Distributional Mellin calculus in , with applications to option pricing
Abstract
We discuss several aspects of Mellin transform, including distributional Mellin transform and inversion of multiple MellinBarnes integrals in and its connection to residue expansion or evaluation of Laplace integrals. These mathematical concepts are demonstrated on several optionpricing models. This includes European option models such as BlackScholes or fractionaldiffusion models, as well as evaluation of quantities related to the optimal exercise price of American options.
Contents
Key words— Mellin transform, Laplace transform, Distributions, Multidimensional Complex Analysis, European option, American option
I Introduction
Mellin transform belongs, together with Fourier transform, Laplace transform and Ztransform, to the most important integral transforms with many applications in applied mathematics, theory of diffusion, quantum mechanics, image recognition, statistics or finance. It can be interpreted as a multiplicative version of twosided Laplace transform and therefore it can be successfully used for description of multiplicative processes and other diffusionlike phenomena as well as for description and analysis of various special functions  MittagLeffler functions, FoxH function [15, 18], stable distributions [17] or option pricing , just to name a few. Within the past few years, the onedimensional Mellin transform has also been applied to various option pricing problems [20, 11]. Although it is an integral transformation in the complex domain, it can be often successfully used in numerical calculations and asymptotic formulas, in particular in analytic number theory [9] or in quantum field theory [1].
Nevertheless, some aspects of Mellin transform and MellinBarnes integral representation have not been satisfactorily discussed. First, generalization of Mellin transform to the domain of generalized functions enables to deal with wider class of functions, including polynomials and logarithmic functions. As a byproduct, we obtain a generalized version of delta function, which is, in analogy with Fourier transform, Mellin transform of a constant; within this framework, we are therefore able to investigate various functions, including powers or powers of logarithms. We call this new approach "distributional Mellin calculus".
Second, generalization of Mellin calculus to two or more dimensions and appropriate techniques for its calculation represents a conceptually new powerful tool for analysis of asymptotic properties of many systems, based on theory of residues in and residue representation. This enables us to perform the complex integrals and end with a (Grothendieck) residue series, which can be used for numerical calculations as well as for description of asymptotic properties. This approach, first formalized in [23, 24] (without distributional aspects) was already used in [2] to estimate analytically some standard model contributions to the muon anomalous moment.
The new and powerful combination of distributional Mellin calculus and residue theory in can be successfully used in many applications, for instance in generalized diffusion or option pricing [3]. In this paper we make more examples, including Gaussian and nonGaussian models as well as an application to the evaluation of Laplace integrals. These integrals appear in various applications as, for instance, the optimal exercise price of an American option [26].
The paper is organized as follows: section 2 discusses basic properties of Mellin transform and MellinBarnes integrals and generalizes it to distributional calculus. As an example, MellinBarnes representation of European call option for BlackScholes model and spacetime fractional model is presented. Section 3 introduces ndimensional Mellin calculus and presents residue theorem for MellinBarnes integral. Consequently, series formula for European BlackScholes call option is calculated. Section 4 discusses applications of multidimensional Mellin calculus to evaluation of Laplace integrals and its application to American options. Section 5 is devoted to conclusions and perspectives.
Ii Distribution theory of the onedimensional Mellin transform
I Definition and properties
See [9] for an excellent overview of the Mellin transform and its applications, and [8] for a dictionnary of frequently used transforms.
The Mellin transform of a locally integrable function on is the function defined by
(1) 
In general, the integral (1) converges on a strip (the socalled fundamental strip of ) in , that is an open subset of the type
(2) 
where and are determined by the asymptotic behaviour of around and . In this framework, the Gamma function, also known as Euler’s integral of the second kind and defined by
(3) 
is the Mellin transform of with fundamental strip . It can be extended to the lefthalf plane thanks to the functional relation
(4) 
unless for negative integers where (4) shows that possesses a pole of residue . We will express this fact by writing
(5) 
where the formal series in the right hand side of (5) is called the singular series of the Gamma function.
Another important Mellin transform follows from the particular case of Beta integral:
(6) 
which means that is the Mellin transform of with fundamental strip . Its singular series follows easily from (5) and from the property for :
(7) 
In Table 1, we list some important properties of the Mellin Transform which follow directly from definition (1).
