Asian Option Pricing in a Lévy Black-Scholes Setting Sergio

Asian Option Pricing in a

Lévy Black-Scholes Setting

Sergio Albeverio, Eva Lütkebohmert

no. 240

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs- gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer- sität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, August 2005 Asian Option Pricing in a L´evy Black-Scholes Setting

S. Albeverio, E. L¨utkebohmert Institute for Applied Mathematics, University of Bonn

Abstract

The price of an Asian Option driven by L´evy type noise is computed. A rather explicit formula is given in the case of a Variance Gamma driving process.

Keywords: Asian Option Pricing, L´evy Processes, Bessel Processes, Variance Gamma Processes, Laplace Transformation, Esscher Transform

AMS-Classiﬁcation: 60G10, 60G51, 44A10, 60H15, 60H30, 90A09, 90A12

1 Introduction

In this paper we discuss the pricing problem of Asian options in a general L´evy Black- Scholes setting. That is we consider a stock prices process St which is driven by a L´evy process Yt with drift:

(1) dSt = σSt dYt + µSt dt, − − having a positive deterministic initial value S0 at time t0. From the Dol´eans-Dade exponential formula we derive a solution of such a stochastic diﬀerential equation which is of the form

(2) S = S exp(Y ∗), t 0 · t where Yt∗ is again a L´evy process. The generating triplet of this process Yt∗ can be derived from the generating triplet of the process Yt. We use an equivalent martingale measure, derived by means of the Esscher transform, such that the discounted stock price is a martingale and Y is still a L´evy process under this new measure. This latter property is important since we only should change the distribution of the asset price but not the property of independent and stationary increments of the driving process. Then we consider an arithmetic Asian option, the value of which at the terminal time T depends on the mean of the whole path of the price process up to time T, i.e.

1 T S dt, t < T. T t t 0 − 0 Zt0 The problem of pricing a path dependent option is to determine the option price at inception time t0. This leads to the problem of computing a discounted condition expectation of the form

T r(T t) 1 CF (K,t)= e− − EP (θ) S0 exp(Yu∗)du K t , · " T t0 t0 − + F # µ − Z ¶ ¯ ¯ ¯ 1 t being the ﬁltration underlying the process (Yt)t 0 at time t. To calculate this, we F ≥ use the method of Laplace transformation introduced by [GemYo] for the classical Black- Scholes setting. Then we can reduce the problem to the task of computing a conditional expectation of the form

λτq E e− exp Yτ∗q = s , h ¯ ³ ´ i ¯ where τq is a stopping time deﬁned by ¯

s τ = inf s 0: A := exp(Y ∗)du>q , q> 0. q ≥ s u ½ Z0 ¾

Finally, we consider the case of a Variance Gamma process Yt. In that situation we derive an explicit pricing formula in terms of an inverse Laplace transform for the price of an Asian option of the stock price process driven by a Variance Gamma process with drift. Here we use the fact that any Variance Gamma process can be expressed as a standard Brownian motion with independent gamma subordinator. Moreover we use a characterization in terms of time changed Bessel processes for the exponential of Brownian motion. This idea was ﬁrst used in [GemYo]. The inverse Laplace transform describing the Asian option price can then be computed numerically.

2 L´evy Processes

In this section we recall basic concepts and results of L´evy processes. For more details and proofs see, e.g. [Ap], [Ber], [IW], [Ru], [Sa].

d Definition 2.1 A stochastic process Xt t 0 on R is called a Lévy process if the { } ≥ following conditions are satisfied:

(i) For any choice of n 1 and 0 t < t < ... < t , the random variables ≥ ≤ 0 1 n X , X X ,...,X X are independent (independent increments); t0 t1 − t0 tn − tn−1

(ii) X0 = 0 a.s.;

(iii) The distribution of Xs t Xs does not depend on s, i.e. it equals the law of − − Y Y = Y (stationary increments); t − 0 t (iv) it is stochastically continuous, i.e. for every t 0 and ǫ> 0, ≥

lim P ( Xs Xt > ǫ)=0; s t → | − |

(v) there is an Ω with P (Ω ) = 1 such that, for every ω Ω , X (ω) is c`adl`ag, 0 ∈ F 0 ∈ 0 t i.e. X (ω) is right-continuous in t 0 and has left limits in t> 0. t ≥

A stochastic process is called a Lévy process in law, if it satisfies (i)-(iv). For any Lévy process in law there exists a unique modification which is càdlàg and which is also a Lévy process. For these concepts and results see, e.g., [Ap], [Sa], [Be]. In this case (i)-(v) hold, and we have a Lévy process.

2 2.1 L´evy-ItˆoDecomposition

Any Lévy process on Rd can be written as the linear combination of a d-dimensional Brownian motion with a deterministic constant drift and a d-dimensional jump process. This property is called the Lévy-Itôdecomposition. We will denote the drift process by γt for some γ Rd and the jump process by M . Then the Lévy process Y can be ∈ t t written in its decomposed form as

Yt = Bt + γt + Mt, where B is a d-dimensional Brownian motion with covariance matrix A Rd d. t ∈ × The value of the jump process Y just before the jump at time t > 0 is denoted by Yt := lims t Ys. The jump of Y at time t is ∆Yt = Yt Yt . Since any Lévy process − ↑ − − is stochastically continuous, one has that for a fixed time t> 0, the jump ∆Yt = 0 a.s.. The Lévy process can, in general, jump infinitely often. However, as we shall see later, there exists only a finite number of jumps of size greater than 1. Thus a Lévy process can only possess, in general, an infinite number of ”small jumps” (i.e. jumps of size less than 1 ).

