Regularizing Fractional Brownian Motion with a View towards Stock Price Modelling

Transcription

1 Diss. ET No. 45 Regularizig Fracioal Browia Moio wih a View owards Sock Price Modellig A disseraio submied o he SWISS FEDERAL INSTITUTE OF TECNOLOGY ZURIC for he degree of Docor of Mahemaics preseed by PATRICK CERIDITO Dipl. Mah. ET bor Jauary 8, 969 ciize of Zürich Z acceped o he recommedaio of Prof. Dr. F. Delbae, examier Prof. Dr. P. Embrechs, co-examier Prof. Dr. A.N. Shiryaev, co-examier Prof. Dr. M. Yor, co-examier

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3 Dak Diese Dokorarbei esad uer der Leiug vo Prof. Freddy Delbae. Ich dake ihm für alles, was er mir beigebrach ha, für sei Ieresse a dieser Arbei ud für die viele Koake, die er mir ermöglich ha. Prof. Paul Embrechs, Prof. Alber Shiryaev ud Prof. Marc Yor dake ich für die Überahme des Korreferas ud die viele Amerkuge, die zu eier Verbesserug der Dokorarbei geführ habe. Sehr wichig für mich war auch der Gedakeausauch mi weiere Miarbeier der ET. Markus Melek war immer berei, über mahemaische Probleme zu diskuiere, ud Prof. Eugee Trubowiz ha mir gezeig, wie mahemaische Physiker Iegralgleichuge löse. Chaal Bueau, Giovai Geile, Iva Jecic, Sve Josse, Fraz Müller, Domiik Schözau ud Michael Suder verdake ich viel Freude währed ud ebe der Arbei. Die Credi Suisse ha eie Teil dieser Arbei gesposer. iii

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5 Absrac There have bee several aemps o remedy some of he shorcomigs of he Samuelso model for sock price movemes usig fracioal Browia moio. I he firs par of his hesis we cosruc arbirage sraegies for wo differe models based o fracioal Browia moio ad show how arbirage ca be ruled ou by puig resricios o he radig sraegies. Sice hese models wih he resriced radig sraegies are icomplee, i is o clear how o price derivaives wihi hem. Aleraively, arbirage ca be excluded from fracioal Browia moio models by regularizig he local pah behaviour of fracioal Browia moio. We iroduce wo differe ways of regularizig fracioal Browia moio ad discuss he pricig of a Europea call opio i regularized fracioal Samuelso models. v

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7 Coes Prelimiaries. Noaio FracioalBrowia moio Weak semimarigales Themarke Arbirage i fracioal Browia moio models 3. Iroducio Theradig sraegies Cosrucioofarbirage Exclusioofarbirage Regularized fracioal Browia moio ad opio pricig Iroducio Regularizig fracioal Browia moio Geeralidea R ϕ ad is semimarigale decomposiio Equivalece of ϕ Rϕ o Browia moio.,t ] 58 vii

8 viii Coes 3.3 Opio pricig wih regularized fracioal Browia moio Naive opio pricig i regularized fracioal Samuelso models Discussio Mixed fracioal Browia moio Iroducio Proof of Theorem 4. for, Proof of Theorem 4. for, 3 4 ] Proof of Theorem 4. for 3 4, ] Opio pricig wih mixed fracioal Browia moio Represeaios of Gaussia processes ha are equivale o Browia moio The represeaios of Shepp ad isuda The Girsaov-isuda represeaio ad relaios beweediffere represeaios The Shepp-represeio of mixed fracioal Browia moio The isuda-represeaio of mixed fracioal Browia moio Bibliography 9

9 Chaper Prelimiaries. Noaio Throughou his hesis,a, P will be a probabiliy space. Le I IR be a ierval ad X I a sochasic process. We call X coiuous, righ-coiuous or càdlàg coiu à droi,limiesà gauche if all pahs have he correspodig propery. If almos all pahs have he propery, we call X a.s. coiuous, a.s. righ-coiuous or a.s. càdlàg. We say X is sochasically righ-coiuous if for all I \ {sup I}, lim s X s = X i probabiliy. By lf X we deoe he filraio geeraed by X, i.e. lf X = F X I,where F X := σ X s : s I, s, I. Le T,. We say ha a filraio lf = F,T ] saisfies he usual assumpios if i is righ-coiuous, F T is complee ad F coais all ull ses of F T.IflF = F,T ] is a arbirary filraio, we deoe by lf = F he smalles filraio ha coais lf ad saisfies he usual,t ] assumpios.. Fracioal Browia moio Defiiio. A fracioal Browia moio wih urs parameer, ], is a coiuous, cered Gaussia process B Cov B, Bs = IR wih + s s,, s IR...

10 Chaper. Prelimiaries These processes were firs sudied by Kolmogorov 94 wihi a ilber space framework. For =, fracioal Browia moio ca be cosruced as follows: B = ξ, IR,.. where ξ is a sadard ormal radom variable. For =, fracioal Browia moio is a wo-sided Browia moio. I ca be cosruced by akig wo idepede oe-sided Browia moios W { B W = if W if <., W ad seig For,, Madelbro ad Va Ness 968 gave he followig cosrucio of fracioal Browia moio: = c B IR ϕ s ϕ s] dw s, IR,..3 where W s s IR is a wo-sided Browia moio, ϕ x = {x } x, x IR,..4 ad c is a ormalizig cosa. If =, i is clear ha for all IR, ϕ s ϕ s] dw s = W. For, IR,, he iegrals IR ϕ s ϕ s] dw s, IR ca be udersood as L -limis or almos sure-limis. I order o defie IR ϕ s ϕ s] dw s for every IR i he L - sese, we defie for sep-fucios f = a k k, k+ ], k= where a,...,a IR ad < < < <, L - IR f sdw s := a k Wk+ W k. k=

11 .. Fracioal Browia moio s s Figure.: Lef: The fucios ϕ s ad ϕ s for = 3 4 ad = 3. Righ: The fucios ϕ s ad ϕ s for = 4 ad = 3. Sice he sep-fucios are dese i L IR ad E L - IR ] f sdw s = IR f sds for all sep-fucios f, L - IR ca be exeded coiuously o a liear, ormpreservig mappig from L IR o L. Y = L - ϕ s ϕ s] dw s IR is he for all IR, a L -limi of liear combiaios of radom variables from {W : IR}. ece, Y is a cered Gaussia process. I is easy IR o see ha i has saioary icremes. Furhermore, we obai for, Var Y = = I follows ha for all, s IR, Cov Y, Ys = Var = Var = Var s s + x x Y Y Y + Var + Var + Var ] ds + s ds ] dx +. Y s Y s Y s Var Var Var Y Y s Y s ] Ys ] ]

12 4 Chaper. Prelimiaries = ece, for ] + x x dx + + s s. c = ] + x x +, c Y is a cered Gaussia process wih covariace.. I coras IR o his, Madelbro ad Va Ness 968 chose o se c = Ɣ +. We could ow apply he Kolmogorov-Česov Theorem compare Theorem..8 of Karazas ad Shreve 988 o obai a coiuous modificaio of c Y IR. Bu we ca also prove ha c Y has a coiuous modificaio by showig ha almos surely, IR ϕ s ϕ s] dw s ca IR be udersood as a improper Riema-Sieljes iegral for all IR. To prove his we eed he followig lemma, which follows from Theorem. of Wheede ad Zygmud 977 ad Remark o page 3 of he same book. Lemma. Le a, b] be a fiie ierval, f Ca, b] ad φ C a, b]. The he Riema-Sieljes iegral RS- b a φsdfs exiss ad equals b Ra f sφ sds + f bφb f aφa, where R- b a f sφ sds is he Riema iegral. Proposiio.3 Le,,, ad le W IR be a wo-sided Browia moio. The here exiss a measurable wih P ] = such ha for each ω he improper Riema-Sieljes iegral irs- ϕ s ϕ s] dw s ω IR exiss for all IR ad is coiuous i. If we se { Z irs- ω := IR ϕ s ϕ s] dw s ω if ω if ω c, Z IR is a coiuous modificaio of Y IR. ece, c Z IR is a fracioal Browia moio.

