Optimal Stopping Time for Holding an Asset

Transcription

1 American Jorna of Operaions Researc,,, p://dxdoiorg/436/ajor46 Pbised Onine November (p://wwwscirporg/jorna/ajor) Opima Sopping Time for Hoding an Asse Pam Van Kan Miiary Tecnica Academy, Hanoi, Vienam Emai: Received Sepember, ; revised Ocober, ; acceped November 3, ABSTRACT In is paper, we consider e probem o deermine e opima ime o se an asse a is price conforms o e Back-Scoe mode b is drif is a discree random variabe aking one of wo given vaes and is probabiiy disribion beavior canges cronoogicay Te res of finding e opima sraegy o se e asse is e firs ime asse price faing ino deerminisic ime-dependen bondary Moreover, e bondary is represened by an increasing and coninos monoone fncion saisfying a noninear inegra eqaion We aso condc o find e empirica opimizaion bondary and simae e asse price process Keywords: Opima Sopping Time; Bondary; Brownian Moion; Back-Scoe Mode Inrodcion In [], Siryaev and Peskir ave considered e probem: V inf W maxw () M were W is sandard Brownian process and ey fond e opima sopping ime as: were S W z z =inf : is e soion of e eqaion s s 4 z z z 3, S max W, x and x e y x d y π e π () x Te simaion for W, bondary z and S W are given in Figres -3 In [], Aber Siryaev, Zoqan and n Y Zo sove e foowing probem: Tere is an invesor oding a sock, and e needs o decide wen o se i for e as ime wi given ime o se T I is obvios a e wans o se a a ime of iges price on e inerva from o T Assme a e disconed sare price compies wi e foowing dynamic eqaion: d d ar d W, on a fiered probabiiy space,,,p were a is e grow rae of e price and is e voaiiy, r is ineres rae, W is e sandard Brownian moion wi W nder e measre P Here, is P- W increasing fier generaed by W Ten, e W Figre A simaion for sopping ime in () Te opima sopping ime is e firs ime e bondary ine (be ine) ies beow e ine describes e process S W (red ine) In is case W is very arge b is no maxw W Figre A simaion for e sopping ime in () In is case W is ess an b is no maxw Copyrig SciRes

2 58 P VAN KHANH W Figre 3 A simaion for e sopping ime in () In is case W is very arge b is no maxw were We define: ar M max s, s a r and Ten, e foowing cases: If, T is an niqe opima seing ime If, or T are opima seing imes If, is an niqe opima seing ime In is paper, we wi find e opima ime o se a sock wen e appreciaion rae is e random variabe aking one of wo given vaes a and a Te Probem of Finding e Opima Seing Time Assme a e asse price process foows a geomeric Brownian moion wi is drif is a random variabe aking one of wo given vaes a or a, e voaiiy is consan, ie d a d d W, () were W is a sandard Brownian moion independen wi a of e probabiiy space,,p Assme,,P is a compee probabiiy space wi nondecrease -fied Sppose a and a saisfy a < r < a, were r is e ineres rae and i is consan and e iniia vae of asses is a posiive consan Invesors oding asses need o decide wen o se i for e as ime wi given ime o se em is T Knowing a a e iniia ime disribion of α as π P a a P a ; a π A ime we p π P a a, were,,t is e compeion of e firaion generaed by Te probem is finding sc a V sp E e r T -sopping ime τ, τ T () were spremm is aken in -sopping ime τ, τ T Te price process and poserior probabiiy process π saisfying e eqaions d a π a ad dw (3) a a dπ π πdw were W, is a P-Brownian moion defined by: d π a πa d W (see [3], eorem 9) Define e process W by: dw π ddw and a new measre T dp P saisfying: T exp π d πdw dp T T exp π d πdw a a were According o e Girsanov eorem, W is a P - Brownian moion Le π, we ave d π d dw Ten, price process and process saisfy e eqaions d a d dw (4) d So, and are geomeric Brownian moions nder measre P Moreover, -fied generaed by W coincides wi e one generaed by We define e ikeiood process exp πs ds πs dw s Copyrig SciRes

