Selling at the ultimate maximum in a regime-switching model

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1 Selling a he ulimae maximum in a egime-wiching model Yue Liu School of Finance and Economic Jiangu Univeiy Zhenjiang P.R. China Nicola Pivaul School of Phyical and Mahemaical Science Nanyang Technological Univeiy 21 Nanyang Link Singapoe Febuay 9, 217 Abac Thi pape deal wih opimal pedicion in a egime-wiching model diven by a coninuou-ime Makov chain. We exend exiing eul fo geomeic Bownian moion by deiving opimal opping aegie ha depend on he cuen egime ae, and pove a numbe of coninuiy popeie elaing o opimal value and bounday funcion. Ou appoach eplace he ue of cloed fom expeion, which ae no available in ou eing, wih PDE agumen ha alo implify he appoach of du Toi & Peki (29 in he claical Bownian cae. Key wod: Opimal opping; ulimae maximum; egime-wiching model; fee bounday poblem; diffuion pocee. Mahemaic Subjec Claificaion (21: 6G4; 35R35; 93E2; 6J28; 91G8. 1 Inoducion Regime-wiching model have been inoduced by Hamilon (1989 in dicee ime and ae among he mo popula and effecive iky ae model. The egimewiching popey i efleced in he change of ae of a Makov chain β, which and fo he influence of exenal make faco. 1

2 Euopean opion have been piced in coninuou-ime egime-wiching model by Yao, Zhang & Zhou (26 via a ecuive algoihm, and in Liu, Zhang Yin (26 uing he fa Fouie anfom. Opimal opping fo opion picing in egimewiching model ha been conideed in Guo (21, Guo & Zhang (25, Le & Wang (21, and in Ly Vah & H. Pham (27 wih opimal wiching. Opimal elling unde hehold ule ha been deal wih in Eloe, Liu, Yauki, Yin & Zhang (28 in an exponenial Gauian diffuion model wih egime wiching. We efe o Shiyaev (1978 and Peki & Shiyaev (26 fo elaed backgound on he chaaceizaion of opimal opping ime and ewad. The poblem of elling a ock a he ulimae maximum ha been conideed by du Toi & Peki (29 a an exenion of he eul of Shiyaev, Xu & Zhou (28. In hi pape we exend he eul of du Toi & Peki (29 o he famewok of Makovian egime wiching. Some of ou eul ae naual exenion of hoe of du Toi & Peki (29 by aveaging ove he egime-wiching componen, howeve he egime-wiching cae peen noable diffeence and addiional difficulie compaed wih he claical Bownian cae. Fo example, he opimal bounday funcion depend on he egime ae of he yem, and hey may no be monoone if he dif coefficien have wiching ign. In addiion we can no longe ely on cloed fom expeion a in du Toi & Peki (29 and inead we ue PDE agumen, cf. e.g. Lemma 4.3, ha alo implify he oiginal appoach. In Lemma 2.1 we wie he opimal value of he poblem a a funcion of ime, he egime ae, and he elaive maximum of he undelying ae. In he geneal cae of eal-valued dif µ(i IR, i M, we idenify he opimal opping ime τ D in Popoiion 3.1, and in Popoiion 3.2 we deemine he ucue of he opimal opping e via i bounday funcion b(, j fo i in he ae pace M of he egimewiching chain. In Popoiion 5.1 we how ha immediae execie i opimal when all dif paamee µ(i ae negaive, i M, while execie a mauiy become opimal when 2

3 µ(i σ 2 (i fo all i M, whee σ(i ae he volailiy paamee. When he dif paamee (µ(i i M of he egime-wiching chain ae nonnegaive we pove he coninuiy and monooniciy of bounday funcion b(, j in Popoiion 5.2, by exending agumen of du Toi & Peki (29 o he egime-wiching eing. Thoe eul ae illuaed in Figue 1 and 2 by he ploing of value funcion ha yield he opimal opping boundaie. In Popoiion 5.3 we deive a Volea ype inegal equaion (5.7 which i aified by he bounday funcion b(, j of he opping e. Such an equaion i difficul o olve becaue, unlike in he claical eing du Toi & Peki (29, i alo elie on he knowledge of he opimal value funcion, cf. Remak 5.4. In addiion he aociaed fee bounday poblem (5.12a-(5.12b coni in a yem of ineacing PDE ha canno be olved wihou addiional aumpion, cf. e.g. Buffingon & Ellio (22 fo a oluion unde an odeing condiion on bounday funcion in he cae of Ameican opion. A eamen of dif coefficien (µ(i i M wih wiching ign ha been popoed in of Liu & Pivaul (217 via a ecuive algoihm ha doe no ely on a Volea equaion. In hi cae i un ou ha he bounday funcion b(, i may no be deceaing in, T. We poceed a follow. In Secion 2 we fomulae he opimal pedicion poblem uing opimal value funcion. In Secion 3 we deive he opimal opping aegie in em of he hiing ime of he bounday funcion of a opping e. Secion 4 i devoed o coninuiy lemma, which ae ued o pove he coninuiy of bounday funcion. In Secion 5 we alo deive he Volea inegal equaion which i aified by he bounday funcion when he dif coefficien ae nonnegaive. Finally we udy paicula execie aegie and we peen a numeical imulaion of bounday funcion. 3

4 2 Poblem fomulaion Given a andad Bownian moion (B IR+ independen of he Makov chain (β IR+, we conide an ae pice (Y IR+ modeled by a geomeic Bownian moion dy = µ(β Y d + σ(β Y db, T, (2.1 wih egime wiching diven by a ime-homogeneou coninuou-ime Makov chain (β IR+ wih ae pace M := {1, 2,..., m} and infinieimal geneao Q = (q ij 1 i,j m, whee µ : M IR, and σ : M (, ae deeminiic funcion. In he equel we le he filaion (F,T be defined fo all, T by F := σ(b B, β :,, T. (2.2 In paicula, (F IR+ i he filaion geneaed by (B IR+ and (β IR+. In hi pape we deal wih he opimal pedicion poblem V = inf E τ T up T Y Y τ F, (2.3 inoduced in du Toi & Peki (29 fo geomeic Bownian moion, in which he infimum of expeced value ove all (F,T -opping ime τ minimize he ege of he opping deciion. The nex Lemma 2.1 how ha he opimal value funcion V in (2.3 can be wien a a funcion of (, β, Ŷ,/Y, whee Ŷ, i defined by Ŷ, := max Y, T. (2.4 Lemma 2.1 The opimal value funcion V in (2.3 ake he fom V = V (, Ŷ,/Y, β, (2.5 whee he funcion V :, T 1, + M IR + i given by 1 ( V (, x, j = inf E max xy, τ T Y Ŷ,T β = j, (2.6 τ T, x 1, j M. 4

