PRICING OF AVERAGE STRIKE ASIAN CALL OPTION USING NUMERICAL PDE METHODS. IIT Guwahati Guwahati, , Assam, INDIA

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1 Internatonal Journal of Pure and Appled Mathematcs Volume 76 No , ISSN: (prnted verson) url: PA jpam.eu PRICING OF AVERAGE STRIKE ASIAN CALL OPTION USING NUMERICAL PDE METHODS A. Kumar 1, A. Wakos 2, S.P. Chakrabarty 3 1,2,3 Department of Mathematcs IIT Guwahat Guwahat, , Assam, INDIA Abstract: In ths paper, a standard PDE for the prcng of arthmetc average strke Asan call opton s presented. A Crank-Ncolson Implct Method and a Hgher Order Compact fnte dfference scheme for ths prcng problem s derved. Both these schemes were mplemented for varous values of rsk free rate and volatlty. The opton prces for the same set of values of rsk free rate and volatlty was also computed usng Monte Carlo smulaton. The comparatve results of the two numercal PDE methods shows close match wth the Monte Carlo results, wth the Hgher Order Compact scheme exhbtng a better match. To the best of our knowledge, ths s the frst work to use the numercal PDE approach for prcng Asan call optons wth average strke. AMS Subject Classfcaton: 91G60 Key Words: Asan opton, Crank Ncolson mplct method, hgher order compact, Monte Carlo smulaton 1. Introducton Optons are one of the most common fnancal dervatves that are traded both n exchanges as well as over the counter [1, 2, 3, 4]. The two most common optons are the European and the Amercan optons. The purchase or sale of the underlyng asset for the opton takes place at the dscreton of the holder of Receved: December 8, 2011 Correspondence author c 2012 Academc Publcatons, Ltd. url:

2 710 A. Kumar, A. Wakos, S.P. Chakrabarty the opton for a prce called the exercse prce or the strke prce at a fxed tme called expraton tme (denoted by T) n case of European opton and any tme n [0,T] n case of Amercan opton. Optons can be further classfed as call or put dependng whether the holder has the rght to buy or the rght to sell the underlyng asset. In addton there are numerous other optons that can broadly be classfed as exotc optons [1, 2, 5, 6]. The specalty of these knds of optons s that the fnal payoff s more sophstcated and sometmes depends on some functon of the path of prces of underlyng asset. One of the common path dependent exotc optons s the Asan opton [1, 2, 5, 6, 7], whose payoff depends on the average hstorcal stock prces. In ths paper, we wll focus on prcng of an Asan call opton wth arthmetc average strke. We begn wth the assumpton that the stock prces S(t) follow a geometrc Brownan moton gven by the stochastc dfferental equaton [1, 2, 5, 6, 7], ds(t) = rs(t)dt+σs(t)dw(t), where r s the average rate of growth of asset prces or drft, σ s the volatlty and W(t)(0 t T) s a Brownan moton or Wener process under the rsk neutral measure P. The payoff for an Asan call opton [1, 2, 5, 6, 7], wth arthmetc average strke s gven by, V(T) = max S(T) 1 T T 0 S(u)du, 0. The prce of the opton at tme t [0,T] (wth fltraton F t ) s gven by the rsk-neutral prcng formula [7], V(t) = E (e r(t t) V(T) F t ). In order to deal wth the challenge of the payoff V(T) beng path-dependent we ntroduce a stochastc process I(t) [2, 7] gven by, I(t) = t 0 S(u)du. The stochastc dfferental equaton for evoluton of I(t) s thus gven by, di(t) = S(t)dt.

