Addressing the bias in Monte Carlo pricing of multi-asset options with multiple barriers through discrete sampling

Transcription

1 Addressng the bas n Monte Carlo prcng of mult-asset optons wth multple barrers through dscrete samplng arxv: v1 [q-fn.cp] 7 Apr 2009 Pavel V. Shevcheno CSIRO Mathematcal and Informaton Scences, Sydney, Australa 27 March 2002 Ths s a preprnt of an artcle publshed n The Journal of Computatonal Fnance 6(3), pp.1-20, Abstract An effcent condtonng technque, the so-called Brownan Brdge smulaton, has prevously been appled to elmnate prcng bas that arses n applcatons of the standard dscrete-tme Monte Carlo method to evaluate optons wrtten on the contnuous-tme extrema of an underlyng asset. It s based on the smple and easy to mplement analytc formulas for the dstrbuton of one-dmensonal Brownan Brdge extremes. Ths paper extends the technque to the valuaton of mult-asset optons wth noc-out barrers mposed for all or some of the underlyng assets. We derve formula for the unbased opton prce estmator based on the jont dstrbuton of the mult-dmensonal Brownan Brdge dependent extrema. As analytc formulas are not avalable for the jont dstrbuton n general, we develop upper and lower based opton prce estmators based on the dstrbuton of ndependent extrema and the Fréchet lower and upper bounds for the unnown dstrbuton. All estmators are smple and easy to mplement. They can always be used to bnd the true value by a confdence nterval. Numercal tests ndcate that our based estmators converge rapdly to the true opton value as the number of tme steps for the asset path smulaton ncreases n comparson to the estmator based on the standard dscrete-tme method. The convergence rate depends on the correlaton and barrer structures of the underlyng assets. Key Words: Monte Carlo smulaton, extreme values, Brownan Brdge, mult-asset barrer opton, mult-varate jont dstrbuton, Fréchet bounds. contact e-mal: Pavel.Shevcheno@csro.au 1

2 1 Introducton Barrer optons ntroduced by Merton (1973) are used wdely n tradng now. The opton s extngushed (noced-out) or actvated (noced-n) when an underlyng asset reaches a specfed level (barrer). A lot of related more complex nstruments for example bvarate barrer, ladder, step-up or step-down barrer optons have become very popular n over-thecounter marets. In general, these optons can be consdered as optons wth payoff dependng upon the path extrema of the underlyng assets. A varety of closed form solutons for such nstruments on a sngle underlyng asset have been obtaned n the classcal Blac-Scholes settngs of constant volatlty, nterest rate and barrer level. See for example Heynen and Kat (1994a), Kuntomo and Ieda (1992), Rubnsten and Rener (1991). If the barrer opton s based on two assets then a practcal analytcal soluton can be obtaned for some specal cases consdered n Heynen and Kat (1994b) and He, Kerstead and Rebholz (1998). In practce, however, numercal methods are used to prce the barrer optons for a number of reasons, for example, f the assumptons of constant volatlty and drft are relaxed or payoff s too complcated. Numercal schemes such as bnomal and trnomal lattces (Hull and Whte (1993), Kat and Verdon (1995)) or fnte dfference schemes (Dewynne and Wlmott (1993)) can be appled to the problem. However, the mplementaton of these methods can be dffcult. Also, f more than two underlyng assets are nvolved n the prcng equaton then these methods are not practcal. In ths paper we focus on a Monte Carlo smulaton method whch s a good general prcng tool for such nstruments. However, fndng the extrema of the contnuously montored assets by samplng assets at dscrete dates, the standard dscrete-tme Monte Carlo approach, s computatonally expensve as a large number of samplng dates and smulatons are requred. Loss of nformaton about all parts of the contnuous-tme path between samplng dates ntroduces a substantal bas for the opton prce. The bas decreases very slowly as 1/ M for M >> 1, where M s the number of equally spaced samplng dates (see Broade, Glasserman and Kou (1997)). Also, extrapolaton of the Monte Carlo estmates to the contnuous lmt s usually dffcult due to fnte samplng errors. For the case of a sngle underlyng asset, t was shown by Andersen and Brotherton-Ratclffe (1996) and Beaglehole, Dybvg and Zhou (1997) that the bas can be elmnated by a smple condtonng technque, the so-called Brownan Brdge smulaton. The method s based on the smulaton of a one-dmensonal Brownan brdge extremum between the sampled dates accordng to a smple analytcal formula for the dstrbuton of the extremum. The technque s very effcent because only one tme step s requred to smulate the asset path and ts extremum f the barrer, drft and volatlty are constant over the tme regon. We extend the technque to the valuaton of mult-asset optons wth contnuously montored noc-out barrers mposed for some or all underlyng assets. We derve the general formula for the unbased estmator based on the jont dstrbuton of the mult-dmensonal Brownan Brdge dependent extrema. In general, however, the analytc formulas are not avalable for ths jont dstrbuton and we develop three based estmators. The upper and lower estmators are based on the Fréchet bounds for the unnown mult-varate jont dstrbuton of the extrema. The thrd estmator (whch s typcally most accurate) s based on the jont dstrbuton of the ndependent extrema. The based estmators can be used to bnd the true opton prce 2

3 by a confdence nterval. Numercal examples ndcate that the bases rapdly decrease as M ncreases n comparson to the bas n the standard dscrete-tme method and the convergence rate depends on the correlaton and barrer structures of the underlyng assets. Fnally we dscuss the applcaton of our based estmators for the valuaton of noc-n, cash-at-ht rebates optons, loobac optons and credt dervatves. 2 Unbased estmator va Brownan Brdge correcton 2.1 Model setup Consder the noc-out opton Q wrtten on the underlyng assets S(t) = (S 1 (t),..., S d (t)). The opton payout at maturty t = T s some functon V ( S(T)) f the underlyng assets never ht the fxed boundares h(t) < S(t) < H(t), t [0, T] and zero otherwse. Here H(t) = (H 1 (t),..., H d (t)) and h(t) = (h 1 (t),..., h d (t)) are the upper and lower barrers. Hereafter, we use vector notaton to smultaneously compare all vector components. For example, A < B s used to denote A < B for all = 1,..., d. Assume that the underlyng assets follow rs-neutral geometrc Brownan moton ds (t)/s (t) = µ (t)dt + σ (t)dw (t), E[dW (t)dw j (t)] = ρ j (t)dt, (1) where S (0) s the -th asset prce today, W (t), = 1,..., d s the d-dmensonal Wener process, µ(t) = (µ 1 (t),..., µ d (t)) and σ(t) = (σ 1 (t),..., σ d (t)) are the drfts and volatltes respectvely. Let us consder the tme slces t m, m = 0,..., M ordered as 0 = t 0 < t 1 < t 2 <... < t M = T, δt m = t m+1 t m and denote S (t m ) = S (m). We assume that the drfts, volatltes, correlaton coeffcents and boundares are pecewse constant functons of tme such that µ (t) = µ (m), σ (t) = σ (m), ρ j (t) = ρ (m) j, H (t) = H (m), h (t) = h (m) f t [t m, t m+1 ), m = 0, 1,..., M 1. Let us also ntroduce the ndcator functon of the barrer ht at dscrete tmes t m, m = 0,..., M by I τ>t = { 1, f h (m) < S (m) < H (m) for m = 0,..., M, 0, otherwse. In the absence of arbtrage the true opton prce at t = 0 can be wrtten as an expectaton (2) Q = H (1) h (1) d S (1)... H (M) h (M) d S (M) V ( S (M) )p( S (1) S (0) )... p( S (M) S (M 1) ), (3) where p( S (m+1) S (m) ) s the rs-neutral probablty densty functon of the asset value S (m+1) at t m+1 gven the asset value S (m) at t m. The functon should satsfy the Kolmogorov forward equaton (also nown as the Foer-Planc equaton) wth the absorbng boundares h(t), H(t), t [0, T], see Cox and Mller (1965). Also we have absorbed the present value dscount factor nto the payoff functon V ( S(T)) and used the short vector notaton for the 3

