Rannacher Time Stepping with ADI Schemes for the Heston Model

Summary

The document asks how to apply Rannacher time stepping to the Heston stochastic volatility model when using the Craig–Sneyd alternating direction implicit (ADI) scheme. In the one-dimensional Black–Scholes setting, the question describes replacing initial Crank–Nicolson steps with implicit Euler steps to improve the numerical treatment of nonsmooth initial data. It asks whether the analogous Heston procedure requires solving a large coupled system or can retain an ADI approach.

The text is a numerical-method question rather than a worked explanation: it gives no implementation, convergence results, or answer about the required linear systems. Its useful context is the distinction between a one-dimensional time-stepping adjustment and a multidimensional PDE discretization with mixed derivatives. Applying the idea requires attention to how the initial damping steps fit the chosen ADI scheme; the document does not specify grid details, boundary conditions, or a particular adaptation.

Key ideas

• Rannacher stepping replaces initial Crank–Nicolson steps with implicit Euler steps in the cited one-dimensional setting.
• The question concerns adapting this initial damping idea to the Heston model.
• Craig–Sneyd is the ADI scheme named for the Heston discretization.
• The document asks whether ADI can be retained but provides no answer or numerical evidence.

Tags

# Rannacher time steps for Heston Model


# Rannacher time steps for Heston Model












I am somewhat confused regarding the Rannacher discretization within the Heston model in the paper “Convergence analysis of Crank–Nicolson and Rannacher time-marching” by Giles and Carter (2005).

In the 1D Black-Scholes model, the initial Crank-Nicolson time steps are replaced by implicit Euler steps. In the Heston model, I use the Craig-Sneyd scheme. If one applies Rannacher time steps in this context, does one have to solve a "huge" system of equations, or are ADI methods also used for this purpose?

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