5 Poof. Given, T, uing he difed Bownian moion +u ( ˆB u µ(β := B u+ B + σ(β σ(β d, 2 u, T, (2.7 we ewie he oluion of (2.1 a and define Y = Y exp ( σ(β u+ d ˆB u,, T, (2.8 Ŝ := up σ(β u+ d ˆB u,, T. (2.9 By he definiion of Ŷ, in (2.4 and expeion (2.8, and fom he condiional independence of ˆB, ( ( ( Ŝ,T wih F,T given β we have, fo any (F,T - opping ime τ wih value in, T, leing a b = max(a, b, Y F Ŷ, E up T Y Ŷ,T F τ Y τ Y τ (Ŷ, e τ σ(β u+ d ˆB u eŝ T τ σ(β u+ d ˆB u F Y (Ŷ, Y Ŷ, Ŷ,T Y τ e τ σ(β u+ d ˆB u β, Ŷ, Y 1 ( max xy, Y Ŷ,T β τ eŝ T τ σ(β u+ d ˆB u x=ŷ,/y β, Ŷ, Y whee he la line follow fom he condiional independence beween Ŷ,/Y and ( ( ( Y τ ( τ τ, Ŷ,T = exp σ(β u+ d Y Y ˆB u, exp ŜT σ(β u+ d ˆB u (2.1 τ given β. Theefoe by definiion (2.3 and expeion (2.6, we obain V = inf E Y F up τ T T Y τ 1 ( = inf E max xy, τ T Y Ŷ,T β τ x=ŷ,/y ( = V Ŷ, Y, β,. 5

6 In he nex lemma we ewie he opimal opping poblem (2.3 in he andad fom (2.12 below, uing he funcion G(, x, i : wih G(T, x, i = x, x 1. ( max x, Ŷ,T /Y β = i,, T, i M, x 1, (2.11 Lemma 2.2 The funcion V :, T 1, + M IR + defined by (2.5 admi he expeion V (, x, j = fo, T, j M, x 1, whee Poof. X,x inf E G ( τ, Xτ,x τ T, β τ β = j, (2.12 := 1 ( max xy, Y Ŷ,,, T, x 1. (2.13 By a condiional independence agumen a in he poof of Lemma 2.1, fo any, T we have E Ŷ,T Y F Ŷ, Ŷ,T Y Ŷ, Ŷ,T F Ŷ,, β Y Y y Ŷ,T β = G ( Y, Ŷ, Y, β y=ŷ,/y. (2.14 Nex, we exend he above elaion (2.14 o (F,T -opping ime τ wien a he limi of a deceaing equence of dicee opping ime by checking ha fo any dicee (F,T -opping ime τ = n i=1 i1 {τ=i }, 1,..., n, T, n 1, by (2.14 we have Ŷ,T n E F Ŷ,T τ 1 {τ=i } Fτ Y τ Y τ = i=1 n E i=1 6 Ŷ,T Y i 1 {τ=i } F i

7 = = n E i=1 Ŷ,T Y i F i ( n G i, Ŷ, i, β i i=1 = G ( Y i τ, Ŷ,τ Y τ, β τ. 1 {τ=i } 1 {τ=i } Taking he condiional expecaion E β = j, Ŷ,/Y = x on boh ide of he above equaliy, we obain E Ŷ,T Y τ β = j, Ŷ, Y = x G ( τ, Ŷ,τ Y τ, β τ β = j, Ŷ, By (2.13 and he condiional independence beween Ŷ,/Y Y and = x given β = j we find 1 ( Ŷ,T E max xy, Y Ŷ,T β = j β = j, Ŷ, = x τ Y τ Y ( G τ, Ŷ,τ, β τ β = j, Ŷ, = x Y τ Y ( G τ, (xy Ŷ,τ, β τ β = j, Ŷ, = x Y τ Y G ( τ, X,x τ, β τ β = j. (2.15 (Y /Y τ, Ŷ,τ/Y τ, (2.16 which complee he poof by ( Sopping e and bounday funcion In hi ecion we apply Coollay 2.9 in Peki & Shiyaev (26 in he famewok of he egime-wiching model (2.1 wih µ(i IR, i M, in ode o pecify he opping e and opimal opping ime aociaed o he opimal opping poblem (2.3, cf. Popoiion 3.1 below. In ode o deal wih he exience of an opimal opping ime fo (2.3 ewien a (2.12, we define he e D := {(, x, j, T 1, M : V (, x, j = G(, x, j}. (3.1 7

8 Fom he elaion V (T, x, j = G(T, x, j = x, j M, x 1, we check ha {T } 1, M D, which i conien wih he fac ha he infimum in (2.3 i ove (F,T -opping ime τ, T. Popoiion 3.1 Le, T. Given β = j M and Ŷ,/Y = x 1,, he (F,T -opping ime τ D (, x, j := inf {, T : (, Ŷ, Y, β D } (3.2 i an opimal opping ime fo (2.3, o equivalenly fo (2.12, povided ha i i a.. finie. Poof. By Coollay 2.9 in Peki & Shiyaev (26 he opimal opping ime fo poblem (2.12 exi and i equal o τ D (, x, j in (3.2 povided ha we check ha fo all, T we have: a G(, x, j i lowe emiconinuou wih epec o x, a follow diecly fom he definiion (2.11 of G(, x, j. b V (, x, j i uppe emiconinuou wih epec o x, a follow fom he coninuiy Lemma 4.5 below. c We have E up T G(, X,x, β <. Indeed, fom (2.13 and (2.8 we have X,x = 1 Y max ( xy, Ŷ, = Y 1 e σ(β u+ d ˆB u max (xy, Y eŝ = e max(log x,ŝ σ(β u+ d ˆB u,, T, x 1, (3.3 whee Ŝ i defined in (2.9. Hence by (2.11 and he condiional independence beween X,x = max (xy /Y, Ŷ,/Y and Ŷ,T /Y given β, we find ha G(, X,x, β X,x Ŷ,T Y y Ŷ,T Y β, X,x β y=x,x 8