4 mult-dmensonal ntegral 1. The explct soluton of the Kolmogorov forward equaton for the transton probablty functon p 0 ( S (m+1) S (m) ) to unrestrcted process (1) over the nterval [t m, t m+1 ] (wthout absorbng boundares) s the d-varate lognormal dstrbuton wth the means: [µ (m) ln S (m), varances: (σ (m) ) 2 δt m, and lnear correlaton coeffcents: ρ (m) The smulaton of S (m+1) S (m+1) from ths dstrbuton s smply = S (m) { exp [µ (m) 0.5(σ (m) ) 2 ]δt m σ (m) 0.5(σ (m) ) 2 ]δt m j, = 1,..., d; j = 1,..., d. δt m Z (m) }, (4) where Z (m), = 1,..., d are random varates from the d-varate Normal dstrbuton wth the lnear correlaton coeffcents ρ (m) j, zero means and unt varances for gven m (the random varates are ndependent for dfferent m). 2.2 Monte Carlo estmators The standard dscrete-tme Monte Carlo approach to estmate the noc-out barrer optons (3) s to assume unrestrcted process between samplng dates t m, m = 0, 1,.., M, then smulate N ndependently drawn asset paths accordng to the teratve equaton (4) and fnally to calculate the opton prce estmate as Q S = V ( S(T))I τ>t. (5) For smplcty, hereafter, we omt the averagng over N paths. Fndng the opton prce accordng to (5) wll ntroduce a bas 2 whch s usually larger than the statstcal error of the Monte Carlo estmates. Ths s because we lose nformaton on the contnuous-tme path between samplng dates. The bas decreases very slowly as Q S Q 1/ M for M >> 1 (see Andersen (1996) and Broade, Glasserman and Kou (1997)). Thus a large number of samplng dates s usually requred to obtan an accurate estmate of the opton prce wth contnuously montored barrers. For example, the bas s stll larger than 1% of the true prce even for 1024 tme steps for the case of the standard down-and-out call, see Table 1. Extrapolaton to the contnuous lmt s complcated by fnte samplng errors of the Monte Carlo estmates. Ths maes the estmator (5) computatonally expensve. Beng manly nterested n elmnatng (reducng) the bases unaffected by the number of paths, hereafter n formulae, we omt the dependence of the Monte Carlo estmates upon N assumng that N s large enough to mae the statstcal errors neglgbly small n comparson to the bases (we wll present the standard errors n the numercal examples). Consder the contnuous-tme asset maxma, M (m) = (M (m) 1,..., M (m) d ), and mnma, L (m) = (L (m) 1,..., L (m) ), over the tme nterval [t m, t m+1 ], where 1 H (m) h (m) d d S (m) = H 1 (m) h (m) 1 ds (m) H d (m) 1... h (m) d ds (m) d 2 The estmator (5) s an unbased estmator of the opton wth dscretely montored barrers f samplng dates match the barrer montorng dates. 4

5 M (m) = max{s (t) : t [t m, t m+1 ]}, L (m) = mn{s (t) : t [t m, t m+1 ]}. (6) If the contnuous barrers are mposed between samplng dates then the correct transton probablty functon (whch s a soluton of the Kolmogorov forward equaton wth absorbng boundares) p( S (m+1), M (m) < H (m), L (m) > h (m) S (m) ), (7) should be used n the opton prce ntegral (3) rather than the transton probablty functon p 0 ( S (m+1) S (m) ) for an unrestrcted process. Once the asset path s smulated 3 accordng to (4) then the unbased estmator for the noc-out opton can be calculated as where Q = V (S(T)) I τ>t M m=1 P (m), (8) P (m) = p( S (m+1), M (m) < H (m), L (m) > h (m) S (m) ) p 0 ( S (m+1) S (m) ) = Pr[ M (m) < H (m), L (m) > h (m) S (m+1), S (m) ] (9) s the probablty that the assets wll not ht the barrers n the tme regon [t m, t m+1 ] condtonal on S (m), S (m+1). Ths probablty s a jont dstrbuton of the extrema L (m) and M (m) condtonal on S (m), S (m+1). After sutable normalzaton 4 P (m) s a jont dstrbuton of the Brownan Brdge extrema (see Karatzas and Shreve (1991) or Borodn and Salmnen (1996)). If ths probablty s easy to calculate then the estmator (8) s trval to mplement. One has to smulate the asset path at t m, m = 0, 1,..., M accordng to the standard procedure (4). If the underlyng assets never ht the barrers at the samplng dates then the opton prce estmator s gven by the dscounted payoff V ( S(T)) weghted wth M P (m) and zero otherwse. However, jont dstrbuton P (m) can be found analytcally for some specal cases only. In the case of barrers dscretely montored at samplng dates, P (m) = 1 and estmator (8) s equvalent to the dscrete-tme estmator (5). 2.3 Margnal dstrbutons of the Brownan Brdge extrema For the case of a sngle barrer for one of the assets at each tme regon the probablty P (m) n (9) can be found analytcally. Usng results from the theory of the Wener process wth absorbng boundares, Cox and Mller (1965), and (9) or the formula for the one-dmensonal Brownan Brdge extremum, Karatzas and Shreve (1991), the margnal dstrbutons of the 3 The asset path should not necessarly be smulated accordng to (4). In general t can be smulated from any dstrbuton. 4 The log-ncrements of S(t) condtonal on S (m) and S (m+1) become a so-called Brownan Brdge on [t m, t m+1 ]. m=1 5

