On the Probability of Hitting a Constant or a Time- Dependent Boundary for a Geometric Brownian. Motion with Time-Dependent Coefficients

Transcription

1 Applied Mahemaical Sciences, Vol. 8, 14, no., HIKARI Ld, hp://d.doi.org/1.1988/ams On he Probabiliy Hiing a Consan or a ime- Dependen Boundary for a Geomeric Brownian Moion wih ime-dependen Coefficiens risan Guillaume Laboraoire hema, Universié de Cergy-Ponoise 33 boulevard du Por 9511 Cergy-Ponoise Cede, France Copyrigh 14 risan Guillaume. his is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac his paper presens eac analyical formulae for he crossing a consan onesided or wo-sided boundary by a geomeric Brownian moion wih imedependen, non-random, drif and diffusion coefficiens, under he assumpion ha he drif coefficien is a consan muliple he diffusion coefficien, as well as approimae analyical formulae for general ime-dependen, non-random, drif and diffusion coefficiens and general ime-dependen, non-random, boundaries. he numerical implemenaion hese formulae is very simple. Mahemaics Subjec Classificaion: 6J65, 6J6; 58J35 Keywords: geomeric Brownian moion; firs passage ime; moving boundary; Fokker Planck equaion; change probabiliy measure. 1 Inroducion he quesion he hiing ime a diffusion process o an absorbing boundary is cenral imporance in many applied mahemaical sciences. I appears as a classical modelling framework in various branches physics bu i is also a he

2 99 risan Guillaume core more recen subjecs such as quaniaive finance. ypically, one needs o compue he probabiliy ha some random dynamics ha can be modelled as a diffusion process will remain under or above some criical hreshold over a given ime inerval. Almos all known eac resuls rely on he assumpion ha boh he coefficiens he diffusion process and he boundary are consan, whereas in pracice ime-dependen diffusion process coefficiens and boundaries are required for modeling purposes. As a resul, slow and edious numerical schemes need o be implemened, he accuracy which may be dubious. Only wo cases ime-dependen boundaries are easily handled from a mahemaical poin view : he crossing a linear barrier by a Brownian moion wih consan drif and diffusion coefficiens (see Doob [7] and Sheike [4] for a sligh generalizaion) and he crossing he eponenial a linear barrier by a geomeric Brownian moion wih consan drif and diffusion coefficiens (see Kuniomo & Ikeda [13]). Some papers provide semi-analyical resuls for boundaries ha are fairly general funcions ime. Durbin [8], Ferebee [9] and Park & Schuurmann [19] hus derive inegral equaions, which mus be numerically solved. Salminen [3] obains a formula involving he epeced value a Brownian funcional requiring o solve a non-ime homogeneous Schrödinger equaion wih boundary and iniial condiions. Nonlinear boundaries are sudied in a general framework in Jennen & Lerche [11], Daniels [6], as well as Alili & Paie [1]. Oher conribuions deal wih specific forms he boundary. Breiman [3] hus sudies he case a square roo boundary and relaes i o he quesion he firs hiing ime a consan boundary by an OU (Ornsein-Uhlenbeck) process, while Groeneboom [1] eamines he case a quadraic funcion ime and shows ha he firs passage densiy can be wrien as a funcional a Bessel process dimension 3. Alernaively, some auhors focus on diffusion processes oher han sandard Brownian moion. Changsun & Dougu [5] hus evaluae he firs passage ime densiies OU process o eponenial boundaries and Brownian bridge o wo linearly shrinking boundaries. Alili & Paie [] sudy he problem he firs hiing ime an OU process in general. Lo & Hui [15] derive a formula for he firs passage densiy a ime-dependen OU process o a parameric class moving boundaries. Finally, a collecion papers focuses on applicaions o financial mahemaics, more specifically o opion pricing, which shifs he focus o geomeric Brownian moion, as he laer serves as a he building block for modeling asse prices. Kuniomo & Ikeda [13] show ha eponenial boundaries are a simple eension o he sandard wo-sided barrier opion pricing problem wih consan boundaries. Robers & Shorland [] obain igh bounds on he prices barrier opions wih general moving boundaries and non consan coefficiens by means a hazard rae angen approimaion involving some numerical inegraion. Rogers & Zane [] use a rinomial ree approach combined wih a ransformaion he saespace and a ime change ha urn smoohly-moving barrier opion pricing problems ino fied barrier problems. heir resul builds on a binomial laice

