MULTIDIMENSIONAL WICK-ITÔ FORMULA FOR GAUSSIAN PROCESSES

MULTIDIMENSIONAL WICK-ITÔ FORMULA FOR GAUSSIAN PROCESSES D. NUALART Department of Mathematics, University of Kansas Lawrence, KS 6645, USA E-mail: nualart@math.ku.edu S. ORTIZ-LATORRE Departament de Probabilitat, Lògica i Estadística, Universitat de Barcelona Gran Via 585, 87 Barcelona, Spain E-mail: sortiz@ub.edu An Itô formula for multidimensional Gaussian processes using the Wick integral is obtained. The conditions allow us to consider processes with infinite quadratic variation. As an example we consider a correlated heterogenous fractional Brownian motion. We also use this Itô formula to compute the price of an exchange option in a Wick-fractional Black-Scholes model. Keywords: Wick-Itô formula. Gaussian processes. Malliavin calculus.. Introduction The classical stochastic calculus and Itô s formula can be extended to semimartingales. There has been a recent interest in developing a stochastic calculus for Gaussian processes which are not semimartingales such as the fractional Brownian motion fbm for short. These developments are motivated by the fact that fbm and other related processes are suitable input noises in practical problems arising in a variety of fields including finance, telecommunications and hydrology see, for instance, Mandelbrot and Van Ness 7 and Sottinen 3. A possible definition of the stochastic integral with respect to the fbm is based on the divergence operator appearing in the stochastic calculus of variations. This approach to define stochastic integrals started from the Supported by the NSF Grant DMS-647

work by Decreusefond and Üstünel3 and was further developed by Carmona and Coutin and Duncan, Hu and Pasik-Duncan 4 see also Hu 5 and Nualart 9 for a general survey papers on the stochastic calculus for the fbm. The divergence integral can be approximated by Riemman sums defined using the Wick product, and it has the important property of having zero expectation. Nualart and Taqqu, have proved a Wick-Itô formula for general Gaussian processes. In they have considered Gaussian processes with finite quadratic variation, which includes the fbm with Hurst parameter H > /. The paper deals with the change-of-variable formula for Gaussian processes with infinite quadratic variation, in particular the fbm with Hurst parameter H /4, /. The lower bound for H is a natural one, see Alòs, Mazet and Nualart. The aim of this paper is to generalize the results of Nualart and Taqqu to the multidimensional case. We introduce the multidimensional Wick-Itô integral as a limit of forward Riemann sums and prove a Wick-Itô formula under conditions similar to those in Nualart and Taqqu, allowing infinite quadratic variation processes. The paper is organized as follows. In Section, we introduce the conditions that the multidimensional Gaussian process must satify and state the Itô formula. Section 3 contains some definitions in order to introduce the Wick integral. In Section 4 we prove some technical lemmas using extensively the integration by parts formula for the derivative operator. The convergence results used in the proof of the main theorem are proved in Section 5. Section 6 is devoted to study two examples related to the multidimensional fbm with parameter H > /4. Finally, in section 7 we use our Itô formula to compute the price of an exchange option on a Wick-fractional Black-Scholes market.. Preliminaries Let X = {X t, t [, T ]} be a d-dimensional centered Gaussian process with continuous covariance function matrix Rs, t, that is, R i,j s, t = E[X i sx j t ], for i, j =,..., d. For s = t, we have the covariance matrix V t = Rt, t. We denote by H be the space obtained as the completion of the set of step functions in A = [, T ] {,..., d} with respect the scalar product i [,s], j [,t] H = R i,j s, t, s, t T, i, j d,

3 where i [,s] = [,s] {i} x, k, x, k A. The mapping i [,t] Xi t can be extended to a linear isometry between the space H and the Gaussian Hilbert space generated by the process X. We denote by let I : h X h, h H this isometry. Let H m denote the mth tensor power of H, equipped with the following scalar product h h m, g g m H m = m h i, g i H, where h,..., h m, g,..., g m H. The subspace of mth symmetric tensors will be denoted by H m. In H m we introduce the modified scalar product given by, H m = m!, H m. In this way, the multiple stochastic integral I m is an isometry between H m and the mth Wiener chaos see Nualart and also Janson 6 for a more detailed discussion of tensor products of Hilbert spaces. We denote by h h m the symmetrization of the tensor product h h m. Now consider the set of smooth random variables S. A random variable F S has the form i= F = f X h,..., X h n, with h,..., h n H, n, and f Cb Rn f and all its partial derivatives are bounded. In S one can define the derivative operator D as DF = n i f X h,..., X h n h i, i= which is an element of L Ω; H. By iteration one obtains D m F = n i,...,i m= which is an element of L Ω; H m. m f x i x im X h,..., X h n h i h im, Definition.. For m, the space D m, is the completion of S with respect to the norm F m, defined by F m, = E[F ] + m E[ D i F H ]. i i=

