Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics

Tanki Motsepa; Taha Aziz; Aeeman Fatima; Chaudry Masood Khalique

doi:10.1515/phys-2018-0006

Article Open Access

Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics

Tanki Motsepa , Taha Aziz , Aeeman Fatima and Chaudry Masood Khalique

Published/Copyright: March 8, 2018

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From the journal Open Physics Volume 16 Issue 1

Abstract

The optimal investment-consumption problem under the constant elasticity of variance (CEV) model is investigated from the perspective of Lie group analysis. The Lie symmetry group of the evolution partial differential equation describing the CEV model is derived. The Lie point symmetries are then used to obtain an exact solution of the governing model satisfying a standard terminal condition. Finally, we construct conservation laws of the underlying equation using the general theorem on conservation laws.

Keywords: CEV model; Terminal condition; Lie point symmetries; Group-invariant solution; Conservation laws

PACS: 02.20.Sv; 02.20.Tw; 02.30.Jr; 02.60.Lj

1 Introduction

An important problem in mathematics of finance is the mathematical modelling of optimal investment-consumption decisions under uncertainty conditions. The investment-consumption problem [1] has been extensively investigated in many works with various modifications and extensions. Cox [2] and Cox and Ross [3] have derived the well-known constant elasticity of variance (CEV) option pricing model and Schroder [4] afterwards broadened the model by stating the CEV option pricing formula in respect of noncentral Chi-square distribution. The CEV model is mostly used to investigate the option and asset pricing formula, as was investigated by Beckers [5], Davydov and Linetsky [6], Emanuel and Macbeth [7] and recently by Hsu et al. [8]. Here, we reconsider the CEV model [9]. Literature survey witness that there have been few studies [10,11,12,13,14,15,16,17] recently reported where its solutions are presented. Nonetheless, to the best of our knowledge, no work has been published on the closed-form solutions of the CEV model, which is the aim of the present work.

The classical Lie symmetry theory was discovered by the Norwegian mathematician Marius Sophus Lie (1842-1899) in the nineteenth century. This theory systematically unites the widely known ad hoc methods to find exact solutions for differential equations. After many years of discovery, the Lie’s theory was popularized by Ovsiannikov in Novosibirsk, Russia in the middle of twentieth century and by Birkhoff and Olver in the West. Lie’s theory is one of the most effective tools to find exact analytical solutions of nonlinear partial differential equations and is established on the analysis of the invariance under one-parameter group of point transformations. See for example, the references [18,19,20,21,22,23,24].

Recently Lie’s theory has been applied to partial differential equations (PDEs) of mathematical finance. One of the earliest study on the subject is [25], where the classical Black-Scholes equation was discussed. The bond-pricing equation via Lie group approach was investigated in [26, 27]. The invariant analysis of some well-known financial mathematics models was presented [28,29,30,31] . More recently the Lie’s theory has been applied to various PDEs of financial mathematics. See for example, Motsepa et al. [32], Lekalakala et al. [33], Nteumagne and Moitsheki [34], Bozhkov and Dimas [35], Caister et al. [36, 37], Naicker et al. [38], Lo [39], Taylor and Glasgow [40], Wang et al. [41] and Pooe et al. [42], are few important studies to mention.

In this paper, we discuss the optimal investment-consumption problem under the CEV model

∂F∂t+λα−δ1−λ+λ2(λ−1)2β−αK2xF+2λ(λ−1)γ(β−α)−2γβx∂F∂x+γ(2γ+1)K2∂F∂x+2γ2K2x∂2F∂x2=0(1)

with the terminal condition

F(T,x)=1,tϵ0,T(2)

from the viewpoint of Lie symmetry approach. The application of Lie’s theory, in general, reduces the partial differential equation in two independent variables to an ordinary differential equation and provides us with the group-invariant solution. Some nontrivial conservation laws are also constructed for the CEV model by employing a general theorem proved in [43].

2 Lie symmetries of (1)

In this section, we compute the Lie symmetries admitted by the model equation (1) and utilised them to obtain the closed-form group-invariant solution of the PDE (1) satisfying the terminal condition (2).

For detailed description on Lie symmetry method and its applications to various disciplines the reader is referred to Refs [18,19,20,21,22,23,24]. However, in this section we give detailed calculations on finding Lie point symmetries of (1).