3 PRICING OF AVERAGE STRIKE ASIAN Thus the Asan call opton prce V(S,I,t) for contnuous arthmetc average strke satsfes the backward PDE [2, 6, 7], V t +rs V S σ2 S 2 2 V S 2 +S V I rv = 0. Note that the above problem s three dmensonal whch leads to greater computatonal costs. Ths motvates the reducton of the problem nto lower dmenson. [2]. For ths purpose, a new varable t 0 R(t) = 1 S(t) S(u)du s defned [2, 6]. Ths n turn motvates the ansatz V(S,I,t) = S H(R,t) for some functon H(R,t). It can be shown that the SDE satsfed by R(t) s [6], dr(t) = ( 1+(σ 2 µ)r(t) ) dt σr(t)dw(t). Also the functon H(R,t) satsfes the PDE [6], H t σ2 R 2 2 H R 2 +(1 rr) H R = 0. The soluton of ths backward PDE requres a fnal condton and two boundary condtons whch are outlned below [2, 6].. Fnal Condton : The fnal payoff for the opton gves the fnal condton, H(R(T),T) = max (1 1T ) R(T),0.. Rght Hand Boundary Condton : The rght hand boundary condton for R can be obtaned by observng that snce the ntegral R(t) s bounded, so S 0 for R. For S 0 the opton s not exercsed renderng t s value to be 0. Hence, H(R,t) = 0 for R.. Left Hand Boundary Condton : The left hand boundary condton for R 0 can be obtaned from the smlarty reducton equaton. The term R H/ R 0 as R 0. Assumng that H s bounded t follows that the term R 2 2 H/ R 2 0 as R 0. Ths leads to the boundary condton, H t + H R = 0 for R 0.

4 712 A. Kumar, A. Wakos, S.P. Chakrabarty The problem to be solved now reduces to, H t σ2 R 2 2 H R 2 +(1 rr) H R = 0. subject to, ( H(R(T),T) = max 1 R(T) ) T,0 H t + H = 0 for R 0 (1) R H = 0,for R. OncethesolutonH(R,t)sobtanedtheprceoftheAsanoptonsdetermned by, V(S(0),R(0),0) = S(0)H(R(0),0), where S(0) s the ntal stock prce. For prcng of optons (especally exotc optons) the standard method used s Monte Carlo smulaton [8, 9, 10]. Ths nvolves smulatng the paths of the underlyng asset and calculatng the opton prce based on ths path. A large number of such smulatons are run and the average of the opton prces from each smulaton s taken to be the opton prce. Several methodologes have been adopted for prcng of optons usng the numercal PDE approach. Some of the most commonly used ones are fnte dfferences of lower order [11, 12, 13, 14] and hgher order compact schemes for Amercan optons [15, 16] and opton prcng n stochastc volatlty model [17]. 2. Crank Ncolson Implct Method The problem of prcng the average strke Asan call opton essentally entals solvng for equaton (1). Whle geometrc mean Asan opton admts closed form solutons [8], the same s not true n case of arthmetc average Asan optons. As such one has to seek a soluton through numercal methods for PDEs [12]. There are several artcles n lterature whch dwell upon numercal PDE approach to Asan opton prcng. One of the frst papers to deal wth numercal PDE prcng of optons s by Rogers and Sh [11]. In ths paper, the authors frst reduce the problem of solvng a parabolc PDE n two varables and present a hghly accurate lower bound. Zvan et al. [13] n ther techncal report, do an extensve analyss of numercal PDE methods of Asan optons. They dscuss the shortcomngs of applyng the usual numercal PDE technques