4 The Wick product F X h between a random variable F D, and the Gaussian random variable X h is defined as follows. Definition.. Let F D, and h H. Then the Wick product F X h is defined by F X h = F X h DF, h H. Actually, the Wick product coincides with the divergence or the Skorohod integral of F h, and by the properties of the divergence operator we can write E [F X h] = E [ DF, h H ]. The Wick integral of a stochastic process u with respect to X is defined as the limit of Riemann sums constructed using the Wick product. For this we need some notation. Denote by D the set of all partitions of [, T ] π = { = t < t < < t n = T } such that π π inf D, where π = max i t i+ t i, π inf D is a positive constant. = min i t i+ t i, and Definition.3. Let u = {u t, t [, T ]} be a d-dimensional stochastic process such that u i t D, for all t [, T ] and i =,..., d. The Wick integral T u t dx t = j= T u j t dx j t is defined as the limit in probability, if it exists, of the forward Riemann sums u j t i X j t i+ X j t i j= i= as π tends to zero, where π runs over all the partitions of the interval [, T ] in the class D.

5 3. Main result We will make use of the following assumptions. Assumptions: has bounded varia- A For all j, k {,..., d} the function t V j,k t tion on [, T ]. A For all k, l {,..., d} i,j= A3 For all j, k {,..., d} i= E[ i X k j X l ], as π. sup s t E[X j s i X k ], as π, where i X j = X j t i+ X j t i, and π runs over all partitions of [, T ] in the class D. Our purpose is to derive a change-of-variable formula for the process fx t, where f : R d R if a function satisfying the following condition. A4 For every multi-index α = α,..., α d N d with α := α + + α d 7, the iterated derivatives α α f f x = x α x xα d d exist, are continuous, and satisfy [ sup E α f X t ] <. 3 t [,T ] Condition 3 holds if det V t > for all t, T ], and the partial derivatives α f satisfy the exponential growth condition α f x C T e c T x, 4 for all t [, T ], x R d, where C T > and c T are such that see Lemma 4.5. < c T < 4 inf <t T x R d, x > x T Vt x x < 5

6 Besides the multi-index notation for the derivatives, we will also use the following notation for iterated derivatives. Let fx,..., x d be a sufficiently smooth function, then i f = f x i, i {,..., d} m i,...,i m f = im im i i f, i k {,..., d}, k =,..., m. The next theorem is the main result of the paper. Theorem 3.. Suppose that the Gaussian process X and the function f satisfy the preceding assumptions A to A4. Then the forward integrals see Definition.3 j f X s dx j s, t T, j =,..., d exist and the following Wick-Itô formula holds: f X t = f X + j= j f X s dx j s + j,k= j,kf X s dv j,k s. Proof. Using the Taylor expansion of f up to fourth order in two consecutive points of a partition π = { = t < t < < t n = t} in the class D we obtain f X ti+ = f Xti + j f X ti i X j + j,kf X ti i X j i X k j= + 3! T π 3 i + 4! T π 4 i, j,k= where and T3 π i = j,k,lf 3 X ti i X j i X k i X l, j,k,l= T4 π i = j,k,l,mfx 4 i i X j i X k i X l i X m, j,k,l,m= X i = λx ti + λ X ti+, λ. By the definition of the Wick product, see Definition., one has j f X ti i X j = j f X ti i X j + D j f X ti, j δ i H,

