## 1. Introduction

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 [3],[8] among others). As a consequence, a substantial body of academic literature provides pricing methods for valuating barrier options, starting from the seminal work of [18], where an exact formula is offered for a down-and-out European call with zero rebate. Further extensions are provided - among others - in [22] for the different types of standard barrier options, in [16] for simultaneous ’down’ and ’up’ barriers with exponential dependence on time, in [10] for two boundaries via Laplace transform, in [12] and [7] for partial barrier and rainbow options, in [17] for multi-asset options with an outside barrier, in [5] in a most comprehensive setting employing the image solution method. Many analytical formulas for barrier options are collected also in handbooks (see [11], 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 [1] and [15], for a survey). Furthermore, a recent literature relaxes the Brownian motion assumption and considers a more general Lévy framework. For example, [4] 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

## 2. Generalizations of the Black-Scholes equation in the multidimensional case: (initial) boundary value problems

Consider in

where

This is the Cauchy problem:

and this is the boundary value problem:

In (1)

Our first step is to make in the non-hypoelliptic PDE

Thus, (1) takes the form:

i.e.

In the case (2) we have

while in the case (3)

Denote

In (6) we make the change of the unknown function

Let us take

and put

where

In conclusion we solve the algebraic system (11):

The Cauchy problem (12) has initial condition

To find a formula (Poisson type) for the solution of the Cauchy problem (12),

One can easily guess that

This way the change

The solution of the Cauchy problem (14) is given by the formula

Going back to the old coordinates

We shall concentrate now on (3),

**Remark 1.** To simplify the things, consider the quadratic form (elliptic)

Then

leads to

Consequently, the transformation

Let us now consider the boundary value problem (8). The above-proposed procedure yields:

The change

Now we use the linear transformation described in Remark 1, that maps the first quadrant

In fact,

From now on we shall make polar coordinates change in (19):

The new change

Then

where

**Remark 2.** One can see that

Formula (21) is given in [21], pages 182 and 166 or in [6], pp.498. The proof of (21) is based on the properties of the Bessel functions and Hankel transform.

**Remark 3.** In the special case when

According to [21]:

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 [24] in order to obtain some numerical results. Let us consider a two-dimensional grid with

[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

b). mainly identical dynamical systems, called cells, which satisfies two properties:

c). most interactions are local within a finite radius

d). all state variables are continuous valued signals.

**Definition 2.** An

*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

where

The first term

It is known [24] 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

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 [18], while the case with rebate is found in [22]. 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

Let us study the properties of

where

According to Lebesgue’s dominated convergence theorem, since

Thus, for fixed

**Remark 4.** Assume that

For this example our CNN model is the following:

where

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

where

Splitting the integral into two integrals and changing to variables

where

Note that

Changing to the variables

In the special case of standard options one has:

where

where

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

and

where

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

**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

Case a).

Case b).

b). (1)

b). (2) Again

b). (3) Then (30) takes the form:

Case c).

Certainly,

c). (1) As

c). (2) As

c). (3) The equation (30) takes the form:

and again a contradiction.

c). (4). Then

We conclude that

The comparison principle is proved.

**Remark 5.** The operator

is non-hypoelliptic. The constants

Consider now the distribution

Of course,

## 5. The approach of Fichera-Oleinik-Radkevic ˘

In this section we revise the results of [9] and [20] 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

where

Then we split

As it is proved in Oleinik and Radkevi

Assume now that

where

and

Assume now that

for any

**Definition 3.** *The functionu∈Lp(Ω), p≥1, is called a generalized solution of the boundary value problem*

if for each test function

**Theorem 2.** (See [20],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*

**Theorem 3.** (See [20], 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 [20], Th. 1.3.3).

Let

**Conclusion.** Assume that

We shall now discuss the problem for existence of a generalized solution of (37) in the Sobolev space

and equip

**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 [20], 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

If

where

**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

**Theorem 6.** (See [20], 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

**Theorem 7.** (See [20], 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

We shall say that the point

**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 [20], 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 [20]. For domains with

In Th. 1.6.9. uniqueness result is proved for the boundary value problem (43) in the class

**Remark 7.** Backward parabolic and parabolic operators satisfy the conditions:

and

where

Having in mind that

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:

As we know from [20] there exists an

According to the Definition 3:

In a similar way there exists

Therefore:

Certainly, there exists

satisfies the identity

We conclude as follows: (a) If

*(b)* In the special case when

The set

### Appendix

One can find results concerning regularity of the generalized solutions of degenerate parabolic operators in cylindrical domains in [14] and [19]. 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

*(i)*

*(ii)* Suppose that if

Moreover, we assume that the following compatibility conditions hold:

*(iii)*

Define now the following parts of the boundary

One can easily see that:

In conclusion,

**Theorem 9.** (see [14]).

*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 [19] 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

**Remark 8.** If

possesses a unique classical solution. As we know,

### Acknowledgement

The authors gratefully acknowledge financial support from CNR/BAS.