The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing When the Horizon Is Infinite

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1 American Journal of Operaions Research, 08, 8, 8-9 hp://wwwscirporg/journal/ajor ISSN Online: ISSN Prin: The Opimal Sopping Time for Selling an Asse When I Is Uncerain Wheher he Price Process Is Increasing or Decreasing When he Horizon Is Infinie Nguyen Khac Minh, Nguyen Thanh Trung, Pham Van Khanh Thang Long Insiue of Mahemaics and Applied Science, Hanoi, Vienam Faculy of Fundamenal Science, Miliary Academy of Logisics, Hanoi, Vienam How o cie his paper: Minh, NK, Trung, NT and Khanh, PV (08) The Opimal Sopping Time for Selling an Asse When I Is Uncerain Wheher he Price Process Is Increasing or Decreasing When he Horizon Is Infinie American Journal of Operaions Research, 8, 8-9 hps://doiorg/0436/ajor Received: January 6, 08 Acceped: March 6, 08 Published: March 9, 08 Copyrigh 08 by auhors and Scienific Research Publishing Inc This work is licensed under he Creaive Commons Aribuion Inernaional License (CC Y 40) hp://creaivecommonsorg/licenses/by/40/ Open Access Absrac Assume ha we wan o shell an asse wih unknown drif bu known ha he drif is a wo value random variable, and he iniial disribuion can be esimaed As ime goes by, his disribuion is updaed and base on he probabiliy of he drif akes he small one gives us he sopping rule Research resuls show ha he opimal sraegy o sell he asse is if he iniial probabiliy ha he drif receives a small value greaer han a cerain hreshold hen liquidaes he asse immediaely, oherwise he asse holder will wai unil he probabiliy of he drif receives a small value passing a cerain hreshold, i is he opimal ime o liquidae he asse Keywords Opimal Sopping Time, Incomplee Informaion, Probabiliy Process Inroducion This paper considers he problem of opimal iming for selling of an asse under incomplee informaion abou is drif The asse price is assumed o follow a geomeric rownian moion wih unknown drif, and an agen who decides o sell a ime for which he epeced value of he discouned asse price is maimized Of course, when informaion is compleed, he corresponding opimal liquidaion problem is rivial Indeed, if he drif of he asse price is larger han he ineres rae, hen on average he asse price grows faser han money in a risk-free bank accoun, and he agen should keep he asse as long as possible DOI: 0436/ajor Mar 9, 08 8 American Journal of Operaions Research

2 N K Minh e al Similarly, if he drif is smaller han he ineres rae, i implies ha he agen should liquidae he asse immediaely and insead deposi he money in he bank In his paper we consider he incomplee informaion problem by modelling he drif as a random variable which akes wo values Then we reduce he wo-dimensional problem ino a one-dimensional opimal sopping problem There are many sudies ha have used various mehods o solve he opimal sopping ime problem For insance, he auhors in [] gave he general heory of opimal sopping ime and considered a se of opimal sopping ime problems in areas such as mahemaical saisics, mahemaical finance, and financial engineering The opimal sopping ime problem wih deerminisic drif is considered in [] in various cases To ge he esimaion of he parameers we use heory in [3] and [4] and use hese esimaed parameers o solve realisic opimal sopping problem This paper is a coninuaion of he sudy in he paper [5] [6] and [7] However, he problem is considered in he case ha he las ime is infinie so he mehod and he resul are differen from he previous resuls The Problem and Is Soluion We assume ha he asse price process is modeled by a geomeric rownian moion (see []) as follows d = µ d+ σ d W, 0 () where { W } is a sandard rownian moion and his process is assumed o be independen of µ on a probabiliy space ( Ω, FP, ) We also assume ha he drif µ is a random variable which can ake wo values µ and µ saisfying he condiion µ < r < µ, where r 0 is consan ineres rae, and 0 is iniial price Assuming he propery owner wans o sell his propery bu does no know he rae of increase of he price is µ or µ and he only knows ha a he iniial ime he probabiliy disribuion of he evens { µ = µ } and { = } as follows (Table ) µ µ The purpose of he propery holder is o sell i for maimum epeced reurns and he purpose of his problem is he same Mahemaically we denoe { } [ 0; ) be he σ-field generaed by he process and propery holders he choose 0; such ha he supremum -sopping ime wih [ ) V = sup E e r () is achieved a a cerain sopping ime For 0 π = P µ = µ is he condiional probabiliy ha he, le { } Table The iniial disribuion of he drif µ µ µ Probabiliy π π DOI: 0436/ajor American Journal of Operaions Research

3 N K Minh e al drif receives small value a ime and herefore π = P { µ = µ } From Theorems 7 and 9 in [3], he price process following sochasic differenial equaion and he belief probabiliy process where ( ) ( ) d = µ π + π µ d+ σ dw π saisfy equaion ( ) dπ = ωπ π dw = and (, ) ω µ µ σ is modeled by he W F is a P-rownian moion defined by ( ) ( π ) µ µ π µ dw = dw + d (3) σ To reduce he dimension of he problem we define a new process W by ( σ ωπ ) dz = + d+ dw and a new probabiliy measure Q wih he Radon-Nikodym derivaive as defined dq = e dp = e T ( σ+ ωπ) d+ ( σ+ ωπ) dw + λ 0 0 T T ( σ+ ωπ) d+ ( σ+ ωπ) dz + λ 0 0 wih respec o measure P where λ is chosen such ha µ λ r < 0 and ( µ λ r) ω > 0 Girsanov s Theorem so ha Z is a rownian moion under measure Q We define a new process T π Φ = Applicaion of Io s formula we obain π ωπ ( π ) ( ( Z π) + ( π + σ) ) dφ = ωπφ d+ dπ = ωπφ d dw ( π) = ωπφ d ωφ dw = ωπφ d ωφ d ω d = σωφ d ωφ dz Epressing of in erms of Z gives ( ) (4) (5) d = σ + µ d+ σ dz (6) Then, under he Q measure, boh and Φ are geomeric rownian moions Moreover, he σ-field generaed by Z and is coincided Now o build he calculaions on he new measure we define he following process Proposiion We have ( σ+ ωπ s) ds ( σ+ ωπs) dzs λ 0 0 θ = e r Q ( µ λ r) E e E e ( ) = + Φ + φ where is a sopping ime adaps o he filer Proof: DOI: 0436/ajor American Journal of Operaions Research

4 N K Minh e al s We have θ = ep ( σ + ωπs) ds ( σ + ωπs) dzs ep{ λ} and here- fore Consider he process 0 0 dθ λd ( ωπ σ ) dz θ = + + K = ( ) ( µ λ e ) + Φ ( + φ ) Using Io s formula, we obain ( µ λ) l e dk = ( + Φ ) λd+ ( ωπ σ ) dz + ( + φ ) ( ωπ σ ) = K λd+ + dz So θ = K almos sure Thus we have: r Q ( µ l λ r) E K e = E e ( + Φ ) + φ This proves he Proposiion From he above Proposiion he value funcion in he measure Q given by We see ha larger Q ( µ l λ r) V = sup E e + + φ ( Φ ) (7) Φ is he less likely ha he gain will increase upon con- 0; such ha he sop- inuaion This suggess ha here eiss a poin ( ) ping ime is opimal in he problem (7) { } : = inf 0 : Φ y opimal sopping heory, he pair ( F, ) is he soluion of he following free boundary problem where ω φ F + ( µ λ r) F = 0 0 < φ ( φ) = + φ φ ( ) = + ( smooh condiion ) '( ) = '( ) 0 < (8) F (9) F (0) F () F = () F = F σωφ F F φ F = 0 is infiniesimal operaor and he condiion ( ) is added o define ( ) This follows ha F is he soluion of he differenial equaion ω φ F σωφ F ( µ λ r) F 0 + = (3) DOI: 0436/ajor American Journal of Operaions Research

5 N K Minh e al One may now recognize his differenial equaion as he Cauchy-Euler equaion and i has characerisic equaion: Le We find ha ω ω σω + + µ λ r = 0 (4) ω ω g ( ) = σω + + µ λ r ( ) g 0 = µ λ r < 0 ; ( ) g = σω µ λ r = µ + µ + µ λ r = µ λ r < 0 Therefore g() has wo soluions < 0< < hus he general soluion F ( φ ) can be wrien as ( ) C C F φ = φ + φ < φ < (5),0 where C and C are consans which are deermined laer The hree condiions (9)-() can be used o deermine C, C and uniquely We have The condiion F ( ) = 0 gives ( φ) φ + φ F = C C C + C = 0 and he condiion F ( ) This follows The condiion F( ) = gives us C + C = C = C = = + is equivalen o C + C = + Subsiue C and C from (6) ino above equaion we have = + (6) We have equaion o deermine as follow ( ) ( ) + + = 0 (7) Theorem The equaion (6) has unique soluion ( ; ) Proof: Consider funcion ( ) : ( ) ( ) ( 0; ) We find ha h + = + + in DOI: 0436/ajor American Journal of Operaions Research

6 N K Minh e al ( ) h = + < 0 and lim h ( ) > 0 since ( ) > 0 So equaion ( ) 0 in ( ; + ) h = has a soluion We see ha h ( ) = ( )( + ) + ( ) ( ) h ( ) = ( )( + )( ) > 0 This follows We have ( )( ) ( ) ( ) = ( ) + ( ) + ( )( + ) = ( ) ( ( ) ) h h ( ) > 0; ( ) > 0; ( r) ( r) µ λ µ λ ω = = > 0 ω ω by choosing λ and herefore h ( ) h ( ) > 0 Thus h ( ) is increasing funcion in ( ) unique saisfies he problem Theorem F ( φ ) + = C + C + φ, ; + This implies here eiss ( C C ) φ + φ,0< φ < wih > is unique soluion off he following equaion ( ) ( ) φ + + = 0 + C and : = inf 0 : Φ is opimal of (7) Proof: Le be he unique soluion o (7), and define funcion G by C defined in (6) and sopping ime { } G ( φ ) + = C + C + φ, Then we have G ( φ) + φ, and G ( φ) ( µ λ r) The process Y e G( ) equaion ( C C ) φ + φ, 0 < φ < φ > + φ if and only if φ < = Φ saisfies he following sochasic differenial ( ) ( ( ) ) { } ( µ λ r) µ λ r µ λ r µ λ r ω G Φ> ( ) dy = e + Φ d e Φ Φ dz Assume ha he soluion of (7) saisfies he condiion 0 Then he drif of Y is always negaive, and herefore Y is a super-maringale and Y Λ is maringale Le is a sopping ime We have inequaliy DOI: 0436/ajor American Journal of Operaions Research

