, 1) for i =1, 2,...,m,andwe let B (H) (t) =(B (H) + t 2Hi s t 2Hi} δ ij (1.2)

Transcription

1 MTHODS AND APPLICATIONS OF ANALYSIS. c 23 Internationa Press Vo. 1, No. 3, pp , September 23 1 MINIMAL VAIANC HDGING FO FACTIONAL BOWNIAN MOTION FANCSCA BIAGINI AND BNT ØKSNDAL Abstract. We discuss the extension to the muti-dimensiona case of the Wic-Itô integra with respect to fractiona Brownian motion, introduced by [6 in the 1-dimensiona case. We prove a mutidimensiona Itô type isometry for such integras, which is used in the proof of the muti-dimensiona Itô formua. The resuts are appied to study the probem of minima variance hedging in a maret driven by fractiona Brownian motions. 1. Introduction. In the foowing we et H = (H 1,H 2,...,H m be an m- dimensiona Hurst vector with components H i ( 1 2, 1 for i =1, 2,...,m,andwe et B (H (t =(B (H 1 (t,...,b m (H (t be an m-dimensiona fractiona Brownian motion (fbm with Hurst parameter H. This means that B (H (t =B (H (t, ω; t, ω Ω is a continuous Gaussian stochastic process on a fitered probabiity space (Ω, F (H t,µ with mean [B (H (t = = B (H ( for a t (1.1 and covariance [B (H i (sb (H j (t = 2{ 1 s 2H i + t 2Hi s t 2Hi} δ ij (1.2 δ ij = { if i j 1 if i = j ; i i, j m, = µ denotes the expectation with respect to the probabiity aw µ of B (H (. In other words, B (H (t consists of m independent 1-dimensiona fractiona Brownian motions with Hurst parameters H 1,...,H m, respectivey. If H i = 1 2 for a i, then B (H (t coincides with cassica Brownian motion B(t. We refer to [11, [13 and [18 for more information about 1-dimensiona fbm. Because of its properties (persistence/antipersistence and sef-simiarity fbm has been suggested as a usefu mathematica too in many appications, incuding finance [1. For exampe, these features of fbm seem to appear in the og-returns of stocs [18, in weather derivative modes [3 and in eectricity prices in a iberated eectricity maret [2. In view of this it is of interest to deveop a powerfu cacuus for fbm. Unfortunatey, fbm is not a semimartingae nor a Marov process (uness H i = 1 2 for a i, so these theories cannot be appied to fbm. However, if H i > 1 2 then the paths have zero quadratic variation and it is therefore possibe to define a pathwise integra, denoted by f(t, ωδb (H (t, eceived January 28, 23; accepted for pubication August 13, 23. Department of Mathematics, University of Boogna, Piazza di Porta S. Donato, 5, I 4127 Boogna, Itay (biagini@dm.unibo.it. Department of Mathematics, University of Oso, Box 153 Bindern, N-316 Oso, Norway (osenda@math.uio.no; Norwegian Schoo of conomics and Business Administration, Heeveien 3, N-545 Bergen, Norway. 347

2 348 F. BIAGINI AND B. ØKSNDAL by a cassica resut of Young from See [12 and the references therein. This integra wi obey Stratonovich type (i.e. deterministic integration rues. Typicay the expectation of such integras is not and it is nown ([12, [15, [16, [19 that the use of these integras in finance wi give marets with arbitrage, eveninthe most basic cases. In fact, this unpeasant situation (from a modeing point of view occurs whenever we use an integration theory with Stratonovich integration rues in the generation of weath from a portfoio. See e.g. the simpe exampes of [4 and [19. Because of this and for severa other reasons it is natura to try other types of integration with respect to fbm. Let L 1,2 f(, : Ω such that f L 1,2 [ f 2 := L 1,2 be the set of (measurabe processes <, ( f(sf(t(s, tds dt + In [6 a Wic-Itô type of integra is constructed, denoted by f(t, ωdb (H (t, 2 D t f(tdt. (1.3 B (H (t is a 1-dimensiona fbm with H ( 1 2, 1. This integra exists as an eement of L 2 (µ for a (measurabe processes f(t, ω such that f L 1,2 <. Here, and in the foowing, (s, t = H (s, t =H(2H 1 s t 2H 2 ; (s, t 2 1, 2 <H<1 (1.4 and D t F = (s, td s Fds (1.5 denotes the Maiavin -derivative of F (see [6, Definition 3.4. If f(t, ω isastep process of the form n f(t, ω = f i (ωx [ti,t i+1(t, t 1 <t 2 < <t n+1, (1.6 and f L 1,2 <, then the integra is defined by f(t, ωdb (H (t = n f i (ω (B (H (t i+1 B (H (t i, (1.7 denotes the Wic product. We have the foowing basic properties of the Wic-Itô integra: [ f(t, ωdb (H (t = for a f L 1,2 (1.8 [( ( f(t, ωdb (H (t g(t, ωdb (H (t = ( f,g for a f,g L 1,2 L 1,2 ( f,g L 1,2 [ = (1.9 ( ( f(sg(t(s, tds dt + D t f(tdt D t g(tdt. (1.1

