Fitted Finite Volume Method for a Generalized Black-Scholes Equation Transformed on Finite Interval

8/13/2019 Fitted Finite Volume Method for a Generalized Black-Scholes Equation Transformed on Finite Interval

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Numerical Algorithms manuscript No.(will be inserted by the editor)

Fitted Finite Volume Method for a Generalized

Black-Scholes Equation Transformed on Finite Interval

Radoslav Valkov

Received: date / Accepted: date

Abstract A generalized Black-Scholes equation is considered on the semi-

axis. It is transformed on the interval (0, 1) in order to make the computa-tional domain finite. The new parabolic operator degenerates at the both endsof the interval and we are forced to use the Garding inequality rather than theclassical coercivity. A fitted finite volume element space approximation is con-structed. It is proved that the time -weighted full discretization is uniquelysolvable and positivity-preserving. Numerical experiments, performed to illus-trate the usefulness of the method, are presented.

Keywords generalized Black-Scholes equation degenerate parabolicequation Garding coercivity fitted finite volume method positivity

1 Introduction

The famous equation, proposed by F. Black, M. Scholes and R. Merton, see[7,14,20], is

V

t +

1

22S2

2V

S2 + r

V

S SV

= 0, S(0, ), t [0, T).

This is a typical example of a degenerateparabolic equation[15]. It is wellknown [14,20, 22] that it can be transformed to the heat equation that allowsus to overcome the degeneracy at S= 0. Many numerical methods, based onclassical finite difference schemes, applied to constant coefficients heat equa-tions, are developed [1, 16]. However, when the problem has space-dependantcoefficients and r one can not transform the Black-Scholes equation to a

Radoslav ValkovFaculty of Mathematics and Informatics, University of Sofia,1000 Sofia, Bulgaria,E-mail: [email protected]

arX

iv:1211.1903v3[math.NA]30Jun2013


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2 Radoslav Valkov

standard heat equation. Finite difference and finite element methods have beenapplied in [2,3,4,5, 6,8,9,11, 17,22] in order to solve this type of generalizedBlack-Scholes equations. In [12] cubic B-splines are implemented. Often, theconvergence of the full discretization is verified by numerical examples only.

An effective method, that resolves the degeneracy, is proposed by S. Wang[21] for the Black-Scholes equation with Dirichlet boundary conditions. Themethod is based on a finite volume formulation of the problem, coupled with afitted local approximation to the solution and an implicit time-stepping tech-nique. The local approximation is determined by a set of two-point boundaryvalue problems (BVPs), defined on the element edges. This fitted techniqueoriginates from one-dimensional computational fluid dynamics [13].

A modification of the discretization, originally presented in [21], was pro-posed by L. Angermann[2]such that the method adequately treats the proper(natural) boundary condition at x = 0. Similar space discretization is derivedin [6]for a degenerate parabolic equation in the zero-coupon bond pricing.

The domain of S is the half real line. For numerical computation it isdesirable to have a finite computational domain. The transformation in the

next section converts S (0, ) to x (0, 1), decreasing significantly thecomputational costs. Also, for a call option, the solution V(S, t) is not boundedand from the numerical methods point of view the problem transformationon a finite interval is better. The resulting equation has variable coefficientsbut this is not an essential difficulty for the numerical computation. However,the transformed equation degeneratesat both ends of the finite interval.

The present paper deals with a degenerate parabolic equation, (5), derivedafter transformation of the generalized Black-Scholes equation (1) to a finiteinterval. The degeneration at the both ends of the interval does not allow theuse of the Poincare-Friedrichs inequality and we are forced to investigate thedifferential problem with the Garding inequality rather than classical coerciv-ity[10].

This paper is organized as follows. The model problem is presented inSection 2, where we discuss our basic assumptions and some properties of thesolution. The space discretization method is developed in Section 3. Section 4is devoted to the time discretization, where we show that the system matrixon each time-level is an M-matrix so that the discretization is monotone.Numerical experiments are discussed in the last section.

Some results, concerning the case of the transformed Black-Scholes equa-tion above, are reported in[18].

2 The transformed problem

We consider the generalized Black-Scholes equation [14,20]:

V

t+

1

22(t)S2

2V

S2+(r(t)SD(S, t))

V

S r(t)V = 0, (S, t) (0, )(0, T),

(1)


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Title Suppressed Due to Excessive Length 3

where = (t) denotes the volatility of the asset, r = r(t) is the risk-freeinterest rate, D = D(S, t) denotes the dividendof the dividend-paying asset.We also introduced = d(S, t) such thatD(S, t) =Sd(S, t), wherethe dividendrate d= d(S, t) is continuously differentiable with respect to S.

