Finite Volume Schemes for Solving Nonlinear Partial Differential Equations in Financial Mathematics

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1 Fnte Volume Schemes for Solvng Nonlnear Partal Dfferental Equatons n Fnancal Mathematcs Pavol Kútk and Karol Mkula Abstract In order to estmate a far value of fnancal dervatves, varous generalzatons of the classcal lnear Black-Scholes parabolc equaton have been made by adjustng the constant volatlty to be a functon of the opton prce tself. We present a second order numercal scheme, based on the fnte volume method dscretzaton, for solvng the so-called Gamma equaton of the Rsk Adjusted Prcng Methodology RAPM model. Our new approach s based on combnaton of the fully mplct and explct schemes where we solve the system of nonlnear equatons by teratve applcaton of the sem-mplct approach. Presented numercal experments show ts second order accuracy for the RAPM model as well as for the test wth exact Barenblatt soluton of the porous-medum equaton whch has a smlar character as the Gamma equaton. 1 Motvaton from Fnancal Mathematcs Black-Scholes lnear model. Modelng fnancal dervatve prces by PDEs has been ntroduced n 1973, when a smple lnear model was derved by Black and Scholes [4] and ndependently by Merton [10]. Its smplcty s obtaned by mposng a couple of lmtng assumptons [8] whch n realty do not always hold. Nevertheless, t s stll consdered as the cornerstone when dervng more general ones. To obtan the governng PDE, Black and Scholes assumed that the underlyng asset S follows a geometrc Brownan moton ds = µ qsdt + ˆσSdW, where µ > 0 s a constant drft, ˆσ > 0 s a constant volatlty parameter of the underlyng asset, q s a constant asset dvdend yeld rate and W s a standard Wener process. Denotng the prce of an opton as V S,t and applyng Ito s lemma to obtan the stochastc dfferental dv, they derved a parabolc partal dfferental equaton for valuaton of optons [14]: V t + ˆσ V V S + r qs rv = 0, 1 S S where r represents the rskless nterest rate. To complete the formulaton of the opton prcng model, we need to prescrbe a termnal pay-off condton at expraton tme T. In the case of an European call opton the termnal pay-off condton s Pavol Kútk, Karol Mkula Department of Mathematcs, Radlnského 11, , Bratslava, Slovak Unversty of Technology, e-mal: kutk@math.sk, mkula@math.sk 1

2 Pavol Kútk and Karol Mkula V S,T = maxs E,0, where E denotes the strke prce. For plan vanlla optons, lke the smple call or smple put opton, an exact soluton to 1- s known [14]. Nonlnear extensons. Note that n equaton 1 the volatlty s constant. However, f we nsert real data nto the model and compute nversely the mpled volatlty, t s not constant [8]. More generally, the volatlty parameter can be defned by σ = σ S V,S,T t, where S V s the so-called Γ of an opton. In fnancal theory and practce varous nonlnear generalzatons of Black-Scholes lnear model exst wth such defned volatlty functon. For nstance, Leland n [9] proposed a model whch takes transacton costs nto account. In order to descrbe opton prcng n ncomplete markets Avellaneda, Levy and Paras n [1] used a jumpng volatlty functon. Barles and Soner n ther model [3] adjusted the volatlty dependng on nvestor s preferences. Illqud market effects due to large traders choosng gven stock-tradng strateges were studed by Frey n [5] and by Schönbucher and Wlmott n [1]. A further nterestng nonlnear model whch we deal wth n ths paper s the so-called Rsk Adjusted Prcng Methodology RAPM model derved by Kratka n [7] and further generalzed by Jandačka and Ševčovč n [6]. Notce that the numercal scheme presented n the next secton can be appled to all the above mentoned models snce they can be represented by a PDE n the general form 6. Interestngly, the nonlnear porous-medum equaton 14 whch we deal wth n the last secton s also a specal case of the Gamma equaton 6. The RAPM model omts the lmtng assumpton of havng no transacton costs. Hence, t assumes that the portfolo s rehedged only at dscrete tmes, snce contnuous rehedgng would lead to nfnte transacton costs. The more often the portfolo s beng rehedged, the hgher the rsk assocated wth transacton costs become. On the other hand, seldom rehedgng mples hgher rsk arsng from ts weak protecton aganst the movement of the underlyng assets s prce. Hence, there exsts an optmal tme step, representng the hedge nterval, for whch the sum of both rsks s mnmal. Usng such deas, the governng PDE for the RAPM model n the followng form s obtaned [6]: V t + 1 ˆσ S 1 + µ S 1 V 3 S V V + r qs rv = 0, 3 S S where µ = 3 C 1 R 3 π, C 0 represents the relatve transacton costs for buyng or sellng on stock and R 0 s the margnal value of nvestor s exposure to rsk. Note that the dffuson coeffcent n 3 s dependent on Γ, thus the equaton s a nonlnear PDE. Snce 1 + µ SΓ 1 3 1, the opton prce computed by ths equaton s slghtly above that from the lnear Black-Scholes model,.e. we obtan a so-called Ask prce. If the dffuson coeffcent s of the form 1 µ SΓ 1 3 1, then we get the lower Bd prce of an opton. Let us note that we can rewrte the equaton 3 as