3 Geomeric Brownian moion wih ime-dependen coefficiens 991 approach by Rogers & Sapleon [1]. Novikov e al. ([16], [17]) compue piecewise linear approimaions for one-sided and wo-sided boundary crossing probabiliies using repeaed numerical inegraion and apply his mehod o he pricing ime-dependen barrier opions. hompson [5] derives upper and lower bounds in he form double inegrals ha prove o be slighly igher bu harder o compue han hose Robers & Shorland []. Lo e al. [14] also provide igh bounds for barrier opion prices by means a mulisage approimaion scheme in erms mulivariae normal disribuion funcions which involve non-rivial numerical inegraion issues. For all his raher abundan published research, here is no known formula for he crossing a ime-dependen absorbing boundary by a diffusion process wih ime-dependen drif and diffusion coefficiens in general. he purpose his aricle is wo-fold : - o provide eac analyical formulae for he crossing a consan one-sided or wo-sided boundary by a geomeric Brownian moion wih ime-dependen, nonrandom, drif and diffusion coefficiens, under he assumpion ha he drif coefficien is a consan muliple he diffusion coefficien - o provide approimae analyical formulae for general ime-dependen, nonrandom, drif and diffusion coefficiens and general ime-dependen, non-random, boundaries he mehod used is a combinaion parial differenial equaions and changes probabiliy measure. All formulae are very easily implemened and yield insananeous numerical values. Secion provides eac analyical resuls under resricive assumpions on he ime-dependen diffusion process coefficiens. Secion 3 provides approimae analyical resuls for general ime-dependen diffusion process coefficiens and boundaries. Secion 4 aims a esing he accuracy he approimaion formulae given in Secion 3. Eac soluions in paricular cases his secion presens wo eac closed form resuls ha hold when he imedependen drif coefficien is a consan muliple he ime-dependen diffusion coefficien and he boundary is consan. he firs resul is called Proposiion 1. I provides he join cumulaive disribuion funcion no hiing a one-sided consan boundary during a finie ime inerval, and being under a given poin a ime. he second resul is called Proposiion. I provides he join cumulaive disribuion funcion no hiing a wo-sided consan boundary during a finie ime inerval, and being above or under a given poin a ime. PROPOSIION 1 Le B be a sandard Brownian moion wih naural filraion.

4 99 risan Guillaume Le s be a piecewise coninuous, non-random, posiive real funcion such ha : s d, : (1) Le m be any real consan. Le X be a geomeric Brownian moion such ha X. Under a given measure, X is driven by he following sochasic differenial equaion : dx ms X d s X db () Le h be a posiive real consan such ha h and k be a posiive real consan such ha k h. Le N. denoe he cumulaive disribuion funcion a sandard normal random variable. hen, he probabiliy ha X will no hi h during he finie ime inerval, and ha i will be below k a ime is given by : sup X h X k k 1 k 1 s d 1 m s d m h h N N h s d s d (3) Pro Proposiion 1 As a consequence he heory absorbed diffusions in general and he Fokker-Planck equaion in paricular (see, e.g., Oksendal [18]), he sough probabiliy, denoed by p,, is he soluion he following IBVP (iniial boundary value problem) (4)-(5) : p p s p ms,, h, h (4) lim p, 1, p, h, p, k (5) where. sands for he Heaviside funcion he following change he space coordinae : y yields a new IBVP (7)-(8) : p 1 p s p h s m,, y y y h lim p, y 1, p, y, p, y k ep y he following change funcion: (6) (7) (8)

