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Technical report, IDE0741 , November 16, 2007

The feedback effects inilliquid markets, hedgingstrategies of large traders

Master’s Thesis in Financial Mathematics

Nadezda Sergeeva

School of Information Science, Computer and Electrical EngineeringHalmstad University

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The feedback effects in illiquidmarkets, hedging strategies of large

traders

Nadezda Sergeeva

Halmstad University

Project Report IDE0741

Master’s thesis in Financial Mathematics, 15 ECTS credits

Supervisor: Prof. Ljudmila A. BordagExaminer: Prof. Ljudmila A. BordagExternal referee: Prof. V. Roubtsov

November 16, 2007

Department of Mathematics, Physics and Electrical engineeringSchool of Information Science, Computer and Electrical Engineering

Halmstad University

Preface

This Masters thesis in Financial Mathematics was written at the depart-ment of Mathematics, Physics and Electrical engineering at the HalmstadUniversity, Sweden.I especially want to thank my supervisor of my master project, Ljudmila A.Bordag, who was always there to help and give creative tips, incentives andcriticism. Without her this thesis would not be what it is.I want to thank Dmitri Kristensson, a good Swedish friend;), who read mythesis, corrected it and discussed it with me, which was of great advantagefor my thesis progress.I even want to thank my laptop for it never let me in the lurch and withwhom I spent a whole lot of hours during these last weeks.Finally I wish You a nice reading and hope you will enjoy my work.Halmstad, 2007.05.29

AbstractThe master thesis is devoted to an analysis of equilibrium or reaction-function models in illiquidity markets of derivatives. The main equationis a nonlinear equation which is a pertubation of Black-Scholes model.By using analitical methods we study invariant and scaling propertiesfor the considered model.

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Contents

1 Introduction 1

2 A short introduction in the Sircar-Papanicolaou model 3

3 The Short introduction to the Lie method 93.0.1 The basic knowledge . . . . . . . . . . . . . . . . . . . 93.0.2 The associated Lie algebra . . . . . . . . . . . . . . . . 123.0.3 A symmetry groups admitted by a differential equation 14

4 The Lie algebraic structure of the main equation 19

5 The symmetry group admitted by the equation (4.1) 33

6 Conclusions 41

Bibliography 43

iii

Chapter 1

Introduction

The financial mathematic is relatively new and developing area of science.One of the first scientists who get the most important analytical resultsin this area were F. Black, M. Scholes and D. Merton. They described themodel defined for the ideal market of securities. They introduced a procedurethat allows investors to calculate the ”fair” price of a derivative securitywhich value depends on the value of underlying security. Now their modeldemands more advanced modifications because of the new kind of traderswhose hedging strategies influence the market value of securities, for example,the programme traders.

Considering the influence of the feedback effect of a hedging strategyfor market prices, we get one of the Black-Scholes model modifications - anequilibrium or reaction-function model - which was described by P. Sircarand G. Papanicolaou [2], P. Frey and A. Stremme [4].

The equation, which describes this model, concerns to the less investi-gated type of equations. The nonlinearity of this partial differential equationincludes the first and the second spatial derivatives and a small parameter.This small parameter measured the influence of an program trader on anoption price.

To get understanding of this equation we can use different mathematicalmethods. In this project we will use the method of Lie groups analysis. Thismethod is sufficiently widespread and used in different research areas. Forexample, in works of L. A. Bordag [8], [9], this method was used for studyingof financial models, in G. I. Barenblat[15] for equations of mathematicalphysics, in S. V. Spichak and V. I. Stogniy [13], V.V. Sokolov [14], M.L.Grandarias, P. Venero and J. Ramirez [16], P.A. Clarkson and E.L. Mansfield[17] for modifications of the heat equation and others.

In this paper there are six Chapters. The first Chapter is Introduction.In the second Chapter the main model and it is deducing is described. Be-

1

2 Chapter 1. Introduction

cause the model were originally introduced by Sircar and Papanicolaou [2] indetail, in this Chapter just the brief review is stated. In the third Chapterthe Lie group analysis is described. This theory is important for the studyof equation (2.6) which is described in Chapter two. In the forth Chapterthe construction of the complete Lie algebra admitted by equation (2.6) ismade and the main results are formulated in form of a theorem (Theorem(4.0.3)). The fifth Chapter is devoted to finding of invariants and to re-duction of the ordinal partial differential equation to a non linear ordinarydifferential equation. In the sixth Chapter, which is the last one, the resultsare represented and discussed. The bibliography and glossary are included.

Chapter 2

A short introduction in theSircar-Papanicolaou model

In this Chapter we describe and derive a nonlinear partial differential equa-tion of the derivative price, and the hedging strategy in case of the equilibriumor the reaction-function model according to the article by K. Ronnie Sircarand George Papanicolaou [2].

Sircar and Papanicolaou [2] write that there are two investment opportu-nities on the market

• a risk-free money market account (B),

• an asset (S, which is a stock value),

where B is perfectly liquid investment opportunity and S is an illiquid asset.In the model the money market account is used as the numeraire. It meansthat Bt ≡ 1, where S is the forward price of the stock, and the interest ratecan be regarded as equal to zero.

Before we get in to details of the reaction-function model, it is importantto define two types of traders which we will use in the following context.

The first type of traders are reference traders, who are the majority in-vesting in the asset expecting gain. Such traders are ”price-takers”. It meansthat they can not influence the price of asset on the market. They trade insuch a way that the equilibrium asset price follows exactly equation

dSt = ν(St, t)dt + λ(St, t)dWt,

where Wt, t is a standard Brownian motion on a filtered probability space(Ω, F, (Ft)0≤t≤T , P ).

In this equation the reference traders distribution of the wealth and theequilibrium asset price modeled by an Ito process (St, t > 0) are independent

3

4Chapter 2. A short introduction in the Sircar-Papanicolaou

model

of each other. We can consider all reference traders as a single aggregatereference trader who represents a total action of all reference traders. Twolimitations for the aggregate reference trader there exist

• An aggregate stochastic income, which is the total income of the allreference traders, can be described by an equation:

dYt = µ(Yt, t)dt + η(Yt, t)dWt,

where Wt is a Brownian motion. Functions µ(s, y) and η(s, y) willnot appear in our model, so that income process need not be known inour model.

• A demand function D(St, Yt, t) is a function of the income and of theequilibrium price process.

The second type of traders are programme or large traders, who trade theasset following a Black-Scholes type dynamic hedging strategy. The mainreason for trading of the assets is to hedge own position against the risk.The programme traders are large enough to change the price of the assetcorresponding to their own trading strategy.

The demand function for a large trader can be represent by followingequation

φ = ςΦ,

where ς is the volume of options being hedged and Φ is a smooth function,which represents the demand per security being hedged.

After we have defined the reference traders and the programme traders,we can define their total relative demand. Sircar and Papanicolaou [2] con-sider that the price process of the asset is determined by a market equilibriumand by the income (Yt). They make an assumption: the supply of the assetS0 is constant and D is the demand of the reference traders relative to thesupply S0(it means D(s, y, t) = S0D(s, y, t)).

The relative demand function of both kinds of traders at time t has thefollowing presentation

G(s, y, t) = D(s, y, t) + ρΦ(s, t), (2.1)

where ρ is equal to the ratio of the volume of options being hedged to thetotal supply of the asset.

By setting demand and supply to 1 at each point in time, market equi-librium will be equal to 1

G(St, Yt, t) ≡ 1. (2.2)

The feedback effects in illiquid markets. 5

By determining the relationship between St and Yt in the function

G(St, Yt, t) ≡ 1,

We assume that the function smooth satisfies the conditions of the implicityfunction theorem.

In this case we obtain that St = Ψ(Yt, t), where Ψ is a some smoothfunction.

This smooth function Ψ defines that the process St follows to the sameBrownian motion as the process Yt.

Sircar and Papanicolaou [2] shows that by using the Ito lemma, we canobtain the stochastic process

dSt =

[µ(Yt, t)

∂Ψ

∂y+

∂Ψ

∂t+

1

2η2(Yt, t)

∂2Ψ

∂y2

]dt + η(Yt, t)

∂Ψ

∂ydWt.

After some transformations the asset price process St has the following rep-resentation

dSt = α(St, Yt, t)dt + υ(St, Yt, t)η(Yt, t)dWt,

where the adjusted volatility is equal to

υ(St, Yt, t) = − Dy(St, Yt, t)

Ds(St, Yt, t) + ρΦs(St, t), (2.3)

and the adjusted drift is given by

α(St, Yt, t) = −(

µGy

Gs

+Gt

Gs

+1

2η2

[Gyy

Gs

− 2GsyGy

G2s

+G2

yGss

G3s

]).

To cover the risk of the derivative, programme traders can sell or buy theasset. How much of the asset they should trade depends on the change ofthe derivative value in comparison to the asset value. An expression for Φ inform of the price of the derivative follows as the result of this assumption.Summing up the usual arguments, we suppose the price of the derivative isgiven by u(s, t), where u(S, t) is a some sufficient smooth function.

We have∂u

∂t+

1

2(υη)2∂2u

∂s2+ r

(s∂u

∂s− u

)= 0.