To specify the jump process Mt we count the jumps of a certain size. Therefore we deﬁne for each ω Ω the counting measure ∈ J(t, A)(ω) := ♯ 0 s t : ∆Y (ω) A = 1 (∆Y (ω)). { ≤ ≤ s ∈ } A s 0 s t ≤X≤

Deﬁnition 2.2 A (Rd 0 ) is bounded from below if 0 = A. ∈ B \{ } 6

A set A (Rd 0 ) which is bounded from below satisﬁes inf x ,x A > 0 and ∈ B \{ } {| | ∈ } thus J(1, A) < a.s.. ∞

Definition 2.3 Let f be a Borel measurable function from Rd to Rd and let A be a set which is bounded from below, then for each t> 0, ω Ω, we may define the Poisson ∈ integral of f on A as a random finite sum by

f(x)J(t,dx)(ω)= f(x)J(t, x )(ω). { } A x A Z X∈ As a consequence we obtain that

xJ(t,dx)(ω)= ∆Ys(ω) A 0 s t Z ≤X≤ defines the height of all jumps of Ys(ω) up to time t. For each ω Ω,t 0, the set function A J(t, A)(ω) is a counting measure on ∈ ≥ 7→ (Rd 0 ) and hence B \{ } E [J(t, A)] = J(t, A)(ω)dP (ω) ZΩ is a Borel measure on (Rd 0 ). We define the intensity measure associated with Y by B \{ } ν := E[J(1, )]. · As already mentioned before a Lévy process can have infinitely many jumps. Thus the intensity measure associated with Y is not necessarily finite, i.e. ν(dx) = . The Rd ∞ R 3 function s Y (ω) has only a finite number of jumps greater than one on [0,t] for each 7→ s fixed ω, since the process has càdlàg paths. Hence there must be an infinite number of small jumps which cause the series of the norm of jumps, 0 s t ∆Ys , to diverge for ≤ ≤ | | all t> 0. Moreover, x <1 xν(dx)= . | | ∞ P One can show, see e.g. [Ap], that ∆Y 2 converges, such that x2ν(dx) < R 0 s t | s| Rd ∞ always holds. This implies that, for all≤ t>≤ 0, ∆Y 2 1 is locally integrable, P 0 s t | s| ∧ R and therefore x 2 1 ν(dx) < . ≤ ≤ Rd | | ∧ ∞ P ¡ ¢ R ¡ ¢ Definition 2.4 Let ν be a Borel measure defined on Rd. It is a Lévy measure if

x 2 1 ν(dx) < . Rd | | ∧ ∞ Z ¡ ¢

Since the Lévy process can have an infinite number of small jumps, in which case x <1 xJ(t,dx) | | does not necessarily exist, we cannot define the jump process Mt by Mt = Rd RxJ(t,dx). We will first split the jump process into two processes, i.e. in one process which has only R jumps larger than one, (2) Mt := xJ(t,dx), x 1 Z| |≥ and a process with jumps less than 1,

(1) ˜ Mt := x(J(t,dx) tν(dx)) =: xJ(t,dx), x <1 − x <1 Z| | Z| | with J˜(t,dx)= J(t,dx) tν(dx). The term xtν(dx) is called the compensator and is − − (1) necessary in order to ensure the existence of the integral in Mt .

Deﬁnition 2.5 For each f L1(A, ν),t 0, deﬁne the compensated Poisson integral by ∈ ≥

f(x)J˜(t,dx)= f(x)J(t,dx) t f(x)ν(dx). − ZA ZA ZA

We write the jump process of the L´evy process as

(1) (2) ˜ Mt = Mt + Mt = xJ(t,dx)+ xJ(t,dx). x <1 x 1 Z| | Z| |≥

Theorem 2.6 (Lévy-ItôDecomposition) Let Y be a Lévy process, then there exists a vector γ Rd, a Brownian motion B with covariance A Rd d and an independent ∈ ∈ × Poisson random measure J on H = (0, ) (Rd 0 ) such that for each t 0, ∞ × \{ } ≥

Yt = Bt + γt + x (J(t,dx) tν(dx)) + xJ(t,dx) x <1 − x 1 Z| | Z| |≥

4 2.2 L´evy-Khintchine Representation

Deﬁnition 2.7 The characteristic functionµ ˆ : Rd C of a probability measure µ on → Rd is deﬁned as i z,y d µˆ(z)= e h iµ(dy), z R , Rd ∈ Z where , denotes a scalar product of two vectors. h· ·i Rd The characteristic function of the distribution PY1 of a random process Yt on is denoted by

ˆ i z,y i z,Y1 Rd PY1 (z)= e h iPY1 (dy)= E e h i =: ΦY1 (z), z . Rd ∈ Z h i In this section we want to derive an expression for the characteristic function of a d- dimensional L´evy process which can be written (as we saw in the previous subsection) as a linear combination of a Brownian motion, a linear process and a jump process. Since all these processes are independent we know that the characteristic function of a L´evy process will be the product of the separate characteristic functions of each component. The characteristic function of the Brownian motion can easily be computed and is given by 1 i z,Bt t z,Az d Φ (z)= E e h i = e− 2 h i, z R . Bt ∈ h i The characteristic function of a linear process γt, γ Rd, is ∈ i z,γt 1 z,γ t E e h i = e h i . h i It remains to compute the characteristic function of the jump process

(1) (2) Mt = Mt + Mt with (1) (2) Mt := x (J(t,dx) tν(dx)) and Mt := xJ(t,dx). x <1 − x 1 Z| | Z| |≥

Theorem 2.8 Let A (Rd) be bounded from below. Then (J(t, A),t 0) is a Poisson ∈ B ≥ process with intensity ν(A).

Proof. See for example [Ap], Theorem 2.3.5, page 88. 2

(2) Lemma 2.9 The characteristic function of Mt is given by

(2) i z,Mt i z,x d Φ (2) (z)= E e = exp t e 1 ν(dx) , z R . M h i h i t Ã x 1 − ! ∈ h i Z| |≥ ³ ´

Proof. Follows from [Ap], Theorem 2.3.8, page 91. See also [Be], Lemma 2, page 11. 2

(1) Lemma 2.10 The characteristic function of Mt is given by

(1) i z,Mt i z,x d Φ (1) (z)= E e = exp t e 1 i z,x ν(dx) , z R . M h i h i t Ã x <1 − − h i ! ∈ h i Z| | ³ ´

5 Proof. Follows immediately from Lemma 2.9 and Theorem 2.3.8 in [Ap], page 91. 2

The above Lemmata show that the characteristic function of the jump process Mt = (1) (2) Mt + Mt is given by

i z,x Rd ΦMt (z) = exp t e h i 1 i z,x 1 x <1 (x) ν(dx) , z . Rd − − h i {| | } ∈ µ Z ³ ´ ¶ 2 i z,x Since the Lévy measure ν integrates ( x 1) the integral Rd e h i 1 i z,x 1 x <1 (x) ν(dx) | | ∧ − − h i {| | } exists and is finite. R ¡ ¢ The characteristic function of the Lévy process Yt is the multiplication of the characteristic functions of each independent component of the decomposition and is thus given by

i z,Yt 1 i z,x E e h i = exp t z,Az + i γ,z + e h i 1 i z,x 1 x <1 (x) ν(dx) . −2h i h i Rd − − h i {| | } h i ½ µ Z ³ ´ ¶¾ Because of the independent increments property the distribution of a Lévy process is uniquely determined by PY1 . Therefore the Lévy process Yt can be characterized by its characteristic function for t = 1. This is known as the Lévy-Khintchine representation (see, e.g. [Ap], [Sa],[Ber]).

