When to Sell an Asset Where Its Drift Drops from a High Value to a Smaller One

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1 American Journal of Operaions Research, 5, 5, Publishe Online November 5 in SciRes. hp:// hp://x.oi.org/.436/ajor When o Sell an Asse Where Is Drif Drops from a High Value o a Smaller One Pham Van Khanh Insiue of Economics an Corporae Group, Hanoi, Vienam Receive 8 June 5; accepe 8 November 5; publishe November 5 Copyrigh 5 by auhor an Scienific Research Publishing Inc. This work is license uner he Creaive Commons Aribuion Inernaional License (CC Y. hp://creaivecommons.org/licenses/by/4./ Absrac To solve he selling problem which is resemble o he buying problem in [], in his paper we solve he problem of eermining he opimal ime o sell a propery in a locaion he rif of he asse rops from a high value o a smaller one a some ranom change-poin. This change-poin is no irecly observable for he invesor, bu i is parially observable in he sense ha i coincies wih one of he jump imes of some exogenous Poisson process represening exernal shocks, an hese jump imes are assume o be observable. The asse price is moele as a geomeric rownian moion wih a rif ha iniially excees he iscoun rae, bu wih he opposie relaion afer an unobservable an exponenially isribue ime an hus, we moel he rif as a wo-sae Markov chain. Using filering an maringale echniques, sochasic analysis ransform measuremen, we reuce he problem o a one-imensional opimal sopping problem. We also esablish he opimal bounary a which he invesor shoul liquiae he asse when he price process hi he bounary a firs ime. Keywors Opimal Sopping Time, Poserior Probabiliy, Threshol, Markov Chain, Jump Times, Maringale, rownian Moion. Inroucion In his paper we consier he following problem: How o fin he opimal sopping ime o sell a sock (or an asse when he expece reurn of a sock is assume o be a consan larger han he iscoun rae up unil some ranom, an unobservable, ime, a which i rops o a consan smaller han he iscoun rae. An invesor wans o hol he posiion as long as he ineria is presen by aking avanage of he rif which is exceeing he iscoune rae (or ineres rae. On he oher han, when he ineria isappears he invesor How o cie his paper: Khanh, P.V. (5 When o Sell an Asse Where Is Drif Drops from a High Value o a Smaller One. American Journal of Operaions Research, 5, hp://x.oi.org/.436/ajor.5.564

2 woul like o exi he posiion by selling he asse. The uner suy problem in his paper was also aresse in [] where he buying problem wih he same assumpion was solve. The resuls of [] showe ha he opimal buying ime was he firs passage ime over some unknown level for he a poseriori probabiliy process π efine below an by simulaing i was foun ha he opimal ime o buy an asse was he ime which he asse price process ha jus passe he rough. The auhor of [] suie a problem of fining an opimal sopping sraegy o liquiae an asse wih unknown rif; more exacly he wane o fin he bes ime o sell a sock when is rif was a iscree ranom variable which ook he given values. The firs ime he poserior mean of he rif passes below a non-ecreasing bounary ha is he unique soluion of a paricular inegral equaion is shown o be opimal. Some classical opimal sopping ime problem has been consiere in [3]. These are applie in mahemaical finance bu hese are basic problem, an i is ifficul o apply in real worl. For relae suies of sock selling problems, see [4] [5] an for suies of basic opimal sopping problems see [3]. The meho we use o suy in his paper is he maringale heory, he ransformaion heory of measuring an he opimal sopping ime is reference in he lieraure [3] [6] [7]. In his paper, he asse price is moele as a linear rownian moion wih a rif ha rops from one consan o a smaller consan a some unobservable ime. This rif is moele as a Markov chain wih wo saes which are enoe by an where is enoe for price ecrease an is enoe for price increase. We efine he asse price moel in Secion, an he opimal selling problem is se up. In Secion 3, we suy he simulaion o examine our suies an finally, Secion 4 is conclusion.. The Moel We ake as given a complee probabiliy space ( Ω, FP, ranom variable wih isribuion ( π ( λ P = =, P > > = e. On his probabiliy space, le he change-poin be a where λ is he inensiy of he ransiion from sae o sae an assume ha λ is posiive an ha belongs o [;. Denoe he rif of he price process a,, can be moele as a Markov chain wih wo saes a l enoe by sae an a h enoe by sae such ha P( a= ah = π ; P( a= al = π a ime, a l < r < a h where r is iscoune rae which is a given consan an process a, can only ransi from sae o sae wih ransiion ensiy marix as follows Q =. λ λ Nex, le W be a rownian moion which is inepen- en of. The asse price process is moele by a geomeric rownian moion wih a rif ha rops from a h o a l a ime. More precisely, = a + σ W an x a = ah ah al I i.e. a = ah if < an a = al if he volailiy σ > is a consan. A he ime of > we efine he a poseriori probabiliy process π by where { } = >, where { al } π = P a = is a filer generae by an. The process π inicae he probabiliy of even ha he T price process ecreases. We consier he opimal sopping problem: Fin -sopping ime, T such ha: r V = sup E e. (. T Similar he buying problem, poserior probabiliy process π saisfying he following sochasic ifferenial (see heorem 9., [6]: a h ahπ + al( π π = λ( π π W σ 55