By using (2.3) and Φ(St, t) = us, and r = 0 we obtain following equationon the function u(S, t)

∂u

∂t+

1

2η2

(Dy(St, Yt, t)

Ds(St, Yt, t) + ρuss(St, t)

)2∂2u

∂s2= 0.


model

Proposition 1 If the income process Yt is a geometric Brownian motionwith the volatility parameter η, then the feedback model reduces to the classicalBlack-Scholes model in the absence of programme traders if and only if thedemand function is of the form

D(s, t) = U

(yγ

s

)

for some smooth increasing function U(·), where γ = ση1

.The proof the Proposition 1 lead us to our main model. Thus, we have

to consider the part of the proof that is interesting for us.The geometric Brownian motion for Yt is given by

dYt = µ1Ytdt + η1YtdWt.

ρΦ = 0 means programme traders are not on the market.From the classical Black-Scholes equation we obtain that λ(s, t) ≡ σs.

Hence,1

2η2

1y2

[Dy

Ds

]2

=1

2σ2s2.

Here D must satisfy to the condition

Dy

Ds

= −γs

y. (2.4)

The left hand side of (2.4) is negative under our hypothesis for the rationaltrading.

Thus, D(s, y) = U(yγ/s) is the general solution of this partial differentialequation for any differentiable function U(·). Finally, if we directly differen-tiate the equation, we get Ds < 0 and Dy > 0 for s, y > 0 if and only ifU ′(·) > 0.

Using definition (see Proposition 1) of a demand function, we can derivethe pricing equation under the feedback effects. The diffusion coefficient isequal to

1

2η2

1y2

[Dy

Ds + ρΦs

]2

=1

2η2

1y2

[γyγ−1

sU ′(yγ/s)

yγ

s2 U ′(yγ/s)− ρΦs

]2

=

=1

2σ2s2

[yγ

sU ′(yγ/s)

yγ

sU ′(yγ/s)− ρsΦs

]2

.


By using (2.1) and (2.2) we obtain

U

(Y γ

t

St

)= 1− ρΦ(St, t).

It means that we eliminate the variable y.Let V (·) be the inverse function of U(·), whose existence is guaranteed

by the strict monotonicity of V . Then we get

yγ

s= V (1− ρΦ) ⇒ 1

2σ2s2

[V (1− ρΦ)U ′(V (1− ρΦ))

V (1− ρΦ)U ′(V (1− ρΦ))− ρsΦs

]2

. (2.5)

After that we made some transformations of the diffusion coefficient (2.5)and we obtain a family of nonlinear feedback pricing equations that are con-sistent with the classical Black-Scholes equation

ut +1

2σ2s2uss

[V (1− ρus)U

′(V (1− ρus))

V (1− ρus)U ′(V (1− ρus))− ρsuss

]2

= 0.

In our case we consider V as the linear function

U(z) = βz, β > 0

to conclude qualitative properties of the feedback effect from the programmetrading. Thus, we get the equation which describes the value of the hedgingstrategy for a programme trader

ut +1

2σ2s2uss

[1− ρus

1− ρus − ρsuss

]2

= 0. (2.6)


model

Chapter 3

The Short introduction to theLie method

3.0.1 The basic knowledge

In this section we introduce a base knowledge of the Lie group analysis forinvestigation of the symmetry properties of partial differential equations.

Consider the Hausdorff space M , an open set U in M and a homomor-phism ϕ : U → Rm which is called map on the manifold M .

Definition 3.0.1 An m-dimension manifold is a set M together with count-able collection of subsets Uα ⊂ M , called coordinate charts and one-to-onefunctions ϕα : Uα → Vα. Vα are connected open subsets, Vα ⊂ Rm, calledlocal coordinate maps which have the following properties

1.⋃α

Uα = M ,

2. for all Uα and Uβ

ϕβ ϕ−1α : ϕα(Uα

⋂Uβ) → ϕβ(Uα

⋂Uβ)

for all α , β, they are smooth functions,

3. If x ∈ Uα and x ∈ Uβ then there exist open subset W ⊂ Vα and V ⊂ Vβ

with ϕα(x) ∈ W and ϕβ(x) ∈ W which satisfied the following property

ϕ−1α (W )

⋂ϕ−1

β (W ) = ∅.

We introduce a definition of a Lie group:

9

10 Chapter 3. The Short introduction to the Lie method

Definition 3.0.2 An r-parametric Lie group is a group G which also carriesthe structure of an r-dimensional smooth manifold in such a way that boththe group operation

m : G×G → G, m(g, h) = g h g, h ∈ G

and the inversioni : G → G i(g) = g−1 , g ∈ G

are smooth maps between manifolds.

Definition 3.0.3 An r-parametric local Lie group consist of connected opensubsets V0 ⊂ V ⊂ Rr containing coordinate origin, and smooth maps

m : V × V → Rr

defining the group operation and a smooth map i : V0 → V - defining thegroup inversion with the following properties

1. If x, y, z ∈ V and m(x, y) , m(y, z) ∈ V than

m(x,m(y, z)) = m(m(x, y), z),

2. for all x ∈ Vm(0, x) = x = m(x, 0),

3. for all x ∈ V0

m(x, i(x)) = 0 = m(i(x), x).

In practice we deal often with groups of transformations on some manifoldM . If to each group element, g ∈ G, there is associated a map from M toitself, then a Lie group G will be represent as a group of transformations ofthe manifold M .

Definition 3.0.4 Let M be a smooth manifold. A local group of transfor-mations acting on M is given by a (local) Lie group G, an open subset U ,with e ×M ⊂ U ⊂ G ×M , definition of the group action, and a smoothmap Ψ : U → M with the following properties

• if (h, x) ∈ U, (g, Ψ(h, x)) ∈ U , and also (g·h, x) ∈ U , then Ψ(g, Ψ(h, x)) =Ψ(g · h, x),

• for all x ∈ M , Ψ(e, x) = x,

• if (g, x) ∈ U , then (g−1, Ψ(g, x)) ∈ U and Ψ(g−1, Ψ(g, x)) = x.


Definition 3.0.5 An orbit of a local transformation group is a minimalnonempty group invariant subset of the manifold M . In other words, O ⊂ Mis an orbit provided it satisfies the conditions

• if x ∈ O , g ∈ G and g · x is defined than g · x ∈ O,

• if O ⊂ O and O satisfies first condition than O = O or O is empty.

We use further notations similar that in the book P. J. Olver [19]. A curveC on a smooth manifold M is parametrized by a smooth map ϕ : I → M ,where I is subinterval of R. In local coordinates, C is defined by m smoothfunctions

ϕ(ε) = (ϕ1(ε), . . . , ϕm(ε)), ε ∈ I.

The curve C has a tangent vector at each point x = ϕ(ε). That means

ϕ(ε) =dϕ

dε= (ϕ1(ε), . . . , ϕm(ε)), is well defined.

By adopting the notation for tangent vectors to C at x = ϕ(ε)

V|x = ϕ(ε) = ϕ1(ε)∂

∂x1+ . . . + ϕm(ε)

∂

∂xm.

By TM |x we denote the tangent space to M at x. The tangent space to M atx means the set of all tangent vectors to all possible curves passing througha given point x in M . M is an m-dimensional manifold, then TM |x is anm-dimensional vector space, with ( ∂

∂x1 , . . . ,∂

∂xm ) as a basis of TM |x in thegiven local coordinates.

We define the tangent bundle of M as the collection of all tangent spacescorresponding to all points x in M . We denote the tangent bundle by

TM =⋃

x∈M

TM |x.

According to P. J. Olver [19] a vector field V on M assigns a tangentvector V|x ∈ TM |x to each point x ∈ M , with V|x varying smoothly frompoint to point. In local coordinates (x1, . . . , xm), a vector field has the form

V|x = ξ1(x)∂

∂x1+ . . . + ξm(x)

∂

∂xm,

where each ξi(x) is a smooth function of x. An integral curve of a vectorfield V is a smooth parametrized curve x = ϕ(ε) whose tangent vector atany point coincides with the value of V at the same point:

ϕ(ε) = V|ϕ(ε)


for all ε. In local coordinates,

x = ϕ(ε) = (ϕ1(ε), . . . , ϕm(ε))

is a solution to an autonomous system of ordinary differential equations (socalled Lie equations)

dxi

dε= ξi(x) , i = 1, . . . , m,

where the ξi(x) are the coefficients of V at x. If the functions ξi(x) aresmooth, the standard existence and uniqueness theorems for the system ofordinary differential equations guarantee that there exists a unique solutionfor each set of the initial data

ϕ(0) = x0.

This in turn implies the existence of a unique maximal integral curve ϕ : I →M passing through a given point x0 = ϕ(0) ∈ M , where “maximal” meansthat it is not contained in any longer integral curve.

Definition 3.0.6 (flow) If V is a vector field, we denote the parametrizedmaximal integral curve passing through x in M by Ψ(ε, x) and call Ψ the flowgenerated by V. Thus for each x ∈ M , and ε in some interval Ix containing0, Ψ(ε, x) will be a point on the integral curve passing through x in M . Theflow of a vector field has the basic properties:

Ψ(δ, Ψ(ε, x)) = Ψ(δ + ε, x) , x ∈ M

for all δ, ε ∈ R such that both sides of equation are defined and

Ψ(0, x) = x,

andd

dεΨ(ε, x) = V |Ψ(ε,x)

for all ε.

