R. Zvan. P.A. Forsyth. K. Vetzal. University ofwaterloo. Waterloo, ON

Transcription

1 Robust Numercal Methods for PDE Models of sa Optos by R. Zva Departmet of Computer Scece Tel: ( ext. 6 Fax: ( rzvayoho.uwaterloo.ca P.. Forsyth Departmet of Computer Scece Tel: ( ext. 445 Fax: ( paforsythyoho.uwaterloo.ca K. Vetzal Cetre for dvaced Studes Face Tel: ( ext. 658 Fax: ( kvetzalwatarts.uwaterloo.ca Uversty ofwaterloo Waterloo, ON Caada NL G

2 bstract We explore the prcg of sa optos by umercally solvg the the assocated partal deretal equatos. We demostrate that umercal PDE techques commoly used ace for stadard optos are accurate the case of sa optos ad llustrate modcatos whch allevate ths problem. I partcular, the usual methods geerally produce solutos cotag spurous oscllatos. We adapt ux lmtg techques orgally developed the eld of computatoal ud dyamcs order to rapdly obta accurate solutos. We show that ux lmtg methods are total varato dmshg (ad hece free of spurous oscllatos for o-coservatve PDEs such as those typcally ecoutered ace, for fully explct, ad fully ad partally mplct schemes. We also modfy the va Leer ux lmter so that the secod-order total varato dmshg property s preserved for o-uform grd spacg.

3 Itroducto sa optos are securtes wth payos whch deped o the average value of a uderlyg stock prce over some tme terval. Such optos have prove to be much more dcult to value tha regular stock optos. Stadard techques ted to be mpractcal, accurate, or slow. For example, tradtoal bomal lattce methods requre such eormous amouts of computer memory (owg to the ecessty ofkeepg track ofevery possble path throughout the tree that they are eectvely uusable. Partal deretal equato (PDE methods, as tradtoally mplemeted the ace lterature, are accurate (see Barraquad ad Pudet (996 for a dscusso. Mote Carlo smulato works well for Europea-style optos (see Kema ad Vorst (99, but s relatvely slow. umber of approxmatos have appeared the lterature (e.g. Turbull ad Wakema (99, Vorst (99, Levy (99, Levy ad Turbull (99, whch are aga sutable oly for Europea-style optos. See also Gema ad Yor (99, who derve the Laplace trasform of the Europea opto prce. Ufortuately, ths trasform s very dcult to vert. Wth regard to merca-style sa optos, there are eve fewer alteratves. Hull ad Whte (99 propose a modcato of the bomal method, but do ot provde ay proof of covergece. Neave (994 uses a frequecy dstrbuto approach o a bomal lattce to derve approxmate values for arthmetc average opto values, but hs method stll requres calculatos of order N 4, where N s the umber of tme-steps the lattce. Barraquad ad Pudet (996 descrbe a forward shootg grd algorthm ad prove that t s ucodtoally coverget. We explore aother possblty: a moded te derece method. I geeral, the prce of a sa opto ca be foud by solvg a PDE two space-lke dmesos (see Igersoll (987 or Wlmott, Dewye, ad Howso (99. Ths PDE has the character of a two dmesoal covecto-duso problem wth o duso oe of the spatal dmesos. s s well-kow computatoal ud dyamcs, stadard cetrally weghted methods for the covectve term are proe to oscllatory solutos. Furthermore, as argued by Barraquad ad Pudet (996, stadard te derece methods (though geerally faster tha ther proposed algorthm are accurate because they troduce \spurous umercal duso" (p. 4. I some cases the prce of a sa opto ca be modeled usg a oe-dmesoal PDE. The two-dmesoal PDE for a oatg strke sa opto ca be reduced to a oe-dmesoal PDE (see Igersoll (987 or Wlmott, Dewye, ad Howso (99. Recetly, Rogers ad Sh (995 have formulated a oe-dmesoal PDE that ca model the prce of both oatg ad xed strke sa optos. However, ths PDE apples oly to the case of Europea-style optos ad s partcularly dcult to solve umercally sce the duso term s very small for values of terest o the te derece grd. We demostrate modcatos to the commo dscretzato methods whch are desged

4 to hadle these problems. I partcular, both the two-dmesoal ad oe-dmesoal cases t s ecessary to solve a problem wth lttle or o duso (.e. secod-order dervatve term a space dmeso. The tradtoal approach computatoal ud dyamcs would be to use rst-order upstream weghtg for the covectve term to elmate the oscllatos caused by cetrally weghted schemes (Roache, 97. However, rst-order upstream weghtg results solutos wth excessve false duso. s a alteratve, we employ a hgh order o-lear ux lmter for the covectve terms. The resultg dscrete o-lear algebrac equatos are solved usg full Newto terato. I addto, we ca also apply the merca early exercse costrat to the algebrac system, ad ths ca be hadled a mplct fully coupled maer. I cases where the model caot be reduced to a problem a sgle space dmeso, the full two-dmesoal problem must be solved. For example, the prce of a xed strke merca-style sa opto must be foud by solvg the two-dmesoal PDE. We apply the above methods (.e. the ux lmter ad full Newto terato to a full two-dmesoal problem. I ths case a teratve method, ILU-CGSTB (D'zevedo et al., 99 va der Vorst, 99, s used to solve the resultg Jacoba matrx. The outle of the paper s as follows. Secto descrbes the opto prcg models to be cosdered. Secto presets a dscretzato aalyss for te derece methods as appled to stadard optos. We cocetrate o stuatos wth extremely low volatlty whch, as oted above, are aalogous to the case of sa optos. We llustrate the types of problems whch ca arse wth commoly appled methods ace ad also how our modcatos mtgate these dcultes, both terms of opto prces ad hedgg parameters. Secto 4 presets applcatos to sa optos, ad the paper cocludes wth a bref summary whch s cotaed Secto 5. The Models We adopt the usual geometrc Browa moto model for the evoluto of a stock prce S: ds = rsdt SdB ( where r deotes the rsk free terest rate, s the volatlty, ad db s a stadard Browa moto. Uder the covetoal assumptos of frctoless markets, the value at tme t of a clam cotget o the stock prce at subsequet tme T may be represeted as: V (S(t t=e ;r(t ;t E t [g(s(t T] ( where g(s(t T deotes the payo fucto for the clam ad E t deotes expectato codtoal o formato avalable at tme t. Famlar examples clude Europea calls (g(s(t T = max(s(t ; K ad puts (g(s T = max(k ; S(T where K s the strke prce of the opto. It s well kow that V solves the followg PDE: V t S V V rs ; rv = ( S S

