Economic Computation and Economic Cybernetics Studies and Research, Issue 2/2016, Vol. 50
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1 Ecoomic Computatio ad Ecoomic Cyberetics Studies ad Research, Issue 2/216, Vol. 5 Kyoug-Sook Moo Departmet of Mathematical Fiace Gacho Uiversity, Gyeoggi-Do, Korea Yuu Jeog Departmet of Mathematics Korea Uiversity, Seoul, Korea Hogoog Kim Departmet of Mathematics, Korea Uiversity, Seoul, Korea hogoog@korea.ac.kr AN EFFICIENT BINOMIAL METHOD FOR PRICING ASIAN OPTIONS Abstract. We costruct a efficiet tree method for pricig pathdepedet Asia optios. The stadard tree method estimates optio prices at each ode of the tree, while the proposed method defies a iterval about each ode alog the stock price axis ad estimates the average optio price over each iterval. The proposed method ca be used idepedetly to costruct a ew tree method, or it ca be combied with other existig tree methods to improve the accuracy. Numerical results show that the proposed schemes show superiority i accuracy to other tree methods whe applied to discrete forward-startig Asia optios ad cotiuous Europea or America Asia optios. Keywords : biomial tree method, cell averagig, Asia optios. JEL Classificatio: G13, C63, C2 1. Itroductio A optio is a fiacial derivative which gives the ower the right, but ot the obligatio, to buy or sell a uderlyig asset for a give price o or before the expiratio date. From the semial papers of Black ad Scholes (1973) ad Merto (1973), the tradig volume of optios has bee icreased ad exotic optios with ostadard payoff patters have become more commo i the over-the-couter market. Amog them, a optio with the payoff determied by the average uderlyig price over some pre-defied period of time is called a Asia optio. A Asia optio has bee popular sice it could reduce the risk of market maipulatio of the uderlyig asset at maturity ad the volatility iheret i the optio. However, these Asia optios based o arithmetic averages caot be priced i a closed-form, ad oe eeds to rely o its umerical approximatio istead. There have bee may approaches to approximate the value of exotic optios, such as biomial tree method, Mote Carlo simulatio, fiite differece method for solvig Black-Scholes partial differetial equatios etc. Both the Mote Carlo method ad fiite differece method suffer from the difficulty to deal with early exercise without bias, whereas the biomial tree method by Cox, Ross ad 151
2 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim Rubistei (1979) are very popular due to its ease of implemetatio ad simple extesio to America type optios. However, due to the averagig ature of Asia optios, the umber of averagig odes i biomial tree grows expoetially. Therefore straightforward extesio of the stadard biomial method to Asia case is ot possible i practice. I order to solve this shortage of biomial method, Hull ad White (1993) cosidered a set of represetative averages at each ode icludig miimum ad maximum average values. Employig this set of represetative averages makes the biomial model feasible for pricig Asia optios, though it still suffers the lack of covergece, see Costabile, Massabo ad Russo (26) ad Forsyth, Vetzal ad Zva (22). For a discrete moitored Asia optio, Hsu ad Lyuu (211) proposed a quadratic-time coverget biomial method based o the Lagrage multiplier to choose the umber of states for each ode of a tree. I this paper, based o the cell averagig approach i Moo ad Kim (213), small bis o the asset price axis, called cells, are defied about each ode of the tree ad the average optio price over each cell has bee computed ad updated i time. See Sectio 2 for details. The biomial method of Hsu ad Lyuu (211) for discrete moitored Asia optios ad that of Hull ad White (1993) for cotiuous Asia optios have bee modified for improvemet with the help of cell averagig method. Numerical experimets i sectio 4 show that the proposed cell averagig biomial method gives more accurate results compared to other existig computatioal methods. The outlie of the paper is as follows. I sectio 2 we explai the