Optimal Stopping Time to Buy an Asset When Growth Rate Is a Two-State Markov Chain

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1 American Journa of Operaions Researc, 4, 4, 3-4 Pubise Onine May 4 in SciRes. p:// p://x.oi.org/.436/ajor Opima Sopping Time o Buy an Asse Wen Grow Rae Is a Two-Sae Markov Cain Pam Van Kan Miiary Tecnica Acaemy, Hanoi, Vienam Emai: van_kan78@yaoo.com Receive 3 Marc 4; revise 3 Apri 4; accepe Apri 4 Copyrig 4 by auor an Scienific Researc Pubising Inc. Tis work is icense uner e Creaive Commons Aribuion Inernaiona License (CC BY. p://creaivecommons.org/icenses/by/4./ Absrac In is paper we consier e probem of eermining e opima ime o buy an asse in a posiion of an upren or ownren in e financia marke an currency marke as we as oer markes. Asse price is moee as a geomeric Brownian moion wi rif being a wo-sae Markov cain. Base on observaions of asse prices, invesors wan o eec e cange poins of price rens as accuraey as possibe, so a ey can make e ecision o buy. Using fiering ecniques an socasic anaysis, we wi eveop e opima bounary a wic invesors impemen eir ecisions wen e poserior probabiiy process reaces a cerain reso. Keywors Opima Sopping Time, Poserior Probabiiy, Treso. Inroucion In [], e auors consier e probem of eermining e opima ime o se a propery wie price grow rae is a ranom variabe a akes e vaue of e given se. Uner e assumpions of e probem consiere in [], grow rae ony ges one of e possibe vaues a o no cange from is vaue o oer vaues, wic means a ransiion probabiiy ensiy is ; bu a a ime I o no know e accuracy of price grow rae an e probabiiy of receiving a cerain vaue of grow rae aso canges over ime. Te auors in [] aso ave muc opima sopping ime in maemaica finance, bu is is e cassica probem an ess common in pracice because of is assumpions. Te auors in [3] consier e opima sopping ime probem wen e grow rae of price process is no a ranom variabe bu in many cases. How o cie is paper: Van Kan, P. (4 Opima Sopping Time o Buy an Asse Wen Grow Rae Is a Two-Sae Markov Cain. American Journa of Operaions Researc, 4, 3-4. p://x.oi.org/.436/ajor.4.433

2 P. Van Kan In [4] e auors consier e probem o fin e opima ime o se wen e price grow rae is Markov cain, owever eir approac is ifferen from our meo in is paper an e resus are aso ifferen. In is paper we consier e price process is escribe by geomeric Brownian moion wen is rif (price grow is e Markov cain wi wo saes an (: ecrease, ownren; : increase, upren. Tere is a penomenon known as e momenum of e sock price, wic means a e sock price as increase or is increasing, i wi be increase in e fuure (usuay near fuure, an a sock price as ecrease or is ecreasing, i wi be ecrease in e fuure. Te invesmen is base on e momenum of price process a is cae e momenum raing or ren foowing raing. By is way if an invesor oing an asse wans o se e asse, e wi ave o wai uni a bu appeare an coninues o o suc asses o e nex price increase (in momenum an wen momenum is no onger avaiabe, i.e. prices going own or jus saring going own, en e ecies o se. Simiary, if a person wans o buy an asse, e wi wai for e appearance of an opporuniy of going own in price an wai for prices o go own furer (in momenum uni no onger faing price, en e ecies o buy. Accoring o is way of invesmen, invesors are expece o buy e propery a e boom of marke an se e propery a e iges poin of e marke. In is paper, we anaye ow o seec buy an se opima sraegies for a momenum raing. More precisey, we seek o maximie e expece profi from a momenum rae. Te meo we use o suy in is paper is e maringae eory, cange of measure an e opima sopping ime is referre in e ieraure [] [5] an [6].. Buying Asse Probem Now we consier e case a,, is a Markov cain wi wo saes an a P a = a = π ; P a = a = π an ony capabe of moving from sae o sae wi ransiion ensiy as λ λ λ foows Q = ( > A e ime of π = = were { } > we pu P{ a a } is e fier generae by. T We now consier e probem: Fin -sopping ime, T suc a: U = inf E e r (. T Accoring o e eorem 9. (view [5], poserior probabiiy process were ( π π saisfying: a aπ + a a a π = λ( π + π W = λ( π + π ( π W σ σ an (, W { } ( π π a a a a W + a π + π + σ = = σ σ { a ( π a + πa } = + W σ W is a Brownian moion an e fier generae by W coincies wi fier Tus e process saisfies e equaion:. = E a + σw = π a + πa + σw Ten e process an e poserior probabiiy process π saisfies e equaion: 33