7 where δ i = t i, t i+ ]. Taking into account that one gets j f X ti i X j = j= D j f X ti = j,kf X ti k [,t, i] k= j f X ti i X j + j= j,k= j,kf X ti k [,t i], j δ i H. Using the definition of, H and adding and subtracting E [ i X j i X k] we have k [,t i], j δ i H = E[X k t i X j t i+ X j t i ] = ϕj,k i E [ i X j i X k], where ϕ j,k i ] = E [X j ti+ X j ti X k ti+ + X k ti This gives Hence, f X ti+ = f Xti + + + i= j,k= j,k= j f X ti i X j j= j,kf X ti { i X j i X k E [ i X j i X k]} j,kf X ti ϕ j,k i + T π 3 i + T π 4 i. [ f X t = f X + f Xti+ f Xti ] = f X + + i= j,k= i= j= j,kf X ti ϕ j,k i + Rπ + 3! Rπ 3 + 4! Rπ 4, j f X ti i X j

8 where R π = i= j,k= R3 π = T3 π i = i= R4 π = T4 π i = i= j,kf X ti { i X j i X k E [ i X j i X k]} i= j,k,l= Note that j,kf X ti ϕ j,k i = j,k= i= j,k,l,m= = + = j= j= k>j= j,k= 3 j,k,lf X ti i X j i X k i X l, 4 j,k,l,mfx i i X j i X k i X l i X m. j,jf X ti ϕ j,j i + k>j= j,jf X ti V j,j t i+ V j,j t i j,kf X ti V j,k t i+ V j,k t i j,kf X ti V j,k t i+ V j,k t i. j,kf X ti ϕ j,k i Using Assumption A it is easy to show the almost sure convergence lim π i= j,k= j,kf X ti V j,k t i+ V j,k t i = j,k= j,kf X s dv j,k s as π. The convergences of R π and R3 π to zero in L Ω as π are proved in Propositions 5. and 5.. The convergence of R4 π to zero in L Ω as π is proved in Proposition 5.3. This clearly implies the convergence in probability lim π i= j= and the result follows. j f X ti i X j = j= j f X s dx j s, Remark 3.. We can also consider a function ft, x depending on time such that the partial derivative f t t, x exists and is continuous. In this case we obtain the additional term f t s, X sds. + ϕ k,j i

9 In order to prove Propositions 5., 5. and 5.3 we need to introduce some technical concepts and prove a number of lemmas. 4. Technical lemmas In this section we establish some preliminary lemmas. The first one is trivial. Lemma 4.. Let F D m+n, and h,..., h m, g,..., g n H. Then D n D m F, h h m H m, g g n H n = D m+n F, h h m g g n H m+n. The next lemmas are based on the integration by parts formula. Lemma 4.. Let F D, and h, g H. Then E [F X h X g] = E[ D F, h g H ] + E [F ] h, g H. Proof. See Nualart and Taqqu, Lemma 6. Lemma 4.3. Let F D,, h, g H, ξ = X h X g h, g H. Then E [F ξ] = E[ D F, h g H ]. Proof. It is an immediate consequence of the preceding lemma. Lemma 4.4. Let F D 4,, h, h, g, g H, ξ = X h X g h, g H and ξ = X h X g h, g H. Then E [F ξ ξ ] = E[ D 4 F, h g h g H 4 ] + E[ D F, h g H ] h, g H +E[ D F, g g +E[ D F, h g H ] h, h H + E[ D F, h h H ] g, g H H ] h, g H + E [F ] h g, h g H. Proof. Applying the last lemma with F replaced by F ξ and ξ by ξ, we get E [F ξ ξ ] = E[ D F ξ, h g H ]. Now, by the Leibniz rule for the derivative operator, where D F ξ = D F ξ + DF Dξ + F D ξ, Dξ = h X g + X h g, D ξ = h g,

and thus D F ξ = D F ξ + X g DF h + X h DF g + F h g Then, = A + A + A 3 + A 4. E [ A, h g H ] = E[ξ D F, h g H ] = E[ D D F, h g H, h g H ] = E[ D 4 F, h g h g H 4 ], where we have applied Lemmas 4.3 and 4. in the second and third equalities respectively. For the term B, we have E [ A, h g H ] = E [X g DF h, h g H ] = [X g DF, h H ] h, g H + [X g DF, g H ] h, h H = E[ D F, h g H ] h, g H + E[ D F, g g H ] h, h H. where we have used the integration by parts formula and Lemma 4.. Analogously, for A 3 we obtain E [ A 3, h g H ] = E[ D F, h h Finally, H ] g, g H + E[ D F, h g H ] h, g H. E [ A 4, h g H ] = E [F ] h g, h g H. Adding up all the terms the result follows. Lemma 4.5. The exponential growth condition 4 implies 3. Proof. The exponential growth assumption 4 implies E[ α f X t ] C T sup E[e c T X t ]. 6 t T For any symmetric and positive definite matrix A we have π e x,ax d / dx =, R A d

