Bergsche Unverstät Wuppertal Fachberech Mathematk und Naturwssenschaften Lehrstuhl für Angewandte Mathematk und Numersche Mathematk Lehrstuhl für Optmerung und Approxmaton Preprnt BUW-AMNA-OPAP 10/13 II Updated verson of Preprnt No. 10/13 I Ekaterna Dremkova and Matthas Ehrhardt A hgh-order compact method for nonlnear Black-Scholes opton prcng equatons of Amercan Optons November 2010 http://www.math.un-wuppertal.de
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed Internatonal Journal of Computer Mathematcs Vol. 00, No. 00, January 2008, 1 18 RESEARCH ARTICLE A hgh-order compact method for nonlnear Black-Scholes opton prcng equatons of Amercan Optons E. Dremkova a,b and M. Ehrhardt c a IDE, Halmstad Unversty, Box 823, 301 18, Halmstad, Sweden; b Volgograd State Unversty, Faculty of Mathematcs and Informatonal Technologes, prospect Unverstetsky 100, Volgograd, 400062, Russan Federaton; c Lehrstuhl für Angewandte Mathematk und Numersche Analyss, Fachberech C Mathematk und Naturwssenschaften, Bergsche Unverstät Wuppertal, Gaußstrasse 20, 42119 Wuppertal, Germany. v3.6 released September 2008 Due to transacton costs, llqud markets, large nvestors or rsks from an unprotected portfolo the assumptons n the classcal Black Scholes model become unrealstc and the model results n nonlnear, possbly degenerate, parabolc dffuson convecton equatons. Snce n general, a closed form soluton to the nonlnear Black Scholes equaton for Amercan optons does not exst even n the lnear case, these problems have to be solved numercally. We present from the lterature dfferent compact fnte dfference schemes to solve nonlnear Black Scholes equatons for Amercan optons wth a nonlnear volatlty functon. As compact schemes cannot be drectly appled to Amercan type optons, we use a fxed doman transformaton proposed by Ševčovč and show how the accuracy of the method can be ncreased to order four n space and tme. Keywords: nonlnear Black-Scholes equaton; compact fnte dfference scheme; Amercan optons; hgh-order methods; fxed doman transformaton; transacton costs AMS Subject Classfcaton: 65M06; 65M12; 65N06 1. Introducton In the recent several years stock opton was one of the most popular fnancal dervatves. There are many types of optons on the market ncludng European Call Put optons, Amercan Call Put optons, Exotc optons, etc.. But t was dffcult to accurately prce optons untl 1973 when Black and Scholes publshed ther Black-Scholes model [5]. There exsts two types of vanlla optons. A Call opton s a contract that gves the rght to the holder to buy the underlyng asset on a partcular date at a specfed value. A Put opton s a contract that gves the rght to the holder to sell the underlyng asset on a partcular date at a specfed value. The prce n the contract s called exercse prce or strke prce. The date n the contract s called expraton date or exercse date, maturty date. Optons are also dvded nto two types accordng to the expraton date T. Whle Amercan optons can be exercsed Correspondng author. Emal: katyapetrovna@gmal.com ISSN: 0020-7160 prnt/issn 1029-0265 onlne c 2008 Taylor & Francs DOI: 0020716YYxxxxxxxx http://www.nformaworld.com
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed 2 E. Dremkova and M. Ehrhardt at any tme before maturty date, European optons can only be exercsed at the maturty date. In ths work we wll focus on Amercan optons. In an dealzed fnancal market, the far prce of an Amercan Call opton can be obtaned by solvng the Black-Scholes partal dfferental equaton PDE V t + σ2 0 2 S2 V SS +r qsv S rv = 0, 0 < S < S f t, t 0,T, 1 wheretsthecurrentdate,σ 0 theconstantvolatlty,s theprceoftheunderlyng asset, r the rsk-free nterest rate, q the dvdend yeld and V the opton prce. In the fnancal sense, the partal dervatves ndcate the senstvty of the opton prce V to the correspondng parameter and are called Greeks. The opton delta s denoted by = V S, the opton gamma by Γ = V SS and the opton theta by θ = V t, cf. [38]. Equaton 1 s a backward-n-tme parabolc PDE and s suppled wth the termnal pay-off condton VS,T = maxs E,0 =: S E +, S 0. 2 Snce the value of an Amercan Call opton equals the value of a European Call opton f no dvdends are pad and the volatlty s constant, we ncluded the contnuous dvdend yeld n 1. Remark 1 dscrete dvdend payments Let us note that also dscrete dvdend payments can be ncluded here, cf. [38]. We assume that there s only one dvdend payment of the dvdend yeld q durng the lfetme of the opton at the dvdend date t q. Neglectng other factors, such as taxes, the asset prce S must decrease exactly by the amount of the dvdend payment q at tme t q. Thus we have the jump condton St + q = 1 qst q, where t q, t + q denote the moments just before and after the dvdend date t q. Ths leads to the followng effect on the opton prce: VS,t q = V1 qs,t + q, 3.e. the value of the opton at S and tme t q s the same as the value mmedately after the dvdend date t q but at the asset value 1 qs. In order to calculate