Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach

By D. Duffy. 2006

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Convergence analysis of Crank-Nicolson and Rannacher time-marching (2006, methodology): “This is avoided here by using the Alternative Direction Implicit (ADI) method which is a factored approximation to the Crank–Nicolson method (Duffy 2006) but has a computational cost that is proportional to the number of grid points, as in one dimension.”
The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines (2009, methodology): “(14) Since the coefficients of the second order derivative terms go to zero as v → 0, we use an upwinding finite difference scheme (see Duffy (2006), Chapter 8) for the first order derivative term, such that, at the grid point (S, vm, τn) we have ∂C ∂v = Cnm+1 − Cnm ∆v if v ≤ α β , Cnm −…”