Constrained Wiener Processes and Their Financial Applications

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1 Journal of Mahemacal Fnance, 08, 8, hp:// ISSN Onlne: 6-44 ISSN Prn: Consraned Wener Processes and her Fnancal Applcaons Andrew Leung Monash Unversy, Melbourne, Ausrala How o ce hs paper: Leung, A. (08) Consraned Wener Processes and her Fnancal Applcaons. Journal of Mahemacal Fnance, 8, hps://do.org/0.436/jmf Receved: Sepember 5, 08 Acceped: November 3, 08 Publshed: November 6, 08 Copyrgh 08 by auhor and Scenfc Research Publshng Inc. hs work s lcensed under he Creave Commons Arbuon Inernaonal Lcense (CC BY 4.0). hp://creavecommons.org/lcenses/by/4.0/ Open Access Absrac he exrema of Wener processes are relevan o he prcng of so-called exoc opons, whch have many fnancal applcaons. he probably denses of such exrema are well known for one dmensonal Wener processes. We employ elemenary mehods o derve analycal expressons for he denses for muldmensonal Wener processes, wh mulple exrema. hese ake he form of (possbly nfne) seres expansons of Gaussan denses. hs s underaken usng he characerzaon of he Wener process by he hea equaon, a well known connecon n mahemacal physcs. Keywords Wener Process, Exoc Opons, Mehod of Images, Hea Equaon. Background I s naural o model many fnancal varables as Wener processes: examples may be found n asse reurns, neres raes, bond yelds as well as nflaon and commody prces. In fac, any saonary varable (and any varable negraed of order zero I ( 0) ) s amenable o such modellng. he markes relang o such varables reflec her varous characerscs n erms of long run behavour, volaly and hgher order momens. Exoc dervaves are desgned o explo he hgher order characerscs hrough her dependence on he evoluon of varables over fne me perods. hs s where he behavour of exrema of Wener processes becomes relevan and mporan. here are well known resuls for he exrema for a one dmensonal Wener processes. For such a process, he exrema are eher he maxmum or mnmum. Bu n pracce fnancal varables canno be vewed o operae n solaon, and s he nerplay beween several varables ha leads o complexy and rchness n OI: 0.436/jmf Nov. 6, Journal of Mahemacal Fnance

2 fnancal modellng. For example we may wsh o model: he behavour of he asse reurns nvolved n an overall porfolo; he nerplay beween bond yelds and neres raes n a fxed neres marke; he dependence of asse behavour on economc varables such as nflaon, employmen and wage growh. Muldmensonal Wener processes are he naural framework for such modellng. However he exrema processes n or hgher dmenson are rcher and more vared han n he one dmensonal case. he purpose of hs paper s o sudy he dsrbuon of hese exrema.. Hea Equaon hee relaonshp beween he Wener process and he hea equaon was frs suded exensvely by [], who approached he soluon of a rchle problem for he hea equaon wh mehods usng probablsc expecaons. In hs paper, we do he reverse, ha s we solve he probablsc problem of he dsrbuon of exrema by expressng as a rchle problem and hen usng elemenary mehods for paral dfferenal equaons (PEs). By focussng on he Wener process, raher han he Black Scholes PE (whch are logcally nerrelaed), we oban smplfcaons n he analyss ha obvae he praccal ssues found n []. Once he dsrbuons of exrema are found, hey may be appled drecly and nuvely o he prcng of exoc opons. 3. Characersaon of Exrema he smples way of vewng he role of exrema s n dscree me, wh a random walk n dmensons. Such a walk sars a 0 and proceeds n seps of ± n one of he possble drecons, all wh equal probably. n hus afer n perods here are ( ) possble pahs. Afer n seps a pon x n s reached, and a pah s denoed x= ( 0x,, x, ) In general we wsh o resrc he possble pahs by mposng m lnear consrans of he form ax y where =,,, m and a, y are specfed vecors. he se of admssble pons s denoed U = { x Ax y } where A = [ a ], a, and y = [ y ], y,. We suppose ha A s of full row rank, so here s no redundancy n he consrans. Whou loss of generaly we may also ake a = for all. Clearly U s a closed convex se, hough possbly nfne, and s boundary s denoed U = { x Ax = y }. enoe by fn ( x ) he probably of reachng x afer n seps under a pah wholly conaned n U. I s clear from he evoluon of pahs ha fn ( x± e) x U f n x = () 0 x U OI: 0.436/jmf Journal of Mahemacal Fnance

