lie symmetry analysis of solution of black-scholes type ... · he largely created the theory of...

OutlineIntroduction

Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models

ConclusionAcknowledgements

References

Lie Symmetry Analysis of Solution ofBlack-Scholes Type Equation in Finance.

Asaph Keikara Muhumuza

Department of Mathematics, Busitema UniversityDivision of Applied Mathematics, Malardalen University, Sweden.

Second Network Meeting for Sida- and ISP-funded PhD Students.Stockholm 26–27 February 2018

1 / 32

OutlineIntroduction



References

My Advisors

Sergei Silvestrov Anatoliy Malyalenko Milica Rancic

Main Supervisor Assistant Supervisor Assistant SupervisorMaladalen University Maladalen University Maladalen University

John Mango Kakuba GodwinAssistant Supervisor Assistant SupervisorMakerere University Makerere University

2 / 32

OutlineIntroduction



References

1 IntroductionBackground

2 Computation Lie Symmetries of Differential EquationsLie Problem FormulationInfinitesimal GeneratorsProlongation of Infinitesimal Generator

3 The Symmetries of Black-Scholes ModelsThe Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors

4 Conclusion5 Acknowledgements6 References

3 / 32

OutlineIntroduction



References

Background

Historic Background

Sophus Lie, 17 December 1842 - 18 February 1899) was aNorwegian mathematician.He largely created the theory of continuous symmetry andapplied it to the study of geometry and differentialequations i.e., Lie [2, 4].Further works have been done for instance:Ovsyannikov [2]: Group properties of differential equations.Bluman and Kumei [3]: New classes of symmetries forpartial differential equations.Olver [4]: Application of Lie groups to differential equations.Ibragimov [1]: Lie group analysis of differential equation.Gazizov and Ibragimov [3]: Lie symmetry analysis ofdifferential equations in finance.

4 / 32

OutlineIntroduction



References

Background

Lie Algebra

A Lie algebra is a vector space g over some field Ftogether with a binary operation [·, ·] : g× g→ g called theLie bracket that satisfies the following axioms:Bilinearity,[ax + by , z] = a[x , z] + b[y , z], [z,ax + by ] = a[z, x ] + b[z, y ]for all scalars a,b in F and all elements x , y , z in g.Alternativity, [x , x ] = 0 for all x in g.The Jacobi identity, [x , [y , z]] + [z, [x , y ]] + [y , [z, x ]] = 0 forall x , y , z in g.Using bilinearity to expand the Lie bracket [x + y , x + y ]and using alternativity shows that[x , y ] + [y , x ] = 0 ∀ x , y ∈ g,Anticommutativity, [x , y ] = −[y , x ], ∀x , y ∈ g.

5 / 32

OutlineIntroduction



References

Lie Problem FormulationInfinitesimal GeneratorsProlongation of Infinitesimal Generator

Problem Formulation

Considering the system of PDEs in p independentsx = (x1, x2, · · · , xp) ⊂ X ⊆ Rp and q− dependentvariables u = (u1,u2, · · · ,up) ⊂ U ⊆ Rq) involvingderivatives up to n,

Fm(x ,u,u(1),u(2), · · · ,u(n)) = 0; m = 1,2, · · · , l (1)

where the notation u(s) stands for a vector in the Euclideanspace U having as coordinates the derivatives

uαj1,j2,··· ,js =∂uα

∂x j1 , ∂x j2 , · · · , ∂x jn, s = 1,2, · · · ,n;

α = 1,2, · · · ,q; jν = 1,2, · · · ,p; ν = 1,2, · · · , s.

6 / 32

OutlineIntroduction



References


Formulation Cont’d

It is said that the system (1) admits a one-parameter localLie group of point-symmetry transformations of the spaceZ = X × U,

x′

= f (ε, x ,u)

u′

= φ(ε, x ,u) (2)

(ε is the group parameter, ε ∈ 4 ⊂ R, 0 ∈ 4), if eachsolution after the transformation of the group remains asolution of the system.

