Open access peer-reviewed chapter

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

By Rossella Agliardi, Petar Popivanov and Angela Slavova

Submitted: November 24th 2011Reviewed: May 17th 2012Published: September 12th 2012

DOI: 10.5772/49936

Downloaded: 1849

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 Tis 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 Radkevic˘and 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 Rt1×Rxnthe following generalization of the Black-Scholes equation:

Lu=ut+i,j=1naijxixjuxixj+i=1nbixiuxi+cu=f(t,x),E1

where 0tTandxj0,1jn.

This is the Cauchy problem:

Lu=f(t,x),u|t=T=u0(x),E2
xj0,1jn0tTE3

and this is the boundary value problem:

Lu=fu|t=T=u0(x)u|xj=aj=gj(t,x)|xj=aj,1jn,E4
0tT0xjaj,aj>01jnE5

In (1)aij=aji=const, bi=const, c=constand

i,j=1naijξiξjc0|ξ|2,c0=const>0.E6

Our first step is to make in the non-hypoelliptic PDE Lthe change of the space variables:

yj=lnxj,1jn,τ=T-tut=-uτ,yjR1E7

uxi=e-yiuyi, 2uxixj=e-yi-yj[2uyiyj-δijuyi], δijbeing the Kronecker symbol.

Thus, (1) takes the form:

L~u=-uτ+i,j=1naij2uyiyj+i=1nuyi(bi-aii)+cu=f,E8

i.e.

uτ=i,j=1naij2uyiyj+i=1nb~iuyi+cu-f;b~i=bi-aiiE9

In the case (2) we have

L~u=f,0τTu|τ=0=u~0(y)=u0(ey1,eyn),yRn,E10

while in the case (3)

L~u=f,0τTu|τ=0=u~0(y)u|yj=a~j=gj|yj=a~jE11

DenoteD={0τT,-<yjlnaj=a~j,1jn}, xj=eyj, 1jnf(t,x)=f(T-τ,ey1,,eyn).

In (6) we make the change of the unknown function u:u=v(τ,y)eαiyi+βτin(τ,y)D. Thus, after standard computations we get:

vτ+βv=i,jaijvyiyj+i,jaij(αivyj+αjvyi)+E12
+i=1nb~ivyi+i,j=1naijαiαjv+i=1nb~iαiv+cv-fe-αiyi-βτ.E13

Let us take

β=i,jaijαiαj+ib~iαi+cE14

and putf1=-fe-iαiyi-βτ. PutA=(aij)i,j=1n, A*=A,α=(α1,,αn). Then the scalar product (Aα,yv)=i,jaijαjvyi=i,jajiαivyj=i,jaijαivyj, i.e. we assume that

2(Aα,yv)+(b~,yv)=0E15
2Aα+b~=0,E16

where b~=(b~1,,b~n)is given,detA0.

In conclusion we solve the algebraic system (11): α=-12A-1(b~)and then we define βby (10). This way (9) takes the form:

vτ=i,j=1naijvyiyj+f1(τ,y)E17

The Cauchy problem (12) has initial condition

v0(y)=v|τ=0=u~0(y)e-iαiyi;u~0u0(ey1,,eyn),yRn.E18

To find a formula (Poisson type) for the solution of the Cauchy problem (12), v|τ=0=v0(y)we must use some auxiliary results from the linear algebra. So letMu=i,j=1naijvyiyj. Then the change of the independent variablesy=Bzz=B-1y, B-1=(βli)l,i=1nleads to2yiyj=k,l=1nβliβkj2zkzl, i.e.

Mu=k,l=1n(i(jaijβkj)βli)2uzkzl.E19

One can easily guess that i(jaijβkj)βli=c~klare the elements of the matrix B-1A(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 changex=Cy. As we know one can find such a matrix Cthat

C*AC=In,E20

Inbeing the unit matrix. Put nowC=(B-1)*C*=B-1. Then C*AC=InB-1A(B-1)*=InMu=k=1n2uzk2.

This way the change y=(C-1)*zz=B-1ytransforms the Cauchy problem (12) to:

vτ=k=1n2vzk2+f~1(τ,z),0τTv|τ=0=v0((C-1)*z)v~0(z),zRn.E21

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

v(τ,z)=1(2πτ)nRnv~0(λ)e-|z-λ|24τdλ+E22
+Rn0τf~1(Θ,λ)[2π(τ-Θ)]ne-|z-λ|24(τ-Θ)dλdΘ,E23

zRn, λRn|z-λ|2=i=1n(zi-λi)2(see [6] or [21]).

Going back to the old coordinates (τ,x)and the old functionu=veαiyi+βτ, we find u(t,x)-the solution of (2);t=T-τ, yj=lnxj,z=B-1y=B-1(lnx1,,lnxn);u=vx1α1xnαneβ(T-t).

