Fractional Brownian motion, random walks and binary market models

Transcription

1 Face Stochast. 5, (21) c Sprger-Verlag 21 Fractoal Browa moto, radom walks ad bary market models Tomm Sotte Departmet of Mathematcs, Uversty of Helsk, P.O. Box 4, 14 Helsk, Flad (e-mal: tomm.sotte@helsk.f) Abstract. We prove a Dosker type approxmato theorem for the fractoal Browa moto the case H > 1/2. Usg ths approxmato we costruct a elemetary market model that coverges weakly to the fractoal aalogue of the Black Scholes model. We show that there exst arbtrage opportutes ths model. Oe such opportuty s costructed explctly. Key words: Fractoal Browa moto, radom walk, stock prce model, bary market model JEL Classfcato: C6, G1 Mathematcs Subject Classfcato (1991): 6F17, 6G15, 9A9 1 Itroducto The fractoal Browa moto s a cotuous zero mea Gaussa process wth statoary cremets. The correlato of the cremets s characterzed by meas of the so-called Hurst dex, H. Ulke the stadard Browa moto the fractoal oe has log-rage depedecy property whe H > 1/2. Ths property makes the fractoal Browa moto a plausble model e.g. telecommucatos ad mathematcal face. I some emprcal studes of facal tme seres t has bee demostrated that the log-returs have ths log-rage depedece, cf. Madelbrot [7] ad Shryaev [12]. (For the use of the fractoal Browa moto telecommucato theory we refer to orros [8].) The fractoal Browa moto s ot, however, a semmartgale whe H /= 1 2. Therefore, oe may suspect that a stock prce model drve by t would admt arbtrage opportutes. Ideed, e.g. Dasgupta [4], Salopek [11] ad Shryaev [13] have costructed Mauscrpt receved: October 1999; fal verso receved: August 2

2 344 T. Sotte such opportutes by usg the stochastc tegrato wth respect to the fractoal Browa moto. Rogers [1] costructed the arbtrage by usg the path propertes of the fractoal Browa moto. I order to gve a very smple example of the arbtrage coected to the fractoal Browa moto we cosder a bary market model that appoxmates the so-called fractoal Black Scholes model,.e. a Black Scholes model where the dyamcs of the stock prces are ot gve by a stadard Browa moto, but a fractoal oe stead. To costruct the approxmatg bary market model we eed a fractoal aalogue of the Dosker s theorem. Ths theorem, or the varace prcple, states that the stadard Browa moto ca be approxmated as a radom walk cosstg of..d. radom varables. I Sect. 2 we show that the fractoal oe ca be approxmated smlarly by a dsturbed radom walk. We clarfy how to use recet represetatos of the fractoal Browa moto our approxmato. Usg the approxmato troduced Sect. 2 we costruct a fractoal bary market model Sect. 3. We show that arbtrage opportutes exst eve ths approxmatg semmartgale model. Oe such opportuty s costructed explctly. The costructo s based o the path propertes of the dsturbed radom walk. Moreover, cotrast to Rogers [1] whose costructo s based o the path formato startg from mus fty, we use formato startg from tme pot zero. 2 Fractoal Browa moto as a lmt of a radom walk 2.1 Fractoal Browa moto The fractoal Browa moto Z wth dex H (, 1) s a cotuous zero mea Gaussa process wth statoary cremets ad covarace fucto EZ t Z s = 1 ( s 2H + t 2H s t 2H ). 2 If H < 1 2 the cremets of the process are egatvely correlated. I case of H > 1 2 they are postvely correlated. Whe H = 1 2 we have the stadard Browa moto W,.e. depedet cremets. We assume that the self smlarty dex H satsfes H > 1 2. I ths case we have the followg kerel represetato of Z wth respect to the stadard Browa moto wth a determstc kerel Z t = t z(t, s) dw s (1) t z(t, s) =c H (H 1 2 )s 1 2 H u H 1 2 (u s) H 3 2 du, where c H s the ormalzg costat s