3 MINIMAL VAIANC HDGING FO FACTIONAL BOWNIAN MOTION 349 See [6 for detais and proofs. This Wic-Itô fractiona cacuus was subsequenty extended to a white noise setting and appied to finance in [9. Later this white noise theory was generaized to a H (, 1 by [7. A the above papers [6, [9 and [7 ony dea with the 1-dimensiona case. In Section 2 of this paper we discuss the extension of this integra to the m-dimensiona case, i.e. we discuss the integra f(t, ωdb (H (t = f i (t, ωdb (H i (t for f =(f 1,...,f m L 1,2 (m B (H (t =(B (H 1 (t,...,b m (H (t is m-dimensiona fbm, =( H1,..., Hm and L 1,2 (m is the corresponding cass of integrands (see (2.5 beow. We prove the m-dimensiona anaogue of the isometry (1.9, which turns out to have some unexpected features (see Theorem 2.1. By combining the muti-dimensiona fractiona Itô formua (Theorem 2.6 with Theorem 2.1 we obtain another fractiona Itô isometry (Theorem 2.7. Finay, we end Section 2 by proving a fractiona integration by parts formua (Theorem 2.9 and Theorem 2.1. In Section 3 we appy the above resuts to study the probem of minima variance hedging in a (possiby incompete maret driven by m-dimensiona fbm. Here we use fractiona mathematica maret mode introduced by [9 and by [7. For cassica Brownian motions (and semimartingaes this probem has been studied by many researchers. See for exampe the survey [17 and the references therein. It turns out that for fbm this probem is even harder than in the cassica case and in this paper we concentrate on a specia case in order to get more specific resuts. 2. Muti-dimensiona Wic-Itô integration with respect to fbm. Let B (H (t =(B (H 1 (t,...,b m (H (t; t, ω Ωbem-dimensiona fbm with Hurst vector H =(H 1,...,H m ( 1 2, 1m, as in Section 1. Since the B (H ( are independent, we may regard Ω as a product Ω = Ω 1 Ω 2 Ω m of identica copies Ω of some Ω and write ω =(ω 1,...,ω m Ω. Let F = F (m,h be the σ-agebra generated by {B (H (s, ; s, =1, 2,...,m} and et F t = F (m,h t be the σ-agebra generated by {B (H (s, ; s t, = 1, 2,...,m}. IfF :Ω is F-measurabe, 1 m, weset D,t F = (s, td,t Fdt (if the integra converges (2.1 =( 1,..., m (2.2 (s, t = H (s, t =H (2H 1 s t 2H 2 ; (s, t 3, =1, 2,...,m (2.3 and D,t F = F ω (t, ω is the Maiavin derivative of F with respect to ω,at(t, ω (if it exists. Let B = B( denote the Bore σ-agebra on. Simiary to the 1-dimensiona case we can define the muti-dimensiona fractiona Wic-Itô integra f(t, ωdb (H (t = =1 f (t, ωdb (H (t L 2 (µ (2.4

