Variable Volatility (Project) For variable volatility (S, t) and constant K, T, r, , PDEs of the type y 1 2 2(x,

Variable Volatility (Project) For variable volatility σ(S, t) and constant K, T, r, δ , PDEs of the type ∂y ∂τ − 1 2 σˆ2(x, τ)  ∂2y ∂x2 − 1 4 y  = 0 are to be solved, with τ = T − t and transformations S ↔ x, V ↔ y from the Black–Scholes model given by (A6.2), (A6.3); consult Appendix A6.

a) For an American put, apply these transformations to derive from V (S, t) ≥ (K − S)+ an inequality y(x, τ) ≥ g(x, τ).

b) Carry out the finite-element formulation for the linear complementarity problem analogously as in Section 5.3.4.

c) Integrals will include local integrals  σ2(x, τ)ϕiϕj dx,  σ2(x, τ)ϕ iϕj dx Apply Simpson’s quadrature rule  b a f(x)dx ≈ b − a 6 " f(a)+4f a + b 2  + f

(b) # to approximate the above local integrals.

d) Set up a finite-element code, and test it with the artificial function [Fen05] σ(S) := 0.3 − 0.2 log(S/K)2 + 1 .