3 P VAN KHANH 59 We know a W, is a P -Brownian moion, so is an -maringae nder measre P We ave: d a d dw exp a ds dw d d dw exp ds dw = expa πs s ds πsdw s Denoe a E is an expecaion operaor wi respec o measre P and e T is an -sopping ime Ten, by e propery -maringae nder measre P of (see [4], eorem ), we ave: r E e r E e T r a r E e E e Lemma were: Proof We ave: can be wrien as: a a e and a a exp a ds dw s and aa e exp e exp ds a a a a exp ds dw s a a exp ds dw s ds dw s (5) (6) dw s exp ds dw s We consider an opima sopping probem as: were: a r, sp e T y E Y Y yexp W, (7) were spremm is aken in -sopping ime, T wi respec o firaion generaed by W I can be seen a e opima sopping ime in (7) can be rned o e opima sopping ime in e probem () Now, we sdy e opima sopping probem (7) We wi prove a exising an increasing and coninos monoone fncion: sc a e sopping ime :,, b T T Y b T, y: inf, : is an opima sopping ime for e probem (7) Y saisfies e eqaion: dy a add W, Y On e oer and, we can wrie Y yz, were Z exp W W i is noaion, we ave: a r, sp e T y E yz Give in (7), we ave, y G y : E Y E y y Define e coninaion region C:,,, :, y G y C y T and e sopping region D:,,, :, D y T y G y Accord ing o genera eory abo opima sopping probems, e sopping ime D inf s T : s, ys D is opima sopping ime probem (7) Ts, deermin- is sfficien o defining e ing e opima sopping ime sopping region D Teorem Tere exiss a rig coninos an d nondecreasing fncion Copyrig SciRes

4 53 P VAN KHANH sc a r a b:, T, a r,,, : C y T y b Frermore, spremm in (7) is acieved wi e sopping ime inf T : Y b D Proof We know a C, y :, y: y, ygy, y, T and y y, assme a C, ere exiss a sopping ime sc a: wi every fix, y So: a r E e a r, E e y And process H ave: yz y y y y yz a r E e yz y y E a r e a r Z a r E e Z e Z is a sbmaringae, so we, y y y y E y y y y a r e Terefore,, y C fncion b:, T, sc a: C, y:, T: y b a r Z Tis proves a remaining a We ave e Y o be a sbmaringae for r a inf s: Ys, so a poins in region a r r a z, : z beong o e coninaion region a r r a Terefore, b Te monooniciy of b foows a r from monooniciy of fncion, y Te rig coniniy of b foows from e con- C is an open se inaion region Teorem Assme a b is e fncion described above wose exisence is prove d in Lemma Define sopping ime: inf : e b T (8) Ten, aains e spremm in () Proof Dedce direcy from Teorem and Lemma by repacing Y by Teorem 3 Te opima sopping bondary b() saisfies e inegra eqaion (see Eqaion (9) beow): x y were x e d y π Proof Fix, T and Y y, Ten, were G y B: Consider: We ave: e e a rt a T e r y a rt E E E G Y G Y T T, Y T a rt a rt E e E e Y T a rt a rt e y e a r a r T a rt e E T, YT e y a r E Z I Y b a re IY b e a r b b T a r T ar T a r b e be a re n b b a re n d (9) Copyrig SciRes

5 P VAN KHANH 53 E, y ZI Y b E, y exp W b n W y I d z z e e e dz e π z d z d e e dz e e π π d b n y b n y en E Z IY b e Nd In a simiar way, we ave, y E I Y b N d, y were d d P y b we aain (9) n b 3 Te Nmerica Soion of e Inegra y were d Eqaion and Simaion Ress Beow we foow [5], devided,t by e poins z Le x and k k, k,,, n wic T n, en Eqaion (8) can be discreized as: ni ar ni a r k k e a r n bik bi i bie a re n k k k bi ni b i k For i n, we ave eqaion De o bt b e a r r a a r b, conine o i n, we obain e foowing eqaion for deermining a r k b a r e n k bi i k k k ar ar bt e e n b b ar n n n b a r n a r bt n e n b, from e above eqaion we deermine b : n a r ar ar n n n bt b T n b a r bn e n b b e b e are n n a r b b a r e n n a r n bn a r bn a r e n b n Copyrig SciRes

