AsianOptions-ArithmeticAverage

8/4/2019 AsianOptions-ArithmeticAverage

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The Arithmetic Average Case

of Asian Options:Reducing the PDE to theBlack-Scholes Equation

Solomon M. Antoniou

SKEMSYS

S cientific Knowledge Engineering

and Management Systems

37 oliatsou Street, Corinthos 20100, [email protected]

Abstract

We consider the path-dependent contingent claims where the underlying assetfollows an arithmetic average process. Considering the no-arbitrage PDE of these

claims, we first determine the underlying Lie Point Symmetries. After

determination of the invariants, we transform the PDE to the Black-Scholes (BS)

equation. We then transform the BS equation into the heat equation and we

provide some general solutions to that equation. This procedure appears for the

first time in the finance literature.

Keywords: Path-Dependent Options, Asian Options, Arithmetic Average Path-

Dependent Contingent Claims, Lie Symmetries, Exact Solutions.


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1. Introduction

A path-dependent option is an option whose payoff depends on the past history of

the underlying asset. In other words these options have payoffs that do not depend

on the assets value at expiry. Two very common examples of path-dependent

options areAsian options and lookback options.

The terminal payoff of an Asian option depends on the type of averaging of the

underlying asset price over the whole period of the options lifetime. According to

the way of taking average, we distinguish two classes of Asian options: arithmetic

andgeometric options.

A lookback option is another type of path-dependent option, whose payoff

depends on the maximum or minimum of the asset price during the lifetime of the

option.

The valuation of path-dependent (Asian) European options is a difficult problem

in mathematical finance. There are only some simple cases where the price of

path-dependent contingent claims can be obtained in closed-form (refs. [1]-[8]).

If the underlying asset price follows a lognormal stochastic process, then its

geometric average has a lognormal probability density and in this case there is a

closed-form solution (ref. [5]). This is not however the case by considering

arithmetic average. In this case there is no closed-form solution. However in both

cases there are some numerical procedures available (ref. [8]).

It is the purpose of this article to provide a closed-form general solution for the

arithmetic path-dependent options.

The paper is organized as follows: In Section 2 we consider the general problem of

pricing the path-dependent contingent claims. In Section 3 we consider the Lie

Point Symmetries of the Partial Differential Equation (equation (3.1)) for the path-

dependent arithmetic average options. Full details of the calculation are provided

in Appendix A. In Section 4, considering two invariants of the PDE, we convert


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the equation into the Black-Scholes equation. In section 5, we transform the BS

equation to the heat equation. We then find the general solution of our PDE

(Theorem 3). In section 6 we provide some solutions to the heat equation.

Because of the complexity of the calculations, we have included a great deal ofcalculation details, supplemented by many Appendices.

2. Path-Dependent Contingent Claims

Suppose that an option pays off at expiration time T an amount that is a function

of the path taken by the asset between time zero and T. This path-dependent

quantity can be represented by an integral of some function of the asset over the

time period 0 :

=T

0

d),S(f)( (2.1)

where )t,S(f is a suitable function. Therefore, for every t, t0 , we have

=t

0

d),S(f)t( (2.2)

or, in differential form,

dt)t,S(fdA = (2.3)

We have thus introduced a new state variable A, which will later appear in our

PDE.

We are now going to derive the PDE of pricing of the path-dependent option. To

value the contract, we consider the function )t,A,S(V and set up a portfolio

containing one of the path-dependent options and a number of the underlying

asset:

S)t,A,S(V = (2.4)

The change in the value of this portfolio is given by, according to Its formula, by

dSS

VdA

A

Vdt

S

VS

2

1

t

Vd

2

222

+

+

+

= (2.5)


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Choosing

S

V

= (2.6)

to hedge the risk and using (2.3), we find that

dtA

V)t,S(f

S

VS

2

1

t

Vd

2

222

+

+

= (2.7)

This change is risk-free and thus earns the risk-free rate of interest r,

dtrd = (2.8)

leading to the pricing equation

0VrSVSr

AV)t,S(f

SVS

21

tV 2

2

22 =+++(2.9)

combining (2.7), (2.8) and (2.4), where in (2.4) has been substituted by (2.6)

for the value of .

The previous PDE is solved by imposing the condition

)A,S()T,A,S(V = (2.10)

where T is the expiration time.