5 PRICING OF AVERAGE STRIKE ASIAN used for standard optons n case of Asan optons. In partcular they adapt flux lmtng technques from computatonal flud dynamcs (CFD) to tackle the problem of spurous oscllatons that arse n Asan optons. Vecer [12] provded a numercal mplementaton of the Asan opton prcng problem usng the θ method. Dubos and Lelevre [14] extend the approach by Rogers and Sh [11] and propose a scheme whch produced fast and accurate results. Whle all these papers [11, 12, 13, 14] do examne the prcng problem from the numercal PDE pont of vew, the focus s mostly on fxed strke optons. Rogers and Sh [11] and Zvan et al. [13] present some results on average strke put optons. Our man objectve n ths paper wll be to use Hgher Order Compact (HOC) scheme for ths purpose. We wll postpone the dscusson on ths untl the next secton. In ths secton we wll present the Crank-Ncolson Implct Method (CNIM) for solvng equaton (1) and compare the results wth those obtaned by Monte Carlo smulaton [18]. CNIM s obtaned by takng the average between Forward-Tme Centered- Space method (FTCS) and Backward-Tme Centered-Space method (BTCS). For ths purpose, let us defne a fnte dfference dscretzaton of the PDE (equaton(1)) wth the unform grd t n = t 1 +(n 1) t,n = 1 : N +1 and R = R 1 +( 1) R, = 1 : M +1, where t and R are the temporal and spatal mesh sze respectvely. The values used n ths paper are t 1 = 0 and t N+1 = T = 1 (.e 1 year opton) wth R 1 = 0 and R M+1 = 5. Let us also defne the varables c(r) = 1 2 σ2 R 2 and d(r) = (1 rr). H(R,t) at the pont (R,t n ) s denoted by H n. The CNIM dscretzaton of equaton (1) s then gven by, [ H n+1 H n + c H n+1 ( +1 +Hn +1 H n+1 +H n ) t 2 2 (2) R 2 2 ] [ ] + Hn+1 1 +Hn 1 + d H n Hn +1 Hn+1 1 +Hn 1 = R 2 2 The above can now be rewrtten as, where, G H n +1 +K H n +J H n 1 = D H n E H n+1 +F H n+1 1 (3) G = c K = 2 2 R d 4 R c 2 R + 1 t J = c d R 4 R D = E = F = c 2 2 R + d 4 R c R t c 2 2 R d 4 R.

6 714 A. Kumar, A. Wakos, S.P. Chakrabarty Let us defne the vector, H (n) = (H n 2,H n 3,H n 4,H n 5,...,H n M) for n = 1 : N +1. The CNIM can now be wrtten n the matrx form as, where, and BH (n) = AH (n+1) +b (n), (4) E 2 D F 3 E 3 D.. 3. A = DM F M E M K 2 G J 3 K 3 G.. 3. B = GM J M K M F 2 H1 n+1 J 2 H n 1 b (n) = D M HM+1 n+1 G MHM+1 n From second order fnte dfferences, the one-sded dfference s gven by, H R = H +2 +4H +1 3H +O( 2 R 2 ). R Therefore, applyng the same on the left boundary along wth backward tme approxmaton we get, H n 1 = (1 3k)H n kH n+1 2 kh n+1 3, where k = t 2 R. The rght boundary HM+1 n = 0 follows from equaton (1). The fnal condton s gven by, ( H N+1 = 1 R ) +. T

7 PRICING OF AVERAGE STRIKE ASIAN The CNIM formulaton above was solved usng MatLab TM. The soluton was obtaned usng an teratve process whch nvolved a tolerance crteron of H new (1,1) H old (1,1) < ǫ. For the purpose of ths mplementaton the number of tme and space grd ponts were taken to be 101 and 501 respectvely. The tolerance level was taken to be ǫ = Hgher Order Compact Scheme Fnte dfference methods have been used for solvng ODE and PDE problems for qute a long tme. They are relatvely easer to set up and solve, but requre structured mesh [19, 20]. Fnte element methods on the contrary are more sophstcated, works well wth rregular domans and are amenable to unstructured meshes. Fnte element methods are relatvely more challengng to mplement. Standard fnte dfference schemes lke the CNIM (used n the prevous secton) are second order accurate. However, n fnancal applcatons lke opton prcng a hgher level of accuracy s desrable. A drect extenson of the central dfference schemes to acheve hgher order accuracy would nvolve more node ponts on the stencl. An nnovatve way of achevng hgher accuracy wth lesser number of nodes n the stencl came by the way of hgher order compact (HOC) schemes. Spotz and Carey [19] and Spotz [20] provde an excellent dscourse on applcaton of ths scheme n case of vscous flow and computatonal mechancs. Despte t s enormous potental of applcaton to fnance problems, HOC schemes have not been used much n ths area. Zhao et al. [21] presented a compact scheme for Amercan opton prcng wth second order accuracy n space. Tangman et al. [15, 16] appled a HOC scheme for the prcng of Amercan put opton. They do a comparatve [15] analyss wth a non-compact fourth order scheme. In ther subsequent paper they descrbe an mprovement of a method suggested by Han and Wu [22]. In ths chapter we shall apply HOC scheme to the setup (1) whch s wrtten n the followng form [18], c(r) 2 H R 2 +d(r) H R = g (5) where c(r) and d(r) are as defned n the prevous secton and g = H t. We now defne notaton δ [19, 20, 23] as follows, δ R f := f R, δ2 R f := 2 f R 2 and so on.