There are various choices for the final (payoff) condition, depending on themodels. In the case of vanilla European option

V(S, T) =

max(S E, 0) for a call option,max(E S, 0) for a put option,

(2)

where E is the strike price. Another example is the bullish vertical spreadpayoff, defined by

V(S, T) =max(S E1, 0) max(S E2, 0), (3)

whereE1 and E2 are two exercise prices, satisfying 0< E1< E2. This repre-sents a portfolio of buying one call option with exercise price E1 and issuingone call option with the same expiry date but a larger exercise price E2. For

detailed discussion on this, we refer to [20].We introduce the transformation [22]

x= S

S+ Pm, u(x, t) =

V(S, t)

S+ Pm, =T t. (4)

The constantPm is called mesh parameter. It controls the distribution of themesh nodes w.r.t. S on the interval (0, ). The higher the value of S, thatwe are interested in, the higher value of Pm should be in order to obtain areasonable accuracy. In the case of a call option, because of the nature of theterminal condition, Pm should be equal to E.

The inverse transformation is S=Pmx/(1 x) and after plugging it in theBlack-Scholes equation, (1), we obtain:

u

t

1

22(t)x2(1 x)2

2u

x2 x(1 x)(r(t) d(x, t))

u

x+ ((1 x)r(t) + xd(x, t))u= 0, (x, t) T = (0, T), = (0, 1).

(5)

We return to the original notation of the variabletfor the sake of simplicity.The initial data for a call option reads

u(x, 0) =u0(x) = max(2x 1, 0). (6)

Being different from the classical parabolic equations, in which the principlecoefficient is assumed to be strictly positive, the parabolic equation(5) belongsto the second-order differential equations with non-negative characteristic form

[15]. The main difficulty of such kind of equations is the degeneracy [19]. Itcan be easily seen that at x = 0 and x = 1 (5) degenerates to

u

t

x=0

=r(t)u(0, t), u

t

x=1

=d(1, t)u(1, t). (7)


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4 Radoslav Valkov

It is well known by the Fichera theory for degenerate parabolic equations[15] that at the degenerate boundaries x = 0 and x= 1 the boundary condi-tions should not be given.

For theoretical analysis of our discrete problem as well as for the construc-

tion of a fitted finite volume mass-lumping discretization we need to considerweak solutions of (5). We shall use the standard notations for the functionspacesCm() andCm() of which a function and its derivatives up to ordermare continuous on (respectively ). The space of square-integrable func-tions we denote by L2() with the usual L2-norm and the inner product(, ). We also use the function space L() with the norm . To handlethe degeneracy in (5), we introduce the following weighted L2w-norm

v0,w := (

10

x2(1 x)2v2dx)1/2

with corresponding inner product (u, v)w. UsingL2() andL2w(), we define

the weighted Sobolev space H1w() := {v : v L2(), v L2w()}, where

v

denotes the weak derivative ofv. Let 1,w be the functional on H1w(),

defined byv1,w = (v20 + v

20,w)1/2. Then it is easy to see that 1,w is a

norm onH1(); it is called weightedH1-norm onH1w(). Furthermore, usingthe inner products inL2() andL2w(), we define a weighted inner product onH1w() by (, )H := (, )+(, )w and, consequently, the pair (H

1w(), (, )H) is

a Hilbert space. Also, H1w() contains the conventional Sobolev space H1()

as a proper subspace.We rewrite (5) in divergent form

u

t

x

x(1 x)

a(x, t)

u

x+ b(x, t)u

+ c(x, t)u= 0,

a(x, t) =

1

2 2

(t)x(1 x), b(x, t) =r(t) d(x, t) + 2

(t)(2x 1),

c(x, t) = (2 3x)r(t) (6x2 6x + 1)2(t) (1 3x)d(x, t) x(1 x)d

x.

(8)

Let us introduce for w, v H1w() the bilinear form

A(w, v; t) := (aw + bw,v) + (cw,v) = (x(1 x)(w), v) + (cw,v).

Here the notation w = wx is used and the function (w) = aw +bw is

the weighted flux density, associated with w. We are in position to state thevariational formulation of problem (5),(6):

Findu(t) H1w(), such that for allv H1w()

u(t)t

, v+ A(u(t), v; t) = 0 a.e. in (0, T) andu(, 0) =u0. (9)The following result provides the weak coercivity and continuity of the

bilinear form A(, , t).


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Lemma 1 Letw, v C1([0, 1]). Then:

1. there exist constantsC1> 0 andC2> 0 such that

|A(w,v,t)| C1wH1w(0,1)vH1(0,1)

|A(w,v,t)| C2wH1(0,1)vH1w(0,1)

for anyt [0, T];2. (Garding inequality) there exist constants > 0 and >0 such that

|A(v,v,t)| v2H1w(0,1) v2L2(0,1),

uniformly with respect to t [0, T].