3 where e are he un vecors n he h drecon n f 0 =. defnes n f x compleely, sarng wh 0. hs recurrence relaon I s well known ha he lm as n of he random walk s he Wener process n dmensons [3]. We explo hs propery o esablsh he characersaon of he consraned Wener process n connuous me. However he random walk mus be scaled for hs o occur; as he sandard random walk produces a varance of n/ afer n seps, we need o employ a scalng facor of n n measurng he level aaned afer n seps. For a gven perod n connuous me, we hus consder n seps o reach a level of x n. (he values of n mus be chosen o gve hs meanng ha s, n/ mus be a perfec square and n mus be an neger. However all hs s possble snce we consder only he behavour as n.) hen he probably dsrbuon ( n4) fn ( x n) mgh be consdered o converge o a densy, wh he facor ( n 4 ) beng requred o allow for he dscree spacng n across each dmenson of x. All levels, ncludng he consrans y need o be so scaled. he followng operaors are used: = =,,, x x x x m dv = r ( ) = x x x m = r ( ) = x x x he dscree me problem n () converges o he hea PE as follows. Proposon. Suppose ha he probably dsrbuon converges o a connuous me densy. hen ha densy s of he consraned Wener process ϕ (, x ) a me, gven by he soluon of he PE ϕ = ϕ wh he boundary condons ϕ = 0 on U. Proof. hough he resul s well known, s proof s very echncal and no fully provded n he leraure. Hence s se ou n Appendx A. he problem wh he spaal boundary condons s known as a rchle problem. Remark. he convergence for he one dmensonal case s guaraneed by he n n e Movre-Laplace lemma, n whch case fn ( x) = and he lm s n+ x provded by n = π e x + ( ) f x n o n unformly over compac ses of x. hs leads drecly o he Gaussan densy for n f n ( x n ). In hgher dmensons he lm may be derved by consderng he Fourer ransform E θ x θ x θ x θx θx ( e ) = ( e + e + e + e + ) n OI: 0.436/jmf Journal of Mahemacal Fnance

4 whch s anamoun o provng he cenral lm heorem n dmensons (ym). Remark. he appeal o he dscree me case as gven above provdes he nuon o he lnk wh he hea equaon. In fac a drec dervaon s possble. If ϕ (,, ) xy denoes he consraned densy under a dmensonal Wener process, hen mus sasfy he funconal relaonshp ( s+ ) = ( ) ( s ) ϕ, xy, ϕ, zy, ϕ, x zy, z d z. akng he laer densy ϕ ( s,, ) x zy z for a small me nerval s = d leads o he hea equaon. Solvng he Hea Equaon Usng Random Walks he above resul suggess ha rchle problems for he hea equaon may be solved numercally wh a suffcen number of seps of a random walk. hs s an effcen alernave o fne dfference mehods for numercal soluon. An example s for he consrans x+ x x 0.5 as se ou n Fgure. In fac he consrans need no be lnear; he same process apples o exponenal consrans of he form as depced n Fgure. + x x e e 3 Fgure. wo consrans. OI: 0.436/jmf Journal of Mahemacal Fnance

5 Fgure. Exponenal consrans. 4. Soluon of he Hea Equaon he equaon for he consraned Wener process may be seen as a rchle problem for he hea equaon, wh lnear boundary condons. hs may be solved by ransformng he boundary condons no nal condons (ha s, a me = 0 ), usng he echnque known n he leraure as Kelvn s mehod of mages (or hea poles [4]). he analogy wh he hea equaon s more han superfcal. I s well known ha he unconsraned hea equaon ϕ = ϕ has as s unque soluon he Gaussan dsrbuon e x ϕ = ( π ) wh a sngle hea source (.e. nal condon) ϕ( 0, x) = δ( x ) where he rgh hand sde s he rac dela funcon. o formalse he soluon, he densy for he consraned Wener process sasfes he rchle (boundary) problem ϕ = ϕ wh boundary condons ϕ = 0 on U () 4.. Unqueness Proposon. If ϕ s a soluon of he hea equaon, hen s he unque soluon. Suppose ha ϕ and ϕ are wo such soluons, and le ψ = ϕ ϕ, whch also sasfes he hea equaon and s zero on he boundary. hen consder he energy negral ψ dx. hs mus be fne as ψ s connuous and absoluely U negrable, and has he me dervave ψ dx = ψ ψ dx = ψ ψd. x However we have he deny U U U OI: 0.436/jmf Journal of Mahemacal Fnance