problem ad itroduce the cell averagig biomial method. I sectio 3 we exted the cell averagig biomial method to discrete ad cotiuous moitored Asia optios. I sectio 4 we compare the accuracy ad efficiecy of the existig tree methods with those of the cell averagig biomial method. We fially summarize our coclusios i sectio Cell Averagig Biomial Methods Let us cosider the price of the uderlyig asset as a stochastic process { S( t), t [, T ]} which satisfies the followig stochastic differetial equatio: ds( t) S( t) dt + S( t) dw ( t), t T, (1) where is a expected rate of retur, is a volatility, T is a expiratio date, ad Wt () is a Browia motio. From the Ito formula i Øksedal (1998), X ( t) l( S( t)) satisfies 2 dx ( t) ( / 2) dt dw ( t), t. I the risk-eutral world, the value of the Europea optio ca be computed by the discouted coditioal expectatio of the termial payoff, ( ) (, ) r T t V x t e E ( X ( T )) X ( t) x, where ( XT ( )) is the payoff at t T. Without loss of geerality, we deote agai the risk eutral process to be Xt () with drift rate equal to the risk-free iterest rate r, istead of i (1). If we cosider a cotiuous divided yield q, the drift rate becomes r q. 152
3 A Efficiet Biomial Method for Pricig Asia Optios 2.1. The biomial model Let us first discretize the time period,t ito N itervals of the same legth t T / N, t t1... tn T. The stadard biomial method by Cox, Ross ad Rubistei (1979) assumes that the asset price St ( ) at t t moves either up to S( t) u for u exp( t) 1 or dow to S( t) d for rt d 1/ u ad, 1,, N 1 with probabilities p ( e d) / ( u d) or 1 p, respectively, or Xt ( ) i log at t t moves either up to X ( t) h or dow to X ( t) h where h l u. Let X X (2 ) h deote the values at t t t for, 1,,, with X() X. The the stadard N N biomial method calculates the payoffs of the optio at expiry, V ( X ) for,1,..., N, ad computes the optio price V V ( X,) by backward averagig, r t V ( x, t) e ( pv ( x h, t t) (1 p) V ( x h, t t)), (2) where x X,,,, ad N 1, N 2,,. The biomial method approximatio coverges to the Black-Scholes value as the umber of time steps, N, teds to ifiity, see the geeral theory i Kwok (1998), Clewlow ad Stricklad (1998), Lyuu (22) ad Higham (24). However, it is widely reported that the covergece is ot mootoe ad the sawtooth patter i the sequece of approximatios makes the biomial approximatio less attractive. 2.2 The cell averagig biomial model I order to reduce the saw-tooth patters i the sequece of approximatios i the stadard biomial method, we employ the cell averagig method. Let us first divide the iterval X ( N 1) h, X ( N 1) h o the X -axis ito N 1 o-overlappig equidistat itervals of legth 2h, called cells, cetered at poits X (2 N) h,,, N, the compute average optio payoffs o each cell cetered at X at expiry t tn, N 1 X h N V ( )d N 2h,, N, (3) X h where () is the payoff fuctio at expiry. If (2) is satisfied at every poit X h, X h i the cell at time t, the the average optio price V 1 X h (, )d 2h V t satisfies the followig backward averagig relatio X h rt 1 V e pv p V 1 (4) 1 1 See Figure 1. Appropriate modificatio will be eeded if (2) does ot hold at 153
4 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim every poit X h, X h. For istace, see Moo ad Kim (213) for the case of barrier optios. The cell averages of the optio values at expiry (3) ca be updated iteratively, which evetually leads to the average of the optio price V o X h, X h at t. Figure 1 : Compariso of backward averagig betwee the stadard tree method (Left) ad the cell averagig tree method (Right). Figure 2 compares the stadard biomial method (dash) ad the cell averagig biomial method (solid) for the Europea up-ad-out barrier put optio price. The figure shows that this cell averagig idea reduces ig-saw oscillatios. Figure 2 : Parameters: The iitial stock price S() 1, the risk-free iterest rate r.5, the volatility.3, strike price K 9, barrier H 15 ad the maturity T 1 for a Europea up-ad-out barrier put optio. Compariso betwee stadard biomial (dash) ad cell averagig biomial (solid) lattice models. Cell averagig produces smoother covergece. 3. Asia Optios Now we exted the cell averagig biomial method i sectio 2 to pathdepedet Asia optios. As it has bee kow, there do ot exist explicit closedform aalytical solutios for arithmetic Asia optios because the arithmetic average of a set of logormal radom variables is ot log-ormally distributed. For that reaso, may umerical approaches have bee proposed. We first cosider discrete moitored Asia optios i Sectio 3.1 ad modify the method of Hsu ad Lyuu (211) to price it. The we improve the method of Hull ad White (1993) i 154
5 A Efficiet Biomial Method for Pricig Asia Optios Sectio 3.2 to price the cotiuous Asia optio. 3.1 The Discrete Moitored Asia optio The discretely moitored Asia optio is ofte foud i practice. The payoff of Asia call optio with strike price K at the expiry date T is give by the followig : 1 max S ti( ) K, (5) i 1 where is the umber of moitor poits ad the payoff of discrete type arithmetic average Asia call optio is moitored at time poits, t t t T. We assume each time iterval betwee two adacet 1 2 moitor poits is partitioed ito I time steps, ad I is called itraday. The we see that the moitor poits are at times, I, 2 I,, I ad the whole umber of time steps is N I. For the stadard Europea-style discrete Asia call, the payoff at expiry is 1 max SiI K, 1 i whereas the forward-startig discrete Asia optio omits the iitial S ad has the payoff 1 max SiI K, i 1 I order to be self-cotaied, we start with explaatio of the biomial method by Hsu ad Lyuu (211) which follows the stadard biomial method suggested by Cox, Ross ad Rubistei (1979). Let N( i, ) deote the ode at time i that results from dow moves ad i up moves. The the price sum to expiry date for a price path ( S, S1,, S i ), P, called the ruig sum, is computed by S SI S for stadard Asia optios i / I I P SI S2 I S for forward-startig Asia optios, i / I I where deotes floor fuctio ad i N. Sice the pricig of Asia optio usig biomial lattice produces 2 N possible paths for each time step N, Hsu ad Lyuu (211) suggested a discrete biomial method for Asia optio pricig, where they proposed the ruig sum P of the form ( 1) K 2( 1) K ( ki 1)( 1) K,,,,,( 1) K if i P ki ki ki (6) S if i= for stadard Asia optio 155
6 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim K 2K ( ki 1) K,,,,, K if i P ki ki ki (7) if i for forward-starig Asia optio where ki 1 represets the umber of states cosidered for each ode N( ii, ). Here 1 3 k i is computed by B( ii,, p) 2 2 ci i ki, (8) 1 2 si B( si, t, p) 3 2 s1 t s i where (,, ) i B i p p (1 p), ad c is the average umber of states per ode. If the 3-tuple ( ii, S, P ) deotes the curret state, the correspodig optio value V ( ii, S, P ii ) ca be computed by I 1 I 2l I 2l V ( ii, S, PiI ) pl V (( i 1) I, Su, PiI Su ) I, R l ii 2 where S Su, R exp( r t), ad the associated brachig probabilities are I I l l pl p (1 p) l for each brach l,, I, ad i,1,,( 1). Whe P ( 1) K, { PiI ( i) S}/ ( 1) K if R 1 V ( ii, S, P ) ( i) I ii ( i) I I 1 R R {( P ) / ( 1) } if R 1. ii SR K I 1 R For forward-startig discrete Asia optios, the similar formulas hold { PiI ( i) S}/ K if R 1 V ( ii, S, P ) ( i) I ii ( i) I I 1 R R {( P ) / } i f 1 ii SR K R I 1 R whe PiI K. Otherwise, liear iterpolatio is computed from the two bracketig ruig sums' correspodig optio values to obtai : I 2l I 2l V(( i 1) I, Su, P Su ) ii ( 1) K ( 1) K, I 2l I 2l lv ( i 1) I, Su,( sl 1) (1 l ) V ( i 1) I, Su, sl ki 1, l ki 1, l ii 156
7 A Efficiet Biomial Method for Pricig Asia Optios where l 1 for l,1,, I. Now let us apply cell averagig algorithm to this discrete model over the cells o the X -axis as i Sectio 2.2. For example, the cell-averaged payoffs at expiry for stadard discrete Asia call optio are computed as follows: * 1 x h 1 x max * P e K, dx 2h x h 1, * I 2l where x l Su for S, the stock price at t ( 1) I t. Algorithm 1 shows the cell averagig tree algorithm for pricig the Europea stadard discrete Asia call optio based o method of Hsu ad Lyuu (211) method The Cotiuous Asia optio Now we cosider the case of a cotiuously moitored Asia optio. The payoff for cotiuously moitored Asia call optio with strike price K at the expiry date T is give by the followig : 157