ProofThe proof is given in [10].

Owing to Lemma1 one can prove the following assertion[10].

Theorem 1 Suppose that u0(x) H1w(). Then the problem (9) has an

unique solution.Theorem 2 (Weak maximum principle) Let u L2

0, T; H1w()

be

such that

1. ut L2 ((0, 1) (0, T));

2. the inequalityu(t)

t , v

+ A(u(t), v; t) 0, v C0 (0, 1) , v 0

holds for a.e. [0, T];3. u|t=0 0.

Thenu 0 a.e. in (0, 1) (0, T).

ProofThe proof is given in [10].

3 Space discretization

In this section we describe the finite volume approximation of (8).Let the interval = [0, 1] be subdivided into Nintervals Ii := (xi, xi+1), i=

0, . . . , N 1, with 0 =: x0 < x1 < < xN := 1. For each i = 0, . . . , N 1we set hi := xi+1 xi and h := maxi=0,...,N1 hi. We also denote xi+1/2 :=xi +hi/2 for i = 0, . . . , N 1, x1/2 := x0 = 0, xN+1/2 := xN = 1 andi := [xi1/2, xi+1/2] for i = 0, . . . , N . Finally, we defineli := xi+1/2 xi1/2fori= 0, . . . , N .

Integrating (8) over the control volumes i we obtainN+ 1 balance equa-tions

i

udx [x(1 x)(u)]xi+1/2xi1/2

+

i

cudx= 0, i= 0, . . . , N .


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(a1/2xv + b1/2v)

=C in (0, x1), v(0) =u0, v(x1) =u1,

whereCis an unknown constant to be determined. We obtain

v(x) =x

u1u0h0

+ u0, 0 0, xh0

(1+0)x u1u0h0 + u0, 0< 0,

0(v) :=

a1/2(1 + 0)x

u1u0h0

+ b1/2u0, 0 0,

b1/2u0, 0< 0. (12)

Case 3Approximation of at xN1/2.We write the flux in the form

(u) :==aN1/2(1 x)

u

x+ bN1/2u,

=a(x) =

2

2x.

The situation is symmetric to Case 2. We can not apply the arguments in Case1 to the approximation of the weighted flux density on IN1 = (xN1, xN)because equation (5) degenerates at x = xN= 1. However the considerations,given in Case 2, should be modified in order to formulate an appropriate two-point BVP. Again, we consider the flux ODE with an extra degree of freedom

in the following form (recall N1= bN1/2/=aN1/2)

((1 x)v + N1v) =C0, x IN1, (13)

v(xN1) =uN1, v(xN) =uN, (14)

whereC0 is an unknown constant to be determined. Integration of (13) yieldsthe first-order linear equation

(1 x)v + N1v= C0x + C1, x IN1, (15)

whereC1 denotes an additive constant. Afterwards we multiply (15) by (1 x)N11

(1 x)N1v

=C0(1 x)

N11x + C1(1 x)N11. (16)

Case 3.1N1> 0, N1= 1.Integrating (16) from xN1 to x IN1 results in

(1 x)N1

v(x) (1 xN1)N1

v(xN1)=C0

(1s)N1+1

N1+1

xxN1

(C0+ C1) (1s)

N1

N1

xxN1

.

Multiplying both sides of the equation by (1 x)N1


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8 Radoslav Valkov

v(x) = (1x)N1

(1xN1)N1 v(xN1) + C0

1xN1+1

C0(1xN1)

N1+1(1x)N1

N1+1

(C0+ C1) 1N1

+ (C0+ C1)(1xN1)

N1(1x)N1

N1.

Letting x xN= 1 and making use of (14)we arrive at v(xN) =uN =C0+C1N1

and finally

v(x) = (1 x)N1

(1 xN1)N1(uN1 uN) + uN

+

N1+ 1(1 x)

1

(1 x)N11

(1 xN1)N11

,

(17)

where = C0 R is a free parameter. Therefore

N1(v) :==aN1/2(x 1)+ bN1/2uN.

Case 3.2N1= 1.Now we solve the following ODE

v

1 x

=C0(1 x)

2x + C1(1 x)2.

Integrating over (xN1, x), x IN1, we obtain

v(x) = 1x1xN1

v(xN1) + C0(1 x)(ln(1 x) ln(1 xN1))

+(C0+ C1)

11x

11xN1

(1 x).

Letting x xN= 1 one gets

v(xN) =uN =C0+ C1= C0+ C1

N1,

v(x) = 1 x

1 xN1(uN1 uN) + uN+ (1 x) ln

1 x

1 xN1.

Since 1 x > 0 we can conclude that this is the result of the limitingprocess N1 1, performed on (17). The flux in both cases 3.1 and 3.2 canbe written in the form

N1(v) ==aN1/2(x 1)+ bN1/2uN.

Case 3.3N1= 0.Integrating over (xN1, x), x IN1, the following ODE

v =C0(1 x)1x + C1(1 x)

1


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we arrive at

v(x) =v(xN1) C0(x xN1) (C0+ C1)(ln(1 x) ln(1 xN1)).

The function v(x) is bounded for x xN when C0+ C1 = 0. ThereforeC0=

uN1uN1xN1

and

v(x) =uN+ (uN1 uN) 1 x

1 xN1.

The flux has the following form

N1(v) ==aN1/2(1 N1)(1 x)

uN uN1hN1

+ bN1/2uN,

where we used that N1= bN1/2= 0.Case 3.4N1< 0.

This time we integrate (16) from x to xN= 1 and obtain

v(x) =C01 x

1 N1+ (C0+ C1)

1

N1

and using the boundary conditions

C0=uN1 uN

1 xN1(1 N1), C1= N1uN

uN1 uN1 xN1

(1 N1).

Therefore

v(x) = 1 x

1 xN+1

(uN1 uN) + uN,

N1(v) ==aN1/2(1 N1)(1 x)

uN uN1hN1

+ bN1/2uN

and these are exactly the same results as in Case 3.3. Finally, a reasonablechoice of the free parameter is 0 and

v(x) =

uN+

1x1xN1

(uN1 uN), N1 0,

uN+ (1x)N1

(1xN1)N1(uN1 uN), N1> 0,

N1(v) = =aN1/2(1 N1)(1 x)

uNuN1hN1

+ bN1/2uN, N1 0,

bN1/2uN, N1> 0.

(18)Let us introduce the finite element space Vhby specifying its basis {i}

Ni=0.

Following[6,18]we introduce the functions


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10 Radoslav Valkov

i(x) =

1xi1

1i1

( 1x1)i1

1xi1

1i1

1xi1

i1 , x (xi1, xi),

1xi+1

1i

(

1

x

1

)

i

1xi+1

1i

1xi1

i , x (xi, xi+1).

On the intervals (0, x1) and (xN1, 1) we define the linear functions

0(x) =

1 xx1 , x (0, x1)

0, otherwise; , N(x) =

xxN11xN1

, x (xN1, 1),

0, otherwise.

Next we define the linear functions 1(x) and N1(x) on the intervals(0, x2) and (xN2, 1)

1(x) = 1 xx1 , x (0, x1), 1x21

1(

1x1)

1

1x21

1 1x11

1 , x (x1, x2);

0, otherwise;

N1(x) =

1xN2

1N2

( 1x1)N2

1xN2

1N2

1xN1

1N2 , x (xN2, xN1),

(1 x)/(1 xN1), x (xN1, 1);0, otherwise.

Then, for any vh Vh, we have the representation vh(x) =N

i=0 vhii,where vhi := vh(xi). Associated with vh, we introduce the natural interpolation

operator Ih : C() Vh by

Ihvh(xi) :=vh(xi) =vhi, i= 0, . . . , N .

Furthermore, using the flux approximations (11),(12),(18), obtained inCases 1, 2 and 3 respectively, we define byh(u) an approximation to(u)|Ii :=i(u), i= 0, . . . , N 1. Coming back to (9), this motivates the following semi-discretization of (8) in the space Vh:

(uh(t), vh)h+ Ah(uh(t), vh; t) = 0 vhVh,

Ah(wh, vh; t) :=N

i=0[x(1 x)h(wh, x , t)]xi+1/2xi1/2

Lhvh(xi) + (c(t)wh, vh)h.

As usual, from (9) an equivalent ODEs system is obtained by setting suc-cessfullyvh = i, i= 0, . . . , N :

uhi(t)li xi+1/2(1 xi+1/2)h(uh(t), xi+1/2, t)

+xi1/2(1 xi1/2)h(uh(t), xi1/2, t) + ci(t)uhi(t)li = 0, ci(t) :=c(xi, t).