6 maxmum M (m) by and mnmum L (m) of the -th asset condtonal on S (m) and S (m+1) are gven where Pr[M (m) Pr[L (m) < H (m) > h (m) S (m), S (m+1) ] = 1 ξ (m) (H (m) S (m), S (m+1) ] = 1 ξ (m) ), (h (m) ), (10) h (m) < mn[s (m+1) ( ξ (m) (X) = exp, S (m) ], H (m) 2 ln X S (m) ln > mn[s (m+1), S (m) ], ) /(σ (m) ) 2 δt m. X S (m+1) Here ξ (m) (h (m) ) and ξ (m) hts by the -th asset n the nterval [t m, t m+1 ] respectvely. The maxmum and mnmum (11) (H (m) ) are the margnal probabltes of the upper and lower barrer can be smulated margnally by M (m) = (ξ (m) ) 1 (1 U) and L (m) = (ξ (m) ) 1 (U), where U s a random varable from the standard Unform dstrbuton 5. Thus, f the only barrer at [t m, t m+1 ] s mposed for the -th asset then P (m) = 1 ξ (m) (h (m) ) for the case of a lower barrer and P (m) = 1 ξ (m) (H (m) ) for the case of an upper barrer. The margnal dstrbutons (10) are vald as long as the nterest rate, asset volatlty and drft are constant n the tme nterval where the constant barrer s mposed on the asset. 2.4 Sngle barrer opton For a sngle underlyng asset and sngle barrer per tme regon t was demonstrated by Andersen and Brotherton-Ratclffe (1996) and Beaglehole, Dybvg and Zhou (1997) that smulaton of the barrer hts n the nterval [t m, t m+1 ] usng (10) elmnates the bas presented n (5). Alternatvely we calculate the opton prce estmator usng (8). We would le to stress that the margnal probabltes (10) can be used to get the unbased opton prce estmator not only for sngle asset barrer opton but also for mult-asset optons f there s a sngle barrer at each tme regon (ths barrer can be mposed for dfferent assets at dfferent tme regons). In Table 1 we present the Monte Carlo results for down-and-out call for the cases of one and two underlyng assets. In the frst case the opton pays max[s 1 (T) K, 0] f S 1 (t) > h 1, t [0, T] and n the second case the opton pays max[s 1 (T) K, 0] f S 2 (t) > h 2, t [0, T]. We have calculated the standard dscrete-tme based estmator, Q S, usng (5) and the unbased estmator, Q, usng (8) versus the number of equally spaced tme steps M. All parameters: volatltes, drfts, barrers, correlatons are assumed constant. Explct formulae for the unbased estmators n these examples are Q = e rt max[s 1 (T) K, 0] I τ>t for the case of a sngle underlyng asset and M m=1 [1 ξ (m) 1 (h 1 )] 5 The nverse functon (ξ (m) ) 1 (U) has two solutons. One soluton s used to fnd the maxmum and the other s used to fnd the mnmum. 6

7 Q = e rt max[s 1 (T) K, 0] I τ>t M m=1 [1 ξ (m) 2 (h 2 )] for the case of two underlyng assets. Beng manly nterested n elmnatng the bases unaffected by the number of smulatons we dd not use any varance reducton technque that can be appled to reduce the statstcal error of the estmates, see e.g. Boyle, Broade and Glassermann (1997). The comparson of the Monte Carlo estmates wth the exact opton prces calculated by analytcal formulae demonstrates that Q s an unbased estmator (the exact value s nsde the 0.95 confdence nterval of the estmates for any M). The dscretely montored barrer opton estmate Q S converges to the contnuous barrer case as M ncreases. However, the convergence s very slow and the bas s larger than 1% of the true prce even for 1024 tme steps. The use of the unbased estmator (8) n the above examples s very effcent because one tme step s enough to obtan the unbased opton prce estmate whle the standard dscrete-tme approach (5) requres an enormous number of tme steps. 3 Based estmators for mult-barrer opton va Fréchet bounds 3.1 Fréchet bounds for dstrbuton of Brownan Brdge extrema The jont dstrbuton P (m) n (9) can be found n closed form va nfnte seres for the case of two dependent extrema (that s two barrers n the same tme regon) usng the results from Andersen (1998) for double barrer on a sngle asset and He, Kerstead and Rebholz (1998) for two barrers mposed on dfferent assets. Then, n prncple, the unbased opton prce estmator can be calculated usng (8). In general, the jont dstrbuton, P (m), of three and more dependent extrema (that s three and more barrers n the same tme regon) s unnown and unbased opton prce estmator (8) can not be calculated. However, the unvarate margnal dstrbutons of the extremes, gven by (10), are nown and very smple. The classes of multvarate dstrbutons wth gven margns are the so-called Fréchet classes. The results on bounds of the dstrbutons wth nown margns and unnown dependence structure can be found n multvarate dstrbuton theory, see for example Joe (1997). The upper and lower bounds, so-called Fréchet bounds, for the unnown jont dstrbuton wth nown unvarate margns are based on smple nequaltes nvolvng probabltes of sets. Theorem (Lemma 3.8 n Joe (1997)) Let A 1,..., A be the events such that Pr(A ) = a, = 1,...,. Then max[0, =1 a ( 1)] Pr(A 1... A ) mn =1,..., a. (12) Let B = {L (m) > h (m) } and A = {M (m) < H (m) } be the events that mnmum s above the lower barrer and maxmum s below the upper barrer for the -th asset on [t m, t m+1 ] 7