9 e ( σ(β u+ d ˆB u e max(log x, Ŝ T Leing Š T : = inf T = inf T e max(log x,up σ(β u+ d ˆB u β, X,x we conclude ha E up G(, X,x, β T xe up E T xe up E eŝ T inf T T = xe eŝ T inf T = xe eŝ T Š T x E e 2Ŝ T E e 2Š T σ(β u+ d ˆB u e up T σ(β u+ d u ˆB β, X,x. (3.4 σ(β u+ d ˆB u (3.5 σ(β u+ db +u + (µ(β u+ σ 2 (β u+ /2du, up E T eŝ T σ(β u+ d ˆB u x e max i M σ 2 (i 2µ(i (T xe e 2Ŝ T σ(β u+ d ˆB u e max(log x, Ŝ T σ(β u+ d ˆB u β, X,x σ(β u+ d ˆB u E e max i M σ 2 (i 2µ(i (T β, X,x e 2Ŝ T E e 2Ŝ T β, X,x <. (3.6 Define F (, x, j := V (, x, j G(, x, j, (3.7 which i nonpoiive by (2.12,, T, j M, x 1, o ha we have D = {(, x, j, T 1, M : F (, x, j = }, (3.8 hence D i cloed fom he coninuiy of (, x V (, x, j and (, x G(, x, j on, T 1,, cf. Lemma 4.5 and Lemma 4.6 below, epecively. The coninuaion 9

10 e C = D c i an open e ha can be wien a C = {(, x, j, T 1, M : F (, x, j < }. (3.9 In he nex Popoiion 3.2 we chaaceize he hape of he opping e D defined in (3.1 in em of he bounday funcion b(, j defined by b(, j := inf{x 1, : (, x, j D}, (3.1 whee we e b(, j := + if {x 1, : (, x, j D} =. Fom he elaion {T } 1, M D we deduce he eminal condiion b(t, j = 1, j M, cf. alo Popoiion 5.2 fo ufficien condiion fo he finiene of b(, j. Popoiion 3.2 Fo any (, x, j, T 1, M uch ha (, x, j D we have {} x, {j} D. (3.11 and D = {(, y, j, T 1, M : y b(, j}. (3.12 Poof. Le y := up{z x, : {} x, z {j} D}. If y < hen we have (, y, j D by he cloedne of D, and fom he monooniciy popey of F (, x, j aed in Lemma 3.3, (, y, j D admi a igh neighbohood of he fom {} x, x + η {j} D (3.13 fo ome η >, which lead o a conadicion. Hence y = + and (3.11 hold. Relaion (3.12 follow fom he equivalence (, x, j D {} x, {j} D x b(, j (3.14 ha follow fom (3.1. The following lemma ha been ued in he poof of Popoiion 3.2. Lemma 3.3 Fo any (, x, j D, we have lim inf ε F (, x + ε, j F (, x, j ε 1. (3.15

11 Poof. We pli he poof ino wo pa. (i Fom (3.4 we have G(, X,x, β X,x Ŷ,T β, X,x Y e max(log x, Ŝ T σ(β u+ d ˆB u which exend o any (F,T -opping ime τ, T a X,x G(τ, Xτ,x, β τ e max(log x, Ŝ T τ σ(β u+ d ˆB u Ŷ,T F Y F,, T, Fτ, (3.16 a in (2.14-(2.15 above. Fo all x 1 and ε >, conide he (F,T -opping ime τ + ε := τ D (, x + ε, j, T (3.17 defined by (3.2, which olve he opimal opping poblem V (, x+ε, j = inf E G ( τ, Xτ,x+ε, β τ β = j τ T ( G τ + ε, X,x+ε τ + ε, β τ + ε β = j (3.18 cf. (2.12. The following agumen elie on he fac ha fo any (, x, j D we have lim τ D(, x + ε, j =, (3.19 ε a will be hown in pa (ii below. Relaion (2.11, (2.12, (3.16 and (3.19 imply V (, x + ε, j V (, x, j lim inf ε ε 1 lim inf ε ε E G(τ ε +, X,x+ε, β τ ε + τ + ε G(τ ε +, X,x, β τ ε + τ + ε β = j 1 = lim inf ε ε E E e log(x+ε Ŝ T τ + ε σ(β u+ d ˆB u e log x Ŝ T τ ε + 1 = lim inf ε ε E e log(x+ε Ŝ T τ + ε σ(β u+ d ˆB u e log x Ŝ T τ ε + 1 = lim inf ε ε E e log(x+ε Ŝ T e log x Ŝ T β = j = x E e max(log x,ŝ T β = j = G (, x, j, x 11 σ(β u+ d ˆB u σ(β u+ d ˆB u, β F = j τ ε + β = j (3.2

12 hence we conclude o (3.15. Hee we ued he dominaed convegence heoem wih he bound 1 e log(x+ε Ŝ T τ + ε σ(β u+ d ˆB u e log x Ŝ T τ ε + σ(β u+ d ˆB u ε elog(x+ε e log x e inf T σ(β +ud ˆB u = e Š T, ε whee Š T of (3.6. i defined in (3.5 and he ighhand ide i inegable a in he deivaion (ii We un o he poof of (3.19. Fom he expeion (2.6 in Lemma 2.1, we have V (, x, j = inf E xy Ŷ,T β = j, F τ T Y τ = inf E e τ σ(β u+ d ˆB ( u x e ŜT β = j, F. (3.21 τ T Fom (3.21 and X,x+ε = e σ(β u+ d ˆB u (x + ε e Ŝ (3.22 cf. (3.3, we obain V (, X,x+ε, β = inf E e τ σ(β u+ d ˆB ( u y e ŜT F τ T = inf E e τ σ(β u+ d ˆB u τ T ( X,x+ε eŝ T y=x,x+ε F. (3.23 Nex, fom he definiion (3.2 of τ D (, x + ε, j and (3.23 we have, on he even {β = j}, τ D (, x + ε, j = inf{, T : (, X,x+ε, β D} (3.24 { = inf, T : inf E e τ σ(β u+ d ˆB ( u X,x+ε τ T eŝ T F X,x+ε eŝ T { inf, T : inf E e τ σ(β u+ d ˆB ( u X,x τ T eŝ T F E X,x+ε eŝ T inf {, T : inf E e τ σ(β u+ d ˆB u τ T whee we applied he inequaliy X,x+ε ( X,x eŝ T F e ε E X,x eŝ T = e σ(β u+ d ˆB ( u e log(x+ε eŝ e σ(β u+ d ˆB ( u+ε e log(x eŝ = e ε X,x, (3.25 } F } F } F, 12