3 P. Van Kan Pu π Φ =. Accoring o Io formua we ave: π ( a π ( a = + a + σw a a π = λ ( π + π ( π W σ a a Φ = π + π 3 ( π σ ( π ( π Definiion process { W } as foows: An e new measure λ π π λ = + + = + Φ + Φ π π π π W π W * P saisfying: ( π σ W = + W P = exp ( σ π + ( σ + π W exp = σ + π + σ + π W P * T T T T a a were =. σ Accoring o Girsanov eorem, We ave: W is a * P -Brownian moion. λ Φ = + π Φ + Φ ( W ( π σ = ( λ + ( λ + σ Φ + Φ W +Φ Φ Φ = Z Φ + λ Zs s ; Z = exp λ + σ + W ( ( ( = π a + π a + σ W π σ = a + σ + σ W σ = exp a + + σw Consier e foowing process: η = exp ( σ π ( σ π W Is a Le = exp ( σ π ( σ π ( W ( π σ = exp ( σ π ( σ π W -maringae an ( π σ η W η = wi η =. ( a λ U is process eermine by U = λuσuw e, we ave U =, U = Consier e foowing process: ( +Φ U Y =, Y =. Accoring o Io formua we ave: +Φ U (. 34

4 P. Van Kan Y ( = U λ + λ + σ Φ + Φ W + ( +Φ λuσuw ΦσU Φ = UΦW ( +Φ σuw = ( Φ ( +Φ σ UW = σ ( +Φ UW +Φ Y = ( π σ YW = ( π σ W Y From is we ave Y = η (a.s., en r r +Φ ( a λr η Y e = e = e +Φ. x Denoe = x, Φ = φ en EPe E e E e E e P P P + φ Probem (. is equivaen o e foowing probem: Pu r r r ( a λr = * ηt = * η = * +Φ F ( a λr ( φ inf E * e ( = +Φ (.3 T ( aλr ( aλr ( ( aλr ( aλr ( ( aλr ( aλr = e ( a r + ( a r Φ + e Φ W ( a λr Z = e +Φ. Accoring o Io formua we ave: aλr ( aλr Z = a λ r e +Φ + e Φ P ( = a λ r e +Φ + e λ + λ + σ Φ + Φ W = e a λ r +Φ + λ + λ + σ Φ + e Φ W ( aλ r ( a r ( a r So, e rif of U is e + Φ, i wi be posiive if r a ( a r + ( a r Φ > Φ > a r, an i wi be negaive if r a Φ < a r opima sopping ime is e firs ime e process is Φ in area [, saisfy e foowing free bounary probem: F + ( a λ r F = < < A F( = + A F ( A = F ( + < w F F F. Differenia Equaion in (.4 as e genera souion as foows: were is infiniesima operaor = + ( λ + ( λ + σ were = + F CF CF + σ + λ λ. Tis suggess o us a e A wi some A. Moreover pair ( AF, + σ + λ F = x e Wiaker W, 4 4λ 4 ( a + a r + + 4( a a + λ λ, ; + σ + λ λ + σ + λ F = x e Wiaker M, 4 4λ 4 ( a + a r + + 4( a a + λ λ, (.4 35