8 716 A. Kumar, A. Wakos, S.P. Chakrabarty Thus rewrtng Equaton (5) n terms of fnte dfference dscretzaton we get, c δ 2 RH +d δ R H = g (6) In the HOC scheme we derve the leadng truncaton error terms n terms of fnte dfference equaton makng use of the orgnal equaton. Denotng these terms by τ, the HOC scheme s obtaned by subtractng τ back to the orgnal fnte dfference dscretzaton. Thus we have, where, τ = 2 R 12 From the ntal PDE (1) we get, c δ 2 R H +d δ R H τ = g (7) ( ) c 4 H H R 4 +2d 3 R 3 +O( 4 R) (8) c 2 H H R R 2 +c 3 R 3 + d H H R R +d 2 R 2 = g R Therefore ths can be rewrtten n δ notaton as, 3 H R 3 = 1 [ δ R c δ c RH 2 +δ R d δ R H +d δrh 2 ]+ 1 δ R g (10) c Dfferentatng the PDE (9) w.r.t. R once more and smplfyng we get, 2 c 2 H c 3 H H R 2 +2 R2 R R 3 +c 4 R d H R 2 R +2 d 2 H H R R 2 +d 3 R 3 = 2 g R 2 (11) whch can be wrtten n δ-notaton as, 4 H c R 4 = δr 2 g +δr 2 c δr 2 H +δr 2 d δ R H +2δ R d δr 2 H + ( δ R g δ R c δrh 2 δ R d δ R H d δrh 2 ) 2δ R c (12) c + ( δ R g δ R c δr 2 H δ R d δ R H d δr 2 H ) d c Makng use of approxmatons n equaton (10) and equaton (12) n the truncaton error (τ ) (equaton (8)) and hence substtutng τ n equaton (7) we obtan, [ δr 2 H c + 2 R d 2 +σ 2 2 R R d + 2 R 12 c 12c 12 σ2 r 2 R 2 Rσ 4 R 2 6 6c (9)

9 PRICING OF AVERAGE STRIKE ASIAN = 2 R [ ]+δ R H d r 2 R d + r 2 R 12c 6 d σ 2 R 6c [ 1+ 2 R 12 δ2 R + 2 R d δ R 2 R(σ 2 R )δ R 12 c 6c ] g σ 2 ] R c (13) Note that, c = 1 2 σ2 R 2 δ Rc = σ 2 R,δ 2 R c = σ 2 and d = 1 rr δ R d = r,δ 2 R d = 0. We defne, A = F1 = R 24 [ c + 2 R d σ 2 2 R R d + 2 R c 12c 12 σ2 r 2 R 6 ] [,B = d r 2 R d + r 2 R 12c 6 d R (σ 2 R ), k1 = t c 12c 2 2 R 2 R 6c d σ 2 R 2 R σ 2 R c, k2 = t 4 R. σ 4 R 2 6c ] We now apply the HOC scheme to the above equaton (recallng that g = H t ) and obtan, k1 ( H n Hn+1 +H n+1 1 +Hn +1 2H n +H n 1) A + k2 ( H n+1 +1 Hn+1 = ( H n R 12 +H n 1 +Hn +1 H n 1) B ) 2 R 12 [ H n +1 2H n +H 1 n ] 2 R F1 [ H n+1 +1 Hn+1 1 [ ] H n Hn+1 +H 1 n+1 2 R ] +F1 [ H n +1 H n 1] (14) The above can now be rewrtten as, where, G H n +1 +K H n +J H n 1 = D H n E H n+1 +F H n+1 1 (15) G = k1a k2b F1, K = 2k1A + 5 6, J = k1a +k2b F1, D = k1a +k2b F1 E = 2k1A F = k1a k2b F1.