where A = deta. As a consequence, E[e c T X t ] = e x,ax dx = π d/ V t / R d d/ V t / A, / with and this gives A = V t c T I d = c T 4c T V t I d, E[e c T X t ] = I d 4c T V t /, provided A is symmetric and positive definite. This matrix is positive definite if and only if for all x R d with x > x T Vt I d x = x T Vt x x >, 4c T 4c T which is implied by 5. Therefore, [ E α f X t ] CT sup t T I d 4c T V t / =: a T, which is finite by condition 5. 5. Convergence Results From now on, C will denote a finite positive constant that may change from line to line. Proposition 5.. Let Then R π = Proof. Set F j,k i then i= j,k= j,kf X ti { i X j i X k E [ i X j i X k]}. = j,k fx t i and lim π E[Rπ ] =. ϕ j,k i = i X j i X k E [ i X j i X k] = X j δ i X k δ i j δ i, k δ i H. E[R π ] = i,i = j,j,k,k = E[F j,k i F j,k i ϕ j,k i ϕ j,k i ],

and by Lemma 4.4 we get the decomposition where E[F j,k i F j,k i B = E[D 4 F j,k i B = E[D F j,k i B 3 = E[D F j,k i B 4 = E[D F j,k i B 5 = E[D F j,k i B 6 = E[F j,k i ϕ j,k i ϕ j,k i ] = B + B + B 3 + B 4 + B 5 + B 6, F j,k i ], j j k k H 4, F j,k i ], j k H j, k H, F j,k i ], k k H j, j H, F j,k i ], j j H k, k H, F j,k i ], j k H j, k H, F j,k i ] j k, j k H. Notice that the terms B h, h =,..., 6, depend on the indices i, i, j, j, k, and k. We omit this dependence to simplify the notation and we set B h = i,i = j,j,k,k = so E[R π ] = 6 h= B h. We have that 4 4 B = E[D p F j,k i p D 4 p F j,k i ], j j k k H 4. p= On the other hand p D p F j,k i = u,...,u d = u + +u d =p Hence, 4 4 p B = p p= u,...,u d = Notice that B h, p! u! u d! u j,kfx ti [,t i] u d [,ti] u d. 4 p v,...,v d = u + +u d =p v + +v d =4 p E [ u j,k fx ti v j,k fx ti ] p! 4 p! u! u d! v! v d! [,t i ] u d [,ti ] u d [,ti ] v d [,ti ] v d, j j k k H 4. [,t i ] u d [,ti ] u d [,ti ] v d [,ti ] v d = w [,s ] w [,s ] w3 [,s 3] w4 [,s 4],

3 where w k {,..., d}, s k {t i, t i }, k =,..., 4. But for any s k t, w k {,..., d}, k =,..., 4, w [,s ] w [,s ] w3 [,s 3] w4 [,s, 4] j j k k H 4 wσ 4!, [,s σ] j H w σ, [,s σ] j H w σ3, [,s σ3] k H w σ4, [,s σ4] k H σ Σ 4 sup s t w d w [,s], j H w [,s], j H w [,s], k H w [,s], k H = sup E[X w s i X j ] E[X w s i X j ] E[X w s i X k ] E[X w s i X k ]. s t w d Furthermore, by Assumption A4, we have E[ u j,k fx ti v j,k fx ti ] at <. Hence, using Cauchy-Schwartz inequality, B Ca T Ca T sup i= s t j,k= w d j,k= i= sup s t w d E[X w s i X j ] E[X w s i X k ] E[X w s i X j ]. The last expression tends to zero as π by Assumption A3. Analogously B = p= p p u,...,u d = p v,...,v d = u + +u d =p v + +v d = p E[ u j,k fx ti v j,k fx ti ] p! p! u! u d! v! v d! [,t i ] u d [,ti ] u d [,ti ] v d [,ti ] v d, j k H j, k Ca T sup s t w d H E[X w s i X j ] E[X w s i X k ] E[ i X j i X k ].

4 Therefore, by Cauchy-Schwartz inequality B Ca T E[X s w i X j ] sup i= j= s t w d i,i = j,k= E[ i X j i X k ] / which tends to zero as π by Assumptions A and A3. The proof for the terms B 3, B 4 and B 5 is almost the same as for the term B. Finally, B 6 = E[F j,k i F j,k i ] j, j H k, k H +E[F j,k i F j,k i ] j, k H k, j H a T E[ i X j i X j ] E[ i X k i X k ] +a T E[ i X j i X k ] E[ i X k i X j ]. Hence, B 6 Ca T Ca T i,i = j,k= j,k= i,i = E[ i X j i X k ] E[ i X j i X k ], which tends to zero as π by Assumption A. Proposition 5.. If then Proof. Setting R π 3 = i= j,k,l= 3 j,k,lf X ti i X j i X k i X l, lim π E[Rπ 3 ] =. i X j i X k i X l = { i X j i X k E [ i X j i X k]} i X l +E [ i X j i X k] i X l,

5 one gets E[R3 π ] E i= j,k,l= +E = C + C i= j,k,l= j,k,lf 3 X ti i X l { i X j i X k E [ i X j i X k]} j,k,lf 3 X ti i X l E [ i X j i X k] To prove the convergence of C to zero, observe that C C E j,k,lf 3 X ti i X l { i X j i X k E [ i X j i X k]}. l= i= j,k= So it suffices to fix l and apply Proposition 5. with the term j,k f X t i replaced by 3 j,k,l f X t i i X l =: g X ti, X ti+ whose exact form does not matter because it satisfies the exponential condition 4. Using Lemma 4., we obtain that C = i,i = j,k,l,j,k,l = E[ 3 j,k,l fx ti 3 j,k,l fx ti i X l i X l ] E [ i X j i X k] E [ i X j i X k] = E + E, where E h = i,i = d j,k,l,j,k,l = E h, for h =,, and E = E[ D 3 j,k,l fx ti 3 j,k,l fx ti, l l δi H ] E [ i X j i X k] E [ i X j i X k], E = E[ 3 j,k,l fx ti 3 j,k,l fx ti ] l, l H E [ i X j i X k] E [ i X j i X k]. Similarly to the preceding proposition, the term E can be bounded by E[X w s i X l ] E[X w s i X l ] E Ca T sup s t w d E [ i X j i X k] E [ i X j i X k]

6 As a consequence, we obtain E Ca T E[X w s i X j ] sup i= j= s t w d i= j,k= [ E i X j i X k], where we have used the Cauchy-Schwartz inequality. This term tends to zero as π by Assumptions A and A3. For the therm E, we have E a T E [ i X l i X l] E [ i X j i X k] E [ i X j i X k], and by Cauchy-Schwartz inequality, [ E Ca T E i X k j X l] i,j= k,l= i= j,k= which converges to as π by Assumption A. [ E i X j i X k] Proposition 5.3. Let X i be a point in the straight line that joins X ti and X ti+ then R π 4 = i= j,k,l,m= 4 j,k,l,mfx i i X j i X k i X l i X m, lim E[ π Rπ 4 ] =. and Proof. We have R4 π L Ω i= j,k,l,m= i= j,k,l,m= 4 j,k,l,mfx i i X j i X k i X l i X m L Ω / E[ j,k,l,mfx 4 i ] E[ i X j i X k i X l i X m ] /. Appliying iteratively the Cauchy-Schwartz inequality one obtains E[ i X j i X k i X l / i X m ] E[ i X j 8 ] /8 E[ i X k 8 ] /8 E[ i X l 8 ] /8 E[ i X m 8 ] /8.

7 Hence, by Assumption A4 R4 π L Ω a T / E[ i X j 8 ] /8 i= j= = a T / k 8 i= C a T / k 8 4 E[ i X j ] / j= i= j= 4 E[ i X j ] 4 7 where we have used that for all p > if ξ is a centered Gaussian variable one has ξ L p Ω = κ p ξ L Ω, where κ p = Γ p+ /p, p >. π And the last term in equation 7 converges to zero as π by Assumption A. 6. Examples 6.. Correlated heterogeneous fractional Brownian motion In this section we give an example where the theory previously developed applies. Let B H = {B,H t,..., B d,h d t, t [, T ]} be a d-dimensional heterogeneous fractional Brownian motion with Hurst parameter H = H,..., H d, d and H H d. That is, B H is a d-dimensional centered Gaussian process with covariance function matrix R H s, t given by R i,j H s, t = δ ijr Hi s, t = δ ij {shi + t Hi s t Hi }, for i, j =,..., d. Set X t = AB H t, where A = a i,j i,j=,...,d is a d d matrix. We call X a correlated heterogeneous fractional Brownian motion. X is a d-dimensional Gaussian process, with the following correlation function