5 P. Van Kan an b Γ( b ( b a ( a + a ba Wiaker M ab,, = e M b a+,+ b, ; M ab,, = e Γ Γ + a ba b ( a Wiaker W ab,, = e U b a+,+ b, ; U ab,, = + e Γ Canging variabes an using some anayic ransformaions we obain an Denoe We ave λu λ σ 8( a λ r ( β 3 8( a λ r λ σ + + ( β + F = e u + u u u 8( a λ r λ λ λ σ σ 3 λ 8( a λ r λ σ σ F = e u u u λ σ 8( λ + ra α β α γ ( β β = >, = + >, = + >. β+ γ 3 γ β+ αu = + = ( β+ γ3 ( ( γ β+ F e u u u an F e u u u Now we consier e propery of e funcion Firs, we cacuae e erivaive of F (. F( + F( F ( = im αu e ( β γ 3 ( ( γ β+. F = u u u + ( ( αu β+ γ 3 γ β+ αu ( β+ γ3 ( γ β+ = im e u u u e u u u + + αu ( β+ γ ( ( ( ( im e 3 γ β+ αu β+ γ u u e u 3 γ β+ = u u + αu ( β+ γ3 ( γ β+ im e u u u + + αu ( β+ γ3 ( ( ( γ β+ ( γ β+ = im e u u u u + αu ( β+ γ3 ( γ β+ im e u u u + 36

6 P. Van Kan + ( ( γ β+ u ( u ( γ β+ + β+ γ3 γ β + e αu β+ γ γβ αu = im e u u = u u u 8( λ + ra Because γ = ( β + > β so F ( <. e γ β 3 αu Accoring o average inegra eorem we ave ( + ( β+ γ were u, F = u u u. Moreover, wi sma enoug we ave + ( β+ γ ( β+ γ ( γ β ( ( γ β β+ γ u < <. Ten we ave: u F > u u = > ε + wen + β + γ because < < an We ave An en F Moreover β + γ <. 4 γ β + e α β+ γ γβ F = + > because r > a so is increasing funcion. 8( λ + ra γ ( β β = + > e α ( β+ γ+ γβ 3 γ β + γ β F = + < because ( λ γ ( β ( β ( λ 8 + r a 8 + a a = + < + 4 ( λ + σ So γ < ( β + = ( β + 4β = β + an F < means a e funcion F ( is increasing an convex funcion on (,. Figure sows e grap of funcion F (, we can ceck e increase an convex properies of i. Te grap of F ( is sown in Figure, we can see a i ens o infinie wen as +. Bu F ( + <, en C = so By (.4 we ave So A is e souion of equaion ( β+ γ3 ( γ β+ F = C e u + u u ( β+ γ3 ( γ β+ C e u + Au u = + A γ β + C u + Au u = e α β+ γ γβ β+ γ 3 γ β+ ( β+ γ γβ u ( + Au u = ( + A( γ β + u ( + Au (.5 e e 37

7 P. Van Kan Figure. Grap of e funcion F (. Figure. Grap of e funcion F (. Lemma.. Equaion (.5 as unique souion. Proof: Equaion (.5 is equivaen o Pu: β+ γ 3 γ β+ ( β+ γ γβ ( γ β e u + xu u + x + e u + xu u = β+ γ 3 γ β+ ( β+ γ γβ α u ( γ β f x = e u + xu u + x + e u + xu u We ave For inegras ( β+ γ 3 ( β+ γ α u ( γ β f + = e u u + e u u u ( 3 4 u 4 u u I e α β γ u u e α β γ u e α + + β+ γ β γ u α e α + = = = + u u β + γ β + γ 4α = β + γ ( β+ γ e u u 38