13 .. Fracioal Browia moio 5 Proof. I follows from he law of he ieraed logarihm see e.g. Theorem.9.4 of Karazas ad Shreve 988 ha here exiss a measurable se wih P ]= such ha for all ω, W ω lim =..5 log Furhermore, i follows from Theorem.9.5 of Karazas ad Shreve 988 ha for all IN, here exiss a measurable se wih P ]= such ha for all ω ad all, ], lim s W ω W s ω =..6 s log s We se = =. I is clear ha P ] =. We assume >. For, he proof is aalogous. Le us firs rea he case,. I follows from Lemma. ha for each ω ad all x,, x ϕ sdw s ω = x R- W s ω s 3 ds + x Wx ω RS- Sice lim x x W x ω = ad he improper Riema iegral x ir- W s ω s 3 ds = lim R- W s ω s 3 ds x exiss, he improper Riema-Sieljes iegral x irs- ϕ sdw s ω = lim RS- ϕ sdw s ω x exiss oo ad equals ir- W s ω s 3 ds...7 To show ha..7 is coiuous i we se for >, f ω s =,]sw s ω s 3, s IR,

14 6 Chaper. Prelimiaries ad observe ha for all T >, he family fω is uiformly iegrable,t wih respec o Lebesgue measure. Therefore, he -coiuiy of..7 follows from a geeralized versio of Lebesgue s Domiaed Covergece Theorem see e.g. Theorem II.6.4.b of Shiryaev 984. Lemma. implies ha for all x >, x RS- x ϕ s ϕ s] dw s ω = R- x + + x x I follows from..6 ha lim x x ] W s ω s 3 s 3 ds ] ] W ω + x x W x ω x + x x ] W ω =. x Moreover, for all x >, + x x x 3. This ogeher wih..5 implies ha lim x + x x ] W x ω =. Furhermore, i follows from..5 ha he improper Riema iegral ] ir- W s ω s 3 s ds = lim R- x x x ] W s ω s 3 s 3 ds exiss. ece, he improper Riema-Sieljes iegral irs- ϕ s ϕ s] dw s ω = lim RS- x ϕ s ϕ s] dw s ω x x

15 .. Fracioal Browia moio 7 exiss oo ad equals ir- ] W s ω s 3 s 3 ds, which is coiuous i by Lebegue s Domiaed Covergece Theorem. irs- ϕ s ϕ s] dw s ω ca ow be defied as irs- I is coiuous i because ϕ s ϕ s] dw s ω + irs- irs- are. Also for,, we defie ϕ sdw s ω. ϕ s ϕ s] dw s ω as ϕ sdw s ω ad irs- irs- ϕ s ϕ s] dw s ω irs- ϕ s ϕ s] dw s ω + irs- ϕ sdw s ω. I ca be deduced from Lemma. ha for each ω ad all x,, RS- x ϕ sdw s ω = RS- x s dws ω W ω = x R- W s ω W ω s 3 ds + x Wx ω W ω + W ω. I follows from..6 ha x Wx ω W ω x,

16 8 Chaper. Prelimiaries ad ha he improper Riema iegral W s ω W ω s 3 ds = lim x R- x W s ω W ω s 3 ds exiss. Therefore he improper Riema-Sieljes iegral ir- irs- exiss oo ad equals x ϕ sdw s ω ir- W s ω W ω s 3 ds + W ω, Tha W ω is coiuous i is clear. The -coiuiy of W s ω W ω s 3 ds ca as before be derived from Theorem II.6.4.b of Shiryaev 984. As i he case,,wehaveforallx >, ϕ sdw s ω = lim x RS- ir- x RS- x ϕ s ϕ s] dw s ω = R- x + + x x As before, x ] W s ω s 3 s 3 ds ] W x ω lim + x x lim x x + x x ] + x x W x ω ] W ω =, x ] W x ω = ad he improper Riema iegral ] ir- W s ω s 3 s 3 ds

17 .. Fracioal Browia moio 9 ] = lim R- x W s ω s 3 s 3 ds x x exiss. ece, he improper Riema-Sieljes iegral irs- exiss oo ad equals ϕ s ϕ s] dw s ω = lim RS- x ϕ s ϕ s] dw s ω x ir- x ] W s ω s 3 s 3 ds, which is coiuous i by Lebegue s Domiaed Covergece Theorem. To show ha Z IR is a modificaio of Y we se for all IR, IR ad for all IN ad s IR, Sice lim f, he same ime f, s = ϕ s ϕ s, s IR, f, s = k= f, = f, i L, Y Z ω = lim k + k, k+ ]s. IR = L -lim f, sdw s ω IR f, sdw s. A for all ω. ece, for all IR, Z is measurable ad Z = Y almos surely. I ca be deduced from.. ha fracioal Browia moios divide io hree differe families. B has idepede icremes. For, ], he covariace bewee wo icremes over o-overlappig ime-iervals is posiive, for, i is egaive. From he represeaios.. ad..3 i ca be see ha fracioal Browia moio has saioary icremes. Furhermore, i ca easily be checked ha B is sochasically self-similar wih self-similariy parameer, i.e. for all a >, a B a IR has he same disribuio as B IR.

18 Chaper. Prelimiaries Figure.: Simulaio of a ypical pah of fracioal Browia moio for =., =.5 ad =.8 Le X be a sochasic process wih saioary icremes. We say ha he icremes of X exhibi log-rage depedece if for all h >, Cov Xh X, X h X h =. = I ca be derived from.. ha for,, ] ad fixed h >, Cov Bh lim, B +h B = h. This implies ha he icremes of B exhibi log-rage depedece if ad oly if,. I he followig lemma we collec some facs abou fracioal Browia moio ha we will eed hroughou he hesis. They are already well-kow. Lemma.4 Le B be a fracioal Browia moio for some, ], ad T, p, q >. The:

19 .. Fracioal Browia moio a For all γ<here exis a cosa δ ad a almos everywhere posiive radom variable ξ such ha ] B P ω : sup ω Bu ω,u,t ]; u γ δ = < u<ξω b p j= B B p B j+ j T T E p ] T i L c p q j= B B p j+ j T T i L d p +q j= B B p j+ j T T i probabiliy, i.e. for all L >, here exiss a such ha for all, P p +q j= B B p ] j+ j T T < L < L Proof. a follows from he Kolmogorov-Česov Theorem see e.g. Theorem..8 of Karazas ad Shreve 988. To prove b we recall ha he sequece B j+t B jt is saioary. j= Sice i is Gaussia ad Cov BT B, B j+t B j jt, i is also mixig. ece, he Ergodic heorem see e.g. Theorem V.3.3 of Shiryaev 984 implies B j+t B jt p B p] E T j= i L...8 O he oher had, i follows from he self-similariy of B ha for all, has he same disribuio as p j= B j+ T B j T B j+t B jt p. j= This ogeher wih..8 implies b. c follows immediaely from b. p

20 Chaper. Prelimiaries To prove d we choose L >. I follows from b ha here exiss a IN such ha P B E p] T p B j+ B p j T T > p] B E T < L j= for all. This implies ha for all, P p B j+ B p j T T < B E p] T < L or, equivalely, P j= p +q j= B j+ T B j T p < q B E p] T < L. This shows ha here exiss a IN such ha P p +q B j+ B p j T T < L < L j= for all, ad d is proved..3 Weak semimarigales The classical oio of a semimarigale sads a he ed of a chai of geeralizaios of Browia moio, each of which exeded he class of sochasic processes ha ca play he role of he iegraor i sochasic iegraio i he Iô-sese see Iô 944 for Iô s cosrucio of he sochasic iegral. I reached is fial form i Doléas-Dade ad Meyer 97. I heir paper a sochasic process X ha is adaped o a filraio lf = F saisfyig he usual assumpios is called a lf-semimarigale if i admis a decomposiio of he form X = X + M + A,.3. where X is a F -measurable radom variable, M = A =, M is a a.s. righ-coiuous local marigale wih respec o lf ad A a a.s. righcoiuous, lf-adaped fiie variaio process. Laer i was foud ha if for