8 respectvely. Then Pr(B ) = [1 ξ (m) (h (m) gves the followng bounds for the jont probablty P (m) n (9) P (m) P (m) L P (m) P (m) U = max[1 d = mn =1,...,d )], Pr(A ) = [1 ξ (m) =1 [ξ (m) [1 ξ(m) (H (m) (H (m) (H (m) )] and the above theorem ) + ξ (m) (h (m) )], 0], ), 1 ξ (m) (h (m) )]. The upper bound P (m) U corresponds to a perfect postve dependence between all events A, B, = 1,..., d. If there are only two events then the lower bound P (m) L corresponds to a perfect negatve dependence between the events. Perfect postve (negatve) dependence between A and A j, B and B j means a perfect postve (negatve) dependence between M (m) and M (m) j, L (m) and L (m) j respectvely. Perfect postve (negatve) dependence between A and B j means a perfect negatve (postve) dependence between M (m) probablty of the ndependent events 7 A, B, = 1,..., d P (m) I = d =1 whch, of course, satsfy P (m) L P (m) I then P (m) I P (m) (P (m) I P (m) ). [1 ξ (m) (H (m) )] [1 ξ (m) 3.2 Fréchet bounds and method of mages and L (m) j (13) 6. Also, consder the jont (h (m) )], (14) P (m) U. If all events are postvely (negatvely) dependent The Fréchet bounds (13) for the jont dstrbuton P (m) n (9) have the followng smple nterpretaton va the method of mages. The jont dstrbuton can be obtaned from a soluton of the Kolmogorov forward equaton wth absorbng boundares usng (9). The soluton for the case of a sngle barrer mposed on the -th asset n [t m, t m+1 ] can be found by the method of mages, see for example Cox and Mller (1965). The method s based on fndng the lnear combnaton of the unrestrcted process solutons p 0 ( S (m+1) S (m) ) started at S (m), the so-called source, and p 0 ( S (m+1) X (m) ) started at X (m), the so-called prmary mage 8, satsfyng the ntal and absorbng boundary condtons. The locaton of the prmary mage after log-scale change s found by reflecton of the source n respect to the boundary. Ths leads to the formula (10), where the contrbuton of the source s represented by 1 and the contrbuton of the prmary mage s represented by ξ (m) (h (m) ) (or ξ (m) (H (m) )) for the case of lower (or upper) barrer. Now t s easy to see that the lower bound P (m) L n (13) s obtaned from the source and all ts prmary mages (one mage for each barrer). The upper bound P (m) U n (13) s obtaned from the source and one of the prmary mages that gves the largest contrbuton. The method of mages cannot be used to fnd the exact transton probablty n the general case where few barrers are mposed on dfferent assets wth arbtrary correlaton. However, formally, prmary 6 The random varables X and Y wth contnuous margnal dstrbutons have perfect postve (negatve) dependence f X = T(Y ) where T(.) s a strctly ncreasng (decreasng) functon. 7 Ths jont dstrbuton has been used by Andersen (1998) and Beaglehole, Dybvg and Zhou (1997) to estmate double barrer and double loobac optons on a sngle asset. 8 In the case of few barrers the prmary mage may create further mages. 8

9 mages can always be ntroduced and used to get approxmate solutons. The numercal results below wll demonstrate that these approxmatons are very effectve. 3.3 Three based estmators for mult-barrer opton Usng the estmators P (m) X, X = L, I, U defned n (13) and (14) for the jont probablty of the extremes (9) we can form three based estmators for the opton prce (8): Q X = V ( S(T)) I τ>t It s easy to see from (8), (13) and (14) that M m=1 P (m) X, X = U, I, L. (15) Q L Q Q U, Q L Q I Q U () and, of course, Q X Q S because P (m) X 1, X = U, I, L. Whle Q I s typcally the most accurate, Q L and Q U are more useful because they can always be used to bnd the true value by the confdence nterval [Q L z 1 α/2 s(q L ) N, Q U + z 1 α/2 s(q U ) N ], (17) where z 1 α/2 s a quantle of the standard Normal dstrbuton, s(.) s the standard devaton, N s the number of smulated paths and α s the sgnfcance level. Then the true value les wthn the nterval wth at least 1 α probablty. The mddle of the nterval, Q (Q L + Q U ) (18) can be used as a pont estmator for the true value and half of the nterval wth z 1 α/2 =1 can be called the standard error of the estmator. Usng the nequalty P (m) I P (m) (P (m) I P (m) ) for negatvely (postvely) dependent events t s easy to fnd, for some smple barrer and dependence structures, that Q I gves better upper or lower estmator (however, n general, we do not now ths a pror). Then the true opton prce can be estmated by Q (Q L + Q I ) or Q (Q I + Q U ) (19) respectvely. The confdence ntervals to bnd the true value n these cases are formed by analogy wth (17). Here we lst some examples. If the barrer structure conssts of two barrers (upper and lower) mposed on one of the assets n each tme regon then Q I Q because the events of the upper and lower barrer hts are always negatvely dependent (that s maxmum and mnmum of the asset are always postvely dependent). If the barrer structure conssts of two upper or two lower barrers mposed on two postvely (negatvely) correlated assets n each tme regon then Q I Q (Q I Q) because the events of the barrer hts are postvely (negatvely) dependent. Also, Q I Q f only lower or only upper barrers are mposed on postvely correlated assets n each tme regon. 9

10 m=1 All three opton prce based estmators Q X, X = U, I, L gven by (15) are trval to mplement because the payoff weghts P (m) X, X = U, I, L are based on smple margnal M dstrbutons (10). As M ncreases, t becomes less and less lely that hts of the dfferent barrers wll occur wthn the same tme nterval and the events of barrer hts (and correspondng extremes) become dsjonted. That s, the probablty dstrbuton P (m) for each of the tme regons [t m, t m+1 ] s one of the unvarate margnal dstrbutons (10) n the lmt t m+1 t m 0. Dsjonng of the extrema means that the opton prce estmator becomes ndependent from the dependence structure (the so-called copula) of the extrema and depends on ther margnal dstrbutons only 9. It mples that Q I, Q L and Q U should converge to the true value. Convergence of Q I and Q U to each other can be used as a wea crteron of the extreme dsjonng. Also note that the standard dscrete-tme based estmator, Q S, gven by (5), always converges to the true value and Q S > Q U Q. In the next Secton we wll demonstrate numercally that all three estmators Q X, X = U, I, L are rapdly convergent to the true value when compared to Q S. The rate of convergence depends on the barrer and asset correlaton structures. In ths paper we do not pursue the analytcal dervaton of the convergence rate but fnd t numercally for some basc cases. The numercal examples presented n the followng Secton demonstrate the effectveness of the estmators (15) to correctly prce barrer optons. 4 Performance of the based estmators To demonstrate the performance (rapd convergence) of the based estmators Q X, X = U, I, L gven n (15) we calculate these estmators and the standard dscrete-tme estmator Q S, see (5), versus the number of equally spaced samplng dates M (the tme step s δt = T/M) for the cases of optons wth two or more noc-out barrers. In addton, we show the results for the pont estmator Q 0 (Q L + Q U )/2, see (18). We assume that all parameters (volatltes, nterest rate, barrers, correlatons) are constant and there are no contnuous dvdends. Beng nterested n the convergence of the bases unaffected by the number of smulatons we dd not use any varance reducton technques that can be appled to reduce the statstcal error of the estmates. In all examples the dscounted payoff (whch s pad at maturty f the assets never ht the barrers) s always determned by the frst asset: V (S 1 (T)) = e rt max[s 1 (T) K, 0]. The convergence rate should be rrelevant to the payoff pad at maturty. 4.1 Double noc-out call on a sngle asset Frst we consder the double noc-out call on a sngle asset wth the lower, h 1, and upper, H 1, barrers. The explct expressons for the probablty bounds (13), (14) used n the based estmators (15) are 9 The phenomenon of maxmum and mnmum decouplng has been noted by Andersen (1998) whle usng ndependently drawn maxmum and mnmum to estmate double barrer and double loobac optons on a sngle asset. 10

