B Solution of the BSM PDE

The diffusion equation (B.4) may be reduced to a simpler form by the substitution

u(τ, x) = eax+bτw(τ, x) (B.5)

for suitable constants a and b. To determine a and b, we calculate the partial derivatives

uτ (τ, x) = e ax+bτ [bw(τ, x) + wτ (τ, x)]

ux(τ, x) = e ax+bτ [aw(τ, x) + wx(τ, x)]

uxx(τ, x) = e ax+bτ [awx(τ, x) + wxx(τ, x)] + ae

ax+bτ [aw(τ, x) + wx(τ, x)]

= eax+bτ [a2w(τ, x) + 2awx(τ, x) + wxx(τ, x)].