Front-Fixing for American Options Apply the transformation := S Sf(t) , y(,t) := V (S,t) to

Front-Fixing for American Options Apply the transformation

ζ := S Sf(t) , y(ζ,t) := V (S,t)

to the Black–Scholes equation (4.1).

a) Show

∂y

∂t + σ2 2 ζ2 ∂2y

∂ζ2 + $

(r − δ) − 1 Sf dSf dt

%

ζ

∂y

∂ζ − ry = 0 (4.63)

b) Set up the domain for (ζ,t) and formulate the boundary conditions for an American call. (Assume δ > 0.)

c) (Project) Set up a finite-difference scheme to solve the derived boundaryvalue problem. The curve Sf(t) is implicitly defined by the above PDE, with final value Sf(T) = max(K, r

δ K).