3.0.2 The associated Lie algebra

Definition 3.0.7 A Lie algebra over P (= R) is a vector (linear) space Jtogether with a bilinear operation [·, ·] : J × J → J called the Lie bracket(Lie multiplication) for J satisfying following axioms


• bilinearity

[v + cv, w] = c [v, w] + c [v, w] , [v, cw + cw] = c [v, w] + c [v, w]

w, w, v, v ∈ J ; c, c ∈ P (= R,C),

• skew-symmetry[v, w] = − [w, v] ,

• Jacobi identity

[v, [ν, w]] + [w, [v, ν]] + [ν, [w, v]] = 0

for all ν, v, w ∈ J .If P = R(C) then J is called real (complex) Lie algebra.

In local coordinates a Lie bracket has the following form

[v, w] =m∑

i=1

m∑j=1

(ξj(x)

∂ηi

∂xj− ηj(x)

∂ξi

∂xj

)· ∂

∂xi,

where

v =m∑

i=1

ξi(x)∂

∂xi, w =

m∑i=1

ηi(x)∂

∂xi.

Definition 3.0.8 A Lie algebra J is called Abelian (or commutative) algebraif [v, w] = 0 for all v, w ∈ J .

In table form we can simple demonstrate the structure of a given Liealgebra. The commutator table for g will be the r × r table whose (i, j)-thentry expresses the Lie bracket [vi, vj], if g is an r-dimensional Lie algebra,and v1, . . . , vr form a basis for g. It is important to note that the table isskew-symmetric since [vi, vj] = − [vj, vi].

Definition 3.0.9 A Lie algebra Lr, r < ∞ is said to be solvable if there is asequence (chain)

L1 ⊂ . . . ⊂ Lr−1 ⊂ Lr

of subalgebras of the respective dimension r, r − 1, . . . , 1 such that Lk is anideal in Lk+1 where 1 ≤ k ≤ r − 1.

Definition 3.0.10 An element V ∈ L is called a central element if [v, ν] = 0for any ν ∈ L. The union Z of all central elements is called the center of theLie algebra L.


3.0.3 A symmetry groups admitted by a differentialequation

Definition 3.0.11 Let G be a local group of transformations acting on amanifold M . A subset S ∈ M is called G-invariant, and G is called a sym-metry group of S, if whenever x ∈ S, and g ∈ G is such that g · x is defined,then g · x ∈ S.

Definition 3.0.12 Let G be a local group of transformations acting on amanifold M . A function F : M → N , where N is another manifold, is calleda G-invariant function if for all x ∈ M and all g ∈ G such that g · x isdefined,

F (g · x) = F (x).

A real-valued G-invariant function ζ : M → R is called an invariant of G.Consider a system S of differential equations involving p independent

variables x = (x1, . . . , xp), and q dependent variables u = (u1, . . . , uq). Wedenote a solution of the system by

uα = fα(x1, . . . , xp) , α = 1, . . . , q.

Rp represent the space of the independent variable, and Rq represent thespace of the dependent variables. A symmetry group of the system S will bea local group of transformations, G, acting on some open subset M ∈ Rp×Rq

in such a way that “G transforms solutions of S to other solutions of S”. (Thearbitrary nonlinear transformations of the both independent and dependentvariables is included in this definition of symmetry).

Definition 3.0.13 Let S be a system of differential equations. A symmetrygroup of the system S is a local group of transformations G acting on anopen subset M of the space of independent and dependent variables for thesystem S with the property that whenever u = f(x) is a solution of S, andwhenever g · f is defined for g ∈ G, then u = g · f(x) is also a solution of thesystem. (By solution we mean any smooth function u = f(x) defined on anysub-domain Ω ⊂ X, which satisfies the system S.)

Definition 3.0.14 Let J = (j1, . . . , jk) is an unordered k-tuple multi-indexof integers, with entries jk , 1 ≤ jk ≤ p. The order of such a multi-indexwhich we denote by #J = |J | ≡ is defined by #J = #j1 + . . . #jk

Let us define the spaces Ui and U (n). We define the spaces Ui as thespaces of the partial derivatives of the function u of the order i. It means

Ui = ω = ∂Ju =∂ku

∂xj1∂xj2 . . . ∂xjk, |J | = i.

We define the space U (n) as U (n) = U1 × U2 × . . .× Un.


Definition 3.0.15 The space Rp×U (n) denoted by M (n), whose coordinatesrepresent the independent variables, the dependent variables and the deriva-tives of the dependent variables up to the order n with the natural contactstrecture is called the n-th order jet bundle of the base space M .

Definition 3.0.16 For smooth function u = f(x), with f : Rp → Rq andcomponents f i, i = 1, . . . , q, we define the n-th prolongation of f to be

pr(n)f : X → U (n),

given by u(n) = pr(n)f(x), uαJ = ∂Jfα(x), where J is a multi-index running

over the space of all possible derivatives.A system S of n-th order differential equations is given as a system of

equations∆ν(x, u(n)) = 0 , ν = 1, . . . , l.

It involves x = (x1, . . . , xp), u = (u1, . . . , uq) and the derivatives u(i), i =1, . . . , l of u.

Similar to P. J. Olver [19] we assume that the functions

∆ν(x, u(n)) =(∆1(x, u(n)

), . . . , ∆l(x, u(n)))

are smooth in their arguments, so ∆ is a smooth map from the jet bundleRp × U (n) to some l-dimensional Euclidean space

∆ : X × U (n) → Rl.

The equality ∆ = 0 determines a sub variety

J∆ = (x, u(n))|∆(x, u(n)) = 0 ⊂ M (n)

of the jet bundle M (n).Now let M ∈ Rp × U be open. A symmetry group of ∆ν is a local group

of transformations G acting on M such that when u = f(x) solves ∆ν , thenu = g ·f(x) solves ∆ν for all g ∈ G where defined. Let X be a vector field onM , and assume X infinitesimally generates the symmetry ground G of ∆ν .Then by projecting X into M via the exponential map, we may construct alocal one-parameter group exp(εX). We may then define the prolongation ofX as

pr(n)X =d

dεpr(n) [exp(εX)] (x, u(n))|ε=0

where exp is the exponential map. We also defined Jacobi matrix of ∆ν tobe

J∆ν (x, u(n)) =

(∂∆ν

∂xi,∂∆ν

∂uαJ

)


and say ∆ν is maximal if the rank of J∆ν is l. The following theorem con-strains the form of coefficients of the n-th prolongation of an infinitesimalgenerator for the symmetry group.

Theorem 3.0.1 (Symmetry Group) Let M be an open subset of X ×U and suppose ∆(x, u(n)) is an n-th order system of differential equationsdefined over M , with corresponding sub variety J∆ ⊂ M (n). Suppose G isa local group of transformations acting on M whose prolongation leaves J∆

invariant, meaning that whenever (x, u(n)) ∈ J∆, we have pr(n)g · (x, u(n)) ∈J∆ for all g ∈ G where defined. Then G is a symmetry group of the systemof differential equations.

Definition 3.0.17 Let

∆ν(x, u(n)) = 0 , ν = 1, . . . , l

be a system of differential equations. The system is said to be of the maximalrank if the l × (p + qpn) Jacobian matrix

J∆(x, u(n)) =

(∂∆ν

∂xi,∂∆ν

∂uαJ

)

of ∆ with respect to all the variables(x, u(n)

)is of rank l whenever

∆(x, u(n)) = 0.

The general prolongation formula, is given by the following definition.

Definition 3.0.18 (The general prolongation formula) Let

X =

p∑i=1

ξi(x, u)∂

∂xi+

q∑α=1

ϕα(x, u)∂

∂uα

be a vector field defined on an open subset M ⊂ X×U . The n-th prolongationof X is the vector field

pr(n)X = X +

p∑α=1

∑J

ϕJα(x, u(n))

∂

∂uαJ

defined on the corresponding jet bundle M (n) = X × U (n), the second sum-mation being over all (unordered) multi-indices J = (j1, . . . , jk) with 1 ≤


ji ≤ p, l ≤ k ≤ n. The coefficient functions ϕJα of pr(n)X are given by the

following formula

ϕJα(x, u(n)) = DJ

(ϕα −

p∑i=1

ξiuαi

)+

p∑i=1

ξiuαJ,i

where uαi = ∂uα/∂xi and uα

J,i = ∂uαJ/∂xi.

The following Theorem is called the fundamental theorem of the Lie groupanalysis:

Theorem 3.0.2 Suppose

∆ν(x, u(n)) = 0 , ν = 1, . . . , l

is a system of differential equations of maximal rank defined over M ⊂ X×U .If G is a local group of transformations acting on M , and

pr(n)X[∆ν(x, u(n))] = 0 , ν = 1, . . . , l, whenever ∆(x, u(n)) = 0 (3.1)

for every infinitesimal generator X of G, then G is a symmetry group ad-mitted by the system. The equation (3.1) is called determining equation forthe group G.