5 subject to the approprate boudary codtos for the call or put. alytc solutos for these cases were derved by Black ad Scholes (97. The early exercse feature for merca put optos ca be corporated by mposg the costrat V (S( max(k ; S( (4 at each pot tme over the lfe of the opto. The hedgg argumets uderlyg ( are stadard ad may also be appled the cotext of sa optos (see Igersoll (987 pp for a dscusso. Such optos deped o the arthmetc average of the stock prce over some tme terval. If we let I(T = Z T S(d the the average s gve by (T =I(T =T. s oted by Igersoll, the value of a sa opto s gve by the followg PDE wth two space dmesos: V t S V V rs S S S V ; rv =: (5 I equvalet formulato terms of the average ( rather tha the rug sum (I s gve equato (.4 of Barraquad ad Pudet (996: V t S V V rs S S (S ; V ; rv =: (6 t ga, deret termal boudary codtos may be used to prce varous deret types of securtes. Examples clude: oatg strke call: g(s(t (T T = max(s(t ; (T oatg strke put:g(s(t (T T = max((t ; S(T xed strke call: g(s(t (T T=max((T ; K xed strke put: g(s(t (T T=max(K ; (T The oly kow aalytc soluto s for the xed strke case whe K =. early exercse costrat smlar to (4 may be appled to value merca-style sa optos. It s mportat to ote that (5 has o duso term the I drecto ad, smlarly, (6 has o duso term the drecto. Ths fact s the source of may umercal dcultes wth stadard te derece methods. s show by Igersoll, for oatg strke optos (5 may be reduced to a oe-dmesoal PDE by makg the chage of varables R = S=I. Ths s because the PDE ad all of the

6 relevat boudary codtos are learly homogeeous S ad I. For xed strke optos, ths homogeety does ot hold for the termal value ad so the reducto caot be appled. alogous to (, solutos to (5 ad (6 may be represeted as: V (S(t I(t t=e ;r(t ;t E t [g(s(t I(T T] ad V (S(t (t t=e ;r(t ;t E t [g(s(t (T T] (7 respectvely. Rogers ad Sh (995 have recetly formulated a alteratve PDE based o the represetato (7 ad a scalg property of geometrc Browa moto. They dee a ew state varable x = K ; R t S((d S t where s a probablty measure wth desty (t ( T. For a xed strke opto, (t ==T. For a oatg strke opto, K = ad (t ==T ; (T ; t, where s a delta fucto. Rogers ad Sh show thatthevalue of a sa opto s govered by the followg PDE: W t W x x ; ((trxw = (8 x The termal codtos for a xed strke call ad oatg strke put are ad W (x T = max( ;x W (x T = max( ;x ; respectvely. The prce of a xed strke call wth exercse prce K ad tal stock prce S s S W ( K S. For a oatg strke put, the prce s S W (. I ths settg, we have a oe-dmesoal PDE for both xed ad oatg strke optos. However, t caot be appled the case of merca-style optos (both the orgal represetato (7 ad the desty (t are deed accordg to exercse occurrg oly at maturty T. Summg up, Europea-style sa optos may be valued usg oe-dmesoal PDEs, ether the Rogers ad Sh framework (for both xed ad oatg strke optos or after a chage of varables (5 or (6 (oly for oatg strke optos. Ths chage also permts the prcg of merca-style oatg strke optos oe-dmeso. To value xed strke optos wth early exercse opportutes, we must solve a two-dmesoal PDE gve by (5 or (6. 4

7 Dscretzato alyss Before addressg the ssue of dscretzg sa opto models, we wll exame several dscretzato techques for the Black-Scholes (97 equato. s oted earler, our ma motvato s to show the types of problems whch ca arse whe stadard methods are used for problems wth very low volatlty as well as how our modcatos may be used to cotrol for these adverse eects. lthough we demostrate the problems for out-of-themoey Europea call optos, the problems are pervasve for sa opto models (-, atad out-of-the-moey.. Europea Optos The prce of a Europea opto ca be determed by solvg equato ( subject to the approprate termal ad boudary codtos. Equato ( s a backward lear parabolc equato ad may also be referred to as a covecto-duso equato (Roache, 97. The value of a Europea call opto ca be determed by solvg ( subject to the termal codto ad boudary codtos V (S(T T = max(s(t ; K V ( t=adv (S(t t S(t ; Ke ;r(t ;t as S(t!: To value a Europea put opto, equato ( must be solved subject to the termal codto ad boudary codtos V (S(T T = max(k ; S(T V ( t=ke ;r(t ;t ad V (S(t t ass(t!: The Black-Scholes equato ca be coverted to a forward equato tme by substtutg t wth t = T ; t whch evolves from exprato to the preset. fter performg the chage of varables, equato ( becomes V t = S V S ; (;rsv S whch s a form that s commo ud dyamcs. The term S V S 5 ; rv: (9

8 -/ / S- S S Fgure : Schematc represetato of the te volume method. s a parabolc duso term. The magtude of the duso s gve by S. I equato (9 (;rs V S s a rst-order hyperbolc covectve term. The covectve term propagates formato wth avelocty of;rs. Sce rs, the formato ows from the S!boudary to the computatoal doma. If the velocty term s large compared to the duso term, the equato (9 s sad to be covecto domated. lthough equato (9 s formally parabolc, whe t s covecto domated the umercal approxmato behaves as f t was hyperbolc ad s therefore much harder to solve accurately. For certa path-depedet optos, such as sa optos, the problem of covecto domated PDEs ca be especally severe. Equato (9 ca be dscretzed usg the te volume approach (see Fgure for a schematc represetato ad Roache (97 for a dervato of the method. The resultg dscretzato wth temporal weghtg for the value at cell at tme-step wrtte geeral form s where V ; V = F t ; F ; F ; F f ( ; F ; ( ; F (; f ; ( = temporal weghtg ( = ux eterg cell at terface ; = ux leavg cell at terface f = source/sk term: For a fully-mplct method we let =,for = wehave the Crak-Ncolso method ad for a fully-explct method we let =. The R.H.S. of equato (9 collects terms volvg 6

9 spatal dervatves to what are referred to computatoal ud dyamcs as ux terms (deoted by F ad source/sk terms (whch do ot clude spatal dervatves ad are dcated by f. I partcular ad F ; F = S 4 (; S = S 4 (; S (V ; V ; S ; (V ; V S (;rs V ; (;rs V 5 ( 5 ( f =(;rv : ( If we use a pot-dstrbuted te volume scheme (.e. cell terfaces are mdway betwee adjacet odes, the ad S = S ; S ; equato (. S = S ; S Note that the ux fuctos ( ad ( allow for o-uform grd spacg. Thus, we ca costruct grds whch wll make the umercal computatos more ecet by havg a e grd spacg ear ad at the exercse prce ad a coarse grd away from the exercse prce. We wll rst exame hadlg the covectve termv followg cetral weghtg scheme equato ( usg the V = V V whch has secod-order accuracy for uform grds. To esure that solutos produced usg cetral weghtg are free of spurous oscllatos, we must satsfy the Peclet codto (Shyy, 994 ad the addtoal codto ( ; t > S S ; S ; 7 > r S (4 S S S r (5

