (t) t L ( ) = ; (s) ds + (s) dw n (s) ; =+1 L ( ) (s) 1+ L ( ) ds (9) whh turns out to be a onsderable path-wse mprovement of the (0);approxmaton and s suggested n [1, 5]. By expandng f f 0 and f 00 as f(x) =x ; x + O(x ) f 0 (x) =1; x + O(x ) and f 00 (x) =;+O(x) respetvely, x = L ( ) and denotng dentty modulo terms of order O(x )ando(x )by ' and = respetvely, wehave Z (1) (t) = Z (0) ; L () ' Z (0) + L ( ) ds +( L ( ) ; L ()) dw (n) s ' L ( )(1 + dw (n) (s) dw (n) s ) Usng Z (1) whle negletng seond order terms leads to another lognormal approxmaton, the (g1);approxmaton: Z ln L(g1) (t) t L ( ) = ; ds+ (s)dw (n) (s); =+1 L ( )(1+ Z s (u)dw u (n) ) (s)ds: (10) The (g1);approxmaton n ts turn mproves the (g);approxmaton sgnantly as wll appear from a omparatve analyss below. Smlarly, wemay nlude also the seond order terms and thus dene a lognormal (g1 0 );approxmaton whh, however, s only slghtly better than the (g1) and s, n fat, subordnate to a nal lognormal approxmaton whh we onstrut below. Instead of L (0) we now plug n L (g) n the rght-hand sde of () and we arrve atthe (); approxmaton whh sgven expltly by, ln L() (t) L ( ) = ; =+1 L (g) (s) (s) ds ; 1 1+ L (g) (s) (s)ds + (s) dw (n) (s): (11) 6

Now onstrut a Gaussan approxmaton for f(l (g) ) asabove. We redene Hene Z now satses the SDE dz = f 0 ( L (g) Z := f( L (g) )= L (g) 1+ L (g) ) L (g) f 0 ( L (g) ) 8 < : ; dw (n) + 1 f 00 ( L (g) k=+1 k L k ( ) k 1+ k L k ( ) L(g) dt )[ L (g) ] dt + wth ntal ondton Z ( )=f( L ( )): Obvously, replang Z by the Pard;0 teraton Z (0) to Z (1) f( L ( )) gves the (g);approxmaton agan, whereas the Pard;1 teraton now leads R R (t) =f( L ( )) + f 0 t ( L ( )) L ( ) dw (n) + 1 f 00 ( L ( )) L (t t 0) ds +f 0 ( L ( )) n ; P k=+1 k L k ( )L ( ) 1+ k L k ( ) 9 = o R t k dt (1) It should be noted that when nstead of L (g) we plug n L (1) or L () n the rght hand sde of (), although we get better and better explt non-lognormal approxmatons, the Gaussan Pard;1 approxmaton for Z = f( L (1) ) and Z = f( L () ) respetvely, s the same as n (1). So we do not get better Gaussan approxmatons n ths way. In fat, we may derve (1) dretly from (). Clearly by negletng seond order terms n (1) we get the (g1);approxmaton agan, whereas by keepng seond order terms we get a new log-normal approxmaton, (g) say, gven by where ln L(g) (t) L ( ) = ; =+1 ~ 1 Z(s) (s)ds ; (s)ds + (s) dw (n) (s) (1) n ; P k=+1 ~Z(t) R = f( L ( )) ; L (t t 0) ds+ k L k ( )L ( ) 1+ k L k ( ) o R t k dt +(1; L ( )) L ( ) R t dw (n) Here we note that the (g1 0 );approxmaton ders from (g) n that the term wth the sum s mssng. It s now nterestng to arry out a omparatve numeral analyss of the derent approxmatons presented. The numeral soluton of the relevant