Fundamental strip  

Example (Heat kernel): Let denote the heat kernel:
(8) 
Using the properties in table 1, it is immediate to see that its Mellin transform writes
(9) 
Ii Inverse MellinBarnes integrals
The inversion of the Mellin transform is performed via an integral along any vertical line in the fundamental strip:
(10) 
and in the case where is a ratio of products of Gamma functions of linear argument, integral (10) is said to be a MellinBarnes integral:
(11) 
The question at stake is whether (11) can be expressed as a sum of residues of the integrand in some region of , that is, if we can close the vertical line of integration in such a way that the integrand does not contribute in this part of the contour when . To answer this question, introduce the characteristic quantity
(12) 
Using the Stirling approximation for the Gamma function, it can be shown that the integrand in (11) decreases exponentially in the halfplane (see [23] for and [24] for ):
(13) 
If , then reduces to:
(14) 
which means that, depending on the sign of , (11) can be expressed as a sum of residues in one halfplane (left to the fundamental strip if , right if ). If , residues summations hold in both halfplanes.
Example 1. Let , then with fundamental strip , and therefore
(15) 
where for the last equality we have used the singular series (5) for the Gamma function; note (15) is the Taylor expansion of around .
Example 2. Let , then with fundamental strip and . One can either left sum the residues of :
(16) 
which is the series expansion of for small x, either right sum the residues:
(17) 
which is the series expansion of for large x. Both residues are directly computed from the singular series (7).
Iii Generalized Mellin transform
The classical Mellin theory does not take into account several elementary functions such as powers or powers of logarithms, because their Mellin transform (1) has an empty fundamental strip. For instance, by definition, the Mellin transform of is the integral
(18) 
which converge for no value of . But, performing the change of variables , the integral (18) becomes:
(19) 
where we have used the fact that the Fourier transform of is the Dirac distribution (see [12]). Note that a formal application of the inversion formula (10) and a change of variables yields
(20) 
where the braket stands for the distributional duality. In particular, the Mellin transform of is and using the "logarithms" rule in table 1, we deduce that the Mellin transform of is
(21) 
This fact can be recovered by considering the integrals:
(22)  
(23) 
(The Fourier transform of integer powers results in derivatives of the Dirac distribution). In table 2 we summarize some useful Mellin transforms, obtained either by the "classical" theory or the distributional one.
Fundamental strip  

The particular values we have obtained for express the fact that the distributionvalued complex function is analytic in the right complex plane (as the Fourier transform of a socalled homogeneous distribution [12, 10]). One can in fact meromorphically extend to negative values of because a simple integration by parts shows that the following functional relation holds:
(24) 
which means that possesses a pole in whose residue is . Iterating the procedure yields:
(25) 
which can be regarded as the distributional analogue to the singular series of the Gamma function (5)
Iv The MellinBarnes representation for an European BlackScholes call
Recall that the BlackScholes price of an European call option can be written as [25]:
(26) 
It is easy to compute the Mellin transform of the payoff [19, 3] and therefore obtain the representation:
(27) 
Introducing the MellinBarnes representation for the heat kernel (9) and a similar representation for (which follows directly from the properties in table 1) gives birth to two more integrals, over complex variables we shall denote by and . We are then left with the integral:
(28) 
Replacing in (26), we have:
(29) 
where (resp. ), corresponding to the fundamental strip of the (resp. ) function. Now, introducing the vector , denoting
(30) 
and using the change of variables , (29) becomes:
(31) 
where , and where is defined by the action of the distribution over a certain test function:
(32) 
To perform the integral (31), we need to extend the concepts we have introduced in this section to the case of , ; this will be the object of the next section.
V The MellinBarnes representation for an option driven by fractional diffusion
BlackScholes formula has been very popular among practitioners, nevertheless its applicability is limited due to simplified assumptions excluding complex phenomena as sudden jumps, memory effects, etc. There have been studied various generalizations of BlackScholes formula, let us mention regime switching multifractal models [6], stochastic volatility models [14] or jump processes [22]. Particularly interesting are generalized diffusion models, especially models with fractional diffusion. These models are also closely related to Mellin transform. Spacetime (double)fractional diffusion equation can be expressed as
(33) 
where is the Caputo fractional derivative and is the RieszFeller fractional derivative. All technical details and a broad mathematical discussion can be found in [15]. Eq. (33) can be solved by in terms of MellinBarnes integral as
(34) 
where Gamma fraction is defined as
The corresponding European call option price has been calculated in [17] as:
(35) 
which is a generalization of the BlackScholes propagator (26). Parameter appearing the “modified payoff” has its origin in the riskneutral (or martingaleequivalent) probability measure, which is for exponential transform obtained by socalled Esscher transform [13]. It can be calculated as
(36) 
Unfortunately, it is not always possible to express analytically. Nevertheless in the degenerate Gaussian case (that is for and ) , we have , and the fractional derivatives in (33) reduce to the usual heat equation.
By very similar manipulations to the ones we have made in the BlackScholes case, it is possible to derive the MellinBarnes representation for European call option driven by spacetime fractional diffusion as
(37) 
where
(38) 
Let us note that this representation slightly differs from the representation of BlackScholes model, because here we have used slightly different MellinBarnes representation of Green function , which for BlackScholes model (Gaussian distribution) is equal to
(39) 
Of course (39) is equivalent to the MellinBarnes representation (9) for the heat kernel: it follows easily from the change of variables and an application of the Legendre duplication formula for the Gamma function. Last, note that the extra term is induced by the fractional nature of the time derivatives and disappears when (natural derivatives). The analytic evaluation of the representation (37) will be the subject of a future research article.
Iii Inversion of muliple MellinBarnes integrals
I An introduction to the calculus of residues in
This brief presentation of fundamental tools is directly inspired by the classical book [16], as well as the doctoral dissertation [1].
i.1 Complex differential forms
Let an integer, and denote
(40) 
and
(41) 
where we use the usual wedge "" to symbolize the properties
(42) 
The vector forms a basis of the cotangent space , meaning that the total differential of an application can be written under the form
(43) 
We will say that is holomorphic in the open set , and denote , if the CauchyRiemann equation
(44) 
is satisfied in . In particular, if , we will denote by the ring of holomorphic functions in . Now, let us introduce complex differential forms on , that is, elements of which have the generic form
(45) 
for some functions and on . If in particular , then by properties (42) the differential of writes
(46)  
If is holomorphic in , then by definition (44) we have and thus ( is closed). Therefore, for any (orientable) subset we have, by Stokes’ theorem
(47) 
This constitutes the analogue to the Cauchy integral theorem in .
i.2 Grothendieck residue
Let a sequence in ; it is called a regular sequence if, for any , , is a nonzero divisor in the quotient ring , where is the ideal generated by .
Example. Let ; then the sequence is a regular sequence, but is not, because the first term of the sequence divides the second one. More generally, the elements of a regular sequence in all vanish in ; this comes from the fact that is a socalled local ring, that is, possesses a unique maximal ideal, and it can be shown that this ideal is generated by . Now, consider applications in and suppose that is a regular sequence; let be the differential form
(48) 
Then, the quantity
(49) 
does not depend on the and is called the Grothendieck residue of in . Of course, it can be extended to any simply by making the change of variables . The subsets for are called the divisors of . In particular, if then, sequentially applying the onedimensional Cauchy formla gives
(50) 
which is known as the Cauchy formula on the polydisks.
Ii Multidimensional Jordan lemma
We denote by . Let a vector in and be the complex differential form: the Euclidean scalar product in
(54) 
Let and denote
(55) 
We want to know whether the dimensional MellinBarnes integral
(56) 
can be expressed as a sum of Grothendieck residues, and, if it is the case, in which region of . To that extent, we introduce the dimensional analogues to (12) and (13):
(57) 
and
(58) 
ii.1 Compatibility of cones in
Let and a linear application; let be the subset of defined by
(59) 
A cone in is a subset of the type
(60) 
where all the are of type (59); its faces are
(61) 
and its distinguished boundary, or vertex, is
(62) 
Let , be the divisors of the form in (56), and . A cone is said to be compatible with if

Its distinguished boundary is ;

Any intersect at most one of its faces :
(63)
ii.2 Residue theorem for multiple MellinBarnes integrals
It is shown in [23] that,if , a Jordan condition is automatically satisfied for the form in every compatible cone in , that is, the growth of the Gamma can be uniformly controlled and be shown to vanish when . In [24], the authors extend their results to the case . The integral over therefore resumes to a sum of residues (in the sense of Grothendieck) in the compatible cone:
If , then there exists a compatible cone into which:
Example. Let ; introducing sequentially the Mellin pair which holds with fundamental strip , we obtain the representation
(65) 
The resulting fundamental "polyhedra" is the direct product of the fundamental strips in each Mellin variable. From the singular behavior of the Gamma function, it is clear that possesses two divisors which turn out to be the two families
(66) 
Computing the characteristic from definition (57):
(67) 
and consequently, from definition (58) the halfplane is
(68) 
which, in the plane, is the halfplane located under the line . If we choose to be the cone
(69) 
it is clear, as shown in Fig. 1, that is compatible with .
Iii Series formula for an European BlackScholes call
We come back to the evaluation of the European call under BlackScholes model; from (31), we introduce the complex differential form
(72) 
so as to write
(73) 
Introducing the notation
(74) 
and using the decomposition