Symmetries of the Black-Scholes equation

We study the algebro-geometrical structure of the Black-Scholes equation. The computation of the symmetry group of a partial differential equation has proven itself to be a very useful tool in Classical Mathematical Physics (see e.g. [Harrison-Estabrook 1971]), as well as in Euclidean Quantum Mechanics (see e.g. [Lescot-Zambrini 2004] and [Lescot-Zambrini 2008]). It was therefore natural to try and use the same method in Financial Mathematics.

After setting the general framework ( §2), and performing some preliminary reductions ( §3), we determine ( §4) the isovectors for the Black-Scholes equation in a way broadly similar to the one used for the backward heat equation with potential term in the second of the two aforementioned joint papers with J.-C. Zambrini. Our computation turns out to suggest Black and Scholes’ original solution method ([Black-Scholes 1973]) of their equation; in particular, the quantities r -σ 2 2 and r + σ 2 2 appear naturally in this context. As corollaries, we determine ( §5) the structure of the Lie algebra of the symmetry group of the equation, then ( §6) we obtain some interesting transformations on the solutions.

Date: January 14, 2011.

We shall be concerned with the classical Black-Scholes equation :

for the price C(t, S) of a call option with maturity T and strike price K on an underlying asset satisfying S t = S (see [Black-Scholes 1973], where σ is denoted by v, C by w, and S by x). As is well-known ([Black-Scholes 1973], p.646), the same equation is satisfied by the price of a put option.

We assume σ > 0, and define r := r -σ 2 2 and

It is useful to remark that r2 2σ 2 + r = s2 2σ 2 . We intend to determine the isovectors for (E), using the method applied, in [Harrison-Estabrook 1971], pp. 657-658 (see also [Lescot-Zambrini 2004], pp.189-192) to the heat equation, and in [Lescot-Zambrini 2008], §3, to the (backward) heat equation with a potential term.

Let us set x = ln(S) and

Equation (E) is therefore equivalent to the following equation in ϕ :

Let us set A = ∂ϕ ∂x and B = ∂ϕ ∂t , and consider thenceforth t, x, ϕ, A and B as independent variables . Then (E 2 ) is equivalent to the vanishing, on the fivedimensional manifold M = R + × R 4 of (t, x, ϕ, A, B), of the following system of differential forms :

Let I denote the ideal of ΛT * (M ) generated by α, dα and β ; as

I is a differential ideal of ΛT * (M ). By definition (see [Harrison-Estabrook 1971]), an isovector for (E 2 ) is a vector field

Using the formal properties of the Lie derivative ([Harrison-Estabrook 1971], p.654), one easily proves that the set G of these isovectors constitutes a Lie algebra (for the usual bracket of vector fields).

In order to determine G, we may use a trick first explained in [Harrison-Estabrook 1971], p.657, that applies in all situations in which there is only one 1-form among the given generators of the ideal I (see also [Lescot-Zambrini 2008], p.211).

Let N ∈ G ; as L N (I) ⊆ I, one has L N (α) ∈ I =< α, dα, β >, whence there is a 0-form (i.e. a function) λ such that L N (α) = λα. Let us define

This can be rewritten as

Whence (letters as lower indices indicating differentiation, as usual)

Using the third equation, we can eliminate λ and obtain

Conversely, the existence of a function F (t, x, ϕ, A, B) such that the above equations hold clearly implies that L N (α) = F ϕ α ∈ I ; but then

and there only remains to be satisfied the condition

The last condition in the previous paragraph can be stated as

for ρ a 1-form, ξ a 0-form and ω a 0-form. Let D denote the coefficient of dϕ in ρ ; replacing ρ by ρ -Dα (which doesn’t affect the validity of (4.1) as α 2 = 0), we may assume that D = 0. Setting then

we shall obtain a system of ten equations in F ; we shall then eliminate R 1 ,…,R 6 .

Identifying the coefficients of, in that order, dtdx, dtdϕ, dtdA, dtdB, dxdϕ, dxdA, dxdB, dϕdA, dϕdB and dAdB yields the following system :

Equation (4.13) gives R 4 = 0 ; then (4.8) defines R 5 . Equation (4.14) is equivalent to N t B = 0 ; if that be the case, then (4.11) holds automatically. Now (4.9) defines R 2 , (4.12) defines R 3 , (4.6) defines R 1 and (4.7) defines R 6 . We are left with equations (4.5) and (4.10) and the condition N t B = 0. Let us begin with the last mentioned ; it is equivalent to F BB = 0 : F is affine in B, i.e. F = c + Bd, where c and d depend only on (t, x, ϕ, A). Now ( * * ) can be rewritten as

From (4.12) follows

and (4.8) yields

Now we can rewrite (4.10) as (4.17)

Comparing the coefficients of B on both sides gives -d A = d A , that is d A = 0, i.e. d depends only upon (t, x, ϕ). Now (4.17) becomes 1 2

the new unknowns being a function c(t, x, ϕ, A) and a function d(t), and we still have to satisfy equation (4.5). Now (4.9) implies R 2 = 0, and (4.6) gives

that is :

) . Both members of (4.24) are second-order polynomials in B ; equating the coefficients of B 2 gives c AA = 0, whence c is affine in A : c = e + Af with e and f functions of (t

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