10 718 A. Kumar, A. Wakos, S.P. Chakrabarty The mplct method can be wrtten n the matrx form, BH (n) = AH (n+1) +b (n), where H (n),a,b and b (n) has already been defned n the prevous secton. As wth the case of CNIM the number of tme and space grd ponts were taken to be 101 and 501 respectvely along wth the tolerance level of ǫ = The scheme was mplemented usng MatLab TM. 4. Results and Dscusson In ths secton we dscuss the results obtaned by usng the CNIM and the HOC schemes as outlned n the prevous two sectons. As already noted we could not fnd any results for average strke Asan call opton usng numercal PDE methods. For the purpose of comparson we used the Monte Carlo (MC) smulaton as the benchmark value. We generated the path of a stock prces usng the geometrc Brownan moton process. We generated such paths and determned the opton prce from each of the paths generated. The average of all these opton prces was taken to be the opton prce, for the purpose of comparson wth the PDE methods. We generated the opton prces usng all the three methods for three values of r = 0.06,0.1,0.2 and fve values of σ = 0.05,0.1,0.2,0.3,0.4. Acomparatve studyofresultsfromthecnimandthemcmethodsshowed a close match. The comparatve results are presented n Table 1 along wth the CPU tme n seconds. The opton prces for the three values of r aganst the fve values of σ have been presented n the graphcal form n Fgures (1), (2) and (3). For r = 0.06 (Fgure (1)), the match was very close except the case where σ = Ths slght dfference n the opton prce s reduced when r = 0.1 (Fgure (2)). The other values for r = 0.1 showed a close match. The results were smlar for the case r = 0.2 except for a very mnmal dfference n the case of σ = 0.1 (Fgure (3)). The CPU tme n case of CNIM was however consderably lower (< 0.5 seconds) as compared wth the Monte Carlo smulaton (> 4 seconds). The results and comparson of the HOC scheme and the MC method ndcates excellent consonance. A comparson of the results from these two methods n terms of values and CPU tme n seconds have been presented n a tabular form n Table 2. The opton prces from both the methods are very close to each other. In fact the results obtaned from the HOC scheme show a better match wth the MC smulaton results as compared wth the CNIM method.

11 PRICING OF AVERAGE STRIKE ASIAN r σ CNIM MC CNIM MC CNIM MC (0.3753) (4.5115) (0.3657) (4.4923) (0.3444) (4.5568) (0.1176) (4.5051) (0.1183) (4.4897) (0.1177) (4.5305) (0.1177) (4.4900) (0.1172) (4.4820) (0.1173) (4.5307) (0.1176) (4.4908) (0.1178) (4.5187) (0.1189) (4.5000) (0.1239) (4.4833) (0.1177) (4.4837) (0.1162) (4.4900) Table 1: Comparson between Monte Carlo Smulaton (MC) and Crank Ncolson Implct Method (CNIM). The values n braces represent the CPU tme n seconds. The ntal stock prce was S(0) = Comparson between Crank Ncolson and Monte Carlo for r= Opton Prce σ Crank Ncolson Monte Carlo Fgure 1: Comparson between Monte Carlo Smulaton (MC) and Crank Ncolson Implct Method (CNIM) for r=0.06 Ths holds for all the values of r and σ and s evdent from the comparatve fgures (Fgures (4), (5), (6)) of HOC and MC. As was the case wth CNIM, the CPU tme taken n case of the HOC scheme s sgnfcantly less (< 0.5 seconds) n contrast to Monte Carlo smulaton whch requres at least 4 seconds.