We shall concentrate now on (3),n=2.

Remark 1. To simplify the things, consider the quadratic form (elliptic)Q=a11ξ2+2a12ξη+a22η2, a11>0, a22>0, a122-a11a22<0,Q=(A(ξη),(ξη)).

ThenQ=1a11(a11ξ+a12η)2+bη2;b=a22-a122a11>0. The change

xy=a11a12a110bξηE24

leads toQ=x2+y2. Moreover, the first quadrantξ0, η0is transformed under the linear transformation with matrixD=a11a12a110b, D-1=1a11-a12a11b01binto angle between the rays (straight lines ) l1:x0y=0and l2:x=a12a11ηy=bη0with openingφ0. Evidently,(D-1)*AD-1=I2.

Consequently, the transformation Dis not orthogonal fora120.

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

vτ=i,j=12aijvyiyj+f1(τ,y)v|τ=0=v0(y)=u0(ey1,ey2)e-αiyiv|y1=a~1=g1(T-τ,ey1,ey2)|y1=a~1e-βτa1-α1e-α2y2g~1(τ,y2)v|y2=a~2=g2(T-τ,ey1,ey2)|y2=a~2e-βτa2-α2e-α1y1g~2(τ,y1)E25
-<yj<lnaj=a~j0τTE26

The change λj=a~j-yj0,j=1,2τ=τin (17) yields:

v~τ=i,j=12aijv~λiλj+f~1(τ,λ)v~|τ=0=v~0(λ)=v0(a~1-λ1,a~2-λ2)e-i=12αi(a~i-λi)v~|λ1=0=g~1(τ,a~2-λ2)v~|λ2=0=g~2(τ,a~1-λ1),E27

Ω={0τT,λj0,j=1,2}, Ωis a wedge with openingπ2.

Now we use the linear transformation described in Remark 1, that maps the first quadrantλ10, λ20onto the angle between the rays l1and l2in the plane 0z1z2and we obtain:

wτ=wz1z1+wz2z2+f(τ,z)w|τ=0=w0(z)w|z1=0=g~1(τ,z1),(τ,z)Ω~w|l2=g~2(τ,z1,z2)|(z1,z2)l2,E28

l1:z1=0z2=λ2b, l2:z1=λ1a11z2=-a12ba11λ1, Ω~is a wedge with openingφ0, i.e.Ω~=[0,T]×Γ, Γbeing the interior of the angle betweenl1,l2.

In fact, λ=Bzz=B-1λand B-1A(B-1)*=I2implies that i,j=12aij2λiλjis transformed in2z12+2z22. According to Remark 1:(D-1)*AD-1=I2. TakingB-1=(D-1)*, i.e. B=D*we obtain that {λ10,λ20}is mapped onto the angle φ0between the raysl1,l2. Of course, there are three possibilities:φ0=π2, 0<φ0<π2,π2<φ0<π.

From now on we shall make polar coordinates change in (19): z1=rcosφz2=rsinφand to fix the ideas let0<φ0<π2, r0π2-φ0φπ2, φ0is the angle between l2andl1.

The new change Φ=φ-(π2-φ0)0Φφ0andΦ=φ. To simplify the notation we shall write again (r,φ)instead of(r,Φ),0Φφ0. Thus we have a wedge type initial-boundary value problem for (19) with unknown functionw(τ,r,φ):

wτ=2wr2+1rwr+1r22wφ2+f(τ,r,φ)w|τ=0=w0(r,φ)w|φ=0=g~~1(τ,r)w|φ=φ0=g~~2(τ,r)E29
r0, 0φφ0, l1:{φ=0,r0}, l2:{φ=φ0,r0}, rξ, φη, 0Θτ, 0ξ, 0ηφ0,0<φ0<π.

Then

w(τ,r,φ)=0τ0φ00f(Θ,ξ,η)G(r,φ,ξ,η,τ-Θ)ξdξdηdΘ+E30
+0τ0g~~1(Θ,ξ)1ξ[ηG(r,φ,ξ,η,τ-Θ)]η=0dξdΘ-E31
-0τ0g~~2(Θ,ξ)1ξ[ηG(r,φ,ξ,η,τ-Θ)]η=φ0dξdΘ+E32
+0φ00w0(ξ,η)G(r,φ,ξ,η,τ)ξdξdη,E33

where G(r,φ,ξ,η,τ)=1φ0τe-(r2+ξ2)4τn=1Inπφ0(rξ2τ)sinnπφ0φsinnπφ0ηand the modified Bessel function w=Iν(z)satisfies the equation:

z2d2wdz2+zdwdz-(z2+ν2)w=0,ν0,E34

Iν(z)=m=0(z2)2m+νm!Γ(m+ν+1)(see [2]).

Remark 2. One can see thatlimτ+00φ00w0(ξ,η)G(r,φ,ξ,η,τ)ξdξdη=w0(r,φ), i.e. formally limτ+0ξG(r,φ,ξ,η,τ)=δ(r-ξ,φ-η)in the sense of Schwartz distributionsD '(R+1×[0,φ0]),R+={ξ0}. Gis the corresponding Green function.

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 a12=0in (16) we obtain (18) and after the changeτ=τ, λj=ajjzj, 1j2(18) takes the form:

v~~τ=2v~~z12+2v~~z22+f~~1(τ,z)v~~|τ=0=v~~0(z)v~~|z1=0=g~~1(τ,z2)v~~|z2=0=g~~2(τ,z1)E35
0τT, zj0,1j2. Certainly,φ0=π2.

According to [21]:

v~~(τ,z)=0τ00f~~1(Θ,ξ,η)G(τ-Θ,z1,z2,ξ,η))dξdηdΘ+E36
+00v~~0(ξ,η)G(τ,z1,z2,ξ,η)dξdη+E37
+0τ0g~~1(Θ,η)[ξG(τ-Θ,z1,z2,ξ,η)]ξ=0dηdΘ+E38
+0τ0g~~2(Θ,ξ)[ηG(τ-Θ,z1,z2,ξ,η)]η=0dξdΘ,E39

where the Green functionG(τ,z1,z2,ξ,η)=14πτ[e-(z1-ξ)24τ-e-(z1+ξ)24τ]×e-(z2-η)24τ-e-(z2+η)24τ].

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 3×3neighborhood system as it is shown on Figure 1.

Figure 1.

3×3neighborhood CNN.

[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 radiusr, and

d). all state variables are continuous valued signals.

Definition 2. An M×Mcellular 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 :

x˙ij=-g(xij,uij,Iijs),E40

wherexijRm, uijis 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:

Iijs=Aij,klxi+k,j+l+E41
+A~ij,kl*fkl(xij,xi+k,j+l)+E42
+B~ij,kl*ui+k,j+l(t).E43

The first term Aij,klxi+k,j+lof (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 Nrneighborhood.

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:

A1=(1,-2,1),E44
  • two-dimensional discretized Laplacian template:

A2=0101-41010.E45

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:

Lu=0        in        Ω=(t,S);0<t<T,0<S<S*u|t=T=u0(S),0SS*u|S=S*=g(t),0tTE46

whereL=t+rSS+12σ2S2S2-r, u0and gare continuous andu0(S*)=g(T). Using the notation of Section 2 and takingα=12-rσ2, β=-r12+rσ2,C=2σwe straightforwardly obtain the following pricing formula (after changing to variablesσ2λ=lnS*-ξ):

u(t,S)=SS*αeβ(T-t)2πσ[1(T-t)0+u0(S*e-ξ)eαξ×E47
×[exp(-[ln(S/S*)+ξ]22σ2(T-t))-exp(-[ln(S/S*)-ξ]22σ2(T-t))]dξ+E48
+lnS*S0T-tg(T-s)(T-t-s)3/2e-βσ2s2exp(-ln2(S/S*)2σ2(T-t-s))ds]E49

Let us study the properties of u(t,S)analytically. Without loss of generality we can assume S*=1and therefore e-β(T-t)u(t,S)=u~(t,S)is written in the form I1+I2+I3with:

I1(τ,y)=-yeαy2π0τg(T-2γσ2)(τ-γ)3/2e-βγexp(-y24(τ-γ))dγE50
I2(τ,y)=eαy2πτ0+u0(e-ξ)eαξexp(-[y+ξ]24τ)dξE51
I3(τ,y)=-eαy2πτ0+u0(e-ξ)eαξexp(-[y-ξ]24τ)dξE52

where y=lnSandτ=σ22(T-t). We shall examine the asymptotics of v~(τ,y)=u~(t,S)for 0<τ<σ22T(i.e.0<t<T) fixed and for y-(i.e.S0+). Puth(ξ)=u0(e-ξ),ξ0. Then:

I2(τ,y)=eτα22πτ0+h(ξ)exp(-[y+ξ-2ατ]24τ)dξ=eτα2πy-2ατ2τ+h(-y+2aτ+2ητ)e-η2dηE53
.

According to Lebesgue’s dominated convergence theorem, since limy-h(-y+2aτ+2ητ)=u0(0)for each fixed ηandτ, one haslimy-I2(τ,y)=eτα2u0(0). On the other hand:

I3(τ,y)const2πτ0+eα(y+ξ)(-(ξ-y)24τ)dξ=E54

=const.

eτα2-y+2ατ2τ+exp[-μ2+2αy-4τα2+2αμτ]dμ=E55

=const.

e2αy-2τα2-y2τ+e-ε2dεE56
.