3 or where W is a P-rownian moion wih respec o W Moreover, in erms of ah al π = λ( π π( π W σ given by { l h } ( π π a + a a l h π a + π a + σ W = = σ σ { a ( π a + π a l h } = + W. σ W we have = E a + σw = π a π l + ah + σ W. Processes an π saisfy he following sysem equaions: = ( al + π ( ah al + σw ah al π = λ( π + π ( π W. σ π a h al Pu Φ = ; = π σ an Io s formula gives ha: We efine new process { W } as follow: an a new measure P y Girsanov heorem, where The price process saisfying: λ Φ = + πφ Φ π W. ( π σ W = + + W T T P = exp ( σ π W σ π P + + T T = exp ( σ + π W. σ + π W is a P -rownian moion. Furhermore, Φ = ( λ + ( λ σ Φ ΦW Φ = Z Φ + λ Zs s ; Z = exp λ + σ W. saisfying he following sochasic ifferenial ( π π σ ( π σ ( = ah + al + W

4 or in erm of W The soluion of his sochasic equaion is ( σ h = a + + σ W. Now we consier he process: exp σ. = a + + σw h η = exp ( σ π ( σ π W hen η is a -maringale an ( π σ η W η = + where η =. Le U is a process which efine by U = λu σu W, we have U Pu ( +Φ ( +Φ U Y =, Y = an accoring o Iô s formula: U From his we have Y = η (a.s., hus: Denoe hen Y Y ( π σ = + W. r r ( ah λ r η = Y = +Φ +Φ e e e. = x, Φ = φ ( a λ r r r x ( ah λ r EPe = E e e e (. P ηt = E η E P = P +Φ + φ To solve he problem (. we solve he following auxiliary problem: G ( φ ( a λ r h = sup E e +Φ. M P h e =, U =. (. Pu ( ah λ r Z = +Φ e. The opimal sopping ime is he firs hiing ime of he process F, saisfying he flowing free bounary problem: Moreover pairs where is infiniesimal generae operaor. ( h λ ( ( G+ a r G = < z< G z = + z z G = G + < Φ o he area [, Differenial equaion in (.3 has he general soluion as follows: G( z CG ( z CG ( z wih some. = + where (.3 57

