WHD Problem 3.6a

Find the most general solution of the Black-Scholes equation that has the special form V=V(S).

(∂V/∂t) + ½ σ² S² (∂²V/∂S²) + r S (∂V/∂S) - rV = 0

Substituting V=V(S) reduces the partial differential equation to an ordinary differential equation.

S² (d²V/dS²) + (2r/σ²) S (dV/dS) - (2r/σ²) V = 0

This is an Euler differential equation x²y'' + axy' - by = 0, the solutions of which can be found with the aid of the characteristic equation.

λ² + (a-1)λ + b = 0

λ² + (2r/σ² -1)λ + (-2r/σ²) = 0

Applying the quadratic equation, one obtains (after a bit of algebra) two roots

λ = 1 and λ = (-2r/σ²)

Using these roots, we determine that the most general solution of the Euler equation is
V(S) = c₁ S + c₂ S ^(-2r/σ2)