10 for all cells. For a precse deto of \spurous oscllatos", ad a dervato of (4 ad (5, refer to ppedx. Whe equato ( s covecto domated (.e. whe r s large relatve to the grd spacg ecessary to satsfy codtos (4 ad (5 becomes prohbtvely e. Note that f S =,thes ; = S at = equato (4. Thus, equato (4 at = becomes S > r S : (6 Codto (6 mples that >, whch may ot be satsed, depedet ofhow e a r grd spacg s used. However, ths does ot preset a problem practce sce the covectve ux leavg cell s very small because the velocty soly;rs. Fgure cotas plots of the prce, delta, ad gamma of a Europea call wth oe year to maturty whe K =5,r =:5 ad =:. The value was calculated usg the Crak-Ncolso method wth a uform grd spacg of S =: adt =:, ad cetral weghtg for the covecto term. The soluto s oscllatory because the grd spacg volated the Peclet codto. lthough the oscllatos are small for the opto value, they crease sgcatly for the sestvtes. Such a terest rate/volatlty structure s clearly urealstc. However, ths example was chose because, geeral, the prce of a cotuously averaged sa opto ca be modeled by a two-dmesoal PDE wth o duso oe of the dmesos. Hece, ths choce of parameters serves to llustrate the dcultes volved solvg sa opto problems. It s sometmes suggested the ace lterature that a log trasform be performed o the Black-Scholes equato (Brea ad Schwartz, 978 Hull ad Whte, 99. Brea ad Schwartz obtaed V t V y (r ; V y ; rv = (7 after performg the followg substtuto of varables y = l(s equato (. Usg the te volume dscretzato ( after covertg equato (7 to a forward PDE, the ux fucto becomes F = y 4 (; (V ; V y (;r V The log trasformed PDE has the coveet property that the covecto ad duso coecets are costat, whch smples the umercal soluto somewhat. Brea ad Schwartz state that explct methods are geerally ustable whe appled to equato ( 5 : 8

11 . Cetral Weghtg Black Scholes.8 Call Value Stock Prce..5.5 Cetral Weghtg Black Scholes.8.5 Delta.6 Cetral Weghtg Black Scholes Gamma Stock Prce Stock Prce Fgure : Europea call prce, delta, ad gamma whe K =5,r =:5, =: ad T ; t =:. Calculated usg cetral weghtg wth S =:, t =: ad =. Plotted agast the Black-Scholes aalytcal soluto. 9

12 drectly, ad that the log trasformato allows for the drect applcato of explct methods to equato (7. These statemets are, fact, somewhat msleadg. If oe esures that the codtos (4 ad (5 are met, a fully explct method wll be stable ad free of oscllatos whe solvg (. Codtos must also be met to prevet spurous oscllatos whe the trasformed PDE s beg solved. For equato (7 we must meet the Peclet codto ad the addtoal codto ( ; t > y ; > y ; r ; (8 y y y r: (9 Notce that for the log trasformato codtos (8 ad (9 are costat for uform grd spacgs, ulke codtos (4 ad (5 whch vary over the grd for equato (. The log trasformato appears to elmate the problem of ot beg able to satsfy the Peclet codto (4 as S! f r, whe the PDE s posed the (S t doma. However, the log trasformato eectvely does ot solve the problem as S! sce ths would mply that y!;. Of course, practce ths problem s avoded because a te computatoal doma s used. It s also terestg to ote that the eectve S spacg s very small for small values of y. Fgure demostrates that oscllatos ca also occur whe the log trasformed equato s solved usg a cetrally weghted covecto scheme, ad codtos (8 ad (9 are ot met. The Peclet codto (4 ca be re-wrtte as (rs S ; S < ( where the L.H.S. s the cell Peclet umber (Shyy, 994. If the grd spacg s ot sucetly e whe the covecto term domates the duso term (.e. whe r s large relatve to, the cell Peclet umber wll exceed codto (. To elmate the eed for excessvely e grd spacg the true duso ca be augmeted by addtoal umercal duso. Oe approach to supplemet the true duso wth umercal duso whch has bee used computatoal ud dyamcs s rst-order upstream weghtg (Roache, 97. The rst-order upstream weghtg scheme for equato (9 s V = V up = ( V f ;rs otherwse V

13 . Cetral Weghtg Black Scholes.8 Call Value Stock Prce..5.5 Delta.8.6 Gamma.5 Cetral Weghtg Black Scholes.4.5. Cetral Weghtg Black Scholes Stock Prce Stock Prce Fgure : Europea call prce, delta, ad gamma whe K =5,r =:5, =: ad T ; t = :. Log trasformed PDE solved usg cetral weghtg wth y = :, t =:4 ad =. Plotted agast the Black-Scholes aalytcal soluto.

14 equato (. I ths case, sce rs, V up = V.Coversely, V s termed the dowstream value. Upstream weghtg correspods to a ud ow problem, where formato should oly ow from upstream to dowstream cells. The artcal duso s troduced through the trucato error of the oe-sded derecg (Roache, 97. The scheme s rst-order accurate for uform grds. The accuracy deterorates for o-uform grds because the dscretzato does ot produce true oe-sded dereces. Ths was ot a ssue for our examples, whch used uform grds. To prevet oscllatos from formg whe upstream weghtg s used we must meet oly the followg codto ( ; t > S S ; S S S rs S r: ( The dervato of ( s aalogous to the dervato of (4 ad (5 ppedx. Fgure 4 demostrates how the solutos are o loger oscllatory. Ufortuately, t s also apparet that rst-order upstream weghtg produces soluto proles that are too duse. To produce oscllato free solutos wthout the excessve duso of rst order upstream weghtg we examed the o-lear va Leer ux lmter (Sweby, 984 Blut ad Rub, 99. For the va Leer ux lmter equato (, where V = V up (q (V dow ; V up ( q = V up ; V up S up ; S up / V ; V dow up ( S up ; S dow (ths formulato allows for o-uform grds refer to ppedx C for a dervato ad (q = q q q I equato ( V up s the secod upstream pot. That s, f V s the upstream pot toode, the V s the secod upstream pot. Coceptually, the scheme oly adds umercal duso at pots where the gradet s steep. The scheme s secod-order accurate away from regos whch are augmeted by umercal duso, ad has the property that t s total varato dmshg (TVD. scheme s TVD whe TV(V TV(V :

15 . Upstream Weghtg Black Scholes.8 Call Value Stock Prce Delta Gamma.5 Upstream Weghtg Black Scholes... Upstream Weghtg Black Scholes Stock Prce Stock Prce Fgure 4: Europea call prce, delta, ad gamma whe K =5,r =:5, =: ad T ; t =:. Calculated usg rst-order upstream weghtg wth S =:, t =: ad =. Plotted agast the Black-Scholes aalytcal soluto.