5 λ λxln x+ σ xln x 4 4λ 4 ( ah al r 4( ah al λ + σ λ λ x G ( x = xe Whiaker W,, x λ λxln x+ σ xln x 4 4λ 4 ( ah al r 4( ah al λ + σ λ λ x G ( x = xe Whiaker M,,. x hen Changing variables an using some analyic ransformaions we obain: We also have ( λ + λ σ 8 r a α = >, β = α >, γ = ( β + h > β ( β+ γ 3 e ( ( γ β+ an G z = u + zu u z αu ( β+ γ 3 γ β+ G z = e u zu. u z β+ γ β+ γ γ β+ 3 γ β+ β+ γ 4 ε 4 G z > z u u = z z > z + β + γ as z + since < z < an We have β + γ <. 4 γ β + G z = + z > e α ( β+ γ γ β 8 λ + r a since r+ λ > ah herefore γ = ( β + h > β an G ( z is an increasing funcion. Moreover γ β + γ β ( 3 e α β+ γ+ γ β G z = + z < since ( λ ( λ σ ( λ 8 + r a 8 + a a γ = ( β + > ( β + h l h 4 = ( β + = ( β + 4β = β +. These mean ha he funcion G ( z is increasing an convex on Figure shows he graph of funcion G graph of,. z, we can check he increase an convex properies of i. The G z is shown in Figure, we can see ha i ens o infinie when z as +. u G ( + < herefore C = an Accoring o (.3 we have ( β+ γ 3 ( γ β+ G z = C e u + zu. u ( β+ γ 3 ( γ β+ C e u + u u = + γ β + C u + u u = ( e α β+ γ γ β 58

6 Figure. Graph of he funcion G ( x. Figure. Graph of he funcion G ( x. So is he soluion of he following equaion: ( 3 β+ γ γ β+ ( β+ γ γ β u ( + u u = ( + ( γ β + u ( + u u (.4 e e. Lemma.. The free bounary Equaion (.4 has unique posiive soluion. Proof: The Equaion (.4 is equivalen o Denoe: ( 3 β+ γ γ β+ ( β+ γ γ β ( γ β e u + xu u + x + e u + xu u =. ( 3 β+ γ γ β+ ( β+ γ γ β ( γ β h x = e u + xu u + x + e u + xu u. lim >, The graph of h( x is shown in Figure 3. We shall prove ha he funcion h( x saisfying h( x lim h x x + We have = an h( x is ecreasing an herefore he equaion x + h x = has unique soluion on, +. 59

7 I follows ha ecause an Figure 3. Graph of he funcion h( x. ( β+ γ 3 ( β+ γ ( γ β h + = e u u + e u u ( β+ γ 3 ( β+ γ ( γ β = e u u + e u u 4 ( β+ γ 3 ( β+ γ = e u ( γ β + e u u β + γ 4 = e u + e u u + e u u β ( β+ γ ( β+ γ ( β+ γ α ( γ β + γ 4α α ( = ( e u β+ γ γ β + u. u β + γ 4α ( β+ γ 4α + ( β γ ( β+ γ h( + = + ( β γ e u u = e u u β + γ β + γ ( λ + r a 8λ 8 h ( β+ γ 8( ah r ( β+ γ = e u u = e u u > β + γ ( β + γ β+ γ γ β ( β+ γ+ γ β ( γ β ( γ β h x = + e u + xu u + e u + xu u γ β ( β+ γ+ ( γ β 3 + x γ β + e u + xu u γ β ( β+ γ+ ( γ β 3 = + x γ β + e u + xu. u ( λ ( λ λ σ + al ah 8 + r ah α = >, β = α = >, γ = ( β + > β 5