3 Fractoal Browa moto, radom walks ad bary market models 345 2H Γ ( 3 2 c H = H ) Γ (H )Γ (2 2H ). The tegral (1) s defed the pathwse sese spte of the sgularty of the kerel z at zero. Ths s possble because of the Hölder cotuty of the paths of Z ad z. For detals see [9]. We terpret z(t, s) to be zero wheever s t. 2.2 Aalogue of the Dosker s theorem Weak covergece to the fractoal Browa moto has already bee vestgated by Bera [1] ad Taqqu [14]. Ther approxmato schemes volve ormal radom varables. Dasgupta [4] proved a approxmato usg bary radom varables ad Madelbrot ad Va ess s [6] represetato of the fractoal Browa moto. Cutlad et. al. [3] also showed a result of ths kd by usg ostadard aalyss. However, usg the tegral represetato (1) we are able to provde a very smple approxmato terms of..d. square tegrable radom varables. Let W be the stadard Browa moto ad ξ ()..d. radom varables wth Eξ () = ad D 2 ξ () =1. Deote W () t := 1 t =1 ξ (), where x deotes the greatest teger ot exceedg x. By Dosker s theorem W () coverges weakly to W the Skorohod space (see e.g. [2]). Set Z () t := t z () (t, s) dw () s = t =1 z( t 1, s) ds 1 ξ (), where z s the kerel that trasforms the stadard Browa moto to a fractoal oe ad for all t the fucto z () (t, ) s a approxmato to z(t, ), vz. z () (t, s) := s s 1 z( t, u) du. Theorem 1 The radom walk Z () coverges weakly to the fractoal Browa moto. Proof The proof cosst of showg that the fte-dmesoal dstrbutos of Z () coverge to those of Z ad the showg that Z () s tght. Let us cosder the lmtg fte-dmesoal dstrbutos. For arbtrary a 1,...,a d R ad t 1,...,t d [, T ] we wat to show that Y () := d k=1 a k Z () t k

4 346 T. Sotte coverges to a ormal dstrbuto wth varace E( d k=1 a k Z tk ) 2. Let us calculate the lmtg varace of Y (). Deote (σ () ) 2 := D 2 Y (). ow (σ () ) 2 = T d a k a l k,l=1 =1 1 By the mea value theorem (2) s equal to d k,l=1 T 1 a k a l z( tk, s () =1 z( tk, s) ds z( tl 1, s) ds (2),k )z( tl, s (),l ) (3) for some s (),k, s(),l ( 1, ]. Sce the fuctos z(t, ) are cotuous ad decreasg (, T ] we obta that the er sum Formula (3) s equal to T 1 =1 z( tk, u () )z( tl, u () ) (4) for some u () [ m(s (),k, s(),l ), max(s(),k, s(),l ) ] ( 1, 1 ]. By usg the fact that the kerel z s cotuous wth respect to both argumets ad that the maps t t coverge uformly to the detty map [, T ] we see that (4) s a Rema type sum. It follows that (3), ad hece (σ () ) 2, coverges to d T d a k a l z(t k, s)z(t l, s) ds = E( a k Z tk ) 2. k,l=1 Let us ow wrte Y () as a sum. T Y () = =1 ξ () d k=1 a k 1 k=1 T z( tk, s) ds =: Y (). Ldeberg s codto s satsfed f for all ε> we have lm T 1 (σ () ) 2 =1 =1 E(Y () ) 2 1 () { Y >εσ () } =. (5) We gve a upper boud for the radom varables (Y () ) 2. By Cauchy Schwartz equalty ad the facts that z(, s) s creasg ad z(t, ) s decreasg we obta