4 35 F. BIAGINI AND B. ØKSNDAL for a B F-measurabe processes f(t, ω =(f 1 (t, ω,...,f m (t, ω m such that f L 1,2 < for a =1, 2,...,m, [ ( 2 f L 1,2 := f (sf (t (s, tds dt + D,t f (tdt. (2.5 Denote the set of a such m-dimensiona processes f by L 1,2 (m. As in the 1- dimensiona case we obtain the isometries [( 2 f db (H = f L 1,2 ; =1, 2,...,m. (2.6 This is intuitivey cear, since we (by independence of B (H 1,...,B m (H can treat the remaining stochastic variabes ω 1,...,ω 1,ω +1,...,ω m as parameters and repeat the 1-dimensiona approach in the ω variabe. It is aso easy to prove (2.6 rigorousy by writing f (t, ω 1,ω 2,...,ω m as a imit of sums of products of functions depending ony on (t, ω and ony on (ω 1,...,ω 1,ω +1,...,ω m, respectivey. In view of this it is cear that if f =(f 1,...,f m L 1,2 (m, then the Wic-Itô integra (2.4 is we-defined as an eement of L 2 (µ and by (2.6 we have fdb (H L2 f L 1,2. (2.7 (µ =1 It is usefu to have an expicit expression for the norm on the eft hand side of (2.7. The foowing formua is our main resut of this section: L 1,2 Theorem 2.1 (Muti-dimensiona fractiona Wic-Itô Isometry I. Let f,g (m. Then ( f,g L 1,2 (m [( fdb (H ( gdb (H = ( f,g (2.8 L 1,2 (m [ m = f (sg (t (s, tds dt + =1,=1 ( D,t f (tdt ( D,t g (tdt. (2.9 emar. Note the crossing of the indices, of the derivatives and the components f,g in the ast terms of the right hand side of (2.9. To prove Theorem 2.1 we proceed as in [6, but with the appropriate modifications: In the 1-dimensiona case, et L 2 be the set of deterministic functions α : such that (α, α := α 2 := α(sα(t (s, tds dt <. (2.1

5 MINIMAL VAIANC HDGING FO FACTIONAL BOWNIAN MOTION 351 If α L 2 then ceary α L 1,2. Hence we can define the Wic (or Doeans-Dae exponentia ( ( (α = exp α(tdb (H (t =exp α(tdb (H (t 1 2 α 2. (2.11 See e.g. [6, (3.1 or [9, xampe 3.1. Simiary, in the mutidimensiona case we put =( 1,..., m and we et L 2 be the set of a deterministic functions α =(α 1,...,α m : m such that α L 2 for =1,...,m.Ifα L 2 we define the corresponding Wic exponentia ( ( m (α = exp α(tdb (H (t =exp ( m =exp α 2 = =1 =1 =1 α (tdb (H (t α (tdb (H (t 1 2 α 2, (2.12 α (sα (t (s, tds dt = Let be the inear span of a (α; α L 2. Then we have and α 2. (2.13 Theorem 2.2. ([6, Theorem 3.1 is a dense subset of L p (F,µ, for a p 1. Theorem 2.3. ([6, Theorem 3.2 Let g i =(g i1,...,g im L 2 for i =1, 2,...,n such that =1 g i g j if i j, =1,...,m. (2.14 Then (g 1,...,(g n are ineary independent in L 2 (F,µ. If F L 2 (F,µandg L 2 we put, as in [6, D,Φ(g F = D,t F g (tdt. (2.15 We ist some usefu differentiation and Wic product rues. The proofs are simiar to the 1-dimensiona case and are omitted. Lemma 2.4. Let f =(f 1,...,f m L 2, g =(g 1,...,g m L 2 (.Then (i D,Φ(g f idb (H i =(f,g, =1,...,m, (f,g = f (sg (t (s, tds dt ; =1,...,m, (2.16 ( (ii D,s f idb (H i = f (u (s, udu ; =1,...,m,