Theorem 2.11 (Lévy-Khintchine representation) The characteristic function of the distribution of a d-dimensional Lévy process is given by (3) i z,Y1 ΦY1 (z) = E e h i h i 1 i z,x Rd = exp z,Az + i γ,z + e h i 1 i z,x 1 x <1 (x) ν(dx) , z , −2h i h i Rd − − h i {| | } ∈ · Z ³ ´ ¸ where A is a symmetric nonnegative-definite d d-matrix, ν is a measure on Rd × satisfying ν( 0 ) = 0 and ( x 2 1)ν(dx) < , and γ Rd. The representation { } Rd | | ∧ ∞ ∈ (3) by A, ν and γ is unique. The measure ν is called the Lévy measure of P and R Y1 (γ, A, ν) is called the generating triplet of PY1 .

Proof. See for example [Sa], Chapter 2.8, Theorem 8.1 (ii), page 37. 2

Example 2.12 (L´evy measure of a Variance Gamma process) The Variance Gamma process is a L´evy process Xt that has a Variance Gamma law VG(σ, ν, θ). Its characteristic function is

t 1 − ν E [exp(iuX )] = 1 iuθν + σ2νu2 , u R, θ R, σ R. t − 2 ∈ ∈ ∈ µ ¶ It can be characterized as a time changed Brownian motion with drift

Xt = θγ(t)+ σWγ(t), where W is a Brownian motion and γ is a Gamma G(1/ν, 1/ν) process with parameters 1/ν and 1/ν (see, e.g., [Ap]). The Variance Gamma process is a ﬁnite variation process, hence it is the diﬀerence of two increasing processes, and in particular (see [MaCaCh]) of two Gamma processes X = G(t; µ , γ ) G(t; µ , γ ), t 1 1 − 2 2

6 where G(t; µ, γ) denotes a Gamma process with parameters µ and γ at time t. The characteristic function can be factorized t t iu − γ iu − γ E [exp(iuX )] = 1 1+ t − ν ν µ 1 ¶ µ 2 ¶ with 1 1 2 2 2 1 1 2 2 2 ν− = θν + θ ν + 2νσ and ν− = θν θ ν + 2νσ . 1 2 2 2 − The L´evy density of³ X isp ´ ³ p ´ 1 1 exp( ν x ) for x< 0 γ x − 1| | | | and 1 1 exp( ν x) for x> 0. γ x − 2

The density of X1 is

1 1 θx 2γ 4 σ2 2 − 2 2e x 1 2 2 σ K 1 1 x θ + 2 σ2 2 γ1/γ√2πσΓ(1/2) θ2 + 2 γ − 2 σ γ Ã γ ! Ã s µ ¶! where Kα is the modiﬁed Bessel function. Stock prices given by (1) respectively (2) when driven by a Variance Gamma process get the special form t σ2ν S = S exp rt + X(t; σ, ν, θ)+ ln 1 θν . t 0 ν − − 2 µ µ ¶¶ 2 From E[eXt ] = exp t ln 1 θν σ ν , we get that S e rt is a martingale. − ν − − 2 t − The L´evy measure of³ a Variance³ Gamma´´ process is

C x(λ− λ+)/2 ν(x)= e − K1/2( x (λ λ+)/2), x 1/2 · | | − − | | where λ , λ+, C are some positive constants and K1/2 is a Bessel function of the third − 1 kind, with index 2 (see, e.g. [Ap], page 290).

Deﬁnition 2.13 The moment generating function of the d-dimensional random variable Yt is deﬁned as θ,Y1 mgfY (θ)= E eh i h i for all θ Rd, provided E e θ,Y1 < . Or equivalently, ∈ h i ∞

£ ¤ t θ,Yt mgfY (θ) = E eh i . h i t Thus mgfY (θ) is the Laplace transform of the probability distribution of Yt evaluated at θ. The following relations between the moment generating function (whenever it exists) and the characteristic function hold E ei z,Y1 = mgf (iz) = Φ (z), z Rd, h i Y Y1 ∈ E £ei z,Yt ¤ = mgf (iz)t = Φ (z), z Rd, h i Y Yt ∈ The moment generating function£ does¤ not necessarily exist. However, the above connection holds for all z Rd since the moment generating function always exists on iRd, i.e., ∈ mgf (iz) < , for all z Rd. Y ∞ ∈

7 d Theorem 2.14 (Exponential Moment) Let Yt be a L´evy process in R with generating triplet (A, γ, ν). Let

d c,x C := c R : eh iν(dx) < . ∈ x >1 ∞ ( Z| | )

(i) The set C is convex and contains the origin.

(ii) c C if and only if E e c,Yt < for some t > 0 or, equivalently, for every ∈ h i ∞ t> 0. £ ¤ (iii) If z Cd is such that Re z C, then ∈ ∈

1 z,x Ψ(z)= z,Az + γ,z + eh i 1 z,x 1 x 1 (x) ν(dx) 2h i h i Rd − − h i {| |≤ } Z ³ ´ is deﬁnable, E e z,Yt < , and h i ∞

£ ¤ z,Yt tΨ(z) E eh i = e , t> 0. h i Proof. See for example [Sa], Theorem 25.17, page 165. 2

3 The Black-Scholes Model in a L´evy Setting

We consider a stock price St driven by a general geometric L´evy process

(4) dSt = σSt dYt + µSt dt, − − having a positive initial value S0 at time t0, where Yt is a Lévy process on the filtered probability space (Ω, , t t 0, P ). Here t t 0 denotes the minimal filtration gen- F {F } ≥ {F } ≥ erated by Y . The drift µ R and the volatility σ > 0 are constant. By Theorem 2.6 t ∈ the Lévy process Yt can be decomposed into

Yt = Bt + γt + Mt = √cWt + γt + Mt where c > 0 is the variance of the one-dimensional Brownian motion Bt and Wt is a standard Brownian motion. Mt is a jump process. Then the stock price process satisﬁes the following stochastic diﬀerential equation

dSt = St (σdBt + σdMt + (σγ + µ) dt. − Applying the Dol´eans-Dade exponential formula we know that the solution of this stochastic diﬀerential equation is given by

1 S = S exp σB + σM + (σγ + µ)t [σB,σB] (1 + σ∆M )exp σ∆M t 0 · t t − 2 t × s {− s} ½ ¾ t0~~t0. This restriction is equivalent to~~