In order to facilitate the calculations when employing the Lie group approach, we rewrite the PDE (1) in the form

∂F∂t+ex∂2F∂x2+(cx+d)∂F∂x+(a+bx)F=0,(3)

where

a=λα−δ1−λ,b=λ(β−α)22(λ−1)2K2,c=2λγ(β−α)(λ−1)−γβ,d=γ(2γ+1)K2,e=2γ2K2.

The operator

X=τ(t,x,F)∂∂t+ξ(t,x,F)∂∂x+η(t,x,F)∂∂F

is a Lie symmetry of PDE (3) if and only if

X[2][Ft+exFxx+(cx+d)Fx+(a+bx)F]∣(3)=0,(4)

where X^[2] denotes the second prolongation of X which is defined by

X[2]=X+ζt∂∂Ft+ζx∂∂Fx+ζxx∂∂Fxx(5)

with

ζt=Dtη−FtDtτ−FxDtξ,ζx=Dxη−FtDxτ−FxDxξ,ζxx=Dxζx−FtxDxτ−FxxDxξ(6)

and the total derivative operators

Dt=∂∂t+Ft∂∂F+Ftt∂∂Ft+Ftx∂∂Fx+⋯,Dx=∂∂x+Fx∂∂F+Fxx∂∂Fx+Ftx∂∂Ft+⋯.(7)

Expanding the symmetry condition (4) with the help of (5)-(7) and then splitting on the derivatives of F leads to

τF=0,ξF=0,ηFF=0,τx=0,ξ+xτt−2xξx=0,(a+bx)η+bFξ+τtF(a+bx)−ηFF(a+bx)+ηx(cx+d)+exηxx+ηt=0,cξ+τt(cx+d)−ξx(cx+d)+2exηxF−exξxx−ξt=0,

which are linear PDEs in τ, ξ and η. Solving the above system gives

τ=C1+2θC3exp⁡(θt)−2θC4exp⁡(−θt),ξ=2C3xexp⁡(θt)+2C4xexp⁡(−θt),η=C2F−1eθC3Fexp⁡(θt)2ae+dθ+cθx−θ2x−cd−1eθC4Fexp⁡(−θt)−2ae+cxθ+cd+dθ+θ2x,

where θ=c2−4be and C₁, …, C₄ are arbitrary constants. Since τ, ξ and η contain four constants, we conclude that the Lie algebra of infinitesimal symmetries of the PDE (3) is spanned by the four vector fields

X1=∂∂t,X2=2θexp⁡(θt)∂∂t+2xexp⁡(θt)∂∂x−1eθFexp⁡(θt)2ae+dθ+cθx−θ2x−cd∂∂F,X3=−2θexp⁡(−θt)∂∂t+2xexp⁡(−θt)∂∂x−1eθFexp⁡(−θt)−2ae+cxθ+cd+dθ+θ2x∂∂F,X4=F∂∂F.

3 Group invariant solution

Many researchers have developed various analytical methods for solving partial differential equations, such as inverse scattering transform [44], the Bäcklund transformation [45], the Hirota bilinear method [46], the Painlevé analysis method [47], the Bell polynomials [48], the homoclinic breather limit method [49].

We now obtain the closed-form group-invariant solution for the PDE (3) by making use of the Lie symmetry algebra calculated in the previous section. Firstly we calculate symmetry Lie algebra admitted by (3) that satisfies the terminal condition (2) [26, 28].

We consider the linear combination of the Lie point symmetries, namely

X=a1X1+a2X2+a3X3+a4X4,(8)

and use the terminal conditions

X(t−T)∣t=T,F=1=0(9)

and

X(F−1)∣t=T,F=1=0.(10)

The condition (9) yields

a1+2a2θexp⁡(θT)−2a3θexp⁡(−θT)=0,(11)

while condition (10) gives

a4+a2cdeθ−2aθ−de+c2xeθ−4bxθ−cxeexp⁡(θT)+a32aθ−cdeθ−de−c2xeθ+4bxθ−cxeexp⁡(−θT)=0.(12)

Splitting Eq. (12) on powers of x yields

a4+a2cdeθ−2aθ−deexp⁡(θT)+a32aθ−cdeθ−deexp⁡(−θT)=0,(13)

a2c2eθ−4bθ−ceexp⁡(θT)+a3−c2eθ+4bθ−ceexp⁡(−θT)=0.(14)

Solving equations (11), (13) and (14), we obtain

a1=a2c(θ−c)beθexp⁡(θT),a3=a2(2be+c(θ−c))2beexp⁡(2θT),a4=−a2(θ−c)(ac−2bd)beθexp⁡(θT).(15)

