Arbitrage in fractional Brownian motion models

Transcription

1 Arbirage i fracioal Browia moio models Parick Cheridio Depareme für Mahemaik, Eidgeössische Techische Hochschule Zürich, 892 Zürich, Swizerlad dio@mah.ehz.ch Ocober 22 Absrac We cosruc arbirage sraegies for a fiacial marke ha cosiss of a moey marke accou ad a sock whose discoued price follows a fracioal Browia moio wih drif or a expoeial fracioal Browia moio wih drif. The we show how arbirage ca be excluded from hese models by resricig he class of radig sraegies. Key words: fracioal Browia moio, arbirage, srog arbirage, exclusio of arbirage JEL Classificaio: C6, G1 Mahemaics Subjec Classificaio 2: 6G15, 6K3, 91B28 1 Iroducio We cosider a marke ha cosiss of a moey marke accou ad a sock ha pays o divideds. All ecoomic aciviy akes place i a ime ierval, T for some T,. Borrowig ad shor-sellig are allowed, he borrowig rae is equal o he ledig rae, ad i is possible o buy ad sell ay fracio of sock shares. Moreover, here exis o rasacio coss ad sock shares ca be be bough ad sold a he same price. We assume ha here exis wo sochasic processes X,T ad Y,T o a probabiliy space Ω, A, P such ha moey i he moey marke accou evolves accordig o X,T ad he sock price follows Y,T. This paper is par of he auhor s docoral disseraio wrie uder he supervisio of Freddy Delbae. Fiacial suppor from Credi Suisse is graefully ackowledged.

2 2 Parick Cheridio A fracioal Browia moio fbm wih Hurs parameer H, 1, is a coiuous, ceered Gaussia process B H R wih covariace Cov B H, B H s 1 = 2H + s 2H s 2H,, s R B 1 2 is a wo-sided Browia moio. The pahs of B 1 are sraigh lies wih a ormally disribued slope. For H 1 2, 1 he correlaio of wo icremes of B H over o-ierlappig ime-iervals is posiive, ad for H, 1 2 i is egaive. I ca easily be see from 1.1 ha E B H Bs H 2 = s 2H. Hece, Kolmogorov s coiuiy crierio applies see e.g. Theorem I.2.1 i Revuz ad Yor 1999, ad i follows ha almos all pahs of B H are locally Hölder coiuous of order α for every α < H. Furhermore, B H has saioary icremes ad is H-self-similar, ha is, for all a >, Ba H R has he same disribuio as a H B H R. More deails abou fbm ca be foud i Secio 7.2 of Samorodisky ad Taqqu We call X = 1, Y = Y + ν + σb H,, T, 1.2 he fracioal Bachelier model ad X = expr, Y = Y exp {r + ν} + σb H,, T, 1.3 he fracioal Samuelso model or fracioal Black-Scholes model. For he mome we assume ha Y, σ are posiive cosas ad ν, r are real cosas, bu all our resuls will also hold rue if ν is replaced by ay fucio i C 1 R + ad r by a arbirary righ-coiuous sochasic process. For a discussio of he empirical evidece of correlaio i sock price reurs see e.g Culad e al or Williger e al ad he refereces herei. For H, , 1, BH is o a semimarigale see e.g. Lipser ad Shiryaev 1989 or Rogers Hece, i immediaely follows from Theorem 7.2 of Delbae ad Schachermayer 1994 ha i boh models 1.2 ad 1.3 here exiss a weak form of arbirage called free luch wih vaishig risk cosisig of simple iegrads ha are predicable wih respec o he smalles filraio ha saisfies he usual assumpios ad coais he filraio geeraed by he discoued sock price process. Rogers 1997, Shiryaev 1998 ad Salopek 1998 give arbirage sraegies for fbm models. Rogers 1997 cosrucs arbirage for he fracioal Bachelier model 1.2. His sraegy cosiss of a combiaio of buy ad hold sraegies ad works for all Hurs parameers H, , 1. However, as selfsimilariy of he process Y is esseial for is cosrucio, Rogers arbirage oly exiss i he case ν =, i.e. Y = Y + σb H. Moreover, Rogers models Y for, ad o geerae a profi o he ime ierval 1,, his arbirage sraegy eeds o kow he whole hisory of Y from ime uil he prese.

3 Arbirage i fracioal Browia moio models 3 I Shiryaev 1998 oly he case H 1 2, 1 is reaed. A iegral wih respec o B H is defied ad i is idicaed how i ca be show ha for regular eough fucios F, he modified Iô formula df, B H = 1 F, B H d + 2 F, B H db H 1.4 holds. Usig his for he fracioal Bachelier model 1.2 wih H 1 2, 1, oe ca choose a c > ad se ϑ := c ν + σb H 2 2cY ν + σb H, ϑ 1 := 2c ν + σb H o obai ϑ X + ϑ 1 Y = ϑ X + ϑ 1 Y + ϑ 1 udy u = c ν + σb H Hece, if coiuous adjusme of he porfolio is allowed, ϑ, ϑ 1 is a self-fiacig arbirage sraegy for he fracioal Bachelier model. For he fracioal Samuelso model 1.3 wih H 1 2, 1, oe ca se for all c >, 2. ϑ := cy { 1 exp 2ν + 2σB H }, I follows from 1.4 ha ϑ 1 := 2c { exp ν + σb H 1 }. ϑ X + ϑ 1 Y = ϑ X + ϑ 1 Y + = cy expr { exp ν + σb H 1 } 2, ϑ udx u + ϑ 1 udy u which shows ha ϑ, ϑ 1 is a self-fiacig arbirage sraegy for he fracioal Samuelso model. More geerally, i follows from Youg s heorem o Sieljes iegrabiliy see Youg 1936 ha if a sochasic process Y is almos surely coiuous ad of bouded p-variaio for some p < 2 his is he case for Y i 1.2 ad 1.3 whe H 1 2, 1, he for a real fucio f o R ha is locally Lipschiz, he pah-wise Riema-Sieljes iegral fy udy u exiss for all ad equals F Y F Y, where F = f. I Salopek 1998 his is used o cosruc a self-fiacig arbirage sraegy for wo fiacial asses whose price processes X ad Y are almos surely coiuous, of bouded p-variaio for some p < 2 ad such ha X Y for all. I his paper we wa o do he followig wo higs: for a class of models ha coais 1.2 ad 1.3 for all H, , 1, firs, cosruc as simple as possible arbirage sraegies ad secodly, fid a as big as possible class of radig sraegies ha does o coai arbirage. The srucure of he paper is as follows: I Secio 2 we iroduce differe classes of radig sraegies ad defie he oios of free luch wih vaishig risk, arbirage ad srog arbirage. I Secio 3 we cosruc arbirage sraegies. As i Rogers 1997 our arbirage sraegies cosis of combiaios of buy ad hold sraegies. Therefore we eed o iegraio

4 4 Parick Cheridio heory for fbm. Moreover, o geerae a profi o he ime ierval, T, our sraegies eed oly kow he hisory of he discoued sock price Y/X from ime o. However, hese sraegies ca oly be performed if i is possible o buy ad sell wihi arbirarily small ime iervals. I Secio 4 we show ha arbirage ca be ruled ou from models of he form 1.2 ad 1.3 by iroducig a miimal amou of ime h > ha mus lie bewee wo cosecuive rasacios. 2 Tradig sraegies I his secio he ime ierval is a arbirary compac ierval a, b. Moey ca be ivesed i a moey marke accou where moey grows accordig o a sochasic process X a,b ad a sock whose price follows a sochasic process Y a,b. We firs give he defiiio of differe oios of arbirage ad specify he radig sraegies aferwards. For he ime beig a radig sraegy is jus a pair ϑ = ϑ, ϑ 1 of sochasic processes ϑ a,b ad ϑ 1 a,b. ϑ X describes he moey i he moey marke accou a ime ad ϑ 1 he umber of sock shares held a ime. Hece, he evoluio of he porfolio value of a sraegy ϑ is give by V ϑ := ϑ X + ϑ 1 Y, a, b. Sice we wa o use X as uméraire, we require i o be posiive. We se Ỹ := Y ad Ṽ ϑ X := V ϑ, a, b. X Defiiio 2.1 Le ξ be a, -valued radom variable wih P ξ > >. a A sequece of radig sraegies {ϑ} =1 is a ξ-flvr ξ-free luch wih vaishig risk if ϑ lim Ṽ b Ṽ a ϑ = ξ i probabiliy, ad lim ϑ Ṽ b Ṽ a ϑ =. {ϑ} =1 is a FLVR if i is a ξ -FLVR for some, -valued radom variable ξ wih P ξ > >. b A radig sraegy ϑ is a ξ-arbirage if Ṽb ϑ Ṽ a ϑ = ξ almos surely. ϑ is a arbirage if i is a ξ -arbirage for some, -valued radom variable ξ wih P ξ > >. c A radig sraegy ϑ is a srog arbirage if here exiss a cosa c > such ha Ṽ b ϑ Ṽ a ϑ c almos surely. I is clear ha we mus pu cerai resricios o a radig sraegy o give i a ecoomic meaig. Firs of all, radig sraegies should oly be based o available iformaio. To describe he evoluio of iformaio we

5 Arbirage i fracioal Browia moio models 5 iroduce a family of σ-algebras F = F a,b. We assume ha a ay ime a, b, X ad Y ca be observed ad o iformaio is los over ime. I oher words, F is a filraio ad F X,Y := σ X u u,, Y u u, F for all a, b. Noe ha F Ỹ := σ Ỹu F X,Y for all a, b. u, Furhermore, we require X ad Y o be progressively measurable wih respec o F. This is i paricular he case whe X ad Y are righcoiuous, ad i esures ha for all F-soppig imes τ, he sopped processes X τ a,b ad Y τ a,b are also progressively measurable wih respec o F. To cosruc arbirage i fbm models of he form 1.2 or 1.3 i is eough o cosider combiaios of buy ad hold sraegies. Defiiio 2.2 a The se of simple predicable iegrads is give by SF := {g 1 {a} + 1 g j1 τj,τ j+1 : 2, a = τ 1... τ = b; all τ j s are F-soppig imes; g is a real, F a -measurable radom variable; ad he oher g j s are real, F τj -measurable radom variables}. The class of simple predicable radig sraegies is give by Θ S F := { ϑ = ϑ, ϑ 1 : ϑ, ϑ 1 SF }. b The se of almos simple predicable iegrads is give by asf := {g 1 {a} + g j1 τj,τ j+1 : a = τ 1 τ 2... b; all τ j s are F-soppig imes; g is a real, F a -measurable radom variable; he oher g j s are real, F τj -measurable radom variables; P j such ha τ j = b = 1}. The class of almos simple predicable radig sraegies is give by Θ as F := { ϑ = ϑ, ϑ 1 : ϑ, ϑ 1 asf }. c For ϑ 1 = g 1 {a} + g j1 τj,τ j+1 asf we defie ϑ1 Y := g jy τj+1 Y τj, a, b. Noe ha his is almos surely a sum of fiiely may erms ad he process ϑ 1 Y a,b is progressively measurable because Y a,b is. Defiiio 2.3 Le ϑ = ϑ, ϑ 1 Θ as F. There exis soppig imes a = τ 1 τ 2... b such ha ϑ ad ϑ 1 ca be wrie i he form ϑ = f 1 {a} + f j 1 τj,τ j+1, ϑ 1 = g 1 {a} + g j 1 τj,τ j