16 where TV(V s the total varato of the soluto ad s deed as TV(V = X V ; V : Thus, f a scheme s TVD the soluto caot cota oscllatos, otherwse the total varato would crease. Stablty ad covergece proofs of TVD methods for coservato laws ca be foud LeVeque (99. mportat compoet covergece proofs s a boud o the varato of the soluto (Sweby, 984. I ppedx B we show that these methods are also TVD for o-coservatve PDEs, such as equato (. For a detaled aalyss of the TVD codtos for equato (, refer to ppedx C. Note that f a lmter s ot used, as Dewye ad Wlmott (995, the the soluto caot be guarateed to be oscllato free uless the grd s very e. Fgure 5 demostrates that the va Leer ux lmter produced oscllato free solutos wthout the excessve smearg of rst-order upstream weghtg. Sce the ux lmter s o-lear the solutos were obtaed usg full Newto terato. The va Leer ux lmter should be used wth the Crak-Ncolso method whe equato ( s covecto domated. I ths stuato, a fully-mplct method geerates a smeared soluto, as Fgure 6 llustrates. Ths s due to the fact that the Crak-Ncolso method s secod-order accurate tme, whle the fully-mplct method s oly rst-order accurate tme.. merca Optos The value of a merca put opto must meet the codto V (S( max(k ; S( (4 at all pots tme over the lfe of the cotract. Wlmott, Dewye ad Howso (99 have show that a merca opto ca be valued by solvg V t S V V rs rv (5 S S subject to costrat (4. If we cosder solvg the dscrete system wth two ukows at each ode, ad V where V s the value of the merca opto, the equatos (4 ad (5 ca be posed dscrete form as ; V = F t ; (V ; V V ; F (V V V f (V ( ; F (V ; ; V V ; ( ; F (V V V (; f (V (6 4

17 . Va Leer Flux Lmter Black Scholes.8 Call Value Stock Prce..5.5 Va Leer Flux Lmter Black Scholes Delta.8.6 Va Leer Flux Lmter Black Scholes Gamma Stock Prce Stock Prce Fgure 5: Europea call prce, delta, ad gamma whe K =5,r =:5, =: ad T ; t =:. Calculated usg the va Leer ux lmter wth S =:, t =: ad =. Plotted agast the Black-Scholes aalytcal soluto. 5

18 . Va Leer Implct Black Scholes.8 Call Value Stock Prce Va Leer Implct Black Scholes Delta.6 Gamma.5.4. Va Leer Implct Black Scholes Stock Prce Stock Prce Fgure 6: Europea call prce, delta, ad gamma whe K =5,r =:5, =: ad T ; t =:. Calculated usg the va Leer ux lmter wth S =:, t =: ad =. Plotted agast the Black-Scholes aalytcal soluto. 6

19 K Moths Bomal Va Leer Table : merca put values whe r =:5 ad =:. The va Leer lmter was used wth =,t =:9 ad a o-uform spatal grd wth S =:5 ear ad at the exercse prce. The bomal results were obtaed from Geske ad Shastr (985. where for a put opto Note that the L.H.S. of equato (6 ca be wrtte as V = max( K ; S (7 ad sce by (7 V t ; V ; V t ; V : t Thus, equato (6 s a dscrete form of the equalty (5. If the system s o-lear, the costrat s appled at each Newto terato. Table cotas results obtaed usg the costrat a mplct fully coupled maer wth the va Leer lmter to prce merca put optos. s the table demostrates, the prces geerated are vrtually detcal to those produced by the bomal method. The tradtoal approach see the ace lterature whe usg PDEs to value merca optos s to apply the merca costrat explctly (Brea ad Schwartz, 977 Geske ad Shastr, 985 Hull, 99. That s, equato ( s solved ad after each tme-step the costrat s appled to the soluto. Ths ders from the mplct fully coupled method whch solves equato (5 drectly. However, we oly otced dereces the rate of covergece for large tme-steps. That s, the mplct applcato of the costrat requred fewer o-lear teratos whe the tme-step was large. These tme-steps were too large to 7

20 acheve covergece to a accurate soluto. For sutable tme-steps, we foud o derece the rate of covergece. Thus, a practcal sese, there s o derece betwee applyg the costrat mplctly or explctly. lthough we have ever expereced stablty problems usg practcal tme-step szes, there s always a possblty that the explct applcato of the costrat wll lead to stablty. Cosequetly, we wll apply the merca costrat mplctly. The addtoal computatoal cost of applyg the costrat mplctly s eglgble. 4 Cotuous rthmetc sa Optos Based o the results from secto, we wll ow exame treatg covecto usg the va Leer ux lmter dscretzatos of oe- ad two-dmesoal PDE models of sa optos. 4. Oe-Dmesoal Models Oe-dmesoal models for prcg cotuously averaged arthmetc sa optos have bee derved by Igersoll (987 ad Rogers ad Sh (995. The Igersoll model caot be used to prce xed strke optos. However, t ca be used to prce merca-style oatg strke optos (Wlmott et al., 99. lthough the Rogers ad Sh (995 model caot hadle the early exercse feature, t ca be used to prce both oatg ad xed strke optos. Cosequetly, wechose to exame ths model. fter covertg (8 to a forward equato, we have the followg ux fucto F = x 4 (; x (W ; W x ( rx W ad source term f =. It should be oted that x wll take o egatve values ad thus, ( rx may take o both egatve ad o-egatve values. The dscretzato of W must take ths to accout. Tables ad cota the results obtaed by usg the va Leer ux lmter to value xed ad oatg strke sa optos, respectvely. Rogers ad Sh (995 examed a umber of deret volatlty/terest rate structures. Tables ad cota oly the results for r = :5, ther most dcult case. Note that the tme-step sze for these examples was selected o the bass of several tral rus, whch dcated that decreasg the tme step sze further had o dscerble eect o the results to four sgcat gures. The results demostrate that sucetly accurate values ca be obtaed for most volatltes uder a average of secods for both xed ad oatg strke sa optos. Suf- cetly accurate results were obtaed for all volatltes uder a average of 6 secods. 8 5