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12 Radoslav Valkov

and if0 0 then

e0,0 = l0tm

+ x1/2(1 x1/2)0.5(a1/2 bm+1/2

) + l0cm+0 ,

e0,1 =x1/2(1 x1/2)0.5(a1/2+ bm+

1/2 ),

F0 =umh0

l0tm

(1 )

x1/2(1 x1/2)0.5(a1/2 bm+1/2

)

+ l0cm+0

+ umh1(1 )

x1/2(1 x1/2)0.5(a1/2+ b

m+1/2 )

;

fori= 1 if0< 0 then

e1,1 = l1tm

+ xi+1/2(1 xi+1/2)bm+i+1/2

ii

ii + l1c

m+1 ,

e1,0 =x1/2(1 x1/2)(bm+1/2

),

e1,2 =xi+1/2(1 xi+1/2)bm+i+1/2

ii+1ii ,

F1 =umh1

l1tm

(1 )

xi+1/2(1 xi+1/2)bm+i+1/2

ii

ii

+l1cm+1

+ umh0(1 )x1/2(1 x1/2)(b

m+i+1/2)

+umh2(1 )xi+1/2(1 xi+1/2)bm+i+1/2

ii+1

ii

and if0 0 then

e1,1 = l1tm

+ xi+1/2(1 xi+1/2)bm+i+1/2

ii

ii

+x1/2(1 x1/2)0.5(a1/2+ bm+i+1/2

) + l1cm+1 ,

e1,0 =x1/2(1 x1/2)0.5(a1/2 b

m+

i+1/2),e1,2 =xi+1/2(1 xi+1/2)b

m+i+1/2

ii+1ii ,

F1 =umh1

l1tm

(1 )

xi+1/2(1 xi+1/2)bm+i+1/2

ii

ii

+x1/2(1 x1/2)0.5(a1/2+ bm+i+1/2) + l1c

m+1

+umh0(1 )x1/2(1 x1/2)0.5(a1/2 b

m+i+1/2)

+umh2(1 )xi+1/2(1 xi+1/2)bm+i+1/2

ii+1

ii ;

fori= 2, . . . , N 2

ei,i = litm+ xi+1/2(1 xi+1/2)bm+i+1/2ii ii

+xi1/2(1 xi1/2)bm+i1/2

i1i

i1i1 + lic

m+i ,

ei,i1 =xi1/2(1 xi1/2)bm+i1/2

i1i1

i1i1 ,


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ei,i+1 =xi+1/2(1 xi+1/2)bm+i+1/2

ii+1

ii ,

Fi =umhi

litm

(1 )

xi+1/2(1 xi+1/2)bm+i+1/2

ii

ii

+xi1/2(1 xi1/2)bm+

i1/2

i1

i

i1

i1 + lic

m+

i +umhi1(1 )xi1/2(1 xi1/2)b

m+i1/2

i1i1

i1i1

+umhi+1(1 )xi+1/2(1 xi+1/2)bm+i+1/2

ii+1

ii ;

fori= N 1 ifN1> 0

eN1,N1 = lN1tm

+ xi1/2(1 xi1/2)bm+i1/2

i1i

i1i1 + lN1c

m+N1,

eN1,N2 =xN1/2(1 xN1/2)bm+N1/2,

eN1,N =xi1/2(1 xi1/2)bm+i1/2

i1i1

i1i1 ,

FN1

=um

hN1 l1tm (1 )xi1/2(1 xi1/2)bm+i1/2i1i i1i1+xN1/2(1 xN1/2)b

m+N1/2+ lN1c

m+N1

+umhN2(1 )xN1/2(1 xN1/2)b

m+N1/2

+umhN(1 )xi1/2(1 xi1/2)bm+N1/2

i1i1

i1i1

and ifN1 0 then

eN1,N1 = lN1tm

+ xi1/2(1 xi1/2)bm+i1/2

i1i

i1i1

+xN1/2(1 xN1/2)0.5(=a

m+

i1/2 bm+i1/2

) + lN1cm+N1,

eN1,N =xN1/2(1 xN1/2)0.5(=

am+

i1/2+ bm+i1/2),

eN1,N2 =xi1/2(1 xi1/2)bm+i1/2

i1i1

i1i1 ,

FN1 =umhN1

lN1tm

(1 )

xi1/2(1 xi1/2)bm+i1/2

i1i

i1i1

+xN1/2(1 xN1/2)0.5(=a

m+

i1/2 bm+i1/2

) + lN1cm+N1

+umhN2(1 )xi1/2(1 xi1/2)b

m+i1/2

i1i1

i1i1

+umhN(1 )xN1/2(1 xN1/2)0.5(=a

m+

i1/2+ bm+N1/2);

fori= N ifN1> 0 then

eN,N = lNtm

+ xN1/2(1 xN1/2)bm+N1/2

+ lNcm+N , eN,N1= 0,

FN =umhN

lNtm

(1 )

xN1/2(1 xN1/2)bm+N1/2+ lNc

m+N


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14 Radoslav Valkov

and ifN1 0 then

eN,N = lNtm

+ xN1/2(1 xN1/2)0.5(=a

m+

N1/2+ bm+N1/2) + lNc

m+N ,

eN,N1 =xN1/2(1 xN1/2)0.5(=am+

N1/2 bm+N1/2),

FN =umhN

lNtm

(1 )

xN1/2(1 xN1/2)0.5(

=a

m+

N1/2

+bm+N1/2) + lNcm+N

+umhN1(1 )xN1/2(1 xN1/2)0.5(

=a

m+

N1/2 bm+N1/2).

Theorem 3 For any givenm= 1, 2, . . . , N t, iftm is sufficiently small, thesystem matrix of (19), Eh, is anM-matrix.

ProofLet us write down the scalar form of (19):

B0um+1h0 + C0u

m+1h1 =F0

A1um+1h0 + B1u

m+1h1 + C1u

m+1h2 =F1

A2um+1h1 + B2u

m+1h2 + C2u

m+1h3 =F2

...............................

Aium+1hi1+ Biu

m+1hi + Ciu

m+1hi+1= Fi,

...............................

ANum+1hN1+ BNu

m+1hN =FN,

fori= 3, 4, . . . , N 1 and

B0= h02tm

+ e0,0, C0= e0,1,

A1= e1,0, B1= l1tm

+ e1,1, C1= e1,2,

Ai = ei,i1, Bi = litm

+ ei,i, Ci = ei,i+1, i= 2, 3, . . . , N 1,

AN =eN,N1, BN= hN12tm

+ eN,N,

F0=

h02tm

(1 )e0,0

umh0+ (1 )e0,1umh1,

F1= (1 )e1,0umh0+

l1tm

(1 )e1,1

umh1+ (1 )e1,2umh2,

Fi = (1 )ei,i1umhi1+

litm

(1 )ei,i

umhi+ (1 )ei,i+1u

mhi+1,

FN= (1 )eN,N1umhN1+ hN12tm (1 )eN,N umhN.Let us first investigate the off-diagonal entries of the system matrix Ai =

ei,i1and Ci = ei,i+1. From the formulas forei,j from the above we have


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ei,j >0, i, j = 1, 2, . . . , N 1, i=j. That is because

bi+1/2

xi+11xi+1

i xi+11xi+1

i xi1xi

i =ai+1/2i

xi+11xi+1

i xi+11xi+1

i xi1xi

i

=ai+1/2i

1xii

>0, 0< xi = xixi+1

1xi+11xi

0 and = O

1

tm

thenB2= O

1

tm

for small tm. ThereforeB2= O 1tm andB2> |C2|.

In a similar way one can eliminate um+1hN1andum+1hN . As a result we obtain

a system of linear algebraic equations with unknowns um+1h2 , . . . , um+1hN2 and

coefficients matrix that is an M-matrix.

While F3,...,FN3 are non-negative, we have to prove thatF2 andFN2are also non-negative. From the formula forF2 it follows that when tm issmallF2 is non-negative since F2 = O 1tm and , 1 are of the sameorder with respect to tm.FN2 is handled by the same way asF2 and alsoconsidered non-negative.

Since the load vector ( F2, F3, . . . , F N3,FN2) is non-negative and thecorresponding matrix is an M-matrix we can conclude that um+1h2 , . . . , u

m+1hN2

are non-negative. Finally, using the formulas for um+1h0 , um+1h1 , u

m+1hN1, u

m+1hN one

can easily check that they are non-negative too iftm is small.

Remark 1 Theorem3shows that the fully-discretized system (19)satisfies thediscrete maximum principle and therefore the above discretization is mono-tone. This guarantees the following: for a non-negative initial function u0(x)the numerical solution umh, obtained via this method, is also non-negative asexpected, because the price of the option is a non-negative number.


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16 Radoslav Valkov

5 Numerical experiments

Numerical experiments, presented in this section, illustrate the properties ofthe constructed method. We solve numerically various European Test Prob-

lems (TP) with different final (initial) conditions and different choices of pa-rameters.

1. (T P1).Call optionwith final condition (2). Parameters:Smax = 700,T =1, r = 0.1, = 0.3,d = 0.04 and E= 400.

2. (T P2). Call optionwith cash-or-nothing payoffV(S, T) =H(S E), S(0, ), whereHdenotes the Heaviside function. Parameters: Smax= 700,T = 1, r = 0.1, = 0.4, d = 0.04 and E= 400.

3. (T P3).Call optionwith final condition (2). Parameters:Smax = 700,T =1, r = 0.1 + 0.02sin(10T t), = 0.4, d = 0.06S/(S+ E) and E= 400.