8 matrix R i,j s, t = E[X i sx j t ] = E = [ k= a i,k a j,k R Hk s, t. k= a i,k B k,h k s l= a j,l B l,h l t ] Proposition 6.. The process X = ABt H with /4 < min H i < satisfies i Assumptions A to A3. Therefore, the Wick-Itô formula applies to X. Proof. We have that V j,k t = R j,k t, t = a j,m a k,m t Hm, m= so Assumption A is fulfilled. Let s check Assumption A. For any k, l we have E [ i X k j X l] = R k,l t i+, t j+ R k,l t i+, t j R k,l t i, t j+ + R k,l t i, t j = a k,m a l,m R k,l H m t i+, t j+ R k,l H m t i+, t j R k,l H m t i, t j+ + R k,l H m t i, t j = m= a k,m a l,m m= t j+ t i Hm + t j t i+ Hm t j t i Hm t j+ t i+ Hm. Therefore A n = i,j= E [ i X k j X l] = B n + C n + D n,

9 where B n = a k,m a l,m t i+ t i Hm, C n = D n = i= m= n i= n 3 m= i= j=i+ a k,m a l,m t i+ t i Hm t i+ t i Hm t i+ t i+ Hm, a k,m a l,m m= t j+ t i Hm + t j t i+ Hm t j t i Hm t j+ t i+ Hm. We have B n C i= m= t i+ t i 4Hm CT π 4Hm, m= which converges to zero as π if H > /4. By a similar argument we obtain the same result for C n. The term D n is more complicated. In Nualart and Taqqu it is proved that, when j i and j i +, Then, t j+ t i Hm + t j t i+ Hm t j t i Hm t j+ t i+ Hm C j i Hm π Hm. D n C m= π 4Hm n h= k= h k 4Hm 4. As n +, one has the following asymptotics n h k 4Hm 4 h= k= Cn 4Hm if H m > 3/4 Cn ln n if H m = 3/4 n if H m < 3/4. Since our partitions are in the class D we have n C π. Therefore, C π if H > 3/4 D n C π ln π if H = 3/4. C π 4H if H < 3/4

and Assumption A is fulfilled. Finally, let us check Assumption A3. One has E [ X k t j X l] = R k,l t, t j+ R k,l t, t j = 4 C m= m= a k,m a l,m t Hm i+ thm i + t t i Hm t t i+ Hm t Hm i+ thm i + t t i Hm t t i+ Hm. As before, the convergence to zero of E [ Xt k j X l] as π is controlled by the term with H. If /4 < H /, one has that t H i+ th i and t t i H t t i+ H are both bounded by t i+ t i H, and the sum of their squares is bounded by C π 4H, which converges to zero as π. If H > /, either term is bounded by C t i+ t i. Hence, the sum of their squares is bounded by C π, which converges to zero as π. So the proof is concluded. 6.. Multimensional fractional Brownian motion The d-dimensional fractional brownian motion with Hurst parameter H, is the centered d-dimensional Gaussian process B H with the following covariance function matrix R H s, t R i,j H s, t = δ ijr H s, t = δ ij {sh + t H s t H }. Obviously, B H is the process X considered in the previous section with parameter H,..., H and A = I d, therefore we have the following result. Proposition 6.. The process B H with /4 < H < satisfies Assumptions A to A3. Therefore, the Wick-Itô formula applies to B H. 7. Application to the pricing of an exchange option The market consists in two risky assets S, S and a risk free asset B. Assume the following form for their dynamics { } St = S exp µ t + σ Xt σ V, t, S >, { } St = S exp µ t + σ Xt σ V, t, S >, B t = B exp {rt}, B >,