Thus, for fixed τ,0<τ<σ22T, andy<<-1, we have

I3(τ,y)const.e2αy-2τα2τ-ye-y24τ, which implies that limy-I3(τ,y)=0.Finally, we observe that:

I1(τ,y)maxg2πyeαy0τe-βτ(τ-γ)3/2exp(-y24(τ-γ))dγas β0implies0-βγ-βτ. The change θ=-y2τ-γyields

I1(τ,y)const.eαy-y2τ+e-θ2dθ, that is

I1(τ,y)const.eαy2τπye-y24τfory-, τfixed. Therefore we get:

limS0+u(t,S)=u0(0)e-r(T-t),
0<t<TE57
.

Remark 4. Assume thatuC2(Ω¯). Then, puttingS=0, U(t)=u(t,0), we getU'(t)=rU, U(T)=u0(0).Evidently, U(t)=u0(0)e-r(T-t)is the only solution of that Cauchy problem. Sou|Σ0, withΣ0=0<t<T,S=0+, is uniquely determined byu0(0).

For this example our CNN model is the following:

dSijdt+rSijA1*Sij+12σ2Sij2A2*Sij-r=0,E58

where *is the convolution operator [24],Mi,jM. We shall consider this model with free-boundary conditions:

uij(x,t)=x-k,uij(x,t)dt=+1,E59
uij(x,t)=k-x,uij(x,t)dt=-1.E60

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:

Figure 2.

CNN simulations for Example 1. (a)r=1, 1≤t≤30,σ=1; (b)r=0.5, 1≤t≤30,σ=1.5.

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 inΩ=(t,S1,S2);0<t<T,0<S1,0<S2<S*:

Lu=0u|t=T=u0(S1,S2)        0S2S*u|S2=S*=0        0tTE61

whereL=t+i=12σi2Si22Si2+ρσ1σ2S1S2S1S22+ri=12SiSi-r, u0is continuous andu0(S1,S*)=0. Assume thatσ1,σ2>0,ρ2<1. Using the notation of Section 2 and taking μi=r-σi22fori, j=1,2, we have αi=-μi+ρμjσi/σjσi2(1-ρ2)fori, j=1,2andij,β=i,j=1,2σiσj2αiαj+i=1,2μiαi-r. Then we have the following pricing formula:

u(t,S1,S2)=S1α1S2α2eβτ4πτR2w0(λ1,λ2)exp[-2lnS1σ11-ρ2-ρ2lnS2σ21-ρ2-λ124τ]E62
exp[-(2lnS2/σ2-λ2)24τ]-exp[-(2lnS2/σ2+λ2)24τ]dλ1dλ2E63

where

w0(λ1,λ2)=exp[-α1σ12(λ11-ρ2+ρλ2)-α2σ22λ2]u0(σ12(λ11-ρ2+ρλ2),σ2λ22)1λ2<2lnS*σ2.E64

Splitting the integral into two integrals and changing to variablesη1=λ11-ρ2+ρλ2-2lnS1σ12τ, η2=λ2-2lnS2σ22τ(η2=λ2+2lnS2σ22τ) in the first (second) integral, one gets:

u(t,S1,S2)=I1-I2E65

where

I1=eβτ2π1-ρ2-+-ln(S*/S2)σ2τexp[-(α1σ1η1+α2σ2η2)τ]u0(S1eσ1τη1,S2eσ2τη2)E66
exp[-(η1-ρη2)22(1-ρ2)-η222]dη1dη2E67
I2=S2eβτ2π1-ρ2-+-ln(S*/S2)σ2τexp[-(α1σ1η1+α2σ2η2)τ]u0(S1eσ1τη1,S2-1eσ2τη2)E68
exp[-(-η1+ρη2-2ρlnS2/(σ2τ))22(1-ρ2)-η222]dη1dη2.E69

Note that(β+r)(1-ρ2)+μ122σ12+μ222σ22-ρμ1μ2σ1σ2=0. Then the first integral (after changing to variablesX1=-η1+μ1σ1τ,X2=η2-μ2σ2τ) is written in the form:

I1=e-rτ2π1-ρ2-+--ln(S2/S*)+μ2τσ2τexp[-12(1-ρ2)(X12+X22+2ρX1X2)]u0(S1eμ1τ-σ1τX1,E70
S2eμ2τ+σ2τX2)dX1dX2E71
.

Changing to the variablesX1=-η1+μ1σ1τ-2ρlnS2σ2τ, X2=η2-μ2σ2τ, the second integral becomes:

I2=e-rτ2π1-ρ2(S2)-2μ2σ22-+-ln(S2S*)-μ2τσ2τexp[-12(1-ρ2)(X12+X22+2ρX1X2)]E72
u0(S1S2-2ρσ1σ2eμ1τ-σ1τX1,S2-1eμ2τ+σ2τX2)dX1dX2E73
.