6 dv ψ = ψ ψ + ψ so ha x x U U U ψ d = ψ d + ψ ψ ds where he surface negral on U follows from he dvergence heorem and equals zero. Snce boh ψ dx and ψ dx are non-negave, we mus have U U ψ = 0, gvng only one soluon o he hea equaon. 4.. An Inal Condon Problem We wll replace he boundary condons under he rchle problem wh nal condons ha lead o he same soluon. Frs we nroduce he reflecon operaor abou a hyperplane ax = y. hs s gven by where I s clear ha x = I aa x + ya = I and leaves he hyperplane nvaran. I s also clear ha ϕ, x sasfes ϕ, x. hs follows mmedaely from he ransformaon leaves he hea equaon nvaran, ha s he hea equaon, so oo does ϕ(, x ) = r ( ) ( ) ϕ = ϕ(, ). I aa I aa x However he ransformed equaon has he nal condon ϕ( 0, ) = δ x x. We may apply he reflecon for each of he consrans, denong by ha correspondng o he h: x = I aa x+ y a he group generaed by = { } m s n general a counably nfne Coxeer group wh a complex srucure. We use o generae nal condons under he followng process. Generang Process for Hea Sources/Snks Le S 0 = { 0 } be he nal sngleon se. hen for =,, le S = S afer removng any pons whch have appeared prevously n S0 S S. hen an alernave nal value problem can formulaed as: ϕ = ϕ wh nal condons ϕ( x) = ( ) δ( x z ) f z S (3) hs mehod of consrucon ensures ha: A pon x S for only one value of ; For any, he dela funcon a x has he oppose sgn o ha a x. By analogy wh he hea equaon, s convenen o refer o he dela funcons a me = 0 n 3 as (hea) sources where he correspondng coeffcen s +, and as (hea) snks where he coeffcen s. he IC problem s clearly defned for he whole of, and we hen have our man resul. OI: 0.436/jmf Journal of Mahemacal Fnance

7 Proposon 3. If he soluon of he rchle problem n can be exended o he whole space, hen he problem n 3 and he rchle problem have he same soluon, and s unque. Proof. We show ha he soluon o he IC problem, f s well posed, sasfes he rchle problem. Consder he operaor. If ϕ (, ) problem, hen so oo does ϕ (, ) wh he sgn reversed. hus ϕ (, x ) and ϕ problem. Bu by unqueness hey mus be equal. x solves he IC x as he nal condons are ransformed,, x boh solve he IC On he hyperplane ax = y we have = ϕ, x = ϕ, x, ϕ, x vanshes on every hyperplane, hereby sasfyng he whch mples ha x x, and hus rchle problem. By unqueness of he soluon, he wo problems have he same soluon n he doman U. Conversely suppose ha he rchle problem can be exended o. hen by unqueness mus have he same soluon as ha of he IC problem, whch nvolves he seres of hea sources and snks descrbed n he above generang process. he proposon exends he mehod of mages used for he hea equaon o many dmensons and ypes of consran. However he mages mehod s bu one of several group heorec mehods, also known as smlary mehods [5]. hey are of praccal mporance as he soluon o he IC problem n many cases can be wren down as he summaon of Gaussan denses, vz 4.3. he Case = ϕ ( x ) = ( ) x z, π e. In he specal case =, we can have a mos wo consrans, beng he maxmum y and mnmum y of he Wener process, wh a =, a =. In hs case x = y x and x = y x. he nal condons are herefore a he pons shown below (leng z = y, z = y for smplcy) Resul of operaor a level 0 z z z z z z z ( z z) z ( z z) 3 z ( z z) z ( z z) z S I s no hard o show ha he pons a he nh level are gven by Level n n( y y) n( y y) n ny n y ny n y Hence he densy s gven by n= ( x ny ) ( ) + ny x ny+ n y ϕ x, = π e e. (4) OI: 0.436/jmf Journal of Mahemacal Fnance