13 x 1, ε,, T. Thi implie lim τ D(, x + ε, j (3.26 ε { lim inf, T : inf E e τ σ(β u+ d ˆB ( u X,x ε τ T eŝ T F e ε E { = inf, T : inf E e τ σ(β u+ d ˆB ( u X,x τ T eŝ T F E X,x eŝ T { = inf, T : inf E e τ σ(β u+ d ˆB ( u X,x τ T eŝ T F X,x eŝ T = inf {, T : (, X,x, β D } =, X,x eŝ T (3.27 ince (, x, j D, β = j and X,x = x. Since τ D (, x+ε, j we conclude o (3.19. } F } F } F 4 Coninuiy lemma The following popey of mooh fi, namely he coninuiy of he funcion y V (, y, j ove he opimal opping bounday C, will be needed in he poof of Popoiion 5.3 below. Lemma 4.1 Fo any (, y, j C, y > 1, we have V V (, y+, j = (, y, j. (4.1 Poof. Fo any ε (, y 1, le τε = τ D (, y ε, j, T, cf. (3.2. Since (, y, j C and D i cloed we have (, y, j D. Similaly o (3.24 o (3.26, τ ε convege o a.. when ε end o. By he ame appoach a in (3.2, eplacing y + ε wih y ε how ha G (, y, j lim inf ε On he ohe hand, ince (, y, j C D, we have V (, y ε, j V (, y, j. (4.2 ε lim up ε V (, y ε, j V (, y, j ε lim ε G(, y ε, j G(, y, j ε = G (, y, j, (4.3 13

14 hence V Finally he fac ha V = G on he cloed e D implie V V (, y, j = G (, y, j = (, y, j. (4.4 G (, y+, j = (, y, j. (4.5 In he nex popoiion, which will be ued in he poof of Popoiion 5.3, we how he nomal eflecion of he fee bounday poblem by poving ha he igh deivaive of he value funcion V (, y, j vanihe a y = 1, cf. alo page 264 of Peki & Shiyaev (26 wihou egime wiching. Lemma 4.2 Fo any, T and j M we have Poof. V (, 1+, j =. (4.6 Fo convenience of noaion we e τ = τ D (, 1, j, and noe ha lim up ε lim up ε = lim up ε = lim up E =, ε V (, 1 + ε, j V (, 1, j ε 1 ε EG(τ, Xτ,1+ε, β τ G(τ, Xτ,1, β τ β = j 1 ε E e log(1+ε Ŝ T τ σ(β + d ˆB e ŜT τ σ(β + d ˆB lim up ε β = j 1 (e log(1+ε Ŝ T τ σ(β + d ˆB e ŜT τ σ(β + d ˆB 1 ε { ŜT <log(1+ε} β = j e log(1+ε Ŝ T τ σ(β + d ˆB e Ŝ T τ σ(β + d ˆB ε 1 { Ŝ T <log(1+ε} β = j ince lim 1 { Ŝ ε T <log(1+ε} =, whee we applied he dominaed convegence heoem a in he poof of Lemma 3.3 wih he ame dominaing funcion a in (3.21. Since V (, y, j i nondeceaing in y 1,, we have lim inf ε V (, 1 + ε, j V (, 1, j ε, (4.7 14

15 which how ha V V (, 1 + ε, j V (, 1, j (, 1+, j = lim ε ε =. (4.8 Nex, we noe ha (, X,x, β,t i a Makov poce, cf. Lemma 1 of Yao, Zhang & Zhou (26, and we conide i infinieimal geneao Lf(, x, j = whee Q = (q ij 1 i,j m ( + x(σ2 (j µ(j x σ2 (jx 2 2 f(, x, j + x 2 m q j,i f(, x, i, i=1 (4.9 i he infinieimal maix geneao of he Makov chain (β,t, fo f a ufficienly diffeeniable funcion of (, y, j, T 1, M, cf. Lemma 4.7 below. The following lemma will be ued in he poof of Popoiion 5.1 below. In Lemma 4.3 we eplace he ue of cloed fom expeion fo LG(, x, j, which ae no longe available in ou eing, wih he diffeenial expeion (4.11. Lemma 4.3 We have and G (, 1+, j =,, T, (4.1 x LG(, x, j = xσ 2 (j G (, x, j µ(jg(, x, j,, T, (4.11 x wih LG(T, x, j = µ(jx, j M, x 1,. In paicula, fo any (, x, j, T 1, M we have LG(, x, j >, when µ(j, LG(, x, j <, when µ(j σ 2 (j. (4.12 In addiion, LG(, x, j i nondeceaing and coninuou in fo all x 1 when µ(j. 15

16 Poof. Fo all j M we le f(, y, z, j := yg (, zy (, j max z, y Ŷ,T β = j,, T, y, z >. Y (4.13 By (2.1 and he Iô fomula we have df (, Y, Ŷ,, f ( β =, Y, Ŷ,, f ( β d + µ(β Y, Y, x Ŷ,, β d f ( + σ(β Y, Y, x Ŷ,, β db σ2 (β Y 2 2 f (, Y, x Ŷ,, β 2 d + f (, Y, Ŷ,, β d Ŷ, + f (, Y, Ŷ,, ( β f, Y, Ŷ,, β, and given ha f (, Y, Ŷ,, β Ŷ,T β, Y, Ŷ, Ŷ,T F,, T, (4.14 i a maingale and ( Ŷ,,T ha finie vaiaion, we find f f (, y, z, j + µ(jy x (, y, z, j σ2 (jy 2 2 f x (, y, z, j + m q 2 j,i f(, y, z, i =, i=1 (4.15 and f (, x, y, j x=y =. Subiuing (4.13 ino (4.15 how ha y G (, zy (, j + µ(jy G (, zy, j + y G (, zy ( x, j zy ( G 2 σ2 (jy (, 2 zy ( x, j zy + z G (, zy 2 y 2 x, j + z2 2 G (, zy y 3 x, j 2 m + q j,i yg (, zy, i =, i=1 which how ha he funcion G(, x, j aifie he PDE µ(jg(, x, j+ G (, x, j µ(jx G x (, x, j+1 2 σ2 (jx 2 2 G x (, x, j+ m q 2 j,i G(, x, i =, i=1 (4.16 and we conclude o (4.11 by (4.9. Nex we noe ha (4.1 follow fom G (Ŷ,T (, x, j = P < x β = j 1, (, x, j, T 1, M, (4.17 x Y 16