Substituting the values of a₁, a₃ and a₄ from Eq. (15) into Eq. (8) and solving X(F) = 0, we obtain the two invariants

J1=xexp⁡(θt)(c+θ)exp⁡(2θt)−2cexp⁡(θ(t+T))+(c−θ)exp⁡(2θT),J2=F(c+θ)exp⁡(θt)+(θ−c)exp⁡(θT)d/e×exp⁡(at+4bexexp⁡(2θt)+x(c−θ)(c+θ)exp⁡(2θT)2e(c+θ)exp⁡(2θt)−2cexp⁡(θ(t+T))+(c−θ)exp⁡(2θT)−cdt2e−dθt2e.

Thus, the group-invariant solution of the PDE (3) is given by J₂ = G(J₁), which yields

F(t,x)=G(z)(c+θ)exp⁡(θt)+(θ−c)exp⁡(θT)−de×exp4bexexp⁡(2θt)+x(c−θ)(c+θ)exp⁡(2θT)2e−(c+θ)exp⁡(2θt)+2cexp⁡(θ(t+T))+(θ−c)exp⁡(2θT)+cdt2e+dθt2e−at,(16)

where z = J₁. Now substituting Eq. (16) into PDE (3) yields a second-order ODE for G(z), viz.,

ezG′′(z)−2exp⁡(θT)2bdG(z)+c2zG′(z)+dG′(z)−8bzG(z)exp⁡(2θT)2be−c2=0.(17)

Solving the reduced ODE (17) for G(z), we obtain

G(z)=C1exp⁡(4bzexp⁡(θT))−C22de−1z1−deexp⁡(4bzexp⁡(θT))−θ2zexp⁡(θT)ede−1×Γ1−de,−2exp⁡(θT)zθ2e,

where C₁ and C₂ are constants and Γ [.,.] is the incomplete Gamma function [50, 51]. The terminal condition dictates that we take C₂ = 0, hence the above solution for G(z) takes the form

G(z)=C1exp⁡(4bzexp⁡(θT)).

Substituting the value of G into Eq. (16), the solution for F(x, t) is written as

F(t,x)=C1exp⁡(4bzexp⁡(θT))(c+θ)exp⁡(θt)+(θ−c)exp⁡(θT)−de×exp⁡[−at+4bexexp⁡(2θt)+x(c−θ)(c+θ)exp⁡(2θT)2e−(c+θ)exp⁡(2θt)+2cexp⁡(θ(t+T))+(θ−c)exp⁡(2θT)+cdt2e+dθt2e].(18)

Finally, making use of the terminal condition F(x, t) = 1 in Eq. (18), we obtain

C1=(θ−c)exp⁡(θT)+(c+θ)exp⁡(2θT)d/eexpaT−cdT2e−dθT2e.

Therefore, the solution of (1+1) evolution PDE (3) satisfying the terminal condition is given by

F(t,x)=2θexp⁡(θT)(c+θ)exp⁡(θt)+(θ−c)exp⁡(θT)de×exp(t−T)(d(c+θ)−2ae)2e−2bxexp⁡(θt)−exp⁡(θT)cexp⁡(θt)−exp⁡(θT)+θexp⁡(θt)+exp⁡(θT).

4 Conservation laws

In this section we derive conservation laws for the (1+1) evolution partial differential equation (3). In classical physics, conservations laws are physical quantities which describe the conservation of energy, mass, linear momentum, angular momentum, and electric charge. One particularly important result concerning conservation laws is the celebrated Noether theorem, which gives us a sophisticated and useful way of constructing conservation laws when a Noether point symmetry connected to a Lagrangian is known for the corresponding Euler-Lagrange equation.

Recently, a theorem due to Ibragimov was proved, which gives a method to construct conservation laws irrespective of the existence of a Lagrangian. We start here by stating the definition of an adjoint equation.

Definition 1

Let

E(x,t,f,ft,fx,fxx)=0(19)

be a second-order PDE with x, t as independent variables and f a dependent variable. Then, its adjoint equation is [43]

E∗(x,t,ft,fx,fxx,g,gt,gx,gxx)=0,(20)

where

E∗=δ(gE)δf,

with

δδf=∂∂f+∑s=1∞(−1)sDi1...Dis∂∂fi1...is

denoting the Euler–Lagrange operator and g is a new dependent variable. Here

Di=∂∂xi+fi∂∂f+fij∂∂fj+...

are the total derivative operators.

