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This paper presents a closed-form solution for the joint probability of the endpoints and minimums of a multidimensional Wiener process for some correlation matrices. This is the only explicit expressions found in the literature for this joint probability. The analysis can only be carried out for special correlation structures as it is related to the fundamentals regions of irreducible spherical simplexes generated by reflections and the link to the method of images. This joint distribution can be used in financial mathematics to obtain prices of credit or market related products in high dimension. The solution could be generalized to account for stochastic volatility and other stylized features of the financial markets.

The paper finds closed-form expressions for the joint density/distribution function of the endpoints and extrema of a

In this paper we generalize the results in [

The paper is organized as follows. Section

Let

instantaneous correlation

Our main objective is to find the joint density/distribution function for the minimum, denoted as

In general

Whenever reasonable and in order to shorten the length of notation, we will write

It is important to realize that if (

It is known (see [

Equations (

To obtain the solution of the system (

Let us define

The last step eliminates the correlations in matrix

Let one define new variables

This follows directly from noticing that the processes associated to

The variables

The method of images (MofI) is now utilized to solve the system (

checking that the differential equation is suitable for MofI and then solving this differential equation for a point source in an infinite medium, but with no boundary conditions except that of good behavior at infinity;

checking that the region of interest is suitable for MofI and then finding the set of image source at each of the reflecting regions;

summing the solution of

In general, step

We will show next that the regions where MofI applies are connected to correlation matrices; therefore feasible regions imply feasible correlations. For the issue of whether the bounded region allows for the method of images, we rely on [

The next result relates these regions and in particular the dihedral angle between these hyperplanes with the correlation matrix

The relationship between the dihedral angle between hyperplanes

The dihedral angle between hyperplanes

Here we use the fact that

Next we use the results in the seminal work of [

The type of correlation matrices

For any

If

For

If

If

If

If

This follows from the dihedral angles provided in Table IV, page 297, in [

In particular, for dimensions 2 and 3, the number of feasible cases and source points reproduces those found in [

Given a collection of source points in

Note that (

In this section we describe a quasi-analytical procedure to find all source points associated to a spherical simplex and therefore to a correlation matrix.

Let us denote

We show next a method to create the image of a hyperplane, the image of a point reflected across a given hyperplane (passing through zero), and the sign of the new point as needed by the method of images; this uses standard concepts from geometry.

New source point: reflecting a point

New hyperplane: reflecting a hyperplane with normal vector

Note the normal vector to both

Sign of new point: assume original point has a positive sign “+” then

The algorithms are based on doing reflections across all originals hyperplanes and then repeating the procedure for all new hyperplanes. This method would lead to repeated values; therefore we also have to check if there are duplicates (hyperplanes or source points). The method stops after the known number of different sources

One of the key application fields for our findings is multidimensional financial derivatives (see [

Two main families of such products are double lookback options (see [

Here

In the past this expression could be evaluated either via Monte Carlo simulations or directly solving the PDE equations, which are highly time consuming and inaccurate approaches for dimensions higher than 3. This price, under the feasible correlations described in Proposition

Another family of products benefiting from this work are credit derivatives, in particular collateralized debt obligation (CDO) and a

This paper describes the correlation matrices for which a closed-form solution for the joint density/distribution of the endpoints and the minimum of a Wiener process can be found. The results are also applicable to other processes like log-normal. The general solution requires a detailed geometrical analysis of certain partitions of the

The method developed in the present paper could also be extended to allow for stochastic volatility and random correlation. It can be applied to maximums and minimums combined, as long as one extreme per dimension is considered. Finally, the solution could be the basis for further approximations like those based on perturbation theory (see [

The authors declare that there is no conflict of interests regarding the publication of this paper.