If f(x, ) 0 in the initial value problem with u(x, 0) = 0, and u 0 as |x| , then

If f(x, τ ) ≥ 0 in the initial value problem

with
u(x, 0) = 0, and u → 0 as |x| → ∞,

then u(x, τ ) ≥ 0. Hence show that if C₁ and C₂ are European calls with volatilities σ₁ and σ₂ respectively, but are otherwise identical, then C₁ > C₂ if σ₁ > σ₂.

Use put-call parity to show that the same is true for European puts.

a2u ne + f(x, ), on - 0, ax2