stohast derental equatons are solved by the Euler sheme. For a orret path-wse omparson, t s neessary to onstrut all the approxmatons n one ommon probablty spae. In the numeral shemes, t s easly aheved by usng one and the same Wener nrements for all approxmatons. In the next tables we show how often the relatve error (n perents) of the orrespondng approxmaton to L 5 L 10 and L 0 les n the relevant perentage ntervals (rst olumns). For nstane, the relatve error between ^L, the numeral soluton to the orgnal equaton () and ^L (1), the numeral soluton to the equaton (7) s dened as ^L (1) (T ) ; ^L (T ) = max 1 ^L (T ) 7

and orresponds, e.g., n the tables to the thrd olumn. The relatve errors to other approxmatons are dened analogously. In all tables we hose unformly 1 = ::: = =0:4 L 1 (0) = ::: = L (0) = 0:061 and =0:8: For nstane, the numbers n the olumns - 7 of table 1 show the fraton of 700000 samples for whh the event shown n the rst olumn happens. From these results we see that among all the path-wse approxmatons, the best one s the ()-approxmaton, whh s however non-lognormal. Among the lognormal approxmatons, the (g);approxmaton shows the best results. Also we onlude that the approxmatons are better when the LIBORs are more de-orrelated. Indeed, de-orrelaton dmnshes the drfts n (). Note that the fat that a path-wse approxmaton s not good enough (e.g., see the (0)-approxmaton n olumn 6) does not mply that the statstal haratersts wll be approxmated not good as well. We wll llustrate ths n the ase of swap and trgger swap, seton (4). 100 5 () (1) (g) (g1) (g) (0) 0:5 % 0.9 0.561 0.0159 0.8059E-0 0.759E-0 0. 0:5% 0.954 0.804 0.0966 0.0591 0.01 0. 0:75 % 0.974 0.895 0.08 0.16 0.098 0. 1% 0.984 0.97 0.4 0.19 0.0781 0. 1:5% 0.99 0.97 0.574 0.95 0.166 0. % 0.996 0.986 0.85 0.606 0.55 0.857E-04 :5% 0.998 0.99 0.89 0.856 0.8 0.5471E-0 % 0.999 0.995 0.91 0.9 0.417 0.987E-0 :5% 0.997 0.90 0.944 0.494 0.01 4% 0.998 0.944 0.955 0.569 0.046 4:5% 0.9987 0.955 0.964 0.64 0.047 5% 0.999 0.964 0.971 0.707 0.0785 6% 0.976 0.981 0.819 0.1597 7% 0.984 0.987 0.891 0.564 8% 0.989 0.991 0.91 0.56 9% 0.99 0.99 0.95 0.450 10 % 0.994 0.995 0.945 0.55 1 % 0.996 0.997 0.961 0.67 14 % 0.998 0.998 0.97 0.769 16 % 0.998 0.999 0.979 0.87 18 % 0.999 0.985 0.884 0 % 0.988 0.917 Table 1. The umulatve dstrbuton of the relatve error 0, for derent approxmatons =0:1, N = 700000 T 1 =1, n =1. 8