6 352 F. BIAGINI AND B. ØKSNDAL (iii D,Φ(g (f =(f (f,g ; =1,...,m, (iv D,s (f =(f f (u (s, udu ; =1,...,m, (v (f (g =(f + g (vi F g db (H = F g db (H D,Φ(g F, =1,...,m, provided that F L 2 (F,µ and D,Φ(g F L 2 (F,µ. (vii [(f (g = exp(f,g. We now turn to the muti-dimensiona case. We wi prove Lemma 2.5. Suppose α L 2, β L 2, D,Φ(β F L 2 (µ and D,Φ(α G L 2 (µ. Then [( ( F α db (H G β db (H = [ (D,Φ(β F (D,Φ(α G+δ FG(α,β, (2.17 δ = { 1 if = otherwise Proof. We adapt the argument in [6 to the muti-dimensiona case: First note that by a density argument we may assume that { } F = (f = exp f(tdb (H (t 1 2 f 2 and { } G = (g = exp g(tdb (H (t 1 2 g 2, for some f L 2, g L2. Choose δ = (δ 1,...,δ m m, γ = (γ 1,...,γ m m and put δ f = (δ 1 f 1,...,δ m f m andγ g =(γ 1 g 1,...,γ m g m. Then by Lemma 2.4 [((f (δ α ((g (γ β (2.18 = [(f + δ α (g + γ β = exp(f + δ α, g + γ β { m } =exp (f i + δ i α i (s(g i + γ i β i (t i (s, tds dt. (2.19 We now compute the doube derivatives 2 δ γ of (2.18 and (2.19 at δ = γ =. We distinguish between two cases:

7 MINIMAL VAIANC HDGING FO FACTIONAL BOWNIAN MOTION 353 Case 1. Then if we differentiate (2.18 we get 2 [ ((f (δ α ((g (γ β δ γ δ=γ= = [( (f (δ α α db (H ((g (γ β γ δ=γ= [( ( = (f α db (H (g β db (H. (2.2 On the other hand, if we differentiate (2.19 we get 2 [ exp(f + δ α, g + γ β δ γ δ=γ= = [ exp(f + δ α, g + γ β α (s(g + γ β (t (s, tds dt γ =exp(f,g α (sg (t (s, tds dt β (sf (t (s, tds dt =exp(f,g (α,g (β,f δ=γ= = [(f (β,f (g (α,g = [ D,Φ(β (f D,Φ(α (g. (2.21 This proves (2.17 in this case. Case 2. =. In this case, if we differentiate (2.18 we get 2 [ ((f (δ α ((g (γ β δ γ δ=γ= = [( (f (δ α α db (H ((g (γ β γ δ=γ= [( ( = (f α db (H (g β db (H. (2.22 On the other hand, if we differentiate (2.19 we get 2 [ exp(f + δ α, g + γ β δ γ δ=γ= = [ exp(f + δ α, g + γ β α (s(g + γ β (t (s, tds dt γ [ =exp(f,g (α,g (β,f + α (sβ (t (s, tds dt δ=γ= = [ D,Φ(β (f D,Φ(α (g+(f(g(α,β. (2.23 This proves (2.17 aso for Case 2 and the proof of Lemma 2.5 is compete. We are now ready to prove Theorem 2.1:

8 354 F. BIAGINI AND B. ØKSNDAL Proof. We may consider f (tdb (H (t as the imit of sums of the form N f (t i (B (H (t i+1 B (H (t i when t i = t i+1 t i, t 1 < t 2 < < t N, N = 2, 3,... Hence [ ( [( fdb(h 2 2 = f db (H is the imit of sums of the form i,j,, =1 [ (f (t i (B (H (t i+1 B (H (t i (f (t j (B (H (t j+1 B (H (t j, which by Lemma 2.5 is equa to i,j,, [( ti+1 t i D,t f (t i dt ( tj+1 When t i this converges to [ m,=1 ( t j D,t f (t j dt ( D,t f (tdt D,t f (tdt +δ t i+1 t i =1 t j+1 t j + f (t i f (t j (s, tds dt. f (sf (t (s, tds dt. (2.24 This proves (2.9 when f = g. By poarization the proof of Theorem 2.1 is compete. Using Theorem 2.1 we can now proceed as in the 1-dimensiona case ([6, Theorem 4.3, with appropriate modifications, and obtain a fractiona muti-dimensiona Itô formua. We omit the proof. Theorem 2.6 (The fractiona muti-dimensiona Itô formua. Let X(t = (X 1 (t,...,x n (t, with dx i (t = j=1 σ ij (t, ωdb (H j (t ; σ i =(σ i1,...,σ im L 1,2 (m ; 1 i n. (2.25 Suppose that for a j =1,...,m there exists θ j > 1 H j such that sup [(σ ij (u σ ij (v 2 C u v θj if u v <δ (2.26 i δ> is a constant. Moreover, suppose that im u,v t u v { sup [(D,u {σ ij(u σ ij (v} 2 =. (2.27 i,j, Let f C 1,2 ( n with bounded second order derivatives with respect to x. Then,