the value of a Call opton wth one dvdend payment we solve the Black Scholes equaton from expry t = T untl t = t + q and use the relaton 3 to compute the values at t = t q. Fnally, we contnue to solve the Black Scholes equaton backwards startng at t = t q usng these values as the ntal data. The boundary condtons, that are dscussed next, do not need to be modfed for ths case. Snce Amercan optons can be exercsed at any tme before expry, we need to fnd the optmal tme t of exercse, known as the optmal exercse tme. At ths tme, whch mathematcally s a stoppng tme, the asset prce reaches the optmal exercse prce or optmal exercse boundary S f t. Ths leads to the formulaton of the problem for Amercan optons by dvdng the doman [0, [ [0,T] of 1 nto two parts along the curve S f t and analyzng each of them see Fg. 1a. Snce S f t s not known n advance but has to be
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed A hgh-order compact method for nonlnear Black-Scholes equatons of Amercan Optons 3 determned n the process of the soluton, the problem s called free boundary value problem. t V S,t T S S f 0 hold exercse S f t 0 S f t 0 S f T S f 0 S S f T E T t a Exercsng and holdng regons. b Schematcal value VS, t. Fgure 1. Amercan Call. For the Amercan Call opton the spatal doman s dvded nto two regons by the free boundary S f t, the stoppng regon S f t < S <, 0 t T, where the opton s exercsed or dead wth VS,t = S E and the contnuaton regon 0 S S f t, 0 t T, where the opton s held or stays alve and equaton 1 s vald wth the followng boundary condtons at S = 0 and S f t V0,t = 0 for 0 t T, VS f t,t = S f t E for 0 t T, V S S f t,t = 1 for 0 t T, 4 Note that we need two condtons at the free boundary S = S f t. One condton s necessary for the soluton of 1 and the other one s needed for determnng the poston of the free boundary S f t tself. The second condton n 4 value matchng condton s the contnuty of the mappng S VS,t snce VS,t = S E + = S E, n the exercse regon S S f t. At S = S f t one requres addtonally that VS, t touches the payoff functon tangentally hgh contact condton,.e. the functon S VS,t/ S should be contnuous at S = S f t. The condtons 4 are jontly referred as the smooth pastng condtons. Note that ths thrd condton can be derved from an arbtrage argument [38]. For the sake of smplcty we wll assume r > q n ths work, and therefore we have S f T = re/q for the Amercan Call. The structure of the value of an Amercan Call can be seen n Fg. 1b, where we notce that the free boundary S f t determnes the poston of the exercse. Ths lnear model s not very realstc [14], snce the Black-Scholes model had been derved under very restrctve assumptons, such as frctonless, lqud and complete market. In a real fnancal market the traders actually work n a dfferent envronment: transacton cost arsng [4, 9], the market s ncomplete, llqud, etc.. Although the Black-Scholes model has been used n practce, t has also caused some crtcsm, for example because the volatlty s not observable. It s often deduced by calculatng the mpled volatlty from sampled opton prces by nvertng the Black-Scholes formula. A wdely observed unque property, the so-called volatlty smle, s that these computed volatltes are not constant. Ths leads to a natural generalzaton of the Black-Scholes model replacng the constant volatl-
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed 4 E. Dremkova and M. Ehrhardt ty σ 0 n the model by a local volatlty functon σ = σe,t, where E denotes the exercse prce and T s the maturty. In practce, transacton costs arse when tradng securtes. Although they are generally small for nsttutonal nvestors, they lead to a notable ncrease n the opton prce. In the past years, dfferent models have been proposed to relax unrealstc assumptons of the Black-Scholes model 1. These models result n fully nonlnear Black-Scholes equatons. Boyle and Vorst [9] derved an opton prce takng nto account transacton costs that s equal to a Black-Scholes prce but wth a modfed volatlty of the form σ = σ 0 1+cA, A = µ σ 0 T, c = 1. 5 Here, µ s the proportonal transacton cost, T denotes the transacton perod, and σ 0 s the orgnal volatlty constant. Leland [26] computed c = 2/π. A more complex model has been proposed by Barles and Soner [4]. In ther model the nonlnear volatlty reads σ 2 = σ 2 01+Ψ[exprT ta 2 S 2 V SS ], 6 where r s the rsk-free nterest rate, T the maturty and a = µ γn where γ s the rsk averson factor and N s the number of optons to be sold. The functon Ψ s the soluton of the nonlnear sngular ntal-value problem Ψ A = ΨA+1 2, A 0, Ψ0 = 0. 