~~1+ σ∆Ms > 0 for all s>t0~~

~~∆M > 1 =: α for all s>t ⇔ s − σ 0 where we assumed that σ = 0. If 0 <σ< 1, then α< 1. The restriction on the jump 6 − part of the L´evy process yields~~

∞ Yt = Bt + γt + x (J(t,dx) tν(dx)) + xν(dx)t α − [α, 1] [1, ) Z Z − ∪ ∞ = B + mt + ∞ x (J(t,dx) tν(dx)) , t − Zα where m = γ + [α, 1] [1, ) xν(dx). The Fourier transform of the L´evy process with restricted jumps is given− ∪ by∞ R 1 Pˆ (z) = exp cz2 + imz + ∞ eizx 1 izx ν(dx) , Y1 −2 − − µ Zα ¶ ¡ ¢ which means that the generating triplet of Yt is (m, c, ν). As proved in [Be], Lemma 4, page 31, the solution of the stochastic diﬀerential equation (4) takes the form

~~(5) St = S0 exp(Yt∗), where Yt∗ is a L´evy process with generating triplet~~

~~2 σ ∞ 2 1 (6) σm + µ c + (ln(1 + σx) σx)ν(dx), σ c, ν f − (γ∗,c∗, ν∗), − 2 − ◦ ≡ µ Zα ¶ where f(x) = ln(1+ σx) and σ~~

~~rt (7) S˜ := e− S = S exp(Y ∗ rt)= S exp(Y˜ ), where Y˜ = Y ∗ rt. t t 0 t − 0 t t t −~~

~~As a straightforward computation shows, Y˜t is a L´evy process with generating triplet~~

~~2 σ ∞ 2 1 (8) σm + µ c r + (ln(1 + σx) σx)ν(dx), σ c, ν f − (˜γ, c,˜ ν˜), − 2 − − ◦ ≡ µ Zα ¶ with f(x) = ln(1 σx). −~~

Definition 3.1 Let L be a Lévy process on some filtered probability space (Ω, , t t 0, P ) F {F } ≥ with an existing moment generating function mgf (θ) = 0 for some θ R. An Esscher L 6 ∈ transform is any change of P to a locally equivalent measure P (θ) with a density process (θ) dP (θ) Zt = dP of the form t ¯F ¯ ¯ (θ) exp(θLt) R Zt = t , θ . mgfL(θ) ∈

~~9 Proposition 3.2 Z deﬁned as above describes a density process for all θ R such that t ∈ E [exp(θL )] < . L is again a L´evy process under the new measure P (θ). t ∞~~

Proof. For the proof see [Ra], Proposition 1.8, page 7. 2 In [Be] an equivalent martingale measure is derived by means of the Esscher transform such that the discounted stock price S˜t is a martingale and Y is still a Lévy process under this new measure. This latter property is important since we only should change the distribution of the asset price but not the property of independent and stationary increments of the driving process. We recall that [GerSh] were the first who proposed to use the Escher transforms. Bernoth in [Be] derived an explicit formula for the characteristic function of the Lévy process under the new probability measure and for the generating triplet of the Lévy process. This result is summarized in the following Proposition.

~~Proposition 3.3 Let Y ∗ be a L´evy process on the ﬁltered probability space (Ω, , t t 0, P ) F {F } ≥ with corresponding characteristic function~~

izY ∗ 1 2 ∞ izx ΦY ∗ (z)= E e 1 = exp c∗z + iγ∗z + e 1 izx ν∗(dx) , z R, 1 −2 − − ∈ ½ Zα ¾ h i ¡ ¢ (θ) where (c∗, γ∗, ν∗) denotes the generating triplet of the process Y ∗. Let P be an equivalent probability measure corresponding to the Esscher transform Z(θ) with θ (a, b 1). ∈ − Then under the measure P (θ) the characteristic function of the L´evy process, denoted by (θ) Φ ∗ , takes the form Y1

~~Φ ∗ (z iθ) (θ) Y1 (9) ΦY ∗ (z)= − . 1 ΦY ∗ ( iθ) 1 −~~

~~Moreover, the generating triplet of the L´evy process Y ∗ under the probability measure P (θ) is given by~~

cθ∗ = c∗, xθ νθ∗(dx) = e ν∗(dx), ∞ xθ γθ∗ = γ∗ + c∗θ 1 e xν∗(dx). − α − Z ³ ´ Proof. The proof of (9) is by a single computation. The rest uses (8) and simple change of variables. Details are given in [Be] (Lemma 7 and Proposition 7, page 40/41). 2

~~4 The Pricing Problem of Asian Options~~

We consider a stock price process St driven by the stochastic differential equation (4) where Yt is a Lévy process as before. As we already mentioned the solution of such a stochastic differential equation is given in form of an exponential Lévy process (5) where the generating triplet of Y ∗ can be constructed from the generating triplet of Y. The problem we will consider in the following is to calculate the price of an Asian Option which is a path dependent option, that means that the value of such an option at maturity time T can depend on the whole path of the price process from inception time up to time T. We will consider only arithmetic Asian options of European type which are defined by the arithmetic mean of the stock price over the time period T. For geometric Asian options there exists a generalized Black Scholes formula, hence a closed form solution for

10 the pricing problem (see for example [Hu], page 467). However, the pricing problem for arithmetic Asian options is still under investigation since no closed form solution is known up to now. The value of an arithmetic Asian Option at the terminal time T depends on the mean of the whole path of the price process up to time T, 1 T S dt, t < T. T t t 0 − 0 Zt0 Consider the process A(τ),τ >t , where { 0} 1 τ A(τ) := S dt, τ >t . τ t t 0 − 0 Zt0 which is a time integral over the stock price process (5). We will sometimes write At instead of A(t). Now the problem of pricing a path dependent option is to determine the option price at inception time t0. We will denote the payoff function of a path dependent option by F (AT ). In the risk-neutral framework of the classical Black-Scholes model (i.e. of an asset following a geometric Brownian motion) the option price at any time t equals the value of a self-financing and replicating portfolio Vt at time t. Under the equivalent martingale measure P (θ) (defined via the Escher transform of the original probability measure P for some θ R ) the discounted value of the portfolio, e rtV is a P (θ) - ∈ − t martingale (see [Be], Theorem 9, Chapter 3). Hence we obtain

rt rT rT e− V (φ) = E (θ) e− V (φ) = E (θ) e− F (A ) t P t |Ft P T |Ft £ ¤ T£ ¤ r(T t) 1 V (φ) = e− − E (θ) F S dt . ⇔ t P T t t |Ft · µ − 0 Zt0 ¶ ¸ For an Asian Call Option the payoﬀ function is 1 T F (S ,t ) = max A K, 0 = max S dt K, 0 , T 0 { T − } T t t − ½ − 0 Zt0 ¾ where K is the strike price. Hence by the previous calculations the price of an Asian Call Option at any time t with strike price K is given by the discounted conditional expectation

T r(T t) 1 (10) CF (K,t) = e− − EP (θ) S0 exp(Yu∗)du K t . · " T t0 t0 − + F # µ − Z ¶ ¯ ¯ ¯ To calculate the conditional expectation in equation (10) we want to derive closed-form expressions for all moments of the arithmetic average 1 T A = S dt. T T t t − 0 Zt0 For this we use the Laplace transformation method introduced by Geman and Yor in [GemYo] in the classical Black-Scholes setting. We will ﬁrst simplify expression (10) above. Set K˜ := K (T t ) and · − 0 T A˜T := Stdt. Zt0 Then e r(T t) (11) C (K,t)= − − C˜ (K,t), F T t · F − 0 11 where

T ˜ ˜ ˜ ˜ (12) CF (K,t) := EP (θ) AT K t = EP (θ) S0 exp(Yu∗)du K t . + − F " t0 − + F # ·³ ´ ¯ ¸ µZ ¶ ¯ ¯ ¯ This can be reduced to a deterministic¯ function. Indeed, we have for t t

Using that the law of exp(Yu∗ Yt∗) is the same as the one of Yu∗ t and exp(Yt∗) is − − independent of exp(Y Y ) we obtain the following equalities (here = means equality u∗ − t∗ L in law): T ˜ AT =L St exp(Yu∗ t)du − Zt0 T t − =L St exp(Yu∗)du t0 t Z − T t 0 − =L St exp(Yu∗)du + St exp(Yu∗)du 0 t0 t Z Z − T t 0 − =L St exp(Yu∗)du + S0 exp(Yu∗+t)du 0 t0 t Z Z − T t ˜ − =L At + St exp(Yu∗)du. Z0 Since Y , which is equal in law to Y Y , is a L´evy process independent of , we u∗ u∗+t − t∗ Ft obtain for equation (12) the following equalities in law:

T ˜ ˜ CF (K,t) =L EP (θ) S0 exp(Yu∗)du K t " t0 − + F # µZ ¶ ¯ T t ˜ ¯ ˜ − K¯ At =L St EP (θ) exp(Yu∗)du − t · "Ã 0 − St ! F # Z + ¯ T t ˜ ˜ ¯ − K At ¯ =L St EP (θ) exp(Yu∗)du − · 0 − St "ÃZ !+# ˜ ˜ T t (here we also used AT =L At + St 0 − exp(Yu∗)du. ) Hence, from (11) and the equation above, we get that the price of an Asian Call Option at time t with strike price K is R given by

~~r(T t) T t K(T t ) t S du e− − − 0 t0 u (13) CF (K,t)= St EP (θ) exp(Yu∗)du − − , T t0 · · 0 − St − "ÃZ R !+#~~

12 where Y ∗ is a L´evy process with generating triplet (γ∗,c∗, ν∗) given by equation (6). To calculate this we need to know the joint distribution of the average At∗ and the underlying asset St, where t At∗ := exp(Yu∗)du. Z0 The moments of At∗ give full information about its law by the moment problem. To compute the moments we use the method of the Laplace transformation in time, following the method used by Geman and Yor in [GemYo] for the case of the continuous Black- Scholes model.

~~Deﬁnition 4.1 Let f be a measurable function from Rd into R respectively from Rd into C. Then the Laplace transform of f, whenever it exists, is deﬁned as~~

~~z,x d L[f](z) := e−h if(x)dx, z C . Rd ∈ Z~~

For the coupling z,x to make sense, x Rd is understood as an element of Cd. For a h i ∈ random variable X on Rd with density function ϕ : Rd R the following connection → between the Fourier transform and the Laplace transform (whenever L exists, and ΦX has an analytic continuation from z Rd to iz iRd ) of X holds ∈ ∈

z,x z,x z,X L[ϕ](z)= e−h iϕ(x)dx = e−h iPX (dx)= E e−h i = ΦX (iz). Rd Rd Z Z h i Relating to Deﬁnition 2.13 we see that for a L´evy process Xt with density function ϕt the Laplace transform of X at a point θ Rd equals the moment generating function t ∈ of X evaluated at θ (provided that both functions exists for X ), i.e. t − t

z,Xt t d L[ϕ ](θ)= E e−h i = (mgf ( θ)) , θ R , t t T. t X − ∈ 0 ≤ ≤ h i We set t K(T t0) Sudu q := − − t0 . St R Recall that we want to compute the value at time t t < T of an Asian Option on an 0 ≤ underlying asset St with initial value S0 at time t0. Hence we can observe the values of S for t u t, and thus we know whether q (deﬁned above) is positive (i.e. q > 0 ) u 0 ≤ ≤ or q 0. In the case q 0, equation (13) reduces to ≤ ≤ r(T t) T t e− − − C (K,t)= S E (θ) exp(Y ∗)du q . F T t · t · P u − − 0 µ ·Z0 ¸ ¶ We have to compute the ﬁrst moment

t EP (θ) [At∗] = EP (θ) exp(Yu∗)du ·Z0 ¸ t (θ) = exp(Yu∗)dudP ZΩ Z0 t = exp(y)ϕu(y)dydu R Z0 Z t = L[ϕ ]( 1)du, u − Z0 13 where ϕ (y) denotes the distribution of Y and L[ϕ ]( 1) the evaluation of L[ϕ ] at t t∗ u − u z = 1. Thus we can compute the Asian Option price for q 0 explicitly and we obtain − ≤ r(T t) e− − CF (K,t) = St EP (θ) AT∗ t q T t · · − − − 0 e r(T t) ¡ t £ ¤ ¢ (14) = − − S L[ϕ ]( 1)du q T t · t · u − − − 0 µZ0 ¶ r(T t) t e− − u = S (mgf ∗ (1)) du q . T t · t · Y − − 0 µZ0 ¶ Thus the valuation problem of an Asian Option reduces to the problem of computing the moment generating function of the L´evy process Y ∗ at 1. If Y ∗ e t ν∗(dx) < , x >1 ∞ Z| | then we can apply Theorem 2.14 (iii) to obtain that

∗ 1 ∗ ∗ x ∗ u Yu u ( 2 c +γ + Rd (e 1 x1{|x|≤1}(x))ν (dx)) mgfY ∗ (1) = EP (θ) e = e · − − . R h i Inserting this into equation (14) yields r(T t) t e− − 1 ∞ x C (K,t)= S exp uc∗ + γ∗u + u (e 1 x) ν∗(dx) du q . F T t · t · 2 − − − − 0 µZ0 ½ Zα ¾ ¶ Since (γ∗,c∗, ν∗) can be constructed from the generating triplet (γ, c, ν) of the original L´evy process Y, we obtained a closed form solution for the price of an Asian Option when q 0 and ≤ Y ∗ e t ν∗(dx) < . x >1 ∞ Z| | Now consider the case q > 0. We want to derive a formula for the Laplace transform with respect to the variable t of

E (θ) (A∗ q) . P t − + By inversion of the Laplace transform and£ inserting the¤ result into equation (13) one would then obtain the desired value of the Asian Option. We recall that we have t 1 ∞ x (15) E (θ) [A∗]= exp uc∗ + γ∗u + u (e 1 x) ν∗(dx) du, P t 2 − − Z0 ½ Zα ¾ where (c∗, γ∗, ν∗) is the generating triplet of the L´evy process Y ∗. Let τ := inf s : A > q . Then q { s∗ } t

~~At∗ = Aτ∗q + exp(Ys∗)ds Zτq on the set A q . Since A = q we obtain { t∗ ≥ } τ∗q~~

~~E (At∗ q)+ τq = E (At∗ q)+ 1 A∗ q + (At∗ q)+ 1 A∗~~

E (At∗ q)+ τq = E 1 A∗ q exp(Ys∗)ds τq { t ≥ } − F " · τq F # h ¯ i Z ¯ ¯ t ¯ ¯ ¯ = E 1 A∗ q exp(Yτ∗ + Ys∗ τ )ds τq { t ≥ } q q " · τq − F # Z ¯ t ¯ ¯ = E 1 A∗ q exp(Yτ∗ ) exp(Ys∗ τ )ds τq { t ≥ } q q " · · τq − F # Z ¯ t ¯ ¯ = exp(Yτ∗ ) E 1 A∗ q exp(Ys∗ τ )ds τq q { t ≥ } q · " · τq − F # Z ¯ t ¯ ¯ = exp(Yτ∗q ) E exp(Ys∗ τq )ds · − "Zτq #

~~= exp(Yτ∗q ) E At∗ τq , · − h i where we used that exp(Yτ∗q ) is τq -measurable. The ﬁrst moment of At∗ τq can be F − computed, as seen before, and we ﬁnally obtain~~

~~t τq − 1 ∞ x E (At∗ q)+ τq = exp(Yτ∗q ) exp uc∗ + γ∗u + u (e 1 x) ν∗(dx) du. − F · 0 2 α − − h ¯ i Z ½ Z ¾ ¯ It follows that¯~~

E (A∗ q) = E E (A∗ q) t − + t − + Fτq ¯ £ ¤ h h t τq ii −¯ 1 ∞ x = E exp(Y ∗ ) ¯ exp uc∗ + γ∗u + u (e 1 x) ν∗(dx) du . τq · 2 − − · Z0 ½ Zα ¾ ¸ Now we want to compute the Laplace transform in t of this. We obtain for λ R, using ∈ Fubini-Tonelli’s Theorem:

∞ λt e− E (A∗ q) dt t − + Z0 £ ¤ t τq ∞ λt − 1 ∞ x = e− E exp(Y ∗ ) exp uc∗ + γ∗u + u (e 1 x) ν∗(dx) du dt τq · 2 − − Z0 · Z0 ½ Zα ¾ ¸ t τq ∞ λt − 1 ∞ x = E exp(Y ∗ ) e− exp uc∗ + γ∗u + u (e 1 x) ν∗(dx) dudt τq · 2 − − · Z0 Z0 ½ Zα ¾ ¸ t λτq ∞ λt 1 ∞ x = E exp(Y ∗ ) e− e− exp uc∗ + γ∗u + u (e 1 x) ν∗(dx) dudt . τq · 2 − − · Z0 Z0 ½ Zα ¾ ¸ Setting

~~t ∞ λt 1 ∞ x f(λ) := e− exp uc∗ + γ∗u + u (e 1 x) ν∗(dx) dudt, λ R, 2 − − ∈ Z0 Z0 ½ Zα ¾ we have obtained~~

~~∞ λt λτq e− E (A∗ q) dt = E exp(Y ∗ ) e− f(λ) . t − + τq · · Z0 £ ¤ h i Remark 4.2 f(λ) could be computed numerically for given values of c∗, γ∗ and given ν∗.~~

~~15 We denote the density at time q of the transition semi group of the process Aq∗ = exp(Yq∗) from 1 at time 0 to s at time q by ρq(1, s). We have:~~

∞ λt ∞ λτq (16) e− E (At∗ q)+ dt = f(λ) s ρq(1, s) E e− exp(Yτ∗q )= s ds. 0 − · 0 · · Z Z h ¯ i £ ¤ ¯ The density ρq(1, s) can be computed from the generating triplet¯ of the process Ys∗. Thus it remains to compute the conditional expectation

λτq (17) E e− exp(Y ∗ )= s , t s T, λ R. τq 0 ≤ ≤ ∈ h ¯ i To compute this, we ﬁrst rewrite¯ the stopping time τ as follows ¯ q τ : = inf s : A∗ > q q { s } q ∗ = id As >u ds 0 { } Z q ∗ = exp( Ys∗) id As >u exp(Ys∗)ds 0 − { } Z q ∗ = exp( Ys∗) id As >u dAs∗ 0 − { } Z q = exp( Ys∗)dτs 0 − Z q = exp( Y ∗ )du, − τu Z0 where we substituted s = τu. Using this identity the conditional expectation in (17) can be computed by expanding around λ = 0 as follows

λτq E e− exp(Yτ∗q )= s h ¯ i ¯ ∞ 1 n = E ¯(τ ) exp(Y ∗ )= s n! q τq n=1 X h ¯ i n ∞ 1 q¯ = E ¯exp( Y ∗ )du exp(Y ∗ )= s n! − τu τq n=1 ·µZ0 ¶ ¸ X n ¯ ∞ 1 q ¯ ¯ q = exp( Yτ∗u )du id ∗ dP exp( Y ∗ )du exp(Y ∗ ). n! − exp(Yτq )=s ( 0 τu ) τq n=1 Ω µ 0 ¶ − ⊗ X Z Z n o R Thus we have to compute the joint probability distribution of q exp( Y )du and 0 − τ∗u exp(Y ). This is often a hard task. However, in some special situations we can explicitly τ∗q R compute the condition expectation (17) and thus derive a pricing formula for Asian options. In the classical Black Scholes model (driven by a Brownian motion) this task has been solved by [GemYo] via a technique using Bessel processes. In the following we will prove a pricing formula for Asian options in a Black-Scholes model driven by a Variance Gamma process. Here we will use the same methods as in [GemYo].

~~4.1 Explicit Computation in the Variance Gamma Case~~

We now consider the case where the stock price process is driven by a Variance Gamma VG(σ, ν, θ) process Yt. As mentioned in Example 2.12 the stock prices dynamics are given by t σ2ν (18) S = S exp rt + Y (t; σ, ν, θ)+ ln 1 θν . t 0 ν − − 2 µ µ ¶¶ 16 Thus our process Yt∗ is deﬁned as

~~t σ2ν Y ∗ = rt + Y (t; σ, ν, θ)+ ln 1 θν . t ν − − 2 µ ¶ Now, going back to equation (13), we can rewrite this by using the identity~~

~~Yt = θγ(t)+ σWγ(t), where W is a Brownian motion and γ is a Gamma G(1/ν, 1/ν) process.~~

r(T t) T t e− − − C (K,t)= S E (θ) exp bu + θγ(u)+ σW du κ , F T t · t · P γ(u) − − 0 ·µZ0 ¶+¸ ¡ ¢ where t 1 σ2ν K(T t0) Sudu b := r + ln 1 θν and κ := − − t0 ν − − 2 St R ³ ´ Using the scaling properties of Brownian motion and Gamma processes we obtain by 2ν 2 1−ν substitution u = cv with c := σ the following

s ¡ ¢ E (θ) exp bu + θγ(u)+ σW du κ P γ(u) − ·µZ0 ¶+¸ cs ¡ ¢ = E (θ) c exp bcv + θγ(cv)+ σW dv κ P · γ(cv) − ·µ Z0 ¶+¸ cs ¡ 1−ν ¢ = E (θ) c exp bcv + θc ν γ(v)+ σW 1−ν dv κ P · c ν γ(v) − ·µ Z0 µ ¶ ¶+¸ cs 1−ν κ ν = c EP (θ) exp bcv + θc γ(v) + 2Wγ(v) dv · 0 − c + ·µZ ³ ´ ¶ ¸ cs 1−ν bc θc ν κ = c EP (θ) exp 2 v + γ(v)+ Wγ(v) dv . · 0 2 2 − c "ÃZ Ã Ã !! !+# Thus we have (19) r(T t) h 1−ν e− − bc θc ν κ CF (K,t)= c St EP (θ) exp 2 v + γ(v)+ Wγ(v) dv , T t0 · · 0 2 2 − c − "ÃZ Ã Ã !! !+# with t 1 σ2ν K(T t0) Sudu b := r + ln 1 θν and κ := − − t0 ν − − 2 St R ³ ´ and 2ν 2 1−ν c := and h := c(T t). σ − µ ¶ What remains to be computed is

~~E (θ) (A q) , P h − + where £ ¤ h 1−ν bc θc ν κ A := exp 2 s + γ(s)+ W ds and q := . h 2 2 γ(s) c Z0 Ã Ã !!~~

~~17 From equation (14) we obtain in case q < 0:~~

r(T t) t e− − u C (K,t) = c S (mgf ∗ (1)) du q F T t · t · Y − − 0 µZ0 ¶ e r(T t) t 1 tu/ν = c − − S du q T t · t · 1 θν + (σ2ν/2) − − 0 ÃZ0 µ − ¶ ! 2 t if (σ ν/2) = 1 θν. Here we used the identity mgf ∗ (iz) = Φ ∗ (z) for z R, where 6 − Y Yt ∈ ∗ ΦYt is the characteristic function of the process Yt∗. Since this characteristic function is given by 1 t/ν Φ ∗ (u)= , Yt 1 iθνu + (σ2ν/2)u2 µ − ¶ 2 2 t which is an analytic function for (σ ν/2)u = 1 iθνu, the identity mgf ∗ (iz) = Φ ∗ (z) 6 − Y Yt also holds for u = i whenever (σ2ν/2) = 1 θν. Thus we have − 6 − t/ν t 1 mgf ∗ (1) = Φ ∗ ( i)= . Y Yt − 1 θν + (σ2ν/2) µ − ¶ We now consider the more complicated case q > 0. For this, we again use the method of Laplace transformation with respect to the variable h. We will ﬁrst write the process X := exp(αs + βγ(s)+ W ), s 0, as s γ(s) ≥ Xs = R(As), where (R(u),u 0) is a Bessel process, starting at R(α,β)(0) = 1. Here we will use the ≥ fact that any Variance Gamma process Y ∗(t) can be expressed as a standard Brownian motion W (T (t)) with independent gamma subordinator T (t). Hence we can write

~~exp(Ys∗) = exp(αs + βγ(s)+ Wγ(s)) = exp(W (T (s)))~~

Moreover we use a result by [Wi] stating that the exponential of Brownian motion with drift is a time-changed Bessel process, i.e. t exp(W (t)+ δt)= R(α) exp(2(W (s)+ αs)ds , t 0, ≥ µZ0 ¶ where (R(δ)(u),u 0) is a Bessel process with index δ. Hence we obtain ≥ T (t) exp(Y ∗) = exp(W (T (t))) = R exp(2W (s))ds , t 0, t ≥ ÃZ0 ! where (R(u),u 0) is a Bessel process with index 0. ≥ Then, returning to equation (17), and considering the fact that q du τ = , q (R(u))2 Z0 we obtain

~~λτq E e− exp(Yτ∗q )= s h ¯ i (20) T (τ ) λ q¯ du q = E e− 0¯ (R(u))2 R exp(2W (u))du = s . " R Ã 0 ! # ¯ Z ¯ Now we can use a Proposition proven¯ in [Yo].~~

18 Proposition 4.3 For every δ R,a> 0,x> 0,t> 0, ∈ 2 t δ ds I δ ax (0) | | Ea exp − 2 Rt = x = , 2 0 Rs I0 t · µ Z ¶ ¯ ¸ ³ ´ ¯ where, on the right-hand side, (f/g)(y) stands¯ for f(y)/g(y) and Iδ resp. I0 denote (0) (0) Bessel functions. Ea denotes the expectation relativ to Pa , the low of the Bessel process R(0) starting at a.