7 / 32

OutlineIntroduction



References


Infinitesimal Generator

Finding the admitted Lie groups of the PDEs is based onthe fundamental correspondence between the Lie groupand the Lie algebras of the infinitesimal generators,

X =

p∑i=1

ξi(x ,u)∂

∂x i +

q∑α

ηα(x ,u)∂

∂xα(3)

with the coefficients

ξi(x ,u) =∂f i(0, x ,u)

∂ε, ηα(x ,u) =

∂φα(0, x ,u)

∂ε;

f = (f 1, f 2, · · · , f p); φ = (φ1, φ2, · · · , φq).

8 / 32

OutlineIntroduction



References


Prolongation

The milestone of the Lie method is the infinitesimal criterionwhich is based on a special technique for prolongation of thegroups and their infinitesimal generators.The system of PDEs (1) is viewed as a submanifold 4F in theprolonged space

Z n = Z × U(1)U(2) × · · · × U(n)

4F = {zn ∈ Z n : Fm (zn) = 0,m = 1, · · · , l} ⊂ Z (n) (4)

The system (1) admits a one parameter group of transformations(2) with the infinitesimal generator V if and only if the followinginfinitesimal condition holds

Pr (n)V [F (z(n))] = 0 for z(n) ∈ 4F (5)

9 / 32

OutlineIntroduction



References


Prolongation Cont’d

Thus the prolongation PrnX becomes

Pr (n)X = X +

p∑i=1

q∑α

ζα=1i

∂

∂uαi+ · · ·

= +

p∑j=1

· · ·p∑

jn=1

q∑α=1

ζαj=1,··· ,jn∂

∂uαj=1,··· ,jn(6)

is the n−th prolongation of the infinitesimal generator V .The coefficients ζαj=1,··· ,jk , k = 1, · · · ,n depend on thefunctions ξ(x ,u), η(x ,u) and can be obtained by therecursive formulae

10 / 32

OutlineIntroduction



References


Prolongation Cont’d

ζαi = Di(ηα)−

p∑s=1

uαs Di(ξs)

ζαj1,··· ,jk = Djk

(ζαj1,··· ,jk

)−

p∑s=1

uαj1,··· ,jk−1,sDjk (ξs) (7)

Di is the operator of total differentiation w.r.t. the variable x i

Di =∂

∂x i +

q∑α=1

uαi∂

∂uα+

p∑j=1

q∑α=1

uαji∂

∂uαj+ · · ·

+

p∑j1=1

· · ·p∑

jn−1=1

q∑α=1

uαj1,··· ,jn−1

∂

∂uαj1,··· ,jn−1

(8)

11 / 32

OutlineIntroduction



References


The Determining Equation

Since the variables x i ,uα,uαj1,··· ,js are supposed to beindependent, and the Equation (6) can be facilitated byequating to zero all the coefficients of the monomials in thepartial derivatives Xα

j1,··· ,js .Thus, a large number of linear homogeneous partialdifferential equations are obtained.They are known as the DSEs of the symmetry groupadmitted by (1) for they serve to determine the unknowncoefficients ξ(x ,u)i , η(x ,u)α of the respective groupgenerator.The solutions of the DSEs constitute the widest admittedLie algebra.

12 / 32

OutlineIntroduction



References

The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors

The Black-Scholes Model

The widely used one-dimensional model (or one statevariable plus time) known as the Black-Scholes model [1]for European option, is described by the equation

ut +12

(σx)2uxx + rxux − ru = 0, (9)

where u(x , t) is the value of option with defined pay off,x ∈ [0,∞) is the price of the underlying asset.

13 / 32

OutlineIntroduction



References


Symmetries

The Lie symmetries for a one parameter B–S model (9) [3]:

X1 =∂

∂t, X2 = X

∂

∂x,

X3 =2tx∂

∂x+ (ln x +Dt)x

∂

∂x+ 2rtu

∂

∂u,

X4 =σ2t2x∂

∂x+ (ln x −Dt)u

∂

∂u,

X5 =2σ2t2 ∂

∂t+ 2σ2tx ln x

∂

∂x+[(ln x −Dt)2 + 2σ2t(rt − 1)

]u∂

∂u,

X6 =u∂

∂t, Xα = φ(t , x)

∂

∂u,

where D ≡ r − σ2/2, α(t , x) is an arbitrary soln of (9).14 / 32

OutlineIntroduction



References


The Two-Dimensional Black-Scholes Model(Feynmann-Kac Model)

Now basing on (9) for two assets x and y both constantrisk-free interest rate r , volatility σ2

i , i = 1,2 and correlationcoefficient ρThen we have the famous Feynman-Kac model in R2

+[0,T ]

ut +12

(σ1x)2uxx +12

(σ2y)2uyy +σ1σ2ρxyuxy +rxux +ryuy−ru = 0(10)

We now proceed to determine the infinitesimal symmetrygroups of equation .

15 / 32

OutlineIntroduction



References


Calculation of Infinitesimal Generators

Based on the method outlined in [3]:

ut − F (t , x , y ,u,u(1),u(2)) = 0, (11)

where u = u(t , x , y), x = (x1, ...., xn), y = (y1, ...., yn) andu(1), u(2) are given:u(1) = (ux1 ,uy1 , ....,uxn ,uyn ),u(2) = (ux1x1 ,uy1y1 ,ux1x2 ,uy1y2 ....uxnxn ,uynyn ).The infinitesimal transformation operator:

X = τ(t , x , y ,u)∂

∂t+ ξi(t , x , y ,u)

∂

∂x i

+ηj(t , x , y ,u)∂

∂y j + φ(t , x , y ,u)∂

∂u.(12)

16 / 32

OutlineIntroduction



References


Calculation of Infinitesimal Symmetries Cont’d

Here X denotes the prolongation of the operator (12) to thefirst and second-order derivatives:

X(2) = τ(t , x , y ,u)∂

∂t+ ξ(t , x , y ,u)

∂

∂x i + ηj(t , x , y ,u)∂

∂u

+ φt ∂

∂ut+ φi ∂

∂ux i+ φj ∂

∂uy j

+ φik ∂

∂ux i xk+ φij ∂

∂ux i y j+ φjk ∂

∂uy j xk,

17 / 32

OutlineIntroduction



References


The Determining Equations

Considering (10) we have the determining vector (i.e., see(3) and (6)) X = ξ ∂

∂x + η ∂∂y + τ ∂∂t + φ ∂

∂u.

and the prolongation

Pr2X = ξ∂

∂x+ η

∂

∂u+ τ

∂

∂t+ φx ∂

∂ux+ φy ∂

∂uy+ φt ∂

∂ut+

φxx∂

∂uxx+ φxy

∂

∂uxy+ φx t

∂

∂uxt+

φyy∂

∂uyy+ φy t

∂

∂uyt+ φt t

∂

∂ut t(13)

18 / 32

OutlineIntroduction



References


The Characteristic Equation for InfinitesimalGenerators

From (7) the characteristic equation of (10) becomes

φ− ξux − ηuy − τut

we compute φx , φy , φt , φxx , φxy .It follows that

φx = Dx (φ− ξux − ηuy − τut ) + ξuxx + ηuxy + τuxt

Dxφ− uxDxξ − uyDxη − utDxτ

19 / 32

OutlineIntroduction



References


Computing Determining Equations

Since Dx = ∂∂x + ux

∂∂u , then

φx =

(∂φ

∂x+ ux

∂φ

∂u

)− ux

(∂ξ

∂x+ ux

∂ξ

∂u

)− uy

(∂η

∂x+ ux

∂η

∂u

)− ut

(∂τ

∂x+ ux

∂τ

∂u

)φx = φx + (φu − ξx )ux − ηxuy − τxut − ξuu2

x − ηuuxuy − τuuxut(14)

20 / 32

OutlineIntroduction



References


Computing Determining Equations Cont’d

Proceeding in this same way we obtain

φy = φy − ξyux + (φu − ηy )uy − τyut − ξuuxuy − ηuu2y − τuuyut

(15)

φt = φt − ξtux − ηtuy + (φu − τt )ut − ξuuxut − ηuuyut − τuu2t

(16)

