Group Classification of a General Bond-Option Pricing Equation

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III I llll I\III ll I1 llIl III I ll

0600433 1OH

North-West University Mafikeng Campus Library

GROUP CLASSIFICATION OF A

GENERAL BOND-OPTION PRICING

EQUATION

by

TANKI MOTSEPA (24602825)

Dissertation submitted for the degree of Master of Science in Applied

Mathematics in the Department of Mathematical Sciences in the Faculty of

Agriculture, Science and Technology at North-West University, Mafikeng

Campus

November 2013

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Declaration

I, TANKI MOTSEPA, student number 24602825, declare that this dissertation for the degree of Master of Science in Applied Mathematics at North-West University, Mafikeng Campus, hereby submitted, has not previously been submitted by me for a degree at this or any other university, that this is my own work in design and execution and that all material contained herein has been duly acknowledged.

Signed...

Mr TANKI MOTSEPA

Date. ...

This dissertation has been submitted with my approval as a University supervisor and would certify that the requirements for the applicable Master of Science degree rules and regulations have been fulfilled.

Signed. ...

PROF C.M. KHALIQUE

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Dedication

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Acknowledgements

I would like to thank my supervisor Professor CM Khalique for his guidance, patience and support throughout this research project. I would also like to thank Dr. M. Molati for support and guidance throughout this research and for his collaboration that he suggested the problem studied in this research. I greatly appreciate the financial support from the North-West University, Mafikeng Campus, through the postgraduate bursary scheme. Finally, my deepest and greatest gratitude goes to all members of my family for their motivation and moral support.

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Abstract

The main purpose of this work is to perform the Lie group classification of a general bond-option pricing equation. The procedure involves finding the Lie symmetries of the said partial differential equation and employing direct method to classify the equation. The optimal systems are then obtained for some cases encountered and group-invariant solutions are found for the optimal systems of one-dimensional subalgebras obtained.

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List of Acronyms

PDE: Partial differential equation

ODE: Ordinary (liflerential equation

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Introduction

The theory of option pricing began in 1900 when the French mathematician Lou's Jean-Baptiste Alphonse Bachelier [2] deduced an option pricing formula based on the assumption that stock prices follow a Brownian motion. The Black-Scholes equation

t +

I

X 2 _{07 2 UXX}_{+ r (xn -0} ₍₁₎

was introduced by Black and Scholes [7] as the general equilibrium theory of option pricing which is particularly attractive because the final formula is a function of observable van-ables. Merton [37] extended the Black-Scholes theory of option pricing by introducing more assumptions and found new explicit formulas for pricing both the call and put options as well as the warrants and the down-and-out options. Merton was the first one to refer to equation (1) as the Black-Scholes equation. Equation (1) is sometimes referred to as the Black-Scholes-Merton equation and because of this work they were awarded the oble prize in 1997 in economics even though Black did not receive it as he had passed away in 1995. The equation is mainly used to find the fair price of a financial instrument (option or derivative) and to find the implied volatility.

The first bond pricing equation

Ut + + k ( - x) - xu = 0 (2)

was introduced by Vasicek [57] with three assumptions. Firstly, the instantaneous (spot) interest rate follows a diffusion process. Secondly, the price of a discount bond depends only on the spot rate over its term. Lastly, Vasicek assumed that the market is efficient. Many other researchers came up with new one factor models which modelled the term structure of interest rates such as [5,6, 11,15,21-23,35,48].

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Many differential equations including financial mathematics equations, involve parameters, arbitrary elements or functions, which need to be determined. Usually, these arbitrary parameters are determined experimentally. However, the Lie symmetry approach through the method of group classification has proven to be a versatile tool in specifying the forms of these parameters systematically (see, for example, [26, 30, 33, 38-42, 52]).

The first group classification problem was investigated by Sophus Lie [34] in 1881 for a linear second-order partial differential equation with two independent variables. The main idea of group classification of a differential equation involving an arbitrary element(s), say, for example, g(n) and f(x), consists of finding the Lie point symmetries of the differential equation with arbitrary functions g(n) and f(x), and then computing systematically all possible forms of g(u) and f(c) for which the principal Lie algebra can be extended.

In the past few decades a considerable amount of development has been made in symmetry methods for differential equations. This is evident by the number of research papers, books and many new symbolic softwares devoted to the subject (see, for example, [3, 4, 12, 14, 19, 20,491).

Semi-invariants for the (1+1) linear parabolic equations with two independent variables and one dependent variable were derived by Johnpillai and Mahomed [31]. In addition, joint invariant equation was obtained for the linear parabolic equation and that the (1+1) linear parabolic equation was reducible via a local equivalence transformation to the one-dimensional heat equation. In [36], a necessary and sufficient condition for the parabolic equation to be reducible to the classical heat equation under the equivalence group was provided which improved on work done in [31].

Goard [18] found group invariant solutions of the bond-pricing equation by the use of classical Lie method. The solutions obtained were shown that they satisfy the condition for the bond price, that is, P(r, T) = 1, where P is the price of the bond. Here r is the short-term interest rate which is governed by a stochastic differential equation and T is time to maturity.

Pooe et al. [47] obtained the fundamental solutions for a number of zero-coupon bond models by transforming the one-factor bond pricing equations corresponding to the bond models to the one-dimensional heat equation whose fundamental solution is well-known. Subsequently,

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the transformations were used to construct the fundamental solutions for zero-coupon bond pricing equations.

Sinkala et al. [54] computed the zero-coupon bonds (group invariant solutions satisfying the terminal condition n(T, T) 1) using symmetry analysis for the Vasicek and CIR equations, given by

Vt + 0-2nxx + K (8 - x) u - = 0,

Vt + GF 2 XUXX + K(8 - x) Ux XU _{= 0,}

respectively. In [53] an optimal system of one-dimensional subalgebras was derived and used to construct distinct families of special closed-form solutions of CIR equation. In [52], group classification of the linear second-order parabolic partial differential equation

nt + p2x2uxx + ( + lx - P)x XU = 0, (3) where c, /3, , p and -y are constants was carried out. Lie point symmetries and group invariant solutions were found for certain values of y. Also the forms where the equation admitted the maximal seven Lie point symmetry algebra, (3) was transformed into the heat equation. Vasicek CIR and Longstaff models were recovered from group classification and some other equations were derived which had not been considered before in literature. Dimas et al. [13] investigated some of the well known equations that arise in mathematics of finance, such as Black-Scholes, Longsaff, Vasicek, CIR and Heath equations. Lie point symmetries of these equations were found and their algebras were compared with that of the heat equation. The equations with seven symmetries were transformed to the heat equation. In this research, we study a general bond-option pricing equation. The partial differential equation which will be investigated is a generalisation of equations (1) and (2) and is given by

Vj ₊aXpUXX + (/

3 - x)n +

=

0, (4) where p and q are arbitrary constants.

The outline of this dissertation is as follows:

In Chapter one, the basic definitions and theorems concerning the one-parameter groups of transformations are presented.

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In Chapter two, Lie symmetry method is employed to find symmetries of the Black-Scholes equation. The symmetries obtained are then used to compute group invariant solutions.