21 .3. Weak semimarigales 3 T,, a filraio lf = F,T ] saisfies he usual assumpios, a a.s. righ-coiuous, lf-adaped sochasic process X,T ] is of he form.3. if ad oly if X fulfils he followig codiio: I X β lf is bouded i L,.3. where β lf = g j j, j+ ] : IN, < < T, j= j, g j is F j -measurable ad g j a.s. }.3.3 ad I X ϑ = g j X j+ X j for ϑ = g j j, j+ ] β lf. j= This resul is usually referred o as he Bicheler-Dellacherie heorem see e.g. Secio VIII.4 of Dellacherie ad Meyer 98 for a proof. For our purposes i is more coveie o work wih codiio.3. ha wih he decomposiio propery.3.. If oe does o require he process o be a.s. righ-coiuous ad he filraio o saisfy he usual assumpios, oe obais a weaker form of he semimarigale propery ha he classical oe. Defiiio.5 A sochasic process X,T ] is a weak semimarigale wih respec o a filraio lf = F,T ] if X is lf-adaped ad saisfies.3.. Le X,T ] be a sochasic process. If lf = F,T ] ad lf = F are wo filraios wih F,T ] F for all, T ],heβ lf β lf. ece, L -boudedess of I X β lf implies L -boudedess of I X β lf. This shows ha if X is o a weak semimarigale wih respec o he filraio geeraed by X, he i is o a weak semimarigale wih respec o ay oher filraio. Therefore i is aural o iroduce he followig defiiio. Defiiio.6 Le X,T ] be a sochasic process. We call X a weak semimarigale if i is a weak semimarigale wih respec o lf X. We call X a semimarigale if i is a semimarigale wih respec o lf X. j=

22 4 Chaper. Prelimiaries Example.7 I is easy o see ha he deermiisic process { for, ] X = for, ], is a weak semimarigale. Bu i is o a semimarigale because i is o a.s. righ-coiuous. owever, he followig proposiio shows ha every a.s righ-coiuous lfweak semimarigale is also a lf-semimarigale. Proposiio.8 Le lf = F,T ] be a filraio. The every sochasically righ-coiuous lf-weak semimarigale is also a lf-weak semimarigale. I paricular, if X is a.s. righ-coiuous, i is a lf-semimarigale. Proof. Defie lf = F N he ull ses of F T,T ] ad se F as follows: Le F T be he compleio of F T, = σ F N,, T ]. Le, T ] ad g L F such ha g almos surely. We se A = {g > E g F ]} ad B = {g < E g F ]}. Sice F = {G : F F such ha G F N }, here exis Ã, B F wih A Ã, B B N. The equaliies g E g F ] dp = g E g F ] dp = ad A B g E g F ] dp = imply P A] = P B] =. ece, Ã B g E g F ] dp = g = E g F ] almos surely..3.4 Le X,T ] be a lf-weak semimarigale. I follows from.3.4 ha for every ϑ β lf here exiss a ϑ β lf wih I X ϑ = I X ϑ almos surely. Therefore, I X β lf = I X β lf i L.

23 .3. Weak semimarigales 5 This shows ha X is also a lf -weak semimarigale. Now le γ = g j j, j+ ] β lf. j= For all, T ], Therefore, F = Fs T. s> γ ε = g j j +ε, j+ ] is i β j= lf.3.5 for all ε wih <ε<mi j j+ j.ifx,t ] is sochasically righcoiuous, he lim I X γ ε = I X γ i probabiliy. ε This, ogeher wih.3.5 ad he fac ha I X β lf is bouded i L, implies ha I X β lf is also bouded i L, ad herefore X is a lf-weak semimarigale. I follows from Lemma.4 d ha for,, B has ifiie quadraic variaio. The ex proposiio shows ha his implies ha B cao be a weak semimarigale if,. Proposiio.9 Le X,T ] be a a.s. càdlàg process ad deoe by τ he se of all fiie pariios = < < < = T, IN, of, T ]. If X j+ X j :,,..., τ j= is ubouded i L, he X is o a weak semimarigale.

24 6 Chaper. Prelimiaries Proof. To simplify calculaios we defie Y = X X,, T ]. The Y,T ] is a lf X -adaped, a.s. càdlàg process wih Y =. I is clear ha I Y = I X ad Y j+ Y j = X j+ X j j= j= for all pariios,,..., τ. To prove he lemma we mus show ha I Y β lf X is ubouded i L.The key igredie i our derivaio of his from he L -uboudedess of Y j+ Y j :,,..., τ is he equaliy j= j= Y j+ Y j = Y T Y j Y j+ Y j,.3.6 which holds for all pariios,,..., τ. Tha Y j+ Y j :,,..., τ j= is ubouded i L meas ha c := lim sup P Y j+ Y j > L >..3.7 L τ We will deduce from his ha lim L ϑ β j= sup lf X P I X ϑ > L] c 4,.3.8

25 .3. Weak semimarigales 7 which implies L -uboudedess of I Y β lf X. To do his we choose L >. Sice Y is a.s. càdlàg, sup,t ] Y < almos surely. Therefore here exiss a N > suchha ] P sup Y > N < c,t ] implies ha here exiss a pariio,,..., τ wih P Y j+ Y j > LN + N > c..3. j= I follows from.3.9 ad.3. ha { } P sup Y > N Y,T ] j+ Y j LN + N j= ] P sup Y > N + P Y j+ Y j LN + N < c,t ] 4. j= ece, { } P sup Y N Y,T ] j+ Y j > LN + N > c 4. j=.3. I is clear ha ϑ = { } Y j N Y j N j, j+ ] is i β lf X ad i ca be see from.3.6 ha o he eve { } sup Y N Y,T ] j+ Y j > LN + N, j=

26 8 Chaper. Prelimiaries we have I Y ϑ = N > N Y j+ Y j Y T j= Togeher wih.3., his implies ha LN + N N = L. P I Y ϑ > L] > c 4. Sice L was chose arbirarily, his shows.3.8, ad he proposiio is proved. Corollary. B,T ] is o a weak semimarigale if,. Proof. I follows from Lemma.4 d ha j= B j+ B j T T i probabiliy. This implies ha B j+ B j : IN T T j= is ubouded i L. Proposiio.9. For, Sice B is coiuous, he corollary follows from, a direc proof of he fac ha B is o a weak,t ] semimarigale seems o be difficul. Bu Proposiio.8 permis us o use already exisig resuls o classical semimarigales. Proposiio. Le X,T ] be a a.s. righ-coiuous process such ha P X,T ] is of fiie variaio ] <.3. ad, for all ε>, here exiss a pariio = < < < = T, IN,

27 .3. Weak semimarigales 9 wih ad max j+ j <ε.3.3 j P X j+ X j >ε <ε..3.4 j= The X is o a weak semimarigale. Proof. Suppose X is a weak semimarigale. By Proposiio.8, X is also a lf X -semimarigale. ece, X is of he form X = X + M + A, where X is a F -measurable radom variable, M = A =, M is a a.s. righ-coiuous local marigale wih respec o lf ad A a a.s. righcoiuous, lf-adaped fiie variaio process. I follows from.3.3,.3.4 ad Theorem II. of Proer 99 ha X, X] = X a.s.,, T ]. ece, M, M] = a.s.,, T ]. Therefore, Theorem II.7 of Proer 99 implies M = a.s.,, T ]. ece, X is a fiie variaio process. This coradics.3.. Therefore X cao be a weak semimarigale. Corollary. B,T ] is o a weak semimarigale if,. Proof. I follows from Lemma.4 d ha B j+ B j T j= T i probabiliy. Therefore, here exiss a sequece k k= of aural umbers such ha k j= B j+ k T B j k T k almos surely.

