An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor wih a long posiion in one sock decides o close he posiion before a given ime. The invesor coninuously observes he sock price performance and has o deermine he poin in ime o close ou he posiion selling sraegy so ha he sock price is as close as possible o he maximum price. The probable proximiy is measured by a probabiliy disance and he problem can be formulaed mahemaically as an opimal predicion problem. 1. Inroducion Suppose ha an invesor has o close a long posiion in a sock wihin a prescribed period of ime. For example, a day rader who mus close ou a posiion on he same day, ha is, he rader has a long posiion and akes an offseing shor posiion laer in he day. The invesor s dream is o close ou he posiion when he sock price is a, or close o, he maximum price over he given ime period sell i near he high so as o maximise he profi minimise he loss. The maximum price is unknown a any poin in ime over he period and is value only becomes known a he erminal dae. I is herefore crucial for he invesor o choose he bes ime o sell. The invesor is faced wih a decision regarding anicipaed marke movemens wihou knowing he exac dae of he opimal occurrence, in oher words he invesor has o make a predicion of fuure price behaviour. More precisely, a ime he invesor sars o coninuously observe he sock price and has o deermine a ime sraegy based only on his available informaion as o when o execue he ransacion. The selling sraegy for sock rading presened in his paper is a soluion o he above problem where he probable proximiy is measured by a probabiliy disance, in aemps o beer predic he maximum price. In his paper, he objecive is o maximise he probabiliy ha he price of he sock is greaer han a given percenage of he maximum price. Since he maximum depends on he fuure, he problem is called an opimal predicion problem. This predicion problem migh be considered as a coninuous ime exension of he secreary problem, ha is, he problem o find sequenially he maximum of a random sequence of fixed lengh, see Gilber & Moseller [5]. Moreover, in bohs problems he opimal sraegies have he same form: he sock is held for a given amoun of ime, and in he remaining ime i is sold a some given even. Opimal predicion problems in he case of coninuous ime were iniiaed by Graversen, Peskir & Shiryaev [6]. They presen 2 Mahemaics Subjec Classificaion. 6G4 and 91B26. JEL Classificaion. G11 and C61. Key words and phrases. Selling sraegy, sock rading, maximum price, opimal predicion, opimal sopping, smooh fi a a single poin. 1
he soluion o he problem of sopping a Brownian pah as close as possible o he unknown maximum heigh of he pah when he closeness is measured by a mean-square disance. These resuls are exended in Pedersen [12] and du Toi & Peskir [4]. The applicaion of opimal sopping problems o he problem of selling asses appear o be of ineres in financial engineering. Taking accoun of given predicive informaion so called predicion disribuion abou he asse, Karlin [9], Boyce [3] and Griffeah & Snell [8] deermine selling sraegies ha maximise he expeced payoff. Mos of his work has been done in he case of discree ime. In a marke where he price process is a geomeric Brownian moion, Øksendal [11, Chaper 1] sudied selling sraegies ha maximise he expeced