5 Geomeric Brownian moion wih ime-dependen coefficiens , ep p y m y s s ds w y, 8 (9) along wih he following change variable : z y h (1) hen yields he new IBVP : w s w,, z z (11) w,,, ep h ep z w z k h z (1) By separaion variables, he following funcion verifies equaion (11) and he boundary condiion in (1) : l w z, A l ep s s ds sin lz dl (13) As a consequence he iniial condiion in (1), he funcion h z f z k h ep z ep (14) can be idenified as he sine Fourier ransform Al. hen, applying Fubini s heorem, classical rigonomeric ideniies and he following Fourier cosine ransform : 1 p a ep cos bz az dz ep, a, b b (15) 4b one can obain : k / h h 1 v v z w z, ep dv p s s ds s s ds k / h h 1 v v z ep dv p s s ds s s ds Sraighforward calculaions yield : k 1 z s s ds h z 1 h w z, ep s s ds N 8 s s ds (16) (17)

6 994 risan Guillaume k 1 z h z 1 h ep s s ds N 8 s s ds I can be checked ha he funcion w z, given in (17) verifies (11) and (1). Revering o he iniial funcion and variables, Proposiion 1 ensues. Hiing imes wo-sided boundaries are considered ne. s s ds PROPOSIION Le B, s and X be defined as in Proposiion 1 and, wihou loss generaliy, le us se m 1 in (3). Le d and u be wo posiive real consans such ha d u and le k be a posiive real consan such ha k d. hen, he probabiliy ha X will hi neiher d nor u during he finie ime inerval, and ha i will be above k a ime is given by : inf, sup, X d X u X k d u / d 1 w np ep s s ds ep sin w dw u / d 8 1 / n u d k / d 1 np np ep s s ds sin u / d u / d d (18) Corollary : inf, sup, X d X u X k u k / u 1 w np ep s s ds ep sin w dw u / d 8 1 / n u d d / u 1 np 1 np ep s s ds sin u / d 8 u / d u (19) Pro Proposiion Following he same seps as in he pro Proposiion 1, one can show ha he probabiliy under consideraion is he funcion w z, solving he following IBVP ()-(1) : w s w u,, z z d ()

7 Geomeric Brownian moion wih ime-dependen coefficiens 995 u w,, w, d,, ep d ep z w z d z k (1) where : z y d, y, y 1, ep p y s s ds w y, 8 () Using separaion variables, he boundary and iniial condiions in (1) and he superposiion principle, one easily obains : 1 np np w z, A ep n s s ds sin z n1 u / d u / d (3) where : d u / d w np An ep sin w dw u / d (4) u / / d k d hen, revering o he iniial variables and funcion, and subsiuing, Proposiion ensues. 3 Approimae analyical soluions for general imedependen, non-random boundary, drif and diffusion coefficiens his secion provides approimae closed form analyical resuls for general nonrandom, ime-dependen boundaries and process coefficiens. he firs resul is called Proposiion 3. I provides an approimae formula for he cumulaive disribuion funcion no hiing a one-sided ime-dependen, nonrandom boundary during a finie ime inerval,. he second resul is called Proposiion 4. I provides an approimae formula for he join cumulaive disribuion funcion no hiing a wo-sided consan boundary during a finie ime inerval, and being below a given poin a ime. he final resul is called Proposiion 5 and provides he join cumulaive disribuion funcion no hiing a one-sided consan boundary during a finie ime inerval, and being under or above a given poin a ime. PROPOSIION 3 Le B and s be defined as in Proposiion 1. Le m be a piecewise coninuous non-random real funcion such ha : m d, : (5) s Under a given measure, he process differenial equaion : X is driven by he following sochasic

8 996 risan Guillaume dx m X d s X db (6) where he usual growh condiions apply o m and s for equaion (6) o be properly defined (Oksendal [18]) Le h be a coninuous, once differeniable real funcion, such ha h and such ha if here is a coefficien muliplying, hen his coefficien is negaive. Under hese assumpions, he probabiliy ha X will no hi h during he finie ime inerval, can be approimaed by he following formula : X h, h s h m h N d s s d s h m h d s h ep s d s h m h h N d s s d Remark : if we denoe by h he firs passage ime he process X o he ime-dependen boundary h in,, ha is : h inf, : X h (8) hen he densiy h can be approimaed by (7) h, d X s h s s (9) where an approimaion X s h s, s is given by (7) Pro Proposiion 3 : As a consequence he heory absorbed diffusions in general and he Fokker-Planck equaion in paricular (see, e.g., Oksendal [18]), he probabiliy