4. (T P4). A portfolio of options. Combinations of different options have stepfinal conditions such as the bullish vertical spread payoff, defined in (3).In this example, we assume that the final condition is a butterfly spread

delta function, defined by

V(S, T) =

1, S (S1, S2),1, S (S2, S3),0, otherwise,

which arises from a portfolio of three types of options with different exerciseprices [20]. Parameters: Smax = 100, T = 1, S1 = 40, S2 = 50, S3 = 60,r= 0.1 + 0.02sin(10T t), = 0.4, d = 0.06S/(S+ E) and Pm = 50.

In the tables below are presented the computed C and L2 mesh norms ofthe error E= uh u by the formulas

EC= maxiuNthi u

Nti , EL2 =

Ni=0

li uNthi uNti 2.The rate of convergence (RC) is calculated using double mesh principle

RC= log2(EN/E2N), EN =uNh u

N,

where . is the mesh C-norm or L2-norm, uN and uNh are respectively the

exact solution and the numerical solution, computed at the mesh with Nsubintervals. We choose the weight parameter with respect to the time variable= 0.5.

In Table1we show the convergence and the accuracy of the constructedscheme, where we numerically solve the model problem with the known exactsolution u(x, t) = exp(x t) and initial data u0(x) = exp(x). We select thisfunction because its feature is similar to that of the exact solution to the calloption problem. The results, corresponding to problems T P1 and T P3 withtm = 0.001, m= 0, . . . , N t 1, are listed in Table 1.


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Table 1

T P1 T P3

N EN

RC EN2

RC EN

RC EN2

RC

80 3.455e-3 - 2.801e-4 - 4.805e-3 - 3.914e-4 -160 1.729e-3 0.998 9.914e-5 1.498 2.405e-3 0.998 1.385e-4 1.498320 8.650e-4 0.999 3.507e-5 1.499 1.203e-3 0.999 4.900e-5 1.499640 4.326e-4 0.999 1.240e-5 1.499 6.015e-4 0.999 1.733e-5 1.499

Figures 1-4 illustrate the numerical solution, computed with = 0.0001on an uniform mesh N= 320 for T P1 and T P3, and the well-known solutionof the classical Black-Scholes equation, computed by the financial toolbox ofMATLAB,blsprice(Price, Strike, Rate, Time, Volatility). Comparison resultsforT P3, whereN= 40 and T= 1, are given in Figure4, while the numericalsolution is visualized in Figure3.

Fig. 1 Numerical Solution T P1 Fig. 2 Black-Scholes Solution T P1

Fig. 3 Numerical Solution T P3

0 100 200 300 400 500 600 7000

50

100

150

200

250

300

S

Solution

blsprice

numprice

Fig. 4 Comparison T P3


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18 Radoslav Valkov

In Table 2 the results are obtained by computations on a power-gradedmesh for the same values of the parameters and exact solution. This meshtakes into account the degeneration at the both endsof the interval (0, 1) andis given by (in the current case p = 2)

xi+1= xi+

i

2N/2

i=0 ip1/pp , i= 1, . . . , N /2,

xi+1= xi+

(N+ 1 i)

2N/2

i=0 ip1/pp

, i= N/2 + 1, . . . , N .

The time step tmis chosen such that tm= min0iNh(i), m= 0, . . . , N t1with T = 0.1.

Table 2

T P1 T P3

N EN

RC EN2

RC EN

RC EN2

RC

20 7.154e-4 - 3.648e-4 - 6.263e-4 - 3.914e-4 -40 1.880e-4 1.927 9.525e-5 1.947 1.650e-4 1.924 8.341e-5 2.23080 4.818e-5 1.964 2.437e-5 1.966 4.226e-5 1.964 2.134e-5 1.966

160 1.220e-5 1.982 6.167e-6 1.982 1.970e-5 1.982 5.401e-6 1.982

We now compute the solutions of the original models T P2 and T P3. Asexact solution we use the numerical solution on a very fine uniform grid, i.e.N = 5120 with tm = 0.0001, m = 0, . . . , N t 1. The results are given inTable3. The numerical solutions ofT P2 and T P4 forN= 640 are plotted inFigures5 and 6.

Table 3

T P2 T P3

N EN

RC EN2

RC EN

RC EN2

RC

80 2.914e-7 - 1.112e-7 - 2.681e-3 - 2.171e-4 -160 9.914e-8 1.555 2.841e-8 1.968 1.321e-3 1.021 7.476e-5 1.538320 5.047e-8 0.974 7.386e-9 1.943 6.393e-4 1.046 2.544e-5 1.554640 2.545e-8 0.987 1.973e-9 1.904 2.984e-4 1.099 8.374e-6 1.603

1280 1.269e-8 1.003 5.42e-10 1.864 1.279e-4 1.222 2.534e-6 1.724

The convergence of the numerical solution, obtained by the method to thesolution of the classical Black-Scholes equation, transformed by (4), is given inTable4.The node x

N= 1 is omitted in the calculations since it corresponds

to the case S = . We use the test parameters in T P1 with d = 0 andtm = 0.0001, m = 0, . . . , N t 1. In the columns 2-5 of Table 4 we showthe overall rate of convergence, while in the last column is given the rate ofconvergence in the strong norm of the numerical solution in a random node


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Fig. 5 Test Problem 2 Fig. 6 Test Problem 4

of the mesh, i.e. the one, corresponding to S = 600. The experiments areperformed on an uniform mesh.

