This work deals with the Dirichlet problem for some PDEs of second order with non-negative characteristic form. One main motivation is to study some boundary-value problems for PDEs of Black-Scholes type arising in the pricing problem for financial options of barrier type. Barrier options on stocks have been traded since the end of the Sixties and the market for these options has been dramatically expanding, making barrier options the most popular ones among the exotic. The class of standard barrier options includes ’in’ barriers and ’out’ barriers, which are activated (knocked in) and, respectively, extinguished (knocked out) if the underlying asset price crosses the barrier before expiration. Moreover, each class includes ’down’ or ’up’ options, depending on whether the barrier is below or above the current asset price and thus can be breached from above or below. Therefore there are eight types of standard barrier options, depending on their ’in’ or ’out’, ’down’ or ’up’, and ’call’ or ’put’ attributes. It is possible to include a cash rebate, which is paid out at option expiration if an ’in’ (’out’) option has not been knocked in (has been knocked out, respectively) during its lifetime. One can consider barrier options with rebates of several types, terminal payoffs of different forms (e.g. power options), more than one underlying assets and/or barriers, and allow for time-dependent barriers, thus enriching this class still further. On the other hand, a large variety of new exotic barriers have been designed to accommodate investors’ preferences. Another motivation for the study of such options is related to credit risk theory. Several credit-risk models build on the barrier option formalism, since the default event can be modeled throughout a signalling variable hitting a pre-specified boundary value (See , among others). As a consequence, a substantial body of academic literature provides pricing methods for valuating barrier options, starting from the seminal work of , where an exact formula is offered for a down-and-out European call with zero rebate. Further extensions are provided - among others - in  for the different types of standard barrier options, in  for simultaneous ’down’ and ’up’ barriers with exponential dependence on time, in  for two boundaries via Laplace transform, in  and  for partial barrier and rainbow options, in  for multi-asset options with an outside barrier, in  in a most comprehensive setting employing the image solution method. Many analytical formulas for barrier options are collected also in handbooks (see , for example).
For analytical tractability most literature assumes that the barrier hitting is monitored in continuous time. However there exist some works dealing with the discrete version, i.e. barrier crossing is allowed only at some specific dates -typically at daily closings. (See  and , for a survey). Furthermore, a recent literature relaxes the Brownian motion assumption and considers a more general Lévy framework. For example,  study barrier options of European type assuming that the returns of the underlying asset follows a Lévy process from a wide class. They employ the Wiener-Hopf factorization method and elements of pseudodifferential calculus to solve the related boundary problem. This book chapter adopts a classical Black-Scholes framework. The problem of pricing barrier options is reducible to boundary value problems for a PDE of Black-Scholes type and with pre-specified boundaries. The value at the terminal time is assigned, specifying the terminal payoff which is paid provided that an ’in’ option is knocked in or an ’out’ option is not knocked out during its lifetime. The option holder may be entitled or not to a rebate. From a mathematical point of view, the boundary condition can be inhomogeneous or homogeneous. While there are several types of barrier options, in this work we will focus on ’up’ barriers in view of the relationships between the prices of different types of vanilla options (see ). Moreover, the case of floating barriers of exponential form can be easily accommodated by substitution of the relevant parameters (see , Chapter 11), thus we confine ourselves to the case of constant barriers. On the other hand, we work within a general framework that allows for multi-asset options, a generic payoff and rebate. Furthermore, we tackle some regularity questions and the problem of existence of generalized solutions. In Section 2 the (initial) boundary value problem is studied in a multidimensional framework generalizing the Black-Scholes equation and analytical solutions are obtained, while a comparison principle is provided in Section 4. Section 3 presents some applications in Finance: our general setting incorporates several known pricing expressions and, at the same time, allows to generate new valuation formulas. Section 5 and the Appendix study the existence and regularity of generalized solutions to the boundary value problems for a class of PDEs incorporating the Black-Scholes type. We build on the approach of Oleinik and Radkeviand adapt the method to the PDEs of interest in the financial applications.
2. Generalizations of the Black-Scholes equation in the multidimensional case: (initial) boundary value problems
Consider in the following generalization of the Black-Scholes equation:
This is the Cauchy problem:
and this is the boundary value problem:
In (1), , and
Our first step is to make in the non-hypoelliptic PDE the change of the space variables:
, , being the Kronecker symbol.
Thus, (1) takes the form:
In the case (2) we have
while in the case (3)
Denote, , .
In (6) we make the change of the unknown function in. Thus, after standard computations we get:
Let us take
and put. Put, ,. Then the scalar product , i.e. we assume that
where is given,.
The Cauchy problem (12) has initial condition
To find a formula (Poisson type) for the solution of the Cauchy problem (12), we must use some auxiliary results from the linear algebra. So let. Then the change of the independent variables, leads to, i.e.
One can easily guess that are the elements of the matrix and of course. On the other hand consider the elliptic quadratic form after the nondegenerate change. As we know one can find such a matrix that
being the unit matrix. Put now. Then .
This way the change transforms the Cauchy problem (12) to:
The solution of the Cauchy problem (14) is given by the formula
Going back to the old coordinates and the old function, we find -the solution of (2);, ,;.
We shall concentrate now on (3),.
Remark 1. To simplify the things, consider the quadratic form (elliptic), , , ,.
Then;. The change
leads to. Moreover, the first quadrant, is transformed under the linear transformation with matrix, into angle between the rays (straight lines ) and with opening. Evidently,.
Consequently, the transformation is not orthogonal for.
Let us now consider the boundary value problem (8). The above-proposed procedure yields:
The change in (17) yields:
, is a wedge with opening.
Now we use the linear transformation described in Remark 1, that maps the first quadrant, onto the angle between the rays and in the plane and we obtain:
, , is a wedge with opening, i.e., being the interior of the angle between,.
In fact, and implies that is transformed in. According to Remark 1:. Taking, i.e. we obtain that is mapped onto the angle between the rays. Of course, there are three possibilities:, ,.
From now on we shall make polar coordinates change in (19): and to fix the ideas let, , is the angle between and.
The new change and. To simplify the notation we shall write again instead of,. Thus we have a wedge type initial-boundary value problem for (19) with unknown function:
where and the modified Bessel function satisfies the equation:
Remark 2. One can see that, i.e. formally in the sense of Schwartz distributions,. is the corresponding Green function.
According to :
where the Green function.
3. Applications to financial options and numerical results via CNN
Here the analysis of Section 2 is applied to some problems arising in option pricing theory. Some known pricing formulas are revisited in a more general setting and some new results are offered. We apply Cellular Neural Networks (CNN) approach  in order to obtain some numerical results. Let us consider a two-dimensional grid with neighborhood system as it is shown on Figure 1.
[htb] One of the key features of a CNN is that the individual cells are nonlinear dynamical systems, but that the coupling between them is linear. Roughly speaking, one could say that these arrays are nonlinear but have a linear spatial structure, which makes the use of techniques for their investigation common in engineering or physics attractive.
We will give the general definition of a CNN which follows the original one:
Definition 1. The CNN is a
a). 2-, 3-, or - dimensional array of
b). mainly identical dynamical systems, called cells, which satisfies two properties:
c). most interactions are local within a finite radius, and
d). all state variables are continuous valued signals.
Definition 2. An cellular neural network is defined mathematically by four specifications:
1) CNN cell dynamics;
2) CNN synaptic law which represents the interactions (spatial coupling) within the neighbor cells;
3) Boundary conditions;
4) Initial conditions.
Now in terms of definition 2 we can present the dynamical systems describing CNNs. For a general CNN whose cells are made of time-invariant circuit elements, each cell is characterized by its CNN cell dynamics :
where, is usually a scalar. In most cases, the interactions (spatial coupling) with the neighbor cell are specified by a CNN synaptic law:
The first term of (26) is simply a linear feedback of the states of the neighborhood nodes. The second term provides an arbitrary nonlinear coupling, and the third term accounts for the contributions from the external inputs of each neighbor cell that is located in the neighborhood.
It is known  that some autonomous CNNs represent an excellent approximation to nonlinear partial differential equations (PDEs). The intrinsic space distributed topology makes the CNN able to produce real-time solutions of nonlinear PDEs. There are several ways to approximate the Laplacian operator in discrete space by a CNN synaptic law with an appropriate -template:
one-dimensional discretized Laplacian template:
two-dimensional discretized Laplacian template:
Example 1 (Single-asset inside barrier options) The case of single-barrier zero-rebate down-and-out options was already priced in , while the case with rebate is found in . A simple method for obtaining analytical formulas for barrier options is the reflection principle that has a long history in Physics and is commonly used in Finance. Here we write down the pricing formula for a general payoff and rebate and study its analytical properties. Let us consider the following boundary value problem:
where, and are continuous and. Using the notation of Section 2 and taking, ,we straightforwardly obtain the following pricing formula (after changing to variables):
Let us study the properties of analytically. Without loss of generality we can assume and therefore is written in the form with:
where and. We shall examine the asymptotics of for (i.e.) fixed and for (i.e.). Put,. Then:
According to Lebesgue’s dominated convergence theorem, since for each fixed and, one has. On the other hand:
Thus, for fixed , and, we have
const., which implies that Finally, we observe that:
as implies. The change yields
const, that is
constfor, fixed. Therefore we get:,
Remark 4. Assume that. Then, putting, , we get, Evidently, is the only solution of that Cauchy problem. So, with, is uniquely determined by.
For this example our CNN model is the following:
where is the convolution operator ,. We shall consider this model with free-boundary conditions:
These are classical first-order contact free-boundary conditions for obstacle problems.
Based on the above CNN model (28) we obtain the following simulations for different values of the parameters:
Example 2. (Multi-asset option with single barrier) Analytic valuation formulas for standard European options with single external barrier have been provided in Heynen-Kat (1994), Kwok-Wu-Yu (1998) and Buchen (2001). Here we give a slightly more general formula in that we allow for any payoff and for both an internal and an external barrier. We confine ourselves to the case of an upstream barrier and zero rebate for simplicity of exposition. Consider the following boundary value problem in:
where, is continuous and. Assume that,. Using the notation of Section 2 and taking for, , we have for, and,. Then we have the following pricing formula:
Splitting the integral into two integrals and changing to variables, () in the first (second) integral, one gets:
Note that. Then the first integral (after changing to variables,) is written in the form:
Changing to the variables, , the second integral becomes:
In the special case of standard options one has:,. Then can be written in the form:
where is the bivariate cumulative normal distribution function, , ,. Similarly is written in the form:
Simulating CNN for multi-asset option with single barrier model, we obtain the following figure with different values of the parameter set:
Example 3. (Two-asset barrier options with simultaneous barriers) While single-asset barrier options have received substantial coverage in the literature, multi-asset options with several barriers have been discussed only in some special cases (e.g. sequential barriers, radial options, etc.). Here we show how the case of two simultaneous barriers can be valued straightforwardly from the arguments in Section 2. Let us confine ourselves to zero-rebate options for simplicity’s sake, although Section 2 deals with the general case too. Then the boundary value problem takes the form:
where ;. Arguing as in the last part of Section 2 and taking, , ,
and as the opening of the angle between and, from (21) we have
where and is the modified Bessel function satisfying (22). Here where. Changing back the variables one obtains.
Simulating CNN for two-asset barrier options with simultaneous barriers model, we obtain the following figure with different values of the parameter set:
4. Comparison principle for multi-asset Black-Scholes equations
For the sake of simplicity consider
where, , are real constants and in the domain,. The boundary of the parallelepiped is split into two parts: Parabolic and free of boundary data part, where, ,. The Dirichlet data are prescribed on:
Theorem 1. (Comparison principle)Assume that uis a classical solution of (30), (31), i.e.u∈C2(D∪Γ-1)∩C0(D-). Let vbe another solution of (30), (31) belonging toC2(D∪Γ-1)∩C0(D-). Suppose thatu|Γ≤v|Γ. Then u≤veverywhere inD-.
Proof. Put Assume that,. Evidently, as.
Case a).. Having in mind that is a strictly elliptic operator in the open rectangle we shall apply the interior parabolic maximum principle ( see A.Friedman, Partial Differential equations of parabolic type, Prentice Hall, Inc. (1964), Chapter II). To do this we shall work in the domain, such that,. Then Th1 from Chapter II of the above mentioned book gives: for, and this is a contradiction with on.
Case b)., , , and a similar case with respect to,. Thus,
b). (1) is interior point of and therefore, , while as it is shown in Friedman book. Obviously, , as. As we know, (30) is satisfied on I -contradiction with,.
b). (2) Again and, as is interior point for the interval. According to (30) : contradiction.
b). (3) Then (30) takes the form: - contradiction.
Case c). ,;, , ,.
Certainly, in each case (1) -(4).
c). (1) As is interior point in the rectangle , ,. According to (30) - contradiction.
c). (2) As,. The contradiction is obvious.
c). (3) The equation (30) takes the form:
and again a contradiction.
c). (4). Then and according to (30) - contradiction.
We conclude that in in.
The comparison principle is proved.
Remark 5. The operator
is non-hypoelliptic. The constants, , are arbitrary. To verify this we recall that the function considered as a Schwartz distribution in satisfies for the following identities:
Consider now the distribution, where, ,. Then satisfies in distribution sense if
Of course, , i.e. is the boundary of the first octant of multiplied by. The nonhypoellipticity is proved. Evidently, under (4) is hypoelliptic in the open domain as it is strictly parabolic there.
5. The approach of Fichera-Oleinik-Radkevi
In this section we revise the results of  and  for the Dirichlet problem for PDEs of second order having non-negative characteristic form; then the method is applied to some PDEs of Black-Scholes type.
To begin with consider the following equation in a bounded domain with piecewise smooth boundary:
where, ,;,. Moreover, , ,. Denote the unit inner normal to by and let be the non-characteristic part of. Define, i.e. and is the characteristic part of. Following Fichera (1956) we introduce on the Fichera function:
Then we split into three parts, namely
As it is proved in Oleinik and Radkevi(1971) the sets, , , are invariant under smooth non-degenerate changes of the variables. More precisely, let in; after the change it takes the form in. Denote the Fichera function for by. Then where and is continuous.
Assume now that and. Then
and,. One can easily see that if we denote the Fichera function for by, then and is defined by (34).
Assume now that, at, and define the following set of test functions:. In view of the Green formula for we get:
for any and. Let us now recall the definitions of generalized solution.
Definition 3. The functionu∈Lp(Ω), p≥1, is called a generalized solution of the boundary value problem
if for each test function the following integral identity holds:
Theorem 2. (See ,Th.1.3.1).Suppose thatc<0, c*<0in Ωandp>1. Then for each f∈Lp(Ω)there exists a generalized solution u∈Lp(Ω)of (37) in the sense of (38) and such that
. The set
Theorem 3. (See , Th. 1.3.2).Let c<0inΩ¯, 1p+1q=1and -c+(1-q)c*>0 inΩ¯. Then for each f∈Lp(Ω)there exists a generalized solution uof (37) satisfying the a-priori estimate (39).
Theorem 4. (See , Th. 1.3.3).
Let in and in,. Then for each there exists a generalized solution of (37) satisfying the estimate (39).
We shall now discuss the problem for existence of a generalized solution of (37) in the Sobolev space with an appropriate weight. Define the following set of test functions:
and equip with the scalar product: The completion of with respect to the norm is a real Hilbert space denoted by. For each two functions we consider the bilinear form, where. According to the Cauchy-Schwartz inequality Therefore, is well defined for and.
Definition 4. Letf∈L2(Ω). We shall say that the function u∈His a generalized solution of (37) if for each v∈Wthe following identity is satisfied:
Theorem 5. (See , Th. 1.4.1).Assume that f∈L2(Ω)and 12∑k(bxkk-∑jaxkxjkj)-c≥c0>0 inΩ¯. Then the boundary value problem (37) possesses a generalized solution u∈H(i.e. a weak solution) in the sense of (40).
Finally we propose the existence of a generalized solution of (37) in the space To fix the ideas we assume that the coefficients of and belong to and is thrice piecewise smooth (i.e. can be split into several parts and each of them is smooth). Consider the boundary value problem:
If is a classical solution of (41) and then according to the Green formula
Definition 5. We shall say that the function u∈L∞(Ω)is a generalized solution of (41) if for each test function v∈Vthe identity (42) is fulfilled.
We point out that and
Theorem 6. (See , Th. 1.5.1).Assume that the coefficient c(x)of Lis such that c(x)≤-c0<0inΩ¯, f∈L∞(Ω), g∈L∞(Σ2∪Σ3)and β(x)≤0in the interior points ofΣ2∪Σ0. Then there exists a generalized solution of (41) in the sense of Definition 5. Moreover,u(x)≤max(supfc0,supg).
Remark 6. In Th.6 it is assumed that in an dimensional neighbourhood of,.
Theorem 7. (See , Th. 1.5.2).Suppose that gis continuous in the interior points ofΣ2∪Σ3. Then the generalized solution uof (41) constructed in Th. 6 is continuous at those points and, moreover, u=gthere.
As we shall deal with (degenerate) parabolic PDEs we shall have to work in cylindrical domains (rectangles in). Therefore will be piecewise smooth. Consider now the bounded domain having piecewise smooth boundary. The corresponding boundary value problem is:
We shall say that the point is regular if locally near to the surface can be written in the form, describing some neighborhood of the projection of onto the plane. The set of the boundary points which do not possess such a representation will be denoted by.
Definition 6. The function u∈L∞(Ω)is called a generalized solution of (43) for f∈L∞(Ω)if for each functionv∈C2(Ω¯), v=0at Σ1∪Σ3∪Bthe following identity holds:
Theorem 8. (See , Th. 1.5.5).Suppose that the boundary Σof the bounded domain Ωis C3 piecewise smooth, f∈L∞(Ω), g=0, c(x)≤-c0<0in Ω¯ and β≤0in the interior points ofΣ0∪Σ2. Then there exists a generalized solution uof (43) in the sense of Definition 6 and such thatu≤supfc0.
We shall not discuss here in details the problems of uniqueness and regularity of the generalized solutions. Unicity results are given by Theorems 1.6.1.-1.6.2. in . For domains with smooth boundary under several restrictions on the coefficients, including, in in the interior points of, at, the maximum principle is valid for each generalized solution in the sense of Definition 5:
In Th. 1.6.9. uniqueness result is proved for the boundary value problem (43) in the class. The existence result is given Th. 8. Regularity result is given in the Appendix.
Remark 7. Backward parabolic and parabolic operators satisfy the conditions:, , and if, i.e.. Put now in (33). Then
Having in mind that we conclude that for and then, uniformly in. So for parabolic (backward parabolic) equations the conditions of Theorems 2, 5 are fulfilled.
We shall illustrate the previous results by the backward parabolic equations:
which is the famous Black-Scholes equation, and
We shall work in the following rectangles:, ,. Under the previous notation for we have:, ,. Certainly, for, another part of the boundary appears,.
As we know from  there exists an solution of the boundary value problem
According to the Definition 3:for each test function,.
In a similar way there exists such that
Therefore:for each test function,.
Certainly, there exists such that for each test function,. Evidently, , , ,. Consequently, and, and thus the function
satisfies the identity, i.e. is a generalized solution of
We conclude as follows: (a) If satisfies
(b) In the special case when, , , , satisfies the identity, we have a uniqueness theorem and therefore.
The set is called interior boundary of.
One can find results concerning regularity of the generalized solutions of degenerate parabolic operators in cylindrical domains in  and . For the sake of simplicity we shall consider only one example from Il’in as the conditions are simple and clear. Consider
in the rectangle and, , ,. Moreover, we assume that in some domain
(i) the function and
(ii) Suppose that if (), , then ().
Moreover, we assume that the following compatibility conditions hold:
Define now the following parts of the boundary:
One can easily see that:, , , i.e., and on, on. Thus, while Therefore, ,. Evidently, , while;.
In conclusion, is free of data as it is of the type; and are of the type, while. Part of is non-characteristic, part of is of type. Data are prescribed on, i.e. on.
Theorem 9. (see ).There exists a unique classical solution uof (51), u|I∪II∪IV=0under the conditions (i), (ii), (iii). More specifically, there exists Lipschitz continuous derivatives:u, ∂u∂t, ∂u∂x, ∂2u∂x2∈C0,α(Q¯),0<α<1.
In  it is mentioned that under several restrictions on the coefficients the boundary value problem
possesses a unique generalized bounded solution which is Lipschitz continuous in. The proof relies on the method of elliptic regularization.
Remark 8. If, , , the Black-Scholes equation (44) is with on, i.e. and the equation
possesses a unique classical solution. As we know, satisfies in classical sense the ODE:
,. Therefore, we can consider the restrictions:, and conclude that they are classical solutions of the respective boundary value problems with data at, respectively at.
The authors gratefully acknowledge financial support from CNR/BAS.