9 Geomeric Brownian moion wih ime-dependen coefficiens 997 under consideraion, denoed by p,, is he soluion he following IBVP (3)-(31) : p p s m p,, h, h (3) lim p, 1, p,, p, 1 (31) h he ne ransformaions he space coordinae : y, z y h (3) yield a new IBVP (33)-(34) : p s h p s p m h z z,, z (33) lim p, z 1, p,, p, z 1 (34) z If he funcion p z, solving (33)-(34) can be obained, hen he sough probabiliy will be equal o p z h /,. Unforunaely, he IBVP (33)-(34) canno be solved eacly by known mehods. A probabilisic approach can neverheless be applied o ge an analyical approimaion he funcion p z,. Se : s h l m (35) h Le a new process Y be sared from he origin, whose moion is driven by : dy l d s db (36) hen, solving IBVP (33)-(34) is he same as calculaing he probabiliy, under he measure, ha Y will remain below he real number y h / during he finie ime inerval,. Le be he measure under which he sochasic differenial Y is given by : dy s dw (37) where W is a sandard Brownian moion. hen, using classical resuls on he hiing imes a sandard Brownian moion (see, e.g., Karazas & Shreve [1]), a simple ime change W yields : y y sup Y y N N (38) s d s d he - measure is equivalen o and is Radon-Nikodym densiy is given by : d l s 1 l s ep dw s ds L d s s s s (39) Eisence he inegrals in (39) is guaraneed by he assumpions made on he funcions

10 998 risan Guillaume l e s. Girsanov s heorem (see, e.g., Karazas & Shreve [1]) allows us o sae ha he process B defined by : l s B W ds (4) s s is a - Brownian moion l he funcion is non-random. Hence, if is fied : s l s l s dw s, ds s s s s (41) where a, b refers o he normal disribuion wih epecaion a and variance b hus, if is fied, he following equaliies hold in law : l s laww l s law f l s dw s ds ds s s s s s s (4) where f is a sandard normal random variable Using (38), (39) and (4), he sough probabiliy sup Y y can now be approimaed as follows : sup Y y L (43) sup Y y y s d 1 ep l l d d s s 1 1 y d p ep ep s d (44) Sraighforward compuaions yield : sup Y y y J y N J epy N J I I I (45) wih : I s s ds, J l s ds (46) s s

11 Geomeric Brownian moion wih ime-dependen coefficiens 999 he resul saed in (45) is jus an approimaion since (4) only holds for fied and no as a sochasic differenial, as shown by he fac ha : W s d l ds l dw s s (47) s I can be checked ha he funcion p z, defined by : z J z p z, N J epz N J I I I (48) saisfies he required boundary and iniial condiions in (34) as well as he following PDE (parial differenial equaion) : p s h p s p m z, h z z, z (49) he funcion z, is a non-zero erm ha can be eplicily compued by subsiuing he funcion p z, given by (48) ino he PDE given by (33). If he soluion (48) were eac, hen z, would be null. he residue z, becomes more and more negligible as he parameer in Proposiion 3 decreases, so ha he proposed analyical approimaion should be very accurae on relaively shor ime inervals, as will be checked in Secion 3. Ne, an approimaion formula is provided when he boundary is consan and wo-sided. PROPOSIION 4 Le B, m, s and X be defined as in Proposiion 3. Le d and u be wo posiive real consans such ha d u and le k be a posiive real consan such ha k d. hen, he probabiliy ha X will hi neiher d nor u during he finie ime inerval, and ha i will be under k a ime can be approimaed by : inf, sup, X d X u X k

