lie symmetry analysis of solution of black-scholes type ... · he largely created the theory of...
TRANSCRIPT
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
Lie Problem FormulationInfinitesimal GeneratorsProlongation of Infinitesimal Generator
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
Lie Problem FormulationInfinitesimal GeneratorsProlongation of Infinitesimal Generator
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
Lie Problem FormulationInfinitesimal GeneratorsProlongation of Infinitesimal Generator
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
Lie Problem FormulationInfinitesimal GeneratorsProlongation of Infinitesimal Generator
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
Lie Problem FormulationInfinitesimal GeneratorsProlongation of Infinitesimal Generator
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
Lie Problem FormulationInfinitesimal GeneratorsProlongation of Infinitesimal Generator
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
Computing Determining Equations Cont’d
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
Computing Determining Equations Cont’d
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
The Black-Scholes ModelThe Infinitesimal GeneratorsThe Determining System of EquationsInfinitesimal Vectors
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
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
Computation Lie Symmetries of Differential EquationsThe Symmetries of Black-Scholes Models
ConclusionAcknowledgements
References
THANK YOU FOR LISTENING
Tack sa mycket!
Thank you!
32 / 32