21 K x Bouds.5..5 Lower Upper Mea Exec. Tme (sec Table : Fxed strke sa call values whe r =:5, S = ad T ; t =. The va Leer lmter was used wth = ad t =:5. o-uform spatal grd was used, where x deotes the spacg the rego [;:5 :5]. Mea executo tmes are for rus performed o a DEC lpha. The bouds were obtaed from Rogers ad Sh (995. By sucetly accurate, we mea that the values are o more tha.5% of S outsde of the bouds derved by Rogers ad Sh. It should be oted that equato (8 wll become covecto domated as x!, ad for short maturtes sce (t = or (t = ; (T ; t. T T lthough Tables ad do ot cota results for short maturty optos, we wll see secto 4. (see Table 4 that accurate results ca be obtaed for short maturtes usg the va Leer lmter. Fgure 7 compares results that we obtaed usg the va Leer lmter ad a cetrally weghted scheme wth Rogers ad Sh's (995 results whch were obtaed usg the method of les. The same coarse spatal grd was used for all the PDE rus. Note that the dscretzato whch uses the va Leer lmter coverges more rapdly tha ether cetral weghtg or the method of les. For ths low volatlty case the bouds that Roger ad Sh derved are very tght. I fact, the bouds were plotted as cross hars because the upper ad lower bouds appear at the same pot whe plotted. It s terestg to ote that sophstcated PDE methods must be used f the problem s covecto domated, whle the PDE problem s easly solved whe the volatlty s large. 9

22 x Bouds.5..5 Lower Upper Mea Exec. Tme (sec Table : Floatg strke sa put values whe r =:5, S = ad T ; t =. Theva Leer lmter was used wth = ad t =:5. o-uform spatal grd was used, where x deotes the spacg the rego [;:5 :5]. Mea executo tmes are for rus performed o a DEC lpha. The bouds were obtaed from Rogers ad Sh ( Cetral Weghtg Rogers & Sh Bouds 7 6 Va Leer Rogers & Sh Bouds 5 5 Call Value 4 Call Value x x Fgure 7: Fxed strke sa opto values whe r =:5, =:5 ad T ; t =:. Note that at the pot tme plotted o the graphs, x = K S. The solutos were computed usg cetral weghtg ad the va Leer ux lmter wth x =:5, t =:5 ad =. The Rogers ad Sh (995 results were calculated usg the method of les. The bouds were obtaed from Rogers ad Sh.

23 4. Two-Dmesoal Models Whe merca-style xed strke sa optos are to be prced, we caot use the oedmesoal models outled secto 4. (Wlmott et al., 99 Barraquad ad Pudet, 996. I these cases a full two-dmesoal PDE must be solved. We chose to exame equato (6 because umercal expermets usg equatos (5 ad (6 dcated that fewer odes were eeded to acheve equvalet accuracy for equato (6 whe compared to (5. Ths was due to the fact that the sgcat part of the computatoal doma appears to be smaller for formulatos usg the average as opposed to the rug sum. Oe may ote that at t = a sgularty exsts equato (6 because of the V (S ; term. However, t ths ca be avodedfwe assume that S = at t =. Thus, equato (6 smply becomes the Black-Scholes equato at t =. where fter covertg equato (6 to a forward PDE, the te volume dscretzato s V ; V = F t ; F j F j; j j ; F ; F j f j ( ; F ; ( ; F ; j j ( ; F j; ; ( ; F j (; f j = j t ( j ; S j V : j s was the case secto 4., order to esure that the upstream pots are deed approprately the dscretzato of V usg the va Leer lmter must take to accout the j fact that ( t j ; S j wlltake o egatve ad o-egatve values. lso, f at j max there exst j max whch are less tha S j max, the a approprate boudary codto must be mposed at those pots. F ad f j j are smlar to ( ad (, respectvely. Sce the PDE s two-dmesoal, we used complete LU decomposto ad the stablzed cojugate gradet method (ILU-CGSTB to solve the resultg system of equatos (D'zevedo et al., 99 va der Vorst, 99. We also corporated a tme-step selector (Forsyth ad Sammo, 986 to our solver. We foud that the qualty of the soluto for certa termal codtos s hghly depedet upo the grd spacg. We cojecture that the problem arses because equato (6 s smlar to the turg pot problem, whch has kow dcultes (scher et al., 988 Tag, 99. Ths s oly a ssue the case of Europea oatg strke calls ad puts. Oe must esure that the S : aspect rato for grd spacgs used these cases s always k :, where

24 5 Call Value 5 5 S 5 5 Fgure 8: merca xed strke callvalue whe K =,r =:, =:, ad T ; t = :5. Calculated usg the va Leer ux lmter wth =. k s a arbtrary teger greater tha or equal to. If oe s terested the values ad sestvtes over a large rego of the grd, t would be best to use a uform grd spacg the dmeso that acheves sucet accuracy. We emphasze, however, that ths ssue s easly avoded etrely sce these cases ca be hadled usg a oe-dmesoal model. Tables 4, 5 ad 6 cota the results of usg the va Leer ux lmter to solve equato (6. Table 4 cotas the results for Europea ad merca xed strke call optos. The results for zero strke sa call optos, whch have a aalytcal soluto, are cotaed Table 5. Table 6 cotas the results for Europea ad merca oatg strke puts. The parameters were chose to allow a comparso wth Barraquad ad Pudet's (996 results. The grd spacg was chose to acheve a accuracy of at least :% of S. To determe the accuracy, the computed lower boud ad the soluto (usg a e grd to the oedmesoal Roger ad Sh (995 model were take to be the true solutos. Fgures 8 ad 9 are plots of a merca xed strke call ad merca oatg strke put, respectvely. Barraquad ad Pudet (996 correctly observe that a explct cetrally weghted scheme for equato (6 s ustable. Note that equato (6 s covecto domated the drecto because there s o duso eect ths dmeso. I partcular, the covectve term the dmeso becomes very large as t!. Barraquad ad Pudet also ote that mplct cetrally weghted schemes wll geerally produce usatsfactory results because of the umercal duso troduced by ths rst-order accurate tme scheme. More m-

25 T-t K Europea merca Lower -D -D B&P -D B&P Mea Executo Tme (sec Table 4: merca ad Europea xed strke callvalues whe r =: ad S =. Lower ad -D refer to the Rogers ad Sh (995 lower boud ad oe-dmesoal PDE, respectvely. B & P refers to the results obtaed by Barraquad ad Pudet (996. -D refers to the value obtaed by solvg the two-dmesoal PDE usg the va Leer lmter wth = ad a o-uform spatal x S grd of 4 x 45. t was set to oe day, two days ad three days for maturtes of three, sx ad twelve moths, respectvely. Mea executo tmes are for rus performed o a DEC lpha.