If A is considered to be an arithmetic average state variable,

=t

0

d)(S (2.11)

then PDE (2.9) becomes

0VrS

VSr

A

VS

S

VS

2

1

t

V

2

222 =

+

+

+

(2.12)

If A is considered to be a geometric average state variable,

=t

0

d)(Sln (2.13)

then PDE (2.9) becomes


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0VrS

VSr

A

V)S(ln

S

VS

2

1

t

V

2

222 =

+

+

+

(2.14)

We list below the payoff types we have to consider along with PDEs (2.12) and

(2.14).

For average strike call, we have the payoff )A,S(

)0,AS(max (2.15)

For average strike put, we have the payoff

)0,SA(max (2.16)

For average rate call, we have the payoff

)0,EA(max

(2.17)

For average rate put, we have the payoff

)0,AE(max (2.18)

where E is the strike price.

This is an introduction to the path-dependent options. Details can be found in

Wilmotts 3-volume set on Quantitative Finance (ref. [3]).

3. Lie Symmetries of the Partial Differential Equation

The technique of Lie Symmetries was introduced by S. Lie [11]-[12] and is best

described in refs [13]-[28]. This technique was for the first time used in partial

differential equations of Finance by Ibragimov and Gazizov [29]. The same

technique was also used by the author in solving the Bensoussan-Crouhy-Galai

equation [30] and the HJB equation for a portfolio selection problem [31]. There is

however a growing lists of papers in applying this method to Finance. For a non

complete set of references, see [29]-[40].

We consider the PDE (2.12)

0VrA

VS

S

VSr

S

VS

2

1

t

V

2

222 =

+

+

+

(3.1)


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We shall determine the Lie Point Symmetries of the previous equation.

We follow closely Olver (ref. [15]).

Let )V,t,A,S( )2( be defined by

VrA

VS

S

VSr

S

VS

2

1

t

V)V,t,A,S(

2

222)2(

+

+

+

= (3.2)

We introduce the vector field X (the generator of the symmetries) by

+

+

=A

)V,t,A,S(S

)V,t,A,S(X 21

V)V,t,A,S(

t)V,t,A,S(3

+

+ (3.3)

We calculate in Appendix A the coefficients 321 ,, and and we find that

the Lie algebra of the infinitesimal transformations of the original equation (3.1)

is spanned by the five vectors

tX1

= (3.4)

A

X2

= (3.5)

AA

SSX3

+

= (3.6)

VV)ArS(

2

AA

SAS2X

2

24

+

+

= (3.7)

V

VX5

= (3.8)

and the infinite dimensional sub-algebra

V)t,A,S(X

= (3.9)

where )t,A,S( is an arbitrary solution of the original PDE (3.1).


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4. Reduction of the equation to the BS equation

We shall now try to transform the pricing differential equation into another by

reducing the number of independent variables. We shall retain the time variable

and try to find a combination of the other two variables S and A. The best

candidate for this job is the generator 4X given by (3.7).

We state and prove the following Theorem:

Theorem 1. Any solution of the partial differential equation

0VrA

VS

S

VSr

S

VS

2

1

t

V

2

222 =

+

+

+

can be expressed into the form

=

A

Sbexp)t,x(uA)t,A,S(V n (4.1)

where the function )t,x(u satisfies the Black-Scholes equation

0uruxrux2

1u xxx

22t =++ (4.2)

with2A

Sx = , brn = and

2

2b = .

Proof. Considering the generator 4X , we can determine two invariants of the

PDE. For this purpose we have to solve the following system

V)ArS(

dV

2

A

dA

AS2

dS 2

2 == (4.3)

The differential equation2

A

dA

AS2

dS= is equivalent to

A

dA2

S

dS= , which can be

integrated to give 12C

A

S= from which we can obtain our first invariant x:

2A

Sx = (4.4)


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The second differential equation

V)ArS(

dV

2

A

dA 2

2 =

can be written as (since 2AxS = )

V

dVdA

A

ArAxb

2

2

=

,

or

V

dVdA

A

rxb =

where 22b = . By integration of this equation we find

VlnC)AlnrAx(b 2 =+

from which there follows that Axbbr eACV= , where C is a constant.