In Chapter three, we carry out Lie group classification on equation (4), that is, we find values of the constants p and q for which the principal Lie algebra extends.

In Chapter four, we obtain optimal systeiñs of one-dimensional subalgebras for two cases of equation (4) and then construct group invariant solutions.

In Chapter five, a summary of the results of the dissertation is presented and future work is discussed.

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Chapter 1 Lie symmetry methods for differential

equations

In this chapter, some basic methods of Lie symmetry analysis of differential equations in-eluding the algorithm to determine the Lie point symmetries of PDEs are given.

1.1 Introduction

Lie group analysis as a method for solving differential equations was developed by Sophus Lie (1842-1899), who showed that the majority of adhoc methods of integration of differential equations could be explained and deduced simply by means of his theory. Lately, many good books have appeared in literature in this field such as, Ovsianikov [44,45], Bluman and Kumei [9], Bluman and Anco [8], Stephani [55], Olver [43], Ibragimov [26-29], Hydon [25], Cantwell [10].

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1.2 Continuous one-parameter (local) Lie group

In this section a transformation will be understood to mean an invertible transformation, that is, a bijective map. Let t and x be two independent variables and n be a dependent variable. Consider a change of the variables t, x and n:

Ta : E=f(t,x,u,a), x =g(t,x,n,a), ut = h(t,x,u,a) (1.1)

where a is a real parameter which continuously ranges in values from a neighbourhood

V' C V C R of a = 0 and f, g and h are differentiable functions.

Definition 1.1 A set C of transformations (1.1) is called a continuous one-parameter (local)

Lie group of transformations in the space of variables t, x and u if

For Ta , T5 C C where a,b C V' CD then TbTa = T C C, c= 0 (a, b) C V (Closure)

To C C if and only if a = 0 such that To Ta = Ta To Ta (Identity)

For Ta C C, a C V' C V, T' = Ta 1 C C, a 1 C V such that

Ta Ta-i = Ti Ta = To (Inverse)

It follows from (i) that the associativity property is satisfied. Also, if the identity transfor-mation occurs at a = a0 0, i.e., Tao is the identity, then a shift of the parameter a

= a

± a0 will give To as above. The group property (i) can be written as

=

g(T, , , b) = g(t, x, u, 5(a, b)),

h(, t, , b) = h(t,x, n, 0 (a, b)). (1.2)

The function

0

_{is termed as the group composition law. A group parameter a is called}

canonical if ql(a, b) = a + b.

Theorem 1.1 For any (a, b) there exists the canonical parameter d defined by

fa _ds

where w(s)=

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Let us now give the definition of a symmetry group for PDEs by considering, for example, evolutionary equations of the second-order, namely

0. (1.3)

&Uxx

Definition 1.2 (Symmetry group) A one-parameter group C of transformations (1.1) is called a symmetry group of equation (1.3) if (1.3) is form-invariant (has the same form) in the new variables f

,x

and i, i.e.,

uE = (1.4)

where the function F is the same as in equation (1.3).

1.3 Infinitesimal transformations

According to the Lie's theory, the construction of the symmetry group C is equivalent to the determination of the coiresponding infinitesimal transformations:

t + a 7 (t, x, n), _{x + a(t, x, n),} _{n + a 77 (t, x, n)} _(1.5)

obtained from (1.1) by expanding the functions f, g and h into Taylor series in a about a 0 and also taking into account the initial conditions

f a=O = X, h =0 Thus, we have - Of I (t,x,u)

-

x, n) = - ahl (1.6) x, u)

-aa

-

aa _a=O 3a _a=O

The vector (, , r) with components (1.6) is the tangent vector at the point (t, x, u) to the surface curve described by the transformed points (E, , ), and is therefore called the tangent

vector field of the group C.

One can now introduce the symbol of the infinitesimal transformations by writing (1.5) as

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where

X = (t, x, u) 3 + (t, r, u) 3 + (t, x, u) 0 . (1.7) OU

This differential operator X is known as the infinitesimal operator or generator of the group

C. If the group C is admitted by (1.3), we say that X is an admitted operator of (1.3) or X

is an infinitesimal symmetry of equation (1.3).

1.4 Group invariants

Definition 1.3 A function F(t, x. u) is called an invariant of the group of transformation

(1.1) if

F(t,

, u)

F(f(t, x, u, a), g(t, x, u, a), h(t, x, u, a)) = F(t, x, u), _(1.8)

identically in t, x, u and a.

Theorem 1.2 (Infinitesimal criterion of invariance) _{A necessary and sufficient}

condi-tion for a funccondi-tion F(t, x, u) to be an invariant is that

OF OF

X F (t, x, u) +(t, x, u) + (t, x, u) OU

OF

It follows from the above theorem that every one-parameter group of point transformations (1.1) has two functionally independent invariants, which can be taken to be the left-hand side of any first integrals

J1 (t, x, u) = c1, J2 (t, x, u) = c2,

of the characteristic equations

dt - dc du

X, u) - _(t,_X,u) = 77(t, x, u)

Theorem 1.3 _{Given the infinitesimal transformation (1.5) or its symbol X, the}

correspond-ing one-parameter group C is obtained by solving the Lie equations

dT d - dii -

- (t, , ), - = (t, , _da _da

u), - (

t, ,

u)

(1.10) da

subject to the initial conditions

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1.5 Construction of a symmetry group

In this section _{we briefly describe the algorithm to determine a symmetry group for a given}

PDE. First we need to give some basic definitions.

1.5.1 Prolongation of point transformations

Consider a second-order PDE

=0, (1.11)

where t and x are two independent variables and u is a dependent variable. Let

0 0 0

X (t, x, + (t, x, u)— + q(t, x, n), (1.12)

t Ox On

be the infinitesimal generator of the one-parameter group C of transformation (1.1). The first prolongation of X is denoted by X~11 and is defined by

X111 = X + 0 (t, x, n, n, u) + (2 (t, in, Ut, OUt O 0 ux where = Dt(g) - uDt() - = D(g)

-

_nD1()

-

and the total derivatives Dt and D are given by

Dt = 0 +nt+ntx+utt+... 0 0 0

t On On Out

Dx = —+ux+nxx+utx+... O 0 0 0 . (1.14)

Ox On On On Likewise, the second prolongation of X, denoted by x21, is given by

91 0 0 0 0 0

X _Out+ (2 - _On +

(—

+ (12 + (22 (1.1 OUtt uu On ,

where

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c12

=

D((1)

-

utD()

c22

=

D((2)

-

uD()

Using the definitions of Dt _{and DX) one can write}

= t+ut utTtiT - ut

c2 +x77 UtTx uuT U xx (1.17)

(11 =i7tt+ 2u + utt77u + - 2itttYt - UtYtt - -

- - 2utxtu - U xuu - (uutt + 2utut). (1.18) c12 = itx + Ux qt u + utrlxu + U xtilu + UtUxOuu - + - uttx -

— UtU(Yt + - - (2uu + - - Ug

U xx t - (2uu

tx

+ Ut xx ) u UtU uu . (1.19)

c

22 xx + 2Ux xu + U xxTlu + uu - 2U XXX UXXX - - 3XUXXU

—u -- UtTxx - - (utu + - utu. (1.20)

1.5.2 Group admitted by a PDE

The operator

X = Y (t,x,U) 0 +(t,x,u) 0 ±(t,x,U) 0 , (1.21)

t Dx

au

is said to be a (generator of) point symmetry of the second-order PDE

(1.22)

if

x

[2]

(E) = 0 (1.23)

whenever E 0. This can also be written as (symmetry condition)

x

2

E _{E=O =} (1.24) where the symbol E0 means evaluated on the equation E = 0.