17 cf. he definiion (2.11 of G. Nex, by (2.11 and Lemma 4.3, fo any (, x, j, T 1, M, we find LG(, x, j = xσ 2 (jp (Ŷ,T Y < x xσ 2 (j1{ŷ,t /Y<x} µ(j ( ( β = j µ(je max x, Ŷ,T /Y β = j β = j x Ŷ,T (4.18 Y ( x(σ 2 (j µ(j1 { Ŷ,T /Y <x} β = j E µ(j x Ŷ,T 1 Y { Ŷ,T /Y x} β = j, which how (4.12, and implie by (4.18 ha LG(, x, j i nondeceaing and coninuou in, T when µ(j. The poof of he nex lemma, which will be ued in Popoiion 5.1 below, exend he agumen of du Toi & Peki (29 page 993 o he egime-wiching eing. Lemma 4.4 We have {(, x, j, T 1, M : LG(, x, j < } C, (4.19 whee C = D c i he coninuaion e. Poof. By Lemma 4.7 below and Lemma 1 in Yao, Zhang & Zhou (26 we have EG(, X,x, β β = j = G(, x, j + E LG(, X,x, β d β = j, (4.2, T. Aume now ha (, x, j, T 1, M i uch ha LG(, x, j <. By he coninuiy of LG(, x, j wih epec o, which follow fom (3.4, he ime homogeneiy of (β IR+ and he pah coninuiy of (Y,T, hee exi an open neighbouhood U, T 1, of (, x, depending on U and uch ha LG(, y, j < fo all (, y U. Subiuing he vaiable in (4.2 wih he fi exi ime τ U of U when (X,x, β,t i aed a (x, j a ime, Relaion (4.2 above how by opional ampling ha EG(τ U, X,x τ U, β τu β = j = G(, x, j + E τu LG(, X,x, β d β = j. 17

18 Since τ U > a.. and LG(, X,x, β < when (, τ U, he igh hand ide i icly malle han G(, x, j, while we have EG(τ U, X,x τ U, β τu β = j V (, x, j, (4.21 howing ha V (, x, j < G(, x, j, which implie ha (, x, j C. Nex we deive he following coninuiy eul wich ha been ued in he poof of Popoiion 3.1. Lemma 4.5 Fo any j M, he mapping (, x V (, x, j i joinly coninuou on, T 1,. Poof. We poceed in wo ep. (i We how ha he mapping V (, x, j i coninuou on, T fo evey fixed x 1 and any j M. By (2.6 we have V (, x, j = inf E (xy Ŷ,T τ T Y τ β = j = inf E x e max T ((µ(β σ2 (β /2+σ(β B τ T β = j whee = inf τ T E e (µ(βτ σ2 (β τ /2τ+σ(β τ B τ U(, τ β = j,, T, j M, x 1,, U(, := x e max T ((µ(β σ2 (β /2+σ(β B e (µ(β σ2 (β /2τ+σ(β B,,, T. (4.22 Fo any (F,T -opping ime τ, T we have E U(, τ U( +, τ β = j ( E e max 2 T ((µ(β σ2 (β /2+σ(β B max e T ((µ(β σ2 (β /2+σ(β B β = j E e 2(µ(βτ σ2 (β τ /2τ 2σ(β τ B τ β = j ( E e max 2 T ((µ(β σ2 (β /2+σ(β B max e T ((µ(β σ2 (β /2+σ(β B β = j 18

19 (T max e 3σ(i 2µ(i i M E e 2σ2 (β τ τ 2σ(β τ B τ β = j ( E e max 2 T ((µ(β σ2 (β /2+σ(β B max e T ((µ(β σ2 (β /2+σ(β B β = j (T max e 3σ(i 2µ(i /2 i M, (4.23 whee we applied he opional ampling heoem. Leing end o on boh ide of (4.23, we ge lim E U( +, τ β = j U(, τ β = j, (4.24 and ince he convegence i unifom on all (F,T -opping ime τ, T, we obain lim inf inf E U( +, τ β = j lim inf τ T inf E U( +, τ β = j τ T = inf lim E U( +, τ β = j = inf E U(, τ β = j. (4.25 τ T τ T Nex, accoding o Popoiion 3.1 hee exi an opimal (F,T -opping ime τ, T uch ha hence we have inf τ T E U(, τ β = j U(, τ β = j, (4.26 inf E U( +, τ β = j inf E U(, τ β = j (4.27 τ T τ T Since U(, i nonnegaive fo any,, T, we have U(, τ (T U(, τ + U(, T = U(, τ + U(, τ + e E U(, τ (T β = j. x e max T ((µ(β σ2 (β /2+σ(β B e (µ(β T σ 2 (β T /2(T +σ(β T B T x e max T ((µ(β σ2 (β /2+σ(β B, inf i M,,T (µ(i σ2 (i/2+ inf (σ(ib i M,,T 19

20 which i inegable by (4.26. By he evee Faou Lemma we have lim up E U(, τ (T β = j E lim up U(, τ (T β = j U(, τ β = j. (4.28 Combining (4.26, (4.27, (4.28 and (4.25 we find lim inf E U( +, τ τ T β = j = inf E U(, τ τ T β = j. (4.29 Similaly we have lim inf E τ T + U(, τ β = j hence V (, x, j i coninuou on, T. = inf E τ T U(, τ β = j, (4.3 (ii We how ha x V (, x, j i coninuou on 1,, unifomly in, T, exending he agumen of du Toi & Peki (29 page 995 o he egime-wiching eing. By Relaion (4.17 and he mean value heoem, fo all y x, hee exi a (andom η X,x +τ, X,y +τ uch ha fo any (F,T -opping ime τ, T we have G( + τ, X,y +τ, β +τ G( + τ, X,x +τ, β +τ = G x ( + τ, η(x,y +τ X+τ,,x β +τ (y x Y Y +τ, (4.31 ince X,y +τ X,x +τ (y xy /Y +τ by (2.13. Le now (, x, j, T 1, M and conide τ x := τ(, x, j given by (3.2. By Lemma 2.2 we have V (, y, j V (, x, j E G( + τ x, X,y +τ x, β +τx G( + τ x, X,x +τ x, β +τx β = j. Since E Y /Y +τ boh ide of (4.31 yield (4.32 β = j i unifomly bounded a in (3.6, aking expecaion on lim E G( + τ, X+τ,,y β +τ G( + τ, X+τ,,x β +τ β = j =, (4.33 y x unifomly in, T and in he (F,T -opping ime τ, T. V (, x, j i inceaing in x 1,, (4.32 and (4.33 yield Since lim y x (V (, y, j V (, x, j, (4.34 2