6 53 P VAN KHANH Js do so ni b Ts, we obain a seqence of vaes b, b,,b n of b and approximae e opima bondaries for e asse iqidaion process We ave e approximae soion of Eqaion (9) by a comper program wrien in Maab sofware, en se e bondaries for e process is B e b Ten e opima ime o se is e firs ime e ine describes e process B ies beow e ine describes e process Tese figres (Figres 4-9) and abes beow ( Tabes -3) israes e sopping ime in (8) and e soions of Eqaion (9) in some cases i, i as been deermined Figre 7 A simaion for e sopping ime in (8) wi parameers = 3; a = ; a = ; r = 5; σ = ; π = 4 Figre 4 Te ine describes b () wi a = ; a = ; r = 5; σ = Figre 8 Te ine describes b () wi a = 9; a = 5; r = ; σ = Figre 5 A simaion for e sopping ime in (8) wi parameers = 3; a = ; a = ; r = 5; σ = ; π = 5 Figre 9 A simaion for e sopping ime in (8) wi parameers = 3; a = 9; a = 5; r = ; σ = ; π = 5 Figre 6 A simaion for e sopping ime in (8) wi parameers = 3; a = ; a = ; r = 5; σ = ; π = In is case π = is sma so is τ Figre A si maion for e sopping ime in (8) wi parameers = 3; a = 9; a = 5; r = ; σ = ; π = 4 Copyrig SciRes

7 P VAN KHANH 533 Figre A simaion for e sopping ime in (8) wi parameers = 3; a = 9; a = 5; r = ; σ = ; π = 3 Figre 5 A simaion for e sopping ime in (8) wi parameers = 3; a = 9; a = 5; r = 3; σ = ; π = 4 Figre A simaion for e sopping ime in (8) wi parameers = 3; a = 9; a = 5; r = ; σ = ; π = Figre 6 A sim aion for e sopping ime in (8) wi parameers = 3; a = 9; a = 5; r = 3; σ = ; π = 5 Figre 3 Te ine describes b () wi a = 9; a = 5; r = 3; σ = Figre 7 A simaion for e sopping ime in (8) wi parameers = 3; a = 9; a = 5; r = 3; σ = ; π = 5 Figre 4 A simaion for e sopping ime in (8) wi parameers = 3; a = 9; a = 5; r = 3; σ = ; π = Figre 8 A simaion for e sopping ime in (8) wi parameers = 3; a = 3; a = 5; r = ; σ = ; π = 4 Copyrig SciRes

8 534 P VAN KHANH Conined T Figre 9 A simaion for e sopping ime in (9) wi parameers = 3; a = 3; a = 5; r = ; σ = ; π = 6 Tabe Te nmerica soions of (9) wi a = ; a = ; r = 5; σ = i b ( i ) i b ( i ) i b ( i ) i b ( i ) i b ( i ) Tabe Te nmerica soions of (8) wi a = 9; a = 5; r = ; σ = i b ( i ) i b ( i ) i b ( i ) i b ( i ) i b ( i ) Tabe 3 T e nm erica so ions of (9) wi a = 9; a = 5; r = 3; σ = i b ( i ) i b ( i ) i b ( i ) i b ( i ) i b ( i ) Concsion T is paper sov es e probem o fin d e opima sopping ime for e oding asse and make a dec ision wen o se asses wi disconed price reacing e greaes expeced vae Te opim a sopping ime is e firs im e e price of e asse i e bondary or b e a e im e T In nex sdy, we wi sdy e disribio ns and caracerisics of e opima sopping ime REFERENCES [] G Peskir and A N S iryaev, Opima Sopping and Free-Bondary Probems (Lecres in Maemai cs ETH Lecres i n Maemaics ETH Züric (Cos ed) ), Birkäse r, Base, 6 [] A N Siry aev, Z and Y Zo, To Sa By and Hod, Qaniaiv e Financ e, Vo 8, No 8, 8, pp [3] R S Lipser and A N Siryaev, Saisics of Random Process: I Genera Teory, Springer-Verag, Berin, Heideberg, Copyrig SciRes

9 P VAN KHANH 535 [4] A N Siryaev, Opima Sopp ing Res, Springer- Verag, Beri n, Heideberg, 978, 8 [5] L Z ang, Nmer ica Anaysis of American Opion Pricing in a Jmp-Diffsion Mode, Maemaics of Operaions Researc, Vo, No 3, 997, pp doi:87/moor3668 Copyrig SciRes