Therefore another invariant of the PDE is the function )t,x(u , related to )t,A,S(V

by

= AS

bexp)t,x(uA)t,A,S(Vn

(4.5)

with2

r2brn == .

Upon substitution of )t,A,S(V given by (4.5) into equation (3.1), we find that the

function )t,x(u satisfies the BS equation

0uruxrux

2

1u xxx

22t =++ (4.6)

The proof is in Appendix B.

5. Transformation of the BS equation to the heat equation.

Under the substitution


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is transformed into the equation

yy2

2

1 = (5.11)

Introducing further a new independent variable by

2

y = (5.12)

equation (5.11) is transformed to the heat equation

ww = (5.13)

where

)y,(),(ww == (5.14)

We thus arrive at the following

Theorem 2. The solution to the BS equation (4.6) is given by

),(we)t,x(ur= (5.15)

where

)x(ln

2y

2 +== (5.16)

t

=(5.17)

and ),(w satisfies the heat equation ww = .

Using Theorems 1 and 2, we get the following

Theorem 3. The differential equation (3.1) has the general solution

= r

A

Sbexp),(wA)t,A,S(V

m (5.18)

where the function ),(w satisfies the heat equation ww = .

The variables and are defined by (5.16) and (5.17) respectively, while

brm = ,2

2b = , is given by (5.3) and x by (4.4).Therefore equation (5.18)

can be written in full notation as


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= r

A

S

2exp),(w

A

1)t,A,S(V

2

r22

where

+

=

2

1r

A

Sln

2

2

2

What we need now is some solutions to the heat equation, which are not singular

at 0 = ( Tt = ). We come to this point next.

6. General solutions of the heat equation

The Lie algebra of the infinitesimal symmetries of the heat equation ww = is

produced by the six vector fields

X1

=

2

X4

+

=

X2

= w

w

2X5

=

w

wX3

= w

w)2(

4

4X22

6

+

+

=


w ),(X =

where ),( is an arbitrary solution of the heat equation (refs. [15], [37]).

6.1. The solution that is invariant with respect to the generator

w2

2

362 w)a2()14(4XaXX +++=++

(a is a constant) is given by ([15], [37])

+

+

+

+=

14

2,

2

aWK

14

2,

2

aWK

14

1),(w

22

214 2

+

+ )2arctan(

2

a

14

exp

2

2

(6.1)


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where )z,a(W is the parabolic cylindrical function of imaginary argument

(Appendix D).

Note. The solution given by Olver [15] (Example 3.17, p.208) contains some

misprints. We have repeated the calculations [28] and we have found the solutiongiven by (6.1).

6.2. The solution that is invariant with respect to the generator

w52 w2XX +=

is given by ([15], [37])

++++=3

2exp})(BiK)(AiK{),(w

32

22

1 (6.2)

where )x(Ai and )x(Bi are the Airy functions (Appendix E).

Both solutions (6.1) and (6.2) we have considered, are not singular for 0 =

( Tt = ). This is because we want to take into account the payoff conditions (2.15)-

(2.18) valid for Tt = ( 0 = ).

Appendix A. Lie Symmetry Analysis of the pricing PDE.

We consider the PDE (3.1) :

0VrA

VS

S

VSr

S

VS

2

1

t

V

2

222 =

+

+

+

(A.1)

We shall determine the Lie Point Symmetries of the previous equation.

We follow closely Olver (ref. [15]).

Let )V,t,A,S()2(

be defined by

VrA

VS

S

VSr

S

VS

2

1

t

V)V,t,A,S(

2

222)2( +

+

+

= (A.2)

We introduce the vector field X (the generator of the symmetries)

by


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+

+

+

=t

)V,t,A,S(A

)V,t,A,S(S

)V,t,A,S(X 321

V)V,t,A,S(

+ (A.3)

The second prolongation is defined in our case by the equation

SS

SS

t

t

A

A

S

S)2(

V

V

V

VXXpr

+

+

+

+= (A.4)

The Lie point symmetries of the equation are determined by

0)]V,t,A,S([Xpr)2()2( = as long as 0)V,t,A,S( )2( = .