5 Fractoal Browa moto, radom walks ad bary market models 347 (Y () ) 2 = (ξ () ) 2 ( d k=1 (ξ () ) 2 A( a k 1 1 z( tk, s) ds) 2 z(t, s) ds) 2 (ξ () ) 2 A z(t, s) 2 ds 1 (ξ () ) 2 A 1 z(t, s) 2 ds = (ξ () ) 2 Aδ (), (6) where A := ( d k=1 a k ) 2 ad δ () := 1/ z(t, s) 2 ds. We obta { { Y () >εσ ()} (ξ () Usg equalty (6) ad the cluso (7) we obta E(Y () ) 2 1 () { Y >εσ () } (σ() ) 2 E(ξ () ) 2 1 D () (ξ () ) 2 Aδ () >ε 2 (σ () ) 2} =: D () (ξ () ). (7) ) =: (σ() ) 2 Eξ 2 1 D (), where ξ := ξ (1) 1, D () := D () (ξ (1) 1 ) ad (σ() ) 2 := D 2 Y (). Usg ths upper boud to the Ldeberg s codto (5) we obta T 1 (σ () ) 2 E(Y () ) 2 1 () { Y >εσ () } =1 (σ() 1 )2 + +(σ () T )2 (σ () ) 2 Eξ 2 1 D () = Eξ 2 1 D (). Sce z(t, ) 2 s tegrable δ (), ad cosequetly Eξ 2 1 D (), teds to zero. Hece (5) holds ad the covergece of the fte-dmesoal dstrbutos follows. It remas to prove the tghtess. Let s < t be arbtrary tme pots. By usg Cauchy Schwartz equalty we obta E(Z () t t Z s () ) 2 = E( = = =1 t ( =1 t =1 t t 1 (z( t 1 1 (z( t z( t z( t, u) z( s, u) du ξ(), u) z( s, u) du)2, u) z( s, u))2 du, u) z( s, u))2 du s 2H ) 2. (8)

6 348 T. Sotte Let ow s < t < u be arbtrary. Usg Cauchy Schwartz equalty aga ad the boud (8) we obta E Z () t Z s () Z u () Z t () ( E(Z () t t u Z s () s s ) 2) 1 2 ( E(Z () H u 2H u Z t () ) 2) 1 2 t H If ow u s 1 we have E Z () t Z s () Z u () Z t () 2(u s) 2H. (9) If o the other had u s < 1 the ether s ad t or t ad u le the same subterval [ m, m+1 ) for some m. Thus the left had sde of (9) s zero. Therefore (9) holds for all s < t < u. Recallg ow that H > 1 2 ad by Theorem 15.6 of Bllgsley [2] we have the tghtess of Z (). ote that the cremets of the radom walk Z () are ot depedet. Also, ote that the approxmatg kerel z () ca be chaged to z () (t, s) := t =1 z( t, s() )1 ( 1, ](s), where s () s are real umbers belogg to the tervals ( 1, ], respectvely. Deote by X ad [X ] the jump ad quadratc varato process of a radom walk X, respectvely,.e. X t := X t lm X s ad [X ] t := ( X s ) 2. s t st Theorem 2 The process [Z () ] coverges to zero L 1 (P Leb), where Leb s the Lebesgue measure o the terval [, T ]. Proof Usg the boud (8) from the proof of the Theorem 1 we obta E( Z t () ) 2 E(Z t () Z () ) 2 2H. t 1 Hece E Z () ] t = st E( Z () t ) 2 t 2H = t 1 2H. Sce [Z () ] s a o-egatve ad creasg process we obta T E[Z () ] t dt T Recall that H > 1 2 ad the clam follows. E[Z () ] T dt T 2 1 2H.

7 Fractoal Browa moto, radom walks ad bary market models 349 Corollary 1 The processes Z () ad [Z () ] coverge to zero probablty. Cosder the process S () defed by S () t := st (1 + Z () s ). Our am s to prove that the process S () coverges weakly to the geometrc fractoal Browa moto. Sce the bg jumps of the process Z () are somewhat of a usace we shall cosder them separately. amely, defe the processes Z (1,) ad Z (2,) as follows Z (1,) t := st Z () 1 { Z () s < 1 2} (1) Z (2,) t := st Z () 1 { Z () s 1 2}. (11) I vew of Corollary 1 ad Theorem 4.1 of Bllgsley [2] the followg lemma s obvous. Lemma 1 The process Z (2,) coverges to zero probablty ad hece Z (1,) coverges weakly to Z. Theorem 3 The process S () coverges weakly to the geometrc fractoal Browa moto e Z. Proof Wrte where S () = S (1,) S (2,), S (,) t := st(1 + Z (,) ) for =1, 2 ad the processes Z (,) are as (1) (11). By Theorem 4.1 ad Problem 1 p. 28 of Bllgsley [2] t s eough to show that S (1,) coverges weakly to e Z ad S (2,) coverges to 1 probablty. Let us frst cosder the process S (2,). Let ε>. The P(sup S t (2,) 1 ε) = P(sup (1 + Z s (2,) ) 1 ε) tt tt st P(sup Z t (2,) > ) tt = P(sup Z t () > 1 2 ). tt Sce the process Z () coverges to zero probablty by Corollary 1 the covergece of S (2,) to oe probablty follows. It remas to prove that S (1,) coverges weakly to e Z. Let us frst prove that log S (1,) coverges weakly to Z. Sce Z t (1,) < 1 2 for all t T the logarthm of S (1,) s well defed. Accordg to Taylor s theorem