11 P (m) U = mn[1 ξ (m) 1 (h 1 ), 1 ξ (m) P (m) I P (m) L 1 (H 1 )], = [1 ξ (m) 1 (h 1 )] [1 ξ (m) 1 (H 1 )], = max[1 ξ (m) 1 (h 1 ) ξ (m) 1 (H 1 ), 0]. Actually, the exact jont dstrbuton P (m) s nown. It s represented by nfnte seres (usually the seres are rapdly convergent) and can be used to calculate the unbased opton prce estmator, see Andersen (1998). Beng manly nterested n the convergence of the based estmators we do not pursue ths calculaton. In Table 2 we show the performance of the tested estmators. The maxmum and mnmum of the asset on the same tme nterval are always postvely dependent (the hts of the lower and upper barrers are negatvely dependent) thus the estmator Q I based on the dstrbuton of the ndependent extremes s always larger than the unbased estmator Q. In ths case Q I s a better upper estmator than Q U. We present the results for Q 1 (Q I + Q L )/2, see (19), whch can be used as a better pont estmator for the true prce nstead of Q 0 (Q U + Q L )/2. All our based estmators Q U, Q I and Q L are rapdly convergent to the true value n comparson to the standard estmator Q S. Comparson of our based estmators wth the exact analytcal result shows that the bases of Q I, Q L and Q U are less than ther standard errors for M 4, M 4 and M 8 respectvely whle the bas of the standard estmator Q S s sgnfcantly larger than ts standard error even for M=1024. The standard errors are less then 1% of the true value. The exact value s always nsde the standard confdence ntervals of the pont estmators Q 1 and Q 0. In the case where a double barrer s mposed on the asset the hts of the upper and lower barrers are physcally dstant (the asset can not be close to the upper and lower barrers at the same tme). Thus ntutvely we expect that the bas Q U Q L should be exponentally small for large M. In Fgure 1 we plot ln(q U Q L ) versus M for the case of the double noc-out call consdered n Table 2. The standard error of the plotted estmates s less than the sze of the symbols. The observed lnear behavour of the graph ndcates that the bas decreases as Q U Q L e α/δt. 4.2 Two asset call wth two noc-out barrers To demonstrate convergence of the estmators for the case where barrers are mposed on dfferent arbtrary correlated assets we consder two asset down-and-out call wth the lower barrers h 1 and h 2 mposed on the frst and second assets respectvely. In ths case the explct expressons for the probablty bounds (13), (14) used n the based estmators (15) are P (m) U P (m) I P (m) L = mn [1 ξ(m) (h )], =1,..,d = d [1 ξ (m) (h )], =1 = max[1 d =1 ξ (m) (h ), 0], where d = 2. We desgned the problem parameters to equate probabltes of the barrer hts ξ (m) 1 (h 1 ) = ξ (m) 2 (h 2 ) f ρ = 1. In ths case P (m) L = max[1 2ξ (m) 1 (h 1 ), 0], P (m) U = 1 ξ (m) 1 (h 1 ) and we expect worst convergence because the events of the barrer hts do not become dsjont (20) 11

12 at M. The exact jont dstrbuton P (m) can be found n a closed form usng the results n He, Kerstead and Rebholz (1998). It s expressed va nfnte seres (usually the seres are rapdly convergent) and can be used to calculate the unbased estmator (8). Agan, beng only nterested n the convergence of the based estmators we do not pursue ths calculaton. In Table 3 we show the exact prces and Monte Carlo estmators for varous asset correlaton values. The exact opton prce for ρ = 0.5, 0.5 has been found va numercal ntegraton of the two-dmensonal densty functon from He, Kerstead and Rebholz (1998). In the cases ρ = 1, 0, 1 the opton value can be expressed va the barrer optons on a sngle asset and the exact solutons are represented va the standard cumulatve Normal functon 10. If the correlaton, ρ, between the assets s postve (negatve) then the mnma of the assets over the same tme nterval are always postvely (negatvely) dependent. Thus the estmator Q I based on the dstrbuton of the ndependent extremes s always larger (less) than the unbased estmator Q f ρ < 0 (ρ > 0). We do not present the results for Q 1 (Q I + Q L )/2 (f ρ < 0) and Q 2 (Q I + Q U )/2 (f ρ > 0) but they can be used as a better pont estmate for the true prce nstead of Q 0 (Q U + Q L )/2. If ρ = 0 then the mnma are ndependent and Q I s an unbased estmator for the true prce because P (m) I s a vald jont dstrbuton. If ρ = 1 then the mnma of the assets have a perfect postve dependence and Q U s an unbased estmator because P (m) U s a vald jont dstrbuton. The obtaned results show that our based estmators Q U, Q I and Q L are rapdly convergent to the true value when compared to the standard estmator Q S. Comparson of our results wth the exact results shows that the bases of Q I, Q L and Q U are less then ther standard errors f M whle the bas of the standard estmator Q S s sgnfcantly larger than ts standard error even for M=1024 (the standard errors are less then 1% of the true value for almost all cases). The only case when the convergence of Q I and Q L s not so rapd s ρ = 1 (n ths case Q U s the unbased estmator). The standard confdence nterval of the pont estmator Q 0 always contans the exact prce. The convergence rate of our based estmators strongly depends on correlaton between the assets. In Fgure 2, Fgure 3 and Fgure 4 we present the graphs ndcatng the convergence rates for ρ = 1, 0, 1. The sze of the symbols used for the graphs s larger than the standard error of the estmates (we have used more smulatons for estmates at large M). The lnear behavour of the graphs at large M ndcates the followng. If ρ = 1 then Q U Q L e β/δt for M >> 1. Ths s smlar to the results for double barrer call n Table 2 and Fgure 1, because the assets have a perfect negatve dependence and the opton s equvalent to a sngle asset noc-out opton wth the flat lower and exponentally growng upper barrers. If ρ = 1 then Q U Q L δt for M >> 1. Ths square root convergence s the worst observed rate. In ths case the asset mnma have a perfect postve dependence and the estmator Q U s the unbased estmator of the true opton prce. That s Q L and Q S converge to the true prce at the same rate from below and above respectvely. If ρ = 0 then Q U Q L δt 2 for M >> 1. We have observed that the convergence rate smoothly deterorates from the best 10 For ρ = 1 the opton s reduced to a double noc-out call on the frst asset wth the exponentally growng upper and flat lower barrers. For ρ = 0 the opton can be represented as a product of a down-and-out call on the frst asset and a down-and-out dgtal opton on the second asset. For ρ = 1 the opton s reduced to a down-and-out call on the frst asset. 12