9 MINIMAL VAIANC HDGING FO FACTIONAL BOWNIAN MOTION 355 for t>, t f t f(t,x(t = f(,x( + (s, X(sds + s t { m 2 f + (s, X(s x i x j i,j=1 = f(,x( + + t t =1 m f s (s, X(sds + n f x i (s, X(sdX i (s } σ i (sd,s (X j(s ds (2.28 j=1 t [ n f (s, X(sσ ij (s, ω x i db (H j Tr [ Λ T (sf xx (s, X(s ds. (2.29 Here Λ= [ Λ ij n m with Λ ij (s = σ i D,s (X j(s ; 1 i n, 1 j m, (2.3 =1 [ 2 f f xx = x i x j 1 i,j n n n (2.31 and ( T denotes matrix transposed, Tr[ denotes matrix trace. If we combine Theorem 2.6 with Theorem 2.1 we get the foowing resut, which aso may be regarded as a fractiona Itô isometry: Theorem 2.7 (Fractiona Itô isometry II. Suppose f =(f 1,...,f m L 1,2 (m. Then, for T>, [( T [ T = ( D T,t f (tdt { f (t t D,t f (tdt D,t f (sdb (H (s+f (t T } D,t f (sdb (H (s dt (s (2.32 Proof. By the Itô formua (Theorem 2.6 we have [( T [ T = ( f db (H T [ T = [ T + δ { f (td,t { f (t t t ( t f db (H f (sdb (H (s + f (td,t D,t f (sdb (H (s+f (t t ( t } f (sdb (H (s dt } D,t f (sdb (H (s dt {f (tf (s+f (tf (s} (s, tds dt, (2.33 we have used that, for u>, ( D u u u,t f (sdb (H (s = D,t f (sdb (H (s+δ f (s (t, sds. (2.34

10 356 F. BIAGINI AND B. ØKSNDAL (See [6, Theorem 4.2. On the other hand, the Itô isometry (Theorem 2.1 gives that [( T ( f db (H T f db (H [( T = D,t f (tdt ( T D,t f (tdt + δ f 2. (2.35 Comparing (2.33 and (2.35 we get Theorem 2.7. We end this section by proving a fractiona integration by parts formua. First we reca Theorem 2.8 (Fractiona Girsanov formua. Suppose γ = (γ 1,...,γ m (L 2 ( m and ˆγ =(ˆγ 1,...,ˆγ m L 2 are reated by γ (t = ˆγ (s (s, tds ; t, =1,...,m. (2.36 Let G L 2 (µ. Then [ ( [G(ω + γ = [G(ωexp ( ω, ˆγ = G(ω ˆγdB (H. (2.37 For a proof in the 1-dimensiona case see e.g. [9, Theorem The proof in the muti-dimensiona case is simiar. If F L 2 (µ andγ =(γ 1,...,γ m (L 2 ( m the directiona derivative of F in the direction γ is defined by F (ω + εγ F (ω D γ F (ω = im, (2.38 ε ε provided the imit exists in L 2 (µ. We say that F is differentiabe if there exists a process D t F (ω =(D 1,t F (ω,...,d m,t F (ω such that D,t F (ω L 2 (dµ dt for a =1,...,m and D γ F (ω = D t F (ω γ(tdt for a γ (L 2 ( m. (2.39 Theorem 2.9 (Fractiona integration by parts I. Let F, G L 2 (µ, γ (L 2 ( m and assume that the directiona derivatives D γ F, D γ G exist. Then [ [D γ F G = F G ˆγdB (H [F D γ G. (2.4 Proof. By Theorem 2.8 we have, for a ε>, [F (ω + εγg(ω = [F (ωg(ω εγexp (ε ω, ˆγ.

11 MINIMAL VAIANC HDGING FO FACTIONAL BOWNIAN MOTION 357 Hence [ 1 [D γ F G = im ε ε {F (ω + εγ F (ω}g(ω [ 1 = im ε ε {F (ω[g(ω εγexp (ε ω, ˆγ G(ω} [ = F (ω d { ( G(ω εγexp ε ˆγdB (H 1 dε 2 ε2 } ˆγ 2 [ = F (ωg(ω ˆγdB (H [F (ωd γ G(ω ε= We now appy the above to the fractiona gradient D t F = D s F (s, tds = D,s F (s, tds = D F (ω (2.41 =1 Theorem 2.1 (Fractiona integration by parts II. Suppose F, G L 2 (µ are differentiabe, with fractiona gradients D t F, D t G. Then for each t, {1,...,m} we have [D,t F G =[F G B(H (t [F D,t G. (2.42 Proof. Choose a sequence ˆγ (j point mass at t, in the sense that if we define (j (s = ˆγ (j (s, rdr then (j ( (,tinl 2 (. Then by Theorem 2.9 [ [D,t F G = im D F G j (j [ = im F G j L 2 ; j =1, 2,..., such that im j ˆγ(j = δ t ( (the = im [D F G j (j ˆγ (j db (H [F D (j = [F G B (H (t [F D,t G. G 3. Appication to minima variance hedging. Consider the mutidimensiona version of the fractiona mathematica maret mode introduced by [9 and by [7, consisting of n + 1 independent fractiona Brownian motions B (H 1 (t,...,b m (H (t with Hurst coefficients H 1,...,H m respectivey ( 1 2 <H i < 1, as foows: (bond price ds (t =r(t, ωdt ; S ( = s, t T (3.1 (stoc prices ds i (t =µ i (t, ωdt + σ ij (t, ωdb (H j (t ; S i ( = s i, (3.2 j=1 i =1,...,n, t T.

12 358 F. BIAGINI AND B. ØKSNDAL Here r(t, ω,µ i (t, ω andσ ij (t, w aref (H t -adapted processes satisfying reasonabe growth conditions. We refer to [7, [9, [14 and [21 for a genera discussion of such marets. We say that g =(g [ 1,...,g m isanadmissibe portfoio if g(t isf (H t -adapted, gσ L 1,2 (m and T n g i(tµ i (t dt <. Here we denote by σ the voatiity matrix [σ i,j ( =σ ij (. Suppose we are ony aowed to trade in some, say, ofthe securities S,...,S n. Let K be the set of i {1,...,n} such that trading in S i is aowed. Then, according to our mode, the weath hedged by an initia vaue z and an admissibe portfoio g(t =(g i (t, ω i K up to time t is V (t =V g z (t =z + i K t g i (uds i (u ; t T. (3.3 Now et F (ω beat -caim, i.e. an F (H T -measurabe random variabe in L 2 (µ. The minima variance hedging probem is to find a z andanadmissibe portfoio g such that [(F V g z (T 2 =inf z,g [(F V g z (T 2. (3.4 This is a difficut probem even in the cassica Brownian motion setting. See e.g. [8, [17 and the references therein. For a recent genera martingae approach see [5. For fractiona Brownian motion marets a specia case is soved in [1 by using optima contro theory. Here we wi discuss the two-dimensiona case ony, and we wi simpy assume that ds (t =, ds 1 (t =db (H 1 (t and ds 2 (t =db (H 2 (t. Assume that ony trading in S and S 1 is aowed. Then the probem is to minimize [( ( T 2 J(z,g 1 = F z + g 1 ds 1 (3.5 over a z and a admissibe portfoios g 1. By the fractiona Car-Haussmann-Ocone formua ([9, Theorem 4.15 we can write T T F (ω =[F + f 1 (tdb (H 1 (t+ f 2 (tdb (H 2 (t (3.6 f i (t =Ẽ[D i,t F F (H t ; i =1, 2. Substituting this into (3.5 we get, by (1.8, [( J(z,g 1 = [F z + T [( T =([F z 2 + T (f 1 g 1 db (H 1 + (f 1 g 1 db (H 1 + T 2 f 2 db (H 2 2 f 2 db (H 2. (3.7