8 35 T. Sotte log(1 + x) =x 1 2 x 2 + r(x)x 2, where r(x) teds to zero as x teds to zero. Hece log S t (1,) = ( Z s (1,) 1 2 st ( Z (1,) s ) 2 + r( Z (1,) s )( Z (1,) s ) 2 ) = Z (1,) t 1 2 [Z (1,) ] t + st r( Z s (1,) )( Z s (1,) ) 2. ow [Z () ] coverges to zero probablty by Corollary 1. Obvously [Z (1,) ] [Z () ]. So, [Z (1,) ] also coverges to zero probablty. Sce Z t (1,) < 1 2 the remader r s uformly bouded. Hece the thrd term coverges also to zero probablty by usg the Corollary 1 aga. Hece we obta, by usg the Theorem 4.1 of Bllgley [2] ad Lemma 1, that log S (1,) coverges weakly to Z. Fally, sce the expoetal s a cotuous fuctoal ( the Skorokhod topology) the theorem follows. 3 Fractoal Browa moto ad bary market models 3.1 Fractoal Black Scholes model Cosder two assets, or securtes, that are traded cotuously over the tme terval [, T ]. Here s the curret date ad the termal date T s fxed. Deote by B the rskless asset, or bod. The dyamcs of the asset B are db t = r t dtb t, (1) where r s a determstc terest rate. The rsky asset, stock, s deoted by S ad has the dyamcs ds t =(a t dt + σdz t )S t, (2) where σ>s a costat ad Z s a fractoal Browa moto wth dex H > 1 2. The fucto a s the determstc drft of the stock. Assume that r ad a are cotuously dfferetable o the terval [, T ] the the solutos of the (stochastc) dfferetal Eqs. (1) (2) are gve by t t B t = B exp( r s ds) ad S t = S exp( a s ds + σz t ), respectvely. For detals we refer to Zähle [15].

9 Fractoal Browa moto, radom walks ad bary market models Bary market models geeral Let us brefly defe what we mea by bary market models. For a detaled treatmet of ths subject we refer to [5] ad Sect. II.1e of Shryaev [12]. Cosder a securtes market whch the two assets are traded at successve tme perods = t < t 1 < t = T. The dyamcs of the bod ad stock are ow gve as B =(1+r )B 1 (3) ad S = (a +(1+X )) S 1, (4) respectvely. Here B ad S are s the values of the bod ad stock, respectvely, over the tme terval [t, t +1 ). Smlarly r ad a are the terest rate ad the drft of the stock, respectvely, the correspodg terval. The sequece X = (X ) s a stochastc process such that at each tme pot the radom varable X has two possble values u ad d where d < u. ote that the values of u ad d may deped o the path of X up to tme 1. So the stock prce S occupes oe of the 2 states at tme. ote that all the states are ot ecessarly dfferet. However, there are 2 dfferet possble paths for the stock prce to evolve up to tme. By Proposto of Dzhapardze [5] a bary market excludes arbtrage opportutes f ad oly f for all =1,..., we have d < r a < u. (5) Ths s coected to the exstece of the so-called equvalet martgale measure the followg way. Let P be the law of X (4). We wat to fd a probablty measure Q equvalet to P such that S B s a Q-martgale. It s easy to see that such Q must satsfy Q(X = u X 1,...,X 1 )= r a d (, 1). (6) u d Obvously, the codtos (5) ad (6) are the same. Moreover, codto (6) defes a uque martgale measure,.e. the bary market models are complete. For detals, see Chap. 3 of Dzhapardze [5] 3.3 Fractoal bary market model We defe a bary model that approxmates the fractoal Black Scholes model descrbed Sect I partcular, we costruct sequeces B ( ) ad S ( ) correspodg to the bod ad stock dyamcs gve Formulas (3) (4) that coverge weakly to the dyamcs gve Formulas (1) (2) as teds to fty. We troduce some otatos. Deote

