Fitting Implied Volatility Surfaces with Moneyness and Maturity

Summary

The document considers whether to fit an options implied-volatility smile using delta or moneyness as the independent variable. Its answer points to research modeling volatility with a quadratic polynomial in both moneyness and maturity. This produces a structured surface using observed variables rather than relying on delta as the coordinate.

The response suggests delta can depend on a volatility measure under the risk-neutral probability framework, which complicates its use as an independent variable. It also mentions the challenge of estimating derivatives of a pricing function. The discussion is brief: it offers no comparison of fit quality, calibration procedure, or empirical evidence that one coordinate is universally superior. The cited modeling example motivates the moneyness approach, but practical choice may depend on the option market and the intended surface model.

Key ideas

• A quadratic specification can model implied volatility as a function of both moneyness and maturity.
• Moneyness and maturity provide observable coordinates for structuring a volatility surface.
• Delta may depend on a risk-neutral volatility measure, complicating its role as an independent variable.
• The document offers a modeling rationale but no empirical comparison showing universal superiority.

Tags

# When Fitting Implied Vol in, implied vol=ax²+bx+c, why is better to use moneyness than delta as independent variable?


# When Fitting Implied Vol in, implied vol=ax²+bx+c, why is better to use moneyness than delta as independent variable?












I am trying to construct a smile curve using Option data, I can either interpolate implied vol vs delta or implied vol vs moneyness.

## Answer by Stéphane (score 1, accepted)

https://quant.stackexchange.com/a/52902

What I have seen in papers such as Christoffersen, Heston and Jacobs (2009) where they look into a two-factor model of volatility is a quadratic polynomial in BOTH moneyness and maturity.

I would assume that the advantage of using this approach is that you get a structured volatility surface using observed variables. Beyond the problem of having to estimate the derivative of a pricing function you do not have, your delta likely will depend on some measure of volatility under Q.

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