13 exponental decay exp( β/δt) at ρ = 1 to a rapd power decay δt 2 at ρ = 0 and slow square root decreasng δt at ρ = 1 as the correlaton s changed between 1 and 1. Note that we have desgned the parameters to get worst convergence at ρ = 1. If we change the parameters to mae the barrer ht probabltes unequal then the extrema should become dsjonted as M even for ρ = 1. For example, f we set the asset spots S 1 (0) = 95, S 2 (0) = 105 and do not change the other parameters then the convergence rate at ρ = 1 s rapd exponental decay exp( β/δt), see Fgure 5, whle for ρ = 0 and 1 the rates do not change (that s δt 2 and exp( β/δt) respectvely). 4.3 Mult-asset call wth multple barrers Fnally, to show that our based estmators wor well for real mult-dmensonal problems we consder, see Table 4, down-and-out call on d underlyng assets wth the lower barrers mposed on all assets for the cases d = 3 and d = 10. The exact (analytcal) result s not avalable for ths problem. Explct expressons for the probablty bounds (13), (14) used n the based estmators (15) are gven by (20). As we have chosen postve correlaton between all assets the asset mnma are postvely dependent. Thus Q I s always less than the unbased estmator Q and we present the results for Q 2 (Q I + Q U )/2 whch s a better pont estmate for the true prce than Q 0. As n the prevous examples, all our based estmators Q U, Q I and Q L are rapdly convergent to each other. Ther bases become less than the statstcal errors for M whle the bas of the standard estmator Q S s larger than ts standard error even for M = Conclusons and dscusson In ths paper we have developed a condtonng technque, that can be called a Brownan Brdge scheme, for Monte Carlo smulaton of a general class of mult-asset optons wth contnuously montored noc-out barrers mposed for some or all underlyng assets. We have derved the general formula (8) for an unbased estmator of the opton based on the jont dstrbuton of the mult-dmensonal Brownan Brdge extrema (9). If the dstrbuton s nown, for example, one barrer mposed on one of the assets (or two barrers mposed on dfferent or the same assets) per tme regon, the scheme provdes a smple unbased estmator. The barrers, drfts and volatltes are requred to be pecewse constant functons of tme. In the case of more than two barrers per tme regon the dstrbuton s unnown n a closed form for arbtrary dependence between the assets and we derved the upper, Q U, and lower, Q L, based opton prce estmators. The estmators are based on the Fréchet lower, P (m) (m) L, and upper, P U, bounds (13) for the unnown jont dstrbuton wth gven unvarate margns. We have also used the estmator Q I based on the jont dstrbuton of ndependent extrema, P (m) I. For some smple barrer and dependence structures Q I can provde better upper or lower bounds. Whle ths estmator s usually more accurate, Q L and Q U are often more useful because they can always be used to bnd the true opton prce by the confdence nterval. As the tme between the samplng dates decreases, the Brownan Brdge extrema become more dsjonted and the based 13

14 estmators Q L, Q I, Q U converge to each other and the true value. In the lmt, the opton prce estmator becomes ndependent of the dependence structure (the so-called copula) of the extrema and depends on ther margnal dstrbutons only. In our numercal examples we showed that the bas Q U Q L s less and convergence s rapd when compared to the bas and convergence of the standard estmator Q S (Q S Q δt for M >> 1, where δt = T/M). Usually, Q U Q L s less than the statstcal error of the estmates for only a few tme steps. In practce, the ntermedate dates are ntroduced due to the nterest rate or volatlty termstructures and the nserton of the addtonal samplng dates to elmnate the bas may not be even requred. The convergence rate depends on the barrer and correlaton structures. We have always observed Q U Q L exp( α/δt) for the case of lower and upper barrers mposed on the same asset. For the case of two lower barrers mposed on two assets the best detected convergence rate s exponental decay exp( β/δt) and the worst detected rate s slow square root decreasng δt. The worst case was obtaned for the unrealstc specal set of parameters (ρ = 1 and dentcal parameters for both assets) whch maes barrer hts always equal each other and Q U to be an unbased estmator of the true prce. The descrbed Brownan Brdge technque s straghtforward to use for the valuaton of the noc-n barrer optons and barrer optons wth constant rebates, R, pad at maturty. The unbased estmators n these cases are gven by and Q = V (S(T))[1 I τ>t M m=1 P (m) ] Q = V (S(T))I τ>t M m=1 P (m) + R [1 I τ>t M m=1 P (m) ] respectvely. The upper and lower based estmators can easly be calculated usng P (m) L and. The technque can easly be appled to effcently estmate dscretely montored barrer P (m) U optons wth a large number of observaton dates usng the method proposed by Andersen (1996). That s, usng our scheme calculate the contnuously montored barrer opton prce Q c and usng the standard Monte Carlo method estmate the opton wth a low frequency montored barrer. Then the nterpolaton formula Q M Q c + λ/ M, where Q M s a barrer opton wth M montored dates, allows for effectve estmaton of the opton wth a hgh frequency montored barrer. Fndng the upper and lower based estmators s not straghtforward for the case of multbarrer optons wth rebates pad at httng tmes. Ths type of problem s also relevant to the valuaton of credt dervatves. To calculate the unbased opton prce estmator, multple httng tmes should be smulated from ther vald jont dstrbuton whch s not nown even for the case of two barrers. However, the httng tmes can easly be smulated from ther nown unvarate margnal dstrbutons, see Anderson (1996). Thus, agan we have a problem of the unnown jont dstrbuton wth the nown unvarate margns. Such optons can be evaluated n the followng way. Calculate one opton estmate Q (1) smulatng the httng 14