~~Applying this to equation (20) yields~~

T (τ ) λ q du q λτq − 0 (R(u))2 E e− exp(Yτ∗q )= s = E e R exp(2W (u))du = s " R Ã 0 ! # h ¯ i ¯ Z ¯ I √2λ s ¯ ¯ = | | . ¯ I q 0 µ ¶ Thus the Laplace transform (16) of the Asian option price is given by

I ∞ λt ∞ √2λ s (21) e− E (A∗ q) dt = f(λ) s ρ (1, s) | | ds. t − + · · q · I q Z0 Z0 0 µ ¶ £ ¤ where ρq(1, s) denotes the density at time q of the Bessel semigroup, with index 0 and starting position 1 at time 0. This density can be computed explicitly and is given by (see [GemYo], Proposition 2.2)

s 1 s ρ (1, s)= exp (1 + s2) I . q q −2q 0 q µ ¶ µ ¶ Hence the Laplace transform of E (A q) with respect to t can be represented by t∗ − + £ ¤2 ∞ λt ∞ s 1 2 s e− E (A∗ q) dt = f(λ) exp (1 + s ) I ds, t − + · q −2q √2λ q Z0 Z0 µ ¶ | | µ ¶ £ ¤ where

t t tu/ν ∞ λt u ∞ λt 1 f(λ) := e− (mgf ∗ (1)) dudt = e− dudt, λ> 0. Y 1 θν + (σ2ν/2) Z0 Z0 Z0 Z0 µ − ¶ By inversion of this Laplace transform for a ﬁxed t we obtain E (A q) and thus t∗ − + by equation (19) the Asian option price. The inversion of the Laplace transform is given £ ¤ by

1 s2 1 s F (t) := ∞ eλt f(λ) ∞ exp (1 + s2) I ds dλ 2π · q −2q √2λ q Z0 µ Z0 µ ¶ | | µ ¶ ¶ 1 1 = L− [f](t) L− [Ψ](t), ∗ 1 where we denote the inverse Laplace transform of a function by L− , thus

~~1 1 ∞ λt L− [f](t) := e f(λ)dλ. 2π Z0 Ψ is the function deﬁned by~~

s2 1 s Ψ= ∞ exp (1 + s2) I ds. q −2q √2λ q Z0 µ ¶ | | µ ¶ 19 Here we used that the inverse Laplace transform of a product f g is the convolution · (denoted by ) of the inverse Laplace transforms of f and g. Hence we have to compute ∗ two inverse Laplace transformations. This can be expressed analytically in terms of power series involving quite complicated expressions. From a practical point of view a direct numerical computation of the inverse Laplace transform might be more eﬃcient. However, since the parameter λ of the inverse Laplace transformation appears in the index of the Bessel function, also a numerical computation is not that easy and there is no standard software for this task.

Remark 4.4 The procedure we used to derive the pricing formula for Asian options on a Variance Gamma driven stock price, can be applied to any Black-Scholes type model with L´evy noise (in the sense of equation (1)), whenever the driving process Yt has a scaling property and the property that it can be written as a time changed Brownian motion with respect to some subordinator.

~~5 Acknowledgements~~

~~We are very grateful to David Applebaum for stimulating discussion, and in particular for the suggestion to look at the Variance Gamma process as an example.~~

~~References~~

[AlRu] Albeverio, S.; Rüdiger, B.: Stochastic integration with respect to compensated Poisson random measures and the LUvy´ ûIto decomposition theorem on separable Banach spaces. Stoch. An. and Appl. ,Vol. 23, Nr. 2, 2005. [Ap] Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge Studies in Ad- vanced Mathematics 93, Cambridge University Press, Cambridge, 2004. [Be] Bernoth, K.: An Extended Multidimensional Black-Scholes Model with Lévy Noises. Diploma Thesis, Bonn, 2003. [Ber] Bertoin, J.: Lévy processes. Cambridge Tracts in Mathematics 121, Cambridge Uni- versity Press, Cambridge, 1996. [GemYo] Geman, H.; Yor, M.: Bessel Processes, Asian Options, and Perpetuities. Mathematical Finance, Vol. 3, No. 4, pp. 349-375, 1993. [GerSh] Gerber, H.U.; Shiu, E.S.W.: Option Pricing by Escher transforms. Trans. Soc. Act. 48, pp.99-191, 1994. [Hu] Hull, J.C.: Options, Futures and other Derivatives. 4th Ed., Prentice Hall, Toronto 1999. [IW] Ikeda, N.; Watanabe, S.: Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. [MaCaCh] Madan, D.B.; Carr, P.; Chang, E.C.: The Variance Gamma Process and Option Pricing. European Finance Review 2, pp.79-105, Kluwer Academic Publishers, 1998. [Ra] Raible, S.: Lévy processes in Finance: Theory, Numerics, and Empirical Facts. PhD Thesis, Universitäts Freiburg, Freiburg, 2000. [Ru] Rüdiger, B.: Stochastic integration for compensated Poisson measures and the Lévy- Itôformula. Proceedings of the International Conference on Stochastic Analysis and Applications, pp. 145–167, Kluwer Acad. Publ., Dordrecht, 2004. [Sa] Sato, K.: Lévy Processes and Infinite Divisibility. Cambridge University Press, Cam- bridge, 1999.

20 [Wi] Williams, D.: Path Decomposition and Continuity of Local Time for One-Dimensional Diﬀusions. Proc. London Math. Soc. (3), 28, pp. 738-768, 1974. [Yo] Yor, M.: Loi de l’indice du lacet brownien, et distribution de Hartman-Watson. Zeitschrift f¨urWahrscheinlichkeitstheorie, 53, pp. 71-95, 1980.

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228. Verleye, Bart; Klitz, Margrit; Croce, Roberto; Roose, Dirk; Lomov, Stepan; Verpoest, Ignaas: Computation of Permeability of Textile Reinforcements; erscheint in: Proceedings, Scientific Computation IMACS 2005, Paris (France), July 11-15, 2005

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231. Albeverio, Sergio; Pratsiovytyi, Mykola; Torbin, Grygoriy: Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of their S-Adic Digits

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233. Hahn, Atle: An Analytic Approach to Turaev’s Shadow Invariant 234. Hildebrandt, Stefan; von der Mosel, Heiko: Conformal Representation of Surfaces, and Plateau’s Problem for Cartan Functionals

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~~237. Albeverio, Sergio; Cebulla, Christof: Müntz Formula and Zero Free Regions for the Riemann Zeta Function~~

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