13 MINIMAL VAIANC HDGING FO FACTIONAL BOWNIAN MOTION 359 Hence it is optima to choose z = z := [F. The remaining probem is therefore to minimize [( T J (g 1 = T (f 1 g 1 db (H f 2 db (H 2. (3.8 From now on we assume that f 1 L 1,2 i for i =1, 2. By a Hibert space argument on L 2 (µ we see that g1 minimizes (3.8 if and ony if [( T T (f 1 g 1 db (H 1 + By Theorem 2.1 (3.9 is equivaent to [ T T ( f 2 db (H T 2 γdb (H 1 = for a adapted γ L 1,2 1. (3.9 ( T ( (f 1 (t g 1 (tγ(s 1 (s, tds dt + D T 1,t (f 1(t g 1 (tdt D 1,t γ(tdt ( T ( + D T 1,t f 2(tdt D 2,t γ(tdt = for a adapted γ L 1,2. (3.1 From this we immediatey deduce Proposition 3.1. The portfoio g 1 (t =g1(t :=f 1 (t minimizes (3.8 if and ony if T D 1,t f 2(tdt = a.s. (3.11 This resut is surprising in view of the corresponding situation for cassica Brownian motion, when it is aways optima to choose g 1 (t =g 1(t =f 1 (t. We aso get Proposition 3.2. Suppose g1(t minimizes (3.8. Then [ T (f 1 (t g1(tdt =. (3.12 Proof. This foows by choosing γ(t deterministic in (3.1.

14 36 F. BIAGINI AND B. ØKSNDAL Now assume that D 1,t (f 1(t and D 1,t (g 1(t are differentiabe with respect to D 1,s and that D 1,t f 2(t is differentiabe with respect to D 2,s for a s [,T. Then we can use integration by parts (Theorem 2.1 to rewrite equation (3.1 as foows: [ T T {(f 1 (t g 1 (tγ(s 1 (s, t+d 1,t (f 1(t g 1 (t D 1,s γ(s + D 1,t f 2(t D 2,s γ(s}ds dt with = T T [(f 1 (t g 1 (t 1 (s, tγ(s+d 1,t (f 1(t g 1 (tγ(sb (H 1 (s D 1,s D 1,t (f 1(t g 1 (tγ(s+d 1,t f 2(tγ(sB (H 2 (s D 2,s D 1,t f 2(tγ(s ds dt [ T = K(sγ(sds =, (3.13 K(s = T G(s, tdt, (3.14 G(s, t =(f 1 (t g 1 (t 1 (s, t+d 1,t (f 1(t g 1 (tb (H 1 (s D 1,s D 1,t (f 1(t g 1 (t + D 1,t f 2(tB (H 2 (s D 2,s D 1,t f 2(t. (3.15 Since γ(s isf s (H -measurabe we get from (3.13 that = = T T [K(sγ(sds = T [ γ(s[k(s F s (H ds = Since this hods for a adapted γ L 1,2 or, using (3.14, T [ [K(sγ(s F s (H ds [ T [ K(s F s (H γ(sds. (3.16 we concude that [K(s F (H s = for a.a. (s, ω. (3.17 { s [f 1 (t g 1 (t 1 (s, t+ s [D 1,t (f 1(t g 1 (tb (H 1 (s s [D 1,s D 1,t (f 1(t g 1 (t + s [D 1,t f 2(tB (H 2 (s s [D 2,s D 1,t f 2(t}dt =, (3.18 we have used the shorthand notation s [ =[ F (H s.

15 MINIMAL VAIANC HDGING FO FACTIONAL BOWNIAN MOTION 361 We have proved: Theorem 3.3. Suppose the caim F represented by (3.6 is such that D 1,s D 1,t f 1(t and D 2,s D 1,t f 2(t exist for a s, t [,T. Supposeĝ 1 (t is an adapted process in L 1,2 such that D 1,t ĝ1(t and D 1,s D 1,t ĝ1(t exist for a s, t [,T. Then the foowing are equivaent: (i ĝ 1 (t is a minima variance hedging portfoio for F, i.e. ĝ 1 (t minimizes (3.8 over a adapted g 1 (t L 1,2 (ii g 1 (t =ĝ 1 (t satisfies equation (3.18. Note that the same method aso appies if we assume a fractiona exponentia dynamics for the asset prices, which represents a more reaistic financia mode. To iustrate this resut we consider the foowing specia case: xampe 3.4. Suppose f 1 (t =and D 1,t f 2(t =h(t, a deterministic function. (3.19 We see a minima variance hedging portfoio g 1(t for the caim In this case (3.18 gets the form T F (ω = T f 2 (tdb (H 2 (t. (3.2 { s [g 1 (t 1 (s, t s [D 1,t g 1(tB (H 1 (s+ s [D 1,s D 1,t g 1(t Let us try to choose g 1 (t such that Then (3.19 reduces to + h(tb (H 2 (s}dt = for a.a. (s, ω. (3.21 T or, since g 1 is adapted, s g 1 (t 1 (s, tdt + T s D 1,t g 1(t =. (3.22 T s [g 1 (t 1 (s, tdt = B (H 2 (s h(tdt (3.23 s [g 1 (t 1 (s, tdt = B (H 2 (s T h(tdt, s [,T. (3.24 In particuar, if we choose s = T we get the equation T T g 1 (t 1 (T,tdt = B (H 2 (T h(tdt, (3.25 which ceary has no adapted soution g 1 (t. (However, it obviousy has a non-adapted soution. Therefore an optima portfoio g 1 (t =g 1(t for the caim (3.2, if it exists, cannot satisfy (3.22.