100 5 () (1) (g) (g1) (g) (0) 0:5 % 0.9478 0.5046 0.1198 0.0491 0.0054 0. 0:5% 0.9859 0.840 0.710 0.1844 0.0466 0. 0:75 % 0.995 0.9417 0.61 0.5 0.1115 0. 1% 0.9980 0.9751 0.8551 0.5064 0.1817 0. 1:5% 0.9996 0.9946 0.9179 0.96 0.16 0. % 0.9998 0.9985 0.9487 0.9660 0.497 0. :5% 0.9999 0.9995 0.967 0.9788 0.5400 0. % 0.9997 0.9801 0.9871 0.68 0.7500E-04 :5% 0.9999 0.9878 0.990 0.75 0.750E-0 4% 0.990 0.9948 0.8 0.975E-0 4:5% 0.9949 0.9964 0.8584 0.158E-01 5% 0.996 0.9976 0.9008 0.0E-01 6% 0.9984 0.9989 0.9444 0.9710E-01 7% 0.999 0.9994 0.9617 0.05 8% 0.9995 0.9996 0.970 0.41 9% 0.9997 0.9997 0.9804 0.46 10 % 0.9997 0.9998 0.9859 0.581 1 % 0.9998 0.9999 0.997 0.7584 14 % 0.9999 0.9999 0.996 0.8668 16 % 0.9979 0.970 18 % 0.9990 0.9607 0 % 0.9994 0.9785 Table. The umulatve dstrbuton of the relatve error 10, for derent approxmatons =0:1 N=40000, T 1 =1, n =1: 100 5 () (1) (g) (g1) (g) (0) 0:5 % 0.9970 0.8810 0.5064 0.157 0.510 0. 0:5% 0.9999 0.999 0.8975 0.5917 0.1799 0. 0:75 % 0.9991 0.945 0.9670 0.079 0. 1% 0.9999 0.9670 0.981 0.408 0. 1:5% 0.9889 0.996 0.6111 0. % 0.996 0.9979 0.7659 0. :5% 0.9985 0.9990 0.878 0.5000E-04 % 0.999 0.9995 0.90 0.8500E-0 :5% 0.9996 0.9999 0.960 0.7400E-0 4% 0.9999 0.9999 0.974 0.947E-01 4:5% 0.9999 0.987 0.8088E-01 5% 0.9879 0.164 6% 0.9947 0.950 7% 0.9974 0.6185 8% 0.9988 0.7864 9% 0.999 0.8846 10 % 0.9997 0.941 1 % 0.9856 14 % 0.9966 16 % 0.9991 Table. The umulatve dstrbuton of the relatve error 5, for derent approxmatons =0:1, N = 40000, T 1 =1, n =1: 9

100 5 () (1) (g) (g1) (g) (0) 0:5 % 0.895 0.545 0.011 0.0066 0.500E-0 0. 0:5% 0.9514 0.79 0.0806 0.050 0.95E-0 0. 0:75 % 0.974 0.8879 0.180 0.1198 0.14E-01 0. 1% 0.988 0.90 0.889 0.1978 0.6515E-01 0. 1:5% 0.9915 0.970 0.514 0.571 0.1451 0. % 0.9956 0.9846 0.7954 0.5410 0.79 0. :5% 0.9974 0.9910 0.880 0.7905 0.04 0.1750E-0 % 0.9985 0.9946 0.9017 0.919 0.74 0.175E-0 :5% 0.9988 0.9966 0.919 0.956 0.4478 0.5475E-0 4% 0.999 0.9978 0.955 0.947 0.519 0.198E-01 4:5% 0.9994 0.9986 0.9469 0.9566 0.5880 0.850E-01 5% 0.9996 0.9989 0.9565 0.9641 0.656 0.510E-01 6% 0.9998 0.999 0.9698 0.9746 0.7695 0.11 7% 0.9998 0.9997 0.9785 0.986 0.85 0.1950 8% 0.9999 0.9998 0.9848 0.9869 0.909 0.857 9% 0.9999 0.9998 0.9887 0.990 0.9 0.774 10 % 0.9999 0.9999 0.9916 0.991 0.954 0.4598 1 % 0.9999 0.9999 0.995 0.9961 0.954 0.6015 14 % 0.9999 0.9969 0.9974 0.9645 0.7090 16 % 0.9979 0.998 0.975 0.7879 18 % 0.9985 0.9986 0.9799 0.845 0 % 0.9988 0.9988 0.9845 0.8857 Table 4. The umulatve dstrbuton of the relatve error 0, for derent approxmatons =0 N=40000, T 1 =0:5, n =1: 100 5 () (1) (g) (g1) (g) (0) 0:5 % 0.9887 0.968 0.079 0.171 0.069 0. 0:5% 0.9979 0.987 0.501 0.80 0.100 0. 0:75 % 0.999 0.9965 0.8094 0.6410 0.448 0. 1% 0.9997 0.9986 0.9075 0.8799 0.49 0.500E-04 1:5% 0.9999 0.9997 0.9516 0.961 0.565 0.7750E-0 % 0.9999 0.974 0.9789 0.70 0.185E-01 :5% 0.9844 0.987 0.810 0.5765E-01 % 0.9909 0.995 0.9054 0.14 :5% 0.9944 0.9951 0.986 0.579 4% 0.9959 0.9967 0.951 0.784 4:5% 0.997 0.9976 0.960 0.49 5% 0.9979 0.998 0.9696 0.595 6% 0.9987 0.9989 0.9806 0.749 7% 0.999 0.9995 0.9880 0.846 8% 0.9995 0.9996 0.991 0.9068 9% 0.9997 0.9998 0.9948 0.946 10 % 0.9998 0.9999 0.9965 0.961 1 % 0.9999 0.9999 0.998 0.9849 14 % 0.999 0.994 16 % 0.9996 0.9968 Table 5. The umulatve dstrbuton of the relatve error 0, for derent approxmatons =0:5 N=40000, T 1 =1, n =1: 10