7 AΨA A In the mathematcal lterature, only a few results can be found on the numercal soluton of nonlnear Black-Scholes equatons. The numercal dscretzaton of the Black-Scholes equatons wth the nonlnear volatlty 6 has been performed usng explct fnte dfference schemes FDS [4]. However, explct schemes have the dsadvantage that restrctve condtons on the dscretzaton parameters for nstance, the rato of the tme and the space step are needed to obtan stable and convergent schemes [36]. Moreover, the convergence order s only one n tme and two n space. Dürng et al. [14, 15] combned hgh-order compact dfference schemes derved by Rgal [32] and technques to construct numercal solutons wth frozen values of the nonlnear coeffcent of the nonlnear Black-Scholes equaton V t + 1 2 σv SS 2 S 2 V SS +r qsv S rv = 0, 0 < S < S f t, t 0,T, 8 to lnearze the formulaton. Snce analytcal solutons to nonlnear Black-Scholes equatons can only be obtaned n rather specal cases [6 8], we wll compute the opton prces numercally. Here, we consder compact FDS for Amercan optons and focus on the transacton cost model of Barles and Soner 6. Instead of solvng the sngular dfferental equaton 7 we propose to use some propertes of Ψ = ΨA descrbed recently n [10]. Theorem 1.1 [10]The nonlnear volatlty correcton functon Ψ, unque soluton of 7 satsfes the followng propertes:
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed A hgh-order compact method for nonlnear Black-Scholes equatons of Amercan Optons 5 Ψ s mplctly defned by A = arcsnh Ψ Ψ+1 + Ψ 2, f Ψ > 0, 9 A = arcsn Ψ Ψ+1 Ψ 2, f 0 > Ψ > 1. 10 Ψ s an ncreasng functon mappng the real lne onto the nterval ] 1,+ [. Many of the developed methods for solvng the Black-Scholes equaton can only be appled to European optons, lke the method of Lao and Khalq [27] and methods derved by Rgal [32]. Hence, for Amercan optons another strategy s needed. Usually, the equaton s transformed nto the heat equaton and the doman s modfed nto a semunbounded one wth a free boundary. Exact analytcal formulas for the free boundary S f t n 8 wth condtons 4 are not known, but there exst several deductons of approxmate formulas for Amercan opton estmaton n the lnear case. Recently, Ševčovč [33] proposed a new method to transform the free boundary problem for the early exercse boundary locaton nto deducton of a tme dependent nonlnear parabolc equaton on a fxed doman, cf. Secton 3. Lao and Khalq [27] offered an uncondtonally stable compact FDS of fourth order both n space and tme appled to European optons. Ths paper s organzed as follows. Frst, n Secton 2 we revew from the lterature a couple of compact hgh order fnte dfference schemes. As compact schemes cannot be drectly appled to Amercan type optons, we wll employ them usng a novel fxed doman transformaton proposed by Ševčovč, cf. [3, 33]. Fnally, we use n Secton 4 the approach n combnaton wth the new method of Lao and Khalq [27] for solvng the nonlnear Black-Scholes equaton 8 wth transacton costs. 2. Compact schemes Here we wll revew from [14, 15] a couple of standard dfference schemes and compact schemes and also present the schemes developed by Rgal [32]. Before we start wth presentng the schemes, we brefly menton a useful transformaton of the nonlnear PDE 8 that wll be the startng pont for the methods. In order to transform the nonlnear Black-Scholes equaton 8 wth volatlty 6 nto a convecton-dffuson problem, we use the followng transformaton [14]: xs = ln S E, τt = σ2 0 2 T t, u = e xv E. Then 8 may be rewrtten n the followng form u τ = 1+Ψ[e Kτ+x a 2 Eu xx +u x ] u xx +u x +Ku x, x R, 11 where 0 τ σ0 2T/2, K = 2r/σ2 0. τ denotes the scaled tme to maturty.
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed 6 E. Dremkova and M. Ehrhardt a b Fgure 2. Solutons of 9, 10 left fgure and ODE 7 rght fgure, for A > 0, where T = 1, K = 10, r = 0.1, q = 0.05. 2.1 Fnte dfference methods All consdered dfference schemes have two tme levels. Let A n and B n be the system matrces of A n U n+1 = B n U n A n = [a 1,a 0,a 1 ], B n = [b 2,b 1,b 0,b 1,b 2 ], where a, b denote the man dagonals, superdagonals and subdagonals. The matrx A n s trdagonal and thus the obtaned lnear systems may be solved effcently usng the Thomas algorthm. We further assume the normalzaton condtons 1 a = = 1 2 b = 1 = 2 and followng [15], we dscretze the volatlty correcton n 6 as [ U s n = Ψ expknk +x a 2 n E 2 2U n +U+2 n ] 4h 2 + Un +1 Un 1. 12 2h Ths formula gves an explct dscretzaton of the nonlnearty and uses a specal stencl for the second dervatve spatal step 2h nstead of h. Another problem les n the ntal condton for ux, 0, as t s nondfferentable n the pont x = 0. Oosterlee et al. [29] solved ths problem of reduced accuracy and proposed a grd stretchng technque, whch s based on an dea of placng more ponts n the neghborhood of the nondfferentable payment condton. Fgure 2 shows the solutons of 9, 10 on the left, and of the ODE 7 on the rght and ther splne nterpolaton. It s easly seen that the plots show ndstngushable results. Let us also ntroduce the followng notatons. λ = 1+K denotes the lnear part of the coeffcent of the convecton term n 11,.e. 11 reads u τ = Ψ[e Kτ+x a 2 Eu xx +u x ]u xx +u x +u xx λu x, x R. 13 Moreover, h = x, k = τ are the step szes n space and tme, α = λh/2 s the cell Reynolds number, ζ = k/h 2 the parabolc mesh rato and µ = k/h the hyperbolc mesh rato.
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed A hgh-order compact method for nonlnear Black-Scholes equatons of Amercan Optons 7 2.2 Classcal fnte dfference schemes Here we summarze the most well-known schemes, cf. [15]. 2.2.1 The Forward-Tme Central-Space explct scheme FTCS Ths scheme s gven by the coeffcents a ±1 = 0, a 0 = 1, b ±1 = ζ ± µ 2 sn λ, b 0 = 1 2ζ ζ 2 sn, b ±2 = ζ 4 sn. It s of order 1 n tme, 2 n space, wth a very strct stablty condton ζ 1/2. The condton α 1 must be satsfed to avod oscllatons. 2.2.2 The Backward-Tme Central-Space sem-explct scheme BTCS Ths scheme treats explctly the nonlnearty and s gven by a ±1 = λ 2 µ ζ, a 0 = 1+2ζ, b ±2 = ζ 4 sn, b ±1 = ± 1 2 µsn, b 0 = 1 ζ 2 sn. Ths scheme s of order 1 n tme and 2 n space. It s uncondtonally stable and f α 1 s satsfed, then t s non-oscllatory. 2.2.3 The Crank-Ncolson scheme CN Ths scheme, wth an explct treatment of the nonlnearty, s gven by a ±1 = ζ 2 µ s n ζ 4 2 ± λ ζ 4 µ, b ±1 = 2 ± µ s n + ζ 4 2 λ 4 µ, a 0 = 1+ζ1+s n, b 0 = 1 ζ1+s n, b ±2 = 0. It s of the order 2 both n tme and space and uncondtonally stable. 2.3 The compact schemes of hgher order Rgal [32] ntroduced several FDS for lnear convecton-dffuson problems and Dürng et al. [14, 15] appled them to the problem 11. These schemes are both compact two-level schemes of order 2 n tme and 4 n space n the lnear case. The nonlnearty s handled sem-mplctly as n the prevous subsecton. 2.3.1 The R3A scheme In ths scheme the coeffcents are chosen as a ±1 = 1 12 ζ 1 α α2 ζ 2 b ±2 = ζ 4 sn, b ±1 = 1 12 + ζ 2 b 0 = 5 6 ζ α2 ζ 3 2α2 ζ 2 ζ 3 2 sn. 6 + α2 ζ 2 3, a 0 = 5 6 +ζ + α2 ζ 3 2α2 ζ 2, 3 1 α α2 ζ 6 + α2 ζ 2 ± 1 6 2 µsn, It s stable n the lnear case s n = 0 f ζ 1/ 2 α and the scheme s nonoscllatory, for arbtrary α, cf. [32].
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed 8 E. Dremkova and M. Ehrhardt 2.3.2 The R3B scheme The coeffcents for ths scheme read a ±1 = 1 12 ζ 1 α α2 ζ 2 6 α3 ζ 2 3 a 0 = 5 6 +ζ + α2 ζ 3 + 4α4 ζ 3, 3 2α4 ζ 3, 3 b ±2 = ζ 4 sn, b 0 = 5 6 ζ α2 ζ 3 4α4 ζ 3 2ζs n, 3 1 b ±1 = 12 ζ 1+α+ α2 ζ 2 6 α3 ζ 2 + 2α4 ζ 3 ζ 3 3 4 1 2 µ s n. It s uncondtonally stable and non-oscllatory n the lnear case s n = 0, cf. [32]. 3. The fxed doman transformaton Compact schemes that many authors appled to the Black-Scholes equaton wth transacton costs have one dsadvantage: these schemes cannot be generalzed to mult-dmensonal problems, and aredrectly applcable to European type optons only. However, wth the fxed doman transformaton ntroduced by Ševčovč [3, 33] we overcome ths second shortcomng. We consder the nonlnear Black-Scholes equaton 8 for an Amercan Call opton,.e. suppled wth the termnal condton 2 and boundary condtons 4. Equaton 8 subject to 2, 4 s a backward-n-tme parabolc free boundary problem. To solve ths free boundary problem numercally, many dfferent methods are developed, for nstance the standard method conssts n the reformulaton to a lnear complementary problem LCP and soluton by a projected SOR method. Alternatvely, penalty and front-fxng methods were developed e.g. n [18], [28]. A dsadvantage of these methods s the change of the underlyng model. A dfferent approach [21] s based on a recursve calculaton of the early exercse boundary, estmatng the boundary by Rchardson nterpolaton. The explct boundary trackng algorthms are for example a fnte dfference bsecton scheme [24] or the front-trackng strategy of Han and Wu [20]. Here we consder the approach of Ševčovč [33] to smplfy the numercal soluton of 8, wth 2, 4 for Amercan call optons and get rd of the explct appearance of the free boundary. To do ths, we need to transform the problem nto a problem posed on a fxed, but unbounded doman addtonally to the forward transformaton n tme. Then, the doman does not depend on the free boundary S f t anymore. All we need s to calculate an algebrac constrant equaton for the poston of the free boundary. To do so, we make the followng substtuton: τ τ = T t, x = ln S = e x τ, τ = S f T τ, S such that x R + and τ [0,T]. The constructed synthetc portfolo wll be Πx,τ = VS,τ SV S S,τ, 14 made by buyng = V S shares S and sellng an opton V.