8 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim 1 T max S ( )d -K, T (9) Let us cosider Hull ad White (1993) biomial model. As explaied above, pricig of Asia optio usig biomial tree produces 2 N possible paths for each time step N. Hull ad White (1993) proposed a biomial model for cotiuous Asia optios to solve this problem. They chose the represetative average values mh of the form Se for each ode, where h is a fixed costat, ad S S, is the kow iitial asset price. Let A i be the represetative averages, ad let mi max A i ad A i be the miimum ad maximum represetative average, respectively at time i t, i 1,2,,. The m is the smallest iteger chose to satisfy by followig iequalities : mi mh 1 mi Ai Se iai 1 dsi 1,, i 1 max mh 1 max Ai Se iai 1 usi 1, i1, i 1 See Costabile, Massabo ad Russo (26) for details. We ca also compute the kh averages of the form Se, where k m 1, m 2,, m 1 for each time step. Hull ad White model also follows the stadard lattice biomial method proposed by Cox, Ross ad Rubistei (1979) ad use the backward iductio procedure rt V ( i,, k) e pv ( i 1, 1, k ) qv ( i 1,, k ). V ( i 1, 1, k u ) is geerated by usig liear iterpolatio as follow. First, ( i 1) A us / ( i 2) i i, is computed ad let k u A be the smallest represetative average greater tha ( i 1) Ai us i, / ( i 2). The V ( i 1, 1, k u ) is the iterpolatio betwee two optio values associated to Ak 1 ad A k. The value V ( i 1, 1, k d ) is derived i a similar way. Now, let us modify Hull ad White model for improvemet by usig cell averages i Sectio 2.2. For istace, we replace the payoff i of Hull ad White model max( A K,) for the represetative averages A at the last time step with kh 1 Se max( x K,) dx kh 2 (1) Se where is fixed costat ad k represets all itegers betwee m ad m. Algorithm 2 shows the cell averagig tree algorithm for pricig the Europea cotiuous Asia call optio based o method of Hull ad White (1993). d 158
9 A Efficiet Biomial Method for Pricig Asia Optios 4. Numerical Experimets This sectio gives umerical results of cell averagig whe applied to exotic optios such as Bermuda optio or path-depedet optios icludig Discrete forward startig Asia optio ad Cotiuous Asia optio. We fid that the cellaveraged values are more accurate tha other schemes i the sese that these values fall i the iterval cotaiig the exact value faster (Bermuda optio) or coverge to a limitig value faster (path-depedet Asia optios). Sectio 4.1 shows that cell averagig ca be used idepedetly to derive a cell averagig tree method. Sectio 4.2 ad Sectio 4.3 show that cell averagig ca be easily combied with other existig tree methods as well. Sectio 4.2 shows that cell 159
10 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim averaged values are so smooth that the Richardso extrapolatio ca be used further to improve the accuracy. Sectio 4.3 shows that cell averagig ca be exteded to price eve America optios with ease Bermuda optio I this sectio, we implemet the cell averagig method to costruct a tree method to price the Bermuda call optio with the iitial stock price S 1, the risk-free iterest rate r.5, the divided q.1, the volatility.2, strike price K 1 ad the maturity 3. Whe there are m 2 exercisig poits, Aderso ad Broadie (24) provided usig a biomial method with 2 time steps a upper boud of 7.23 ad a lower boud of 7.8 for the optio price whose exact value is See Table 1. See also Aderso ad Broadie (24) for the computatioal results by Aderso ad Broadie ad the explaatios o them. The proposed cell-averaged values fell ito this boud with as low as 2 time steps. I additio, the boud of Aderso ad Broadie has the width of.15 with 2 time steps while the variatio of cellaveraged values for 4 2 is oly.3, which implies that the proposed cell averagig scheme coverges fast. Similar results are observed for the Bermuda optio with m 1 exercisig poits ad the America optio. Table 1 : Parameters: The iitial stock price S() 1, the risk-free iterest rate