Smulaton of a log-normal random eld (DST) The results of seton lsted n Tables 1-5 learly show that lognormal models (g), (g1), (g) and (0) are good approxmatons to the soluton of SDE (). Ths suggests the followng dret smulaton tehnque (DST): onstrut lognormal random eld models whose rst two statstal moments are onsstent wth those of the approxmatons (g), (g1), (g) (0): The motvaton of DST s lear: n ontrast to numeral soluton of stohast derental equatons there s no need for takng small tme steps n DST, t s possble to onstrut the soluton dretly at the desred ponts, e.g., at the ponts of the gven tenor struture 0 <T 1 < T <:::T n. Therefore, DNT takes generally muh less omputer tme. To be more spe, let us onstrut the dret smulaton algorthm onsstent wth the (g)-approxmaton. We thus have to onstrut a lognormal random eld L (g) ( t) =expf (g) ( t)g (14) wth gaussan (g) ( t), = 1 ::: n ; 1, t T, whose mean and ovaraton struture onde wth that of ln(l (g) (t)=l ( )), t T, =1 :::n; 1 n the IP n ; measure: From (9) we see that h (g) ( t) = hln (g) L (t) L ( ) (g) L h (g) ( 1 t 1 ) (g) ( t ) = hln h (g) ( t) (g) ( t)=; =+1 L ( ) 1+ L ( ) 1 (t 1 ) L 1 ( ) (15) ln L (g) (t ) L ( ) 1 (s)ds ; : (16) (s)ds (17) h (g) ( 1 t 1 ) (g) ( t )ov (g) ( 1 t 1 ^ t )+ (g) ( 1 t 1 ) (g) ( t ) (18) where ov (g) ( 1 t)= (s) (s)ds: In prate, one usually evaluates LIBOR dervatves whh depend on the values L (T ), =1 ::: n; 1, =1 :::. Therefore, we have to onstrut numerally the desred random eld L (g) ( T ) = 1 ::: n ; 1 = 1 :::. To do ths, we ould smulate the gaussan vetor wth the gven ovarane struture by aonventonal smulaton tehnque. However the spe tme orrelaton suggests a derent smulaton algorthm, []. Indeed, n the rst step, we smulate a n ; 1-dmensonal gaussan vetor ( (g) (1 T 1 ) ::: (g) (n ; 1 T 1 )) as (g) ( T 1 )= (g) ( T 1 )+ K 1 h (1) k (1) k k=1 where the postve nteger number K 1 and the entres h (1) k K 1 k=1 =1 ::: n; 1 (19) are hosen so that h (1) k h(1) k = ov(g) ( T 1 ) =1 :::n; 1 11

and f (1) k gk 1 s a set of ndependent standard gaussan random numbers. k=1 In the l-th step ( l n ; 1) we have: (g) ( T l )= (g) ( T l;1 )+ (g) ( T l ) ; (g) ( T l;1 )+ The postve nteger number K l and the entres h (l) k where (l) l K l k=1, (l) l+1 Thus after n ; 1 steps we nd K l h (l) k (l) k k=1 are hosen so that = l : : : n ; 1: (0) h (l) k h(l) k = ov(g) ( T l;1 T l ) = l : : : n ; 1 (1), :::, (l) s a set of ndependent standard gaussan