6 6 Parick Cheridio We se τ = a 1 ad call ϑ self-fiacig for X, Y if for all j 1, k = 1,..., j ad l, { } a.s. 1 {τj k <τ j k+1 =τ j+l <τ j+l+1 } fj+l f j k X τj + g j+l g j k Y τj =. 2.2 Noe ha propery 2.2 is idepede of he represeaio 2.1 of ϑ. Furhermore, we se Θ S sff := { ϑ Θ S F : ϑ is self-fiacig } ad Θ as sf F := { ϑ Θ as F : ϑ is self-fiacig }. Proposiio 2.4 Le ϑ = ϑ, ϑ 1 Θ as F. The he followig are equivale: i ϑ is self-fiacig for X, Y ii almos surely, V ϑ iii ϑ is self-fiacig for 1, Ỹ iv almos surely, Ṽ ϑ = V ϑ a + ϑ X + ϑ 1 Y for all a, b = Ṽ ϑ a + ϑ 1 Ỹ for all a, b Proof Le a = τ 1 τ 2... b be a icreasig sequece of F-soppig imes such ha ϑ = f 1 {a} + f j 1 τj,τ j+1, ϑ 1 = g 1 {a} + g j 1 τj,τ j+1. i ii: I follows from i ha here exiss a measurable Ω Ω wih P Ω = 1 such ha for each ω Ω, equaio 2.2 holds for all j 1, k = 1,..., j ad l simulaeously. For = a, he equaio i ii holds for all ω Ω. Furhermore, here exiss a measurable Ω Ω wih P Ω = 1 such ha for all ω Ω, here exiss for every a, b, a j N, such ha τ j, τ j+1, ad = a + ϑ X + ϑ 1 Y = f j 1 X τ1 + g Y τ1 + f i Xτi+1 X τi V ϑ i=1 j 1 +f j X X τj + g i Yτi+1 Y τi + gj Y Y τj i=1 j X τi f i 1 f i + i=1 j Y τi g i 1 g i + f j X + g j Y = ϑ X + ϑ 1 Y. i=1 This shows ii. ii i: Le j 1, k = 1,..., j ad l. O {τ j k < τ j k+1 = τ j+l < τ j+l+1 } we have f j+l f j k X τj + g j+l g j k Y τj = f j+l X τj+l+1 + g j+l Y τj+l+1 fj k X τj + g j k Y τj

7 Arbirage i fracioal Browia moio models 7 f j+l Xτj+l+1 X τj gj+l Yτj+l+1 Y τj = ϑ τj+l+1 X τj+l+1 + ϑ 1 τj+l+1 Y τj+l+1 ϑ τj X τj + ϑ 1 τj Y τj j+l ϑ ax a + ϑ 1 j+l ay a + f i Xτi+1 X τi + g i Yτi+1 Y τi i=1 j 1 + ϑ ax a + ϑ 1 j 1 a.s. ay a + f i Xτi+1 X τi + g i Yτi+1 Y τi =, i=1 which proves i. The equivalece of i ad iii is rivial, ad he equivalece of iii ad iv ca be show i he same way as he equivalece of i ad ii. Remark 2.5 I follows from Proposiio 2.4 ha for all ϑ Θsf as F, almos surely, ϑ = Ṽ a ϑ + ϑ 1 Ỹ ϑ 1 Ỹ, a, b. This shows ha if we ideify idisiguishable processes, he map ϑ = ϑ, ϑ 1 i=1 i=1 Ṽ ϑ a, ϑ 1 is a bijecio from Θsf asf o L F a asf. I paricular, here exiss for all ξ, ϑ 1 L F a asf, a uique ϑ asf such ha ϑ = ϑ, ϑ 1 is i Θsf as ϑ F ad Ṽa = ξ. I Θsf asfỹ here exis so called doublig sraegies which ca creae srog arbirage eve i he sadard Samuelso or Black-Scholes model 1.3 wih H = 1 2. I was oiced by Harriso ad Pliska 1981 ha hey ca be ruled ou by puig a admissibiliy codiio o he radig sraegies. We use he admissibiliy codiio of Delbae ad Schachermayer I is more liberal ha he oe of Harriso ad Pliska 1981 bu resricive eough o exclude arbirage i he sadard Samuelso model. Defiiio 2.6 For c, we call ϑ Θsf as F c-admissible if almos surely, ϑ if Ṽ a,b Ṽ a ϑ = if ϑ 1 Ỹ c. a,b We call ϑ admissible if i is c-admissible for some c. Furhermore, we se Θ S sf,admf := { ϑ Θ S sff : ϑ is admissible } ad Θ as sf,admf := { ϑ Θ as sf F : ϑ is admissible }.

8 8 Parick Cheridio 3 Cosrucio of arbirage Theorem 3.1 Le B H be a fbm. Le T,, ν C 1, T ad σ >. The i all four cases i H 1 2, 1, Ỹ = ν + σb H,, T ii H 1 2, 1, Ỹ = exp ν + σb H,, T iii H, 1 2, Ỹ = ν + σb H,, T iv H, 1 2, Ỹ = exp ν + σb H,, T, here exiss for every cosa c > ad all N, a ϑ 1 SFỸ such ha a P ϑ 1 Ỹ = c > 1 1 ad T b if,t ϑ 1 Ỹ 1. I paricular, he sraegies ϑ = ϑ, ϑ 1 Θ S sf,adm FỸ, N, where ϑ is give by ϑ = ϑ 1 Ỹ ϑ 1 Ỹ,, T, N, form a c-flvr. I he cases iii ad iv, ϑ 1 ca be chose such ha also c ϑ 1 1. Theorem 3.2 I all four cases i-iv of Theorem 3.1 here exiss for every cosa c >, a 1 c -admissible c-arbirage ϑ ΘaS sf,adm FỸ. I he cases iii ad iv, ϑ ca be chose such ha ϑ 1 1 c. For he proofs of Theorems 3.1 ad 3.2 we eed he followig hree lemmas. Lemma 3.3 Le Z a,b be a coiuous sochasic process. If P Z b = Z a =, 3.1 ad for all ε >, here exis F Z -soppig imes a = τ... τ = b such ha 1 P 2 max Zτj+1 Z τj ε < ε, 3.2 a,b he here exiss for all M >, a β SF Z such ha a P β Z b < M < 1 M ad b if a,b β Z 1 M.

9 Arbirage i fracioal Browia moio models 9 Proof Le M >. I follows from 3.1 ad 3.2 ha here exis a ε > such ha P Z b Z a 2 < ε < M ad F Z -soppig imes a = τ... τ = b, such ha 1 P 2 ε max Zτj+1 Z τj a,b M 2 < M. 3.4 Sice Z is coiuous, 1 ξ := if a, b : 2 ε Zτj+1 Z τj M se if = b 3.5 is a F Z -soppig ime see e.g. Proposiio I.4.5 i Revuz ad Yor 1999, ad 3.4 implies P ξ < b < 1 2M. 3.6 Furhermore, β := 2 ε β Z = M + 1 M ε M + 1 M 1 Zτj Z a 1τj,τ j+11,ξ is i SF Z, ad for all a, b, 1 Z ξ Z a 2 2 Zτj+1 ξ Z τj ξ. 3.7 This ogeher wih 3.5 implies b. From 3.7, 3.6 ad 3.3 i follows ha P β Z b < M = P M + 1 M ε Z ξ Z a 2 P 1 Z ξ Z a 2 < ε P ξ < b + P 2 Zτj+1 ξ Z τj ξ < M Z b Z a 2 < ε < 1 M. This shows a, ad he lemma is proved. Lemma 3.4 Le Z a,b be a coiuous sochasic process. If for all L > here exis F Z -soppig imes a = τ... τ = b, such ha 1 P 2 Zτj+1 Z τj < L < 1 L, 3.8

10 1 Parick Cheridio he here exiss for all M >, a β SF Z such ha a P β Z b < M < 1 M, b if a,b β Z b 1 M ad c β 1 M. Proof Le M >. Sice Z is coiuous, ξ N := if { a, b : Z Z a N} we se if = b 3.9 is for all N > a F Z -soppig ime ad {ξ N < b}, as N. Therefore here exiss a N 2, such ha P ξ N < b < 1 2M. 3.1 By assumpio 3.8 here exis F Z -soppig imes a = τ... τ = b, such ha 1 P Zτj+1 Z 2 τ j < N 2 M < 1 2M I is easy o see ha β := 2 MN 2 1 Zτj Z a 1τj,τ j+11,ξn is i SF Z ad saisfies c. For all a, b, β Z = Zτj+1 ξ MN 2 N Z τj ξ N Z ξn Z a This ogeher wih 3.9 implies b. From 3.12, 3.1 ad 3.11 i follows ha P β Z b < M = P Zτj+1 ξ MN 2 N Z τj ξ N ZξN Z a 2 < M 1 P 2 Zτj+1 ξ N Z τj ξ N < M 2 N 2 + N 2 1 P ξ N < b + P 2 Zτj+1 Z τj < N 2 M < 1 M. This shows a, ad he lemma is proved.