3 Fnte Volume Schemes n Fnancal Mathematcs 3 V V + SβSΓ + r qs rv = 0, 4 t S where βh = 1 1 ˆσ + µh 1 3 H. Snce the equaton 4 contans the term SΓ t s convenent to ntroduce the followng transformaton: Hx,τ = SΓ = S SVS,t, 5 where the new varables, x and τ, are obtaned by transformng the orgnal ones usng the standard substtuton: x = ln S E, x R and τ = T t, τ 0,T. If we take the second dervatve of equaton 4 wth respect to x t turns out that the functon Hx, τ s a soluton to the followng nonlnear parabolc dfferental equaton, the so-called Gamma equaton [13]: Hx,τ τ = βh + βh + r q Hx,τ qhx,τ. 6 Notce that unlke n equaton 3, all terms contanng spatal dervatves n the Gamma equaton 6 are n dvergent form, thus t s sutable to use fnte volume method dscretzaton whch follows. Concernng the boundary condtons, snce the second dervatve of V S,t wth respect to S tends asymptotcally to zero as S 0, respectvely S, from 5 t follows that the transformed Drchlet boundary condtons are H,τ = H,τ = 0. Fnte Volume Approxmaton Schemes The most general form of the Gamma equaton s as follows: Hx,τ τ Notce that = βh,x,τ βhx,τ,x,τ + βh,x,τ = + f x Hx,τ + gxhx,τ, 7 β HH,x,τ Hx,τ + β xh,x,τ, 8 where β H H,x,τ and β xh,x,τ are partal dervatves of the functon βhx,τ,x,τ by H and x, respectvely. Moreover, f x Hx,τ = f xhx,τ Hx,τ f xx. 9 Insertng 8 and 9 nto 7 and ntegratng over the fnte volume wth center pont denoted by x, we get x 1,x + 1,

4 4 Pavol Kútk and Karol Mkula x+ 1 H x+ x 1 τ dx = 1 β H H x 1 + β x + β + f xh dx x+ 1 + gx f x x H dx. 10 x 1 Usng central spatal dfferences, Newton-Lebnz formula, the md-pont rule and notatons β + 1 = βh + 1,x + 1,τ, β x + 1 = β xh + 1,x + 1,τ, β H + 1 = β H H + 1,x + 1,τ, we obtan the followng general numercal scheme for solvng 7: h H j+1 H j k = β H H H h β H H 1 H 1 β 1 + f H x +1 +H + 1 f H x +H hh h + β x + 1 β x 1 + β + 1 gx f xx, 11 where H j represents the approxmate value of the soluton n pont x at tme τ j and { j, j + 1} represents the chosen tme layer. Dependng on n whch tme we evaluate the terms on the rght-hand sde n 11 we obtan three dstnct frst-order schemes. Explct scheme s obtaned by takng all terms from the old tme layer,.e. = j. Sem-mplct scheme s obtaned by takng all lnear terms from the old tme layer,.e. = j, and all nonlnear terms from the new tme layer,.e. = j + 1. The soluton s found by solvng a trdagonal system of lnear equatons by the Thomas algorthm. Fully-mplct scheme s obtaned f all terms are taken from the new tme layer,.e. = j + 1. We get a system of nonlnear equatons. The algorthm for solvng such a system s based on teratve soluton of the sem-mplct scheme. We start the teratve process by assgnng the old tme step soluton vector to the startng teraton soluton vector for the new tme step. Then, n each teraton, we nsert the soluton vector nto the nonlnear terms, to get ther actual teraton. If we collect all unknowns from the soluton vector,.e. the lnear terms from the new layer, on the left-hand sde and all remanng terms,.e. the nonlnear terms and the lnear term from the old layer, on the rght-hand sde we obtan a lnear trdagonal system for determnng next teraton of the soluton vector. The whole process s termnated when the successve soluton vectors are close enough []. New second order scheme s of the Crank-Ncolson type and s obtaned by the arthmetc average of the explct and the fully-mplct scheme. The system of nonlnear equatons has a smlar structure to that from the fully-mplct scheme, thus we solve t usng the same prncples. As notced above, the lnear systems arsng n our schemes are solved by the Thomas algorthm. Its numercal stablty s guaranteed by the strct dagonal dom-

