Solution of the Black-Scholes model is obtained through a transformation into a heat equation. The general one-dimensional heat equation is given by
where α > 0 is a constant.
a) Derive the algorithm for the FTCS (forward in time centered in space) scheme and the fully implicit scheme for (1).
b) Hence, deduce the Crank-Nicolson scheme.
c) Use the Taylor series expansion to show that the Crank-Nicolson scheme is consistent and has a truncation error of 0(Δt^{2} , Δx^{2}).
d) Apply the von Neumann stability analysis to show that the Crank Nicolson scheme is unconditionally stable.
e) Explain how the Crank-Nicolson scheme is an improvement from the FTCS and fully implicit scheme.