random numbers. L (g) (T )=L ( )L (g) ( T )=L ( )expf (g) ( T )g =1 ::: n; 1 =1 ::: : () Here we presented smulaton of a lognormal random eld onsstent wth the g-approxmaton. Analogously, the same ould be easly done for the lognormal approxmatons (0). Indeed, the smulaton formulae (19)- () reman the same, but the funtons (g) and ov (g) should be replaed wth (0) ( t 1 )=0 and ov (0) ( 1 t)=ov (g) ( 1 t) for the (0);approxmaton. For the g1-approxmaton we may dene, From(10)wederve ln L(g1) (t) L ( ) and thus nd (g1) ( t)=; =+1 = (g1) ( t)+ Cov[ (g1) ( 1 t 1 ) (g1) ( t )] = R t 1^t h 1 L ( ) (s)ds ; 4 1 ; =+1 L ( ) h P 1 (s) 1 ; P R 1 ; k= t +1 kl k ( ) s s (s)ds: (u)du5 (s) dw s = 1 +1 L ( ) R t 1 k (u)du ds s 1 (u)du and smlar expressons for the (g)-approxmaton an be derved from (1). However, unfortunately the ovarane funtons of (g1 ) have not the speal struture as n the ase of the (0) and (g); approxmaton, so the smulaton of the orrespondng random elds mght be slower. Remark Note that the ost of the smulaton algorthm used for the (g);approxmaton has the order O(n 4 ) sne n the l-th step, we apply the Cholesky deomposton (1) whose ost has the order O(n ). The onventonal dret method would take abouto(n 6 ) operatons. It should be noted also that f the fator loadngs funtons do not depend on tme, then the ost of our algorthm s O(n ), sne we apply the Cholesky deomposton only one, at the rst step. There s one nterestng feature of our algorthm whh s to be stressed: n prate, one often spees the model not by the fator loadngs, but through the quanttes Cov( t) 1 (s)ds

whh an be determned from the Cap/Swapton markets, see also [5]. In our random eld approxmatons ust these quanttes are only relevant and an thus be plugged n dretly, whereas n ase the approxmatons are obtaned by numeral soluton of the relevant SDE, t s needed to alulate the fator loadngs by Cholesky deomposton of the tme dervatves of Cov( t) generally, n eah ntegraton step. Ths an be very tme onsumng, espeally when the fator loadngs are tme dependent. 4 Valuaton of swaptons and trgger swaps We now present some test results on the valuaton of two typal LIBOR dervatves: the swapton and the trgger swap. For a dervaton of the several valuaton formulas, see e.g. [5] The value of a swapton, an opton to swap LIBOR aganst a xed oupon at the settlement dates T ::: T n an be represented n the IP n measure by Swpn(t) = =1 B n (t)ie n B+1 (T 1 ) B n (T 1 ) 1 A(L (T 1 ) ; ) F t () In (), A denotes the F T1 measurable event fs(t 1 ) >g where the swaprate S(T 1 ) s gven by S(T 1 ):= 1 ; B n (T 1 ) P ;1+Q k=1 kb k+1 (T 1 ) = (1 + k=1 kl k (T 1 )) P Q k=1 k (1 + =k+1 L (T 1 )) and B +1 (T 1 )=B n (T 1 ) an be expressed n the LIBORs by B +1 (T 1 ) B n (T 1 ) = Y =+1 (1 + L (T 1 )): In a trgger swap ontrat wth speed trgger levels K 1 ::: K n as soon as L (T ) >K one has to swap LIBOR aganst a xed oupon for the remanng perod [T T n ] wth settlement dates T +1 ::: T n : The value of the trgger swap n the IP n measure an be expressed by T rswp(t) = p=1 B n (t)ie n 4 1[ =p] 1 B n (T p ) 0 @ 1 ; Bn (T p ) ; =p B +1 (T p ) 1 A Ft 5 (4) where the trgger ndex, s gven by := mn 1p<n fp L p (T p ) > K p g see [5]. In (4) the expresson nsde the expetaton an be expressed n LIBORS only and we thus have Trswp(t) = p=1 B n (t)ie n 4 1[ =p] 0 Y @ ;1+ =p Y (1 + L (T p )) ; =p =+1 (1 + L (T p )) 1 A Ft (5) We now smulate the pres of swaptons and trgger swaps for the LIBOR traetores smulated n the tables 1, and 4, where the strke s taken to be the ntal swaprate = 0:06045 and all trgger levels equal to the strke K p = for every p: The Monte Carlo errors are based on three standard devatons. Note that the dsrepany between an opton value smulated wth L and a value smulated wth some approxmaton should be nterpreted as a systemat error aused by the approxmaton sne the traetores of L and the approxmaton are onstruted wth one and the same Wener nrements. 5 : 1

smulaton swapton M:C: error trg: swap M:C: error L 0.4400E-01 0.90E-0 0.4747E-01 0.44E-0 L () 0.496E-01 0.67E-0 0.4745E-01 0.440E-0 L (1) 0.487E-01 0.56E-0 0.479E-01 0.47E-0 L (g) 0.44E-01 0.77E-0 0.4749E-01 0.47E-0 L (g1) 0.4414E-01 0.749E-0 0.475E-01 0.465E-0 L (g) 0.4579E-01 0.4519E-0 0.4940E-01 0.455E-0 L (0) 0.5188E-01 0.5441E-0 0.5450E-01 0.451E-0 Table 6. Swapton and trgger swap values for derent approxmatons =0:1, N = 700000 T 1 =1, n =1. smulaton swapton M:C: error trg: swap M:C: error L 0.489E-01 0.1441E-0 0.4799E-01 0.1810E-0 L () 0.484E-01 0.14E-0 0.4796E-01 0.1808E-0 L (1) 0.476E-01 0.147E-0 0.479E-01 0.1807E-0 L (g) 0.441E-01 0.1561E-0 0.480E-01 0.18E-0 L (g1) 0.4404E-01 0.155E-0 0.4791E-01 0.188E-0 L (g) 0.4568E-01 0.1770E-0 0.500E-01 0.185E-0 L (0) 0.5176E-01 0.08E-0 0.55E-01 0.197E-0 Table 7. Swapton and trgger swap values for derent approxmatons =0:1 N = 40000, T 1 =1, n =1: smulaton swapton M:C: error trg: swap M:C: error L 0.80E-01 0.7791E-0 0.4487E-01 0.7704E-0 L () 0.88E-01 0.7764E-0 0.4486E-01 0.7704E-0 L (1) 0.86E-01 0.775E-0 0.4488E-01 0.7701E-0 L (g) 0.844E-01 0.8171E-0 0.4488E-01 0.7705E-0 L (g1) 0.88E-01 0.8147E-0 0.4494E-01 0.7705E-0 L (g) 0.89E-01 0.884E-0 0.4517E-01 0.775E-0 L (0) 0.19E-01 0.978E-0 0.4508E-01 0.797E-0 Table 8. Swapton and trgger swap values for derent approxmatons =0 N= 40000, T 1 =0:5, n =1: smulaton swapton M:C: error trg: swap M:C: error L 0.691E-01 0.769E-0 0.1995E-01 0.187E-0 L () 0.691E-01 0.767E-0 0.1994E-01 0.186E-0 L (1) 0.690E-01 0.765E-0 0.199E-01 0.186E-0 L (g) 0.69E-01 0.789E-0 0.1998E-01 0.195E-0 L (g1) 0.689E-01 0.78E-0 0.1991E-01 0.194E-0 L (g) 0.715E-01 0.766E-0 0.048E-01 0.107E-0 L (0) 0.880E-01 0.768E-0 0.65E-01 0.18E-0 Table 9. Swapton and trgger swap values for derent approxmatons =0:5 N = 40000, T 1 =1, n =1: 14