26 T-t alytc -D Table 5: alytc ad umercal soluto of the two-dmesoal PDE model for the zero strke sa call whe r =: ad S =. The va Leer lmter was used wth = ad a o-uform spatal x S grd of 4 x 45. t was set to oe day, two days ad three days for maturtes of three, sx ad twelve moths, respectvely. The alytcal results were obtaed from Barraquad ad Pudet (996. T-t Europea merca Lower -D -D B&P -D B&P Mea Executo Tme (sec Table 6: merca ad Europea oatg strke putvalues whe r =: ad S =. Lower ad -D refer to the Rogers ad Sh (995 lower boud ad oe-dmesoal PDE, respectvely. B & P refers to the results obtaed by Barraquad ad Pudet (996. -D refers to the value obtaed by solvg the two-dmesoal PDE usg the va Leer lmter wth = ad a o-uform spatal x S grd of 4 x 45. t was set to oe day, two days ad three days for maturtes of three, sx ad twelve moths, respectvely. Mea executo tmes are for rus performed o a DEC lpha. 4

27 S 5 5 Fgure 9: merca oatg strke put value whe r =:, =:, ad T ; t =:5. Calculated usg the va Leer ux lmter wth =. portatly, Barraquad ad Pudet fal to meto that solutos geerated usg a cetrally weghted scheme for equato (6 caot be esured to be free of oscllatos. For example, Fgure demostrates the severe oscllatos that resulted from prcg a Europea-style xed strke call opto usg cetral weghtg. The grd spacg used was detcal to that used to obta our results wth the va Leer lmter. Usg the va Leer ux lmter cojucto wth the Crak-Ncolso scheme gves us a method that s oscllato free ad secod-order accurate tme. The careful reader wll ote that as t! the CFL codtos establshed ppedx C caot be satsed. However, ths does ot appear to have aected the qualty of our results. If oe wated to esure that the method was TVD eve as t!, the ux the drecto ca be swtched from a Crak-Ncolso scheme to a fully mplct scheme for specc cells whe the CFL codto s volated. lthough ths wll result addtoal umercal duso, we foud that the values dered by o more tha $:5 from the results reported here. Barraquad ad Pudet (996 state that because the correlated varables ad S are take to be depedet equato (6, umercal schemes for solvg equato (6 wll ot ecessarly coverge to a soluto. However, as the results Tables 4, 5 ad 6 demostrate, we dd ot experece ay falures to coverge to a approprate soluto durg our study. stroger demostrato of covergece usg successve grd reemets s cotaed Table 7. 5

28 Call Value S 8 4 Fgure : Europea xed strke call opto whe K =, r = :, = :, ad T ; t =:5. Calculated usg cetral weghtg wth =. K Spatal Grd ( x S 4 x 45 8 x 89 6 x Europea merca Table 7: Successve grd reemets demostratg the covergece of the two-dmesoal PDE for a xed strke sa opto whe S =, r =:, =: ad T ; t =:5. 6

29 T-t Mea Exec. Tme (sec. B&P -D -D Table 8: Mea executo tmes for obtag Europea xed strke call values. B & P refers to the forward shootg grd algorthm. -D ad -D refer to results of smlar accuracy obtaed usg oe-dmesoal ad two-dmesoal PDEs, respectvely. Rus were performed o a DEC lpha. The results Tables 4, 5 ad 6 demostrate that the desred level of accuracy of :% of S ca be obtaed all cases uder a average (for all maturtes cosdered of 5 secods. We mplemeted Barraquad ad Pudet's (996 forward shootg grd algorthm ad foud that the desred level of accuracy could be obtaed uder a average (for all maturtes cosdered of secods. More speccally, 9 tme-steps were requred for the three ad sx moth maturtes usg tme-step szes of oe ad two days, respectvely. The average tme requred for 9 tme-steps was uder secods. lthough the parameters were the same as those used by Barraquad ad Pudet, our tme of secods s slghtly hgher tha the Barraquad ad Pudet result of 5 secods. By performg the rus o a deret DEC lpha from the oe used for our aalyss, we were able to obta a average tme of 5 secods. For the oe year maturty wth a tme-step sze of three days, tme-steps were requred. I ths case the average tme requred was uder 54 secods. The substatal crease tme requred s due to the fact that the forward shootg grd algorthm has a tme complexty ofo(n, where N s the umberoftme-steps.for a gve grd sze, the tme complexty for a PDE method s lear, that s O(T. Thus, as the umber of requred tme-steps creases, the tme requred for the Barraquad ad Pudet algorthm wll grow dramatcally. Table 8 demostrates the cubc tme complexty of the forward shootg grd algorthm ad the lear tme complexty (for a gve grd sze of PDE methods. For example, to value a two year xed strke call opto wth a tme-step sze of three days requred a average of 485 secods usg the Barraquad ad Pudet algorthm. Ths compares to a average of 8 secods for the two-dmesoal PDE ad 5 secods for the oe-dmesoal PDE. Both methods have polyomal spatal complextes wth degrees (of two for the oe asset case that grow wththeumber of assets, ad thus ca oly be used to solve problems wth a low umber of spatal dmesos. 7

30 5 Coclusos The ave applcato of cetral dereces to the umercal treatmet of certa partal dfferetal equatos ca result a umber of dcultes. Wehave demostrated that treatg covecto usg cetral dereces the dscretzato of PDEs wth low duso relatve to covecto, such as sa opto models, ca produce solutos cotag spurous oscllatos. s oted by Barraquad ad Pudet (996, explct cetrally weghted schemes are ustable whe appled to equatos (5 ad (6. Furthermore, oe caot esure that mplct cetrally weghted schemes wll be free of oscllatos whe appled to equatos (5 ad (6. To remedy the problem of spurous oscllatos produced by cetrally weghted schemes whle stll matag hgh order accuracy, we employed the use of a hgh order ux lmter. We treated covecto usg the secod-order accurate va Leer ux lmter. The lmter s secod-order accurate away from regos wth steep gradets where t augmets the true duso wth umercal duso. The va Leer lmter has the property that t s total varato dmshg ad thus produces oscllato free solutos. Usg the va Leer lmter cojucto wth the Crak-Ncolso scheme gves us a method that s secod-order accurate tme ad oscllato free. We have demostrated that the applcato of the va Leer lmter to oe-dmesoal sa opto models (Europea ad merca oatg strke, ad Europea xed strke leads to the rapd computato of accurate solutos (.e. wth a average of secods for most volatlty/terest rate structures for maturtes of up to oe year. For the most extreme volatlty/terest rate structures a accurate soluto ca be obtaed wth a average of 6 secods. Whe the full two-dmesoal model must be used, as s the case for merca xed strke optos, the computato tme aturally creases. However, accurate solutos ca be computed uder a average of 5 secods (DEC lpha for the maturtes that we cosdered. ccurate solutos were obtaed for both Europea- ad merca-style sa optos usg the two-dmesoal model. PDE methods have O( d spatal complextes, where s the umber of cells a dmeso ad d s the umber of dmesos. Thus, PDE methods ca oly be used to solve problems wth a low umber of spatal dmesos. Ths s also the case for Barraquad ad Pudet's (996 forward shootg grd algorthm. However, the forward shootg grd algorthm has a cubc tme complexty compared to the lear tme complexty of PDE methods. Thus, the forward shootg grd algorthm s less desrable for large umbers of tme-steps. For example, the prcg ad hedgg of log-term (e.g. 5 year sa optos s a problem of practcal terest to surace compaes sellg guaratees o vestmet auty products. The applcato of the va Leer lmter s ot lmted oly to PDE prcg models for 8