Again using the characteristic function we compute

φxx = Dx (φx − ξuxx − ηuxy − τuxt ) + ξuxxx + ηuxxy + τuxxt

= Dxφx − uxxDxξ − uxyDxη − uxtDxτ

Again using Dx = ∂∂x + ux

∂∂u ,

21 / 32

OutlineIntroduction



References



This leads us to

φxx =

(∂φx

∂x+ ux

∂φx

∂u

)− uxx

(∂ξ

∂x+ ux

∂ξ

∂u

)− uxy

(∂η

∂x+ ux

∂η

∂u

)− uxt

(∂τ

∂x+ ux

∂τ

∂u

)φxx = φxx + (2φxu − ξxx )ux − ηxxuy − τxxut + (φuu − 2ξxu)u2

x

− 2ηxuuxuy − 2τxuuxut + (φu − ξx )uxx − 2ηxuxy

− 2τxuxt − ξuuu3x − ηuuu2

x uy − τuuu2x ut − 3ξuuxuxx

− ηuuyuxx − τuutuxx − 2ηuuxuxy − 2τuuxuxt (17)

22 / 32

OutlineIntroduction



References



Similarly we compute:

φyy = φyy − ηyyux + (2φyu − ηyy )uy − τyyut − 2ηyuuxuy

+ (φuu − 2ηyu)u2y − 2τyuuyut + (φu − ηy )uyy

− 2ξyuxy − 2τyuyt − ηuuu3y − ξuuuxu2

y − τuuu2y ut

− 3ηuuyuyy − ξuuxuyy − τuutuyy − 2ξuuyuxy − 2τuuyuyt(18)

φxy = φxy + (φyu − ξxy )ux + (φxu − ηxy )uy − τxyut

+ (φuu − ηyu − ξxu)uxuy − τyuuxut − τxuuyut − ξyuu2x

− ηxuu2y − ξyuxx − ηxuyy + (φu − ηy − ξx )uxy − τyuxt

− τxuyt − ξuuxuxy − ξuuxxuy − ηuuxuyy − ηuuuxu2y

− τuuxuyt 2ηuuyuxy − τuuyuxt − ξuuxyut − τuuuxuyut .23 / 32

OutlineIntroduction



References


Solving Determining System for Infinitesimal Vectors

Substituting φt = ut , φx = ux , φ

y = uy , φt = ut , φ

xx =uxx , φ

xy = uxy , φyy = uyy as obtained into Equation (10).

Equating corresponding terms we obtained a system ofpartial differential called the determining system ofequations (DES).This was then solved for τ(t), ξ(x , y , t), η(x , y , y), andφ(x , y , t ,u).The corresponding results were as shown below

24 / 32

OutlineIntroduction



References


τ(t) = C4

[12

t2 + t]

+ C0 (19)

ξ(x , y , t) = C5

[12σ1

ρσ2x ln y +

12

xt ln x]

+ C1 (20)

η(x , y , t) = C6

[12

1σ1ρ

y (2σ1ρ ln y − σ2 ln x) +12

yt ln y]

+ C2(21)

25 / 32

OutlineIntroduction



References


φ(x , y , t ,u) =

C4uΓ

[ (−1

2σ2

2 ln x + σ1σ2ρ ln y)

ln(x)

]+

C4uΓ

[t(1− ρ)

(12σ2

1σ22 + ρσ1σ2(

12σ2

2 − r)− rσ22

)ln x]

+

C4uΓ

[(−1

2σ2

1 ln y + (σ31σ2(

12ρt + ρ2 − 1

2)

)ln y]

+

C4uΓ

[(σ2

1(12σ2

2 − r)(t − ρ) + σ1σ2r(ρt + 2ρ2 − 1)

)ln y]

+

26 / 32

OutlineIntroduction



References


C4uΓ

[t(t + 2ρ)[−1

8σ4

1σ22 −

12ρσ3

1σ2(−12σ2

2 + r)

]+

C4uΓ

[(−1

8σ4

2 + σ21σ

22(ρ2 + r − 1))− 1

2σ2

1r2 + σ21σ

22t(1− ρ2)

]+

C4uΓ

[ρσ1σ2r(−1

2σ2

2 + r)− 12σ2

2r2]