1 4 7 10 1 16 19 5 8 1 1.00 4 0.8 1.00 7 0.69 0.84 1.00 10 0.59 0.7 0.86 1.00 1 0.51 0.6 0.74 0.86 1.00 16 0.44 0.54 0.64 0.75 0.87 1.00 19 0.9 0.47 0.56 0.65 0.76 0.87 1.00 0.4 0.41 0.49 0.57 0.67 0.77 0.88 1.00 5 0.0 0.6 0.4 0.51 0.59 0.67 0.77 0.88 1.00 8 0.6 0. 0.8 0.45 0.5 0.60 0.68 0.78 0.88 1.00 Table 9. Forward log LIBOR orrelatons (ln L (T 1 ) ln L (T 1 )) for =0:8 and =0:1. 1 4 7 10 1 16 19 5 8 1 1.00 4 0.6 1.00 7 0.15 0.4 1.00 10 0.07 0.19 0.46 1.00 1 0.0 0.09 0. 0.48 1.00 16 0.0 0.05 0.11 0.4 0.49 1.00 19 0.01 0.0 0.06 0.1 0.5 0.51 1.00 0.00 0.01 0.0 0.06 0.1 0.6 0.5 1.00 5 0.00 0.01 0.0 0.0 0.07 0.14 0.7 0.5 1.00 8 0.00 0.00 0.01 0.0 0.04 0.07 0.15 0.8 0.54 1.00 Table 10. Forward log LIBOR orrelatons (ln L (T 1 ) ln L (T 1 )) for =0:8 and =0:5. 5 Conluson For pratal relevane, smulaton pres of dervatves should be well wthn so alled bdask spreads: A bd-ask spread an be estmated roughly by the hange of the lam pre due to an overall LIBOR-volatlty movement of 5% up and 5% down. By experment we found out that for the examples above ths omes down to desre a relatve auray of about 5% both for the swapton and the trggerswap. So, from table 6, where the Monte Carlo error (dened as standard devatons) s muh smaller than the spread we may onlude that the (g1);approxmaton performs exellent whereas the (g);approxmaton performs tolerable. The (0);approxmaton, however, produes relatve errors of more than 6%: From tables 6 and 7 we see also that 40000 payo smulatons are suent to reah a Monte Carlo error below 5% and when smulated wth the DSM method appled to the (g); approxmaton ths takes a few seonds for the swapton and a few mnutes for the trggerswap respetvely. However, smulaton of these pres by solvng the SDE for L L (1) or L () by usng small tme steps takes muh longer, for nstane, a few hours for the trggerswap. Referenes [1] Brae, A., Gatarek, D., Musela, M.: The Market Model of Interest Rate Dynams. Mathematal Fnane, 7 (), 17-155, (1997). [] Jamshdan, F.: LIBOR and swap market models and measures. Fnane and Stohasts, (1), 9-0, (1997). 15

[] Sabelfeld, K.K.: Monte Carlo methods n boundary value problems. Sprnger-Verlag, (1991). [4] Shoenmakers, J.G.M., Heemnk, A.W.: Fast Valuaton of Fnanal Dervatves. Journal of Computatonal Fnane, rst ssue, (1997). [5] Shoenmakers, J.G.M., Coey, B.: LIBOR rate models, related dervatves and model albraton. Preprnt no. 480, Weerstrass Insttute Berln, (1999). 16