28 Chaper. Prelimiaries ece, P B,T ] O he oher had, Lemma.4 c shows ha j= B j+ B T T j T ] is of fiie variaio =. il. ece, B saisfies he assumpios of Proposiio.. Therefore,T ] i is o a weak semimarigale..4 The marke Throughou his hesis we will 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 bough ad sold a he same price. We assume ha moey i he moey marke accou evolves accordig o a sochasic process S ad he sock price follows a sochasic,t ] process S. Sice we wa o use S as a uméraire, we require i,t ] o be posiive. By S we deoe he discoued sock price S/ S. To make clear how derivaive prices deped o he explici modellig of S, S, we will aalyse he price of a Europea call opio o he sock. Such a opio is specified by is mauriy T ad he srike price K. I has a radom pay-off a ime T which is give by + S T K. The firs coiuous-ime sochasic model for a fiacial asse appeared i he hesis of Bachelier 9. e proposed modellig he price of a sock as follows: S = S + µ + σ B, where S,µad σ are cosas ad B is a Browia moio. The drawbacks of his model are ha S ca become egaive ad he relaive reurs are lower for higher sock prices.

29 .4. The marke Samuelso 965 iroduced he more realisic model { } S = S exp µ σ + σ B,.4. where S,µ ad σ are cosas ad B is a Browia moio. Black ad Scholes 973 oiced ha if S is as i.4. ad here is a cosa r such ha S = expr, he he pay-off of a Europea call opio o S ca be replicaed by coiuous radig i S ad S, ad hey derived a explici formula for he price of such a opio. owever, he Samuelso model also has deficiecies ad up o ow here have bee may effors o build beer models. Culad e al. 995 discuss he empirical evidece ha suggess ha log-rage depedece should be accoued for whe modellig sock price movemes ad prese a fracioal versio of he Samuelso model. For cosas S >,ν,σ >adr, we call S =, S = S + ν + σ B,, T ],.4. he fracioal Bachelier model ad S = expr, S = S exp {r + ν} + σ B,, T ],.4.3 he fracioal Samuelso model or, aleraively, he fracioal Black-Scholes model.

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31 Chaper Arbirage i fracioal Browia moio models. Iroducio I Secio.3 we showed ha for,,, B is o,t ] a weak semimarigale. I paricular, i is o a lf B -semimarigale, eiher is S = S/ S i he models.4. ad.4.3. Therefore, i follows immediaely from Theorem 7. of Delbae ad Schachermayer 994 ha.4. ad.4.3 admi a free luch wih vaishig risk cosisig of simple predicable iegrads adaped o lf B. Rogers 997, Shiryaev 998 ad Salopek 998 eve give arbirage sraegies for fracioal Browia moio models. Rogers 997 cosrucs arbirage for he fracioal Bachelier model.4.. is sraegy cosiss of a combiaio of buy ad hold sraegies ad works for all urs parameers,,. owever, as selfsimilariy of S is esseial for is cosrucio, Rogers arbirage oly exiss i he case ν =, i.e. S = S + σ B. Moreover, Rogers models S for, ] ad o geerae a profi o he ime ierval,, his arbirage eeds o kow he whole hisory of S from ime uil he prese. I Shiryaev 998 oly he case, is reaed. A iegral wih respec o B is defied ad i is idicaed how i ca be show ha for regular 3

32 4 Chaper. Arbirage i fbm models eough fucios F, he modified Iô formula df, B = F, B d + F, B db..,, holds. Usig his for he fracioal Bachelier model.4. wih oe ca choose a c > adse ϑ = c ν + σ B c S ν + σ B, ϑ = c o obai ϑ S + ϑ S = ϑ S + ϑ S + ν + σ B ϑu d S u = c ν + σ B. ece, if coiuous adjusme of he porfolio is allowed, ϑ,ϑ is a selffiacig arbirage sraegy for he fracioal Bachelier model. For he fracioal Samuelso model.4.3 wih,, oe ca se for all c >, ϑ = c S exp ν + σ B, ϑ = c exp ν + σ B. I follows from.. ha ϑ S + ϑ S = ϑ S + ϑ S + ϑu d S u + ϑu d S u = c S expr exp ν + σ B, which shows ha ϑ,ϑ is a self-fiacig arbirage sraegy for he fracioal Samuelso model. More geerally, i is show i Salopek 998 ha if a sochasic process X is almos surely coiuous ad of bouded p-variaio for some p < his is he case for he processes S ad S i.4. ad.4.3 whe,, he for a real fucio f o IR ha is locally Lipschiz, ad a sequece of pariios = < < < J =, IN wih lim j+ j =, max j he fiie sums J j= f X j X j+ X j

33 .. The radig sraegies 5 almos surely coverge o a limi f X udx u ad f X u dx u a.s. = FX FX, where Fx = x f udu, x IR. This is used i Salopek 998 o cosruc a self-fiacig arbirage sraegy for wo fiacial asses X ad Y ha are boh almos surely coiuous, of bouded p-variaio for some p < adsuch ha X Y almos surely for all. I his chaper we cosruc arbirage sraegies for a class of fracioal Browia moio models ha coais.4. ad.4.3 for all,,, ad we show how arbirage ca be excluded from hese models by puig resricios o he class of radig sraegies. I Secio we defie he oios of free luch wih vaishig risk, arbirage ad srog arbirage. The we iroduce differe classes of radig sraegies. I Secio 3 we cosruc arbirage sraegies. As i he case of Rogers 997 our arbirage sraegies cosis of combiaios of buy ad hold sraegies. Therefore we eed o iegraio heory for fracioal Browia moio. Moreover, o geerae a profi o he ime ierval, T ], our sraegies eed oly kow he hisory of S = S/ S o, T ]. owever, o perform hese sraegies i mus be allowed 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.4. ad.4.3 by iroducig a miimal amou of ime h > ha mus lie bewee wo cosecuive rasacios.. The radig sraegies I his secio he ime ierval is a arbirary closed ierval a, b]. Moey ca be ivesed i a moey marke accou where moey grows accordig o a posiive sochasic process S ad a sock whose price follows a a,b] sochasic process S. A radig sraegy is a pair ϑ = ϑ,ϑ of a,b] sochasic processes ϑ a,b] ad ϑ a,b]. ϑ S describes he moey i he moey marke accou a ime ad ϑ he umber of sock shares held a ime. ece, he evoluio of he porfolio value of a sraegy ϑ is give by Ṽ ϑ = ϑ S + ϑ S, a, b].

34 6 Chaper. Arbirage i fbm models We se V ϑ = Ṽ ϑ S = ϑ + ϑ S, a, b]. Defiiio. Le ξ be a, ]-valued radom variable wih Pξ > ] >. a A sequece of radig sraegies {ϑ} = is a ξ-flvr ξ-free luch wih vaishig risk if V ϑ b Va ϑ = ξ i probabiliy ad lim lim V ϑ b Va ϑ =. {ϑ} = is a FLVR if i is a ξ -FLVR for some, ]-valued radom variable ξ wih Pξ > ] >. b A radig sraegy ϑ is a ξ-arbirage if V ϑ b V ϑ a = ξ almos surely. ϑ is a arbirage if i is a ξ -arbirage for some, ]-valued radom variable ξ wih Pξ > ] >. c A radig sraegy ϑ is a srog arbirage if here exiss a cosa c > such ha Vb ϑ V 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 iroduce a family of σ -algebras lf = F a,b]. We assume ha a ay ime a, b], S ad S ca be observed ad o iformaio is los over ime. I oher words, lf is a filraio ad F S, S Noe ha := σ S u u,], S u u,] F for all a, b]. F S := σ S u u,] F S, S for all a, b]. Furhermore, we require S ad S o be progressively measurable wih respec o lf. This is i paricular he case whe S ad S are righ-coiuous, ad i

35 .. The radig sraegies 7 esures ha for all lf-soppig imes τ, he sopped processes S τ ad a,b S τ are also progressively measurable wih respec o lf. To cosruc a,b arbirage i fracioal Browia models of he form.4. or.4.3 i is eough o cosider combiaios of buy ad hold sraegies. We sar our discussio of differe classes of combiaios of buy ad hold sraegies by recallig he defiiio of he class SlF of simple predicable iegrads ad iroducig he class aslf of almos simple predicable iegrads. Defiiio. a SlF := {g {a} + g j τ j,τ j+ ] :, a = τ τ = b; all τ j s are lf-soppig imes; g is a real, F a -measurable radom variable; ad he oher g j s are real, F τ j -measurable radom variables} b aslf := {g {a} + g j τ j,τ j+ ] : a = τ τ b; all τ j s are lf-soppig imes; g is a real, F a measurable radom variable; he oher g j s are real, F τ j -measurable radom variables; P j such ha τ j = b ] = } c For ϑ = g {a} + g j τ j,τ j+ ] aslf we defie ϑ S := g j S τ j+ S τ j, a, b]. Noe ha his is almos surely a sum of fiiely may erms, ad he process ϑ S is progressively measurable a,b] because S a,b] is. Remark.3 For ϑ = g {a} + g j τ j,τ j+ ] aslf we ca defie he ses A = {τ < b} {τ + = b}, IN. The P ] = A =, he fucio N : IN defied by is F b -measurable ad {, ω A Nω :=, ω = A ϑ = g {a} + g j τ j,τ j+ ] = g {a} + N g j τ j,τ j+ ] almos surely. If a ivesor buys ad sells sock shares accordig o ϑ, he will almos surely carry ou oly fiiely may rasacios. Bu he does o kow from he

36 8 Chaper. Arbirage i fbm models begiig how may. Noe ha if we ake a arbirary F b -measurable fucio N : IN, a icreasig sequece of lf-soppig imes a = τ τ b, a real, F a -measurable fucio g ad real, F τ j -measurable fucios g j, j IN,he N g {a} + g j τ j,τ j+ ] = g {a} + eed o be i aslf. { j N } g j τ j,τ j+ ] Defiiio.4 { } S lf := ϑ : ϑ,ϑ SlF, as lf := { } ϑ : ϑ,ϑ aslf. Defiiio.5 Le ϑ = ϑ,ϑ as lf. There exis lf-soppig imes a = τ τ b such ha ϑ ad ϑ ca be wrie i he form ϑ = f {a} + f j τ j,τ j+ ], ϑ = g {a} + g j τ j,τ j+ ]... We se τ = a ad call ϑ self-fiacig for S, S if for all j, k =,..., j ad l, { {τ j k <τ j k+ =τ j+l <τ j+l+ } f j+l f j k S τ j + } a.s. g j+l g j k S τ j =... Noe ha he propery.. is idepede of he represeaio.. of ϑ. } S sf {ϑ lf := S lf : ϑ is self-fiacig. } as sf {ϑ lf := as lf : ϑ is self-fiacig. Proposiio.6 Le ϑ = ϑ,ϑ as lf. The he followig are equivale: i ϑ is self-fiacig for S, S a.s. = Ṽa ϑ + ϑ S ϑ + S ii iii ϑ is self-fiacig for, S iv Ṽ ϑ V ϑ a.s. = V ϑ a + ϑ S for all a, b] for all a, b]

37 .. The radig sraegies 9 Proof. Lea = τ τ b be a icreasig sequece of lf-soppig imes such ha ϑ = f {a} + f j τ j,τ j+ ], ϑ = g {a} + g j τ j,τ j+ ]. i ii: For = a, ii is rivially saisfied. So le us assume a, b]. For almos all ω, here exiss a j IN, such ha τ j,τ j+ ],ad Ṽa ϑ + ϑ S ϑ + S = = f S τ + g S τ + j i= f i S τ i+ S τ i + f j S S τ j j + g i S τi+ S τi + g j S S τ j i= j S τ i f i f i + i= j i= S τi g i g i + f j S + g j S = ϑ S + ϑ where he las iequaliy follows from i ad he fac ha f j = ϑ, g j = ϑ. ii i: Le j, k =,..., j ad l. O { τ j k <τ j k+ = τ j+l <τ j+l+ } S, we have f j+l f j k S τ j + g j+l g j k S τ j = f j+l S τ j+l+ + g j+l S τ j+l+ f j k S τ j + g j k S τ j f j+l S τ j+l+ S τ j g j+l S τ j+l+ S τ j = ϑτ j+l+ S τ j+l+ + ϑτ j+l+ S τ j+l+ ϑτ j S τ j + ϑτ j S τ j ϑa S a + ϑ a S j+l j+l a + f i S τ i+ S τ i + g i S τi+ S τi + ϑa S a + ϑ a S a + j i= i= i= j f i S τ i+ S τ i + g i S τi+ S τi a.s. =, i=

38 3 Chaper. Arbirage i fbm models where he las iequaliy follows from ii. 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.7 I follows from Proposiio.6 ha for all ϑ as sf lf, ϑ a.s. = Va ϑ ϑ + S ϑ S, a, b]...3 This shows ha if we ideify idisiguishable processes, he map ϑ = ϑ,ϑ V a ϑ,ϑ is a bijecio from as sf lf o L F a aslf. I paricular, here exiss for all ξ,ϑ L F a aslf, a uique ϑ aslf such ha ϑ = ϑ,ϑ is i as sf lf ad V ϑ a = ξ. I as sf lfs here exis so called doublig sraegies which ca creae arbirage eve i he sadard Samuelso model, where S = S exp ν + σ B,, T ], for cosas S >, ν, σ ad a Browia moio B. I was oiced by arriso ad Pliska 98 ha hey ca be ruled ou by puig a admissibiliy codiio o he radig sraegies. We use he admissibiliy codiio of Delbae ad Schachermayer 994. I is more liberal ha he oe of arriso ad Pliska 98 bu resricive eough o exclude arbirage i he Samuelso model. Defiiio.8 Le c. Wecallϑ as sf lf, c-admissible if if V ϑ Va ϑ a,b] = if ϑ S a,b] c almos surely. We call ϑ admissible if i is c-admissible for some c. { } S sf,adm lf := ϑ S sf lf : ϑ is admissible. as sf,adm lf := { ϑ as sf lf : ϑ is admissible }.

39 .3. Cosrucio of arbirage 3.3 Cosrucio of arbirage Theorem.9 Le B be a fracioal Browia moio. Le T,, ν C, T ] ad σ>. The i all four cases: i,, S = ν + σ B,, T ] ii,, S = exp ν + σ B,, T ] iii,, S = ν + σ B,, T ] iv,, S = exp ν + σ B,, T ] here exiss for every cosa c > ad all IN,aϑ SlF S such ha a P ϑ S T = c] > ad b if,t ] ϑ S. I paricular, he sraegies ϑ = ϑ, ϑ S sf,adm lf S, IN, where ϑ is give by ϑ ϑ = S ϑ S,, T ], IN, form a c-flvr. I he cases iii ad iv, ϑ ca be chose such ha also c ϑ. Theorem. I all four cases i-iv of Theorem.9 here exiss for every cosa c >, a c -admissible c-arbirage ϑ as sf,adm lf S. I he cases iii ad iv, ϑ ca be chose such ha ϑ c. I order o prove Theorems.9 ad. we eed he followig wo lemmas. Lemma. Le Z a,b] be a coiuous sochasic process. If P Z b = Z a ] =,.3. ad for all ε>here exis deermiisic imes a = < < = bsuch ha P max Z j+ Z j ε <ε,.3. a,b] j= he here exiss for all M > a γ SlF Z such ha a Pγ Z b < M] < M ad b if a,b] γ Z M.

