Finite-Difference Grid Design for Quantitative Finance PDEs

Summary

The document asks how to choose grid boundaries and spacing when solving Black–Scholes-type partial differential equations with finite differences. It points to sparse grids and methods that avoid conventional boundary conditions, and raises applications such as stochastic local volatility and models with stochastic interest rates. The central practical concern is how domain limits and grid spacing affect numerical results.

No grid construction method is explained or evaluated in the text. The author mentions papers by Peter Austing and Jherek Healy as useful leads, but provides no descriptions of their approaches, comparisons, or numerical evidence. The document is therefore a research question and a request for references rather than a tutorial. Its scope is quantitative finance PDEs, and any conclusions about which grids work best, how to set boundaries, or how performance changes across models require consulting the cited work or other sources.

Key ideas

• Finite-difference solutions to pricing PDEs depend on how the computational grid is constructed.
• The author seeks guidance on setting upper and lower domain limits and choosing grid spacing.
• Potential applications include stochastic local volatility and models with stochastic interest rates.
• Sparse grids and approaches without standard boundary conditions are mentioned as related techniques.
• The document points to papers by Peter Austing and Jherek Healy but gives no method details or numerical results.

Tags

# State-of-the-art grid construction techniques


# State-of-the-art grid construction techniques












I am wondering what the state-of-the-art regarding grid definition and construction, for solving PDEs using finite differences. I know some techniques are described in Duffy's Finite difference methods in financial engineering. I am aware of sparce grids and finite difference methods without boundary conditions.

However, my question or seek for advice is if someone knows some good references or techniques that are well-known to work, specially in the context of quantitative finance, i.e. give good results when applied to solve BS PDE equations (let it be SLV, a stock together with stochastic rates, etc).

I'd appreciate for example some references with discussions on how to set an upper/lower limit on the grid or how the spacing should be defined.

Thanks a lot!

Edit: @Frido was very helpful by pointing out this paper by Peter Austing which is along the lines of what I am looking for. Another paper that present some ideas is this one by Jherek Healy

Do you know some other techniques or papers such as these?

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