10 352 T. Sotte k(, ) :=k ( ) (, ) := 1 z(, s) ds, where z s the kerel troduced Sect. 2. Defe the stochastc process X Formula (4) by X := X ( ) := σ Z ( ) T/, where Z ( ) s the approxmato of fractoal Browa moto defed Sect. 2. The costat σ>s the volatlty of the stock the Formula (2). Let us wrte the process X by usg the kerel k. X = σ (k(, ) k( 1, )) ξ, (7) =1 where the radom varables ξ = ξ ( ) Settg are..d. ad bary,.e. for all P(ξ =1) = 1 2 = P(ξ = 1). 1 f 1 (x 1,...,x 1 )=σ (k(, ) k( 1, )) x to deote the cotrbuto of the 1 frst jumps of the radom walk ad =1 g (x) =σk(, )x to deote the cotrbuto of the last jump we ca wrte X = f 1 (ξ 1,...,ξ 1 )+g (ξ ), whe 1 usg the coverso f =. Sce the ξ s are a bary we obta u = u (ξ 1,...,ξ 1 )=f 1 (ξ 1,...,ξ 1 )+g (1) d = d (ξ 1,...,ξ 1 )=f 1 (ξ 1,...,ξ 1 )+g ( 1). Defe the determstc sequeces r ( ) ad a ( ) the Formulas (3) (4) by r ( ) := 1 r T/ ad a ( ) := 1 a T/, respectvely, where r ad a are as the fractoal Black Scholes model. I vew of Theorem 3 t s ow clear that ths model approxmates the fractoal Black Scholes model the followg sese. Theorem 4 The prce processes B ( ) ad S ( ) coverge weakly to the correspodg prce processes B ad S the fractoal Black Scholes model. Let us ow show that eve ths approxmatve model s ot free of arbtrage.

11 Fractoal Browa moto, radom walks ad bary market models 353 Theorem 5 The fractoal bary market admts arbtrage opportutes. Proof It s eough to show that codto (5) does ot hold,.e. the codto f 1 (ξ 1,...,ξ 1 )+g ( 1) < r a < f 1 (ξ 1,...,ξ 1 )+g (1) fals for some sequece (ξ 1,...,ξ ). I partcular, by symmetry t s eough to show that f 1 (1,...,1) g (1) for some 2or 1 =1 1 z(, s) z( 1, s) ds 1 z(, s) ds. (8) Let us frst gve a lower boud for the tegral term uder the sum equalty (8). Deote C := c H (H 1 2 ) ad α := H z(, s) z( 1, s) ds = C 1 s α 1 u α (u s) α 1 du ds C ( ) α { 1 ( ) ( ) α α } 1 s α α 1 s s ds C ( ) α {( ) α ( ) α } 1 1 s α ds α 1 C ( ) α {( ) α ( ) α } (9) α By usg the lower boud (9) we obta 1 =1 1 z(, s) z( 1, s) ds 1 =1 C α ( ) α {( ) α ( ) α } C 1 +1 α ( ) α 1 1 ( 1) α {( +1 ) α ( ) α } C = α α 1 ( α 1). (1) We gve a upper boud for the latter term the rght had sde of (8). =1