discouned ne sale amoun in he long run. Chang [17] obained an opimal selling sraegy in order o maximise an expeced uiliy funcion of he invesor when he model of he sock is exended o swiching geomeric Brownian moions. In he wo laer papers he selling sraegies are specified wih a arge price and a sop-loss limi. These sraegies only use pas informaion o show ha prices have always been in his given inerval. A consequence of his is ha, in a rising marke, using hese sraegies migh lead o he posiion being closed ou premaurely. The selling sraegy presened in his paper Theorem 2.1 uses he informaion of pas prices hrough he running maximum price. The remainder of his paper is consruced as follows. In he nex secion he opimal selling sraegy is saed and an example is given. The opimal predicion problem is solved in Secion 3. Secion 4 presens he derivaion of he densiy of a firs hiing ime and a ime-dependen version of Iô-Tanaka formula is presened in Secion 5. 2. The opimal selling sraegies The invesor wih he long posiion in one sock decides a ime = o close ou he posiion before ime = T. The dynamics of he price process S T is assumed o be geomeric Brownian moion saisfying he sochasic differenial equaion ds = µs d + σs dw, S = s > wih σ > he volailiy, µ R he drif, and W T a sandard Brownian moion he sandard framework of Black-Scholes model for opion valuaion [1] and [1]. Assume ha he invesor can sell he sock a any poin in ime. A selling sraegy τ is hen he amoun of ime in he given period [, T] ha he invesor holds he posiion and is based on he informaion accumulaed o dae and no on fuure prices. Thus, τ is resriced o be a sopping ime for S saisfying τ T. The invesor s objecive is o maximise, over all sraegies, he probabiliy ha he selling price of he sock is greaer han a given percenage of he maximum price in [,T]. Mahemaically, i is an opimal predicion problem which is o compue he value funcion 2.1 V p T = supp S τ p max u T S u τ T for p, 1 and o find an opimal sraegy for which he supremum is aained. For m s, he process max u S u m/s is a diffusion saring a y = m/s 1 wih sae space [1, where 1 is an insananeous reflecion boundary poin see [16]. The variable m is included o conrol he saring poin of his diffusion. Lévy s disribuional heorem for Brownian moion wih drif implies ha logmax u S u m/s /σ is a refleced Brownian moion wih drif see nex secion for deails. For p, 1 le τ p = inf{ > : S = p max u S u m } 2
be he firs hiing ime of max u S u m/s o he fixed level 1/p. Wih he noaion EX;A = EX1 A define he funcion V p,y = E s F T τ p T logp/σ ; p max u τp T S u m S τp T for y = m/s 1 where = µ/σ σ/2 and ε ε 2.2 F ε = Φ e 2ε Φ, >. Here ϕx = 1 2π e x2 /2 and Φx = x ϕu du denoe he densiy and disribuion funcion respecively of a sandard normal variable. The funcion V p,y plays a key role in consrucing he value funcion V p. Noe ha y V p,y is coninuous a y = 1/p for < T, bu in general is no differeniable. The soluion o problem 2.1 he opimal selling sraegy can now be saed in he heorem below. Theorem 2.1. Consider problem 2.1 where S T is a geomeric Brownian moion and p, 1. Assume ha T is chosen such ha here is a mos one poin in he ime inerval [,T] where y V p,y is differeniable a y = 1/p and if y V p,y is no differeniable a y = 1/p for any [,T] se =. Then he opimal selling sraegy τ is o defer selling he sock unil he remaining ime is T and hen sell he sock he firs ime ha S is equal o p max u S u see Figure 1 for illusraions of he opimal selling sraegy, ha is, τ = inf{ < T : S = p max u S u } inf = T. The selling sraegy enables he invesor o sell he sock for a price ha is greaer han a percenage p of he maximum price wih probabiliy V p T given by V p T = E V p, max u S u /S. max u S u /S S p max u S u 1.4 14 1/p 1 τ T 1. τ 1. T Figure 1. Two drawings of he opimal selling sraegy when p =.8 and, in his case,.61. Proof. The calculaions in he nex secion show ha problem 2.1 is equivalen o problem 3.3. The heorem is hen a consequence of Theorem 3.1 wih V p logp/σ T = W, and = µ/σ σ/2. 3
Example 2.2. Suppose ha he drif µ =.5, he volailiy σ =.3 and erminal ime T = 1. See Figure 2 for an illusraion of and V p 1 as funcions of p. From Figure 2 one can see ha if, for example, p =.8 hen.61 and V.8 1.76. Conversely, if he predicion of he maximum price is done wih a given probabiliy, say 9%, ha is V p =.9, hen p.75 and.36. The selling sraegy is illusraed in Figure 1 wih p =.8. 1. 1. V p 1.5.5 p p.5 1..5 1. Figure 2. On he lef a drawing of and on he righ a drawing of he value funcion V p 1 as a funcion of p wih drif µ =.5, volailiy σ =.3 and ime horizon T = 1. In order o perform hese compuaions, noe ha E y F T τp logp/σ ; τ p < T + P y τ p > T for 1 y < 1/p V p,y = E y F T τp logp/σ ; τ p < T for y 1/p. The densiy of τ p is given in Proposiion 4.2 wih z = logy/σ, ε = logp/σ and = µ/σ σ/2. This closed-form expression is very complex for 1 y 1/p and is difficul o evaluae, bu i is possible, a leas in principle, o calculae V p and derive an equaion for. Also, a formula for he value funcion V p T can be derived from he formula of V p and he ransiion densiy of max u S u /S given in Proposiion 4.2. In summary, is a soluion o an equaion and here is a formula for he value funcion V p T, bu hese expressions are no convenien for numerical compuaions. One way for he numerical compuaion of V p,y when 1 y 1/p is based on a parial differenial equaion. The infiniesimal generaor of he diffusion max u S u /S in 1, is L = σ 2 µy y + σ2 2 y2 2 y 2. 4
Then, according o general Markov process heory, V p solves he following sysem + L V p,y = for < T and 1 < y < 1/p V p,y y=1/p = F T logp/σ for < T V p,y =T = 1 for 1 y 1/p V p y,y y=1 = for < T. Numerical echniques for parial differenial equaions can hen be used for he numerical compuaions of V p,y when 1 y 1/p. The oher erms can be compued direcly from he closed-form formulas menioned above. 3. The opimal sopping problem The soluion o predicion problem 2.1 is esablished in his secion. The approach o solving he problem is he following. Firs, a represenaion of a condiional expecaion of he gain process is derived in order o reduce he original problem o an opimal sopping problem. Nex, Lévy s disribuional heorem for Brownian moion wih drif is used o reduce he dimension of he problem by one. Lasly, he principle of smooh fi a a single poin is applied o solve he problem. Recall, ha he price process S is modelled as a geomeric Brownian moion given by S = s exp σ W + µ/σ σ/2 = s exp σx where X = W + is a Brownian moion wih drif = µ/σ σ/2. Then V p T = supp S τ p max u T S u = sup P max u T X u X τ ε τ T τ T where ε = logp/σ >. Firs, he above predicion problem is ransformed ino an equivalen ordinary sopping problem, ha is, he gain process is adaped o he filraion. For < T, he saionary and independen incremens give ha 1 [,ε] max u T X u X F E = E 1 [,ε] max u X u { max u T X u X + X } X F = E 1 [,ε] s { max u T X u + x} x s=max u X u x=x = E 1 [,ε] s { max u T X u + x} x s=max u X u x=x = P max u T X u ε 1 [,ε] max u X u X = FT ε1 [,ε] max u X u X 5
where he disribuion funcion FT of max u T X u is given in 2.2. Using his equaliy he predicion problem can be wrien as T = supe E 1 [,ε] max u T X u X τ Fτ V p τ T = supe FT τε ; max u τ X u X τ ε. τ T Le Z T be he srong soluion of he sochasic differenial equaion 3.1 dz = sgnz d + dw. The process Z is a refleced Brownian moion wih drif see [7, Theorem 2] and he infiniesimal generaor of Z is given by 2 z 2. 3.2 L Z = sgnz z + 1 2 Lévy s disribuional heorem for Brownian moion wih drif [7, Theorem 1] gives ha he wo processes max u X u X T and Z T are idenical in law. Therefore he predicion problem is equivalen o V p T = supe FT γε ; Z γ ε γ T where he supremum is aken over all sopping imes γ T of Z. To calculae V p T define he opimal sopping problem 3.3 W ε,z = sup E,z F T γ ε ; Z +γ ε +γ T wih Z = z under P,z. This is a finie-horizon sopping problem wih a disconinuous gain funcion g,z = FT ε1 [,ε] z. Noe ha V p T = W ε, when ε = logp/σ. Denoe he firs hiing ime by γ ε = inf{ + u > : Z +u = ε }. Le γ T be an arbirary sopping ime of Z and le γ be he firs hiing ime afer ime γ o he level ε of Z, ha is, γ = inf{ + u > + γ : Z +u = ε }. Then γ is a sopping ime of Z and in accordance wih he form of he gain funcion g,z, i is clear ha E,z F T γ ε ; Z +γ ε E,z F T + γ T ε ; Z + γ T ε. I is herefore only opimal o sop if Z +γ = ε on he se { + γ < T }. For z ε hen E,z F T +γε Tε ; Z +γε T ε = E,z F T γε ε ; + γ ε < T + E,z 1 ; + γε T F T ε g,z. The inequaliy is rivial for z > ε. These facs indicae ha he opimal sopping sraegy is of he form γ ε = inf{ < + u T : Z +u = ε } inf = T where is o be found. The principle of smooh fi a a single poin see [12] provides a mehod o deermine. Define he funcion 3.4 W ε,z = E,z F T +γε Tε ; Z +γε T ε. For fixed, he funcion z W ε,z is in general only coninuous a z = ε and no differeniable. Le be he larges poin in he ime inerval [,T] where z W ε,z is 6
differeniable a z = ε and if z W ε,z is no differeniable a z = ε for any [,T] se =. Define he funcion W ε,z = E,z F T γ ε ε ; Z +γ ε ε. Clearly, W ε,z = W ε,z for T and he Markov propery of Z shows ha W ε,z = E,z W ε,z for <. Markov heory implies ha boh W ε,z and W ε,z solve he parial differenial equaion 3.5 + L Z W,z = for, z / [,T] {ε} wih L Z as in 3.2. If > hen z W ε,z is C 1 a z = ε. Hence, Iô s formula implies for < ha W ε,ε = E,ε W ε,z + = W ε,ε + E,ε L Z W ε,z +u 1 { Z+u =ε} du = W ε,ε E ε ε,z u du where he laer equaliy follows from 3.5. Nex, differeniae wih respec o and le o deduce ha This shows ha W ε ε,ε = is C 1 a,ε and hence W ε and space a a singular poin. Hence, γ ε following heorem shows ha γ ε ε =,ε = F T ε =. saisfies he principle of smooh fi in ime is a candidae for he opimal sopping ime. The is in fac opimal. Theorem 3.1. Consider he opimal sopping problem 3.3. Le T be given and fixed such ha z W ε,z is differeniable a z = ε for a mos one poin in he period [,T]. Se = if z W ε,z is no differeniable a z = ε for any [,T]. Then he opimal sopping ime is γ = inf{ < + u T : Z +u = ε } inf = T. The associaed value funcion is given by W ε,z = E,z W ε,z for which here can be obained a closed formula by using he ransiion densiy p u z given in Proposiion 4.2 and he densiy of γ ε given in Proposiion 4.1. Proof. The problem is o verify ha W ε,z W ε,z = E,z F T γ ε ε ; Z +γ ε ε. By he definiion of he opimal sopping problem i hen follows ha γ = γ ε will be he opimal sopping ime. The firs sep is o verify ha W ε,z dominaes he gain funcion g,z for T and z R. From he discussion above i is clear ha W ε,z g,z and hence i is enough o verify ha W ε,z W ε,z. If, hen by he definiions of he funcions i follows ha W ε,z = W ε,z. If <, hen W ε W ε,z is a bounded coninuous 7
funcion ha solves he parial differenial equaion 3.5 for, z / [, ] {ε}. For any + s < T, he ime-dependen Iô-Tanaka formula see Proposiion 5.1 shows ha W ε W ε + s,z +s = W ε W ε,z + M s + + 1 2 where L ±ε s s s ε W ε + u,ε+ z = W ε W ε,z + M s 1 s ε 2 z + u,ε+ + L Z W ε ε ε is he local ime of Z +s a ±ε and M s = s ε W ε + u,z +u1 { Z +u =ε} du W ε z z + u,ε + u,ε d L ε u + L ε u W ε + u,z z +u1 { Z +u =ε} dw u d L ε u + L ε u is a local maringale. Since ε / z,ε+ > ε / z,ε for <, he local maringale M s is bounded from below and herefore is a supermaringale. Choosing s such ha + s >, i is eviden ha = W ε W ε + s,z +s W ε W ε,z + M s and by aking expecaion i follows ha W ε,z W ε,z for T. The nex sep is o verify ha W ε + s,z s +s T is a supermaringale. Using he ime-dependen Iô-Tanaka formula as above gives W ε + s,z +s = W ε,z + M s + 1 s ε ε + u,ε+ 2 z z + u,ε d L ε u + L ε u where M s is a supermaringale by similar argumens as above. Furhermore, due o he inequaliy ε / z,ε+ < ε / z,ε for < T i follows ha s ε ε + u,ε+ + u,ε d L ε u + L ε u z z v ε ε + u,ε+ + u,ε d L ε u + L ε u z z for +s < +v < T. This inequaliy implies ha W ε +s,z +s +s T is a supermaringale. Finally, he supermaringale propery and he fac ha W ε dominaes he gain funcion g yield W ε ε,z E,z W + γ,z +γ E,z F T γ ε ; Z +γ ε for any sopping ime γ saisfying + γ T. Taking supremum i follows ha W ε,z W ε,z. The conclusion is ha W ε,z is he value funcion for he sopping problem 3.3 and hence γ ε is he opimal sopping ime. 8
4. The densiy of he firs hiing ime γ ε Recall he noaion of γ ε = inf{ > : Z = ε } for ε > where Z is he soluion o sochasic differenial equaion 3.1. The process Z is hen refleced Brownian moion wih drif. Le Z = z under P z. General Markov process heory shows ha for α > he Laplace ransform l α z = E z e αγ ε is he soluion of he differenial equaion L Z l α z = α l α z for z ε,ε wih he boundary condiion l α ±ε = 1 and L Z is given in 3.2. In view of his, i is no difficul o check ha l α z = E z e αγ ε = a 2 e a 1 z a 1 e a 2 z a 2 e a 1ε a 1 e a 2ε for z < ε and where a 1,2 = 2 + 2α. Insead of invering he Laplace ransform o derive he densiy of γ ε, he approach is o use he Girsanov heorem see e.g. [15, Chaper VIII]. Le L W denoe he local ime of W a zero see [15, Chaper VI] given by and se ρ ε = inf{ > : W = ε }. L 1 W = lim δ 2δ 1 [,δ W u du Proposiion 4.1. The densiy of he firs hiing ime by a refleced Brownian moion wih drif o a level ε > is given by z ε z ε P z γε d ϕ d for z > ε 3/2 d = e ε z 2 /2 e y P z L ρε W dy,ρ ε d dy for z < ε where P z L ρε W dy,ρ ε d is given in Lemma 4.3 below. Proof. For z > ε, he process canno hi zero before sopping and hence he process behaves like a Brownian moion wih drif and he resul is provided in [2, 2 2..2]. Fix z < ε. The process Z becomes a P z -Brownian moion by Girsanov heorem where d P z = exp sgnzu 2 du dp z Then = exp P z γε d = Ẽz exp = E z exp sgnz u dz u + 1 2 sgnz u dz u + 2 2 dp z. sgnw u dw u 2 2 = E z exp W L W z sgnz u dz u 2 2 γ ε = P z γε d ρ ε = P z ρε d 2 2 ρε = P z ρε d = e ε z 2 /2 E z exp L W ρε = P z ρε d 9
where he second las equaliy follows from W = z + sgnw u dw u + L W. Since E z exp L W e y ρε = = P z ρ ε d P zl ρ ε W dy,ρ ε d he above equaion yields he desired resul where he join densiy of L ρ ε W and ρ ε is given in Lemma 4.3. If = hen for z < ε he densiy of γ ε has a more simple expression given by see [2, 3 2..2] P z γ ε d = P z ρ ε d = cc z,ε d where cc z,ε is defined in Remark 4.4. Proposiion 4.2. The ransiion densiy of Z is given by p u z = 1 u z e u z 2 /2 ϕ + e 2 u Φ + u + z. Proof. An inspecion of he above proof shows ha he ransiion densiy can also be derived by using he Girsanov heorem. Indeed, as above p u z du = P z Z du = e u z 2 /2 E z exp L W ; W du. The las erm of he righ hand-side of he equaliy see [2, 1 1.3.7] is given by E z exp L W { 1 u z ; W du = ϕ + e u + z +2 /2 Φ } u + z + du. To complee his secion, he join densiy of L ρ ε W and ρ ε is given in he following lemma. Lemma 4.3. If z < ε, hen L ρ ε W,ρ ε has he join densiy P z L ρε W dy,ρ ε d = ss ε z,ε s ε h 1,y gs 2 ε,y dy d for y, >. See Remark 4.4 for he definiions of he funcions appearing in he formula. Proof. Noe ha [2, 3 2.3.3] see also [15, Exercise 4.9, Chaper VI] gives E z exp αρ ε βl ρ ε W 2α coshz 2α + β sinhz 2α = 2α coshε 2α + β sinhε 2α and hen see [2, 3 2.3.4] E z exp αρ ε ; L ρε W dy = sinhε z 2α sinhε 2α 2α sinhε 2α exp Using he formulas in Remark 4.4 he resul follows. y 2α 2y exp ε 2α 2α expε 2α exp ε 2α dy. Remark 4.4. The convoluion g h of wo Borel funcions g and h is defined by g h = guh u du. Furhermore, g k = g g k 1 and g = δ is he Dirac δ-funcion. Le L 1 α be he inverse Laplace ransform wih respec o α. From [2, Appendix 2] he following formulas are available: 2α s v = L 1 α sinhv = 2 2α k= 2k + 1 2 v 2 5/2 1 2k + 1v ϕ, v >
sinhu 2α ss u,v = L 1 α sinhv = 2α k= h 1,v = L 1 α exp v 2α = v v ϕ 3/2 ĉs v = L 1 α 2α exp 2v 2α 1 exp 2v = 2α v u + 2kv 3/2 k=, v > 4k 2 v 2 5/2 gs v,z = L 1 α exp z 2α exp 2v 2α 1 exp 2v 2α cc u,v = L 1 α coshu 2α coshv = 2α k= = v u + 2kv ϕ, u < v 2kv ϕ, v > 1 k k= k u + 2k + 1v 1 5. Time-dependen version of Iô-Tanaka formula 3/2 k! ĉs k v, v > u + 2k + 1v ϕ, u < v. This secion presens a resul ha may be hough of as a ime-dependen version of Iô-Tanaka formula which fis ino he seing of he proof of Theorem 3.1. The formula is a special case of he Iô formula in Peskir [13]. However, he curve considered in his paper is quie rivial and local ime-space calculus is no needed. A simple direc proof of he formula based on Iô-Tanaka formula is presened. Le Y be a coninuous semimaringale and le L a Y denoe he local ime of Y a a given by see [15, Chaper VI] L a 1 Y = lim δ δ 1 [a,a+δ Y u d Y,Y u. Le h : [, R R be such ha h,y is C 1 and y h,y is coninuous. Suppose moreover ha y h/ y,y and y 2 h/ y 2,y exis on R \ {a} and are coninuous, and ha he limis h/ y,a± = lim y a± h/ y,y and 2 h/ y 2,a± = lim y a± 2 h/ y 2,y exis and are finie. In oher words, he funcion h is C 1,2 everywhere bu a y = a i is only coninuous. Then follows an exension of Iô-Tanaka formula. Proposiion 5.1. Then for every >, h h,y = h,y + y u,y h u dy u + u,y u du + 1 2 h 2 y u,y u1 2 {Yu a} d Y,Y u + 1 h h u,a+ 2 y y,a dl a uy. Proof. If f : R R is a coninuous funcion such ha f and f exis on R \ {a} and are coninuous, and he limis f a± = lim y a± f y and f a± = lim y a± f y exis and are finie, hen Iô-Tanaka formula reads as fy = fy + f Y u dy u + 1 2 f Y u 1 {Yu a} d Y,Y u + 1 2 f a+ f a L a Y. 11
Le h be of he form h,y = gfy where g : [, R is a C 1 -funcion. The above formula and he inegraion by pars formula yield ha h,y = gfy + + 1 2 guf Y u dy u + guf Y u 1 {Yu a} d Y,Y u + 1 2 g ufy u du f a+ f a gu dl a uy. This is he formula conained in he proposiion wih h/ yu, a+ h/ y, a = f a+ f a g. The proof for any h given in he lemma follows from he same argumens as hose which prove he Iô formula see he proof in [15, Chaper IV] and need no herefore be deailed here. References [1] Black, F. and Scholes, M. 1973. The pricing of opions and corporae liabiliies. J. Poli. Econ. 81 637-659. [2] Borodin, A.N. and Salminen, P. 22. Handbook of Brownian moion - facs and formulae. Second ediion. Birkhäuser. [3] Boyce, W.M. 197. Sopping rules for selling bonds. Bell J. Econom. and Managemen Sci. 1 27-53. [4] du Toi, J. and Peskir, G. 25. The rap of complacency in predicing he maximum. To appear in Ann. Probab. [5] Gilber, J.P. and Moseller, F. 1966. Recognizing he maximum of a sequence. J. Sais. Assoc. 61 35-73. [6] Graversen, S.E., Peskir, G. and Shiryaev, A.N. 21. Sopping Brownian moion wihou anicipaion as close as possible o is ulimae maximum. Theory Probab. Appl. 45 41-5. [7] Graversen, S.E. and Shiryaev, A.N. 2. An exension of P. Lévy s disribuional properies o he case of a Brownian moion wih drif. Bernoulli 6 615-62. [8] Griffeah, D. and Snell, J.L. 1974. Opimal sopping in he sock marke. Ann. Probab. 2 1-13. [9] Karlin, S. 1962. Sochasic models and opimal policy for selling an asse. Suidies in Applied Probabiliy and Managemen Science, Sandford Univ. Press 148-158. [1] Meron, R.C. 1973. Theory of raional opion pricing. Bell J. Econom. and Managemen Sci. 4 141-183. [11] Øksendal, B. 1998. Sochasic differenial equaions. An inroducion wih applicaions. Fifh ediion. Springer. [12] Pedersen, J.L. 23. Opimal predicion of he ulimae maximum of Brownian moion. Soch. Soch. Rep. 75 25-219. [13] Peskir, G. 25. A change-of-variable formula wih local ime on curves. Theore. Prob. 18 499-535. [14] Peskir, G. and Shiryaev, A. 26. Opimal Sopping and Free-Boundary Problems. Birkhäuser. [15] Revuz, D. and Yor, M. 1999. Coninuous Maringales and Brownian Moion. Third ediion. Springer. [16] Salminen, P. 2. On Russian opions. Theory Soch. Process. 622 161-176. [17] Zhang, Q. 21. Sock rading: An opimal selling rule. SIAM J. Conrol Opim. 4 64-87. Jesper Lund Pedersen Laboraory of Acuarial Mahemaics, Universiy of Copenhagen Universiesparken 5, DK-21 Copenhagen, Denmark E-mail: jesper@mah.ku.dk hp://www.mah.ku.dk/ jesper 12