Table 4

T P1 S= 600

N EN

RC EN2

RC EN

RC

80 3.7473e-4 - 6.7765e-5 - 1.8848e-5 -160 1.8939e-4 0.985 2.0388e-5 1.733 4.7877e-6 1.977320 9.5196e-5 0.992 6.4913e-6 1.651 1.2016e-6 1.994640 4.7722e-5 0.996 2.1574e-6 1.589 3.0070e-7 1.999

1280 2.3892e-5 0.998 7.3723e-7 1.549 7.5196e-8 1.999

The benchmark of numerical methods in computational finance is theCrank-Nicolson second-order centered space difference scheme (CSDS). It iswell known that it produces spurious oscillations [1,7,16]in the solution and

its spatial derivatives, i.e. = V/S, that are financially unrealistic and arenot tolerable. The Figures7 and8illustrate the problem for the vanilla calloption T P1, computed on an uniform mesh for = 0.0001 and parametersSmax = 700, T = 1, r = 0.1, = 0.01, d= 0 and E= 400. We compare theMATLAB function, blsdelta(Price, Strike, Rate, Time, Volatility), with thefirst derivative of the numerical solution. In the Figures 9and10,generatedby the our difference scheme (ODS), no oscillations are observed.

A realistic situation in financial engineering occurs when the convectionand diffusion terms have opposite signs. For example, situations, similar tor d < 0, arise in the bond-pricing models, that are also of Black-Scholestype [7, 14,22], where the parameters are interpreted in different context. InFigures11and12we show the numerical solutions, generated by our differencescheme and by the second-order centered space difference scheme respectively,applied to (5), for initial condition as in T P2 with parameters T = 2, =0.1, r = 0, d = 0.4x and N = 40, = 0.001. In Figures 13 and 14 aregiven the numerical solutions for initial condition as in T P3 with parametersT = 2, = 0.1, r= 0, d= 2x and N = 40, = 0.001.


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20 Radoslav Valkov

0 100 200 300 400 500 600 7000

50

100

150

200

250

300

350

x

Solut

ion

blsprice

numprice

Fig. 7 Black-Scholes price CSDS

0 100 200 300 400 500 600 7000

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Delta

blsdelta

numdelta

Fig. 8 Black-Scholes CSDS

0 100 200 300 400 500 600 7000

50

100

150

200

250

300

350

x

Solution

blsprice

numprice

Fig. 9 Black-Scholes price ODS

0 100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Delta

blsdelta

numdelta

Fig. 10 Black-Scholes ODS

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8x 10

4

x

Solution


0 0.2 0.4 0.6 0.8 11

0

1

2

3

4

5

6

7

8

9x 10

4

x

Solution


As seen in Figures 11-14, monotonicity and stability are not guaranteedfor the centered space difference scheme if the convection and diffusion coeffi-cients are of different signs. Simple calculations show that the discrete maxi-mum principle is violated. We do not observe such problems in the numericalsolution, generated by our numerical method.


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0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

x

Solution


0 0.2 0.4 0.6 0.8 15

0

5

10

15

20x 10

3

x

Solution


6 Conclusion

In this article we present a fitted FVM for the generalized Black-Scholes equa-

tion (1). The method is applicable to more general Black-Scholes models, forexample when = (S, t) and r = r(S, t). We may also use any interval(0, l), l >0 (here we took l = 1 for simplicity) to solve the transformed prob-lem. The main advantage of the developed numerical algorithm is reductionof the computational costs as well as positivity-preserving.

The conducted experiments show first order of convergence of the pro-posed scheme on a quasi-uniform mesh and second order of convergence on aparticular graded mesh. Moreover, they also indicate better stability and un-conditional (w.r.t. to the space step) monotonicity in comparison with otherknown schemes.

In a forthcoming paper we study the stability and the convergence of theproposed finite volume method.

Acknowledgement:The author is supported by the Bulgarian NationalFund under Project DID 02/37/09 and by the European Social Fund throughthe Human Resource Development Operational Programme under contractBG051PO001-3.3.06-0052 (2012/2014).

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Fitted Finite Volume Method for a Generalized Black-Scholes Equation Transformed on Finite Interval

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