15 tmes margnally wth perfect postve dependence 11. Another estmate Q (2) can be found by smulatng the events margnally wth perfect negatve dependence n the case of two barrers or smulatng the events ndependently f there are more than two barrers. The dfference between Q (1) and Q (2) can be used as a wea crteron for the dsjonted events to justfy ther margnal smulatons. That s, f the dfference s less than the statstcal error then the events can be assumed dsjonted enough (margnal smulaton s justfed) and (Q (1) + Q (2) )/2 can be used as a pont estmate for the true prce. Otherwse addtonal samplng dates need to be nserted. Loobac type optons wth payoff dependent on a few contnuous extrema can be evaluated n a smlar way. We wll consder these problems n further research. In all our examples we have assumed the lognormal dffuson process (1) to allow for comparson wth analytcal results. However, the Brownan Brdge scheme dscussed here s stll applcable for more general dffuson processes. For example, t s applcable f drfts and volatltes are state-dependent. To solve these models approxmate dscretsaton schemes freezng the drfts and volatltes to the left pont of each smulaton step are used, see for example Kloeden and Platen (1992). Then Brownan Brdge schemes can be used because the requrements of pecewse constant drfts and volatltes are satsfed. Ths wll elmnate the bas due to dscrete underestmaton of the contnuous extrema (bas due to the dscretsaton scheme wll not be removed). In ths paper we have focused on the applcaton of the technque for the valuaton of smple noc-out mult-asset optons. However, practcal use of the technque les n a broad range of problems. For example, the technque can potentally be used for credt and maret rs problems where valuaton of a mult-asset payoff wth some barrer levels mposed on the underlyng assets s very essental. Acnowledgements The author thans Volf Frshlng and Fran de Hoog for stmulatng dscussons and helpful advce. References Andersen, L., and Brotherton-Raclffe, R. (1996). Exact Exotcs. Rs, 9(10), Andersen, L. (1996). Monte Carlo smulaton of Barrer and Loobac Optons wth contnuous or Hgh-Frequency Montorng of the Underlyng Asset. Worng paper, General Re Fnancal Products. Andersen, L. (1998). Monte Carlo smulaton of Optons on Jont Mnma and Maxma. Worng paper, General Re Fnancal Products. Beaglehole, D. R., Dybvg, P. H., and Zhou, G. (1997). Gong to extremes: Correctng Smulaton Bas n Exotc Opton Valuaton. Fnancal Analyst Journal, January/February, The random varables X, = 1,..., wth contnuous margnal dstrbutons F and perfect postve dependence can be smulated as X = F 1 (U), where U s a random varable from U(0, 1). If X, = 1, 2 have a perfect negatve dependence then X 1 = F1 1 (U), X 2 = F2 1 (1 U). If X, = 1,..., are ndependent then X = F 1 (U ), where U, = 1,..., are ndependent random varables from U(0, 1). 15

16 Borodn, A., and Salmnen, P. (1996). Handboo of Brownan Moton-Facts and Formulae. Brhauser Verlag, Basel. Boyle, P., Broade, M., and Glassermann, P. (1997). Monte Carlo Methods for Securty Prcng. Journal of Economc Dynamcs and Control, 21, Broade, M., Glasserman, P., and Kou, S. (1997). A contnuty correcton for dscrete barrer optons. Mathematcal Fnance, 7, Cox, D., and Mller, H. (1965). The Theory of Stochastc Processes. Chapman and Hall. Dewynne, J., and Wlmott, P. (1994). Partal to Exotc. Rs, December, He, H., Kerstead, W. P., and Rebholz, J. (1998). Double Loobac. Mathematcal Fnance, 8(3), Heynen, R., and Kat, H. (1994a). Partal Barrer Optons. Journal of Fnancal Engneerng, 3(3/4), Heynen, R., and Kat, H. (1994b). Crossng barrers. Rs, 7(6), Hull, J., and Whte, A. (1993). Effcent Procedures for Valung European and Amercan Path-Dependent Optons. Journal of Dervatves, Fall, Joe, H. (1997). Multvarate Models and Dependence Concepts. Chapman & Hall. Kat, H., and Verdon, L. (1995). Tree Surgery. Rs, February, Karatzas, I., and Shreve, S. (1991). Brownan Moton and Stochastc Calculus. Sprnger Verlag. Kloeden, P., and Platen, E. (1992). Numercal Soluton of Stochastc Dfferental Equatons. Sprnger Verlag. Kuntomo, N., and Ieda, M. (1992). Prcng Optons wth Curved Boundares. Mathematcal Fnance, 4, Merton, R. (1973). Theory of Ratonal Prcng. Bell Journal of Economcs and Management Scence, 4, Sprng, Rubnsten, M., and Rener, E. (1991). Breang down the barrers. Rs, 4(8),

17 One asset down and out call. The exact value s Two asset down-and-out call. The exact value s M Q(std.err.) Q S (std.err.) M Q(std.err.) Q S (std.err.) (0.02) 8.80(0.02) 8.80(0.02) 8.79(0.02) 8.80(0.02) 8.80(0.02) 8.80(0.02) 10.91(0.02) 10.66(0.02) 10.32(0.02) 9.74(0.02) 9.33(0.02) 9.08(0.02) 8.94(0.02) (0.02) 8.26(0.02) 8.27(0.02) 8.27(0.02) 8.28(0.02) 8.28(0.02) 8.28(0.02) 14.93(0.03) 13.62(0.03) 12.35(0.03) 10.52(0.02) 9.47(0.02) 8.90(0.02) 8.59(0.02) Table 1: One asset down and out call: S 1 (0) = 100, K = 100, h 1 = 90, σ 1 = 0.3, r = 0.1, T = 0.5, smulatons. Two asset down-and-out call wth a sngle barrer: S 1 (0) = S 2 (0) = 100, K = 100, h 2 = 90, r = 0.1, T = 1, σ 1 = σ 2 = 0.3, ρ = 0.5, smulatons. M Q U (std.err.) Q I (std.err.) Q L (std.err.) Q S (std.err.) Q 1 (std.err.) Q 0 (std.err.) (0.01) 2.21(0.01) 1.84(0.01) 1.79(0.01) 2.41(0.01) 1.89(0.01) 1.79(0.01) 1.79(0.01) 1.11(0.01) 1.72(0.01) 1.79(0.01) 12.23(0.04) 9.60(0.04) 7.41(0.03) 5.73(0.03) 4.50(0.02) 3.06(0.02) 2.40(0.02) 2.08(0.02) 1.76(0.66) 1.80(0.09) 1.79(0.01) 1.79(0.01) 2.06(0.96) 1.97(0.26) 1.81(0.04) 1.79(0.01) Table 2: Double noc-out call on a sngle asset. h 1 = 900, H 1 = 1100, S(0) = K = 1000, σ 1 = 0.2, r = 0.1, T = 0.5, smulatons. The exact value of the contnuously montored opton s Fgure 1: Double noc-out call on a sngle asset consdered n Table 2. 17