11 Arbirage i fracioal Browia moio models 11 Lemma 3.5 Le B H be a fbm ad T, p, q >. The: a ph 1 q 1 BH j+1 T BH j p i L 1 T b ph 1+q 1 BH j+1 T BH j p i probabiliy, T i.e. for all L > here exiss a such ha for all, P ph 1+q 1 BH j+1 T BH j T p < L < 1 L Proof See Lemma 2.1 i Cheridio 21b. Proof of Theorem 3.1 By self-similariy of B H i is eough o prove Theorem 3.1 for T = 1. i H 1 2, 1, Ỹ = ν + σb H,, 1: I is clear ha Ỹ,1 saisfies 3.1. Sice he fucio ν is Lipschiz ad almos all pahs of B H,1 are Hölder coiuous of order α for every α 1 2, H, i follows ha 1 max,1 2 Ỹ j+1 Ỹ j almos surely This shows ha Ỹ,1 also saisfies 3.2. Thus, i follows from Lemma 3.3 ha for all N, here exiss a β SFỸ such ha a P β Ỹ < c < 1 ad 1 b if,1 β Ỹ 1. For every N, ξ := if { } : β Ỹ = c we se if = 1 is a F Z -soppig ime, ad for ϑ 1 := β1,ξ SF Z we have a P ϑ 1 Ỹ = c > 1 1 ad 1 b if,1 ϑ 1 Ỹ 1. ii H 1 2, 1, Ỹ = exp ν + σb H,, 1 : Ỹ,1 saisfies 3.1, ad i is clear ha 3.13 sill holds rue. Hece, Ỹ,1 also saisfies 3.2, ad he proof ca be cocluded as i case i. iii H, 1 2, Ỹ = ν + σb H,, 1: To show ha Ỹ,1 saisfies 3.8 we choose a L >. I follows from Lemma 3.5 a ha 1 1 B H j+1 B H j i L 1.

12 12 Parick Cheridio Hece, 1 2 ν j+1 ν j σb H j ν 1 σ B H j+1 B H j σb H j i L 1. I paricular, here exiss a 1 N, such ha for all 1, 1 P 2ν j+1 ν j σb H j+1 σb H j > L < 1 2L. O he oher had, Lemma 3.5 b implies ha here exiss a 2 N, such ha for all 2, 1 2 P σb H j+1 σb H j < 2L < 1 2L. Hece, for all max 1, 2, 1 2 P Ỹ j+1 Ỹ j < L 1 2 P σb H j+1 σb H j + 2ν j+1 ν j σb H j+1 σb H j < L 1 2 P σb H j+1 σb H j < 2L 1 +P 2 ν j+1 ν j σb H j+1 σb H j > L < 1 L. This shows ha Ỹ,1 saisfies 3.8. By Lemma 3.4 here exiss for all N, a β SF Z such ha a P β Ỹ < c < 1 1 b if,1 β Ỹ 1 c β 1. Havig show his, we ca cosruc ϑ 1 as i i. By c we ge ϑ 1 1.

13 Arbirage i fracioal Browia moio models 13 iv H, 1 2, Ỹ = exp ν + σb H,, 1 : Sice Ỹ,1 is posiive ad coiuous, mi,1 Ỹ >. Therefore, here exiss a ε > such ha P mi Ỹ ε < 1,1 2L. I follows from wha we have show i he proof of iii ha here exiss a N, such ha 1 2 P l Ỹ j+1 l Ỹ 1 j < ε 2 L < 1 2L. Sice for all j, we obai 1 P Ỹ j+1 Ỹ j+1 Ỹ j Ỹ j l mi Ỹ Ỹ j+1,1 2 < L l Ỹ j, 1 2 P mi Ỹ ε + P l Ỹ j+1 l Ỹ 1 j <,1 ε 2 L < 1 L. This shows ha Ỹ,1 saisfies 3.8. Thus, ϑ 1 ca be cosruced as i iii. Agai, ϑ 1 1. This complees he proof of he heorem. Proof of Theorem 3.2 Sice B H is self-similar, i is eough o prove he heorem for T = 1. We spli, 1 io he subiervals I := a = 1 2 1, b = 1 2, N. By Ỹ we deoe he resricio of Ỹ o I ad by FỸ = F Ỹ I he filraio geeraed by Ỹ. Noe ha F Ỹ F Ỹ for all N ad I. Sice B H has saioary icremes, i follows from Theorem 3.1 ha here exiss for all N, a β SFỸ such ha a P β Ỹ < c + 1 c < 1 b b if I β Ỹ 2 c. For β := β1 I, =1

14 14 Parick Cheridio { } ξ := if, 1 : β Ỹ = c we se if = 1 is a FỸ -soppig ime. I follows from a ad b ha P ξ < 1 = 1. Therefore, ϑ 1 := β1,ξ belogs o asfỹ ad ϑ, ϑ 1 wih ϑ := ϑ 1 Ỹ ϑ 1 Ỹ,, T, is a 1 c -admissible c-arbirage i ΘaS sf,adm FỸ. I he cases iii ad iv all β s ca be chose such ha β 1 c. The, ϑ 1 1 c oo, ad he heorem is proved. Remarks More geerally, coclusios a ad b of Theorem 3.1 hold wheever he discoued sock price Ỹ,T saisfies codiios 3.1 ad 3.2 of Lemma 3.3. If Ỹ,T fulfils codiio 3.8 of Lemma 3.4, he a, b ad c of Theorem 3.1 are valid. I paricular, codiio 3.2 is fulfilled by all processes wih vaishig quadraic variaio ad codiio 3.8 by all processes wih ifiie quadraic variaio. For differe geeralizaios of Lemma 3.5 see e.g. Shao 1996, Takashima 1989 or Kôo ad Maejima Shao 1996 coais resuls o p-variaio of Gaussia processes wih saioary icremes. Takashima 1989 gives sample pah properies of ergodic self-similar processes, ad i Kôo ad Maejima 1991, resuls o Hölder coiuiy of sample pahs of some self-similar sable processes ca be foud. 2 I a model X, Y,T wih srog arbirage i is possible o superreplicae a Europea call opio wih ime-t pay-off C T = Y T K +, K >, wihou iiial edowme i he followig way: A ime oe borrows moey from he moey marke accou o buy oe sock share. The oe applies a srog arbirage sraegy o geerae he amou of moey eeded o pay back he deb wihou sellig he sock share. A ime T oe ows a sock share ad has o debs. This hedges he opio. The followig example shows ha C T ca have a posiive super-replicaio price if he model X, Y,T oly admis arbirage: Le B,1 be a Browia moio ad B H,1 a fbm wih Hurs parameer H, , 1. Le ξ be a radom variable ha is idepede of B ad B H ad such ha P ξ = = P ξ = 1 = 1 2. Le r, ν ad σ >, be cosas. The, he model X = expr, Y = exp { r + ν + σ 1 ξb + ξb H },, 1, has arbirage bu o srog arbirage i Θsf,adm as FY. Is is clear ha he super-replicaio of C 1 wih a sraegy from Θsf,adm as FY coss a leas he Black-Scholes price.

15 Arbirage i fracioal Browia moio models 15 4 Exclusio of arbirage The arbirage sraegies ha we cosruced i Secio 3 ac o ever smaller ime iervals. They ca be excluded by iroducig a miimal amou of ime h > ha mus lie bewee wo cosecuive rasacios. Defiiio 4.1 Le F = F,T be a filraio ad h >. We defie 1 S h F := g 1 {} + g j 1 τj,τ j+1 SF : j, τ j+1 τ j + h ad Θ h sff := { ϑ = ϑ, ϑ 1 Θ S sf : ϑ, ϑ 1 S h F }. I he followig we will show ha oe of he models i-iv of Theorem 3.1 has a arbirage i h> Θh sf FỸ. Lemma 4.2 Le B be a Browia moio ad H, , 1. Le Z be a coiuous versio of he process sh 1 2 db s. The, for all c ad all h ad T such ha < h T, P if Z c = P sup Z c >. h,t h,t Proof Le c ad < h T. Clearly, Z has he same disribuio as Z. Hece, P if Z c = P sup Z c. h,t h,t We deoe by µ W he Wieer measure o C, T, B, where B is he σ- algebra geeraed by he cylider ses. I follows from Lévy s modulus of coiuiy for Browia moio see e.g. Theorem I.2.7 i Revuz ad Yor 1999 ha µ W ˆΩ = 1, where ˆΩ := ω ωs ω C, T : ω = ad, T, lim = s s log 1. s For every ω ˆΩ, sh 1 2 dωs ca for all, be defied as a Riema-Sieljes iegral which is coiuous i his ca be proved like Proposiio 1.3 i Cheridio 21a. Hece, we have = µ W P if Z c h,t ω ˆΩ : if h,t Le us firs assume H 1 2, 1. I his case we se m := H + { 1 2 c + T H 1 h H+ 1 2 ad A m := ω ˆΩ : 2 s H 1 2 dωs c. sup ω m 1,T },

16 16 Parick Cheridio where ω m is give by ω m := ω m,, T. By Girsaov s Theorem here exiss a probabiliy measure µ m o ˆΩ ha is equivale o µ W such ha ω m,t is a Browia moio uder µ m. I is well kow ha µ m A m >. Equivalece of µ W ad µ m implies ha also For all ω ˆΩ ad, T, µ W A m >. 4.1 s H 1 2 dωs = ωsh 1 2 sh 3 2 ds = H 1 2 ω m s s H 3 2 ds + H 1 2 = H 1 2 ω m s s H 3 H ds + m H ms s H 3 2 ds For ω A m, we obai for all h, T he followig esimaes: H 1 2 ad, by our choice of m, Hece, ω m s s H 3 2 ds H 1 2 m H+ 1 2 H I follows ha = s H 3 2 ds T H 1 2, H+ 1 2 c + T H 1 2 c + T H 1 2. h s H 1 2 dωsds T H c + T H 1 2 = c. { A m if h,t } s H 1 2 dωs c, which ogeher wih 4.1 proves he lemma for H 1 2, 1. For H, 1 2, he proof is slighly more delicae. Sice all ω ˆΩ are Hölder coiuous of order α for every α < 1 2, here exiss a cosa δ > such ha µ W A 1 2, δ >, where A {ω 1 2, δ := ˆΩ : sup ω 1,T 2 ad We se m := H h H+ 1 2 } sup ω ωs δ,s,t s 1 2 H 2 c H2δ H T H hh 1 2,