where X = X, X is the following correlated heterogeneous fractional Brownian X t = B,H t, X t = ρb,h t + ρ B,H t, where ρ, and H H. Note that V, t = E[ X t ] = t H, V, t = E[ X t ] = ρ t H + ρ t H, V, t = E[X t X t ] = ρt H. Suppose that H > /4, hence the Wick-Itô formula applies to X and we obtain that ds t = µ S t dt + σ S t dx t, ds t = µ S t dt + σ S t dx t. Our aim is to price at time t [, T ] the contingent claim S T S T +, which is known as an exchange option. Assume that the price process for this option has the form C t, S t, S t, where C t, x, y is a function of class C,, and satisfies the exponential growth condition 4. Then the Wick-Itô formula yields C t, St, St = C, S, S t C + u +µ C x u, S u, Su du u, S u, S u S u du + σ C t +µ u, S y u, Su S u du + σ + σ + σ +σ σ C x u, S u, S u C y u, S u, S u C x y u, S u, S u S u V, u du S u V, u du C u, S x u, Su S u dxu C u, S y u, Su S u dxu S u Su V, u du. 8 The price C t, S t, S t should coincide with the value at time t of a portfolio which replicates the contingent claim ST + S T. Let Πt denote the amuount of this portfolio invested in the risk free asset B t an h t, h t the amount of stocks S and S, respectively. Then, C t, S t, S t = Πt + h t S t + h t S t.

We will consider portfolios satisfying the following Wick self-financing type condition C t, St, St = C, S, S t + r Π u du +µ h usudu + σ h usu dxu 9 +µ h usudu + σ h usu dxu. We also suppose that the portfolio is admissible, that is, T Π t dt <, T h i t dt < and { h i usu} i is Wick forward integrable on any interval [, t], i =,. Choosing h t = C x t, S t, St and h t = C y t, S t, St and comparing equations 8 and 9 we get that C t, x, y must satisfy the partial differential equation rc = C t + r C x x + r C y y + σ with terminal condition and boundary conditions C x x V, t + σ C y y V, t C + σ σ x y C T, x, y = x y + C t,, y =, C t, x, = x. xyv, t, Reasoning as Margrabe, 8 C t, x, y is homogeneous of degree in x and y. Therefore, thanks to Euler s theorem for homogeneous functions, we have that C t, x, y C t, x, y C t, x, y = x + y x y and equation simplifies to C t + σ C x x V, t + σ C y y V, t C, + σ σ xyvt =. x y Using again the homogeneity of C t, x, y we can define Ct, z := C t, x, y /y where z = x/y and find the following partial differential equation for C C t + z {σ V, t + σv, t σ σ V, t } C =, z

3 with terminal condition and boundary condition Define C T, z = z + C t, =. θ t := σ V, t + σ V, t σ σ V, t, then the solution to equation is where C t, z = znd Nd, d := ln z + T θ s ds T t, d := d θ s ds, T θ s ds t t and N x is the N, cumulative distribution function. Finally, taking into account the values of V, t, V, t and V, t, we get C t, S t, S t = S t N d S t N d, where d and d are obtained from d and d making z = S t /S t T t θ s ds = σ + σ ρσ σ T H t H +σ ρ T H t H. and References. E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab. 9, 766 8.. P. Carmona and L.Coutin, Stochastic integration with respect to fractional Brownian motion, Ann. Inst. H. Poincaré 39, 7 68 3. 3. L. Decreusefond and A. S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Analysis, 77 4 998. 4. T. E. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion I. Theory, SIAM J. Control Optim. 38, 58 6. 5. Y. Hu, Integral transformations and anticipative calculus for fractional Brownian motions, Memoirs of the AMS 75 5. 6. S. Janson, Gaussian Hilbert Spaces Cambridge University Press, Cambridge, 997. 7. B. B. Mandelbrot and J. W.Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 4 437 968.

4 8. W. Margrabe, The value of an option to exchange one asset for another, The Journal of Finance 33, 77-86 978. 9. D. Nualart, Stochastic integration with respect to fractional Brownian motion and applications, Contemporary Mathematics 336, 3 39 3.. D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin, 6.. D. Nualart and M.S. Taqqu, Wick-Itô formula for regular processes and applications to the Black and Scholes formula, Stochastics and Stochastics Reports, to appear.. D. Nualart and M.S. Taqqu, Wick-Itô formula for Gaussian processes, J. Stoch. Anal. Appl. 4, 599 64 6. 3. T, Sottinen, Fractional Brownian motion in finance and queueing, Ph.D. Thesis, University of Helsinki 3.