18 ρ M Q U (std.err.) Q I (std.err.) Q L (std.err.) Q S (std.err.) Q 0 (std.err.) Q c (0.03) 3.78(0.04) 3.70(0.04) 3.66(0.04) 3.65(0.04) 3.64(0.04) 3.65(0.03) 3.66(0.04) 3.66(0.04) 3.65(0.04) 3.65(0.04) 3.64(0.04) 2.27(0.02) 3.62(0.04) 3.65(0.04) 3.65(0.04) 3.65(0.04) 3.64(0.04) 11.76(0.07) 6.84(0.06) 5.92(0.05) 5.27(0.05) 4.81(0.05) 3.93(0.05) 3.64(1.41) 3.70(0.12) 3.67(0.06) 3.65(0.05) 3.65(0.04) 3.64(0.04) (0.05) 6.71(0.06) 6.61(0.06) 6.57(0.06) 6.55(0.06) 6.54(0.06) 2.57(0.02) 1.47(0.02) 1.42(0.02) 1.41(0.02) 1.39(0.02) 1.38(0.02) 11.36(0.06) 11.36(0.07) 11.37(0.07) 11.35(0.07) 11.34(0.07) 11.33(0.07) 0.415(0.002) 0.018(0.001) 0.014(0.001) 0.014(0.001) 0.013(0.001) 0.013(0.001) 5.84(0.04) 6.48(0.05) 6.53(0.06) 6.54(0.06) 6.54(0.06) 6.54(0.06) 1.70(0.01) 1.41(0.02) 1.40(0.02) 1.40(0.02) 1.39(0.02) 1.38(0.02) 8.05(0.05) 10.22(0.07) 10.63(0.07) 10.84(0.07) 10.98(0.07) 11.24(0.07) 0.7(0.001) 0.014(0.001) 0.013(0.001) 0.014(0.001) 0.013(0.001) 0.013(0.001) 4.22(0.04) 6.41(0.05) 6.51(0.06) 6.53(0.06) 6.54(0.06) 6.54(0.06) 0.67(0.01) 1.40(0.02) 1.40(0.02) 1.40(0.02) 1.39(0.02) 1.38(0.02) 6.31(0.05) 10.00(0.07) 10.49(0.07) 10.74(0.07) 10.91(0.07) 11.22(0.07) 0(0) 0.014(0.001) 0.013(0.001) 0.014(0.001) 0.013(0.001) 0.013(0.001) 14.97(0.08) 10.28(0.07) 9.27(0.07) 8.52(0.07) 7.98(0.06) 6.93(0.06) 7.86(0.05) 3.63(0.04) 2.88(0.03) 2.45(0.03) 2.09(0.03) 1.55(0.03).79(0.08) 14.35(0.08) 13.63(0.08) 13.06(0.07) 12.63(0.07) 11.69(0.07) 2.839(0.018) 0.476(0.008) 0.250(0.06) 0.137(0.004) 0.080(0.003) 0.023(0.002) 6.00(1.82) 6.56(0.20) 6.56(0.10) 6.55(0.07) 6.55(0.06) 6.54(0.06) 1.62(0.96) 1.43(0.06) 1.41(0.03) 1.41(0.02) 1.39(0.02) 1.38(0.02) 8.84(2.58) 10.68(0.74) 10.93(0.51) 11.04(0.37) 11.12(0.29) 11.28(0.12) 0.207(0.209) 0.0(0.003) 0.014(0.001) 0.014(0.001) 0.013(0.001) 0.013(0.001) Table 3: Two asset down-and-out call wth lower barrers for the frst and second assets. h 1 = h 2 = 90, S 1 (0) = S 2 (0) = 100, K = 100, σ 1 = σ 2 = 0.3, r = 0.1, T = 1, smulatons. Q c s the exact value of the opton wth contnuously montored barrers. 18

19 d M Q U (std.err.) Q I (std.err.) Q L (std.err.) Q S (std.err.) Q 2 (std.err.) Q 0 (std.err.) (0.07) 8.26(0.07) 7.83(0.07) 7.65(0.07) 7.60(0.08) 7.60(0.08) 7.60(0.08) 7.60(0.08) 6.69(0.06) 7.20(0.07) 7.43(0.07) 7.51(0.07) 7.56(0.08) 7.59(0.08) 7.59(0.08) 7.60(0.08) 5.13(0.06) 6.76(0.07) 7.31(0.07) 7.47(0.07) 7.54(0.08) 7.58(0.08) 7.59(0.08) 7.60(0.08) 14.96(0.10) 13.27(0.09) 11.81(0.09) 10.76(0.09) 9.96(0.09) 9.29(0.08) 8.80(0.08) 7.91(0.08) 7.83(1.20) 7.73(0.60) 7.63(0.27) 7.58(0.14) 7.58(0.10) 7.59(0.08) 7.59(0.08) 7.60(0.08) 7.04(1.97) 7.51(0.82) 7.57(0.33) 7.56(0.) 7.57(0.11) 7.59(0.09) 7.59(0.08) 7.60(0.08) (0.05) 3.56(0.05) 2.98(0.05) 2.80(0.05) 2.71(0.05) 2.67(0.05) 2.65(0.05) 2.65(0.05) 1.19(0.02) 1.97(0.03) 2.39(0.04) 2.60(0.05) 2.64(0.05) 2.65(0.05) 2.64(0.05) 2.65(0.05) 0.21(0.01) 1.33(0.03) 2.20(0.04) 2.54(0.05) 2.61(0.05) 2.64(0.05) 2.64(0.05) 2.65(0.05) 10.36(0.09) 7.92(0.08) 6.13(0.07) 5.09(0.07) 4.37(0.06) 3.84(0.06) 3.48(0.06) 2.86(0.05) 2.90(1.75) 2.77(0.84) 2.68(0.34) 2.70(0.15) 2.67(0.08) 2.66(0.06) 2.65(0.05) 2.65(0.05) 2.41(2.23) 2.45(1.) 2.59(0.44) 2.67(0.18) 2.66(0.10) 2.66(0.07) 2.64(0.06) 2.65(0.05) Table 4: Knoc-out calls on three and ten assets wth lower barrers for each of the asset. S (0) = 100, h = 80, ρ j = 0.5 f j, σ = 0.4 (, j = 1,..., d), T = 1, K = 100, r = 0.05, smulatons. Fgure 2: Two asset down-and-out call consdered n Table 3 wth ρ = 1. 19

20 Fgure 3: Two asset down-and-out call consdered n Table 3 wth ρ = 0. Fgure 4: Two asset down-and-out call consdered n Table 3 wth ρ = 1. Fgure 5: Two asset down-and-out call consdered n Table 3 wth ρ = 1 and S 1 (0) = 95, S 2 (0) =