12 354 T. Sotte 1 z(, s) ds = C = C C α C α(α +1) s α s s α ( u α (u s) α 1 du ds ) α s (u s) α 1 du ds ( s α ) α ( ) α s ds ( ) α α 1. (11) 1 Sce (1) teds to fty ad (11) teds to α(α+1) α 1 we obta (8) for all H ad the theorem follows. ote that H grows to fty as H teds to 1 2. I partcular, whe H =.6 we have H 35. Ths estmate was obtaed by usg the bouds (9) ad (11). Let us costruct oe arbtrage opportuty explctly. Suppose frst that the dscouted drft ã := a r satsfes ã < for some H. I ths case do as follows: f the stock prce oly takes jumps dow up to tme 1, sell short M stocks ad put the moey to the bod. Sce ow u < by (8) we have MS +1 < MS (1 + ã ) < MS. Hece your wealth at tme + 1 s postve. If o the other had ã remas o-egatve, costruct the strategy as follows: f the stock prce has take oly upward jumps up to tme H 1, buy M stocks. ow d > by (8). We obta MS +1 > MS. Your wealth s postve a tme +1. Fally, ote that up to tme H there are at least two paths admttg arbtrage opportutes, vz. ( 1,..., 1) ad (1,..., 1). Sce these paths do ot deped o, provded H, we obta that # abrtrage paths lm 2 C 2 2 H 1 = 2 2 H >. 4 Coclusos Our bary approxmato of the fractoal Black Scholes model admts arbtrage opportutes. Ths s due to the log-rage depedece of the fractoal Browa moto wth H > 1 2. Heurstcally, f the stock prce had a upward tred log eough t wll keep creasg for some tme. ote that usg (5) oe ca always determe from the data whether there s a arbtrage opportuty at had or ot. I order to get a arbtrage free fractoal Black Scholes model we must troduce e.g. trasacto costs or costly formato. Oe ca also troduce a sutable predctable terest rate. For some dscusso we refer to [11].

13 Fractoal Browa moto, radom walks ad bary market models 355 Ackowledgemets. I would lke to thak Esko Valkela for suggestg the problem studed ths paper ad for hs patet gudace ad helpful commets throughout the preparato of ths work. The author was supported by the Academy of Flad project Fractal processes telecommucato, grat Refereces 1. Bera, J.: Statstcs for log-memory processes. ew York: Chapma & Hall Bllgsley, P.: Covergece of probablty measures. ew York: Chapma & Hall Cutlad,.J., Kopp, P.E., Wllger, W.: Stock prce returs ad the Joseph effect: a fractoal verso of the Black Scholes model. Progr. Probabl. 36, (1995) 4. Dasgupta, A.: Fractoal Browa moto: ts propertes ad applcatos to stochastc tegrato. Uversty of orth Carola: Ph.D. Thess Dzhapardze, K.: Itroducto to opto prcg a securtes market. CWI Syllabus 47 (2) 6. Madelbrot, B. B., Va ess, J.W.: Fractoal Browa motos, fractoal oses ad applcatos. SIAM Rev. 1, (1968) 7. Madelbrot, B.: Fractals ad scalg face, dscotuty, cocetrato, rsk. Berl Hedelberg ew York: Sprger orros, I.: O the use of the fractoal Browa moto the theory of coectoless etworks. IEEE J. Sel. Ar. Commu. 13, (1995) 9. orros, I., Valkela E., Vrtamo J.: A elemetary approach to a Grsaov formula ad other aalytcal results o fractoal Browa moto. Beroull 5(4) (1999) 1. Rogers, L. C. G.: Arbtrage wth fractoal Browa moto. Math. Face 7, (1997) 11. Salopek, D. M.: Tolerace to arbtrage. Stoch. Proc. Appl. 76, (1998) 12. Shryaev, A..: Essetals of stochastc face: facts, models, theory. Sgapore: World Scetfc Shryaev, A..: O arbtrage ad replcato for fractal models. Research Report 2, MaPhySto, Cetre for Mathematcal Physcs ad Stochastcs (1998) 14. Taqqu, M. S.: Weak covergece to fractoal Browa moto ad to the Roseblatt process. Z. Wahrsch. Verw. Geb. 31, (1975) 15. Zähle, M.: Itegrato wth respect to fractal fuctos ad stochastc calculus. Prob. Theory Rel. Felds 111, (1997)