5 Fnte Volume Schemes n Fnancal Mathematcs 5 nance of the system matrx whch can be always acheved by a sutable choce of tme step k n 11. Another mportant ssue s the study of stablty whch s usually related to the approxmaton of dffuson and advecton terms. Inspectng the Gamma equaton 6, one can see that the dffuson coeffcent s gven by β H whle the speed of the advecton s proportonal to β H +r q and thus n the studed applcaton they are comparable ˆσ r q. The fully explct scheme gves oscllatons for the couplng k h due to volatng the CFL condton n approxmaton of the dffuson term. On the other hand, all other schemes are mplct and we dd not observe any oscllatons, manly due to the fact that the advecton does not domnate the dffuson. 3 Numercal Experments Three dfferent numercal experments were made. The frst two are concerned wth the approxmate soluton to the RAPM Gamma equaton and the last one deals wth the numercal soluton to a nonlnear porous-medum PDE. RAPM Gamma equaton experments. As no comparatve exact soluton to such an equaton s known, a natural choce s to take the exact soluton of the lnear Black-Scholes model. Clearly, to mantan the equalty n the Gamma equaton we have to add a resdual term Resx,τ nto 6 whch balances the dfference between the Black-Scholes soluton and the hgher Ask prce of the RAPM model: H τ = β + β + r q H qh + Res, 1 where βh = ˆσ 1 + µh 1 3 H. The frst two experments dffer from each other n two man aspects: the coeffcent µ and the ntal condton. Followng parameters were set for both cases the same: ˆσ = 0.30, r = 0.03, q = 0.01, E = 5. In all numercal experments we mpose boundary condtons Hx L,τ = Hx R,τ = 0, where x L and x R are boundares of the space nterval. The ntenton of the frst experment s to show how well the proposed numercal schemes can handle the nonlnearty n the Gamma equaton 1. We put the coeffcent µ = 0., hence the functon βh s nonlnear. As the ntal condton Hx,τ 0 we consder Black-Scholes soluton V S,T τ 0 transformed by 5, n tme τ 0 = 1. Measurements of the estmated error e m n L are done by comparson wth the exact soluton Hx,τ to 1 for τ > τ 0. Snce all frst-order schemes exhbted very smlar features, we show here outputs just for the sem-mplct scheme. The reason for excluson of the explct scheme was ts nstablty usng couplng k = h. Regardng the fully-mplct scheme, experments show that the accuracy of the sem-mplct scheme s very close to the fully-mplct scheme, thus t s suffcent to use just the former one whch s less tme consumng. The experment was done on the tmespace doman x,τ = [,] [1,]. Tables 1 and ndcate that for ths type of problem the sem-mplct scheme s frst order accurate whle the Crank-Ncolson type scheme s second order accurate.

6 6 Pavol Kútk and Karol Mkula Table 1 Outputs obtaned by solvng the RAPM Gamma equaton 1 τ 0 = 1, k = h usng the sem-mplct scheme: estmated error e m n L, CPU-tme and EOC wth respect to e m n L. n h e m n L CPU EOC k h Table Outputs obtaned by solvng the RAPM Gamma equaton 1 τ 0 = 1, k = h usng the Crank-Ncolson type scheme: estmated error e m n L, CPU-tme and EOC wth respect to e m n L. n h e m n L CPU EOC k h In the the second experment we set µ = 0 and we show how the regularzaton of the transformed ntal condton and the backward transformaton of the Gamma equaton soluton affects the total accuracy of the method. In ths case the soluton of the Gamma equaton concdes wth the transformed soluton Hx, τ of the lnear Black-Scholes equaton 1 whch mples that the resdual term n 1 s zero. The ntal condton Hx,τ 0 s consdered for τ 0 = 0. Hence the transformed payoff functon, see and 5, s the Drac delta functon, Hx,0 = δx, x R. In order to get a sutable ntal condton for our computaton, we consder ts regularzaton gven by the functon Hx,0 = N d ˆσ, where τ τ > 0 s suffcently small, Nd s the cumulatve dstrbuton functon of the normal dstrbuton and d = x+r q ˆσ /τ ˆσ τ [13]. The backward transformaton of numercal soluton s done by usng formula V S k,t τ j = h n = n maxs k Ee x,0h j = h k = n S k Ee x H j k = hs k H j k he e x H j = hs k F k he G k, 13 = n = n where F k = F k 1 + H j k, G k = G k 1 + e x kh j k and S k = Ee x k. Formula 13 s obtaned by ntegraton of 5. Measurements of the estmated error e m n L are done by comparson wth the Black-Scholes soluton V S, t. However, n practce, dong computatons wth such an ntal condton s not as straghtforward task as n the frst experment. The problem s that we do not know a pror the optmal value of τ for a gven tme-space mesh. We consder the optmal value of τ as a value for whch the estmated error of the numercal soluton s mnmzed. Numercal outputs for the dscretzed tme-space doman x, τ = [, ] [0, 1] are summarzed