21 which how he coninuiy of x V (, x, j, unifomly in, T, fo all j M. Fom (i and (ii we conclude o he join coninuiy of (, x V (, x, j on, T 1, by claical agumen. Lemma 4.6 The mapping (, x G(, x, j i joinly coninuou on, T 1,. Poof. By Relaion (4.17 and he mean value heoem, fo all y x, hee exi an η x, y uch ha fo any, T we have G(, y, j G(, x, j = (y x G (, η, j y x, (4.35 x which how he coninuiy of x G(, x, j, unifomly in, T. On he ohe hand, we have by (2.11 ha G(, x, j i coninuou on, T fo evey x 1. We conclude o he join coninuiy of (, x G(, x, j on, T 1, by a claical agumen. We cloe hi ecion wih he following hee lemma. Lemma 4.7 The Makov poce (, X,x, β,t ha he infinieimal geneao Lf(, y, j = ( + y(σ2 (j µ(j σ2 (jy 2 2 f(, y, j + 2, T, j M, y 1,, fo f Dom (L aifying f (, 1+, j =. Poof. Leing Z,x := log x Ŝ, T, x 1, fom (3.3 we have X,x (Ŝ,T i nondeceaing i ha finie vaiaion, hence m q j,i f(, y, i, i=1 (4.36 σ(β u+ d ˆB u, (4.37 = exp (Z,x,, T, x 1. Since d Z,x, Z,x = σ 2 (β d ˆB, d ˆB = σ 2 (β d B, B = σ 2 (β d, (4.38 which how ha dx,x = X,x dz,x X,x d Z,x, Z,x 21

22 = X,x dz,x = X,x σ2 (β X,x d d(log x Ŝ σ(β X,x d ˆB σ2 (β X,x d. (4.39 Given ha f (, 1+, j = fo (, y, j, T 1, M, we have f (, X,x, β d(log x Ŝ = f = f =, ince d(log x Ŝ = when Z,x f (, X,x, β dx,x = f (, X,x, β and we conclude he poof by Iô calculu. (, X,x (, X,x, β 1 {X,x >1} d(log x Ŝ, β 1 {Z,x >} d(log x Ŝ >,, T. Fom (4.39 hi how ha ( σ(β X,x d ˆB σ2 (β X,x d, The nex wo lemma will be ued in he poof of Popoiion 5.2 below. Lemma 4.8 Le j M uch ha µ(j. The funcion h(, j defined by (4.4 h(, j := inf{x 1, : LG(, y, j, y x, }, (4.41 i noninceaing and coninuou in, T and aifie h(t, j = 1, fo all j M. Poof. By Lemma 4.3 he funcion LG(, x, j i nondeceaing in fo all x 1 ince µ(j and i follow fom he definiion (4.41 of h(, j ha h(, j i noninceaing in, T. Fo any, T and deceaing equence ( n n 1 (, T conveging o fom he igh hand ide we have lim h( n, j h(, j and n lim h( n, j h( k, j fo any k 1, hence lim h( n, j h(, j a by he coninuiy n n of LG(, x, j we have LG(, lim n h( n, j, j = lim k LG( k, lim n h( n, j, j, (4.42 and hi pove ha lim h(, j = h(, j. On he ohe hand we have h(, j := lim h(, j h(, j fo any, T, j M. In cae h(, j > h(, j we have LG(, x, j fo all x h(, j,. In addiion, fo any, and 22

23 x h(, j, h(, j we have LG(, x, j < ince h(, j h(, j, hence LG(, x, j = fo all x h(, j, h(, j by he coninuiy of LG(, x, j. By Lemma 4.3 we would have xσ 2 (j G x (, x, j = µ(jg(, x, j, x h(, j, h(, j, (4.43 which how ha G(, x, j = C(, jx µ(j/σ2 (j, whee C(, j depend only on and j M. Thi i a conadicion ince G( x, x, j = P (Ŷ,T /Y < x β = j = C(, jµ(jx 1+µ(j/σ2 (j /σ 2 (j fo x h(, j, h(, j canno hold when µ(j < σ 2 (j, and moe geneally Ŷ,T /Y canno have a powe law, even locally. Similaly o (3.22-(3.23 in du Toi & Peki (29, we now how ha F (, x, j defined by (3.7 i nondeceaing in, T fo all j M and x 1,, a in he following Lemma 4.9 which will be ued fo Popoiion 5.2, and whoe poof follow he line of du Toi & Peki (29 page 994. Lemma 4.9 Unde he condiion µ(j fo all j M, he funcion F (, x, j = V (, x, j G(, x, j (4.44 i nondeceaing in, T, fo any (j, x M 1,. Poof. Fo any,, T wih <, le τ := τ D (, x, j, T by he definiion (3.2 of τ D. Replacing wih τ and wih in he fomula (4.2 and uing opional ampling, we have F (, x, j = V (, x, j G(, x, j (4.45 EG( + τ, X,x +τ, β +τ β = j G(, x, j +τ LG(v, Xv,x, β v dv β = j τ LG(v +, Xv+,,x β v+ dv β = j Combining (4.45 wih τ LG(v +, Xv,x, β v dv β = j, < T. F (, x, j = V (, x, j G(, x, j G( + τ, X,x +τ, β +τ β = j G(, x, j 23