The 2-tuple vector T = (T^t, T^x), is a conserved vector of (19) if T^t and T^x satisfy

DtTt+DxTx|(19)=0.(21)

We now state the following theorem.

Theorem 1

[43] Any Lie point, Lie–Bäcklund or non-local symmetry

X=ξi(x,f,f(1),...)∂∂xi+η(x,f,f(1),...)∂∂f,

of Eq. (19) gives a conservation law D_i(Tⁱ) = 0 for the system (19)–(20). The conserved vector is given by

Ti=ξiL+w∂L∂fi−Dj∂L∂fij+Dj(w)∂L∂fij,

where w and 𝓛 are determined as follows:

w=η−ξjfj,L=gE(x,f,f(1),...,f(s)).

We now apply the above theorem to our problem. The adjoint equation to the equation (3) is

w(a+bx−c)−wx(cx+d−2e)+exwxx−wt=0,(22)

and hence the second-order Lagrangian of the equation (3) and its adjoint equation (22) is written as

L=w[Ft+exFxx+(cx+d)Fx+(a+bx)F].

We now apply the above theorem to each Lie point symmetry of Eq. (3). We start with X₁ = ∂/∂t. Corresponding to symmetry X₁ we obtain the conserved vector with components

T1t=w(a+bx)F+Fx(cx+d)+exFxx+Ft−Ftw,T1x=−Ftw(cx+d−e)+exwx−exwFtx.

The symmetry X₂ provides us with the conserved vector whose components are

T2t=−wexp⁡(tθ)eθFdθ+cxθ+2bex−c2x−cd−2eFxxc−θ+d+exFxx,T2x=exp⁡(tθ)eθ[e2exwxxFxθ+Ft−wxFx2ae+dθ+cxθ+4bex−c2x−cd+2Ft−xθ+cx+d−e+2exFtx+Fwc−2ex(a+2bx)−d2xθ+e+d2+2aexθ−d+e−d2θ+c2x2d−xθ+deθ+2bex2θ−4bdex+c3x2+exwx2ae+dθ+cxθ+4bex−c2x−cd],

where θ=c2−4be.. Likewise, the conserved vectors associated with the symmetries X₃ and X₄ are given by

T3t=−wexp⁡(−θt)eθ[Fcxθ+d+dθ−2bex+c2x+2eFxxθ+c+d+exFxx],T3y=exp⁡(−θt)eθ[2ewFtxθ+c+d−e+eFFtx−xFx−2e(a+2bx)+cxθ+d+dθ+c2x+2exwxxFxθ−Ft+2Fwcex(a+2bx)+de−2xθ−d2+2aexθ+d−e−d2θ−c2xxθ+2d+deθ+2bex2θ+4bdex−c3x2+exwx−2e(a+2bx)+cxθ+d+dθ+c2x],T4t=wF,T4y=Fw(cx+d−e)−exwx+exwFx,

respectively.

5 Concluding remarks

The evolution (1+1) PDE (1) describing the optimal investment-consumption problem under the CEV model [9] satisfied the classical Black–Scholes–Merton equation with boundary condition which differ from those often used in the most common cases. It is well-known that the evolution (1+1) PDE (1) is related to the heat equation via the equivalence transformations and thus its general solution can be obtained. However, in this paper, for the first time, we have solved the PDE (1) subject to the terminal condition (2) by utilizing the Lie group method. This demonstrates the usefulness of Lie’s theory. We found a four-dimensional Lie symmetry algebra for evolution PDE (1). Using the nontrivial Lie point symmetry operator, we have shown that the governing PDE can be transformed into a second-order variable coefficient ODE. The reduced ODE is solved to obtain a new exact closed-form solution of the CEV model which also satisfy the terminal condition. Thus for the first time with the application of Lie’s theory closed-form solution of (1) is derived. Finally, we constructed conservation laws corresponding to the four Lie point symmetries by employing a general theorem on conservation laws. This is the first time that the evolution PDE (1) for optimal investment-consumption problem has been considered from the view point of group theoretical approach and the conservation laws have been derived in the literature.

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Conflict of interest
Conflict of Interests: The authors declare that there is no conflict of interests regarding the publication of this paper.

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Received: 2017-08-07

Accepted: 2017-10-28

Published Online: 2018-03-08

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/phys-2018-0006

Keywords for this article

CEV model; Terminal condition; Lie point symmetries; Group-invariant solution; Conservation laws

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