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed A hgh-order compact method for nonlnear Black-Scholes equatons of Amercan Optons 9 Dfferentatng ths artfcal portfolo Π wth respect to x and τ gves us Π x = V S S x S x V S SV SS S x = S 2 V SS 15 Π τ = V S S τ +V t t τ S τ V S SV SS S τ +V St t τ = V t τ τ S2 V SS +SV St = V t τ τ Π x S S V t. 16 Equaton 15 wll be used e.g. to reformulate the nonlnear volatlty correcton of Barles and Soner also n terms of functon Π Substtutng 15, 16 nto 8 we get Π τ = 1 σ 2 x σ 2 2 Π x + 2 bτ Π x rπ, x R +, τ 0,T, 17 where the tme-dependent coeffcent bτ reads bτ = τ log τ+r q. 18 The ntal condtons2 and boundary condtons4 after substtuton14 transform to { E for S > E x < ln r Πx,0 = VS,T SV S S,T = q 19 0 otherwse Πx,τ = 0 as x, 0 τ T, Π0,τ = E for 0 τ T. 20 Wth the assumpton r q we obtan the constrant equaton τ = 1 2q σ2 Π x 0,τ+ re q wth 0 = re q, 21 where 0 τ T and the modfed volatlty functon becomes σ 2 = σ0 2 1+Ψ[e rτ a 2 Π x ]. 22 Our transformed problem 17 subject to 19 21 wth the volatlty functon 22 can be solved by the splt-step FDS proposed by Ševčovč [33]. However, we wll propose another scheme recently developed by Lao and Khalq [27]. After solvng the transformed problem wth some sutable method, we calculate the value of the Amercan call opton VS, t by transformng 14 back to the orgnal varables. Snce we know that Πx,τ S 2 = VS,t S 2 V SS,t S = S VS,t, S
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed 10 E. Dremkova and M. Ehrhardt we ntegrate the above equaton from S to S f t wth the boundary condton VS f t,t = S f t E and we obtan VS,T τ = S τ E + τ ln τ S 0 e x Πx,τdx. 23 yeldng the prce of an Amercan call VS,t n the presence of transacton costs. 4. The method of Lao and Khalq Lao and Khalq [27] proposed a new effcent fourth-order compact scheme based on the Padé approxmaton. The method was appled to the nonlnear Black-Scholes equaton for European optons. In ths secton we wll brefly revew ther work and show n the followng Secton 5 how t can be appled to Amercan optons. Let us consder the followng one-dmensonal tme dependent convectondffuson equaton u t = βu xx +λu x ru, 24 where β, λ and r are constants. Instead of solvng a sngle convecton-dffuson equaton 24, Lao and Khalq transform t nto a system of two equatons. The followng new unknown functon vx,t = u x x,t s ntroduced, hence we consder u t = βu xx +λv ru 25 v t = βv xx +λu xx rv. 26 We state for ux,t the ntal condtons ux,0 = u 0 x and boundary condtons u0,t = b 0 t, u1,t = b 1 t. For vx,t the ntal condton s smply the x-dervatve of ux,t: vx,0 = u 0 x. In the sequel we wll make frequent use of the standard central dfference operator 0 h u := u +1 u 1 = 2hD 0 h u and the standard second order dfference operator 2 h u := u +1 2u +u 1 = h 2 D 2 h u Suppose now the spatal grd s unform,.e. N sub-ntervals form the nterval [0,1] and h = 1/N. The second order approxmaton vh,t = u x h,t D0 h uh,t = 0 h 2h uh,t.
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed A hgh-order compact method for nonlnear Black-Scholes equatons of Amercan Optons 11 can be mproved to fourth order f 0 h s replaced by 0 h /1+ 1 6 2 h vh,t = 0 h 2h1+ 1 6 2 h uh,t +Oh4. Dong so, we obtan a fourth order approxmaton v0,t = 3 h u2h,t u0,t 4vh,t v2h,t. Hence, we can approxmate the rght boundary condton of v at x = 1 as v1,t = 3 h u1 h,t u1 2h,t 4v1 h,t v1 2h,t. Let us consder a more general system u t = βu xx +fu,v, 27 v t = λu xx +βv xx +gu,v, 28 where the term λv s ncluded n the general functon fu,v and there s only one dffuson term βu xx n 27. The method starts from the Crank-Ncolson scheme u n+1 k u n = 1 2 β h 2 2 h un+1 + β h 2 2 h un +f n+1 +f n, 29 v n+1 k v n = 1 2 λ h 2 2 h [un+1 +u n ]+ β h 2 2 h [vn+1 +v n ]+g n+1 +g n, 30 where f n+1 = fu n+1,v n+1, f n = fu n,v n, g n+1 = gu n+1,v n+1, g n = gu n,v n, We mprove ths second order approxmaton to the fourth order by usng nstead of the prevous approxmaton the Padé approxmaton u xx 2 h h 2 1+ 1 31 12 2 h. We apply ths Padé approxmaton n 29 30, multply both sdes by 1+ 1 12 2 h : 1+ 2 h 12 βζ 2 2 h u n+1 = 1+ 2 h 12 βζ 2 2 h u n + k 1+ 2 h f n+1 +f n, 2 12 32
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed 12 E. Dremkova and M. Ehrhardt 1+ 2 h 12 βζ 2 2 h v n+1 = 1+ 2 h 12 βζ + k 2 2 2 h v n +λ ζ 2 2 h un+1 +u n 1+ 2 h g n+1 +g n, 12 33 wth the parabolc mesh rato ζ = k/h 2. The truncaton error of 32 33 s C 1 k 2 +C 2 k 4 +C 3 h 4 and snce we solely have even powers w.r.t. the tme step k, a Rchardson extrapolaton technque can mprove the approxmaton to fourth order n tme. Suppose the solutons of 32 and 33 after l teratons are denoted by u n+1l and v n+1l fu n+1,v n+1 respectvely. To get u n+1l+1 = fu n+1l,v n+1l + f and v n+1l+1, we frst expand f n+1 : un+1l u,v n+1l u n+1 u n+1l 34 and nsert t nto 32, then solve the followng equaton for u n+1l+1 1+ 1 12 2 h βζ 2 2 h k 2 1+ 1 12 2Ĵn+1l h u n+1l+1 + k 2 1+ 1 12 2 h fu n+1l = 1+ 1 12 2 h βζ 2 2 h,v n+1l u n Ĵn+1l u n+1l +fu n,v n, 35 where Ĵn+1l = f un+1l u,v n+1l. Once we have found u n+1l+1, we expand g n+1 gu n+1,v n+1 = gu n+1l+1 Substtutng 36 nto 34 we get:,v n+1l + g un+1l+1 u,v n+1l v n+1 v n+1l. 36 1+ 1 12 2 h βζ 2 2 h k 2 1+ 1 + k 2 1+ 1 12 2 h fu n+1l = 12 2 h J n+1l v n+1l+1 1+ 1 12 2 h βζ 2 2 h u n,v n+1l n+1l J u n+1l +fu n,v n, 37 where J n+1l = g un+1l v,v n+1l. We then solve 37 for v n+1l+1. The two steps are repeated alternatvely untl convergence occurs. Lao and Khalq [27] made a careful von Neumann stablty analyss for ther scheme and proved that ths hgh order compact scheme s uncondtonally stable.
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed A hgh-order compact method for nonlnear Black-Scholes equatons of Amercan Optons 13 Fgure 3. Unform grd for Amercan optons. 5. The Numercal Soluton of the Amercan Opton problem In ths secton we wll explan the steps, how to solve the nonlnear Black-Scholes equaton 8 n case of Amercan optons. We confne the unbounded doman x R + and τ [0,T] to x 0,R wth R > 0 large enough see [33]. For the calculaton Ševčovč chooses to take R = 3, snce ths s equvalent to S S f te R,S f t and yelds a good approxmaton for S 0,S f t as the transformaton was S = S f te x. We also take h > 0 as spatal step sze, k > 0 as temporal step sze, x = h, [0,N], R = Nh, τ n = nk, n [0,M], T = Mk, cf. [3] see Fg. 3. Our computatons rely on the transformaton technque of Secton 3 and thus we compute numercally values of the synthetc portfolo Π = Πx,τ and use n the sequel the standard notaton Π n Πx,τ n, b n bτ n In all our computatons the nonlnearty s dscretzed explctly n the scheme,.e. now s n denotes the nonlnear volatlty correcton of Barles and Soner 6 wrtten n terms of Π: s n = Ψ [ e rτn a 2 D + ] h Πn We emphasze that another advantage of ths synthetc portfolo formulaton s that, compared to 12, we do not need to evaluate here a second dervatve,.e. we can use a smaller stencl. Now, the synthetc portfolo equaton 17 can be wrtten as a sem-dscretzed equaton 38.e. Π τ = 1 σ 2 xσ01+s 2 n 2 Π x + 0 2 1+sn b n Π x rπ, 39 that has the form Π τ = σ2 0 σ 2 2 1+sn Π xx + 0 2 1+sn + x s n b n Π x rπ, 40 Π τ = βπ xx +λπ x rπ, 41 and we wll consder FDS for solvng 41 wth arbtrary parameters β,r > 0 and λ R. Especally the scheme of Lao and Khalq [27] fts to ths equaton and wll be used.
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed 14 E. Dremkova and M. Ehrhardt We can sketch our procedure for the tme advancement as follows: Π n n b n Π n+1. Hence, assume Π n s gven, we descrbe n the next part how to determne n and the coeffcent b n. We consder the free boundary 21 n the pont x = 0 and approxmate the dervatve n space by forward dfferences: n = 1 2q σ2 0 1+s n 0 D + h Πn 0 + re q wth 0 = re q, 42 where D + h Πn 0 = Πn 1 Πn 0 /h s the forward dfference quotent n pont x = 0. Clearly, ths fnte dfference approxmaton s only of frst order, but ths equaton s just to determne the approxmate locaton of the free boundary. We expect that our detaled numercal tests n [17] wll show that the order of accuracy of the overall scheme s not reduced. Anyway, the approxmaton error can easly be ncreased by usng one-sded fnte dfference approxmatons of hgher order. Next, the coeffcent b n s readly obtaned by dscretng 18 b n = log n log n 1 k +r q, n = 2,3,... 43 Only, n the frst step, to compute b 1 we use wth very hgh accuracy an asymptotc expanson for the free boundary. Remark 1 For the Amercan Call opton n contrast to the Amercan Put opton t s possble to derve a seres for the locaton of the optmal exercse boundary close to expry usng standard asymptotc analyss [38]. Ths local analyss of the free boundary S f t yelds S f t S f T 1+ξ 0 1 2 σ2 T t+..., as t T, 44 where ξ 0 = 0.9034... s a unversal constant of Call opton prcng. Equaton 44 can be rewrtten as τ 0 1+ξ 0 1 2 σ2 τ+..., as τ 0. 45 Wth only very few terms we get a farly accurate result for the free boundary and thus equaton 45 wll serve us as a check for the case of a constant volatlty σ 2 = σ 2 see Fg. 4. Note that ths result s especally useful n the frst tme levels of a numercal calculaton where rapd changes n τ nfluence the whole soluton regon. Fgure 4 shows the dfference between the free boundary and the asymptotc free boundary taken wth exercse date T = 1 year, exercse prce E = 10 and dvdend yeld q = 0.05. Fnally we can compute the soluton π n+1 on the next tme level by solvng the sem dscretzed equaton 41 subject to the ntal condtons 19 and boundary
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed A hgh-order compact method for nonlnear Black-Scholes equatons of Amercan Optons 15 Fgure 4. The free boundary sold and the asymptotc free boundary dashed for nonlnear Black-Scholes equaton 8 wth a nonlnear volatlty 6 wth T = 1, E = 10, σ 0 = 0.2, r = 0.1, q = 0.05, a = 0.02, h = 0.1, k = 0.1 condtons 20: Π 0 = Πx,0 = Π n 0 = E, Π n N = 0. { E for x < ln 0 E = ln r q, 0 otherwse 46 Sometmes hgh order methods need morenumercal boundary condtons to close the system of equatons. In these cases on uses hgh order extrapolaton technques, cf. [19] for the necessary order of accuracy. Recall that at each tme level n, we can calculate VS,t n = Ve x n,t τ n wth these values and proceed to the next tme level n+1. From 23 we then know that: where j=0 VS,t n = e x n E +I, 47 1 x I = I k + e x Πx,τdx x 1 1 = I k + x x 1 e x 1 Π n 1 +e x Π n. 2 j=0 Here, we use the trapezodal rule n order to approxmate the ntegral 23. Fgure 5 shows the computed value VS,0 of an Amercan Call opton determned from the nonlnear Black-Scholes equaton 8 usng the dscretzaton steps h = 0.1, k = 0.1 and the parameters: exercse date T = 1 year, exercse prce E = 10 and dvdend yeld q = 0.05. wth T = 1, E = 10, σ 0 = 0.2, r = 0.1,
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed 16 E. Dremkova and M. Ehrhardt Fgure 5. Prce of the Amercan Call opton VS,0 n the presence of transacton costs n case of nonlnear Black-Scholes equaton. q = 0.05, a = 0.02. 5.1 The algorthm descrpton The calculatons of the prce VS,t for Amercan optons n the presence of transacton costs lead us to the followng algorthm. Algorthm 1 Computaton of the prce VS,t for the Amercan opton. Input parameters: σ, r, q, E, a, R, T, h, k, M, N, γ, α, β 1 Use formula 22 for the Barles and Soner volatlty model 6 and nterpolate the solutons 2 ntalze Π 0, 0, VS,T 3 calculate Π n+1 for each tme level teratvely a calculate from 38 the volatlty correcton s n for the tme step τ n usng results from the prevous tme step b calculate the free boundary n for the tme step τ n usng 42 c calculate the coeffcent b n from 43 d calculate the soluton Π n+1 usng an FDS for solvng 41, e.g. method of Lao and Khalq [27] 4 transform Π nto V 5 plot V for each tme level and each stock prce Conclusons and Future Work In ths work we consdered the numercal soluton of nonlnear Black-Scholes equatons n the presence of transacton costs n case of Amercan optons. Whle we focused n ths paper on standard optons known as plan vanlla optons of Amercan type, our future work wll deal wth extensons: forward and future contracts, optons on futures, more general pay off functons e.g. cash or nothng call wth transacton costs and nstalment optons. We presented several hgh-order compact schemes and dscussed how to ncrease the order of accuracy to be fourth order both n tme and space. It turned out n
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed REFERENCES 17 our prelmnary tests that the method of Lao and Khalq [27] performs better than other methods descrbed n Secton 2.2. We showed that the compact methods can be used very effcently to prce Amercan optons usng the fxed doman transformaton technque of Ševčovč [34]. In a second follow-up paper [17] we wll present concsely the numercal results of our comparson study ncludng detals about the obtaned rates of convergence. Our future research wll be drected towards the mplementaton of dscrete artfcal boundary condtons, cf. [16, 37], snce 17 s posed on an unbounded doman. Acknowledgements The authors thank Ljudmla Bordag of Halmstad Unversty for her hosptalty. The authors are grateful to the anonymous referees for ther frutful comments that mproved the content of ths manuscrpt. References [1] J. Ankudnova, The numercal soluton of nonlnear Black-Scholes equatons. Dploma Thess, Technsche Unverstät Berln, Berln, Germany, 2008. [2] J. Ankudnova and M. Ehrhardt, On the numercal soluton of nonlnear Black-Scholes equatons, Comput. Math. Appl. 56 2008, pp. 799 812. [3] J. Ankudnova and M. Ehrhardt, Fxed doman transformatons and splt-step fnte dfference schemes for Nonlnear Black-Scholes equatons for Amercan Optons, Chapter 8 n M. Ehrhardt ed., Nonlnear Models n Mathematcal Fnance: New Research Trends n Opton Prcng. Nova Scence Publshers, Inc., Hauppauge, NY 11788, 2008, pp. 243 273. [4] G. Barles and H.M. Soner, Opton prcng wth transacton costs and a nonlnear Black-Scholes equaton, Fnance Stochast. 2 1998, pp. 369 397. [5] F. Black and M.S. Scholes, The prcng of optons and corporate labltes, J. Poltcal Economy 81 1973, pp. 637 654. [6] L.A. Bordag and R. Frey, Nonlnear opton prcng models for llqud markets: scalng, propertes and explct solutons, Chapter 3 n M. Ehrhardt ed., Nonlnear Models n Mathematcal Fnance: New Research Trends n Opton Prcng. Nova Scence Publshers, Inc., Hauppauge, NY 11788, 2008, pp. 103 129. [7] L.A. Bordag, Study of the rsk-adjusted prcng methodology model wth methods of geometrcal analyss, Quanttatve fnance papers, arxv.org, 2010. [8] L.A. Bordag and A.Y. Chmakova, Explct solutons for a nonlnear model of fnancal dervatves, Int. J. Theor. Appl. Fnance 11 2007, pp. 1 21. [9] P. Boyle and T. Vorst, Opton replcaton n dscrete tme wth transacton costs, J. Fnance 47 1973, pp. 271 293. [10] R. Company, E. Navarro, J.R. Pntos and E. Ponsoda, Numercal soluton of lnear and nonlnear Black-Scholes opton prcng equatons, Comput. Math. Appl. 56 2008, pp. 813 821. [11] R. Company, L. Jódar and J.R. Pntos, Consstent stable dfference schemes for nonlnear Black- Scholes equatons modellng opton prcng wth transacton costs, ESAIM: Math. Model. Numer. Anal. 43 2009, pp. 1045 1061. [12] R. Company, L. Jódar, J.R. Pntos and M.