17 Arbirage i fracioal Browia moio models 17 ω m := ω m,, T ad defie µ m as before. Furhermore, we se A m 1 2, δ := { ω ˆΩ : ω m A 1 2, δ }. Sice ω m,t is a Browia moio uder µ m, µ m A m 1 2, δ = µ W A 1 2, δ >, ad herefore, µ W A m 1 2, δ >. 4.2 For ω ˆΩ ad h, T, we ca wrie s H 1 2 dωs = s H 1 2 d ωs ω = 1 2 H ω ωs s H 3 2 ds + H 1 2 ω = 1 2 H ω m ω m s s H 3 2 ds + H 1 2 ωm + m H+ 1 2 H If ω A m 1 2, δ ad h, T, we ca esimae he hree precedig erms as follows: 1 2 H ω m ω m s s H 3 2 ds 1 2 H H 1 2 ωm 1 2 hh 1 2, m H+ 1 2 H = δ s H 2 1 ds 1 2 H2δ H T H 2, H+ 1 2 c + 1 h 2 H2δ H T H hh 1 2 c H2δ H T H hh 1 2. I follows ha s H 1 2 dωs c. This ad 4.2 prove he lemma for H, 1 2.

18 18 Parick Cheridio Theorem 4.3 Le B H be a fbm wih H, , 1. Le T,, σ > ad ν :, T R be a measurable fucio such ha sup,t ν <. Cosider he wo cases If i Ỹ = ν + σb H,, T ii Ỹ = exp ν + σb H,, T 1 ϑ 1 = g 1 {} + g j 1 τj,τ j+1 h> S h FỸ ad here exiss a j {1,..., 1} wih P g j >, he i case i, P ϑ 1 Ỹ c > for all c, T ad i case ii, P ϑ 1 Ỹ < >. T Proof For oaioal simpliciy we give he proof for Ỹ = B H ad Ỹ = expb H. The geeralizaios o he cases i ad ii are obvious. To prove he heorem for Ỹ = B H we fix a h >, ad cosider a 1 ϑ 1 = g 1 {} + g j 1 τj,τ j+1 S h F BH, such ha here exiss a j {1,..., 1} wih P g j >. If he k = max {j {1,..., 1} : P g j > }, ϑ1 B H T = k g j B H τ j+1 B H τ j almos surely. Le c. I is clear ha k P g j Bτ H j+1 Bτ H j c 4.3 k 1 P g j Bτ H j+1 Bτ H j + sup g k B H τk + B H τ k c. h,t Le ω ωs ˆΩ := ω CR : ω = ad R, lim = s s log 1, s

19 Arbirage i fracioal Browia moio models 19 B he σ-algebra of subses of ˆΩ ha is geeraed by he cylider ses ad P he Wieer measure o ˆΩ, B. Wihou loss of geeraliy we ca assume ha B H is defied o ˆΩ, B, P by he improper Riema-Sieljes iegrals B H ω = s H 1 2 1{s } s H 1 2 dωs, 4.4 see Proposiio 1.3 i Cheridio 21a. We defie he filraio F ˆΩ = F ˆΩ,T by {{ := σ ω ˆΩ } } : ωs a : < s, a R. F ˆΩ I is clear ha F ˆΩ coais he filraio F BH by F BH := σ { B H s : s }. = F BH,T, which is give Therefore he F BH -soppig imes τ 1,... τ k, are also F ˆΩ-soppig imes. I he followig we spli each fucio ω ˆΩ a he ime poi τ k ω. We se ad le π 1 ωs := ωs1,τk ωs, s R, π 2 ωs := ωτ k ω + s ωτ k ω, s, Ω 1 := { π 1 ω R R : ω ˆΩ }, B 1 he σ-algebra of subses of Ω 1 ha is geeraed by he cylider ses, { Ω 2 := π 2 ω C, : ω ˆΩ } ad B 2 he σ-algebra of subses of Ω 2 ha is geeraed by he cylider ses. I ca easily be checked ha he mappig π 1 : ˆΩ, B Ω 1, B 1 is F ˆΩ τk -measurable. O he oher had, i follows from Theorem I.32 of Proer 199 ha π 2 ωs s is a Browia moio uder P which is idepede of F ˆΩ τk. I ca be see from 4.4 ha for all ω ˆΩ ad h, T, k 1 g j B H τ j+1 B H τ j where for ω 1 Ω 1, ω 2 Ω 2 ad h, T, + g k B H τk + Bτ H k ω = U π 1 ω, π 2 ω U ω 1, ω 2 := U ω 1 + g k ω 1 U 1 ω 1 + U 2 ω 2, k 1 U ω 1 := g j Bτ H j+1 Bτ H j ω 1, U 1 ω 1 := U 2 ω 2 := τk ω 1 τ k ω 1 + s H 1 2 τk ω 1 s H 1 2 dω 1 s, s H 1 2 dω2 s.