7 N K Minh e al This follows F( φ) G( φ) And if Therefore, ( ) ( ) ( µ λ r) φe + Φ φy φy0 = G φ (8) =, we see ha he inequaliies in (8) are equaliies I follows ha F( φ) G( φ) ( φ) = = ( φ) F sup Y Y G φ φ = π Corollary Le φ = The value funcion is given by: π + +,0< φ < V = C + C + φ, φ ( ) ( Cφ Cφ ) Moreover, sopping ime : = inf 0 : π + () Theorem 3 Value funcion V is decreasing in π Proof: V When φ = 0 If φ φ ( + ) V Cφ + C φ = C φ + C + φ = C = C C 0, we have is opimal of problem ( + ) + ( ) + ( ) C + C ( + φ ) ( + ) ( ) ( ) + + φ + C ( + φ ) φ φ φ φ φ φ φ If φ he funcion + ( ) : ( ) + = + ( ) h φ φ φ φ is increasing funcion so φ < follows h( φ ) 0 and if 0< φ < we have + ( ) ( ) : = + + ( ) ( ) φ + + ( ) φ = ( ) + ( ) φ = ( ) φ < 0 h φ φ φ φ his gives ha V is decreasing in φ and φ is increasing funcion in π We complee he proof 3 Simulaion In his secion we simulae he price process, Poserior probabiliy process and he hreshold for selling he asse The parameers are DOI: 0436/ajor American Journal of Operaions Research

8 N K Minh e al µ = 005; µ = 0; r = 008; λ = 005; σ = 05; ( ) ( ) ω = µ µ σ; = 3; φ = (9) Le, are he soluions of he following equaion ω ω σω + + µ λ r = 0 (0) wih he parameers are given in (9), equaion (0) becomes: = or = 0 and we have = and = And is he soluion of ( ) ( ) + + = 0 () + We use Malab sofware o solve Equaion () and ge = 798 In he Figure, we see ha he poserior probabiliy process canno pass hrough he red hreshold, so he holder of he propery will no be able o sell i However, since he ime in simulaion is finie, he propery owner will sell he asse a he end ha is = And in he Figure below he poserior probabiliy shows ha he fall in prices is very fas so he opimal sopping ime is very close o he iniial ime Figure The sopping ime and corresponding asse price in a simulaion = ; ( ) = DOI: 0436/ajor American Journal of Operaions Research

9 N K Minh e al Figure The sopping ime and corresponding asse price in a simulaion Conclusion = ; ( ) = This paper sudies he opimal sraegy o liquidae an asse when i is uncerain wheher he asse price is rising or falling (high or low drif) To solve he problem we have o change he iniial measure o he new measure and under his measure he price process is maringale We also have o solve a nonlinear equaion o find he hreshold of he probabiliy ha he drif receives he smaller value The resuls show ha he value funcion is decreasing in he iniial probabiliy ha he drif is low Simulaion resuls are consisen wih proven heory References [] Peskir, G and Shiryaev, AN (006) Opimal Sopping and Free-oundary Problems (Lecures in Mahemaics ETH Zürich) irkhäuser, asel [] Shiryaev, AN, u, Z and Zhou, Y (008) Thou Shal uy and Hold Quaniaive Finance, 8, hps://doiorg/0080/ [3] Lipser, RS and Shiryaev, AN (00) Saisics of Random Process: I General Theory Springer-Verlag, erlin, Heidelberg [4] Shiryaev, AN (008) Opimal Sopping Rules Springer-Verlag, erlin, Heidelberg [5] Khanh, P (0) Opimal Sopping Time for Holding an Asse American Journal of Operaions Research,, hps://doiorg/0436/ajor0406 [6] Khanh, P (04) Opimal Sopping Time o uy an Asse When Growh Rae Is a DOI: 0436/ajor American Journal of Operaions Research

10 N K Minh e al Two-Sae Markov Chain American Journal of Operaions Research, 4, 3-4 hps://doiorg/0436/ajor [7] Van Khanh, P (05) When o Sell an Asse Where Is Drif Drops from a High Value o a Smaller One American Journal of Operaions Research, 5, hps://doiorg/0436/ajor DOI: 0436/ajor American Journal of Operaions Research