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(Δt² , Δx²).

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.

Posted Date: 11/14/2013 5:57:13 AM | Location : United States

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