r.5, the divided q.1, the volatility.2, strike price K 1 ad the maturity T 3 for Bermuda call optio ad m is the umber of discrete exercisig poits. AB represets the lower ad upper bouds for the optio price by Aderso ad Broadie (24) usig a biomial lattice with 2 time steps ad f 1.6. f gives the ratio of the critical exercise price uder the suboptimal policy to the optimal critical exercise price (See Aderso ad Broadie (24) for details). Aderso ad Broadie also preseted exact values (MC-exact) i Aderso ad Broadie (24) Discrete Asia optio respectively. Sice cell averagig reduces oscillatios as explaied i Sectio 2.2, the cell averaged values ca be later improved by the Richardso extrapolatio. Table 2 shows the compariso of several umerical schemes by Večeř, Tavella ad Radall (TR), Curra, Hsu ad Lyuu (HL) with the cell 16
11 A Efficiet Biomial Method for Pricig Asia Optios averagig modificatio of HL (HL-CA) usig the Algorithm 1, ad the Richardso extrapolatio of HL-CA method ( HL-CA ) for the iitial price S 95,1,15 ad various values of the umber of moitor poits. The umber of time steps N I. See Hsu ad Lyuu (211) for detailed explaatios o those schemes icludig the parameters used for the simulatios. I Table 2, for S 95, the variatios i HL-CA method from 25 to 5 ad from 5 to 1 are oly.2 ad.1, respectively. O the other had, correspodig variatios i other methods are about.2 ad.1. Similar differeces are observed for S 1 or S 15. Thus, the applicatio of cell averagig results i sufficietly smooth covergece ad the additioal applicatio of the Richardso extrapolatio ( HL-CA ) produces very rapid covergece. The proposed method is, i this sese, very competitive with may other existig methods. Table 2 : Parameters: A discrete forward startig Asia call optio is cosidered with the strike price K 1, the time to maturity 1, the volatility.4, the iterest rate r.1, ad the divided rate is ot cosidered. Parameters for umerical schemes: itraday period I 1 ad the average umber of states per ode c 5 for Hsu ad Lyuu (211) are cosidered. The table shows the 161
12 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim compariso of various umerical schemes by Večeř, Tavella ad Radall (TR), Curra, Hsu ad Lyuu (HL) with the cell averagig modificatio of HL (HL-CA) usig the Algorithm 1, ad the Richardso extrapolatio of HL-CA method (HL- ( CA for the iitial price S 95,1,15 ad various values of the umber of moitor poits (ad the umber of time steps N I ) 4.3. Cotiuous Asia optio I this sectio, we ow apply the cell averagig method to value aother pathdepedet optio, a cotiuous Asia call optio, based o the Hull ad White (1993) method ad the exted it to value a America optio. The iitial stock price S 5, strike price K 5, risk-free iterest rate r.1, volatility.3, expiry date T 1, ad divided rate q for Europea Asia call optio. The parameters for umerical scheme by is h.1. Figure 3 shows the errors i the Hull ad White biomial method ad the cell averagig Hull ad White method as the umber of time steps icreases, whe.5 is used for the Algorithm 2. The solutio computed by 4 Mote Carlo simulatios based o 1 8 time steps ad 1 simulatio rus is used whe the error is measured, which meas that Mote Carlo value has statistic 4 error of O (1 ). The figure shows that the error from the cell averaged values decreases to zero faster. Table 3 shows the optio values from Hull ad White biomial method (HW) ad the cell averagig HW method (HW-CA) usig the Algorithm 2 with.5,.6,.7, ad.8 as the umber of time steps,, icreases. The values i parethesis are the errors. We see that the covergece of the cell averagig HW-CA values is faster tha that of HW values. Figure 3 : Parameters: the iitial stock price S 5, strike price K 5, risk-free iterest rate r.1, volatility.3, expiry date T 1, ad divided rate q for Europea Asia call optio. The parameters for umerical scheme by Hull ad White (1993) ad Algorithm 2 are h.1 ad.5. The figure shows the errors i optio values from Hull ad White 162