8 we obain 8( λ+ r ah 8( λ+ al ah γ ( β ( β ( β β β = + > + = + 4 = + γ β ( β+ γ+ h ( x ( x ( e u ( xu ( γ β = + γ β u < x >. We will prove ha lim h( x since x + =. Inee, wih he large enough x we have + xu xu so ( β+ γ 3 γ β+ ( β+ γ γ β ( γ β h x = e u + xu u + x + e u + xu u ( γ β+ ( β+ γ 3 ( γ β+ γ β β+ γ γ β ( γ β x e u u u + x + x e u u u ( γ β+ ( β+ γ 3 ( γ β+ γ β+ γ ( β γ ( γ β = x e u u u + x e u u = + ( γ β+ γ x e u u,, e αu γ + β γ < γ β + > u u = cons >. Consequenly, lim h( x >, lim h( x = an h( x is ecreasing so he equaion x + x + soluion on (, +. The heorem is prove. Theorem.. Sopping ime = inf { : Φ } Proof: Le an we will prove ha L( x Now, we examine he funcion g( x Take he erivaive we obain ( x is he opimal sopping ime for (.. + G ( x x < L( x = G ( + x x > + x x<, inee + x G ( x > + x + > +. G G G x + x =. G ( x h x = has unique β+ γ 3 γ β+ γ β + ( β+ γ ( γ β g x = e u xu u x e u xu u F h x h + + x = > = g( > g( x >. F x F G G x This follows Using he Dynkin s formula o he process Y L( x + x x. ( a λ r e h = L Φ we have: ( { } ( ah λ r ah λ r h l Φ> Y = e a r+ a r Φ e Φ L Φ W. 5

9 ah r ecause saisfying : = r a Y is maringale. y opional heorem we have: y l so he rif of Y is posiive an herefore Y is super maringale an ( +Φ Y Y = L G L ( ah λ r φe φ φ φ φ φ. = we have Y = Y = L moreover φ φ φ Q Q ( φ = sup = ( φ. So G( φ L( φ G Y Y L y y =. ah r We will show ha saisfy he coniion: : =. Inee, by general opimal sopping heory all r al poins saisfy he form wih posiive value will be in coninuaion area ( + a r( + = a r+ ( a r λ φ φ h h l { φ: G( φ φ} C = > +. The opimal sopping ime is he firs hiing ime of { Φ } o he area: D= { φ: G( φ = + φ} or = { Φ } Thus funcion G saisfy he following coniion: We efine he funcion: Now, we assume G G : Φ inf :. ( ah λ r G( φ = sup e +Φ. G φ + G ( φ φ < = + φ φ ( φ G ( φ <, because h( x is a ecreasing funcion so h( < ( ( > ( +. Specially, Φ is super maringale an ( a λ r G ( e h ( h λ G φ < + φ wih some. Then he lef erivaive of φ <. This follows ( e ah λ r G Φ maringale. For he sopping ime saisfying Φ we have: a r ( a λ r h φe +Φ = φe G Φ G φ G φ G φ. This conraics o he exisence of φ such ha G ( φ < + φ when G ( φ achieve. The opimal sopping ime u G ( φ + φ φ herefore by his, we have finishe he provemen. 3. Simulaion Suy is he firs hiing ime of { } { φ: G( φ φ} D= = +. { } = inf : Φ, Φ o he area + φ φ. Finally, we To make visual for he above heory we simulae he asse price process, he poserior probabiliy process π π process Φ = (noice ha Φ is an increasing funcion of π an he selling hreshol. Some π 5

10 parameers is use in our simulaing are a l =.5 ; r =.6 ; a h =. ; σ =.4 ; λ =.5 an he ime inerval is [, ]. As can be seen in he figures from 4 o 8 if he price is increasing hen he Φ an π are ecreasing an conversely. Figure 4 shows he price process has increase since he ime. so he Φ ecrease from he respecive ime an i can no hi he re line enoe he hreshol, herefore he opimal selling ime in his case is. Figure 5 simulae a price process which is flucuae from ime o.4 an ecrease ramaically a he ime.4 so he process Φ increase sharply from his an crossover he hreshol, i follows ha he opimal ime o liquiae he asse is abou.7. A his ime he price of he asse is lower han he origin bu if we hol i we will sell i a a much more loss in he fuure. Anoher simulaion is shown in Figure 6. Clearly, whenever he price process is increasing, he Φ an he poserior probabiliy process are ecreasing an he liquiae ime is.77. A his ime he price is no he highes bu i is significanly higher han he original value which is.5. Figure 4. A simulaion of asse price process, he poserior probabiliy process, process Φ(, he hreshol probabiliy an he opimal sopping ime. In his case, he process Φ( always uner he hreshol probabiliy so he opimal sopping ime is he final ime. Figure 5. A simulaion of asse price process, he poserior probabiliy process, process Φ(, he hreshol probabiliy an he opimal sopping ime. In his case, he firs ime ha he process Φ( over passes he hreshol probabiliy a he ime.7 so he opimal sopping ime is.7. 53

