Arbitrage-Free Volatility Smoothing Versus Interpolation

Summary

The note explains a distinction in methods for fitting an implied volatility surface. Interpolation passes through each observed market volatility point, while smoothing seeks a fitted surface that balances agreement with the observations against constraints such as absence of arbitrage. As a result, smoothing can produce fitted values that differ from individual raw quotes, so the input quotes themselves need not form an arbitrage-free set.

The explanation is a brief interpretation of a statement in Fengler’s paper, offered by a respondent who says they only skimmed the relevant section. It gives no equations, implementation details, empirical comparison, or fuller account of the paper’s smoothing procedure. The key practical takeaway is conceptual: a constrained fit can accommodate noisy or mutually inconsistent observations while enforcing desirable properties on the resulting surface. The note does not establish how to choose smoothing strength or constraints, nor whether a particular fit is reliable for pricing or risk management.

Key ideas

• Interpolation reproduces the observed implied volatilities at the input points.
• Smoothing fits a surface that may not pass through every raw quote.
• Arbitrage constraints can be imposed on the fitted surface even when market inputs are inconsistent.
• The explanation is a brief interpretation rather than a detailed account of Fengler’s method.

Tags

# Arbitrage free smoothing of implied volatility surface, by Fengler


# Arbitrage free smoothing of implied volatility surface, by Fengler












I'm reading this paper link and have came across the below statement. Can someone shed some light on it.

"The approach we propose here builds on smoothing rather than interpolation. Therefore, the input data do not need to be arbitrage-free." Thanks you.

## Answer by FinanceGuyThatCantCode (score 3, accepted)

https://quant.stackexchange.com/a/33761

I think they mean that by interpolation, the smile goes exactly through the implied vols of the raw market data. By smoothing, it means they are attempting a best fit subject to arbitrage constraints and the fit may not actually go exactly through the raw vol data points.

Disclaimer - I only skimmed that section of the paper rather than reading thoroughly.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.