13 A Efficiet Biomial Method for Pricig Asia Optios biomial method ad the cell averagig HW method usig the Algorithm 2 as the umber of time steps,, icreases. Table 3: Parameters: the iitial stock price S 5, strike price K 5, riskfree iterest rate r.1, volatility.3, expiry date T 1, ad divided rate q for Europea Asia call optio. The parameters for umerical scheme by Hull ad White (1993) ad Algorithm 2 are h.1 ad.5,.6,.7,.8. The table shows the optio values from Hull ad White biomial method (HW) ad the cell averagig HW method (HW-CA) usig the Algorithm 2 as the umber of time steps,, icreases. The values i parethesis are the errors. The solutio computed by Mote Carlo simulatios (MC) based o time steps ad 1 simulatio rus is used whe the errors are measured. A simple extesio of the Algorithm 2 for a exercise of the optio results i Table 3. Table 4: Parameters: the iitial stock price S 5, strike price K 5, riskfree iterest rate r.1, volatility.3, expiry date T 1, ad divided rate q for America Asia call optio. The parameters for umerical scheme by Hull ad White (1993) ad Algorithm 2 are h.1 ad.5,.6,.7,.8. The table shows the optio values from Hull ad White biomial method (HW) ad the cell averagig HW method (HW-CA) usig the Algorithm 2 as the umber of time steps,, icreases. The cell averagig method ca be easily exteded to America optios. Table 4 cosiders a America Asia call optio with the same parameters as those for a Europea Asia call optio above usig the Hull ad White method ad the cell averagig Hull ad White model. 5. Coclusios We propose the cell averagig method for pricig the exotic optios, i particular path-depedet Asia optios. Cell averagig reduces the oscillatios of 163
14 Kyoug-Sook Moo, Yuu Jeog, Hogoog Kim the tree method ad thus improves the accuracy. It ca be used to derive a idepedet tree scheme or to be combied with existig methods. It ca be eve combied with the extrapolatio as i Sectio 4.2 to ehace the accuracy usig the fact that the correspodig result is smooth or it ca be easily exteded to value America path-depedet optios as i Sectio 4.3. Algorithms 1 ad 2 show that the itroductio of cell averagig does ot icrease computatioal loads much, while umerical experimets validate that cell averagig improves the accuracy of Hsu ad Lyuu method ad Hull ad White method pricig path-depedet Asia optios. For istace, cell averagig gives better represetative averages tha those proposed by Hull ad White. We are curretly workig o mathematical aalysis o the order of covergece of HL-CA ad HW-CA methods for path-depedet Asia optios. REFERENCES [1] Aderso, L. ad Broadie, M. (24),Primal-Dual Simulatio Algorithm for Pricig Multidimesioal America Optios. Maagemet Sciece 5, 9, ; [2] Black, F ad Sholes, M. (1973), The Pricig of Optios ad Corporate Liabilities. The Joural of Political Ecoomy 81, 3, ; [3] Clewlow, L. ad Stricklad, C. (1998),Implemetig Derivatives Models; Joh Wiley & Sos, Chichester, UK; [4] Costabile, M. Massabo, I. ad Russo, E. (26),A Adusted Biomial Model for Pricig Asia Optios. Rev Quat Fia Acc 27, ; [5] Cox, J. Ross, S. ad Rubistei, M. (1979), Optio Pricig : A Simplified Approach. Joural of Fiacial Ecoomics 7, ; [6] Forsyth, P. A., Vetzal, K. R. ad Zva, R. (22), Covergece of Numerical Methods for Valuig Path-Depedet Optios Usig Iterpolatio. Rev Derivatives Res 5, ; [7] Higham, D. J. (24), A Itroductio to Fiacial Optio Valuatio. Cambridge Uiversity Press; [8] Hsu, W. Y. ad Lyuu, Y. D. (211), Efficiet Pricig of Discrete Asia Optios. Applied Mathematics ad Computatio 217, ; [9] Hull, J. ad White, A. (1993), Efficiet Procedures for Valuig Europea ad America Path-Depedet Optios. Joural of Derivatives 1, 21-31; [1] Øksedal, B. (1998), Stochastic Differetial Equatios. Spriger, Berli; [11] Kwok, Y. K. (1998), Mathematical Models of Fiacial Derivatives. Spriger, Sigapore; [12] Lyuu,Y. D. (22), Fiacial Egieerig ad Computatio. Cambridge; [13] Merto, R. C. (1973), Theory of Ratioal Optio Pricig. Bell Joural of Ecoomics ad Maagemet Sciece 4, 1, ; [14] Moo, K. S. ad Kim, H. (213),A Multi-dimesioal Local Average Lattice Method for Multi-asset Models. Quatitative Fiace 13,
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