Time-Level Choices in an Implicit Finite-Difference Scheme

Summary

The document asks how to assign time indices when discretizing a two-factor pricing partial differential equation with an implicit finite-difference scheme. The equation includes first and second derivatives in the asset and auxiliary state variables, a mixed derivative, drift terms, a time derivative, and a rate-dependent value term. The specific questions concern whether the discounting term uses the known or unknown time level, and which time level supplies boundary values when rearranging the system.

The response describes the Euler implicit convention: the time derivative links the known solution at the current step to the next step, while the spatial and other non-time terms are evaluated at the next time level. Under that convention, both the rate term and boundary contributions use the next-level values. This is a concise answer rather than a full derivation. It does not show the discretized matrix, define the index direction relative to calendar time, or discuss stability and boundary-condition choices, so those details must be checked against a particular implementation.

Key ideas

• An implicit Euler step evaluates spatial terms at the unknown next time level.
• The known solution enters through the time derivative at the current level.
• The rate-dependent value term uses the next-level value under this convention.
• Boundary contributions in the rearranged system are also taken at the next level.

Tags

# Finite Difference implicit scheme


# Finite Difference implicit scheme












I'm trying to solve the following PDE numerically using an implicit FD scheme:

\begin{equation} \frac{\sigma_s^2}{2}\frac{\partial^2 V}{\partial S^2} + \rho \sigma_S \sigma_\alpha\frac{\partial^2 V}{\partial S \partial \alpha} + \frac{\sigma_\alpha^2}{2}\frac{\partial^2 V}{\partial \alpha^2} + \mu_s \frac{\partial V}{\partial S} + \mu_\alpha \frac{\partial V}{\partial \alpha} + \frac{\partial V}{\partial t} - rV \end{equation}

This raises the following two questions I have not been able to find out yet:

- When substituting the derivatives with FD approximations, is the part $rV$ replaced by $rV_{i,j,k}$ or $rV_{i,j,k+1}$?

- When rewriting FD formula in the form of $V_{k}=AV_{k+1} - C$, are the boundary values needed to calculate $C$ taken from $V_{k+1}$ or $V_k$?

## Answer by Yian Pap (score 0, accepted)

https://quant.stackexchange.com/a/40328

When using the (Euler) Implicit scheme, the only thing that's taken at the previous time level (the one for which you have the solution already), is the $V_{i,j,k}$ that comes from the time derivative. Everything else in the discretized equation is taken at the next time level. So, for both your questions, it's k+1.

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