16 362 F. BIAGINI AND B. ØKSNDAL FNCS [1 F. Biagini, Y. Hu, B. Øsenda and A. Suem, A stochastic maximum principe for processes driven by fractiona Brownian motion, Stoch. Proc. and their App., 1 (22, pp [2 F.Biagini, B.Øsenda, A.Suem, N.Waner, An introduction to White noise theory and Maiavin cacuus for fractiona Brownian motion Proceedings of the oya Society of London (to appear. [3 D. Brody, J. Syroa and M. Zervos, Pricing weather derivative options, Quantitative Finance, 2 (22, pp [4 A. Dasgupta, Fractiona Brownian Motion: Its properties and appications to stochastic integration, Ph.D. Thesis, Dept. of Statistics, University of North Caroine at Chape Hi [5 G. Di Nunno, Stochastic integra representations, stochastic derivatives and minima variance hedging, Stochastics and Stochastics eports, 73 (22, pp [6 T.. Duncan, Y. Hu and B. Pasi-Duncan, Stochastic cacuus for fractiona Brownian motion, I. Theory. SIAM J. Contro Optim., 38 (2, pp [7.J. iott and J. van der Hoe, A genera fractiona white noise theory and appications to finance, Math. Finance, 13 (23, pp [8 H. Fömer and M. Schweizer, Hedging of contingent caims under incompete information, In M.H.A. Davis and. iott (eds: Appied Stochastic Anaysis, Gordon and Breach 1991, pp [9 Y. Hu and B. Øsenda, Fractiona white noise cacuus and appications to finance, Infinite dimensiona anaysis, quantum probabiity and reated topics, 6:1 (23, pp [1 B. Mandebrot, Fractas and Scaing in Finance: Discontinuity, Concentration, is, Springer-Verag, [11 B.B. Mandebrot and J.W. van Ness, Fractiona Brownian motions, fractiona noises and appications, SIAM ev., 1 (1968, pp [12. Norvai sa, Modeing of stoc price changes. A rea anaysis approach, Finance and Stochastics, 4 (2, pp [13 I. Norros,. Vaeia and J. Virtamo, An eementary approach to a Girsanov formua and other anaytic resuts on fractiona Brownian motions, Bernoui, 5 (1999, pp [14 B. Øsenda, Fractiona Brownian motion in finance, Preprint, Department of Mathematics, University of Oso 28/23. [15 L.C.G. ogers, Arbitrage with fractiona Brownian motion, Math. Finance, 7 (1997, pp [16 D.M. Saope, Toerance to arbitrage, Stoch. Proc. and their Appications, 76 (1998, pp [17 M. Schweizer, A guided tour through quadratic hedging approaches, In: Jouini., Cvitanic J., Musiea M.(eds Option pricing, interest rates and ris management.cambridge, UK:Cambridge University Press 21a, pp [18 A. Shiryaev, ssentias of Stochastic Finance, Word Scientific, [19 A. Shiryaev, On arbitrage and repication for fracta modes, In A. Shiryaev and A. Suem (eds.: Worshop on Mathematica Finance, INIA, Paris, [2 I. Simonsen, Anti-correations in the Nordic eectricity spot maret, Manuscript, NODITA Copenhagen, May 21. [21 N. Waner, Fractiona Brownian Motion and Appications to Finance, Thesis, Phiipps- Universität Marburg, March 21.