-D. Roselló, Computng opton prcng models under transacton costs, Comput. Math. Appl. 59 2010, pp. 651 662. [13] R. Company, L. Jódar and J.R. Pntos, Numercal analyss and computng for opton prcng models n llqud markets, Math. Comput. Modellng 52 2010, pp. 1066 1073. [14] B. Dürng, Black-Scholes Type Equatons: Mathematcal Analyss, Parameter Identfcaton and Numercal Soluton, Dssertaton, Unverstät Manz, Germany, 2005. [15] B. Dürng, M. Fourné and A. Jüngel, Hgh-order compact fnte dfference schemes for a nonlnear Black-Scholes equaton, Int. J. Theor. Appl. Fnance 6 2003, pp. 767 789. [16] M. Ehrhardt and R.E. Mckens, A fast, stable and accurate numercal method for the Black-Scholes equaton of Amercan optons, Int. J. Theor. Appl. Fnance 11 2008, pp. 471 501. [17] E. Dremkova and M. Ehrhardt, A numercal comparson study of hgh-order compact methods for nonlnear Black-Scholes equatons, n preparaton. [18] P.A. Forsyth and K.R. Vetzal, Quadratc Convergence for Valung Amercan Optons usng a Penalty Method, SIAM J. Sc. Comput. 23 2002, pp. 2095 2122. [19] B. Gustafsson, The convergence rate for dfference approxmaton to mxed ntal boundary value problems, Math. Comput. 29 1975, pp. 396 406. [20] H. Han and X. Wu, A fast numercal method for the Black-Scholes equaton of Amercan optons, SIAM J. Numer. Anal. 41 2003, pp. 2081 2095. [21] J.-Z. Huang, M.G. Subrahmanyam and G.G. Yu, Prcng and Hedgng Amercan Optons: A Recursve Integraton Method, Rev. Fnancal Studes 9 1996, pp. 277 300. [22] M. Jandačka and D. Ševčovč, On the rsk-adjusted prcng-methodology-based valuaton of vanlla optons and explanaton of the volatlty smle. J. Appl. Math. 3 2005, pp. 235 258.
November 21, 2010 15:1 Internatonal Journal of Computer Mathematcs DremkovaEhrhardt revsed 18 REFERENCES [23] L. Jódar, R. Company and J.R. Pntos, A semdscretzaton method for solvng nonlnear Black- Scholes equatons: numercal analyss and computng, Chapter 5 n: M. Ehrhardt ed., Nonlnear Models n Mathematcal Fnance: New Research Trends n Opton Prcng, Nova Scence Publshers, Inc., Hauppauge, NY 11788, 2008, pp. 149 171. [24] T. Km and Y. Kwon, Fnte Dfference Bsecton Algorthms for free boundares of Amercan Call and Put Optons, Preprnt, Pohang Unversty of Scence and Technology, South Korea, 2002. [25] M. Kratka, No mystery behnd the smle, Rsk 9 1998, pp. 67 71. [26] M. Leland, Opton prcng and replcaton wth Transacton Costs, J. Fnance 5 1985, pp. 1283 1301. [27] W. Lao and A.Q.M. Khalq, Hgh order compact scheme for solvng nonlnear Black-Scholes equaton wth transacton cost, Int. J. Comput. Math. 86 2009, pp. 1009 1023. [28] B.F. Nelsen, O. Skavhaug and A. Tveto, Penalty and front-fxng methods for the numercal soluton of Amercan opton problems, J. Comp. Fnance 5 2002, pp. 69 97. [29] C.W. Oosterlee, C.C.W. Leentvaar and A. Almendral, Prcng optons wth dscrete dvdends by hgh order fnte dfferences and grd stretchng, ECCOMAS 2004, P. Nettaanmäkk et al. eds., CD Rom, Jyväskylä, Fnland. [30] J.R. Pntos, Numercal analyss and computng of nonlnear opton prcng models, PhD Thess, Unversdad Poltecnca de Valenca, Span, 2010. [31] O. Pronneau and F. Hecht, Mesh adapton for the Black and Scholes equatons, East-West J. Numer. Math. 8 2000, pp. 25 35. [32] A. Rgal, Hgh Order Dfference Schemes for Unsteady One-Dmensonal Dffuson-Convecton Problems, J. Comp. Phys. 114 1994, pp. 59 76. [33] D. Ševčovč, An teratve algorthm for evaluatng approxmatons to the optmal exercse boundary for a nonlnear Black-Scholes equaton, Canad. Appl. Math. Quarterly 15 2007, pp. 77 79. [34] D. Ševčovč, Transformaton methods for evaluatng approxmatons to the optmal exercse boundary for lnear and nonlnear Black-Scholes equatons, Chapter 6 n M. Ehrhardt ed., Nonlnear Models n Mathematcal Fnance: New Research Trends n Opton Prcng. Nova Scence Publshers, Inc., Hauppauge, NY 11788, 2008, pp. 173 218. [35] D. Ševčovč, B. Stehlíková and K. Mkula, Analytcal and Numercal Methods for Prcng Fnancal Dervatves, Frst edton, Nova Scence Publshers, 2010. [36] W.F. Spotz and G.F. Carey, Extenson of hgh-order compact schemes to tme dependent problems, Numer. Meth. Partal Dff. Eqs. 17 2001, pp. 657 672. [37] M. Uhlark, Operator Splttng Methods and Artfcal Boundary Condtons for nonlnear Black- Scholes equaton, Masters Thess, Halmstad Unversty, Sweden, 2010. [38] P. Wlmott, S. Howson, and J. Dewynne, The Mathematcs of Fnancal Dervatves, A Student Introducton, Cambrdge Unversty Press, 2002.