In the special case of standard options one has:u0(S1,S2)=max(ω(S1-K),0),ω=±1. Then I1can be written in the form:

ωS1N2(ωd+,e+;-ρω)-ωKe-rτN2(ωd-,e-;-ρω)E74

where N2is the bivariate cumulative normal distribution function, d±=ln(S1K)+(r±σ122)τσ1τ, e-=-ln(S2S*)+μ2τσ2τ,e+=e--ρσ1τ. Similarly I2is written in the form:

ωe-2μ2σ22ln(S2S*)[e-2ρσ1σ2ln(S2S*)S1N2(ωd^+,e^+;-ρω)-Ke-rτN2(ωd^-,e^-;-ρω)E75

whered^±=d±-2ρσ2τln(S2S*),

e^±=e±+2σ2τln(S2S*)E76
.

Simulating CNN for multi-asset option with single barrier model, we obtain the following figure with different values of the parameter set:

Figure 3.

CNN simulations for Example 2. (a)r=1, T=60days, σ=1,ρ=0.05; (b)r=0.5, T=120days, σ=1.5,ρ=0.06.

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:

Lu=0        in        Ωu|t=T=u0(S1,S2)u|S1=S1*=0        and,  u|S2=S2*=0        0tTE77

where L=t+i=12σi2Si22Si2+ρσ1σ2S1S2S1S22+ri=12SiSi-r,Ω={(t,S1,S2);0<t<T,0<S1<S1*,0<S2<S2*}. Arguing as in the last part of Section 2 and taking

D=σ1ρσ201-ρ2σ2, ρ2<1, σ1>0,
σ2>0E78

and φ0as the opening of the angle between x0y=0andx=ρσ2η,η0y=1-ρ2σ2η, from (21) we have

w(τ,r,φ)=0φ00w0(ξ,η)G(r,φ,ξ,η,τ)ξdξdη,E79

where G(r,φ,ξ,η,τ)=1φ0τe-(r2+ξ2)4τn=1Inπφ0(rξ2τ)sinnπφ0φsinnπφ0ηand Ivis the modified Bessel function satisfying (22). Here w0(r,φ)=v~0(D*z)|z1=rcosφ,z2=rsinφwherev~0(λ)=u0(S1*e-λ1,S2*e-λ2)e-Σαi(lnSi*-λi). Changing back the variables one obtainsu(t,S1,S2).

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

Figure 4.

CNN simulations for Example 3. (a)r=1, T=120days, σ=1,ρ=0.05; (b)r=0.5, T=180days, σ=1.5,ρ=0.06.

4. Comparison principle for multi-asset Black-Scholes equations

For the sake of simplicity consider

ut+i,j=12aijxixjuxixj+i=12bixiuxi+cu=f,E80

where(aij)*=(aij), (aij)>0, aij,bi,care real constants and c<0in the domainD:0<t<T0<xj<aj,j=1,2,aj=const>0. The boundary of the parallelepiped Dis split into two parts: Parabolic Γ={x1=a1,0<x2<a2,0<t<T}{x2=a2,0<x1<a1,0<t<T}{t=T,0<xj<aj,j=1,2}and free of boundary data partΓ1=IIIIII, whereI={0<xj<aj,j=1,2;t=0}, II={x1=0,0<x2<a2,0<t<T},III={x2=0,0<x1<a1,0<t<T}. The Dirichlet data are prescribed onΓ:

u|Γ=gE81

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 w=u-v.Assume thatmaxw=w(t0,x0)=M>0,P0=(t0,x0)D-. Evidently, (t0,x0)DΓ1asw|Γ0.

Case a).(t0,x0)D. Having in mind that aijxixjwxixjis a strictly elliptic operator in the open rectangle {0<xj<aj,j=1,2}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 domainD1:0<t<T0<εj<xj<aj,j=1,2, such thatx0Π=(ε1,a1)×(ε2,a2),0<t0<T. Then Th1 from Chapter II of the above mentioned book gives: wM>0forTtt0, xΠ-and this is a contradiction with w0ont=T.

Case b).(t0,x0)It0=0, (1)    0<x10<a10<x20<a2, (2)      0<x10<a1x20=0, (3)      x10=0x20=0and a similar case with respect tox200,a2),x10=0. Thus,

b). (1) x0is interior point of (0,a1)×(0,a2)and thereforewxj(P0)=0, j=1,2, while i,j2aijxi0xj02wxixj(P0)0as it is shown in Friedman book. Obviously, wt(P0)0, asw(0,x0)=M=maxD-w. As we know, (30) is satisfied on I 12aijxi0xj02wxixj(P0)+cw(P0)+wt(P0)=0-contradiction withc<0,w(P0)>0.

b). (2) Again wt(P0)0andwx1(P0)=0, wx1x1(P0)0as P0is interior point for the interval(0,a1). According to (30) : a11x1022wx12(P0)+b1wx1(P0)+cw(P0)+wt(P0)=0contradiction.

b). (3) Then (30) takes the form: cw(P0)+wt(P0)=0- contradiction.

Case c). (t0,x0)II0t0<T,x10=0;(1)      0<t0<T0<x20<a2, (2)      t0=00<x20<a2, (3)      t0=0x20=0,(4)      T>t0>0x20=0.

Certainly, wt(P0)0in each case (1) -(4).

c). (1) As P0is interior point in the rectangle {0<t<T}×{0<x2<a2}wt(P0)=0, wx2(P0)=0,wx2x2(P0)0. According to (30) a2x202wx2x2(P0)+b2x20wx2(P0)+cw(P0)+wt(P0)=0- contradiction.

c). (2) Asx20(0,a2)wx2(P0)=0,wx2x2(P0)0. The contradiction is obvious.

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

cw(P0)+wt(P0)=0E82

and again a contradiction.

c). (4). Then wt(P0)=0and according to (30) cw(P0)+wt(P0)=0- contradiction.

We conclude that M=supD-w0u-v0in D-uvinD-.

The comparison principle is proved.

Remark 5. The operator

Lu=ut+i,j=1naijxixjuxixj+i=1nbixiuxi+cuE83

is non-hypoelliptic. The constantsaij, bi, care arbitrary. To verify this we recall that the function s+a=sa,s>00,s0considered as a Schwartz distribution in D '(R1)satisfies for Re  a>1the following identities:

ss+a=s+a+1,ddss+a=as+a-1,d2ds2s+a=a(a-1)s+a-2.E84

Consider now the distributionu=eλtu1(x1)un(xn), whereλ=const, uj(xj)=xjdjD '(Rxj1),Redj>1. Then uD '(Rn+1)satisfies in distribution sense Lu=0if

λ+ijnaijdidj+i=jnaiidi(di-1)+i=1nbidi+c=0E85

Of course, sing  supp  u={xRn:xj0,1jn}, i.e. sing  supp  uis the boundary of the first octant of Rxnmultiplied byRt1. The nonhypoellipticity is proved. Evidently, under (4) Lis hypoelliptic in the open domain {xj>0,1jn}as it is strictly parabolic there.

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 ΩRmwith piecewise smooth boundaryΣ:

L(u)=k,j=1,...,makj(x)uxkxj+k=1,...,mbk(x)uxk+c(x)u=f(x)E86

wherek,j=1,...,makj(x)ξkξj0, xΩ¯,ξRm;akj(x)=ajk(x),xΩ. Moreover, akjC2(Ω¯), bkC1(Ω¯),cC0(Ω¯). Denote the unit inner normal to Σby n=(n1,...,nm)and let Σ3=xΣ;k,j=1,...,makj(x)nknj>0be the non-characteristic part ofΣ. DefineΣ0=xΣ;k,j=1,...,makj(x)nknj=0, i.e. Σ=Σ0Σ3and Σ0is the characteristic part ofΣ. Following Fichera (1956) we introduce on Σ0the Fichera function:

β(x)=k=1,...,m(bk(x)-j=1,...,maxjkj(x))nk,xΣ0E87

Then we split Σ0into three parts, namely

Σ1=xΣ0;β(x)>0E88
,
Σ2=xΣ0;β(x)<0E89
,
Σ0=xΣ0;β(x)=0E90
.

As it is proved in Oleinik and Radkevic˘(1971) the setsΣ0, Σ1, Σ2, Σ3are invariant under smooth non-degenerate changes of the variables. More precisely, let L(u)=finΩ; after the change y=F(x)it takes the form L~(u~)=f~inΩ~. Denote the Fichera function for L~(u~)=f~byβ~. Then β~=β.Awhere A>0and Ais continuous.

Assume now that uC2(Ω)andvC0(Ω). Then

ΩL(u)vdx=ΩuL*(v)dxE91
,

where

L*(v)=k,j=1,...,makj(x)vxkxj+k=1,...,mb*k(x)vxk+c*(x)vE92

andb*k=2j=1,...,maxjkj-bk,c*=k=1,...,m(j=1,...,maxkxjkj-bxkk)+c. One can easily see that if we denote the Fichera function for L*(v)byβ*, then β*=-βand βis defined by (34).

Assume now thatuC2(Ω¯), u=0atΣ2Σ3, and define the following set of test functions:V=vC2(Ω¯);v=0  at    Σ1Σ3. In view of the Green formula for Lwe get:

Ω(L(u)v-L*(v)u)dx=0ΩL(u)vdx=ΩuL*(v)dxE93

for any uandvV. 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

L(u)=fin      Ωu=0at      Σ2Σ3E94

if for each test function vVthe following integral identity holds:

Ωfvdx=ΩuL*(v)dx.E95

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
infu0Zu+u0Lp(Ω)KfLp(Ω)E96

K=const>0. The set Z=u0Lp(Ω):Ωu0L*(v)dx=0,vV.

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 c*<0in Ω¯and -c+(1-q)c*>0inΩ¯,1p+1q=1. Then for each fLp(Ω)there exists a generalized solution uof (37) satisfying the estimate (39).

Conclusion. Assume thatc<0. Then (37) is solvable in the sense of Definition 1 for p>>1as p+q1. On the other hand, c*<0implies the solvability of (40) forp1, p1as p1q+.

We shall now discuss the problem for existence of a generalized solution of (37) in the Sobolev space H1(Ω)with an appropriate weight. Define the following set of test functions:

W={vC1(Ω¯);v|Σ3=0}E97

and equip Wwith the scalar product: (u,v)H=Ω(k,jakjuxjvxk+uv)dx+Σ1Σ3uvβdσ.The completion of Wwith respect to the norm uHis a real Hilbert space denoted byH. For each two functions u,vWwe consider the bilinear formB(u,v)=-Ω[k,jakjuxjvxk+k(ulkvxk+(lxkk-c)uv)]dx-Σ1uvβdσ, wherelk=bk-jaxjkj. According to the Cauchy-Schwartz inequality B(u,v)const[vH1(Ω)+vL2(Σ1)]uH.Therefore, B(u,v)is well defined for vWanduH.

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:

Ωvfdx=B(u,v).E98

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 L(Ω).To fix the ideas we assume that the coefficients of Land L*belong to C1(Ω¯)and Σis thrice piecewise smooth (i.e. Σcan be split into several parts and each of them is C3smooth). Consider the boundary value problem:

L(u)=fin      Ωu=gon      Σ2Σ3E99

If uC2(Ω¯)is a classical solution of (41) and vVthen according to the Green formula

ΩL*(v)udx=Ωfvdx-Σ3gvνdσ+Σ2βgvdσ,E100

whereν=(ν1,,νm), νk=jakjnj,

ν=kνkxkE101
.

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 fL(Ω)and gL(Σ2Σ3).

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 k,j=1,...,makj(x)ξkξj0in an m-dimensional neighbourhood ofΣ0,ξRm.

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 inR2). Therefore Σ=Ωwill be piecewise smooth. Consider now the bounded domain Ωhaving piecewise C3smooth boundaryΣ. The corresponding boundary value problem is:

L(u)=fin      Ω,u=0on      Σ2Σ3E102

We shall say that the point PΣis regular if locally near to Pthe surface Σcan be written in the formxk=φk(x1,...,xk-1,xk+1,...,xm), (x1,,xk-1,xk+1,,xm)describing some neighborhood of the projection of Ponto the planexk=0. The set of the boundary points which do not possess such a representation will be denoted byB.

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:

ΩuL*(v)dx=Ωfvdx.E103

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 C3smooth boundary under several restrictions on the coefficients, includingc(x)-c0<0, c*<0in Ω¯,β0in the interior points ofΣ0Σ2, β*=-β<0atΣ1, the maximum principle is valid for each generalized solution uin the sense of Definition 5:

umaxsupΩfc0,supΣ3Σ2gE104
.

In Th. 1.6.9. uniqueness result is proved for the boundary value problem (43) in the classL(Ω). The existence result is given Th. 8. Regularity result is given in the Appendix.

Remark 7. Backward parabolic and parabolic operators satisfy the conditions:akm=0, k=1,...,m, and bm=±1ifx=(x1,...,xm-1,t), i.e.t=xm. Put now u=veαtin (33). Then

L1(v)=k,j=1,...,makjvxkxj+k=1,...,mbkvxk+(c+α)v=fe-αtE105

and

L1*(w)=k,j=1,...,makjwxkxj+k=1,...,mb*kwxk+c1*wE106

wherec1=c+α, b*k=2j=1,...,maxjkj-bk,

c1*=k,j=1,...,maxkxjkj-k=1,...,mbxkk+c+αE107
.

Having in mind that cc~=constwe conclude that for bm=±1and αthenc1-, c1*-uniformly in(x1,...,xm-1,t)Ω. 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:

L(u)=ut+12σ2x22ux2+rxux-ru=f(t,x)E108

which is the famous Black-Scholes equation, and

M(u)=ut+x22ux2+b(x)ux+c(x)u=f(t,x)E109

We shall work in the following rectangles:Ω1=(t,x):0<t<T,0<x<a1, Ω2=(t,x):0<t<T,a2<x<0,Ω=(t,x):0<t<T,a2<x<a1. Under the previous notation for Ωwe have:Σ1=t=0, Σ2=t=T,Σ3=x=a1x=a2. Certainly, forΩ1, Ω2another part of the boundary appears,Σ0=x=0.

As we know from [20] there exists an Lp(Ω1)solution of the boundary value problem

L(u1)=f      Ω1u1=0on      Σ2(1)Σ3(1)E110

According to the Definition 3:Ω1u1L*(v1)dx=Ω1fv1dxfor each test functionv1C2(Ω¯1),v1|Σ1(1)Σ3(1)=0.

In a similar way there exists u2Lp(Ω2)such that

L(u2)=fin      Ω2u2=0on      Σ2(2)Σ3(2)E111

Therefore:Ω2u2L*(v2)dx=Ω2fv2dxfor each test functionv2C2(Ω¯2),v2|Σ1(2)Σ3(2)=0.

Certainly, there exists uLp(Ω)such that ΩuL*(v)dx=Ωfvdxfor each test functionvC2(Ω¯),v|Σ1Σ3=0. Evidently, vC2(Ω¯), v|Σ1Σ3=0vC2(Ω¯i), v|Σ1(i)Σ3(i)=0,i=1,2. Consequently, Ω1u1L*(v)dx=Ω1fvdxandΩ2u2L*(v)dx=Ω2fvdx, and thus the function

W=u1in      Ω1u2in      Ω2Lp(Ω)E112

satisfies the identityΩfvdx=Ω1fvdx+Ω2fvdx=ΩWL*(v)dx, i.e. Wis a generalized Lp(Ω)solution of

L(W)=fin      ΩW=0on      Σ2Σ3E113

We conclude as follows: (a) If uisatisfies

L(ui)=fin      Ωiui=0on      Σ2(i)Σ3(i)E114

i=1,2, then (48) satisfies (49).

(b) In the special case whenfL(Ω), uiL(Ωi), i=1,2, uL(Ω), usatisfies the identityΩfvdx=ΩuL*(v)dx, we have a uniqueness theorem and thereforeu=W.

The set Σ0is called interior boundary ofΩ.

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

N(u)=ut+h(t,x)2ux2+g(t,x)ux+c(t,x)u=F(t,x)E115

in the rectangle Q=(t,x):0<t<T,a2<x<a1andh, g, c,FC3(Q-). Moreover, we assume that in some domain

(i) Q'Q-the function h0and hC2(Q').

(ii) Suppose that if h(t,a1)=0(h(t,a2)=0), 0tT, then g(t,a1)>0(g(t,a2)<0).

Moreover, we assume that the following compatibility conditions hold:

(iii)Dt,xαF(T,a1)=Dt,xαF(T,a2)=0,

α2E116
.

Define now the following parts of the boundaryQ:

I={(t,x):0<t<T,x=a2},II={(t,x):0<t<T,x=a1),E117

III={(t,x):a2<x<a1,t=0}and

IV={(t,x):a2<x<a1,t=T}E118
.

One can easily see that:Σ3=(t,x)III:h(t,x)>0, Σ0={(t,x)III:h(t,x)=0}(t,x)IIIIV, β=gn1+n2-hxn1, i.e.(t,x)Σ0, (t,x)IIIh(t,x)=0hx=0and n=(1,0)onI, n=(-1,0)onII. Thusβ|IΣ0=gn1=g<0, while β|IIΣ0=-g<0.Therefore, IΣ0Σ2,IIΣ0Σ2. Evidently, β|III=n2=1IIIΣ1, whileIVΣ2;Σ0=.

In conclusion, IIIis free of data as it is of the typeΣ1; (III)Σ0and IVare of the typeΣ2, whileΣ3=(III)h>0. Part of IIIis non-characteristic, part of IIIis of Σ2type. Data are prescribed onΣ2Σ3, i.e. onIIIIV.

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

N(u)=0u|IIIIV=0E119

possesses a unique generalized bounded solution which is Lipschitz continuous inQ¯. The proof relies on the method of elliptic regularization.

Remark 8. Ifa2<x<a1, a2<0, a1>0, the Black-Scholes equation (44) is with h(t,x)=σ22x2>0onIII, i.e. Σ3=IIIand the equation

L(u)=f      in      Qu|IIIIV=0E120

possesses a unique classical solution. As we know, u|x=0=U(t)satisfies in classical sense the ODE:

U'(t)-rU(t)=f(t,0),U(T)=0. Therefore, we can consider the restrictions:u|x>0, u|x<0and conclude that they are classical solutions of the respective boundary value problems with 0data atΣ2(1)Σ3(1), respectively atΣ3(2)Σ2(2).

Acknowledgement

The authors gratefully acknowledge financial support from CNR/BAS.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Rossella Agliardi, Petar Popivanov and Angela Slavova (September 12th 2012). Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach, Risk Management - Current Issues and Challenges, Nerija Banaitiene, IntechOpen, DOI: 10.5772/49936. Available from:

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