31 sa optos. It ca be appled to other acal PDE models that have the problem of covecto domace. Furthermore, sce the method s o-lear t ca be easly exteded to solve o-lear opto models such as that of Peszek (995. ppedces Preveto of Spurous Oscllatos I ths appedx we wll derve the codtos uder whch cetrally weghted schemes for equato (9 wll ot produce spurous oscllatos. The same argumets ca be used to derve codtos for other schemes, such as, rst-order upstream weghtg. Usg the pot-dstrbuted te volume dscretzato V ; V = F t ; ; F f wth cetral weghtg space ad fully mplct tme-steppg for equato (9 gves us V ; V = t Lettg k = S 4 (; S S ; S 4 (; S V ad a = rs, ad smplfyg ; V = ; (V ; V ; (V (;rs S ; (V ; V (V (;rs S k t S S ; k t S S ; k t S S ; a t V ; S ; k t S S a t S ; V V ; rt V 5 5 ; rv : V : (8 fter corporatg a temporal weghtg factor, (where = s a fully mplct scheme, = s the Crak-Ncolso method ad = s a fully explct scheme, equato (8 becomes V ; V = k t S S ; ; a t V ; S 9

32 k t ; S S ; ( ; k t S S k t ; k t S S a t S S S ; ( ; k t ; S S ; ( ; Regroupg terms equato (9 gves ; ; = k t S S ; k t S S ; k t S S ( ; ; ( ; ad the rearragg ; ( ; k t S S ; k t k t S S ; a t V ; S k t S S V S a t S S ; k t S S ; k t S S ( rt V ; a t V ; S V ; a t S ; rt V V ; ; k t S S a t S rt V ; ( ; a t S V ; k t ; ( ; S S (; a t V S k t S S ; k t S S ; rt V V : (9 ; ( ; rt V V

33 ; k t S S a t V S = ( ; k t S S ; ( ; ( ; rt V ( ; k t S S ; ( ; a t S V ; k t ;( ; ; ( ; S S ; (; a t S k t S S V V : ( I order to solate the behavor of the soluto from the spatally depedet expoetal decay (whch s due to the;rv term equato (9, we wll elmate the expoetal decay term by substtutg V = where the superscrpt for ;(;rt rt whch result o-oscllatory behavor for Substtutg ( to equato ( gves ; k t S S ; ( rt ; k t S S ; k t S S ; a t ; S! ; ( ; rt ( rt s a expoet. We wll determe the codtos ; ( ; rt rt k t S S a t S the followg. ; ( ; rt rt!! ; ( ; rt rt ; ( ; rt rt!! = ( ; k t S S ; ; ( ; a t S ; ; ( ; rt rt! ;

34 ( ; ( ; rt ;( ; ( ; By lettg =( ;(;rt rt ; k t S S ; k t S S k t S S ; ; = k t S S ; ( ; rt rt! ; ; ( ; k t S S (; a t S ; ( ; rt rt! ; ; ( ; rt rt! ; : ( ad dvdg by rt equato ( we obta ; a t ; S rt k t S S ; a t S k t ( ; S S ; k t ;( ; S S ; ( ; Dvdg by gves ; ; = ( rt k t S S ( rt ( rt k t S S ; k t S S rt ; ( ; a t S ; ; ( ; (; a t S k t ; S S ; k t S S ( rt rt! ; rt k t S S! ; rt a t ; S k t ( rt S S ( rt a t S :! ; rt

35 Regroupg terms = ( ; ( ; ( ; rt ; ( ; ( ; ( ; rt ( ; ( ; ( ; rt ( rt ( ; ( ; ( ; rt ; ( ; k t S S ; k t ; S S ; k t S S k t S S ; ( ; ( ; rt ( ; ( ; ( ; rt ( rt ( rt k t S S ; k t S S k t S S ; ( ; ( ; ( ; rt ( ; ; ( ; ( ; rt ( ; ( ; ( ; rt ( rt k t ; S S ; k t S S ; k t S S ( ; ( ; ( ; rt a t ; S k t S S a t S : a t ; S ( ; k t ; ( ; ( ; rt S S ( ; ( ; ( ; rt ( rt ( rt a t ; S a t S a t S : ( If we requre all the coecets of ( to be postve, the we must esure that ad ( ; ( ; ( ; rt ( rt k t S S ; k t S S ; ; ; ( ; ( ; ( ; rt ( rt a t > (4 S a t > (5 S (. becomes fter recallg that k = S ad a = rs, ad smplfyg, codto (4 S ; > r S : (6

36 Codto (6 s also kow as the Peclet codto. Note that f codto (6 s met, the both (4 ad (5 wll be satsed. We also must meet the addtoal codto ; ( ; ( ; ( ; rt fter substtutg k = S k t S S ; ( ; t > S ( ; ; ( ; ( ; rt k t S S > : (7 ad a = rs, ad smplfyg, codto (7 becomes S ; S S S r: (8 If codtos (6 ad (8 are met, the all the coecets of ( are postve ad we ca employ the maxmum prcple. By deg max such that the equato ( ca be wrtte as ( rt ( ; ( ; ( ; rt ; ( ; max =max( ; ; k t S S ; ( ; ( ; rt ( ; ( ; ( ; rt ( rt ( rt k t S S ; k t S S k t S S ; ( rt k t ; S S ; k t S S ; k t S S ( ; ( ; ( ; rt ( ; ; ( ; ( ; rt ( ; ( ; ( ; rt ( rt ( rt a t max S a t max S k t S S a t max S max a t max : (9 S Smplfyg ( rt k t S S ; ( rt k t S S 4

37 ( rt k t S S ; ( rt k t S S max : Thus, Smlarly, f we employ the mmum prcple by deg the max : (4 m = m( ; ; m : (4 Hece, (4 ad (4 mply that o ew local maxma or mma ca occur the umercal soluto for,whch s a precse deto of a o-oscllatory soluto. Sce V = ;(;rt rt ad > ( ; rt by codto (8, the soluto for V s multpled by a spatally depedet postve decay term. Thus, the soluto for V wll ot cota spurous oscllatos f codtos (6 ad (8 are met. B TVD Schemes for PDEs No-Coservatve Form Flux lmtg schemes are ofte expressed as fully explct schemes the lterature (va Leer, 974 Sweby, 984 for PDEs that are the followg form V t = ;(av S (4 where a s the covectve velocty. Equato (4 s sad to be coservatve form. Blut ad Rub (99 have show that partally mplct ad fully mplct schemes are TVD for coservatve equatos. I ths appedx we wll show that fully explct, partally mplct ad fully mplct ux lmtg schemes are TVD for PDEs o-coservatve form, ad derve the crtera uder whch these schemes are TVD. The argumets are smlar to those foud Blut ad Rub (99. Cosder the scalar covecto equato V t = ;a(sv S (4 5

38 whch s o-coservatve form. We make the smplfyg assumpto that we are solvg (4 o a te rego, so that the eect of boudary codtos may be gored. Ths assumpto s usually made TVD aalyss (Sweby, 984 Blut ad Rub, 99. For ux lmtg schemes, V s ofte expressed as (Blut ad Rub, 99 Yag ad Przekwas, 99 V = V up where (q s the lmter fucto ad (q (V dow ; V up (44 q = V ; V ; : (45 V ; V Usg a ux lmtg scheme, the fully mplct te volume dscretzato of (4 for a s V ; V Smplfyg V ; V = a t S whch s equvalet to V t 4 (V = a S 4 V ; ; a 4 V S ; V ; ; V = ; V ; (q ; (q ; (q (q ; 4 ; (V ; V ; (V ; V 5 5 : (V ; V ; (q ; (q where = a t S ad V ; =(V ; V ;. Notg that q V ; V = ; V ; (q ; 4 ; V ; V V ; V ; = V ;V ; V ;V (q q 5 : (V ; V! 5 Icorporatg a temporal weghtg factor, (where = s a fully mplct scheme ad = s a fully explct scheme, V ; V = ; V ; (q 4 ; ; (q ; V;( ; 4 ; ; 6 (q q 5 (q q 5 5 : (46

39 Smlarly, Lettg ad V ; (q V = ; V 4 ; (q ; V ( ; 4 ; c = (q ; ( ; 4 ; ; c (q ; ; = 4 ; (q q (q q (q 5 q (q q : (47 the equatos (46 ad (47 ca be wrtte as ad V ; V = ;c respectvely. Subtractg (48 from (49 V ; V ; ; c ; V ; (48 V ; V = ;c V ; c V (49 ; V = ;c V ; c V ; c V ; c ; V ; : If we mpose that ad for all, the ( c V =(; c V c ; V ; c ;V;: ( c V ( ; c jv c (5 c (5 j c ; 7 V ; ; c ; V :

40 Summg over all ad otg that P c jv j = P c ; V X V X jv j : ; gves us Thus, the scheme wll be TVD f codtos (5 ad (5 are met. Hece, we requre that (q ( ; 4 ; ; (q q 5 (5 ad (q 4 ; ; (q q 5 (5 for all ad, for the scheme to be TVD. Fully mplct (.e. = ux lmtg schemes wll be TVD f we esure that codto (5 s met. If addto to codto (5 we esure that codto (5 s met, the fully explct (.e. = ad Crak-Ncolso (.e. = ux lmtg schemes wll be TVD. Thus, f codtos (5 ad (5 are satsed, the scheme wll be TVD for o-coservatve PDEs. I fact, these codtos are smlar to those requred to esure that ux lmtg schemes are TVD for PDEs coservatve form (Blut ad Rub, 99. Note that equato (4 does ot cota a duso term, thus the codtos wll be overly strget for equatos such as(9. C The Flux Lmter Fucto I ths appedx we wll rst exame the propertes that ux lmter fuctos,, possess for uform grds. We wll the modfy the va Leer ux lmter fucto to accout for o-uform grds. The aalyss for uform grd spacg s smlar to that Blut ad Rub (99. We clude the aalyss for uform grds order to provde the reader wth sucet backgroud to uderstad the aalyss for o-uform grd spacg. The examato of ux lmter fuctos for o-uform grds has ot, to the best of our kowledge, appeared the lterature. s ppedx B, the results ths appedx wll perta to scalar covecto PDEs, such as, equatos (4 ad (4. However, the results ca be exteded to other PDEs, such as, equato (9. For smplcty ad clarty we may at tmes omt superscrpts ad/or subscrpts whe referrg to the ux lmter argumet (.e. q deed equato (45 ad ode values. 8

41 C. Uform Grd Spacg I appedx B we derved codtos (5 ad (5 whch ux lmtg schemes must meet order to be TVD. If we mpose the codtos (q ad (q, the codto (5 q ca be restated as ; (q! : (54 Thus, f (q (55 the codto (54 wll be satsed ad fully mplct (.e. = ux lmtg schemes wll be TVD. Notg that (q ad (q, the codto (5 ca be restated as q ( ; (q q! (56 where, deed ppedx B, s the CFL umber (Roache, 97 Shyy, 994. If (q q (57 the codto (56 wll be satsed for fully explct schemes (.e. =whe.for Crak-Ncolso schemes (.e. = codto (56 wll be satsed whe. Equatos (55 ad (57 dee a rego whch the ux lmter fucto must le order for the scheme to be TVD. The shaded rego gure deotes the TVD rego (Sweby, 984. Note that the codtos (q ad (q mply that (q vashes q whe q<. Referrg to gure, alog the le (q = the ux lmtg scheme (44 reverts to a secod-order accurate cetrally weghted scheme. That s, V = V V whe (q =. log the le (q =q the ux lmtg scheme reverts to V ; V ; V = V whch s a secod-order accurate two-pot upstream weghted scheme. We do ot oly requre that the scheme be TVD, but that t be secod-order spatally accurate wheever possble. Ths ca be acheved by makg the ux lmtg scheme a 9

42 φ=q φ=q, two-pot upstream weghtg φ(q φ=, cetral weghtg q Fgure : TVD rego for uform grd spacg. weghted covex average of a cetrally weghted scheme ad a two-pot upstream weghted scheme (Sweby, 984. Thus, f the lmter fucto s the rego show gure, the the scheme wll be secod-order accurate. Oe such fucto s the va Leer ux lmter (q = jqj q jqj (58 (va Leer, 974 Sweby, 984. Note that whe q thescheme reverts to a rst-order accurate upstream weghted scheme. C. No-Uform Grd Spacg For o-uform grds, a pot-dstrbuted ux lmtg scheme wll ot revert to a two-pot upstream scheme whe (q = q, but rather whe (q = q, where = S ; S S ; S ; for q gve by equato (45. We wll later see that ths mples that we must modfy the ux lmter fucto order to esure that the scheme wll be secod-order accurate wheever possble. Furthermore, we must rst expad the rego that we requre to be TVD. Fgure demostrates that the le represetg the two-pot upstreamscheme wll fall outsde the TVD rego for certa magtudes of grd sze chages. Speccally, f the magtude of the grd sze chage,, s greater tha or equal to, the two-pot upstream scheme wll fall outsde the TVD rego. 4