40 3 Chaper. Arbirage i fbm models Proof. LeM >. I follows from.3. ad.3. ha here exis a ε > such ha ] P Z b Z a <ε <.3.3 M ad a pariio a = < < = b, such ha P ε max Z j+ Z j a,b] M < + M..3.4 j= Sice Z is coiuous, ξ = if a, b] : ε Z j+ Z j M + j=.3.5 we se if =b is a lf Z -soppig ime see e.g. Problem..7 of Karazas ad Shreve 988 ad.3.4 implies P ξ <b] < M..3.6 Furhermore, γ = M + Z ε M j Z a j, j+ ],ξ].3.7 is i SlF Z ad a calculaio shows ha for all a, b], γ Z = M + M Z ξ Z a Z ε j+ ξ Z j ξ. j=.3.8 This ogeher wih.3.5 implies b. From.3.8,.3.6 ad.3.3 i follows ha Pγ Z b < M] = P M + M Zξ Z ε a j= Zξ ] P Z a <ε P ξ <b] + P Z j+ ξ Z j ξ < M ] Z b Z a <ε < M. This shows a, ad he lemma is proved.

41 .3. Cosrucio of arbirage 33 Lemma. Le Z a,b] be a coiuous sochasic process. If for all L > here exis deermiisic imes a = < < = b, such ha P Z j+ Z j < L < L,.3.9 j= he here exiss for all M > a γ SlF Z such ha a Pγ Z b < M] < M, b if a,b] γ Z b M ad c γ M. Proof. Le M >. Sice Z is coiuous, ξ N = if { a, b] : Z Z a N} we se if =b.3. is for all N > alf Z -soppig ime ad {ξ N < b},asn. Therefore here exiss a N, such ha P ξ N < b] < M..3. By assumpio.3.9 here exiss a pariio a = < < = b, such ha P Z j+ Z j < N M + < M..3. j= I is easy o see ha γ = MN Z j Z a j, j+ ],ξn ] is i SlF Z ad saisfies c. As i he proof of Lemma. a calculaio shows ha for all a, b], γ Z = Z MN j+ ξ N Z j ξ N Z ξn Z a. j=.3.3

42 34 Chaper. Arbirage i fbm models This ogeher wih.3. implies b. From.3.3,.3. ad.3. follows ha P γ Z b < M ] = P Z MN j+ ξ N Z j ξ N ZξN S a < M j= P Z j+ ξ N Z j ξ N < M N + N j= P ξ N < b] + P Z j+ Z j < N M + < M. j= This shows a ad he lemma is proved. Remark.3 The coclusios of Lemmas. ad. remai rue if.3. or.3.9 are saisfied for geeral soppig imes a = τ τ = b isead of deermiisic imes a = < < = b. owever, for he proof of Theorems.9 ad. he versios wih deermiisic imes are sufficie. Proof of Theorem.9 By self-similariy of B i is eough o prove Theorem.9 for T =. i,, S = ν + σ B,, ]: I is clear ha S,] saisfies.3.. I follows from Lemma.4 a ad he fac ha ν is Lipschiz ha max,] j= S j+ S j almos surely..3.4 This shows ha S,] saisfies.3.. Thus, i follows from Lemma. ha for all IN, here exiss a γ SlF S such ha a Pγ S < c] < ad b if,] γ S. For every IN, ξ = if { : γ S = c } we se if =

43 .3. Cosrucio of arbirage 35 is a lf Z -soppig ime ad for ϑ = γ,ξ ] SlF Z we have a P ϑ S = c] > ad b if,] ϑ S. ii,, S = exp ν + σ B,, ] : I is clear ha S,] saisfies.3.. Tha S,] saisfies.3. follows from S S u max S v l S l S u, u,, ], v,] ad.3.4. Now he asserio ca be deduced from Lemma. as before. iii,, S = ν + σ B,, ]: To show ha S,] saisfies.3.9 we choose a L >. I follows from Lemma.4 c ha j= B j+ B j il. ece, j + ν ν j= ν σ B j+ j= j σ B j+ B j σ B j il. I paricular, here exiss a IN, such ha for all, P j + ν ν j= j σ B j+ σ B j > L < L. O he oher had, Lemma.4 d implies ha here exiss a IN, such ha for all, P j= σ B j+ σ B j < L < L.

44 36 Chaper. Arbirage i fbm models ece, for all max,, P S j+ S j < L P j= + ν j + ν P j= +P j + ν ν j= j= j σ B j+ σ B j+ σ B j j σ B j+ σ B j+ σ B j σ B j < L σ B j ] < L > L < L. This shows ha S,] saisfies.3.9. By Lemma. here exiss for all IN, a γ SlF Z such ha a Pγ S < c] < b if,] γ S c γ. avig show his, we ca cosruc ϑ as i i. By c we ge ϑ. iv,, S = exp ν + σ B,, ] : Sice S,] is posiive ad coiuous, mi v,] S v >. Therefore, here exiss a ε>such ha ] P mi S v ε < v,] L. I follows from wha we have show i he proof of iii ha here exiss a pariio = < < =, such ha P l S j+ l S j < ε L < L. Sice for all j, j= S j+ S j mi v,] S v l S l S j+ j,

45 .3. Cosrucio of arbirage 37 we obai P S j+ S j < L j= ] P mi S v ε + P l S j+ l S j < v,] ε L < L. j= This shows ha S,] saisfies.3.9. Thus, ϑ ca be cosruced as i iii. Agai ϑ. This complees he proof of he heorem. Proof of Theorem. Sice B is self-similar, i is eough o prove he heorem for T =. We spli, ] io he subiervals I he filra- I = a =, b = ], IN. By S we deoe he resricio of S o I ad by lf S = F S io geeraed by S. Noe ha F S F S for all IN ad I. Sice B has saioary icremes, i follows from Theorem.9 ha here exiss for all IN, a γ SlF S such ha a Pγ S b < c + c ] < b if I γ S c. For γ = γ I, = ξ = if {, ] :γ S = c } we se if = is a lf S -soppig ime. a ad b imply P ξ <] =. Therefore, ϑ = γ,ξ] belogs o aslf S ad ϑ,ϑ wih ϑ = ϑ S ϑ S,, ], is a c -admissible c-arbirage i as sf,adm lfs. I he cases iii ad iv, all γ s ca be chose such ha γ c. The ϑ c oo, ad he heorem is proved.

46 38 Chaper. Arbirage i fbm models Remarks.4. I a marke model S,T, S wih srog arbirage i is ],T ] possible o super-replicae a Europea call opio wih ime-t pay-off C T = +, S 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 oes debs 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 a Europea call opio ca have a posiive super-replicaio price if he model S,T, S oly admis ],T ] arbirage: Le, A, P be a probabiliy space wih a Browia moio B ad a idepede fracioal Browia moio B,,,. Furhermore, le ξ be a radom variable o, A, P ha is idepede of B ad B ad such ha Pξ = ] =Pξ = ] =.Ler,ν ad σ>, be cosas. The model S,T, S wih ],] { } S = expr ad S = exp r + ν + σ ξb + ξ B,, ], has arbirage bu o srog arbirage i as sf,adm lf S. Is is clear ha he superreplicaio of C wih a sraegy from as sf,adm lf S coss a leas he Black- Scholes price.. As we meioed i he iroducio, i is show i Salopek 998 ha a sochasic process Z which is almos surely coiuous ad of bouded p- variaio for some p <, ca be iegraed pah-wise wih respec o iself, ad Z u Z dz u = Z Z for all, T ]..3.5 The process.3.7, which is he buildig block for our arbirage sraegy i he cases i ad ii of Theorem.9, is a muliple of a discree versio of he iegrad i I is clear ha Theorem.9 cao oly be applied o models S,T, S ],T ]

47 .4. Exclusio of arbirage 39 wih S = ν + σ B or S = exp ν + σ B, bu o all models S, S such ha S,T ] saisfies codiios.3. ad.3. of Lemma. or codiio.3.9 of Lemma,T ],T ].. I paricular, codiio.3. is fulfilled by all processes wih vaishig quadraic variaio, ad all processes wih ifiie quadraic variaio saisfy codiio.3.9. For differe geeralizaios of Lemma.4 see e.g. Shao 996, Takashima 989 or Kôo ad Maejima 99. Shao 996 coais resuls o p-variaio of Gaussia processes wih saioary icremes. Takashima 989 gives sample pah properies of ergodic self-similar processes, ad i Kôo ad Maejima 99, resuls o ölder coiuiy of sample pahs of some self-similar sable processes ca be foud..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.5 Le lf = F,T ] be a filraio ad h >. S h lf := g {} + g j τ j,τ j+ ] SlF : τ j+ τ j + h, j. } h sf {ϑ lf := S sf : ϑ,ϑ S h lf..4. I he followig we will show ha oe of he models i-iv of Theorem.9 has a arbirage i h> h sf lf S. Lemma.6 Le,, ad B a oe-sided Browia moio. Le Z be a coiuous versio of s 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 ]

48 4 Chaper. Arbirage i fbm models Proof. Lec ad< h T. ] ] P if c = P sup Z c h,t ] h,t ] follows from he fac ha Z has he same disribuio as Z.Theorem.9.5 of Karazas ad Shreve 988 shows ha for all IN, here exiss a measurable se wih P ]=such ha for all ω ad all, ], lim s B B s =..4. s log s For = =, P ] =, ad.4. holds for all ω ad. ece, B iduces Wieer measure Q W o ˆ, B,where ω ωs ˆ = ω C, : ω =, lim s =, s log s ad B is he σ -algebra of subses of ˆ geeraed by he cylider ses. Noe ha for all ω ˆ, s dωs ca for all, be defied as a improper Riema-Sieljes iegral which is coiuous i. ece, ] ] P if Z c = Q W if s dωs c. h,t ] h,t ] Le us firs assume,. I his case we se ad m = + h + ] c + T, ω m = ω m,, T ] { A m = ω ˆ : sup ω m,t ] }. By Girsaov s Theorem here exiss a probabiliy measure Q m ha is equivale o Q W such ha ω m,t ] is a Browia moio uder Q m.iiswell kow ha Q m A m ] >. Equivalece of Q W ad Q m implies ha also Q W A m ] >..4.3

49 .4. Exclusio of arbirage 4 For all ω ˆ ad, = s dωs = ωs s 3 ds, ω m s s 3 ds + m s s 3 ds = ω m s s 3 + ds + m + For ω A m, we obai for all h, T ] he followig esimaes: ω m s s 3 ds s 3 ds ad, by our choice of m, = T m + + = + c + T c + T. h ece, I follows ha s dωsds T + c + T = c. { A m if h,t ] } s dωs c. This ad.4.3 prove he lemma for,. For,, he proof is slighly more delicae. I follows from ] Q W sup ω >,T ] ad Lemma.4 a ha here exis cosas ε, h ad δ>suchha Q W A,ε,δ ] >,

50 4 Chaper. Arbirage i fbm models where A,ε,δ= ω ˆ : We se m = + h + sup ω,t ] ad sup ω ωs,s,t ]; s < s<ε ] c + ε + δ ε + h, ω m = ω m,, T ] ad Q m as before. Furhermore, we defie A m,ε,δ= { ω ˆ : ω m A,ε,δ }. Sice ω m,t ] is a Browia moio uder Q m, Q m A m ],ε,δ = Q W A ],ε,δ >. δ ece, also Q W A m,ε,δ ] >..4.4 For ω ˆ ad h, we ca wrie s dωs = s d ωs ω] = ω ωs] s 3 ds + ω = ω m ω m s] s 3 ds + m s ds + ωm + m + = ε ω m ω m s] s 3 ds + ω m ω m s] s 3 ds + ωm + m + ε +.

51 .4. Exclusio of arbirage 43 If ω A m,ε,δad h, T ], we ca esimae he four precedig erms as follows: ε ω m ω m s] s 3 ds ε s 3 ds = ε + ε, ε ε ω m ω m s] s 3 ds δ s ds = δ ε, ωm h ad m + + = + ] c + ε + h δ ε + h ece, c + ε + δ ε + h. s dωs ε δ ε h +c+ε + δ ε + h = c. This ad.4.4 prove he lemma for,. Theorem.7 Le B be a fracioal Browia moio wih,,. LeT,, σ>ad ν :, T ] IR be a measurable fucio such ha sup,t ] ν <. Cosider he wo cases i ii S = ν + σ B,, T ] S = exp ν + σ B,, T ] If ϑ = g {} + g j τ j,τ j+ ] h> S h lf S

52 44 Chaper. Arbirage i fbm models ad here exiss a j {,..., } wih P g j ] >, he i case i, ] P ϑ S T c > for all c, ad i case ii, P ] ϑ S T < >. Proof. For oaioal simpliciy we give he proof for S = B ad S = exp B. The geeralizaios o he cases i ad ii are obvious. To prove he heorem for S = B we fix a h >, ad ake a ϑ = g {} + g j τ j,τ j+ ] S h lf B, such ha here exiss a j {,..., } wih P g j ] >. If he k = max { j {,..., } : P g j ] > }, ϑ B T = k Le c. I is clear ha k P P g j B τ j+ B τ j almos surely. k g j Bτ j+ Bτ j c.4.5 g j Bτ j+ Bτ j + sup g k h,t ] Le ˆ = ω CIR : ω = ; lim s ω ωs s log s Bτ k + B τ k c. =, IR, B he σ -algebra of subses of ˆ ha is geeraed by he cylider ses ad P he Wieer measure o ˆ, B. Wihou loss of geeraliy we ca assume ha

53 .4. Exclusio of arbirage 45 B is defied o ˆ, B, P by he improper Riema-Sieljes iegrals ] B ω = s {s } s dωs,..4.6 We defie he filraio lf ˆ = F ˆ,T by ] F ˆ {{ } ω ˆ : ωs a = σ I is clear ha lf ˆ is bigger ha he filraio lf B = give by { } F B = σ Bs : s. } : < s, a IR. F B,T ],whichis Therefore he lf B -soppig imes τ,...τ k,arealsolf ˆ -soppig imes. I he followig we spli each fucio ω ˆ a he ime poi τ k ω. Wese π ωs = ωs,τk ω]s, s IR, ad le π ωs = ωτ k ω + s ωτ k ω, s, = { } π ω IR IR : ω ˆ, B he σ -algebra of subses of ha is geeraed by he cylider ses, { } = π ω C, : ω ˆ ad B he σ -algebra of subses of ha is geeraed by he cylider ses. I ca easily be checked ha he mappig π : ˆ, B, B is F ˆ τk -measurable. O he oher had, i follows from Theorem I.3 of Proer 99 ha π ωs s is a Browia moio uder P which is idepede of F ˆ τk. I ca be see from.4.6 ha for all ω ˆ ad h, T ], k g j B τ j+ B τ j + g k B τ k + B τ k ω = U π ω,π ω

54 46 Chaper. Arbirage i fbm models where for ω, ω ad h, T ], U ω,ω = U ω + g k ω U ω + U ω, ad k U ω = g j Bτ j+ Bτ j ω, τk U ω ] ω = τ k ω + s τk ω s dω s, U ω = s dω s. Sice U h,t ] is a coiuous sochasic process o, B B, he se { } A = ω,ω : sup h,t ] U ω,ω c is B B -measurable. I follows from Proposiio A..5 of Lambero ad Lapeyre 996 ha for almos every ω ˆ, ] E A π,π F ˆ τk ω = φ π ω, where φ : IR is defied by φ ω = E A ω,π ],ω. Sice U ω is for all ω coiuous i, sup h,t ] U ω is for all ω fiie. Therefore ad sice π ω is a Browia moio uder P, i follows from Lemma.6 ha for all ω wih g k ω, ] φω = P sup h,t ] U ω,π c ] P U ω + sup g k ω U + sup g k ω U c h,t ] h,t ] >. Sice P g k π ] >,