Chapter 4

The Lie algebraic structure ofthe main equation

Consider the following equation

∆ = ut +1

2σ2S2uSS

(1− ρuS)2

(1− ρuS − ρSuSS)2= 0, (4.1)

which was introduced in the paper [2] and described in Chapter 2. Letus study the Lie symmetry admitted by this partial differential equation(PDE). The general form of the symmetry group G we obtain if we have acorresponding Lie algebra. Let

X = ξ∂

∂S+ τ

∂

∂t+ ϕ

∂

∂u. (4.2)

be a vector field. The second prolongation pr(2)X is given by

pr(2)X = ξ∂

∂S+τ

∂

∂t+ϕ

∂

∂u+ϕS ∂

∂uS

+ϕt ∂

∂ut

+ϕSS ∂

∂uSS

+ϕSt ∂

∂uSt

+ϕtt ∂

∂utt

,

where coefficients ϕt, ϕS, ϕSS etc. are defined by Definition 3.0.18 (see aswell in Ovsiannikov, L.V. [6]).

ϕS = DS(ϕ− ξuS − τut) + ξuSS + τutS,

ϕt = Dt(ϕ− ξuS − τut) + ξuSt + τutt,

ϕSS = D2S(ϕ− ξuS − τut) + ξuSSS + τuSSt,

ϕSt = DSDt(ϕ− ξuS − τut) + ξuSSt + τuSSt,

ϕtt = D2t (ϕ− ξuS − τut) + ξuStt + τuttt.

19

20Chapter 4. The Lie algebraic structure of the main equation

In our case, (4.1) is of the second order and if we a general vector field X inform (4.2) we have the following form for the second prolongation

pr(2)X = ξ∂

∂S+ ϕS ∂

∂uS

+ ϕt ∂

∂ut

+ ϕSS ∂

∂uSS

,

where

ϕS = ϕS + uS(ϕu − ξS)− τSut − τuuSut − ξuu2S,

ϕt = ϕt + ut(ϕu − τt)− ξtuS − ξuuSut − τuu2t ,

ϕSS = ϕSS + 2ϕSuuS + ϕuuSS + ϕuuu2S − ξSSuS

−2ξSuSS − 2ξSuu2S − 3ξuuSuSS − ξuuu

3S − τSSut

−2τSutS − 2τSuuSut − 2τuuSuSt − τuuSSut − τuuu2Sut.

Now we have to find unknown functions ξ(t, S, u), τ(t, S, u), ϕ(t, S, u) usingthe determining equation (3.1). If some symmetry group is admissible it hasto satisfy the following condition mod(∆ = 0). This condition means thatpr(2)X(∆) = 0 should be satisfied just on solutions of the equation (4.1). Itmeans we should make following substitution

ut = −1

2σ2S2uSS

(1− ρuS)2

(1− ρuS − ρSuSS)2(4.3)

in the determining equation

pr(2)X(∆)| ≡ 0, mod(∆ = 0).

On our case it means that we replace ut by (4.3) and obtain

pr(2)X(∆) =1

(1− ρuS − ρSuSS)3(ξσ2SuSS(1− ρuS)3 + (ϕS + uS

(ϕu − ξS)− τSut − τuuSut − ξuu2S)rho2σ2S3u2

SS(1− ρuS)

+(ϕt + ut(ϕu − τt)− ξtuS − ξuuSut − τuu2t )(1− ρuS

−ρSuSS)3 + (ϕSS + 2ϕSuuS + ϕuuSS + ϕuuu2S − ξSSuS

−2ξSuSS − 2ξSuu2S − 3ξuuSuSS − ξuuu

3S − τSSut − 2τSutS

−2τSuuSut − 2τuuSuSt − τuuSSut − τuuu2Sut)

1

2σ2S2

(1− ρuS)2(1− ρuS + ρSuSS).


Correspondingly, after substitution (4.3) we obtain

pr(2)X(∆)|∆=0 =1

(1− ρuS − ρSuSS)3(ξσ2SuSS(1− ρuS)3 + (ϕS + uS(ϕu

−ξS)− ξuu2S)ρ2σ2S3u2

SS(1− ρuS) +1

2(ϕt − ξtuS)(1− ρuS

−ρSuSS)3 + (ϕSS + (2ϕSu − ξSS)uS + (ϕuu − 2ξSu)u2S

+(ϕu − 2ξS)uSS − 3ξuuSuSS − ξuuu3S)σ2S2(1− ρuS)2

(1− ρuS + ρSuSS) +1

2(τS + τuuS)

σ4S5ρ2u3SS(1− ρuS)3

(1− ρuS − ρSuSS)2

+1

4σ4S4uSS(1− ρuS)4 1− ρuS + ρSuSS

(1− ρuS − ρSuSS)2(τSS + 2τSuuS

+τuuSS + τuuu2S)− (τS + τuuS)uStσ

2S2(1− ρuS)2(1− ρuS

+ρSuSS)− 1

2σ2S2uSS(ϕu − τt)− ξuuS)(1− ρuS)2

(1− ρuS − ρSuSS)− 1

4τu

σ4S4u2SS(1− ρuS)4

1− ρuS − ρSuSS

).


After some transformations we obtain

pr(2)X(∆)|∆=0 =1

(1− ρuS − ρSuSS)5(ξσ2S(uSS − 3ρuSuSS + 3ρ2u2

SuSS

−ρ3u3SuSS)(1− 2ρuS + ρ2u2

S − 2ρSuSS + 2ρ2SuSuSS

+ρ2S2u2SS) + (ϕSu2

SS + (ϕu − ξS − ρϕS)uSu2SS − (ξu

+ρ(ϕu − ξS))u2Su2

SS + ξuρu3Su2

SS) +1

2(ϕt − ξtuS)(1− 5ρuS

+10ρ2u2S − 10ρ3u3

S + 5ρ4u4S − ρ5u5

S − 5ρSuSS + 20ρ2SuSuSS

−30ρ3Su2SuSS + 20ρ4Su3

SuSS − 5ρ5Su4SuSS + 10ρ2S2uSS

−30ρ3S2uSu2SS + 30ρ4S2u2

Su2SS − 10ρ5S2u3

Su2SS − 10ρ3S3u3

SS

+20ρ4S3uSu3SS − 10ρ5S3u2

Su3SS + 5ρ4S4u4

SS − 5ρ5S4uSu4SS

−ρ5S5u5SS) + σ2S2(ϕSS + (2ϕSu − ξSS)uS + (ϕuu − 2ξSu)u

2S

+(ϕu − 2ξS)uSS − 3ξuuSuSS − ξuuu3S)(1− 5 + 10ρ2u2

S

−10ρ3u3S + 5ρ4u4

S − ρ5u5S + 4ρ2SuSuSS − 6ρ3Su2

SuSS

+4ρ4Su3SuSS + (ρ4 − 2ρ5S)u4

SuSS − ρ2S2u2SS − 3ρ3S2uSu2

SS

−3ρ4S2u2Su2

SS + ρ5S2u3Su2

SS + ρ3S3u3SS − 2ρ4S3uSu3

SS

+ρ5S3u2Su3

SS) +1

2σ4S5ρ2(τSu3

SS + τuuSu3SS)(1− 3ρuS

+3ρ2u2S − ρ3u3

S) +1

2σ2S4uSS(τSS + 2τSuuS + τuuSS + τuuu

2S)

(1− 5ρuS + 10ρ2u2S − 10ρ3u3

S + 5ρ4u4S − ρ5u5

S + ρSuSS

−4ρ2SuSuSS + 6ρ3Su2SuSS − 4ρ4Su3

SuSS + ρ5Su4SuSS)

−σ2S2(τSuSt + τuuSuSt)((1− 5 + 10ρ2u2S − 10ρ3u3

S + 5ρ4u4S

−ρ5u5S + 4ρ2SuSuSS − 6ρ3Su2

SuSS + 4ρ4Su3SuSS − ρ5Su4

SuSS

−ρ2S2u2SS + 3ρ3S2uSu2

SS − 3ρ4S2u2Su2

SS + ρ5S2u3Su2

SS

+ρ3S3u3SS − 2ρ4S3uSu3

SS − ρ5S3u2Su3

SS)− 1

4τuσ

4S4u2SS

(1− 5ρuS + 10ρ2u2S − 10ρ3u3

S + 5ρ4u4S − ρ5u5

S − ρSuSS

+4ρ2SuSuSS − 6ρ3Su2SuSS + 4ρ4Su3

SuSS − ρ5Su5SuSS)

−1

2σ2S2((ϕu − τt)uSS − ξuuSuSS)(1− 5ρuS + 10ρ2u2

S

−10ρ3u3S + 5ρ4u4

S − ρ5u5S − 3ρSuSS + 12ρ2SuSuSS

−18ρ3Su2SuSS + 12ρ4Su3

SuSS − 3ρ5Su4SuSS + 3ρ2S2u2

SS

−9ρ3S2uSu2SS + 9ρ4S2u2

Su2SS − 3ρ5S2u3

Su2SS − ρ3S3u3

SS

+2ρ4S3uSu3SS − ρ5S3u2

Su3SS))).

We have to solve these system of equations pr(2)X(∆)|∆=0 ≡ 0 by assump-


tions that the variables S, t, u, us . . . are independent variables in M (2). Weshould demand that all coefficients of each monomial containing derivativesuSS, uS . . . of the function u must be equal to zero. It immediately gives us thefollowing system of the differential equations on the functions ξ, τ, ϕ whichis listed in Table 4.1 and Table 4.3. From the Table 4.1 we can see that τ

uSt τS = 0,

uSuSt 5ρτS − τu = 0.

Table 4.1: The system of the differential equations on the function τ(S, t, u).

should not depend on u and S, i.e. τ(S, t, u) = τ(t). Using this fact we obtainthe following system of equations on the functions ξ(s, t, u), τ(t), ϕ(s, t, u)(see Table 4.3). From the Table 4.3 we get

ξuu = 0, (4.4)

ξu = 0, (4.5)

ξt = 0, (4.6)

ϕt = −2σ2S2ϕSS, (4.7)

ϕt = 0. (4.8)

Hence ξ does not depend on u and t and is a function of the variable S only,ξ(S, t, u) = ξ(S). Compare the equations (4.7) and (4.8), we obtain ϕSS = 0.Insert our results in the Table 4.3 we obtain a new system (see Table 4.4).We will insert now in the general system Table 4.4, τ = τ(t), ξ = ξ(S),ϕSS = 0 then from the Table 4.4 ((22) and (28) equations) we obtain

ϕuu = 2ξS,

2ϕSu = ξSS.


We as well immediately obtain

ξ + 12Sϕu + τt = 0, (4.9)

−2ξ + 12Sϕu + 2SξS + SρϕS − 3

2Sτt = 0, (4.10)

−4ξ − Sϕu − 9SξS − 3SρϕS + 6Sτt = 0, (4.11)

−12ξ − 3SρϕS + 6Sϕu + 3Sξs − 9Sτt = 0, (4.12)

8ξ − 3Sϕu − 3SξS − SρϕS = 0, (4.13)

−4ξ + Sϕu + 2SξS − 3Sτt = 0, (4.14)

2ξ − 4SρϕS − 3Sϕu + 3Sτt = 0, (4.15)

−6ξ − Sϕu + 8SρϕS + 16SξS = 0, (4.16)

6ξ − Sϕu − 8SξS − 4SρϕS + 9Sτt = 0, (4.17)

−2ξ − Sϕu + 4SξS − 3Sτt = 0, (4.18)

2ρϕS + 3ϕu − 4ξS − τt = 0, (4.19)

ρϕS + ξS − τt = 0, (4.20)

2ξS − ϕu + τt = 0. (4.21)

If we multiply (4.20) by two and subtract from equation (4.19) we obtain

3ϕu − 6ξs + τt = 0. (4.22)

Multiply (4.21) by three and subtract from (4.22) we obtain

4τt = 0.

It means that τ(t) = k where k is constant. After simplifications we obtainthe following reduced system of determining equations

ξ + 12Sϕu = 0, (4.23)

−2ξ + 12Sϕu + 2SξS + SρϕS = 0, (4.24)

−4ξ − Sϕu − 9SξS − 3SρϕS = 0, (4.25)

−12ξ − 3SρϕS + 6Sϕu + 3Sξs = 0, (4.26)

8ξ − 3Sϕu − 3SξS − SρϕS = 0, (4.27)

−4ξ + Sϕu + 2SξS = 0, (4.28)

2ξ − 4SρϕS − 3Sϕu = 0, (4.29)

−6ξ − Sϕu + 8SρϕS + 16SξS = 0, (4.30)

6ξ − Sϕu − 8SξS − 4SρϕS = 0, (4.31)

−2ξ − Sϕu + 4SξS = 0, (4.32)

2ρϕS + 3ϕu − 4ξS = 0, (4.33)

ρϕS + ξS = 0, (4.34)

2ξS − ϕu = 0. (4.35)


From (4.23),(4.34) and (4.35) equations we have

ξ = −12Sϕu, (4.36)

ξS = −ρϕS, (4.37)

ξS = 12ϕu. (4.38)

Let us substitute equation (4.38) to (4.36). We obtain ξ = SξS. Solving thisequation we obtain ξ(S) = CS, where C=const. Hence ξS = C.Substitute this result in equations (4.38) we obtain

ϕ = 2Cu + g(S). (4.39)

HenceϕS = gS(S)

but

ϕS = −C

ρ.

From this equations we obtain

gS(S) = −C

ρ⇒ g(S) = −C

ρS + d, where d ≡ const.

Substitute the function g(S) in equations (4.39) we obtain a following struc-ture of the function ϕ(S, t, u)

ϕ(S, u) = 2Cu− C

ρS + d.

Summarize the results of the previous calculations we obtain following valuesfor the coefficients τ, ξ, ϕ of the vector field X (4.39)

τ = k,

ξ(S) = CS,

ϕ(S, u) = 2Cu− CρS + d,

where C, k, d are arbitrary constants.


Theorem 4.0.3 The equation

ut +1

2σ2S2uSS

(1− ρuS)2

(1− ρuS − ρSuSS)2= 0

admits a three dimensional Lie algebra L3 with the following generators

U1 =∂

∂t,

U2 =∂

∂u, (4.40)

U3 = S∂

∂S+

(2u− S

ρ

)∂

∂u.

The algebra L3 has a two dimensional Abelian subalgebra L2 spanned by in-finitesimal generators U1, U2, i.e. L2 =< U1, U2 >.

U1 U2 U3

U1 0 0 0U2 0 0 2U2

U3 0 −2U2 0

Table 4.2: The commutator table for the algebra L3 (4.40).


Table 4.3: The determining system of equations for the Lie algebra L. Ad-mitted by (4.1)

uSu5SS

12ρ5S5ξt = 0

u5SS −1

2ρ5S5ϕt = 0

u3Su4

SS σ2S5ρ5ξu − 3σ2S5ρ5ξu − 12σ2S5ρ5ξu = 0

u2Su4

SS σ2ρ4S5ξu + 12σ2ρ5S5(ϕu − τt) + σ2ρ5S5(ϕu − 2ξS) + 6σ2ρ4S5ξu

+52S4ρ5ξt − σ2S5ρ4(ξu + ρ(ϕu − ξS)) = 0

uSu4SS σ2ρ4S5(ϕu − ξS − ρϕS)− 5

2S4ρ4ξt − 5

2S4ρ5ϕt − 2σ2ρ4S5(ϕu − 2ξS)

−3σ2ρ3S5ξu − 12σ2S5ρ3ξu − σ2S4ρ4(ϕu − τt) = 0

u4SS σ2S5ρ4ϕS + 5

2S4ρ4ϕt + σ2S5ρ3(ϕu − 2ξS) + 1

2σ2S5ρ3(ϕu − τt) = 0

u4Su3

SS32S4ρ5ξu − σ2S5ρ5(ϕuu − 2ξS) + 3σ2S4ρ5ξu + 2σ2S5ρ4ξuu

−2σ2S4ρ5ξu = 0

u3Su3

SS −σ2S3ρ5ξ − 2σ2S4ρ4ξu − 2σ2S4ρ4(ξu + ρ(ϕu − ξS)) + 5σ2S3ρ5ξt

+σ2S5ρ5(2ϕSu − ξSS) + σ2S4ρ5(ϕu − 2ξS)− 2σ2S5ρ4(ϕuu − 2ξS)+9σ2S4ρ4ξu − σ2S5ρ3ξuu + 9σ2S4ρ4ξu + 3

2σ2S4ρ5(ϕu − τt) = 0

u2Su3

SS 3σ2S3ρ4ξ + 2σ2S4ρ4(ϕu − ξS − ρϕS) + 2σ2S4ρ3(ξu − ρ(ϕu − ξS))−10S3ρ4ξt − 5S3ρ5ϕt + σ2S5ρ5ϕSS − 2σ2S5ρ4(2ϕSu − 2ξSS)−3σ2S4ρ4(ϕu − 2ξS) + σ2S5ρ3(ϕuu − 2ξS) + 9σ2S4ρ3ξu+σ2S4ρ2ξuu − 9

2σ2S4ρ3ξu − 9

2σ2S4ρ4(ϕu − τt) = 0

uSu3SS −3σ2S3ρ3ξ + 5S3ρ3ξt + 10S3ρ4ϕt − 2σ2S5ρ4ϕSS + σ2S5ρ3(2ϕSu

−2ξSS)− 3σ2S4ρ3(ϕu − 2ξS) + 3σ2S4ρ2ξu − 2σ2S4ρ3(ϕu − ξS

−ρϕS) + 2σ2S4ρ4ϕS + 92σ2S4ρ3(ϕu − τt) = 0

u3SS σ2S3ρ2ξ − 2σ2S4ρ3ϕS − 5σ2S3ρ3ϕt + σ2S5ρ3ϕSS

−32σ2S4ρ2(ϕu − τt) = 0

u5Su2

SS σ2S3ρ2(ϕuu − 2ξS)− 3σ2S2(ρ4 − 2Sρ5)ξu − σ2S4ρ5ξuu

−35σ2S3ξu + σ2S3ρ5ξu = 0


u4Su2

SS −2σ2S2ρ5ξ − σ2S3ρ4(ξu + ρ(ϕu − ξS))− 2σ2S3ρ4ξu + 5S2ρ5ξt

+σ2S4ρ5(2ϕSu − 2ξSS) + σ2S2(ρ4 − 2Sρ5)(ϕu − 2ξS)−3σ2S4ρ4(ϕuu − 2ξS)− 12σ2S3ρ4ξu − 3σ2S3ρ4ξuu + 6Sρ4ξu

+32σ2S3ρ5(ϕu − τt) = 0

u3Su2

SS 8σ2S2ρ4ξ + σ2S3ρ4(ϕu − ξS − ρϕS) + 2σ2S3ρ3(ξu + ρ(ϕu − ξS))−15S2ρ4ξt − 5S2ρ5ϕt + σ2S4ρ5ϕSS − 2σ2S4ρ4(2ϕSu − ξSS)+4σ2S3ρ4(ϕu − 2ξS)− 3σ2S4ρ3(ϕuu − 2ξS) + 18σ2S3ρ3ξu

+σ2S4ρ2ξuu − 9σ2S3ρ3ξu − 6σ2S3ρ4(ϕu − τt) + σ2S3ρ3ξu = 0

u2Su2

SS −12σ2S2ρ3ξ + σ2S3ρ4ϕS − 2σ2S3ρ3(ϕu − ξS − ρϕS)−σ2S3ρ2(ξu + ρ(ϕu − ξS)) + 5S2ρ3ξt + 15S2ρ5ϕt − 3σ2S4ρ4ϕSS

−3σ2S4ρ3(2ϕSu − ξSS)− σ2S4ρ2(ϕuu − 2ξS)− 6σ2S3ρ2ξu

+9σ2S4ρ3(ϕu − τt) = 0

uSu2SS −4σ2S2ρ2ξ + σ2S3ρ2(ϕu − ξS − ρϕS)− 2σ2S3ρ3ϕS − 5S2ρ2ξt

−15S2ρ3ϕt − 3σ2S4ρ3ϕSS − σ2S4ρ2(2ϕSu − ξSS) + 4σ2S3ρ2(ϕu

−2ξS) + 3σ2S3ρξu − 32σ2S3ρξu − 6σ2S3ρ2(ϕu − τt) = 0

u2SS −2σ2S2ρξ + σ2S3ρ2ϕS + 5S2ρ2ϕt − σ2S4ρ2ϕSS − σ2S3ρ(ϕu − 2ξS)

+32σ2S3ρ(ϕu − τt) = 0

u6SuSS (ρ4 − 2ρ5S)σ2S2(ϕuu − 2ξS)− 4σ2Ssρ4ξuu − 1

2σ2S2ρ5ξu

+15σ2S2ρ5ξu = 0

u5SuSS −σ2Sρ5ξ + 5

2ρ5Sξt + (ρ4 − 2ρ5S)σ2S2(2ϕSu − ξSS)− σ2S2ρ5(ϕu

−2ξS) + 4σ2S3ρ4(ϕuu − 2ξS)− 15σ2S2ρ4ξu − 6σ2S3ρ3ξuu

+52σ2S2ρ4ξu − 1

2σ2S3ρ5ξu = 0

u4SuSS 5σ2Sρ4ξ − 10ρ4Sξt − 5

2ρ5Sϕt + (ρ4 − 2ρ5S)σ2S2ϕSS

+4σ2S3ρ4(2ϕSu − ξSS) + 5σ2S2ρ4(ϕu − 2ξS)− 6σ2S2ρ3(ϕuu

−2ξS) + 30σ2S2ρ3ξu − 4σ2S2ρ2ξuu − 5σ2S2ρ3ξu − 52σ2S2ρ4(ϕu

−τt) = 0

u3SuSS −10σ2Sρ3ξ + 10Sρ4ϕt + 15Sρ3ξt + 4σ2S3ρ3ϕSS + 4σ2S3ρ4(2ϕSu

−ξSS)− 10σ2S2ρ3(ϕu − 2ξS) + 4σ2S3ρ2(ϕuu − 2ξS)− 30σ2S2ρ2ξu

+σ2S3ρξuu + 5σ2S2ρ2ξu + 5σ2S2ρ3(ϕu − τt) = 0

u2SuSS 10σ2Sρ2ξ − 15Sρ3ϕt − 10Sρ2ξt − 6σ2S3ρ3ϕSS + 4σ2S3ρ2(2ϕSu

−ξSS) + 10σ2S2ρ2(ϕu − 2ξS)− σ2S3ρ(ϕuu − 2ξS) + 15σ2S2ρξu

−52σ2S2ρξu − 5σ2S2ρ2(ϕu − τt) = 0


uSuSS −5σ2Sρξ + 52Sρξt + 10Sρ2ϕt + 4σ2S3ρ2ϕSS − 5σ2S2ρ(ϕu − 2ξS)

−3σ2S2ξu − 52σ2S2ρ(ϕu − τt)− σ2S3ρ(2ϕSu − ξSS) + 1

2σ2S2ξu = 0

uSS σ2Sξ − 52Sρϕt − σ2S3ρϕSS + σ2S2(ϕu − 2ξS)− 1

2σ2S2(ϕu − τt) = 0

u8S σ2S2ρ5ξuu = 0

u7S −σ2S2ρ5(ϕuu − 2ξS)− 10σ2S2ρ3ξuu = 0

u6S

12ρ5ξt − σ2S2ρ4(2ϕSu − ξSS) + 5σ2S2ρ4(ϕuu − 2ξS)

+10σ2S2ρ3ξuu = 0

u5S −5

2ρ4ξt − 1

2ρ5ϕt − σ2S2ρ5ϕSS + 5σ2S2ρ4(2ϕSu − ξSS)

−10σ2S2ρ3(ϕuu − 2ξS)− 10σ2S2ρ2ξuu = 0

u4S 5ρ3ξt + 5

2ρ4ϕt + 5σ2S2ρ4ϕSS − 10σ2S2ρ3(2ϕSu − ξSS)

+10σ2S2ρ2(ϕuu − 2ξS) + 5σ2S2ρξuu = 0

u3S −5ρ3ϕt − 5ρ2ξt − 10σ2S2ρ3ϕSS + 10σ2S2ρ2(2ϕSu − ξSS)

−5σ2S2ρ(ϕuu − 2ξS)− σ2S2ξuu = 0

u2S

52ρξt + 5ρ2ϕt + 10σ2S2ρ2ϕSS − 5σ2S2ρ(2ϕSu − ξSS) + σ2S2(ϕuu

−2ξS) = 0

uS −12ξt − 5

2ρϕt − 5σ2S2ρϕSS + σ2S2(2ϕSu − ξSS) = 0

1 12ϕt + σ2S2ϕSS = 0


Table 4.4: The reduced system of the differential equations on the functionsτ(t), ξ(S), ϕ(S, u).

1 12(ϕu − τt)− ξS = 0

2 S(ϕu − ξS − ρϕS)− 2S(ϕu − 2ξS)− (ϕu − τt) = 0

3 ρϕS + 52(ϕu − 2ξS) + 1

2(ϕu − τt) = 0

4 ϕuu − 2ξS = 0

5 −ρξ − 2Sρ(ϕu − ξS) + S2ρ(2ϕSu − ξSS) + Sρ(ϕu

−2ξS)− 2S2(ϕuu − 2ξS) + 32Sρ(ϕu − τt) = 0

6 3ρξ + 2Sρ(ϕu − ξS − ρϕS) + 2Sρ(ϕu − ξS)−2S2ρ(2ϕSu − ξSS)− 3Sρ(ϕu − 2ξS) + S2(ϕuu − 2ξS)−9

2Sρ(ϕu − τt) = 0

7 −3ξ + S2(2ϕSu − 2ξSS)− 3S(ϕu − 2ξS)−2S(ϕu − ξS − ρϕS) + 2SρϕS + 9

2S(ϕu − τt) = 0

8 ξ − 2SρϕS − 32S(ϕu − τt) = 0

9 ϕuu − 2ξS = 0

10 −2ρξ − Sρ(ϕu − ξS) + S2ρ(2ϕSu − ξSS) + (1−2Sρ)(ϕu − 2ξS)− 3S2(ϕuu − 2ξS) + 3

2Sρ(ϕu − τt) = 0

11 8ρξ + Sρ(ϕu − ξS − ρϕS) + 2Sρ(ϕu − ξS)−3S2ρ(2ϕSu − ξSS) + 4Sρ(ϕu − 2ξS)− 3S2(ϕuu − 2ξS)−6Sρ(ϕu − τt) = 0

12 −12ρξ + Sρ2ϕS − 2Sρ(ϕu − ξS − ρϕS)− Sρ(ϕu − ξS)−3S2ρ(2ϕSu − ξSS)− S2(ϕuu − 2ξS) + 9S2ρ(ϕu − τt) = 0

13 −4ξ + S(ϕu − ξS − ρϕS)− 2SρϕS − S2(2ϕSu

−ξSS) + 4S(ϕu − 2ξS)− 6S(ϕu − τt) = 0

14 −2ξ + SρϕS − S(ϕu − 2ξS) + 32S(ϕu − τt) = 0


15 (1− 2ρS)(ϕuu − 2ξS) = 0

16 −ρξ + (1− 2ρS)S(2ϕSu − 2ξSS)− Sρ(ϕu − 2ξS)+4S2(ϕuu − 2ξS) = 0

17 5ρξ + 4S2ρ(2ϕSu − ξSS) + 5Sρ(ϕu − 2ξS)− 6S(ϕuu

−2ξS)− 52Sρ(ϕu − τt) = 0

18 −10ρξ + 4S2ρ2(2ϕSu − ξSS)− 10Sρ(ϕu − 2ξS)+4S2(ϕuu − 2ξS) + 5Sρ(ϕu − τt) = 0

19 10ρξ + 4S2ρ(2ϕSu − ξSS) + 10Sρ(ϕu − 2ξS)− S2(ϕuu

−2ξS)− 5Sρ(ϕu − τt) = 0

20 ξ + S(ϕu − 2ξS) + 12S(ϕu − τt) = 0

21 ξ + S(ϕu − 2ξS)− S2ρ(2ϕSu − ξSS)− 12S(ϕu − τt) = 0

22 ϕuu − 2ξS = 0

23 −2ϕSu + ξSS + 5(ϕuu − 2ξS) = 0

24 ρ(2ϕSu − ξSS)− 2(ϕuu − 2ξS) = 0

25 −ρ(2ϕSu − ξSS) + (ϕuu − 2ξS) = 0

26 2ρ(2ϕSu − ξSS)− ϕuu + 2ξS = 0

27 −5ρ(2ϕSu − ξSS) + ϕuu − 2ξS = 0

28 2ϕSu − ξSS = 0

Chapter 5

The symmetry group admittedby the equation (4.1)

Let us now find the symmetry group admitted by the equation (4.1)

ut +1

2σ2S2uSS

(1− ρuS)2

(1− ρuS − ρSuSS)2= 0.

According to the fundamental theorem of the Lie group analysis (seeTheorem 3.0.2) to find the global representation of the group symmetryadmitted by this equation we have to solve the following system of ordinarydifferential equations with the corresponding initial conditions

dtεdε

= k , tε|ε=0 = t,

dSε

dε= CSε , Sε|ε=0 = S,

duε

dε= 2Cuε − CSε

ρ+ d. , uε|ε=0 = u.

(5.1)

To study different particular cases we will consider action of different sub-groups. We will consider following particular cases of the system (5.1)

1. C 6= 0, k 6= 0,

2. C 6= 0 , k = 0,

3. C = 0, k 6= 0,

4. C = 0, k = 0.

Now we study all listed cases and the corresponding subgroups as well as,their invariants. Using these invariants we will reduce the main equation(4.1) to ordinary differential equations.

33

34Chapter 5. The symmetry group admitted by the equation

(4.1)

Case 1. C 6= 0, k 6= 0 in (5.1).

It means that complete symmetry group G3 admitted by the equation (4.1)is studied, i.e., all of generators of the Lie algebra G3 acting on the solutionsmanifold L∆ of our equation is present. We solve the first equation of thesystem (5.1) and obtain

tε = C1 + kε,

if we insert the initial condition, we obtain the value of the constant C

C1 + kε|ε=0 = t,

C1 = t,

consequently the transformation of the variable t under the action of thegroup G3 has the following form

tε = kε + t, (5.2)

where k is an arbitrary constant. The second equation of the system (5.1)describe the change of the independent variable Sε under the action groupG3 and it is general solution has a form

Sε = C2ebε.

The initial condition defines the value of the constant d

C2ebε|ε=0 = S,

C2 = S,

consequently the transformation of the variable t under the action of thegroup G3 has the following form

Sε = SeCε. (5.3)

The last equation of the system (5.1) defines the transformation of the de-pendent variable u under the action of the group G3.

To solve the linear non-homogeneous equation

duε

dε= 2Cuε − CSε

ρ+ d, (5.4)


we first solve the corresponding homogeneous equation

duε

dε= 2Cuε,

and obtain

uε = C3e2Cε,

Sε = SeεC

then using the method of the variation of the parameters we represent C3 asa function of ε, i.e. C3 = C3(ε).We insert C3(ε) in the inhomogeneous equation (5.4) and obtain

C ′3(ε) =

(−CSeCε

ρ+ d

)e−2Cε.

After integration by ε we obtain the expression for C3(ε)

C3(ε) =S

ρe−Cε − d

2C+ C4.

Finally, the expression for uε takes the form

uε =

(S

ρe−Cε − d

2C+ C4

)e2Cε,

where the constant C4 is defined by the initial condition

uε|ε=0 = u =S

ρ− d

2C+ C4,

consequently, C4 is equal to

C4 = u− S

ρ+

d

2C

and

uε =S

ρeCε − de2Cε

2C+

(u− S

ρ+

d

2C

)e2Cε. (5.5)

Now we can describe the invariants of the symmetry group G3 (5.1), (5.2)and (5.5). From equation (5.2) we can find that

ε =tε − t

k, (5.6)


(4.1)

by the substitution to the expression (5.3) we obtain

Sε = SeCk

(tε−t) ⇒ Sεe−C

ktε = Se−

Ck

t.

We see that this expression is unaltered under the action of the group G3. Itmeans that this expression is an invariant of the group G3, i.e.

inv1 = ln S − bt, where b =C

k. (5.7)

Remark 5.0.1 Any function of an invariant is an invariant, it means werepresent one of the possible forms of the first invariant of G3.

From the equation (5.3) we obtain

ε =1

Cln

Sε

S

if we insert this expression in (5.5) then

uε =S

ρeC 1

Cln Sε

S − d

2C+

(u− S

ρ+

d

2C

)e2C 1

Cln Sε

S

consequently we obtain

uε − Sε

ρ+

d

2C=

(u− S

ρ+

d

2C

)(Sε

S

)2

and finally (uε − Sε

ρ+

d

2C

)1

S2ε

=

(u− S

ρ+

d

2C

)1

S2.

Hence we can take as the second invariant

inv2 =

(u− S

ρ+

d

2C

)1

S2. (5.8)

Let us introduce new variables in equation (4.1). We denote a new inde-pendent variable by z where z = inv1 = ln S − bt, and suppose that a newdependent variable is

w(z) = inv2 =

(u− S

ρ+

d

2C

)1

S2.

We reduce the number of independent variables from two to one and corre-spondingly obtain an ordinary differential equation whose solutions are group


invariant solutions of the original equation. Let us find representation for thedependent variables using invariants (z, w)

u = wS2 +S

ρ− d

2C.

We represent now the partial derivatives of u in the following way

ut = −bw′S2,

uS = Sw′ + 2Sw +1

ρ,

uSS = 3w′ + w′′ + 2w.

Substitute these expressions to equation (4.1) we obtain a non-linear secondorder ordinary differential equation

bw′ − 1

2σ2 (3w′ + w′′ + 2w)

(w′ + 2w)2

(4w′ + w′′ + 4w)2 = 0

Case 2. C 6= 0 , k = 0

We take a two dimensional Lie subalgebra L2 which is spanned by generators

U1 =∂

∂u,

U2 = S ∂∂S

+ (2u− Sρ) ∂

∂u.

The system (5.1) describing the global group representation has now the form

dt

dε= 0 ⇒ tε = t,

dSε

dε= CSε, Sε|ε=0 = S,

duε

dε= 2Cuε − CSε

ρ+ d, uε|ε=0 = u.

(5.9)

As in the previous case we can see that the transformation of the variable Scan be described as follows

Sε = SeCε. (5.10)

From the equation (5.10) follows

ε =1

Cln

Sε

S,


(4.1)

and consequently

uε =S

ρeC 1

Cln Sε

S − d

2C+

(u− S

ρ+

d

2C

)e2C 1

Cln Sε

S ,

after transformations

uε − Sε

ρ+

d

2C=

(u− S

ρ+

d

2C

)(Sε

S

)2

,

and finally (uε − Sε

ρ+

d

2C

)1

S2ε

=

(u− S

ρ+

d

2C

)1

S2.

We can take as a first invariant the expression

inv1 =

(u− S

ρ+

d

2C

)1

S2. (5.11)

Let us denote a new independent variable z = t, and as a new dependentvariable

w(t) = inv2 =

(u− S

ρ+

d

2C

)1

S2,

then the old dependent variable u is now presented by the expression

u = wS2 +S

ρ− d

2C.

The partial derivatives of u can be represented in the following form

ut = S2w′,

uS = 2Sw +1

ρ,

uSS = 2w.

Substitute to equation (4.1) we obtain a non-linear first order ordinary dif-ferential equation

w′ − 1

4σ2w = 0

This equation describes invariant solutions to (4.1) variable t is unalteredunder action of the group G2, i.e. all solutions of the type:

u(s, t) = S2w(t) +S

ρ− d

2C.


Case 3. C = 0, k 6= 0

In this case we have subalgebra of L3 which is spanned by generators

U1 =∂

∂t,

U2 =∂

∂u.

The Lie equations have now the form

dS

dε= 0, ⇒ Sε = S

dtεdε

= k, tε|ε=0 = t,

duε

dε= d, uε|ε=0 = u.

(5.12)

We can see that the global transformations of the variables t and S can berepresent in the following form

tε = kε + t, Sε = S. (5.13)

The last equation of the system (5.12) defines transformation of on the vari-able u. To find the global form we have to solve a linear inhomogeneousequation

duε

dε= d, (5.14)

after transformations we obtain

uε = dε + C6.

uε|ε=0 = u = dε + C2, (5.15)

consequently,C2 = u (5.16)

and after the substitution (5.16) in (5.15) we obtain

uε = dε + u. (5.17)

From equation (5.13) we obtain ε = tε−tk

and insert this result in equation(5.17). We obtain

uε =d

k(tε − t) + u.


(4.1)

or in symmetrical form

uε − d

ktε = u− d

kt.

Hence we can take as a first invariant the expression

inv2 = u− d

kt. (5.18)

Let us denote a new independent variable by z = s, and a new dependentvariable by ω

w(z) = inv2 = u− d

kt.

In new variables

ut = w +d

k,

uS = w′,

uSS = w′′.

Substitute the results to equation (4.1) we obtain a non-linear second orderordinary differential equation

w +d

k− 1

2σ2S2w′′

[1− ρw

1− ρw′ − ρSw′′

]2

= 0

In this case the solutions depend only in trivial way on t

u(S, t) = −d

kt + f(S),

where d and k are constants.

Case 4. C = 0, k = 0

We have one dimensional Lie algebra L1 spanned by generator

U1 =∂

∂u,

The Lie equations has now the form

duε

dε= d, uε|ε=0 = u, ⇒ uε = u + εd.

This case is trivial. When both variables S, t to be invariants of the corre-sponding group G1 we do not have any transformations.

Chapter 6

Conclusions

In this work we have considered the equilibrium or reaction-function model(2.6) by analytical methods. Because the main equation (2.6) is nonlinear,it is difficult to chose the appropriate method to find a solution.

By using the methods of Lie groups analysis, the complete symmetryalgebra was found which is admitted by the following equation

ut +1

2σ2S2uSS

(1− ρuS)2

(1− ρuS − ρSuSS)2= 0.

The main result is represented in the Theorem 4.0.5.The structure of the model lead to the three parametrical group of sym-

metry G3, given by global representations (5.2), (5.3), (5.5). By using thisresult we obtain a number of invariants, which were used to obtain the fol-lowing reductions of the main equation

1.

bw′ − 1

2σ2 (3w′ + w′′ + 2w)

(w′ + 2w)2

(4w′ + w′′ + 4w)2 = 0,

where

z = ln S − bt, w(z) =

(u− S

ρ+

d

2C

)1

S2,

2.

w′ − 1

4σ2w = 0,

where

z = t, w(z) =

(u− S

ρ+

d

2C

)1

S2,

41

3.

w +d

k− 1

2σ2S2w′′

[1− ρw

1− ρw′ − ρSw′′

]2

= 0,

where

z = S, w(z) = u− d

kt.

The listed ordinary differential equations considerably more simple as thanthe main model (2.6). The solutions of these equations will give rise to theinvariant solutions of the main equation. This task is out of the frame of thepresent work.

42

Bibliography

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[2] R. Sircar, G. Papanicolaou, General black-scholes models accounting forincreased market volatility from hedging strategies, Appl. Math. Finance 5,45-82.

[3] E. Platen, M. Schweizer, On feedback effects from hedging derivatives,Math. Finance 8, 67-84.

[4] R. Frey, A. Stremme, Market volatility and feedback effects from dynamichedging, Math. Finance 7(4), 351-374

[5] P. Bank, D. Baum, Hedging and portfolio optimization in financial mar-kets with a large trader, Math. Finance, (2004), 14, pp. 1-18

[6] L. V. Ovsiannikov, Group analysis of differential equation, AcademicPress, New York, 1982 (Moscow, Nauka, 1978, in Russian).

[7] L. A. Bordag, A. Y. Chmakova, Explicit solutions for a nonlinear modelof financial derivatives, International Journal of Theoretical and AppliedFinance (JTAF), (2007), 10(1), pp. 1-21.

[8] L. A. Bordag, R. Frey, Nonlinear option pricing models for illiquid mar-kets: scaling properties and explicit solutions,11 Aug, (2007).

[9] L. A. Bordag, On the valuation of a fair price in case of a non perfectlyliquid market, Workshop on Mathematical Control Theory and Finance,Lisbon, 10-14 April, (2007), pp. 1-18

[10] L. A. Bordag, New types of solutions for nonlinear Black-Scholes model,in Abstracts of the twelfth General Meeting of EWM, Volgograd, Septem-ber 18-24, 2005, Volgograd State University, 2005, pp. 23-26.

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[11] A. Y. Chmakova, Symmetriereduktionen und explicite Losungen fur einnichtlineares Modell eines Preisbildungsprozesses in illiquiden Markten.PhD thesis, BTU Cottbus, 2005.

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44

Glossary

Asset. Items of value owned by an individual. Assets that can be quicklyconverted into cash are considered ”liquid assets.” These include bank ac-counts, stocks, bonds, mutual funds, and so on. Other assets include realestate, personal property, and debts owed to an individual by others.

Bond. A debt instrument, issued by a borrower and promising a specifiedstream of payments to the purchaser, usually regular interest payments plusa final repayment of principal. Bonds are exchanged on open markets includ-ing, in the absence of capital controls, internationally, providing a mechanismfor international capital mobility.

Black-Scholes formula. An equation to value a call option that usesthe stock price, the exercise price, the risk-free interest rate, the time tomaturity, and the standard deviation of the stock return.

C0 = S0N(d1)−Xe−rT N(d2),

where

d1 =ln S0

X+ (r − δ + σ2

2)T

σ√

T,

d2 = d1 − σ√

T

and whereC0 - Current call option value.S0 - Current stock price.N(d) - The probability that a random draw from a standard normal

distribution will be less than d.X - Exercise price.r - Risk-free interest rate.T - Time to maturity of option, in years.σ - Standard deviation of the annualized continuously compounded rate

of return of the stock.Classical Black-Scholes equation. An Ito process for the price of the

underlying is taken as given:

dSt = ν(St, t)dt + λ(St, t)dWt

where Wt, t > 0 is a standard Brownian motion on a probability space(Ω,F,P), and ν and λ satisfy Lipschitz.

∂C

∂t+

1

2λ2(s, t)

∂2C

∂s2+ r

(s∂C

∂s− C

)= 0, fors > 0, t > 0

45

which is known as the classical Black-Scholes equation and which we used inthis work.

Common stock. Equities, or equity securities, issued as ownershipshares in a publicly held corporation. Shareholders have voting rights andmay receive dividends based on their proportionate ownership.

Demand function. The mathematical function explaining the quantitydemanded in terms of its various determinants, including income and price;thus the algebraic representation of the demand curve.

Disequilibrium

1. Inequality of supply and demand.

2. An untenable state of an economic system, from which it may be ex-pected to change.

Equilibrium.

1. A state of a balance between offsetting forces for change, so that nochange occurs.

2. In competitive markets, equality of quantity supplied and quantity de-manded.

Equilibrium price. The market price at which the supply of an itemequals the quantity demanded.

Fair price. In anti-dumping cases, the price to which the export price iscompared, which is either the price charged in the exporter’s own domesticmarket or some measure of their cost, both adjusted to include any trans-portation cost and tariff needed to enter the importing country’s market. Seedumping.

Financial assets. Financial assets such as stocks and bonds are claims tothe income generated by real assets or claims on income from the government.

Hedging. Investing in an asset to reduce the overall risk of a portfolio.Hedging demands. Demands for securities to hedge particular sources

of consumption risk, beyond the usual mean-variance diversification motiva-tion.

Income.

1. The amount of money (nominal or real) received by a person, house-hold, or other economic unit per unit time in return for services pro-vided or goods sold.

2. National income.

46

3. The return earned on an asset per unit time.

Interest rate. The amount of money charged as a fee for lending moneyor the price of borrowing money.

Liquidity. The capacity to turn assets into cash, or the amount of assetsin a portfolio that have that capacity. Cash itself (i.e., money) is the mostliquid asset.

Market equilibrium. Equality of quantity supplied and quantity de-manded. See equilibrium.

Money market. The money market, in macroeconomics and interna-tional finance, refers to the equilibration of demand for a country’s domesticmoney to its money supply. Both refer to the quantity of money that peoplein the country hold (a stock), not to the quantity that people both in andout of the country choose to acquire during a period in the exchange market,mostly for the purpose of then using it to buy something else.

Numeraire. The unit in which prices are measured. This may be acurrency, but in real models, such as used trade models, the numeraire isusually one of the goods, whose price is then set at one. The numeraire canalso be defined implicitly by, for example, the requirement that prices sumto some constant.

Programme trader. Trader who trade the asset following a Black-Scholes type dynamic hedging strategy.

Program trading. Coordinated buy orders and sell orders of entireportfolios, usually with the aid of computers, often to achieve index arbitrageobjectives.

Reference trader. Trader who is the majority investing in the the assetexpecting gain.

Risk-free asset. An asset with a certain rate of return; often taken tobe short-term Treasury Bills.

Volatility. The extent to which an economic variable, such as a price oran exchange rate, moves up and down over time.

Volatility risk. The risk in the value of options portfolios due to unpre-dictable changes in the volatility of the underlying asset.

47

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