8 P. Van Kan So: Because 4α ( β+ γ 4α + ( β γ ( β+ γ f ( + = + ( β γ e u u = e u u β + γ β + γ 8λ 8( λ + ra 8 u β γ ( r a α + ( β+ γ = e u u = e u u < β + γ ( β + γ β+ γ γβ ( β+ γ γβ ( γ β ( γ β β ( β+ γ+ 3 f x = + e u + xu u + e u + xu u γ ( e ( ( γβ + x γ β + u + xu u γ β ( β+ γ+ ( γβ3 = + x γ β + e u + xu u 4 ( λ + σ 8( λ + ra γ ( β ( β β β γ ( β β < + = + 4 = + ; = + > an γ β ( β+ γ+ so f ( x ( x e u ( xu ( γβ = + γ β u >, x > We wi prove im f ( x x + = + Inee, for arge enoug x we ave + xu xu an ( β+ γ 3 γ β+ ( β+ γ γβ ( γ β f x = e u + xu u + x + e u + xu u ( γ β+ α u ( β+ γ3 ( γ β+ γβ β+ γ γβ ( γ β x e u u u + x + x e u u u ( γ β+ α u ( β+ γ3 ( γ β+ γ β+ α u γ ( β γ ( γ β = x e u u u + x e u u = + ( γ β+ γ x e u u > Tus, im f ( x <, im f ( x = + và f ( x is increasing funcion so x + x + f x = ave unique expe- rience. So e eorem is prove. Te grap of f ( x is sown in Figure 3, i is an increase funcion an im f ( x, im f ( x x + x + Teorem.. Sopping ime = inf { : Φ A} is e opima sopping ime for (.. D A < = +. Proof: Accoring o e genera eory of opima sopping ime is e poin of making e expression: ( + a λ r( + φ = a r+ ( a r φ is posiive wi be uner e coninuaion area: C = { φ: F( φ > + φ}. An sopping ime wi be opima if i is e firs ime a e { } = { φ: F( φ = + φ} or A = inf { : Φ A}. In Figure 4, we see a simuaion of processes: Asse Price process, process Φ poserior probabiiy process an e opima sopping ime wi a =.; a =.; σ =.4 ; r =.5; λ =.3. Φ is e sopping regions, reso probabiiy, 39

9 P. Van Kan Figure 3. Grap of e funcion f ( x. Figure 4. A simuaion of asse price process, e poserior probabiiy process, process Φ, e reso probabiiy an e opima sopping ime (e opima ime o buy. We fin a * =.63 an e asse price (iscoune o buy is.4.5 < =. In Figure 5, we simu-, reso probabiiy poserior probabiiy process an e ae 4 processes: Asse Price process, process Φ opima sopping ime wi parameers We fin a a =.; a =.; σ =.4 ; r =.5; λ =.3. * =.58 an e asse price (iscoune o buy is.7.5 < =. 4

10 P. Van Kan Figure 5. A simuaion of asse price process, e poserior probabiiy process, process Φ e opima sopping ime (e opima ime o buy. 3. Concusion, e reso probabiiy an In is paper, we consier e probem of buying an asse wen e asse price is moee by e geomeric Brownian moion wic as a cange poin, were price grow rae is e Markov process wi wo saes a escribes e ecreasing an increasing of asse prices process on e marke. For buying probem we assume a e price wi be a sif from ecreasing o increasing prices an a buying ecision is mae wen e probabiiy of ecreasing sae surpasse some cerain reso. We ave o simuae e price process wi a number of parameers an conuc numerica souion o e experimena buying reso. In e nex suy we wi consier e probem wi assumpions a are coser o reaiy an more consrains. Acknowegemens Tis researc is fune by Vienam Naiona Founaion for Science an Tecnoogy Deveopmen (NAFOSTED uner gran number 3-.7 References [] Kan, P. ( Opima Sopping Time for Hoing an Asse. American Journa of Operaions Researc,, p://x.oi.org/.436/ajor..46 [] Peskir, G. an Siryaev, A.N. (6 Opima Sopping an Free-Bounary Probems (Lecures in Maemaics ETH Lecures in Maemaics. ETH Züric (Cose. Birkäuser, Base. [3] Siryaev, A.N., u, Z. an Zou,.Y. (8 Tou Sa Buy an Ho. Quaniaive Finance, 8, p://x.oi.org/.8/ [4] Guo,. an Zang, Q. (5 Opima Seing Rues in a Regime Swicing Moe. IEEE Transacions on Auomaic Conro, 5, [5] Lipser, R.S. an Siryaev, A.N. ( Saisics of Ranom Process: I. Genera Teory. Springer-Verag, Berin, Heieberg. [6] Siryaev, A.N. (978, 8 Opima Sopping Rues. Springer Verag, Berin, Heieberg. 4