7 Fnte Volume Schemes n Fnancal Mathematcs 7 H Opton prce V Fg. 1 A comparson of Bd and Ask opton prces computed by means of the RAPM model for a call opton n tme T t = 1. Left rght fgure presents the results before after the backward transformaton. The blue dashed fne-dashed curve ndcates the Ask Bd prce of a call opton. Green curve represents the opton prces computed by the lnear Black-Scholes model and the red lne s the payoff functon. Parameters: n = 80, h = 0.05, m = 160, k = , τ = , ˆσ = 0.30, µ = ±0., r = 0.011, q = 0.0, X = 5. n the table 3. Snce the total error s nfluenced not only by the dscretzaton error, but also by the error related to the regularzaton and backward transformaton, the Crank-Ncolson method exhbts EOC slghtly below the second order. Fnally, n fgure 1 we present the numercal soluton of the RAPM model for a call opton usng parameter τ obtaned by the above descrbed strategy but consderng nonzero µ. Such an experment s of partcular nterest also for practcal applcatons. Table 3 Outputs obtaned by solvng numercally Gamma equaton 1 τ 0 = 0, k = h/4 usng the Crank-Ncolson type scheme and usng formula 13 for backward transformaton. n h τ e m n L EOC k h CPU Gamma CPU Total Experment wth an exact Barenblatt soluton. The goal of the thrd experment was to nvestgate the accuracy of the proposed Crank-Ncolson type scheme usng exact soluton of the followng porous-medum type equaton: vx,t = 1 ωt max [0,1 t v = x v ω, x R, t > 0, ω > 1 14 x ωt CPU Transform whch s a specal case of the Gamma equaton 6. The exact soluton has the form ] 1 ω 1 [ ] 1, where λt = ωω+1 ω+1 ω 1 t + 1 represents a sharp nterface of the soluton s fnte support. EOC of the Crank-Ncolson type scheme n L 1 -norm, whch s used due to the sngularty n the exact soluton, s equal to, see table 4.

8 8 Pavol Kútk and Karol Mkula Table 4 Numercal approxmaton of the Barenblatt exact soluton usng Crank-Ncolson type scheme. n h e m n L1 CPU EOC k h Conclusons In ths paper we proposed a new nonlnear second order Crank-Ncolson type numercal scheme based on the fnte volume method. Our man goal was to provde an effcent and precse numercal soluton to nonlnear PDEs arsng n fnancal mathematcs. Varous experments have shown such propertes of the new scheme. Acknowledgement. Ths work was supported by grants APVV and VEGA 1/0733/10. References 1. Avellaneda, M., Levy, A., and Paras, A.: Prcng and hedgng dervatve securtes n markets wth uncertan volatltes. Appled Mathematcal Fnance, Balažovjech M., Mkula K.: A Hgher Order Scheme for the Curve Shortenng Flow of Plane Curves. Proceedngs of ALGORITMY, STU Bratslava, Barles, G., and Soner, H. M.: Opton prcng wth transacton costs and a nonlnear Black- Scholes equaton. Fnance Stoch., Black, F., and Scholes, M.: The prcng of optons and corporate labltes. The Journal of Poltcal Economy 81, Frey, R.: Market llqudty as a source of model rsk n dynamc hedgng n model rsk. In RISK Publcatons. R. Gbson Ed., London, Jandačka, M., Ševčovč, D.: On the rsk-adjusted prcng-methodology-based valuaton of vanlla optons and explanaton of the volatlty smle. J. Appl. Math. 3, Kratka, M.: No mystery behnd the smle. Rsk 9, Kwok, Y. K.: Mathematcal models of fnancal dervatves. Sprnger-Verlag Sngapore, Sngapore Leland, H. E.: Opton prcng and replcaton wth transacton costs. Journal of Fnance 40, Merton, R.: Theory of ratonal opton prcng. The Bell Journal of Economcs and Management Scence, Mmura, M., Tomoeda, K.: Numercal approxmatons to nterface curves for a porous meda equaton. Hroshma Math. J. 13, Hroshma Unversty, Schönbucher, P. J., and Wlmott, P.: SIAM J. Appl. Math. 61, 1, Ševčovč, D., Stehlíkova, B., Mkula, M.: Analytcal and numercal methods for prcng fnancal dervatves. Nova Scence Publshers, Hauppauge NY Wlmott, P., Dewynne, J., Howson, S.: Opton Prcng: Mathematcal Models and Computaton. Oxford Fnancal Press, Oxford 1993