12 720 A. Kumar, A. Wakos, S.P. Chakrabarty 12 Comparson between Crank Ncolson and Monte Carlo for r= Opton Prce σ Crank Ncolson Monte Carlo Fgure 2: Comparson between Monte Carlo Smulaton (MC) and Crank Ncolson Implct Method (CNIM) for r= Comparson between Crank Ncolson and Monte Carlo for r= Opton Prce σ Crank Ncolson Monte Carlo Fgure 3: Comparson between Monte Carlo Smulaton (MC) and Crank Ncolson Implct Method (CNIM) for r= Concluson In ths paper we examned several ways of computng the prce of an average strke Asan call opton, namely Monte Carlo smulaton and the numercal PDE approach. In case of opton prcng, the benchmark generally used s Monte

13 PRICING OF AVERAGE STRIKE ASIAN r σ HOC MC HOC MC HOC MC (0.3933) (4.5115) (0.3855) (4.4923) (0.3676) (4.5568) (0.1275) (4.5051) (0.1301) (4.4897) (0.1251) (4.5305) (0.1214) (4.4900) (0.1219) (4.4820) (0.1256) (4.5307) (0.1233) (4.4908) (0.1251) (4.5187) (0.1255) (4.5000) (0.1252) (4.4833) (0.1209) (4.4837) (0.1201) (4.4900) Table 2: Comparson between Monte Carlo Smulaton (MC) and Hgher Order Compact Scheme (HOC). The values n braces represent the CPU tme n seconds. The ntal stock prce was S(0) = 100. Comparson between Hgher Order Compact and Monte Carlo for r= Opton Prce σ Hgher Order Compact Monte Carlo Fgure 4: Comparson between Monte Carlo Smulaton (MC) and Hgher Order Compact Scheme (HOC) for r=0.06 Carlo smulaton whch suffers from some severe drawbacks lke computatonal costs and a certan amount of uncertanty of prcng. In contrast, the usage of numercal PDE approaches that we have taken results n lesser computatonal costs and also provdes an unque answer. The numercal PDE approach n prcng the average strke Asan call opton s by and large an unexplored area,

14 722 A. Kumar, A. Wakos, S.P. Chakrabarty Comparson between Hgher Order Compact and Monte Carlo for r= Opton Prce σ Hgher Order Compact Monte Carlo Fgure 5: Comparson between Monte Carlo Smulaton (MC) and Hgher Order Compact Scheme (HOC) for r=0.1 Comparson between Hgher Order Compact and Monte Carlo for r= Opton Prce σ Hgher Order Compact Monte Carlo Fgure 6: Comparson between Monte Carlo Smulaton (MC) and Hgher Order Compact Scheme (HOC) for r=0.2 snce ths approach appled to Asan opton s mostly concentrated on the case of fxedstrke. Inthspaper,wetakethePDEapproachtotheprcngproblemand present two schemes to accomplsh ths numercally. Frstly, we use the Crank- Ncolson Implct Method (whch s second order) to solve the PDE and hence prce the opton. Then, we present a Hgher Order Compact scheme (fourth

15 PRICING OF AVERAGE STRIKE ASIAN order) to solve the problem. Fnally we make a comparson of results obtaned from the PDE approach wth that of Monte Carlo. The results obtaned usng the two PDE technques were n excellent agreement wth the Monte Carlo results. The results obtaned usng Hgher Order Scheme are closer to the Monte Carlo results as opposed to Crank-Ncolson Implct method vs-a-vs Monte Carlo. Ths s more so n case of lower values of σ. To the best of our knowledge, ths s the frst work to use the numercal PDE approach for prcng Asan call optons wth average strke. We beleve ths work would fnd more applcatons n the area of opton prcng through the PDE approach. 6. Acknowledgments The authors express ther deep grattude to Dr. Jten C. Kalta and Prof. Anoop K. Dass for ther valuable gudance and suggestons durng the preparaton of the manuscrpt. References [1] J.C. Hull, Optons, Futures and Other Dervatves, Prentce Hall of Inda (2006). [2] P. Wlmott, S. Howson, J. Dewynne, The Mathematcs of Fnancal Dervatves, Cambrdge Unversty Press (1995). [3] S. Roman, Introducton to the Mathematcs of Fnance: From Rsk Management to Optons Prcng, Sprnger (2004). [4] M. Capnsk, T. Zastawnak, Mathematcs for Fnance: An Introducton to Fnancal Engneerng, Sprnger (2003). [5] D.J. Hgham, An Introducton to Fnancal Opton Valuaton: Mathematcs, Stochastcs and Computaton, Cambrdge Unversty Press (2004). [6] R. Seydel, Tools for Computatonal Fnance, Sprnger (2006). [7] S. Shreve, Stochastc Calculus for Fnance, Volume II: Contnuous-Tme Models, Sprnger-Verlag (2004). [8] P. Glasserman, Monte Carlo Methods n Fnancal Engneerng, Sprnger (2003).

16 724 A. Kumar, A. Wakos, S.P. Chakrabarty [9] P.P. Boyle, Optons: A Monte Carlo approach, Journal of Fnancal Economcs, 4, No. 3 (1977), [10] M. Broade, P. Glasserman, Estmatng securty prce dervatves usng smulaton, Management Scence, 42, 2 (1996), [11] L.C.G. Rogers, Z. Sh, The value of an Asan opton, Journal of Appled Probablty, 32, No. 4 (1995), [12] J. Vecer, A new PDE approach for prcng arthmetc average Asan optons, Journal of Computatonal Fnance, 4, No. 4 (2001), [13] R. Zvan, P. A. Forsyth, K. Vetzal, Robust Numercal Methods for PDE Models of Asan Optons, Techncal Report, Cherton School of Computer Scence, Unversty of Waterloo (1996). [14] F. Dubos, T. Lelevr, Effcent prcng of Asan optons by the PDE approach, Journal of Computatonal Fnance, 8, No. 2 (Wnter 2004/05), [15] D.Y. Tangman, A. Gopaul, M. Bhuruth, Numercal prcng of optons usng hgh-order compact fnte dfference schemes, Journal of Computatonal and Appled Mathematcs, 218, No. 2 (2008), [16] D. Y. Tangman, A. Gopaul, M. Bhuruth, A fast hgh-order fnte dfference algorthm for prcng Amercan optons, Journal of Computatonal and Appled Mathematcs, 222, No. 1 (2008), [17] B. Durng, M. Fourne, Hgh-order compact fnte dfference scheme for opton prcng n stochastc volatlty models, SSRN , d= [18] A. Kumar, A. Wakos, Prcng of Asan optons usng hgher order compact scheme, B. Tech. project, Department of Mathematcs, Indan Insttute of Technology Guwahat (2011). [19] W.F. Spotz, G. F. Carey, Hgh-order compact fnte dfference methods wth applcatons to vscous flows, Techncal Report, Texas Insttute for Computatonal and Appled Mathematcs (1994). [20] W.F. Spotz, Hgh Order Compact Fnte Dfference Schemes for Computatonal Mechancs, Ph.D. Thess, Unversty of Texas at Austn (1995).

17 PRICING OF AVERAGE STRIKE ASIAN [21] J. Zhao, M. Davson, R. M. Corless, Compact fnte dfference method for Amercan opton prcng, Journal of Computatonal and Appled Mathematcs, 206, No. 1 (2007), [22] H. Han, X. Wu, A fast numercal method for the Black-Scholes equaton of Amercan optons, SIAM Journal of Numercal Analyss, 41, No. 6 (2003), [23] J.C. Kalta, D.C. Dalal, A.K. Dass, A class of hgher order compact schemes for the unsteady two-dmensonal convecton-dffuson equaton wth varable convecton coeffcents, Internatonal Journal for Numercal Methods n Fluds, 38, No. 12 (2002),

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