]+ C3u + Cαα(x , y , t) (22)

whereΓ =

116ρσ2

1σ22(ρ2 − 1)

27 / 32

OutlineIntroduction



References

Conclusion

The Lie Symmetry Method solves for the exact solutions ofPDEs.In our special case of the two-dimensional Black-ScholesPDE infinitesimal symmetries obtained can help todetermine all the exact solution of the equationPart of the results were presented in ASMDA2017Conference in London, an article was submitted andaccepted for publication in the conference proceedings.Our future research direction is to extend further ourresults obtained so far and also use numerical approachbased on Wavelets and Vandermonde determinant tosolve similar equations.

28 / 32

OutlineIntroduction



References

Acknowledgements

We acknowledge the financial support for this research bythe Swedish International Development Agency, (Sida),Grant No.316, International Science Program, (ISP) inMathematical Sciences, (IPMS).We are also grateful to the Division of AppliedMathematics, Malardalen University for providing anexcellent and inspiring environment for research educationand research.May The Almighty God Richly Bless You.

29 / 32

OutlineIntroduction



References

References

F. Black and M. Scholes. The pricing of option andcorporate liabilities. Journal of political Economy, 81,637–654, 1973.

S. Lie. On integration of a class of linear partial differentialequations by means of definite integral. Archive formathematik og naturvidenskab 6, 3, 328–368, 1881 [inGerman]. reprinted in S., Lie Collected Works, Vol.6, paperIII, 139-223.

R.K. Gazizov and N.H. Ibragimov. Lie symmetry analysis ofdifferential equations in finance. Nonlinear Dynamics,Kluwer Academic Publishers, Netherlands, 17, 4 387–407.1998.

P.J. Olver. Application of Lie groups to differential equations.Second Edition, Springer-Verlag New York, Inc., 1993.

N.H. Ibragimov. CRC Handbook of lie Group Analysis ofDifferential Equations. (ed) Vol.1, 1994, Vol. 2, 1995; Vol.3,1996, CRC Press, Boca Raton, FL.

L.V. Ovsyannikov. Group properties of differentialEquations. USSR Academy of Sciences, Siberian Branch,Novosibirsk, 1962 [in Russian].

G.W. Bluman, S. Kumei and G. Reid. New classes ofsymmetries for partial differential equations. Journal ofMathematical Physics 29, 806–811, 1988.

S. Lie. Genreal Studies on differential equations admittingfinite continuous groups. Mathematische Annalen, 25, 1,71–151, 1885 [in German]. reprinted in S. Lie. GesammelteAbhandlundgen, Vol.3, paper XXXV. (English translationpublished in N. H., Ibragimov (ed), CRC Handbook of lieGroup Analysis of Differential Equations, Vol.2, 1995, CRCPress, Boca Raton, FL.)

30 / 32

OutlineIntroduction



References

References Cont’d

N.H. Ibragimov. CRC Handbook of lie Group Analysis ofDifferential Equations. (ed) Vol.1, 1994, Vol. 2, 1995; Vol.3,1996, CRC Press, Boca Raton, FL.

L.V. Ovsyannikov. Group properties of differentialEquations. USSR Academy of Sciences, Siberian Branch,Novosibirsk, 1962 [in Russian].

G.W. Bluman, S. Kumei and G. Reid. New classes ofsymmetries for partial differential equations. Journal ofMathematical Physics 29, 806–811, 1988.

S. Lie. Genreal Studies on differential equations admittingfinite continuous groups. Mathematische Annalen, 25, 1,71–151, 1885 [in German]. reprinted in S. Lie. GesammelteAbhandlundgen, Vol.3, paper XXXV. (English translationpublished in N. H., Ibragimov (ed), CRC Handbook of lieGroup Analysis of Differential Equations, Vol.2, 1995, CRCPress, Boca Raton, FL.)

31 / 32

OutlineIntroduction



References

THANK YOU FOR LISTENING

Tack sa mycket!

Thank you!

32 / 32

lie symmetry analysis of solution of black-scholes type ... · he largely created the theory of...

Documents