8 hs resul was orgnally derved n a sochasc seng by [6], and ndrecly n an opon prcng seng n []. However from a much older hsorcal perspecve, hs problem s dencal o ha of fndng he emperaure n a rod of lengh, where boh ends a x = ± are held a zero emperaure and wh an nal un source a x = 0 : ϕ = ϕ x and ϕ (, ± ) = 0, ϕ( 0, x) = δ( x) he classcal soluon [7] derved from Fourer seres s ( n+ ) π y ( ) ( ) (5) n=0 ϕ x, = y e sn ny π ysn nx π y y where y = y y. he form of hs soluon s more complex han he Gaussan form above, as conans oscllaory elemens, and s hus less preferred for calculaon purposes. he equaly of (4) and (5) s known as Jacob s deny [7]. Where only a maxmum s relevan, he densy reduces o he well known resul 5. Analyc Soluons x ( y x) ϕ x, = π e e. he examples gven above demonsrae ha analyc soluons of he exremum problem can always be found for =. In hgher dmensons hs s no always so. hs may be because: he rchle problem may no be exensble o he whole of [n hs case mulple shees of may be useful [4]; or from a numercal vewpon, he hea sources and snks cluser whn a lmed doman n and compuaonal accuracy becomes an ssue. Where hese ssues arse, s always possble o resor o he numercal procedure of solvng he hea equaon by usng random walks as presened n secon 3.. In addon somemes happens ha a slgh varaon n he consrans wll lead o an analyc soluon. he possbles are llusraed n hs secon. As a general rule, he mos successful cases where he IC approach leads o an analyc soluon are where he hea sources and snks become nfnely dspersed under he Coxeer reflecon group, as evden for he exrema case for =. he mos serous dffcules are where he Coexer group s fne [8], as hs may lead o he rchle problem beng nexensble, n conras o he asserons gven n [9]. Remark. For compuaonal purposes, s convenen o ransform he affne reflecons nvolved no lnear reflecons. hs can be done by mbeddng + no a hyperspace of projecve space where he las coordnae becomes. (6) OI: 0.436/jmf Journal of Mahemacal Fnance

9 hus he lnear reflecon correspondng o he affne reflecon x = I aa x + ya becomes x I aa ya x = 0 he example below are all gven for =, as hs should suffce o llusrae wha s possble. In he fgures below, he densy of he conour ndcae he level of he Wener densy funcon. Hea sources are ndcaed as red pons, and hea snks as blue. he capon of each fgure ndcaes he consrans mposed. 5.. he Case m = An example of a sngle consran for = s shown n Fgure 3. For hs possbly, all denses are well behaved and are smlar o he case for =. 5.. he Case m = An example of wo consrans for = s shown n Fgure 4. Mos of he cases are well behaved. However he consrans x+ x x 0.5 lead o a fne Coxeer group, and s easy o show ha he addonal consran x 0.5 s necessarly mpled by he oher wo. hs s an example of an nexensble rchle problem. Addng hs addonal consran, however, leads o a well posed rchle soluon as shown n Fgure 5. Fgure 3. A sngle consran. OI: 0.436/jmf Journal of Mahemacal Fnance

10 Fgure 4. wo consrans. Fgure 5. An nvsble consran. However he orgnal problem, whou he addonal consran, mgh be approxmaed by varyng he consran levels o = 0 4 as shown n Fgure 6. Whch can be compared wh Fgure. I also of neres o approxmae non-lnear consrans wh a se of lnear ones. Here s an example whch s relevan o evaluang exoc opons as dscussed n Secon 7, and shown n Fgure 7. OI: 0.436/jmf Journal of Mahemacal Fnance

11 Fgure 6. A slghly vared suaon wh wo consrans. Fgure 7. An approxmaon o exponenal consrans he Case m 3 Examples of mulple consrans are shown n Fgure 8 and Fgure 9. No dffcules are evden, excep ha he larger he number of consrans, he greaer he possbly of cluserng and herefore of compuaonal problems. Slgh varaons n he consrans may resolve hese dffcules, as n Fgure 6. OI: 0.436/jmf Journal of Mahemacal Fnance

12 Fgure 8. hree consrans. Fgure 9. Four consrans. 6. me of Frs Breach he densy ϕ (, ) x clearly depends on he level of he exrema y and s hus a cumulave dsrbuon wh respec o he exrema; hs may be recognzed by max Φ, xy, of expressng as (,, ) ϕ xy, wh ( ) y = ax. he jon densy OI: 0.436/jmf Journal of Mahemacal Fnance

13 he Wener process x and s exrema y may hen be found from Φ (, xy, ) = ϕ (, xy, ). y In parcular, f k denoes a specfc exremum, he densy of he process whch obeys all he consrans apar from k, whch s breached a some me, s: Φ (, xy, \ y ) Φ(, xy, ) k where he symbol\denoes excluson of he relevan consran. I s no dffcul o ncorporae me of breach of exremum k n hs analyss. hs s useful n prcng opons (such as of he Parsan ype), where he mng of breach and re-enry are relevan. For hs purpose, consder he suaon where he me of frs breach of exremum k occurs afer me τ. hen he process mus sasfy he rchle condon unl me τ a a level z y sasfyng all exrema, and breach exremum k afer me τ. he densy of he process a level x s herefore ( τ zy) ( τ x zy ) Φ,, Φ,, \ y k. Summaon over all possble mes 0 τ and levels z y herefore provdes he densy of meeng he exrema, apar from k and breachng k afer me τ bu before me. hs can be obvously generalzed o breaches of mulple exrema a specfed mes. hs provdes a sraghforward generalzaon of he resul n [0] for =, m =. 7. Applcaon o Opon Prcng We now conclude wh our nal movaon for consderng exrema of Wener processes opon prcng. hs s based, no on asse prces or varables beng modelled as Wener processes, bu more commonly as asse reurns beng such dpp= µ d + Σdx where x s he oucome of a Wener process as before. hs resuls n ( µ σ ) + Σx P = e where σ = r ( Σ ). he exponen n he above prce s a Wener process wh drf µ σ. he general expresson for call opons wh srke K on he asses s (,0 ) C= E P K P U s he exponen appearng n he prce s of he form. wo dffcules arse. Frs, he exponen conans he drf erm s µ σ + Σx. For µ σ, whch means ha he consrans hemselves conan drf. hs can easly be deal wh usng Grsanov s heorem, or alernavely (and equvalenly) by ncludng drf n he hea equaon. hs s dealed n Appendx B. Second, lnear consrans on asse prces P do no ranslae exacly no lnear consrans for he Wener process. hs s because lnear consrans for OI: 0.436/jmf Journal of Mahemacal Fnance

14 prces of he form ap y mply a non-lnear consran on he exponens, ( ) ha s µ σ + Σx a e y. hs s deal wh nex. 7.. Non-Lnear Consrans for Wener Processes Where he me perod nvolved does no allow he exponenal of a Wener process o be approxmaed, wo approaches can be consdered. One of hese s drec o solve he hea equaon wh non-lnear consrans. he random walk of Secon 3. shows hs can be handled numercally and effcenly. he second approach s o approxmae he non-lnear consrans wh an envelope of lnear ones. he example of Secon 3. shows how hs mgh be feasble compuaonally. 7.. Example [] use a smlar echnque o ha employed above for fndng analyc soluons for exoc opon prcng for =. he echnque s also called `he mehod of mages. However he reason for hs s less obvous han ha of hs paper as he auhors consder, no he PE for he Wener process, bu ha for he opon prce V under he Black Scholes PE, namely V τ + S V S + rs V S rv = σ 0 where r and σ relae o he characerscs of he sock prce S and τ = s he me of expraon. Is s sraghforward o show hs PE can be reduced o he hea equaon u = u for = usng a ransformaon of he form xx V = Kue S = Ke x = ( τ) σ α 4 + α+ αx α = r σ. hs can be accomplshed n many ways by choosng he value of he consan K; hs possbly leads o nvarance of he PE under he mages nvolved. Consder he example where he opon has a payoff of f ( P ) a me = provded he spo prce does no exceed he celng H. he spo prce can be solved o gve P = Se λ + σ x where value of he payoff f ( P ) a me s he negral uxy y ln H σ x y ln H σ λ = r σ and S s s nal prce. he λ ( + σx) C= f Se uxy, dd xy, dy s jus he densy of he Wener process where x ( ln H x) he maxmum s ln H σ. From 6 hs s gven π σ e e. Hence he expresson for C becomes OI: 0.436/jmf Journal of Mahemacal Fnance

15 x y ln H σ λ+ σx x ( σ ) ln H x C= π f Se e e dd xy Now brng n a furher ransformaon o remove drf: le λ+ σx = σz for a new Wener process z. he ransformaon may be wren z = x+ λσ, so ha he change of measure under Grsanov s heorem from x o z s e λσz ( λσ). he above negral may hen be wren as r σ z x λσz ( λσ) C = e π f ( Se ) e e dz x ln H σ x ln H σ σ z ln H σ z λσz λσ f ( Se ) e e dz r σ z ( z λσ) = e π f ( Se ) e dz x ln H σ λσ σz z ln H σ λσ H f ( Se ) e dz x ln H σ r σ z ( z λσ) = e π f ( Se ) e dz x ln H σ λσ σz z λσ H f SH e e dz x ln H σ hus he value of he pah-resrced call opon s hus gven by he dfference beween wo pah-unresrced opons. hs resul s conssen wh [], who refer o he second negral as an mage of he frs. Clearly smlar examples nvolvng boh exrema can be found. 8. he Case of Seel Anoher applcaon of a mul-dmensonal Wener process s n he case of commodes, e.g. seel (s). he major npus are ron ore (), coal (c), labor and energy. he frs wo npus and he oupu may be measured n US. hese oupus may be analyzed by he Augmened ckey Fuller es n solaon o be saonary,.e. I ( 0), excep possbly for he labor and energy npus. For smplcy, we assume ha he full process s saonary; me rends can be be deal wh by nroducng drf as n Appendx B. he smplfcaon s ha commodes do no have an nrnsc reurn, compared o fnancal asses. aa was provded for 600 observaons durng he perod /6/06 o 8/8/06 []. he observed means and he covarance marx of SIC = ( s,, c) are as follows: and SIC = OI: 0.436/jmf Journal of Mahemacal Fnance

16 4, Σ= he commodes can be modelled as a hree dmensonal Wener process SIC = Ax wh x beng a 3 dmensonal Wener process, and wh A beng gven by he Cholesky decomposon of A : A = For an example of an exoc opon, consder a down-and-ou opon, where SIC s subjec o he parcular floors over a one year perod as follows: Ax + SIC (7) Snce he commodes are saonary n her own rgh, he prce of a call opon a 500 s gven by Call = f [ s] max ( 0, s 500) ds where f [ s] s he densy of he seel prce f he varable SIC obeys he floors, or equvalenly he consrans 7. hus remans o assess he densy f ( x ) under he consrans, and when s exceeds 500 n s payoff. he poles o be assessed under he consraned process are he sources: and he snks hus he densy of x s f ( x ) 0 = = , x p x q = e e. 3 ( π) p q From hs expresson he he densy of s = e Ax can be compued. where e = [,0,0]. If s seel (and s labor and energy npus) were subjec o me e κ κ rends, hen drf as n Appendx B can be employed usng a facor for he rend κ n seel s. We now compue he densy of SIC = Ax. Snce s has an hsorc mean of Noe ha poles dfferng by more han 0 9 have been elmnaed on he bass of compuaonal naccuracy. OI: 0.436/jmf Journal of Mahemacal Fnance

17 495, we requre only ha ( π) e Ax 0. he value of he barrer opon s hen 3 f x e Ax x max 0, 0 d = I s more convenen o evaluae hs negral numercally n he resul above. In conras he opon prce whou consrans s π4776 s> 500 [ s ] ( ) 4776 ( s 495.) e ds = e ( ) Φ = π 4776 hus he mposon of floors on he npus durng he one year process has a srong effec n reducng he varably of he seel prce, and hus narrowng he opon prce. Of course more exreme examples are possble. 9. Concluson hough he connecon beween he Wener process and he hea equaon has long been known, here has been lle research o explo. he conex of consraned processes provdes a naural seng for dervng compuaonal, and even analyc, soluons for hese problems. he power of group heorec mehods, emboded n Kelvn s mehod of mages, s only ouched on n hs paper, bu s evden hey have a much greaer role o play n mahemacal fnance. Conflcs of Ineres he auhor declares no conflcs of neres regardng he publcaon of hs paper. References [] oob, J.L. (955) A Probably Approach o he Hea Equaon. ransacons of he Amercan Mahemacal Socey, hps://do.org/0.090/s [] Buchen, P. and Konsandaos, O. (009) A New Approach o Prcng ouble-barrer Opons. Appled Mahemacal Fnance, 6, hps://do.org/0.080/ [3] Marche,. and da Slva, R. (999) Brownan Moon Lm of Random Walks n Symmerc Nonhomogeneous Meda. Brazlan Journal of Physcs. [4] Sommerfeld, A. (949) Paral fferenal Equaons. Academc Press. [5] Bluman, G.W. and Cole, J. (969) he General Smlary Soluon of he Hea Equaon. Journal of Mahemacs and Mechancs, 8. [6] Freedman,. (983) Brownan Moon and ffuson. Sprnger Verlag. hps://do.org/0.007/ [7] ym, H. and McKean, H.P. (97) Fourer Seres and Inegrals. Academc Press. [8] Coxeer, H. (935) he Complee Enumeraon of Fne Groups of he Form. Jour- OI: 0.436/jmf Journal of Mahemacal Fnance

18 nal of he London Mahemacal Socey, s-0. [9] Murhead, S. (0) Prcng Mul-Asse Barrer Opons Usng he Generalsed Reflecon Prncple. Maser s hess, Mahemacs and Sascs, Unversy of Melbourne. [0] Shepp, L.A. (979) he Jon ensy of he Maxmum and Is Locaon for a Wener Process wh rf. Journal of Appled Probably, 6. [] radng Economcs Group. Seel. hps://radngeconomcs.com/analycs/feaures.aspx?source=/commody/seel OI: 0.436/jmf Journal of Mahemacal Fnance

19 Appendx A: Convergence of he Random Walk o he Wener Process he recurrence relaon n () afer seps may be wren as follows or n + ( x ) = ( ) ( x e e ) f + n f n± ± n n j, j ( x ) n ( x ) f n f n = ± ± + ± j f ( x n e e ) f ( x) f ( x n e ) f ( x) enong ψ = fn ( n) ψ + h n j n n n x x and h= n hs may be expressed as ( x) ψ ( x) h = 6 ψ( x± he ± hej) ψ( x) h + ψ( x± he) ψ( x) h j he aylor seres expansons are: 3 ( x± he ± he ) ( x) = ± he ± he + h e e + O( h ) ψ ψ ψ ψ ψ j j j 3 ( x± he ) ( x) = ± he + h e e + O( h ) ψ ψ ψ ψ here are ( ) of he frs of he above relaons, and of he second. In he above summaons, he aggregae coeffcens of e ψ and e ψ e j for j are zero, as for any erm x± he ± he j here s he opposng erm x he he j. he only non-zero erms are e ψ e, each wh 8 + 8= 8. hus as n, or equvalenly h 0, we oal coeffcen have he lm ϕ ψ = lm h 0 x and ϕ = r ϕ = ϕ whch s he hea equaon n dmensons. he consrans smlarly scaled as Ax n y n. Appendx B: Allowance for rf n Wener Processes Ax (8) y are he sochasc approach o nroducng (or removng) drf no Wener processes s va Grsanov s heorem, whch allows he use of a Radon Nkodỳm change of measure for he process. Snce we are dealng drecly wh denses under he PE approach o such processes, s no surprsng ha a more drec reamen s possble under hs approach. hs can be acheved by consderng he hea equaon wh lnear drf, namely where usual hea equaon u v + κ v = v κ s a specfc drf vecor. hus f (, ) = u, hen v (, ) = u (, + ) u x s a soluon of he x x κ s he soluon o OI: 0.436/jmf Journal of Mahemacal Fnance

20 ha wh drf. I also follows ha f u s consraned o vansh on surfaces of he form ax= y, hen mus v vansh on he drf-adjused surfaces a x κ = y. hus we can solve problems where he consrans hemselves drf wh me. Forunaely here s an easy adjusmen ha allows us o connec he soluons of hea equaons wh or whou drf. If u s a soluon of he drfless equaon, hen consder he funcon We have x κ x x κ κ v = e e u = e u. and hus x κ κ v = e κ u+ u x κ κ [ κ ] v = e u+ u x x κ κ xx = e κ + κ x + xx v u u u v + κ v = v sasfes he hea equaon wh drf. he facor x e x κ κ s precsely he change of measure prescrbed by Grsanov s heorem, bu here appears as a ransformaon for lnear drf. Clearly hs resul may be generalzed o nonconsan drf. OI: 0.436/jmf Journal of Mahemacal Fnance