We implement next the equation 0)]V,t,A,S([Xpr)2()2( = .We have

0)]V,t,A,S([Xpr)2()2( =

0VrVSVSrVS2

1VXpr ASSS

22t

)2( =

+++

++++ )r(]VVrVS[ ASSS2

1

0S2

1)S()Sr(

SS22tAS =++++ (A.5)

We have the following expressions for the coefficients S , A ,t

and SS

+= tS32SV1SS1VS

S VVV)(

ASV2tSV3AS2 VVVVV

+= tA32AV2AA2VA

A VVV)(

tAV3SAV1SA1 VVVVV

+= St12tV3tt3Vt

t VVV)(

tSV1tAV2At2 VVVVV


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++= 2SSV1VVtSS3SSS1SVSSSS

V)2(VV)2(

+ SSS1Vt2SVV3

3SVV1tSSV3

V)2(VVVVV2

StSV3SSAV2SSSV1SAS2

VV2VVVV3V2

SStV3SASV2StS3SASV2 VVVV2V2VV2

ASS2A2SVV2 VVV

Using the previous expressions for S , A ,t and SS we get from

the equation (A.5) that

+++ r]VVrVS[ ASSS2

1

++ AS2tS32SV1SS1VS VVVV)({Sr

+ }VVVV ASV2tSV3

++ SA1tA32AV2AA2VA VVVV)({S

+ }VVVV tAV3SAV1

++ At2St12tV3tt3Vt VVVV)(

+ tSV1tAV2 VVVV

+++ 2SSV1VVtSS3SSS1SVSS22 V)2(VV)2({S

2

1

+ SSS1Vt2SVV3

3SVV1tSSV3

V)2(VVVVV2

StSV3SSAV2SSSV1SAS2 VV2VVVV3V2

SStV3SASV2StS3SASV2 VVVV2V2VV2

0}VVV ASS2A2SVV2 = (A.6)

We have now to take into account condition 0)V,t,A,S()2( = .

For this purpose we substitute tu by


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VrVSVSrVS2

1ASSS

22 +

into (A.6). We thus derive the equation

+++ r]VVrVS[ ASSS2

1

+++ }VVVVV)({Sr ASV2AS22SV1SS1VS

+

++ VrVSVSrVS

2

1}V{Sr ASSS

22SV3S3

+++ }VVVVV)({S SAV1SA12AV2AA2VA

+

++ VrVSVSrVS

2

1}V{S ASSS

22AV3A3

+++ VrVSVSrVS

2

1)( ASSS

22t3Vt

+

+ At2St1

2

ASSS22

V3 VVVrVSVSrVS2

1

+

++ VrVSVSrVS

2

1}VV{ ASSS

22SV1AV2

++++ }V)2(V)2({S21 2SSV1VVSSS1SVSS22

+ }VV2{S2

1 2SVV3SSV3SS3

22

+

+ VrVSVSrVS

2

1ASSS

22

++ SAS2SSS1V3SVV1

22V2V)2(V{S

2

1

StSV3SSAV2SSSV1 VV2VVVV3

+ }VV2V2VV2 SASV2StS3SASV2

+

++ VrVSVSrVS

2

1}V{S

2

1ASSS

22SSV3

22

0}VVV{S2

1ASS2A

2SVV2

22 =+ (A.7)


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Equate all the coefficients of the partial derivatives of the function V to zero. We

are then going to obtain a system of partial differential equations of the unknown

coefficients 321 ,, and of the vector field (A.3). Before that, and just for the

economy of the calculations, we observe that we can get some simple expressions,

by first looking at the coefficients of the mixed partial derivatives.

In fact the coefficient of the derivative SAV is )2(S2

1S2

22 and that of

SAS VV is )2(S2

1V2

22 which means that 0 S2 = and 0 V2 = .

Therefore the coefficient 2 does not depend either on S or V:

)t,A( 22 = (A.8)

The coefficient of the derivative StS VV is )2(S2

1V3

22 and that of StV is

)2(S2

1S3

22 which means that 0 V3 = and 0 S3 = . Therefore the

coefficient 3 does not depend neither on V nor on S:

)t,A( 33 = (A.9)

Because of (A.8) and (A.9), equation (A.7) becomes

+++ r]VVrVS[ ASSS2

1

+++ }VV)({Sr 2SV1SS1VS

+++ }VVVV)({S SAV1SA1AA2VA

+

++ VrVSVSrVS

2

1)(S ASSS

22A3

+++ VrVSVSrVS

2

1)( ASSS

22t3Vt

+

++ VrVSVSrVS

2

1)V(VV ASSS

22SV1At2St1


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++++ }V)2(V)2({S2

1 2SSV1VVSSS1SVSS

22

0}VV3V)2(V{S2

1SSSV1SSS1V

3SVV1

22 =++ (A.10)

The coefficient of SSSVV is the sum of the terms

22V1 S

2

1)( and )3(S

2

1V1

22

and therefore 0 V1 = , which means that

)t,A,S( 11 = (A.11)

Therefore (A.10) becomes

+++ r]VVrVS[ ASSS2

1

+++++ }VV)({S}V)({Sr SA1AA2VASS1VS

+

++ VrVSVSrVS

2

1)(S ASSS

22A3

+++ VrVSVSrVS

2

1)( ASSS

22t3Vt

++++ }VV)2({S2

1VV 2

SVVSSS1SVSS

22

At2St1

0V)2(S2

1SSS1V

22 =+ (A.12)

From (A.12) we get the following coefficients, which have to be zero:

Coefficient of 2SV :

0S2

1VV

22 = (A.13)

Coefficient of AV :

0)(SS t212t332 =++ (A.14)

Coefficient of SSV :


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0S2

1)2(S

2

1S A3

32S1t3

221

2 =++ (A.15)

Coefficient of SV :

+++ A32A1S1t31 SrS)(Srr

0)2(S2

1t1SS1SV

22 =+ (A.16)

Zero-th order coefficient

++++ t3AS VSrSSrr

0S2

1V)(r SS

22t3V =++ (A.17)

We have now to solve the system of PDEs (A.13)-(A.17).

From (A.13) we get 0VV = and therefore is a linear function with respect to

V:

)t,A,S(V)t,A,S( += (A.18)

From (A.15) we get

A3t31S1 S2

1

2

1

S

1 += (A.19)

which is a linear differential equation with unknown function 1 .

The solution of the previous differential equation is

)t,(SS2

1)SlnS(

2

1 A3

2t31 ++= (A.20)

where )t,( is a function to be determined.

From (A.14), using the expression (A.20) for the function 1 , we derive the

equation

++ )SlnS(2

1S

2

3t3

23

0S)]t,([ t22t3 =++ (A.21)


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Since the previous equation should hold for every S, each one of the coefficients

has to be zero. Therefore we obtain the following four equations:

0 3 = (A.22)

0 t3 = (A.23)0)t,( 2t3 =+ (A.24)

0 t2 = (A.25)

From the equations (A.22) and (A.23) and taking into account (A.9),we get that 3

is a constant:

13 a = (A.26)

From equation (A.25) there follows that 2 is independent of the time t and so

)A(f2 = (A.27)

Therefore equation (A.24) gives us

)A(f)t,( = (A.28)

The function 1 becomes, as follows from (A.20),

)(fS1 = (A.29)

From the above equation we get the following expressions to be used later on.

)(f S1 = (A.30a)

0 SS1 = (A.30b)

)A(fS A1 = (A.30c)

0 t1 = (A.30d)

Equation (A.16), using the expressions (A.18)-(A.20), takes the form

0S)A(fS)A(fSr)A(fSr S222

=+Solving the previous equation with respect to S , we find

)A(f S2 = (A.31)

and therefore


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)t,A(S)A(f2 += (A.32)

where )t,A( is a function to be determined.

We also find from (A.31) that

0S SS2 = (A.33)

an expression we shall need later on.

From (A.32) we also get

)t,A( tt2 = (A.34)

and

)t,A(S)A(f AA2 += (A.35)

From (A.17), using (A.18) we obtain the equation

++++++ )V(S)V(Sr)V(r ASS

0)V(S2

1V)(r)V( SSSS

22t3tt =+++++ (A.36)

Equating the coefficients of V to zero, we get from the previous equation the

following two equations:

+++ AS SSrr

0S2

1)(r SS

22t3t =+++ (A.37)

and

0S2

1SSrr SS

22tS =++++ (A.38)

Equation (A.38) expresses the fact that the function introduced in (A.14) is a

solution of the original PDE.Equation (A.37) can be expressed as

+++ )}t,A(S)A(f{

1S)A(f

1Sr A22


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0)t,A(

1t2

=+ (A.39)

The previous equation can be written in equivalent form

0S)A(fS)}A(fr)t,A({)t,A(2

At =+++ (A.40)

Since the previous equation should hold for any value of S, we get the following

equations:

0)A(f = (A.41)

0)A(fr)t,A(A =+ (A.42)

0)t,A( t = (A.43)

From (A.41) we conclude that )A(f is a second degreepolynomial with respect to A:

2

432 AaAaa)A(f ++= (A.44)

From (A.43), we obtain that )t,A( does not depend on t. Therefore

)A(g)t,A( = (A.45)

and so equation (A.42) becomes, because of (A.45) and (A.44)

0a2r)A(g4=+

from which we determine the function )A(g :

54 aAar2)A(g += (A.46)

Therefore

54 aAar2)t,A( += (A.47)

Le us collect now everything together.

We find from (A.29) and (A.44) that

)Aa2a(S 431 += (A.48)

From (A.27) and (A.44) we find that

2

4322 AaAaa ++= (A.49)

We also have from (A.26)


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Using the transformation (4.5), we can express the various partial derivatives of

the function )t,A,S(V in terms of the partial derivatives of the function )t,x(u .

We find that

=

ASbexpAuV ntt

+=

A

Sbexp)AubAu(V 1n2nxS

++=

A

Sbexp)AubAub2Au(V 2n23nx

4nxxSS

+=

A

Sbexp)ASubAunASu2(V 2n1n3nxA

Substituting the above expressions into the original equation (3.1), we obtain

++++ )AubAub2Au(S2

1uA 2n23nx

4nxx

22t

n

+++ )AubAu(Sr 1n2nx

++ )ASubAunASu2(S 2n1n3nx

0Aur n =

The previous equation can be put into the form

++ xx4n22

tn u)AS(

2

1uA

+

++ x

3n22n3n22 u)AS(2)AS(r)AS(b22

1

+++

)AS(n)AS(br)AS(b2

1 1n1n2n222

] 0uAr)AS(b n2n2 = (B.1)We now have to simplify the expressions appearing within the brackets of the

previous equation. We have


24/33


25/33

25

C1. Lie Symmetry Analysis of the equationAfter performing a Lie symmetry analysis, we have found that the above equation

has the six symmetry generators (six vector fields)

tX1

= (C.1)

uutc

a4

bx

a4

b

tt

xx

2

1X

2

2

22

+

+

= (C.2)

+

+

= 2xa8

1t2x

a4

b

tt

xtxX 2

22

23

uut

2

1tc

a4

b 22

2

(C.3)

xX4

= (C.4)

uut

a2

bx

a2

1

xtX

225

+

= (C.5)

uuX6

= (C.6)


u)t,x(X

= (C.7)

where )t,x( is an arbitrary solution of the original equation (ref. [37])

C.2. Transformation of the equation to the heat equation

We shall consider a linear combination of generators such as to convert the BS

equation to the heat equation. For this reason, we consider

uu

x

1

t

1X

+

+

= (C.8)

where and are constants to be determined.

In order to determine the invariants corresponding to the symmetry (C.8), we

consider the system


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26

u

du

1

dx

1

dt == (C.9)

From the equation1

dx

1

dt = we get by integration xCt 1 =+

or 1Ctx += and hence one invariant of the equation is

txy = (C.10)

From the equationu

du

1

dt = we get by integration )uCln(t 2=

or uCe 2t = and hence another invariant is

teu= (C.11)

Therefore

teu = (C.12)

Using (C.12) and (C.10) we find that

tytt e)(u +=

tyx eu =

tyyxx eu =

Therefore the equation ucubuau xxx2

t ++= transforms to the equation

(divided by the factort

e )

cba yyy2

yt ++=+

from which we get

)c()b(a yyy2t +++= (C.13)

We determine now and such as to become zero the coefficients

of y and :

0b =+


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27

0c =

Therefore we should have b = and c= , and hence the two

invariants will be

tbxy += and ue tc = Therefore we arrive at the following

Lemma

The substitution

)btyx,t(ue tc ==

converts the equation ucubuau xxx2

t ++= to the heat equation

yy2

t a =

Appendix D. Webers Differential Equation

Abramowitz and Stegun ([41], 19.17) define as standard solutions of Webers

equation

yx4

1ay 2

=

the functions )x,a(W where

)yG2yG(2

)a(cosh)x,a(W 2311

4/1 =with

+= ai

2

1

4

1G1

+= ai

2

1

4

3G3

and 1y , 2y are defined by

+

+

++=

!6

xa

2

7a

!4

x

2

1a

!2

xa1y

63

42

2

1

+

++

++

!8

xa

4

211a25a

!8

x

4

15a11a

835

824


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28

and

+

+

++=

!7

xa

2

13a

!5

x

2

3a

!3

xaxy

73

52

3

2

+

++

++

!11

xa

4

531a35a

!9

x

4

63a17a

1135924

respectively.

In the previous expressions, the non-zero coefficients na of!n

xn

are connected by

2nn2n a)1n(n4

1

aaa + =

It is also worth noting that )x,a(W can be expressed in terms of the Confluent

Hypergeometric functions ([41], 19.25)

= 2

3

14/3 x4

1,a

2

1,

4

3H

G

G2)x,a(W

21

3 x

4

1,a

2

1,

4

1Hx

G

G2

where

=++= )xi2;2m2;ni1m(Fe)x,n,m(H 11xi

)xi2;2m2;ni1m(Me xi ++=

Note. Webers equation can be obtained by separation of variables for the Laplace

equation in parabolic cylindrical coordinates (ref [42], Chapter 10).

Appendix E. The Airy equation

The differential equation ofAiry

0wzw =


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29

has as solutions the linearly independent solutions

)z(Ai and )z(Bi

(Abramowitz and Stegun [41], 10.4).

These functions are defined by

)z(gc)z(fc)z(Ai 21 =

)]z(gc)z(fc[3)z(Bi 21 +=

where

!)k3(

z

3

13z

!9

741z

!6

41z

!3

11)z(f

k3

k0k

k963

=

=+

+

++=

and

!)1k3(

z

3

23z

!10

852z

!7

52z

!4

2z)z(g

1k3

k0k

k1074

+

=+

+

++=

+

=

respectively.

The constants 1c and 2c are given by

)3/2(3

1

3

)0(Bi)0(Aic

3/21===

and

)3/1(3

1

3

)0(iB)0(iAc

3/12=

==

They also can be expressed in terms ofBessels functions of fractional order:

)(K3

z

1)z(Ai 3/1= ([41], 10.4.14)

)}(I)(I{3z)z(Bi 3/13/1 += ([41], 10.4.18)

where zz3

2 =

References.


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30

[1] M. Rubinstein: Exotic Options,

University of California at Berkeley, 1991

[2] P. G. Zhang: Exotic Options

Second Edition, World Scientific 1998[3] P. Wilmott: Quantitative Finance, Three-Volume Set

Second Edition, Wiley 2006

[4] A G Z Kemna and A C F Vorst : A pricing method for options

Journal of Banking and Finance 14 (1990) 113-129

[5] J. Angus: A note on pricing Asian Derivatives with continuous

geometric averaging, Journal of Future Markets 19 (1999) 845-858

[6] Y. Z. Bergman: Pricing path contingent claims

Research in Finance 5 (1985) 229-241

[7] H. Geman and M. Yor: Bessel Processes, Asian Options and perpetuities.

Mathematical Finance 3 (1993) 349-375

[8] E. Barucci, S. Polidoro and V. Vespri: Some results on partial differential

equations and Asian Options

University of Pisa preprint

[9] L. Jiang and M. Dai: Convergence Analysis of Binomial Tree Method

for American-type Path-dependent Options

Preprint, Peking University

[10] E. G. Haug: Option Pricing Formulas

Second Edition, McGraw Hill 2007

[11] S. Lie: Gesammelte Werke, Vols. 1-7,

F. Engel and P. Hergard (Eds), Teubner, Leipzig 1899

[12] S. Lie: Vorlesungen ber Differentialgleichungen mit bekannten

infinitesimalen Transformationen

Teubner, Leipzig 1912

[13] J. M. Page: Ordinary Differential Equations with an Introduction to


31/33

31

Lies Theory of the Group of one parameter

McMillan and Co. New York 1897

[14] A. Cohen: An Introduction to the Lie Theory of one-parameter groups:

With Applications to the solution of Differential EquationsD. C. Heath and Co. Boston 1911

[15] P. J. Olver: Applications of Lie Groups to Differential Equations

Graduate Texts in Mathematics, vol.107

Springer Verlag, New York 1993

[16] H. Stephani: Differential Equations: Their Solutions Using Symmetries

Cambridge University Press, 1989

[17] G. Bluman and S. Kumei: Symmetries and Differential Equations

Springer Verlag, 1989

[18] L. V. Ovsiannikov: Group Analysis of Differential Equations

Academic Press, New York 1982

[19] N. H. Ibragimov: Elementary Lie Group Analysis and Ordinary

Differential Equations

Wiley, New York

[20] L. Dresner: Applications of Lies Theory of Ordinary and Partial

Differential Equations

IOP Publishing, 1999

[21] R. L. Anderson and N. H. Ibragimov: Lie-Bcklund Transformations

in Applications SIAM, Philadelphia 1979

[22] N. H. Ibragimov (ed.): CRC Handbook of Lie Group Analysis of

Differential Equations Vol.1 1994, Vol.2 1995, Vol.3 1996

CRC Press, Boca Raton, FL.

[23] W. Hereman: Review of Symbolic Software for the Computation of Lie

Symmetries of Differential Equations

Euromath Bulletin 2, No.1 Fall 1993


32/33

32

[24] P. E. Hydon: Symmetry Methods for Differential Equations:

A Beginners Guide

CUP 2000

[25] G. Emanuel: Solution of Ordinary Differential Equations by ContinuousGroups

Chapman and Hall 2001

[26] B. J. Cantwell: Introduction to Symmetry Analysis

CUP 2002

[27] Steeb Willi-hans: Continuous Symmetries, Lie Algebras, Differential

Equations and Computer Algebra

World Scientific 2007

[28] F. Schwarz: Algorithmic Lie Theory for Solving Ordinary Differential

Equations

Chapman and Hall/CRC, 2008

[29] N. H. Ibragimov and R. K. Gazizov: Lie Symmetry Analysis of

differential equations in finance

Nonlinear Dynamics 17 (1998) 387-407

[30] S. M. Antoniou: A General Solution to the Bensoussan-Crouhy-Galai

Equation appearing in Finance, SKEMSYS preprint

[31] S. M. Antoniou: Closed-Form Solutions to the Hamilton-Jacobi-Bellman

Equation, SKEMSYS preprint

[32] J. Goard: New Solutions to the Bond-Pricing Equation via Lies Classical

Method, Mathematical and Computer Modelling 32 (2000) 299-313

[33] L. A. Bordag: Symmetry reductions of a nonlinear model of financial

derivatives, arXiv:math.AP/0604207

[34] L. A. Bordag and A. Y. Chmakova: Explicit solutions for a nonlinear

model of financial derivatives

International Journal of Theoretical and Applied Finance 10 (2007) 1-21


33/33

[35] L. A. Bordag and R. Frey: Nonlinear option pricing models for illiquid

markets: scaling properties and explicit solutions

arXiv:0708.1568v1 [math.AP] 11 Aug 2007

[36] G. Silberberg: Discrete Symmetries of the Black-Scholes EquationProceedings of 10th International Conference in Modern Group Analysis

[37] S. M. Antoniou: Lie Symmetry Analysis of the partial differential

equations appearing in Finance. SKEMSYS Report

[38] W. Sinkala, P. G. L. Leach and J.G. OHara: Zero-coupon bond prices

in the Vasicek and CIR models: Their computation as group-invariant

solutions

Mathematical Methods in the Applied Sciences, 2007

[39] C. A. Pooe, F. M. Mahomed and C. Wafo Soh : Fundamental Solutions

for Zero-Coupon Bond Pricing Models

Nonlinear Dynamics 36 (2004) 69-76

[40] P. G. L. Leach, J.G. OHara and W. Sinkala: Symmetry-based solution

of a model for a combination of a risky investment and a riskless

investment

Journal of Mathematical Analysis and Applications, 2007

[41] M. Abramowitz and I.A. Stegun: Handbook of Mathematical Functions

Dover, 1972

[42] N. N. Lebedev: Special Functions and their Applications

Dover 1972

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