24 we have τ LG(v +, Xv,x, β v dv β = j, F (, x, j F (, x, j (4.46 τ τ E LG(v +, Xv,x, β v dv β = j E LG(v +, Xv,x, β v dv β = j τ LG(v +, Xv,x, β v LG(v +, Xv,x, β v dv β = j. Since by (4.18 he funcion LG(, x, i i nondeceaing in when µ(i, we find ha he igh hand ide of (4.46 i nonnegaive, heeby F (, x, j i nondeceaing in, T. 5 Soluion of he fee bounday poblem In hi ecion we un o he oluion of he fee bounday poblem (2.6. Fi, we noe ha he opping e D ha a imple fom in wo pecial iuaion. Popoiion 5.1 We have he following pecial cae of opimal opping e D. i Immediae execie. Unde he condiion µ(j fo all j M, we have D =, T 1, M. ii Execie a mauiy. Unde he condiion µ(j σ 2 (j fo all j M, we have Poof. we find D = {T } 1, M. Replacing in (4.2 wih τ D defined in (3.2 and uing opional ampling, V (, x, j = G(, x, j+e τd(,x,j whee (X,x,T i defined in (2.13. LG(, X,x, β d β = j,, T, j M, (5.1 i In cae µ(j fo all j M, by Lemma 4.3, we have LG(, x, i > fo all (, x, i, T 1, M, hence (5.1 implie τ D (, x, i = a.., ohewie i conadic he fac ha V (, x, i G(, x, i becaue of (5.1. Thi implie, T 1, M D. 24

25 ii In cae µ(j σ 2 (j fo all j M, by Lemma 4.3 we have LG(, x, i < fo all (, x, i, T 1, M, and applying Lemma 4.4, we ee ha, T 1, M C, which mean D = {T } 1, M. Nex, we povide ufficien condiion on he dif coefficien (µ(j j M fo he bounday funcion b(, j defined by (3.1 o be noninceaing and coninuou in, T. Popoiion 5.2 Aume ha µ(j fo all j M. Then he bounday funcion b(, j defined by (3.1 i noninceaing in, T. If in addiion µ(j (, σ 2 (j fo all j M, hen b(, j i finie and coninuou in, T. Poof. (i Monooniciy. Le (, x, j D and, T. We have F (, x, j = and F (, x, j = ince F (, x, j i nondeceaing in by Lemma 4.9, hence, T {x} {j} D, (5.2 howing ha (, x, j D, T {x} {j} D. Then fo any (, T, we have (, b(, j, j D ince (, b(, j, j D. By Popoiion 3.2 and noing ha (, b(, j, j D, we conclude ha b(, j b(, j. (ii Finiene. Since he funcion h(, j defined in Lemma 4.8 i noninceaing and coninuou in, T wih h(t, j = 1, we can epea he agumen on page of du Toi & Peki (29 a by (4.18 and he condiion µ(j (, σ 2 (j fo all j M, he funcion LG(, x, j aifie lim x LG(, x, j =,, T, j M, and inf { LG(, x, j : (, x, j, T 1, M } >. (5.3 (iii Righ coninuiy. Given (, b(, j, j D, conide a icly deceaing equence ( n n 1 uch ha lim n n =. By pa (i above we know ha b( n, j b(, j, n 1, and lim n b( n, j b(, j. Nex, by Popoiion 3.2 we have, T b(, j, {j} D, (5.4 ( and ince ( n, j, b( n, j D, n 1, and D i cloed, we have, lim b( n, j, j n D, hence lim b( n, j b(, j. n 25

26 (iv Lef coninuiy. Similaly o poin (ii above we can apply he agumen of du Toi & Peki (29 page 998 baed on he fac ha fom Lemma 4.8 he funcion h(, j i noninceaing and coninuou in, T fo all j M. Figue 1 illuae he eul of Popoiion 5.2 by applying he ecuive algoihm of Liu & Pivaul (217 in ode o plo he value funcion V (, a, j and G(, a, j in he cae µ(j (, σ 2 (j, j M. In Figue 1 we ake he poiive dif µ(1 =.15, µ(2 =.5, wih σ(1 =.5, σ(2 =.3, T =.5, n = 1, δ n = T/n =.5, and Q =. 2 2 V(,a,1 G(,a,1 b(,1 V(,a,2 G(,a,2 b(, D D a a (a Value funcion fom ae 1. (b Value funcion fom ae 2. Figue 1: Value funcion in he wo-ae cae a funcion of ime and he undelying. Figue 1 alo allow u o viualize he opping e D and he coninuaion e C = { (, a, j, T 1, M : V (, a, j < G(, a, j }. (5.5 The numeical inabiliie obeved ae due o he neceiy o check he equaliy V (, a, j = G(, a, j when V (, a, j and G(, a, j ae vey cloe o each ohe. The bounday funcion ae ploed in Figue 2 baed on Figue 1, wih pline moohing. We obeve ha aing fom ae 1 i i bee o execie ealie han if we a fom ae 2 which ha a lowe dif. Thi i due o he poibiliy o wich fom ae 1 o ae 2 afe he aveage ime 1/q 1,1 =.4 and o ay a ae 2 fo he emaining ime T 1/q 2,2 =.5, in which cae he dif ake he lowe value µ(2 =.5. The oppoie occu if we a fom ae 2, fo which he bounday 26

27 gaph i highe han if we a fom ae b(,1 b(, a Figue 2: Bounday cuve a funcion of ime in he wo-ae cae. We noe ha wihou he condiion µ(i fo all i M, he funcion F (, x, j in Lemma 4.9 may no be nondeceaing in, T, in which cae he equivalence (, x, j D, T {x} {j} D in he poof of Popoiion 5.2 doe no hold, and in hi iuaion he bounday funcion b(, j may no be deceaing in, T, cf. e.g. Figue 4 in Liu & Pivaul (217. Nex, we deive a Volea ype equaion (5.7 below aified by he funcion b(, j defined in (3.1, fo he bounday cuve {(, x, T 1, : x = b(, j} (5.6 of he opimal opping e D in (3.1, fo any j M. Popoiion 5.3 Aume ha µ(j (, σ 2 (j fo all j M. Then he bounday funcion b(, j aifie he Volea ype equaion G(, b(, j, j = J(, b(, j, j T T, wih eminal condiion b(t, j = 1, j M, whee and K(,, b(, j, jd, (5.7 J(, x, j :X,x T β = j, (5.8 K(,, x, j : LV (, X,x, β 1 {X,x fo T and x >b(,β } β = j, (5.9

28 Poof. Noing ha V (, x, j G(, x, j fo all (, x, j, T 1, M by (2.12, he coninuaion e C := D c i given by C = D c = {(, x, j, T 1, M : V (, x, j < G(, x, j}. (5.1 Accoding o Popoiion 3.1, fo any (, x, j C, we have V (, x, j G ( τ D, Xτ,x D, β τd β = j, (5.11 whee τ D = τ D (, x, j i defined by (3.2. Given ha V (, 1+, j = by Lemma 4.2, by he applicaion of Peki & Shiyaev (26, Chape III, 7.1.1, a in du Toi & Peki (29 3.5, page 996, he funcion V in (5.11 i C 1,2 in he coninuaion e C in (5.1 and i olve he Cauchy-Diichle fee bounday poblem LV (, y, j =, (, y, j C, (5.12a V (, y, j = G(, y, j, (, y, j C, (5.12b hence C D, whee C denoe he bounday of he open e C. By he local ime change of vaiable fomula of Peki (25, and by Lemma 4.7 below wih he popey V (, 1+, j = hown in Lemma 4.2 above, we have EX,x T β = j V (T, X,x T, β T β = j T = V (, x, j + E LV (, X,x, β 1 {X,x ( V E T (, X,x +, β V (, X,x, β whee we applied he equaliy V (T, X,x T b(,β d β } = j 1 {X,x =b(,β } dlb (X,x (5.13 β = j,, β T = X,x, and (lb (X,x,T denoe he local ime of X,x on he (piecewie coninuou and noninceaing by Popoiion 5.2 cuve b(, β. By he mooh fi popey hown in Lemma 4.1 above, he la em in (5.13 vanihe. By Popoiion 3.2 above and he definiion (3.1 of b(, j, Relaion (5.12a can be ewien a T LV (, y, j1 {y<b(,j} =,, T, j M, y 1, (

29 which implie T T E LV (, X,x, β d β = j Hence, combining (5.13 and (5.15, we obain EX,x T β = j = V (, x, j+ T LV (, X,x E LV (, X,x, β 1 {X,x and ubiuing x wih b(, j in (5.16 above we find ha, β 1 {X,x >b(,β } >b(,β d β } = j. (5.15 β = j d, (5.16 G(, b(, j, j = V (, b(, j, j T X,b(,j T β = j E = J(, b(, j, j T LV (, X,b(,j K(,, b(, j, jd, whee he funcion J, K ae defined by (5.8-(5.9., β 1 {X,x b(,β d β } = j Remak 5.4 Noe ha he equaion (5.7 alo involve he opimal value funcion V (, y, j and no only he funcion G(, y, j. Indeed, when m 2 he equaliy V (, y, j = G(, y, j in (5.15 fo a given (, y, j = (, X,x, β D doe no imply LV (, y, j = LG(, y, j (5.17 a in du Toi & Peki (29 becaue we may no have V (, y, i = G(, y, i fo all i = 1,..., m in he ummaion ove he ae of (β,t in he definiion (4.9 of L. In Buffingon & Ellio (22, hi iue i deal wih via an odeing aumpion on he bounday funcion (b(, j,t in he wo-ae cae j = 1, 2, ee Aumpion 3.1 heein, howeve hi mehod applie pecifically o Ameican opion and no o ulimae maximum poblem, which have a moe complex payoff ucue. Moeove, uch an odeing condiion may no be aified in ou cuen eing, cf. Figue 4 of Liu & Pivaul (217. In he abence of egime wiching wih Y eplaced by = Y e (µ σ2 /2+σB, Relaion (3.1 i b( = inf{x IR + : (, x D},, T, (

30 and he bounday equaion (5.7 become T G(, b( X,b( T E LG(, X b( 1 { X,b( >b( } d, (5.19 which ecove (3.5 in du Toi & Peki (29, wih ( LG(, x = + x(σ2 µ x σ2 x 2 2 G(, x,, T, x IR x 2 +. (5.2 Since he Volea ype equaion (5.7 canno be olved by andad mehod unde egime wiching, we have applied he ecuive algoihm of Liu & Pivaul (217 in ode o plo Figue 1 and 2. Refeence 1 J. Buffingon & R.J. Ellio (22 Ameican opion wih egime wiching. In. J. Theo. Appl. Finance, 5(5: J. du Toi & G. Peki (29 Selling a ock a he ulimae maximum. Ann. Appl. Pobab., 19(3: P. Eloe, R. H. Liu, M. Yauki, G. Yin & Q. Zhang (28 Opimal elling ule in a egimewiching exponenial Gauian diffuion model. SIAM J. Appl. Mah., 69(3: X. Guo (21 An explici oluion o an opimal opping poblem wih egime wiching. J. Appl. Pobab., 38(2: X. Guo & Q. Zhang (25 Opimal elling ule in a egime wiching model. IEEE Tan. Auoma. Conol, 5(9: J.D. Hamilon (1989 A new appoach o he economic analyi of non-aionay ime eie. Economeica, 57: H. Le & C. Wang (21 A finie ime hoizon opimal opping poblem wih egime wiching. SIAM J. Conol Opim., 48(8: R H. Liu, Q. Zhang & G. Yin (26 Opion picing in a egime-wiching model uing he fa Fouie anfom. J. Appl. Mah. Soch. Anal., page A. ID 1819, Y. Liu & N. Pivaul (217 A ecuive algoihm fo elling a he ulimae maximum in egime-wiching model. Pepin axiv: , 19 page. 1 G. Peki (25 A change-of-vaiable fomula wih local ime on cuve. J. Theoe. Pobab., 18(3: G. Peki & A.N. Shiyaev (26 Opimal opping and fee-bounday poblem. Lecue in Mahemaic ETH Züich. Bikhäue Velag, Bael. 12 A.N. Shiyaev (1978 Opimal opping ule. Spinge-Velag, New Yok, NY. 13 A.N. Shiyaev, Z. Xu & X.Y. Zhou (28 Thou hal buy and hold. Quan. Finance, 8(8:

31 14 V. Ly Vah & H. Pham (27 Explici oluion o an opimal wiching poblem in he woegime cae. SIAM J. Conol Opim., 46(2: D.D. Yao, Q. Zhang & X.Y. Zhou (26 A egime-wiching model fo Euopean opion. Inenaional Seie in Opeaion Reeach and Managemen Science, 94: