Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

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-riskmodels 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 T 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 [25]).Moreover, the case of floating barriers of exponential form can be easily accommodated by substitution of the relevant parameters (see [25], 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 Radkevič and adapt the method to the PDEs of interest in the financial applications.

Generalizations of the Black-Scholes equation in the multidimensional case: (initial) boundary value problems
Consider in R 1  t × R n x the following generalization of the Black-Scholes equation: where 0 ≤ t ≤ T and x j ≥ 0, 1 ≤ j ≤ n.This is the Cauchy problem: and this is the boundary value problem: Our first step is to make in the non-hypoelliptic PDE L the change of the space variables: δ ij being the Kronecker symbol.
In the case (2) we have while in the case (3) Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach In (6) we make the change of the unknown function u : u = v(τ, y)e ∑ α i y i +βτ in (τ, y) ∈ D. Thus, after standard computations we get: Let us take and put 2Aα where b =( b1 ,..., bn ) is given, detA = 0.
To find a formula (Poisson type) for the solution of the Cauchy problem (12), v| τ=0 = v 0 (y) we must use some auxiliary results from the linear algebra.So let Mu = ∑ n i,j=1 a ij v y i y j .T h e n the change of the independent variables One can easily guess that ∑ i (∑ j a ij β kj )β li = ckl are the elements of the matrix B −1 A(B −1 ) * and of course (B −1 ) * =(B * ) −1 .On the other hand consider the elliptic quadratic form (Ax, x)= (C * ACy, y) after the nondegenerate change x = Cy.As we know one can find such a matrix I n being the unit matrix.Put now C =( This way the change y =(C −1 ) * z ⇒ z = B −1 y transforms the Cauchy problem (12) to: The solution of the Cauchy problem ( 14) is given by the formula [6] or [21]).Going back to the old coordinates (τ, x) and the old function u = ve ∑ α i y i +βτ ,w efi n d u(t, x)-the solution of ( 2 We the rays (straight lines ) l 1 : x ≥ 0 y = 0 and l 2 : Consequently, the transformation D is not orthogonal for a 12 = 0.
Let us now consider the boundary value problem (8).The above-proposed procedure yields: Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach The change , Ω is a wedge with opening π 2 .Now we use the linear transformation described in Remark 1, that maps the first quadrant λ 1 ≥ 0, λ 2 ≥ 0 onto the angle between the rays l 1 and l 2 in the plane 0 z 1 z 2 and we obtain: being the interior of the angle between l 1 , l 2 .
In fact, i.e.B = D * we obtain that {λ 1 ≥ 0, λ 2 ≥ 0} is mapped onto the angle ϕ 0 between the rays l 1 , l 2 .Of course, there are three possibilities: From now on we shall make polar coordinates change in (19): , ϕ 0 is the angle between l 2 and l 1 .

Remark 2. O n ec a ns e et h a tlim
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 a 12 = 0 in ( 16) we obtain (18) and after the change τ = τ, λ j = √ a jj z j ,1≤ j ≤ 2 (18) takes the form: Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach According to [21]: where the Green function ].

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 3 × 3 neighborhood system as it is shown on Figure 1.
(1, 1) (2, 2) [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 n-dimensional array of b). mainly identical dynamical systems, called cells, which satisfies two properties: c). most interactions are local within a finite radius r, and d
). all state variables are continuous valued signals.

Definition 2. An M × M 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 C(ij) is characterized by its CNN cell dynamics : where x ij ∈ R m , u ij is usually a scalar.In most cases, the interactions (spatial coupling) with the neighbor cell C(i + k, j + l) are specified by a CNN synaptic law: The first term A ij,kl x i+k,j+l 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 N r neighborhood.
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 A-template: -one-dimensional discretized Laplacian template: -two-dimensional discretized Laplacian template: Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach 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: r , u 0 and g are continuous and u 0 (S * )=g(T).U s i n gt h e notation of Section 2 and taking α σ we straightforwardly obtain the following pricing formula (after changing to variables σ )ds] Let us study the properties of u(t, S) analytically.Without loss of generality we can assume S * = 1 and therefore e −β(T−t) u(t, S)= u(t, S) is written in the form I 1 + I 2 + I 3 with: where y = ln S and τ = σ 2 2 (T − t).We shall examine the asymptotics of v(τ, y)= u(t, S) for 0 < τ < σ 2  2 T (i.e.0 < t < T)fixedandfory −→ − ∞ (i.e. S → 0 + ).Put h(ξ)=u 0 (e −ξ ), ξ ≥ 0. Then: According to Lebesgue's dominated convergence theorem, since lim y−→ − ∞ h(−y + 2aτ + 2η √ τ)=u 0 (0) for each fixed η and τ, one has lim y−→ − ∞ I 2 (τ, y)=e τα 2 u 0 (0).On the other hand: Thus, for fixed τ,0< τ < σ 2 2 T,andy << −1, we have , which implies that lim y−→ − ∞ I 3 (τ, y)=0.Finally, we observe that: Remark 4. Assume that u ∈ C 2 (Ω).Then, putting S = 0, U(t)=u(t,0),wegetU ′ (t)=rU, U(T)=u 0 (0).Evidently ,U(t)=u 0 (0)e −r(T−t) is the only solution of that Cauchy problem.So u | Σ 0 ,withΣ 0 = 0 < t < T, S = 0 + , is uniquely determined by u 0 (0).
For this example our CNN model is the following: where * is the convolution operator [24], M ≤ i, j ≤ M. 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   taking for i, j = 1, 2 and i = j, Then we have the following pricing formula: ] dλ 1 dλ 2 where Splitting the integral into two integrals and changing to variables ) in the first (second) integral, one gets: where Changing to the variables √ τ, the second integral becomes: In the special case of standard options one has: Then I 1 canbewrittenintheform: where N 2 is the bivariate cumulative normal distribution function, ).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: Arguing as in the last part of Section 2 and taking and ϕ 0 as the opening of the angle between where ( rξ 2τ )sin nπ ϕ 0 ϕsin nπ ϕ 0 η and I v is the modified Bessel function satisfying (22).Here Changing back the variables one obtains u(t, S 1 , S 2 ).
We conclude that The comparison principle is proved.

Remark 5. The operator
is non-hypoelliptic.The constants a ij , b i , c are arbitrary.To verify this we recall that the function s a + = s a , s > 0 0, s ≤ 0 considered as a Schwartz distribution in D ′ (R 1 ) satisfies for Re a > 1 the following identities: Consider now the distribution u = e λt u 1 (x 1 ) ... u n (x n ),whereλ = const, u j (x j )=x Of course, sing supp u = ∂{x ∈ R n : x j ≥ 0, 1 ≤ j ≤ n},i.e.sing supp u is the boundary of the first octant of R n x multiplied by R 1 t .The nonhypoellipticity is proved.Evidently, under (4) L is hypoelliptic in the open domain {x j > 0, 1 ≤ j ≤ n} as it is strictly parabolic there.

The approach of Fichera-Oleinik-Radkevič
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 Ω ⊂ R m with piecewise smooth boundary Σ: where ∑ k,j=1,...,m a kj (x)ξ k ξ j ≥ 0,∀x ∈ Ω, ∀ξ ∈ R m ; a kj (x)=a jk (x), ∀x ∈ Ω.M o r e o v e r , Denote the unit inner normal to Σ by − → n =( n 1 , ..., n m ) and let Σ 3 = x ∈ Σ; ∑ k,j=1,...,m a kj (x)n k n j > 0 be the non-characteristic part of Σ.D e fi n e Σ 0 = x ∈ Σ; ∑ k,j=1,...,m a kj (x)n k n j = 0 ,i.e.Σ = Σ 0 ∪ Σ 3 and Σ 0 is the characteristic part of Σ.Following Fichera (1956) we introduce on Σ 0 the Fichera function: Then we split Σ 0 into three parts, namely As it is proved in Oleinik and Radkevič (1971)   Assume now that u ∈ C 2 (Ω), u = 0atΣ 2 ∪ Σ 3 , and define the following set of test functions: In view of the Green formula for L we get: Assume that f ∈ L 2 (Ω) and 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 L ∞ (Ω).T ofi x the ideas we assume that the coefficients of L and L * belong to C 1 (Ω) and Σ is thrice piecewise smooth (i.e.Σ can be split into several parts and each of them is C 3 smooth).Consider the boundary value problem: is a classical solution of (41) and v ∈Vthen according to the Green formula 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.Remark 6.In Th.6 it is assumed that ∑ k,j=1,...,m a kj (x)ξ k ξ j ≥ 0i na nm−dimensional neighbourhood of Σ 0 , ∀ξ ∈ R m .
Suppose that g is continuous in the interior points of Σ 2 ∪ Σ 3 .Then the generalized solution u of (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 R 2 ).Therefore Σ = ∂Ω will be piecewise smooth.Consider now the bounded domain Ω having piecewise C 3 smooth boundary Σ.The corresponding boundary value problem is: We shall say that the point P ∈ Σ is regular if locally near to P the surface Σ can be written in the form x k = ϕ k (x 1 , ..., x k−1 , x k+1 , ..., x m ), (x 1 ,...,x k−1 , x k+1 ,...,x m ) describing some Suppose that the boundary Σ of the bounded domain Ω is C 3 piecewise smooth, f ∈ L ∞ (Ω),g= 0, c(x) ≤− c 0 < 0 in Ω and β ≤ 0 in the interior points of Σ 0 ∪ Σ 2 .Then there exists a generalized solution u of (43) in the sense of Definition 6 and such that |u| ≤ sup | f | c 0 .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 C 3 smooth boundary under several restrictions on the coefficients, including c(x) ≤−c 0 < 0, c * < 0inΩ, β ≤ 0 in the interior points of Σ 0 ∪ Σ 2 , β * = −β < 0atΣ 1 ,the maximum principle is valid for each generalized solution u in the sense of Definition 5: In Th. 1.6.9.uniqueness result is proved for the boundary value problem (43) in the class L ∞ (Ω).The existence result is given Th. 8. Regularity result is given in the Appendix.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: Ω 1 = {(t, x) :0< t < T,0 < x < a 1 }, Ω 2 = {(t, x) :0< t < T, a 2 < x < 0}, Ω = {(t, x) :0< t < T, a 2 < x < a 1 }.U n d e r t h e p r e v i o u s notation for Ω we have: Σ 1 = {t = 0}, Σ 2 = {t = T}, Σ 3 = {x = a 1 } ∪ {x = a 2 }.Certainly, for Ω 1 , Ω 2 another part of the boundary appears, Σ 0 = {x = 0}.
Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

Simulating
CNN for two-asset barrier options with simultaneous barriers model, we obtain the following figure with different values of the parameter set:

Definition 6 .
Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach neighborhood of the projection of P onto the plane x k = 0.The set of the boundary points which do not possess such a representation will be denoted by B. The function u ∈ L ∞ (Ω) is called a generalized solution of (43) for f ∈ L ∞ (Ω) if for each function v ∈ C 2 (Ω),v= 0 at Σ 1 ∪ Σ 3 ∪ B the following identity holds:Ω uL * (v)dx = Ω fvdx.Theorem 8. (See[20], Th. 1.5.5).
Using the notation of Section 2 and 41 Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach