Interpolating Implied Volatility Across Option Delta Quotes

Summary

The document considers how to estimate implied volatility at a target delta when option quotes share an expiry but have different deltas and implied volatilities. It presents cubic splines as a possible first approach when the volatility-versus-delta relationship is smooth and continuous. Visual inspection is recommended before choosing a method, since sharp changes can make a spline unsuitable; linear or polynomial interpolation are mentioned as alternatives.

The discussion also raises arbitrage consistency as a constraint. A common route is to convert delta quotes into strike or moneyness, smooth the volatility skew or surface there using an arbitrage-aware method, then map the result back to delta. A direct delta-space method for foreign-exchange options is mentioned as a further avenue. The document gives no dataset, numerical comparison, or proof that any method will preserve arbitrage freedom, so the suggested quick spline should not be treated as a validated surface construction.

Key ideas

• Cubic splines can estimate volatility at an intermediate delta when the relationship is sufficiently smooth.
• Inspect the quoted volatility curve before selecting an interpolation method.
• Sharp variation in quotes may make a cubic spline unreliable; simpler alternatives may be worth considering.
• Interpolation in strike or moneyness followed by conversion back to delta is one proposed workflow.
• Arbitrage constraints matter because a smooth interpolated curve can still imply arbitrage.

Tags

# Interpolation of IV based on delta


# Interpolation of IV based on delta












I have a dataset with options, all the same date and time to maturity but different IV and delta. Now, I want to find the IV for certain delta values (e.g 0.5) through interpolation. Do you think that cubic spline interpolation is an appropriate method here or is there a better way to do that?

## Answer by Sane (score 1)

https://quant.stackexchange.com/a/79124

Cubic spline should work, if the relationship between IV and delta is relatively smooth and continuous. However, if there are sharp changes in the IV values as delta varies, cubic spline interpolation may not be the best approach. In such cases, interpolation methods such as linear interpolation or polynomial interpolation could be more effective.

Visually inspect the relationship between IV and delta before selecting interpolation method.

## Answer by Frido (score 1)

https://quant.stackexchange.com/a/81718

There are several arb-free interpolations/extrapolations. Usually they are expressed in terms of strike or moneyness. So what you could do is convert the (discrete) delta quotes into strike quotes, interpolate/extrapolate in strike space and then convert back to delta space once you've smoothed the skew/surface.

I did find the following paper (which I'm planning to read myself) which could be of interest to you as it works directly in delta space:

Arbitrage-free smile construction on FX option markets using Garman-Kohlhagen deltas and implied volatilities

Cubic spline might work for a first quick run, but my understanding is that cubic spline can result in arbitrage.

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