12 1 risan Guillaume k u d n N I u I1 d I n ep n I1 d u d n N I I1 d u d I n ep (5) n I1 k d u d n N I I1 d u d n N I I1 where : I s d s m I d s (51) (5) Pro Proposiion 4: Following he same seps as in he pro Proposiion 3, he probabiliy under consideraion, denoed by p y,, is he soluion he following IBVP (53)-(54): p s p s p d u m,, y y y (53) d p, where y Se :, p u,, p, y k ep y (54)

13 Geomeric Brownian moion wih ime-dependen coefficiens 11 s b m (55) Le a new process Z be sared from he origin, whose moion is driven by : dz b d s db (56) Solving IBVP (53)-(54) is he same as calculaing he probabiliy, under he measure, ha Z will remain below he real number u / and above he real number d / during he finie ime inerval, and ha Z will be smaller han he real number k a ime. Le be he measure under which he sochasic differenial Z is given by : dz s dw (57) where W is a sandard Brownian moion. Using he classical mehod images (Carslaw & Jaeger [4]), one can obain : inf Z d /, sup Z u /, Z k / k / n u / d / N s d (58) n d / n u / d / N s d k / d / n u / d / N s d n d / n u / d / N s d he - measure is equivalen o and is Radon-Nikodym densiy is given by : d b s 1 b s ep dw s ds L d s s s s (59)

14 1 risan Guillaume Along he lines he pro Proposiion 3, i can hen be shown ha he sough probabiliy can be approimaed by : inf Z d /, sup Z u /, Z k / L (6) inf Z d /, sup Z u /, Z k / where he epecaion operaor in (6) is epanded ino inegral form using (58) and he following equaliy in law, for fied : b s s s law b s dw s f ds, f,1 (61) s s Performing he necessary calculaions, he formula in (5) is hen obained. Evenually, an approimaion formula is saed when he boundary is consan and one-sided. PROPOSIION 5 Le B, m, s and X be defined as in Proposiion 3. Le h be a posiive real consan such ha h and k be a posiive real consan such ha k h. hen, he probabiliy ha X will no hi h during he finie ime inerval, and ha i will be below k a ime can be approimaed by : sup, X h X k k s m N d (6) s s d s m k d s m s h h ep N d s s d s d Corollary : Le h be a posiive real consan such ha h and k be a posiive real consan such ha k h. hen, he probabiliy ha X will no hi h during he finie ime inerval, and ha i will be above k a ime is given by :

15 Geomeric Brownian moion wih ime-dependen coefficiens 13 inf X h, X k s m k N d (63) s s d s m h d s m s k h ep N d s s d s d he pro is similar o ha Proposiion 3, so he deails are omied. 4 Numerical resuls In his final secion, he accuracy he analyical approimaions provided by Proposiion 3, Proposiion 4 and Proposiion 5, is numerically esed, by comparing obained values wih seleced benchmarks. he firs sage he procedure is o randomly draw parameers for he diffusion process X defined in Proposiion 3. hree differen funcional forms coefficiens m and s are seleced. he linear form reads : m a1 b1, s a b (64) he quadraic form reads : m a1 b1 c1, s a b c (65) he square roo form reads : m a1 b1, s a b (66) he real consans a1, a, b1, b, c1, c are randomly drawn wihin wide predefined ranges. For insance, if he linear form is prescribed for he volailiy funcion s, he iniial volailiy s a may vary from 5% o 5%, while he coefficien b, measuring he increase in volailiy over ime, may vary from 3% o 8%. Depending on he random value parameer, final values he volailiy funcion, s, up o 85% were observed during he numerical eperimen. he iniial value he process, X, is se a 1. o es Proposiion 4 and Proposiion 5, he parameers u, d, h and k are randomly drawn. he predefined ranges for u, h and d are X 1%;4%. he parameer k