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Definition 1.4 Equation (1.24) is called the determining equation of (1.22), because it determines all the infinitesimal symmetries of equation (1.22).

The theorem below enables us to construct some solutions of (1.22) from known one.

Theorem 1.4 A symmetry of equation (1.22) transforms any solution of (1.22) into another solution of the same equation.

Proof: It follows from the fact that a symmetry of an equation leaves invariant that equation.

1.6 Lie algebras

Let X1 and X2 be any two operators defined by

X1 r1(t,w,n) 0 +i(t.x,n) 0 — +i(t,x,n) 0

—

Ox On and 0 0 3

=

Ox On

Definition 1.5 (Commutator) The commutator of X1 and X2 , written as [X1,X2], is defined by the formula [X1, X2

₁

= X1 (X2 ) - X2 (X1).

Definition 1.6 (Lie algebra) A Lie algebra is a vector space L of operators such that, for all X1,X2 EL, the commutator [X1,X2] EL.

The dimension of a Lie algebra is the dimension of the vector space L. It follows that the commutator is

Bilinear: for any X, Y, Z E L and a, b E IR,

[aX ± bY, Z] = a[X, Z] ± b[Y, Z], {X, aY ± bZ] = a[X, Y] + b[X, Z];

Skew-symmetric: for any X, Y E L,

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3. and satisfies the Jacobi identity: for any X, Y, Z E

[[X, Y], Z} + [[Y, Z], X] + [{Z, X], Y] 0.

Theorem 1.5 The set of all solutions of any determining equation forms a Lie algebra.

1.7 Conclusion

In this chapter we gave a brief introduction to the Lie group analysis of PDEs and pre-sented some results which will be used throughout this work. We also gave the algorithm to determine the Lie point symmetries of PIDEs.

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Chapter 2 Symmetry analysis of the

Black-Scholes equation

In this section we consider the Black-Scholes equation, which arises in financial mathematics and calculate its symmetry Lie algebra. We also find group-invariant solutions under three symmetry generators of the Black-Scholes equation. Black-Scholes equation (1) was first investigated from the point of view of Lie point symmetry analysis by Gazizov and Ibragimov [17], who found its symmetries and used two different transformations to transform it to the heat equation and the latter was used to solve the initial value problem. The invariance principle was used to construct the fundamental solution that could be used for general analysis of an arbitrary initial value problem.

Pooe et al. [46] obtained two classes of optimal systems of the one-dimensional subalgebras for the Black-Scholes equation using the two transformations by Gazizov and Ibragimov [17] that transform Black-Scholes to the heat equation. Sukhomlin and Ortiz [56] obtained solutions for the Black-Scholes equation and the diffusion equation by ansatz using similarities between the two equations. Also in [56], the equivalence group for the Black-Scholes equation was established and the largest set of transformations each of which converts the Black-Scholes equation to the diffusion equation was obtained.

In [16], two potential symmetries were found and used to obtain new solutions to the Black-Scholes equation. First, the equation was written in conserved form which required the

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conservation laws. Conservation laws were found by the method of Kara and Mahomed [32], which uses symmetries to directly compute the conservation laws. Many other researchers also studied the Black-Scholes equation from the point of view of Lie symmetry analysis and pricing of contingent claims, see for example [13, 24,30, 50, 51, 58]

2.1 Calculation of symmetries of the Black-Scholes equa-

tion

Consider the Black-Scholes equation

ut + A2x2u + Bxu - Cu 0. (2.1)

This equation admits the one-parameter Lie group of transformations with infinitesimal generator

X = T (t, x, u) 3 + (t, x, u) 0 - + (t, x, u) 0 - (2.2)

if and only if

X{2J(ut ₊ A2x2u _{± Bxu} - Cu) = 0. (2.3)

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Using the definition of X121 from Chapter one, we obtain

(

+ A2xu + (22A2x2 + Bu + Bx(2 -pC 0, (2.4)

2

(2.1)

where _(,(2 and (22 are given by equations (1.16), (1.17) and (1.20) respectively. Substituting

the values of (, (2 and (22 in equation (2.4) (and replacing u1

by A2x2 I Cu - Bxu - ut])

we obtain

Iqt +

UtPu - UtTt - uT - - _utu1± A2x

[AT2 (Cu - Bxu - ut)] +

A2x2 ₊ ₊

- ux~xx -2uX - - 2utx7x -

2

ut7 - 2utuY - 2uu7 - utur ±

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(A2 x +q. _{- 2} _3u _-_utT.

)]

=

0. (2.5)

Since T_{, and depend only on t, x and u and are independent of the derivatives of u,}

the coefficients of like derivatives of u can be equated to yield the following over determined system of linear PDEs:

= 0, _(2.6) UxUtx TU = 0, (2.7) = 0, _(2.8) = 0, _(2.9) 2 - - - 0, _(2.10) x A2x 2 - + Bx + A2x2 2 - B = 0, (2.11) 1 : +Bxo—C+ A22 2 2Cu —2Cu= 0. (2.12)

Equations (2.6) and (2.7) imply that

= a(t), _(2.13)

where a(t) is an arbitrary function of t. Equation (2.8) gives

= b(t, x), _(2.14)

where b(t, x) is an arbitrary function of t and x. Integrating equation (2.9) twice with respect to u, we obtain

= c(t, x)u + d(t, x), (2.15) where c(t, x) and d(t, x) are arbitrary functions of t and x. Substituting this value of in (2.10), we obtain

= a'(t) ± b(t,x). Solving the above equation we obtain

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where e(t) is an arbitrary function of t. Thus

- a'(t)xinx+xe(t) _(2.17) Substituting the values of and 77 _{in (2.11), we obtain}

1 / e'(t) d(t) Ba'(t)

2xA2 (2.18)

Integrating the above equation with respect to x, we obtain

c - a"(t) (in x)2 - Ba'(t) (in x) ₊e'(t) (in + a'(t)(ln ₊ _(2.19)

, - 4A2 2A2 42 4 1'

where f(t) is an arbitrary function of t. Substituting the values of and ij in (2.12), we obtain

cu + d -- Bx(cu + d) - Cd(t, x) ± — A2 2 ---(c1 u + d) -

2Cu(t) (m x) _{+ e(t)) + Cua'(t) mx}

0. _(2.20)

Separating (2.20) with respect to u, we obtain

A2 x2 c

Ct ± Bxc + -

2 Ca'(t) = 0 (2.21)

A2 x 2d

d + Bxd - Cd(t, x) + = 0. _(2.22)

Substituting the value of c into (2.21), we obtain

a"'(t) (inx)2 - Ba"(t) (in x) e"(t) in x a" (t) in x

4A2 2A2 + A2 + + f'(t) +

N

t ( t)mnx- Ba'(t) e'(t) a'(t) A2 x 2

r

a"(t) a"(t) mx

Bx _

A2x _{2A2x + 42 + 4] + 2 [2A2 - 2A2x2 +}

Ba'(t) - e'(t)

_{ic1 -}

2A2 x2 A2x2 - 4x2 I Ca'(t) = 0. (2.23) Separating (2.23) with respect to mx, we obtain

(

(inx)2 aI,,

t) = 0 _(2.24)

(mx) : e"( t) = 0 (2.25)

- B2a'(t) Be'(t) Ba'(t)

1 : p(t)2A2 _{+ 2 ±}

a"(t) - e'(t) - A2a'(t)

- Ca'(t) = 0. (2.26)

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Integrating (2.24) with respect to t three times, we obtain

A1t 2

a(t)

-v

- + A2t + A3. _(2.27) Now integrating (2.25) gives

e(t) A4t + A5. _(2.28)

Substituting the values of a(t) and e(t) in (2.26), and integrating gives

f(t)= B2 A1t 2 B2 A2t BA4t BAt2 BA2t A1t _4A2

-

2A2 A2 4 2 4

A4t A2 A1t 2 A2 AA CA1t 2

_____

16 8 2 CA2t A6-- . (2.29)

Substituting the values of e(t), a(t) and f(t) into (2.19) we obtain

c(t,x) - Ai(lnx)2

-

Blnx(Ai t+A2 ) A4 lnx lnx(Ait+A2)

-

_4A2 _2A2 +A2

+

₄

+

B2 A1t 2 B2 A2t BA4t BA1t 2 BA2t A1t A4t

4A2

+

2A2

-

A2

-

4

A2 A1t 2 A2 A9t CA1t 2

+

- +

2

±

CA2t A6. 16 8 (2.30) Thus A1t2

₊

_A2t

₊

_A3 _(2.31)

+

A2 )xlnx

+

x(A4t

+

A5) _(2.32) (Ai (lnx)2 Blnx(Ai t

±

A2 ) A4 1nx 1n(A1 t

+

_A2)

4A2

-

2A2

+

A2

+

4

B2 A1t 2 _{B2 A2t} BA4t BA1t 2 .BA2t A1t A4t

+

_4A2

+

2A2

-

A2

-

4 2

A2 A1t 2 A2 A2t CA1t 2

+

16 +

8 +

2 ± CA2t +As )n+d(t) (2.33)

and so the infinitesimal symmetries of the Black-Scholes equation are

x

1

= _, _(2.34)

Ot

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X3 = 2A2t a _{+ A2xlnx} _{+ (2A2Ct + D2t -Dlnx) u,} _(2.36)

X4 = A2tx a

_{+ (mx -}

₍

2.37) = 2A2t2

a

+ 2A2txlnx _{± (2A2 Ct 2 - A2t ± (}

_{mx -}

_{Dt) 2) u,}

(2.38)

au

X6 = and _(2.39) Ou Xd

=

d(t,x), _(2.40)

au

where D B - A2/2 and d is an arbitrary solution of (2.1). Furthermore, X1, , X6 are operators which generate six parameter group and Xd generates an infinite group.

2.2 _{Invariant solutions of the Black-Scholes equation}

In this section, we construct group invariant solutions under some of the symmetry operators of the Black-Scholes equation. We start with the operator X1.

Example 2.1 Let us calculate the invariant solution under the symmetry operator X1. The operator X1 is given by

x1

==. The characteristic equations are

dt dx du 1 = 0 = 0'

which provide the two invariants J1 = x and J2 =u.Thus, the invariant solution is given by J2 = (J1 ), i.e.,

Substituting this value of u in (2.1), we obtain

A2 x' + Bx' - C 0. This is a Cauchy-Euler equation and its solution is given by

(x) =

+c2x'2 _\/2A4_8A2B+16A2+_2\/B}/{2v/A2}

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where c1 and c2 are arbitrary constants. Hence the invariant solution of (2.1) under X1 is

U(t, x)

+c2x B}/{2}. _(2.42)

Example 2.2 Let us calculate the invariant solution under the operator X4, namely

X4 = A2Lx a + (mx - Dt)u ax

,

where D = B - A2 / 2. Now X4 _{J 0}

01

+ A2tx

+ (

ai in x - Dt)u ai

=

0. (2.43)

The characteristic equations are

dt dx - d'u

0 - A2 tX - (

mx -

Dt)n

Thus, one invariant is J1 = t. The other is obtained from the equation

dx dn

42 _{- (lnx -}

I

₁

and is given by J2 = 'u/ exp

(m x -

Dt) 2 j 2A2 t

Consequently, the invariant solution of (2.1) under X4 is J2 = çl(J1), i.e.,

( (mx -

Dt) 2

1

= exp

2A2t (t), (2.44)

where 0 is an arbitrary function of t. Substituting (2.44) into equation (2.1), gives

' +(

c) = o.

This is a first-order variables separable equation and its solutions is given by

K

ç(t) = eCt, VIt-

where K is an arbitrary constant, and hence the invariant solution of the Black-Scholes equation under the operator X4 is

K ((lnx—Dt)2

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Example 2.3 Let us find the invariant solution under the operator X5, namely

X5 = 2A2t2 + 2A2txlnx

0 +

I

(In x - Dt) 2 + 2A2Ct 2 A2t] u.

The characteristic equations are

dt dx du

in X

[(in x - Dt) 2 + 2A2Ct2 - A2t1 By considering

dt dx 2A2t2 = 2A2tx in x

and integrating, we obtain one invariant as equation

The other is obtained from the

dt d'u

2A2t 2

[On x - Dt)2 + 2A2 Ct 2 A2 t]

and is given by J2 = u/exp {(lnx - Dt)2 + Ct}.

2A2t

Consequently, the invariant solution under X5 is J2 = ç5(Ji ), i.e.,

U = exp + Ct 1 I(inx—Dt)2 J,(Inx ). (2.45) 2A2t

Substituting u, u, u., and u in (2.1) and simphfying yieids

= 0. _(2.46)

Solving equation (2.46) we obtain (J1

) = K1

J1 ± K2 where K1 and K2 are arbitrary

constants of integration. Hence equation (2.45) becomes

(

mx K2 '\ 1(lnx—Dt)2

u(t, x) = K1 — + ) exp

2A2t ± Ct

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2.3 Conclusion

In this chapter we obtained the symmetry Lie algebra for Black-Scholes equation. This equation arises in mathematics of finance. We then constructed group-invariant solutions under some infinitesimal generators of the Black-Scholes equation.

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Chapter 3 Group classification of a general

bond-option pricing equation

3.1 Introduction

In this chapter we study the general bond-option pricing equation

ut + axpu,, + B - x)u + = 07 (3.1) where A, 3. -y, c are constants with \, 'y, a different from zero and p > 0, q > 0. Here t is the time, x is the stock (share or equity) price or instantaneous short-term interest rate at current time t and u(t, x) is the current value of the option or bond depending on the form of (3.1).

Equation (3.1) is a generalisation of the Black-Scholes, the Cox-Ingersoll-Ross and the Va-sicek equations because it reduces to the Black-Scholes equation when p = 2, q = 0, 3 = 0,

0 2/ 2 \ = -r and 'y -r, to Cox-Ingersoll-Ross when a= a2/ 2 p _{1, q = 1 and}

7=—i and toVasicek when n=a2 /2,p=0, q= land'y=—l.

We also note that when q = 0, equation (3.1) is the option pricing equation and it is the bond pricing equation when q = 1.

Here we perform Lie group classification of (3.1). We follow the workings of Sinkala et al. [52].

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3.2 Determination of classifying equations of (3.1)

The Lie point symmetries for (3.1) are given by the vector field

if and only if

where

X

=

(

t,xu) a +(t,x,u) a - +17(t,xn)

a

-

(3.2)

(rq + axPuz, +

8

- x)u + xn)

_{(3.1) =} (

3.3)

x 21 =X+(1 a +(2

a

+(22

a

3Ut

aL

OUXX

Here (j's are given by

=

- utDt() -

(2 = D() - uD() -

(22 = D((2) - utD() -

uxx

F

I

L

where the total derivatives

Dt

and D are defined as

a a a a

IZv

Dt =

at an an an

a

D

= + Ux ± + +

ax

an

To perform the group classification of (3.1) it turns out that we need to consider two cases

of p separately; p 2 and p = 2.

3.2.1 Classifying equation of (3.1) for p 2

Expanding the determining equation (3.3), we obtain

-

₊

177Xq+77t

- tux - + -

— _Y2U2

Tu X

2q +

₊

_Ttu -

Axn -

₊

-

+

A2 x2uy

+ 22

u7 -

22xu

-

+

ipx 1u

-

+

(33)

x

_{(tu -} _- ₊ ₊ _{wu +}

+UTX

-

- aXpUgxx

+

+ 2X2LTuuLxx

- 2x

2y ± aoAxP

(u uu + yp+q +

2

x2 u (2u

+ Txx)

-

+

8AXn XTXX =

0. (3.4)

Separating (3.4) with respect to the derivatives of

u,

since the functions

,

and

77

do not

depend on them, leads to the following linear PDEs:

P + X (i

- 2) = 0,

L7X7] + _Xflx_{qux1 -} - Xxx - _{- -}

0,

A +

+ x +

t

+

/\8

- x)

-

= 0

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

To solve the above system of equations, we first observe from equations (3.5) and (3.8) that

i_

_{does not depend on x and u, which means that r is a function of t. Thus}

= i_(t).

_(3.12)

Equation (3.6) implies that depends on both t and x but not on u. Thus

=

(t, x).

_(3.13)

Integration of equation (3.7) with respect to u twice gives

=

A(t,x)u + B(t,x),

_(3.14)

where

A(t,

x) and

B(t,

x) are arbitrary functions of t and x. Using the expressions for

i_ 4

nd

into (3.9) and integrating with respect to x, leads to

(t,

x) =

c(t)x2

-

xi_'(t) p

2,

(3.15)

p-2'

(34)

where c(t) is an arbitrary function of t.

Using (3.12), (3.14) and (3.15) in (3.11) yields

8x

1 _{A(t, x) (2 - p) + c(t)px1+3+2 (p 2)2 - 2c(t)x'}

2 _{(p - 2)2 4x21(t)}

Xpl

- 2) + 4x+2xc'(t) (p - 2)

+ (4x(p— 2) +4x(1 —p))'(t) = 0.

(3.16)

Integrating (3.16) with respect to x and solving for A we obtain

x

-p-1

A(t,x) = d(t)

+ 4(p - 2)2 {( - 2)

2 x 2 c(t) (px + 2x(x - )) - 2x3

T}

1

2 4(p - 2)2 2x

(2(p - 2)x 2c'(t) + (2 p)'(t)( - x)) },

(3.17)

where d(t) is an arbitrary function of integration.

Using (3.12) - (3.15) and (3.17) in (3.10) gives

16(p - 2)

2 x

2

/2

+

+ B + OAB,

_{- xB) +}

- 2)x22

(27(p - q - 2)x - A(p 2)(p - 1))

Tt

- 8u 2 (p 2)2x42 Tt

+8u(p - 2) {(p - 1)px'

_{(p - 1)x2}

/7 + (

2p - )X3-pl2Tt

- 2) (2(p - 2)dx22

+ (p - 1)x22 - 2x3ctt) + 8x4_+2T ttt

+a(p - 2)2c(t)x1u{cp3c - 60p2x + 8cpx

8+\px + 16yqx

2

1 +42(p - 2)2xuc(t)(x - )(— p + px - 2x) = 0.

(3.18)

Since the functions B, c, d and do not depend on u, we split (3.18) with respect to u and

get

1 :

_{axPBxx +}

_{+ B + /}

_{3AB - AxBx}

₌

₀

_(3.19)

8(p - 2)x22

_{(27(p - q - 2)x - (p - 2)(p - 1)) - 82(p - 2)2x472 Tt}

+8(p - 2) ((p - l)px'2 -

_{(p - 1)x22r ± A(2p - 3)x3/2)}

+ (8(p - 2) (2(p - 2)dx22

+ a(p

-

1)x22 - 2 3ctt) + 8x412

)

+a(p - 2) 2c(t)x'

(p3x

- 6p2 x + 8px - 8px + 16qx 2)

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It is clear that

B

satisfies the original equation (3.1). Rewriting

(3.20)

yields our classifying

equation as

ho(t)x 52 + h1(t)x+2+3P/2

+ h

2

(t)x' 3 /2

_{+ h}

3

(t)x4

+ h

4

(t)x 3 + h5(t)x 2 + h6(t)xP+±3 4-h7(t)x 3

± h

8

(t)x 32

± h

9

(t)x 22 ± hio(t)x42

± h

11

(t)x 5

0,

(3.21)

where

ho(t)

=

₂

(p-4)(p-2)3pc(t), h1(t)

=

_16(p

-

2)2qc(t), h2(t)

=

—8nA(p

-

2)2 pc(t), h3(t)

=

8 2(p-2)(2p-3)'(t), h4(t)

=

_—8022(p

_-

2)(p

-

1)7-'(t), h5(t)

=

8cfiA(p

_-

2)(p

-

1)p'(t), h6(t)

=

16cy(p

_-

2)(p

-

q

-

2)'(t), h7(t)

=

8Q(p —2)

_((p

-

1)"(t)

-

A(p

_-

2)(p

-

1)'(t)

+

2(p

-

2)d'(t)), h8(t)

=

—8A2(p -2 2)2(p )2(p

-

1)c(t), h9(t)

=

4 22(p

-

2)2 pc(t), h10(t)

₌

4(p

-

2) (2p2c(t)

-

4 2 pc(t)

+

4A2c(t)

-

h11(t)

=

—8 (2p2T1(t)

-

4A2 p '(t)

+

421(t)

-

T... (t) ) I

3.2.2 Classifying equation of (3.1) for p = 2

In the case when p

2

in (3.1), we proceed as above and get the determining equation as

qux' + ₊ - ± + - 272TX2 + - - + + U/TX 2 + ± + n2uY11x4 - - -2nAyx3 + A 2uYx2 - - 2nyut x2 + n7x2 + - - 2nux2 + n uyx2 +

(36)

+ux2 -

- 2n

XLx2

-

x 2

+

x2

+

2uux2

+ 77xxX2

- ux - 2

2 uyx + Aux Ax + 2ux

—'\u + OAux

7t + 22u

_{- u}

+ 77t

+

- 0.

(3.22)

As before, splitting the above equation (3.22) on derivatives of u and simplifying leads to

TU

- 0,

(3.23)

= 0,

(3.24)

= 0,

(3.25)

-Tx

= 0,

(3.26)

2x3 -

-

+ x -

tX +

Ax - 23 = 0,

(3.27)

2c - 2 -

0,

(3.28)

- 2) + x(y - n

+ 2n)) + x 2 ix + xi(

x)

+ Xt

= 0.

(3.29)

Equations (3.23) and (3.26), imply that

= T

(t),

_(3.30)

whereas (3.24) means that does not depend on u, that is,

= (t, x)

(3.31)

and (3.25) gives

x, u) = A(t, x)u + B(t, x),

(3.32)

after integrating twice by u and for some functions A and B. Substituting (3.30) and (3.31)

into (3.28), we get a linear first order ODE in which can be easily integrated with respect

to x to give

(t, x) = xe(t) + x'(t)

mx,

(3.33)

where e(t) is an arbitrary function of integration. If we substitute (3.30), (3.33) and (3.32)

into (3.27) and solve the resulting differential equation for A, we get

A(t, x) =

[fx[4e' (t) + lnx"(t)] + 2r'(t)[x( + ) -

_]}

mx -

4Ae(t)]

(37)

where

f

i

b

i

iuing (3.34) and (3.33)n(3.29) we obtain

82x4B1

-

8Ax 3 B + 8

aO

+ 8x2 B + 8x 2B +

(-4xe'(t)

-

4Ax 2e'(t) + 4e(t) (27qx 2 + A(A

- x(2 + A))) +

4x2e"(t)lnx + 8Qx2 f'(f) + 8c x 2 r'(f) +

- 2 2 A2 '(t)—

2a.

- 4Ax2 '(t) + 2ax2 "(t) - 2A2 x2 r'(t) +

x 23 (t)

in2

x + 4A2 x'(t) +8Axr'(t)

- 4Ax'(t)1nx

+

2 2 A21(t)inx - 2A2 x'(t)1nx) = 0.

(3.35)

Splitting equation (3.35) on u yields

1 :

B + + ax2 B +

( -

x)AB

= 0

(3.36)

(4x 2e"(t) + 47qx 2T' (t) +

2 2 A2 '(t) -

4Ax'(t)

- 23A2 x'(t)) in

—4x2e'(t)

- 4Ax2 e'(t) + 4e(t) (27qx 2 +

/A(1A

- x(2 + A))) + 8cx2 f'(t)

+8x 2 '(t) - 2 2 A2 '(t) - 2 2 x2 '(t) -

4Ax 2 '(t) +

2ce

—2A2 x 2 '(t) +

x2 "(t)1n2

x + 8Ax'(t) +

4A2 x'(t) = 0.

(3.37)

Rewriting (3.37) we get our classifying equation as

bo

(t) + b1 (t)

mx + x (52 (t) +53(t) mx) + x2 (b4 (t) +

b5 (t)

mx + b6(t) 1n2 )

(57(t) + b8 (t) mx) = 0,

(3.38)

where

bo

(t)

= 432 A2 e(t) - 2/ 2 A2 '(t),

b(1) = 2/ 2 A2

71

(t),

b2

(t)

= —4A(2 a + A)(e(t) -

b3

(t)

= —2A(2c+A)7'(t),

b4

(t)

= 8f'(t) - 4( +

A)e'(t)

- 2( + A)2 '(t) + 2"(t),

b5

(t)

=

4e"(t),

b6

(t)

=

b7

(t)

=

4cy(2qe(t) +

2'(t)),

b8

(t)

=

4cyq ' (t).

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3.3 Results of group classification

\vb note that our classifying equations (3.21) and (3.38) are satisfied if we choose

c(t) = = 0, d(t) f(t) = c2 and

for some constants c1 and c2. Thus using these values, for both cases, the coefficients of the mfinitesiinal operator are

c1 , = 0 and i = c2u + B(t,x), where B(t, x) is any solution of equation (3.1).

Case 0. ci, 77 A. 3, p, q arbitrary

We obtain the following Lie point symmetries:

x

1 = -, X2 = u— X

a

B B(t, )—, a _(3.39)

at

a

, x 3tz

where the symmetry associated with B is said to be a solution symmetry. Lie symmetries (3.39) generate what is called the principal Lie algebra.

By equating the powers of x in equation (3.21)and solving for p we infer that possible extensions of the principal Lie algebra are possible for the following values of p:

34688 0, 1,-, -, , , —,3,4,6.

23o.3

In this project, we consider p = 0, 1 and 21 as these values of p provide us with very important equations in mathematics of finance. For example, when p = 2 and q 0, we have the Black-Scholes equation. We obtain the Vasicek equation when p 0 and q = 1 and CIR equation when p 1 and q = 1.

We now discuss equation (3.1) for three different values of p: p = 0, p = 1 and p = 2.

Case 1 p=O

The classifying equation (3.21) for p = 0 gives

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where go (t) —16 ( ~'A 2 71 (t) - 2 2c(t) - 4nd'(t) - 2n'(t) - 91 (t) = 32c"(t) - 32A2 c(t) ± 48/3. 2 T'(t), 92 (t) = 8" (t) - 32 2 Y(t), 93 (t) = 32n(q+2)'(t), g4 (t) = 64n7qc(t).

By examining powers of x in (3.40), we find that the possible values of q are 0, 1, 2 and 3. Thus, we now look at the following four cases for q:

Case 1.1 q = 0

When q = 0, (3.40) becomes

(go (t) + 93 (t))x3 + gi (t)x4 + 92(t)x5 = 0.

For the above equation to hold, gj's should all identically be zero that is, we need to solve the following system of equations:

32A2 c(t) + 64nd'(t) + 64n77'(t) ± 32n,\7'(t) + 16"(t) —16 22 '(t) = 0, _(3.41) 32c"(t) - 32A2 c(t) + 48X2 '(t) ₌0, (3.42) 83(t) - 327\ 2 '(t) = 0. _(3.43) Solving (3.43), we obtain c e2V c e2t r(t) = - 2 + c3, (3.44)

where c1, c2 and c3 are arbitrary constants of integration. Using (3.44) into (3.42) leads to 32c"(t) + 48c1A2 e2t + 48c22e2t - 32 2 c(t) = 0. (3.45)

Integrating (3.45) with respect to t and solving for c(t) yields

c(t) = _cie2t

(40)

where c4 _{and c5 are arbitrary constants. Substituting (3.46) and (3.44) into (3.41) and} integrating with respect to t gives

d(t)

= 2 c1 e2 t 7cie2t c1e2At ₊ c2e 2A 2c2 Ac 2 t

4n 2\ 2 2)\ 4c I3c4et _/c5e 2c - 2n + +c6,

where c6 is a constant of integration. Substituting expressions for

_{c, d}

_{and back into (3.14)} and (3.15) we get the coefficients of the general Lie symmetry as

c1e 2\ c2e

(t) = 2..\

(t,

x) =

cie2(x — ) + c2e2(x - ) ± ce + c5e_At,

x,

n) = {ciue2t (— _-

_{c + A}

_{2(5 — x)2) } +} c4ue At

(x —

+

+c6

n+B(t,x).

It follows that the corresponding Lie point symmetries that extend the principal Lie algebra are given by 62A1 2t 2t (B2 x2

8Ax

1

8

l - --1—+ne (--+—--j—, 2/\ at 2 2)

0x

2c 2/\ 2u n

2j 3u

e

—+—n(x—)

(9), IOX

c

Ou X6

0x

which generates an infinite dimensional Lie algebra.

Case 1.2

q=1

Continuing in the same manner as in Case 1.1, we find that in this case the principal Lie algebra extends by the following Lie point symmetries:

e2t3 2t ( O7

a

X3

-- +e _{j--—-+—)T-} /32 2 _{Ax2 3Ax 37x i\ 3} 2At 2 _[- + ((x - 1 2 ) - 2)

- ± - (A x - —

2 3x A2 3Uj

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= eAt [ ± _{(-7 - BA2 + A2 x)}

x6

= e- At 'yn 0 1.

IOX AOu

It should be noted that this case results in the Vasicek equation [54].

Casel.3 q = 2

This case has two suhcases. 1.3.1 A2

The Lie point symmetries are given by

e 0 _{2ct (1} 3A2 \\ 3 ne2Kt - -e —x— ₎ + (-23'yA 2kOt 2 2k) k + A) ((kx - A) (k2x — 2) 2 +( _- (9n' e e2t 3 -2Kt 2x A2 ) X4 = - ___- + - _{2k Ot} _2k2 — — _— ( A (282 A ± k(k + A)) Ox 4cO 0 427k - 2 A3 (k + A) + k3x2(k - A) + kAx(k — A)9) OU = e k r3 1 n (k + A) (x + A) k LOx 2nk OU 1 0 X6 = e [Ox 2an n — A) (xk — A) 1 where ic 1.3.2 c=— A2 47

The Lie point symmetries are

= t + (ot2) 0 + n (3 27A2t3 + 32'yAt2 - At 'yx2 B'yX 0

— _{— 313'ytx +}

,

X4 = t2 +

Ot (tx — A2t3) Ox + n (2₄ 7A2t4 ± 2'yAt3 - ThAt3x 482 v /,2 - _{— 3(f}₂₊2'ytx2 — 2/S'ytx — t 'yx 3

2 A A 2 A2 ) On' O 7 2'yx " 0

₎

X5 _{= — + u} _{— 2/37t —,} Ox A On 0 ' ₂ 28'yt 2"vtx 2'yx'\ 0 X6 _{= t—+n-7t—__±__+__)_}

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Case 1.4 q = 3

This case does not extend the principal Lie algebra. We now consider the case when p = 1.

Case 2 p==l

Substituting p = 1 into (3.21), we obtain

ao (t)x9/ 2 _{+ ai (t)x} 72 + a2 (t)x5/ 2 + a3x4 + a4X

+4

+ a5 (t)x5 = 0, (3.47) where ao (t) = 16c"(t) - 4)\ 2c(t),

ai (

t) = 16c'yqc(t), a2 (t) = c(t)

(

- 2/3,\)(3 - 2),

a3

(t) = 16d'(t) + 83A2Y(t), a4 (t) = 16(q+1)7'(t), I" a5 (t)= 8T (t) - 8 '(t).

We deduce from (3.47) that q takes the values 0, , 1 and . Thus, we need to consider the following four cases for q:

Case 2.1 q=O

This leads to three subcases. 2.1.1

The principal Lie algebra is extended by

=

_e

At

+—+u

3 (Ax3A) 3]

-

)\3t 0x ,\ &u tr X4 = L t Dx DuJ 2.1.2 /3= — Ce 2,\

The Lie point symmetries are X3 =

_e

At[

--+x—+u

13 3 (/\171)3]

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e 0 0 31

=

ox

On] At 0 Au\/e 0 = _____ Ox At ₀

x6

Ox 2.1.3

The Lie point symmetries that extend the principal Lie algebra are

X 3 + + eA (Ax 7 3\ 0 Ox e t ₀ xe _ t - ₊ 0 X4 Ox A _On X 5 = - ( - - Ox , AtO + e At7A/ 1) 8 2/ On X6 = e i — — 0 Ox 2/0n Case 2.2 q= 1

This has three subcases.

3a 2.2.1

This case gives the same symmetries as the principal Lie algebra. 2.2.2

This case results in the following extra Lie point symmetries:

x3

= + eAt( - 2ay 0 UeAI, (Ax 37/ c7 2 i 0 A + Ox A3 2 ) On e 8

x4

= __+ e _t (x_2a + ue _t 7v' 72\ 3 jOx AtO _At(AVIx7\₀ - ne - _- -x5 = Ox C A)Ou' Xt = 7Ue 0 Ox+ _{A •} 2.2.3 2A

(44)

The extra Lie point symmetries are 3 e At, 0 _Ai.

₍

=

-j-+e -+e 2) a

,

(2 3

_y

Ax a \\ A2 --+—+--- )u x4 e_t 0

=

(

0 +e

(

2 ₍₇ 3

+

) —, A0t \X A2

₎

A3+A2\[ _{A j On} X5

₌

\/ex—

₊

ex At 1)

(—

0 Ox \/\ 2/ 0 (AT 1 0 Ox c Case 2.3 q = 1

This leads to six subcases.

2.3.1 _and _and

The principal Lie algebra is extended by

e tK a _tK 0 uet ( 8A2 0

= _+

)

x----+Ax--8A — ic at -tk +____ (2 a at 2n - + Ax xK)

where K= A2 - 4a. _{We note that this case gives us the CIR equation}_{[53, 54].}

A2 _3ce

2.3.2

The extra symmetries that extend the principal Lie algebra are given by

= t +X a +7U t ax _{( A} _{) -ju}- I a a (4tx _{2t2 431 a}

x4

= t 2 —+2tx+7n rn -- at Ox A A2 A ) au A2 2.3.3 o=—,3=-- 4y 2A

Additional Lie point symmetries are

X _ a a (-A 2x13 ax 4) an' a a '47tx 1 2 1 4-yx'\ a aUl

x4

= t 2 -+2tx---i--u_--_ At t Ox A - -t+---)_ 2yn/ a ax A On'

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x6

= t+2n(2 t\ a

ax

A2

3 2.3.4

Additional Lie point symmetries to the principal Lie algebra are given by

X3 = t—+x—+n

a

( A

—At

0

4 2/x

3)

at

ax On

= t-- + 2tx--+ u

7 4ytx 3

4'yx

At2'\ a

at ax

_t+_4 },

X

5

=

a

/2'y/

1\a

ax

_{A —-)'}

t

_{47v'\ 0}

ax '

A A2 2.3.5

2/\

The extra Lie point symmetries are

= _{+ xe— + netk /x} 1 A Ax\

a

kat

Ox

e_t

_a

_tK + (A

1 Ax xc) a

X K

at

ax

ne ---+---

_{4 2 2}

4

_On

=

a

Ox

2ci

On' = _{e t} _{+ne t ___}

a

Ox

2a

On'

where ic \/A2 —

3o

A2

2.3.6

3=—,c-

2A47

The principal Lie algebra is extended by

e

0 =

+xe 0

+

e't

/ix

3A

3 Ax)

0 at

ax

_On

Xi = —+xe

e

_t 0

- + -

0 t

(3A

—

— — + —

3 Ax

--)u —

0

it dt

ax

\\ 4,c

4

2 2nJ On

.

X

5 =

a

t( 2 a

— +

e

A

1 '\

u

0 ax

o

+

---

_—

₇₌₎

X

6 =

_0x

₂

7Aj

_{__-__)n-}

1 \

0

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where n

=

-

4 Case 2.4 q=-

2

This does not lead to any extension of the principal Lie algebra.

Case 3

p-2

We can conclude from (3.38) that q can only take the value 0. Proceeding as before, we find that the principal Lie algebra extends for the case when ₀_{= 0 by the following symmetry} operators: 1 3 — +—xlnx+u+—t—t+—At+ .A 2t 1 1 ,\lnx +lnx 1

)

0

( 4a t 2 Dx 4

,

x4

₌

—+txlnx—+u(+

a

( 2t2 1 t 2 7t 2' At 2't t Ox \ 4O 4 ,\tlnx 1 ln2 x 0

+

—tInx+

) —

2o 2

x5

=

Ox ,\t 1

=

0 (2a tx — +n—+—t±) lnx\ 0 Ox 2 2ciOr[

This case gives us the Black-Scholes equation [17].

3.4 Conclusions

Group classification of the general bond-option pricing PDE (3.1) was performed for the values p = 0, 1 and 2. The principal Lie algebra was found to be Xi = Ot, X2 = uO and

XB _{B(t, x)3. These values of p resulted in 16 cases, which extended the principal Lie}

algebra. We presented the Lie point symmetries for each case. Three cases gave us the option pricing equations, which were given by Cases 1.1, 2.1.1 and 3. In the last case, Black-Scholes equation was recovered. Seven bond pricing equations were obtained and they were Case 1.2 and Cases 2.3.1-2.3.6. Cases 1.2 and 2.3.1 were found to be the Vasicek and CIR equations, respectively.

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Chapter 4 Optimal systems and classification of

group invariant solutions for some

bond-option pricing equations

4.1 Introduction

In this chapter, we first obtain optimal systems of one-dimensional subalgebras for Cases 1.2 and 2.1.1 of Chapter three. Subsequently, we perform symmetry reductions and construct group invariant solutions using each element of the optimal systems of the corresponding case.

4.2 Classification of group invariant solutions of Case

To find the optimal system of one-dimensional subalgebras, we first, compute the commutator and adj oint representation tables.

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4.2.1 Computation of commutators

For the two symmetries X1 and X2 its commutator is given by [Xi , X2

1 =

X1 (X2 ) —X2 (X1).

We now compute the commutator table for the Case 1.2. Recall that the symmetries for

Case 1.2 are (omitting the solution symmetry

_XB)

2.\t 3

_e

_{2 t}

2 +

2A2

( 2x - 2n - A2 )

Dx

+

(272 + 2yA2 -

3nyA2x

+ 4(

- x)2

2c\ 3

)

,

2Atr

₀

₁

___

31

e + ₍

_{2( -}

) - 2n)

±

(A2x - Ce

&u]

x

3

= _, ₍_4.1) Ot

[3

u

X4

= e

LOx

+

ceA

(—

n7 - 2 + 2 x)

Ou]

[0 7iO1

= e LOx0?i]'

3 .

We show detailed work of computing nonzero commutators and later write them in a

tabu-lated form. Bear in mind also that the -commutator is skew-symmetric, [X,X] =

and that the diagonal elements in the commutator table are all zero, that is, {X, X1] = 0.

As an illustration, we compute one nonzero commutator [X3,X4

1.

[X31 X41

=

X3 (X4)

-

X4 (X) (e t ) + e

— (--cry — NA

U

22x))

3

- at t —

cA

3 au _eAt [(i) + _{(_} -

_{2 + 2}

x ) Ou

J

= At +

(—n7 -

+

2x) Dx

ceA

Ou

=

(

,- At + e— (—n7 - Dx

ceA

+ A2 x)

OU)

= x

4 .

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Table 4.1: The commutator table of subalgebras

[X,X] X1 X2 X3 X4 X5 X6

X1 0 A X3

-

vX6 —2AX1 0 --14 0

X 2 _{—X 3 +1'X6} 0 2AX —X5 0 0

X3 2AX —2AX2 0 AX —AX5 0

X4 0 X5 —AX4 0 —X6 0 X5 X4 0 AX5 ACk X6 0 0 Xe 0 0 0 0 0 0 where LI = (2n 2 + 2

0-

yA 2 + A 3) 2A3

4.2.2 Adjoint representation

To compute all the entries in the adjoint table we give an illustration by computing the adjoint representation of X1 and X2. With assistance from Table 4.1 we proceed as follows:

00 _n

Ad(exp(cX1))X2 -(adX1)(X2)

= X2 - E[Xi,X2] + [X1, [Xi,X2]1 -

= X2 _E(X3 _LIXe_)+[X1 X3 _uXe]_...

= x2 _ E (x3LIx6)_ E2

Xl

o... = —E 2X1 + X2 - E X3 + LIX6.

The remaining adjoint representation table entries are obtained in the same manner and are tabulated in Table 4.2

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CD CD CD CD CD CD

+

Cr) Cr] rr) Cr)

+

Cr) Cr)

+

-C IJ Cr) Cr)

+

_cq _Ui

+

Ui Cr) C" CD

uj

Cr) ,_cIN Cr1-c I CD U.)

+

Ui CD Ui Cr1-c Ui U.)

—

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4.2.3 Optimal system of one dimensional subalgebras

The Lie algebra

L6

spanned by symmetries

(4.1)

provides a possibility to find invariant

solutions of Eq.

(3.1)

with p = 0 and q = 1 which is based on any one-dimensional subalgebra

of L6

. In light

of

these we can write an arbitrary operator from L6 as

X = 01X1 + 02X2 + 03X3 ± a4X4 + 05X5 + NX6,

which depends on the six arbitrary constants

a,, ••• , c.

To construct the optimal system of one-dimensional subalgebra, we follow the method by

Olver

[43].

First, we need to find the invariant of the full adjoint map as it places limits on

how far we can expect to simplify operator X. The composition of adjoint X1 and X2 on

Group Classification of a General Bond-Option Pricing Equation

III I llll I\III ll I1 llIl III I ll