55 .4. Exclusio of arbirage 47 we have k P g j Bτ j+ Bτ j + sup g k h,t ] Bτ k + B τ k c ]] = E A π,π ] = E E A π,π F ˆ τk = E φ π ] >. This ad.4.5 prove he heorem i he case S = B. If S = exp B, le us assume here exiss a h > ada ϑ = g {} + g j τ j,τ j+ ] S h lf B such ha ϑ S almos surely ad here exiss a j {,..., } T wih P g j ] >. If k = mi l : P g l ] > ad l g j e B τ j+ e B τ j a.s., he eiher or g = =g k = almos surely k P g j e B τ j+ e B τ j < >. I boh cases, P C] > for k C = g j e B τ j+ e B τ j, g k. Wih he same mehod ha we used i he firs par of he proof oe ca deduce from Lemma.6 ha for almos all ω C, k P g j e B τ j+ e B τ j + sup g k e B τ k + e B τ k < h,t ] F ˆ τk ω >.

56 48 Chaper. Arbirage i fbm models ece, k P k = E P k E C P P k g j e B τ j+ e B τ j < g j e B τ j+ e B τ j + sup g k h,t ] g j e B τ j+ e B τ j + sup g k h,t ] g j e B τ j+ e B τ j + sup g k e B τ k + e B τ k < h,t ] e B τ k + e B τ k < e B τ k + e B τ k < F ˆ τk ]] F ˆ τk >. This coradics our assumpio ad he heorem is proved. I follows from Theorem.7 ha i boh cases i S = ν + σ B,, T ], ad ii S = exp ν + σ B,, T ], he model S,T, S has o arbirage i h> ],T ] h sf lf S.Moreover, i case i here exis o o-rivial admissible sraegies i h> h sf lfs. A ispecio of he proof of Theorem.7 shows ha i case ii, a ϑ h> h sf lfs ca oly be admissible if ϑ is almos surely o-egaive. Clearly, he class S sf lfs is bigger ha h> h sf lf S. I is a ope problem wheher or o models of he form i ad ii have arbirage i S sf lfs or S sf,adm lf S. I follows from similar argumes o he oes i he proof of Theorem.7 ha i boh cases i ad ii he cheapes way o super-replicae a Europea call opio wih a sraegy ϑ h> h sf lfs is o buy he sock. I paricular, i boh cases i ad ii of Theorem.7 he model S,T, S ] is icomplee whe radig sraegies are resriced o h> h sf lf S.,T ]

57 Chaper 3 Regularized fracioal Browia moio ad opio pricig 3. Iroducio For simpliciy we will from ow o cosider marke models S,T, S ],T ] wih S = e r,, T ], forsomer >. I his case, lf S = lf S,adhe model is specified if he evoluio of he discoued sock price S is give. A way o make he fracioal Browia moio models S = S + ν + σ B,, T ], 3.. S = S exp ν + σ B,, T ], 3.. arbirage-free wihou resricig he radig sraegies is idicaed i he las secio of Rogers 997. Rogers 997 regularizes fracioal Browia moio by chagig he covoluio kerel ϕ..4 i he Madelbro-Va Ness represeaio..3 of fracioal Browia moio. e gives a class of fucios ϕ such ha he sochasic process R ϕ = ϕ s ϕ s] dw s,,

58 5 Chaper 3. Regularized fbm ad opio pricig is a Gaussia semimarigale wih he same log-rage depedece as fracioal Browia moio ad proposes o use such a process for modellig a discoued sock price. owever, he semimarigale propery of he process 3..3 is o eough o esure ha he model R ϕ S = S exp ν + σ ϕ,, T ], 3..4 R where S >,ν,σ > are cosas, is arbirage-free. Defiiio 3. Le C, T ], B be he space of coiuous fucios wih he σ -algebra geeraed by he cylider ses. If Y,T ] is a a.s. coiuous sochasic process, we deoe by Q Y he measure iduced by Y o C, T ], B. We call wo a.s. coiuous sochasic processes Y,T ] ad Z,T ] equivale if Q Y ad Q Z are equivale. The mai resul of his chaper is ha for a larger class of fucios ϕ ha he oe i Rogers 997, he process R ϕ, give by 3..3, is o oly,t ] a semimarigale bu also equivale o Browia moio. This implies ha he model 3..4 has a uique equivale marigale measure. ece, i is arbirage-free ad complee. I Secio we cosruc for each,, a class of processes whose fiie-dimesioal disribuios are close o hose of B ad which have a uique equivale marigale measure. I Secio 3 we use hese processes o build regularized fracioal Samuelso models. Sice hese models have a uique equivale marigale measure, opio prices ca be obaied by calculaig codiioal expecaios. We discuss he pricig of a Europea call opio i such a framework. 3. Regularizig fracioal Browia moio 3.. Geeral idea I his subsecio we give some heurisic argumes ha idicae why for,,, he behaviour of he fucio ϕ..4 ear zero is resposible for he exisece of arbirage i he models 3.., 3.. ad how ϕ ca be regularized o yield a process ha ca be used o build a arbirage-free sock price model wih log-rage depedece. The arbirage sraegies i Secio.3 cosis of combiaios of buy ad hold sraegies ha ac o ever smaller ime iervals. For,,hey

59 3.. Regularizig fracioal Browia moio 5 exploi he fac ha B has ifiie quadraic variaio. For, hey use ha B is a o-cosa process wih vaishig quadraic variaio. To exclude hese arbirage sraegies we vary he local pah behaviour of fracioal Browia moio i such a way ha we obai a process wih o-zero, fiie quadraic variaio. To skech how his ca be achieved we firs show ha he quadraic variaio of B ] B is relaed o he rae of covergece of E o, as. Sice fracioal Browia moio has saioary icremes, we have for all ads, E Bs+ B s ] = E For,, we ge for every pariio = < < = T, of, T ], he esimae E B j B j = ] B =. j j max j j j T. ] This shows ha E B j B j coverges o zero as he grid size of he pariio goes o zero. ece, B has vaishig quadraic variaio for,. O he oher had, if,,he E B j T B T j T =, for. This idicaes ha B has ifiie quadraic variaio, for,. We have show his rigorously i he proof of Lemma.4. To see which par of he fucio ϕ..4 accous for he behaviour of B ] E for small >, we fix a small δ>, ad wrie = c δ = E ] B = c ϕ s ϕ s] ds ϕ s ϕ s] ds + c δ ϕ s ϕ s] ds

60 5 Chaper 3. Regularized fbm ad opio pricig If =,he If δ, δ,,he ϕ s ϕ s] ds = ϕ s ϕ s] ds =. δ + x x ] dx ] x 3 dx = δ δ. 3.. This shows ha for all,, forsmall >, he esseial coribuio B ] o E comes from he erm c δ ϕ s ϕ s] ds. ece, he behaviour of he fucio ϕ ear zero deermies he rae of covergece of E B ] o, as. To chage B io a process wih similar disribuio bu o-zero, fiie quadraic variaio, we vary ϕ i a eighbourhood of zero so ha he resulig fucio ϕ saisfies δ ϕ s ϕ s] ds, as, where we wrie for wo fucios f ad g, f g, as,ifhere f exiss a cosa c, such ha lim g = c. To give a cocree example for he sor of fucios we have i mid we se for,, a IR ad b >, ϕ a,b { a + x := ϕ b a b x x, b] ϕ x x, b,. As ϕ, he fucios ϕ a,b IR ϕ a,b saisfy s ϕa,b s ] ds <, for all IR.