16 14 risan Guillaume may vary beween X 1% and X 1%. o es Proposiion 3, a imedependen boundary h is also randomly picked ha can ake on he funcional forms (64)-(65). Finally, he widh he ime inerval may vary from o 1. he ne sage in he procedure is o define a benchmark ha is assumed o be he rue value. Possible benchmarks are he numerical soluions he iniial boundary value problems obained in (33)-(34) and (53-54). his approach was aemped by means classical Crank-Nicolson finie differences and led o quie unreliable resuls, even wih fine meshes. Alernaively, one can resor o he Mone Carlo simulaion diffusion process X - and ime dependen boundary h as far as Proposiion 3 is concerned. his approach is boh simple and robus. I was implemened using a sandard Euler scheme for he discreizaion he sochasic differenial equaion (6) wih a imesep equal o 1/16. A raher fine imesep such as he laer is needed, oherwise oo many boundary crossings would be missed. Sill, plain Mone Carlo simulaion is nooriously inaccurae. Besides increasing he number simulaions and using a reliable random number generaor, i is advisable o have eac analyical benchmarks a disposal, wih which one can es he accuracy he Mone Carlo simulaion. his can be achieved by means Proposiion 1 and Proposiion for he special case in which m s. So, our numerical procedure is performed wice for each se process coefficiens and boundaries : (i) drawing a general m in he firs place o es he accuracy Proposiion 3, Proposiion 4 and Proposiion 5, in comparison wih a Mone Carlo approimaion used as a benchmark ; (ii) seing m s and using Proposiion 1 and Proposiion as eac benchmarks o es he accuracy he Mone Carlo approimaion iself. he second sage he procedure is carried ou using he same random numbers ha are generaed o compue a Mone Carlo approimaion for a general m in he firs place. ables 1-4 repor he obained resuls when conducing he numerical eperimen above described. A sample 1, collecions randomly drawn parameers is used in each able. hree main indicaors are repored : - he observed mean absolue value he error - he observed highes absolue value he error - he observed proporions errors ha are : less han.5%, beween.5% and 1%, beween 1% and %, beween % and 3%, and over 3% In able 1, he error under consideraion is he difference beween he analyical approimaions provided by Proposiion 4, Proposiion 5 and Proposiion 3 on he one hand, and Mone Carlo approimaions used as benchmarks on he oher hand. he error is epressed as a percenage he numerical value he Mone Carlo approimaion. In able, he error under consideraion is he difference beween a Mone Carlo approimaion and eac analyical benchmarks provided by Proposiion 1 and Proposiion. able 1 shows ha he mean absolue value he error enailed by he use Proposiion 4, Proposiion 5 and Proposiion 3, is always less han 1%. Bu, in

17 Geomeric Brownian moion wih ime-dependen coefficiens 15 pracice, i is obviously imporan o have an idea he order magniude he greaes possible error incurred by a single use an analyical approimaion. Over he fairly large and uniformly disribued sample random parameers ha was used, i urns ou ha he maimum absolue value he error is always less han 3% for Proposiion 4 and 5, and always less han 5% for Proposiion 5. hese resuls are he all he more reassuring as one mus bear in mind ha he Mone Carlo approimaions used as benchmarks in able 1 are relaively inaccurae, despie he high number simulaions and he fine imesep. Indeed, able shows ha he absolue error enailed by he use a Mone Carlo approimaion as a benchmark may be as high as 1.67%. his suggess ha he maimum absolue value error enailed by Proposiion 4, Proposiion 5 and Proposiion 3 may acually be smaller han he figures repored in able 1. As poined ou in Secion 3, he magniude he errors enailed by he analyical approimaions is a funcion he widh he ime inerval,. So, in ables 3 and 4, hree ranges for are eamined:.1,.1,.5, and.5,1. Only Proposiion 4 and Proposiion 3 are esed, as he resuls obained for Proposiion 5 are very similar o hose obained for Proposiion 4. I urns ou ha, when.1, which includes ime inervals ha may conain as many as 1,6 ime seps, he maimum absolue value he error enailed by Proposiion 4 is only.58%, while 93% observed errors are less han.5%. he figures are roughly he same for Proposiion 3. hese numerical resuls sugges ha our analyical approimaions should be quie relevan for pracical purposes in a variey applied sciences. Indeed, no only do hese analyical approimaions provide a framework in which he impac and he ineracions variables can be mahemaically analyzed, bu hey are also eremely efficien compared wih he slowness a Mone Carlo simulaion. o ge a single Mone Carlo esimae in able 1, one mus draw beween 8,, and 8,,, uniform random numbers as he parameer varies beween.1 and 1, which may ypically ake up o half an hour on a single PC. Furhermore, he issue he reliabiliy he random number generaor becomes more acue as he number draws increases. In conras, i always akes less han one second o obain an esimae using Proposiion 3, Proposiion 4 or Proposiion 5, as he funcions involved are sandard and he number funcion evaluaions implied by he numerical inegraions is very modes. Evenually, one can also wonder wheher he magniude he errors enailed by he analyical approimaions may vary according o he funcional forms he diffusion coefficiens m and s, as well as o he funcional form he ime dependen boundary h. Numerical ess were implemened in his regard and showed no significance ha facor, as he orders magniude remained he same for all differen funcional forms he diffusion process coefficiens and he ime dependen boundary.

18 16 risan Guillaume able 1 Numerical assessmen he error enailed by he use he analyical approimaions provided by Proposiion 4, Proposiion 5 and Proposiion 3 mean absolue value error maimum absolue value error Proporion error.5% Proporion error.5%;1% Proporion error 1%;% Proporion error %;3% Proporion error 3% Proposiion 4 Proposiion 5 Proposiion 3.85%.77%.81%.76%.6% 4.87% 39% 36% 4% 37% 41% 38% 15% 13% 1% 9% 1% 11% % % 6% Noes : he error is measured as he divergence from a Mone Carlo approimaion used as a benchmark, and epressed as a percenage he value he Mone Carlo approimaion. A sample 1, differen ses parameers for he process X is considered. Deails abou he way he parameers are drawn can be found a he beginning Secion 4. For each se parameers, a oal 5, Mone Carlo simulaions are carried ou o obain a Mone Carlo benchmark, using a imesep equal o 1 / 16 for he discreizaion scheme he sochasic differenial X. able Numerical assessmen he error enailed by he use a Mone Carlo approimaion as a benchmark Proposiion 1 Proposiion mean absolue value error.54%.58% maimum absolue value error 1.59% 1.67% Proporion 59% 61% error.5% Proporion 7% 4% error.5%;1% Proporion 14% 15% error 1%;% Proporion % % error %;3% Proporion % % error 3% Noes : he error is measured as he difference beween he eac analyical resuls provided by Proposiion 1 and Proposiion and heir Mone Carlo approimaions, and epressed as a percenage he eac values. Deails abou he seleced sample and he way Mone Carlo simulaion is carried ou are he same as in he noes able 1

19 Geomeric Brownian moion wih ime-dependen coefficiens 17 able 3 Numerical assessmen he error enailed by Proposiion 4 as a funcion he widh he ime inerval, mean absolue value error maimum absolue value error Proporion error.5% Proporion error.5%;1% Proporion error 1%;% Proporion error %;3% Proporion error 3%.1.1,.5.5,1.14%.35% 1.8%.58% 1.1%.97% 93% 78% 38% 7% 18% 39% % 4% 13% % % 1% % % % Noes : same as in able 1 able 4 Numerical assessmen he error enailed by Proposiion 3 as a funcion he widh he ime inerval, mean absolue value error maimum absolue value error Proporion error.5% Proporion error.5%;1% Proporion error 1%;% Proporion error %;3% Proporion error 3%.1.1,.5.5,1.18%.41% 1.36%.71% 1.45% 5.4% 86% 66% 16% 14% % 9% % 1% 8% % % 13% % % 14% Noes : same as in able 1 References [1] Alili L., Paie P., On he firs crossing imes a Brownian moion and a family coninuous curves, C.R.A.S. (5), 34, 5-8

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21 Geomeric Brownian moion wih ime-dependen coefficiens 19 [3] Salminen, P., On he firs hiing ime and las ei ime for a Brownian moion o/from a moving boundary, Advances in Applied Probabiliy (1988),, [4] Sheike.H., A boundary crossing resul for Brownian moion, Journal Applied Probabiliy (199), 9, [5] hompson G.W.P., Bounds on he value barrier opions wih curved boundaries, Working Paper () Received: November 1, 13