20 2 Parick Cheridio Sice U h,t is a coiuous sochasic process o Ω 1 Ω 2, B 1 B 1, he se { } A := ω 1, ω 2 Ω 1 Ω 2 : sup U ω 1, ω 2 c h,t is B 1 B 2 -measurable. I follows from Proposiio A.2.5 of Lambero ad Lapeyre 1996 ha for almos every ω ˆΩ, E 1 A π 1, π 2 F ˆΩ τk ω = φ π 1 ω, where he mappig φ : Ω 1 R is give by φω 1 := E 1 A ω 1, π 2, ω 1 Ω 1. Sice U 1 ω 1 is for all ω Ω 1 coiuous i, sup h,t U 1 ω 1 is for all ω 1 Ω 1 fiie. Therefore ad sice π 2 ω is a Browia moio uder P, i follows from Lemma 4.2 ha for fixed ω 1 Ω 1, P φω 1 = P U ω 1 + sup U ω 1, π 2 c h,t sup h,t g k ω 1 U 1 ω 1 + sup h,t g k ω 1 U 2 π 2 c Therefore, k 1 P g j Bτ H j+1 Bτ H j + sup g k B H τk + B H τ k c h,t = E 1 A π 1, π 2 = E E 1 A π 1, π 2 F ˆΩ τk = E φ π 1 >. This ad 4.3 prove he heorem i he case Ỹ = B H. If Ỹ = expb H, le us assume here exiss a h > ad a 1 ϑ 1 = g 1 {} + g j 1 τj,τ j+1 S h F BH >. such ha here exiss a j {1,..., 1} wih P g j > ad ϑ 1 Ỹ T almos surely. If l k = mi l : P g l > ad g j e BH τ B j+1 e H τ j a.s., he eiher k 1 g j e BH τ B j+1 e H τ j = almos surely, or

21 Arbirage i fracioal Browia moio models 21 k 1 P g j e BH τ B j+1 e H τ j < >. I boh cases, P C > for k 1 C := g j e BH τ B j+1 e H τ j. Wih he same mehod ha we used i he firs par of he proof oe ca deduce from Lemma 4.2 ha for almos all ω C, k 1 P g j e BH τ B j+1 e H τ j + sup g k e BH τ k + e BH τ k < h,t F ˆΩ τk ω >, which implies ha k P g j e BH τ B j+1 e H τ j < k 1 P g j e BH τ B j+1 e H τ j + sup g k e BH τ k + e BH τ k < >. h,t This coradics our assumpio ad herefore, complees he secod par of he proof. I follows from Theorem 4.3 ha if radig sraegies are resriced o he class h> Θh sf FỸ, he i case i here exis o o-rivial admissible sraegies ad i paricular o FLVR, ad i case ii here exiss o arbirage. A ispecio of he proof of Theorem 4.3 shows ha i case ii, a ϑ h> Θh sf FỸ ca oly be admissible if ϑ 1 is almos surely o-egaive. I follows from similar argumes o he oes i he proof of Theorem 4.3 ha i boh cases i ad ii he cheapes way o super-replicae a Europea call opio wih a sraegy ϑ h> Θh sf FỸ is o buy he sock. I paricular, i boh cases i ad ii of Theorem 4.3 he model X, Y,T is icomplee whe radig sraegies are resriced o h> Θh sf FỸ. Refereces 1. Cheridio, P.: Regularizig fracioal Browia moio wih a view owards sock price modellig. Diss. ETH No. 1451, dio 21a 2. Cheridio, P.: Mixed fracioal Browia moio. Beroulli 76, b 3. Culad, N.J., Kopp P.E., Williger W.: Sock price reurs ad he Joseph effec: A fracioal versio of he Black-Scholes model. Progress i Probabiliy 36,

22 22 Parick Cheridio 4. Delbae, F., Schachermayer, W.: A geeral versio of he fudameal heorem of asse pricig. Mah. A. 3, Harriso, J.M., Pliska, S.P.: Marigales ad sochasic iegrals i he heory of coiuous radig. Soch. Proc. Appl. 11, Kôo, N., Maejima, M.: Hölder coiuiy of sample pahs of some self-similar sable processes. Tokyo Joural of Mahemaics 141, Lambero, D., Lapeyre, B.: Sochasic calculus applied o fiace. Chapma & Hall Lipser, R.Sh., Shiryaev A.N.: Theory of marigales. Kluwer Acad. Publ., Dordrech Proer, P.: Sochasic iegraio ad differeial equaios. Berli: Spriger- Verlag Revuz, D., Yor, M.: Coiuous marigales ad Browia moio. Spriger Rogers, L.C.G.: Arbirage wih fracioal Browia moio. Mahemaical Fiace 7, Salopek, D.M.: Tolerace o arbirage. Sochasic Processes ad heir Applicaios 76, Samorodisky, G., Taqqu, M.S.: Sable o-gaussia radom processes. Sochasic models wih ifiie variace. Chapma & Hall, New York Shao, Q.M.: p-variaio of Gaussia processes wih saioary icremes. Sudia Sci. Mah. Hugar. 31, Shiryaev, A.N.: O arbirage ad replicaio for fracal models. Research Repor 2, MaPhySo, Deparme of Mahemaical Scieces, Uiversiy of Aarhus, Demark Takashima, K.: Sample pah properies of ergodic self-similar processes. Osaka J. Mah. 26, Williger, W., Taqqu, M.S., Teverovsky, V.: Sock marke prices ad lograge depedece. Fiace Sochas. 31, Youg, L.C.: A iequaliy of he Hölder ype, coeced wih Sieljes iegraio. Aca Mah. Swede 67,