GROUP CLASSIFICATION FOR A GENERAL NONLINEAR MODEL OF OPTION PRICING1

We consider a family of equations with two free functional parameters containing the classical Black–Scholes model, Schönbucher–Wilmott model, Sircar–Papanicolaou equation for option pricing as partial cases. A five-dimensional group of equivalence transformations is calculated for that family. That group is applied to a search for specifications’ parameters specifications corresponding to additional symmetries of the equation. Seven pairs of specifications are found.


Introduction
In the paper a nonlinear model from the theory of financial markets is considered.In the case of v ≡ 0 it is generalized Black-Scholes equation [1], if, besides, w(t, x) = σ 2 x 2 (0.1) is the classical Black-Scholes model [2].
For arbitrary v and w(t, x) = σ 2 x 2 (0.1) is the Sircar-Papanicolaou nonlinear feedback pricing equation [1].If v is arbitrary, w(t, x) = σ 2 x 2 and r = 0, it is the equilibrium pricing model or Schönbucher-Wilmott nonlinear feedback pricing model [3][4][5][6].The last two models take into account a feedback effect of the presence of two types of traders.The programm traders are the portfolio insurers and the reference traders are the Black-Scholes uploaders.The aim of the paper is to obtain a group classification [7] of equation (0.1) with free parameters v and w.The group of equivalence transformations [7,8] of equation (0.1) will be found.By means of this group symmetries for the equation with all specifications will be calculated.Further these results will be applied to the theory of financial markets, particularly, they will allow to calculate various exact solutions of equation (0.1).
The groups of classical Black-Scholes model and their accordance to the groups of the heat equation were found in [9].Research of symmetries of Schönbucher-Wilmott model and of some other nonlinear pricing models was made in [10][11][12][13].

Group of the equivalence transformations
Let us find the continuous group of equivalence transformations of equation (0.1) for the applying to the search of specifications of the functions v = v(u x ), w = w(t, x) in the equation, that corresponds to additional symmetries for the symmetries of the kernel of principal Lie group for the equation.We rewrite equation (0.1) in the form where v, w are the additional variables, depending on t, x, u, u t and u x .Generators of a continuous group of equivalence transformations will be searched in the form where the functions τ, ξ, η depend on t, x, u, and µ ν depend on t, x, u, u t , u x , v, w.For brevity hereafter ∂ ∂t ≡ ∂ t and similar notations are used.We add to (1.1) the equations ) meaning that in the statement of the problem the function v depends only on u x and the function w depends on t, x.We consider the system of equations (1.1)-(1.3)as a manifold N in an expanded space of corresponding variables.Let us act on the left-hand side of system (1.1)-(1.3)by the extended operator we restrict a result of the action on N and we obtain the equations ) ) From (1.2) and (1.3) it follows that Therefore, equations (1.5) and (1.6) have the form ) ) By equality (1.1) equations (1.7)-(1.9)can be rewritten in the form ) ) By means of the equality ) is rewritten as (1.15) We differentiate the last equations with respect to u tx and obtain w(1 .17) We multiply by 2(1 − xvu xx ) 3 the last equation, then (1.20) Equation (1.20) for the case v ̸ = 0 has at u 3 xx multiplier after its splitting with respect to u x , we obtain two equations After the splitting with respect to u x of the multiplier at u xx in zero degree it follows that and by (1.21), (1.22) The last equality implies that Then from (1.23) it follows that From (1.16) it follows that ν = w x (F e −rt x + A 0 )u + S(t, x, u x , v, w).
The coefficient at u xx in equation (1.20) is equated to zero and we obtain the equation Let us substitute in it the expressions for ξ, η, ν that were found before, and splitting with respect to the variable u leads to the equations −2F e −rt w + w x (F e −rt x + A 0 ) = 0, (1.24) The last of them implies the equalities ν v = S v = 0, consequently, by (1.16) we obtain Thus, Analogous calculations are made with the coefficient at u 2 xx in equation (1.20), we obtain the equation ) .Therefore µ u = µ w = 0, and for the case v ′ ̸ = 0 obtain G = 0 from equation (1.19).Then equation (1.18) implies that µ x = 0, hence J = 0. From equation (1.17) it follows that µ t = 0, it corresponds to the resulting formula µ = (H − P )v.Thus, τ (t) is an arbitrary function, Let us formulate the result in the form of theorem.

Theorem 1. The Lie algebra of infinitesimal generators of the equivalency transformations groups for equation (0.1), is generated by operators
when v ′ , w are identically unequal to zero.Remark 1.It is easy to check that the infinitely-dimensional part of the Lie algebra from Theorem 1 consists of operators of the form Y 5 only.
The extensions of the operators Y k , k = 1, 2, 3, 4, 5, are (1.25) Therefore, the kernel of the principal Lie algebras for equation (0.1) is one-dimensional with the basis Y 2 , because the corresponding group only doesn't transform the additional variables v, w and their arguments t, x, u x .
Corollary 1.The kernel of the principal Lie algebras for equation (0.1) is spanned by the operator X 1 = e rt ∂ u when v ′ , w are identically unequal to zero.

Group classification
Consider Lie algebra of projections of operators (1.25) on the subspace of the variables t, x, u x , v, w, i. e. the algebra generated by It is the direct sum of subalgebras ⟨Z 1 , Z 2 ⟩ and ⟨Z 3 , Z 4 ⟩ that corresponds to two different functions v and w and their different arguments.Therefore, the subalgebras can be considered separately.
Nonzero structure constants of ⟨Z 1 , Z 2 ⟩ are c 1 12 = −1, c 1 21 = 1.Therefore, the inner automorphisms are E 1 : ē1 = e 1 −e 2 a 1 , E 2 : ē1 = e 1 e a 2 .Here e i , i = 1, 2 are the coefficients at Z i respectively in the basis decomposition of Z.If e 2 ̸ = 0, then e 1 = 0 by the acting of E 1 .Therefore the optimal system of one-dimensional subalgebras consists of subalgebras with bases Z 1 and Z 2 .
In the subalgebra ⟨Z 3 , Z 4 ⟩ there are no nontrivial inner automorphisms, consequently, the optimal system of one-dimensional subalgebras has a form For operators Z from optimal systems we calculate the expressions Note, that if Z contains Z 1 with a nonzero coefficient and doesn't contain Z 2 , then v ′ = 0.Such case doesn't correspond to the conditions of Theorem 1.If an operator Z has nonzero coefficients at Z 1 and at Z 3 , then by E 1 the coefficient at Z 1 can be equated to zero for equivalent operator to Z. Therefore, the operator Z 1 can be excluded from further considerations.We have for arbitrary function D(t).Finally, for arbitrary functions φ ̸ = 0, τ ̸ = 0.
Optimal system of two-dimensional subalgebras consists of ⟨Z 2 , Z 3 ⟩, ⟨Z 2 , bZ 3 + Z 4 ⟩, ⟨Z 3 , Z 4 ⟩.In the first two cases we have the simultaneous specifications for v and w that are already known.In the last one specification we have the form W = γx 2 /τ (t).
For every basis operator from the optimal systems calculate the projection of the corresponding generator of the group of equivalency transformations on the space of the variables t, x, u.Then Z 2 corresponds to pr (t,x,u) (Y 4 − Y 3 ) = −u∂ u , for the operator Z 3 it will be pr