11 In Figure 7, we can see he same scenario wih he simulaion in Figure 6. The ime o liquiae in his case is.795, he price is abou.75 whereas he sare price was.5. We can see r Se =.75 e =.668 >.5 = S i means ha we benefi by his rae affair. The same scenario wih he simulaion in Figure, he simulaion resuls in Figure 8 show he price illusraes an upren from ime o he en ha he process Φ can no pass over he selling hreshol, consequenly, he opimal ime o sell in his siuaion is. 4. Conclusion This research consiers he problem of how o fin he opimal ime o liquiae an asse when he asse price is moele by he geomeric rownian moion which has a change poin. In paricular, he rif of he process rops from a high value o a smaller one an his rif process can be moele as wo-sae Markov process. The resuls of his research inicae ha a opimal selling ecision is mae when he probabiliy of ownren surpasse some cerain hreshol. We also simulae he price process wih a number of parameers an conuc Figure 6. A simulaion of asse price process, he poserior probabiliy process, process Φ(, he hreshol probabiliy an he opimal sopping ime. In his case, he firs ime ha he process Φ( over passes he hreshol probabiliy a he ime.77 so he opimal sopping ime is.77. Figure 7. A simulaion of asse price process, he poserior probabiliy process, process Φ(, he hreshol probabiliy an he opimal sopping ime. In his case, he firs ime ha he process Φ( over passes he hreshol probabiliy a he ime.795 so he opimal sopping ime is

12 Figure 8. A simulaion of asse price process, he poserior probabiliy process, process Φ(, he hreshol probabiliy an he opimal sopping ime. In his case, he process Φ( always uner he hreshol probabiliy so he opimal sopping ime is he final ime, he same wih he case in Figure 4. numerical soluion o he experimenal selling hreshol. In nex suies, we will consier problems in which he price growh rae is a Markov process which has more han saes an esablish some properies as well as isribuion of sopping ime. Acknowlegemens This research is fune by Vienam Naional Founaion for Science an Technology Developmen (NAFOSTED uner gran number References [] Khanh, P. (4 Opimal Sopping Time o uy an Asse When Growh Rae Is a Two-Sae Markov Chain. American Journal of Operaions Research, 4, 3-4. hp://x.oi.org/.436/ajor [] Khanh, P. ( Opimal Sopping Time for Holing an Asse. American Journal of Operaions Research, 4, hp://x.oi.org/.436/ajor..46 [3] Peskir, G. an Shiryaev, A.N. (6 Opimal Sopping an Free-ounary Problems (Lecures in Mahemaics ETH Lecures in Mahemaics. ETH Zürich (Close. irkhäuser, asel. [4] Shiryaev, A.N., u, Z. an Zhou,.Y. (8 Thou Shal uy an Hol. Quaniaive Finance, 8, hp://x.oi.org/.8/ [5] Guo,. an Zhang, Q. (5 Opimal Selling Rules in a Regime Swiching Moel. IEEE Transacions on Auomaic Conrol, 9, hp://x.oi.org/.9/tac [6] Lipser, R.S. an Shiryaev, A.N. ( Saisics of Ranom Process: I. General Theory. Springer-Verlag, erlin, Heielberg. [7] Shiryaev, A.N. (978 Opimal Sopping Rules. Springer Verlag, erlin, Heielberg. 55