Arbitrage-Free Volatility Surfaces and When to Recalibrate a Slice

# Arbitrage-free interpolation


# Arbitrage-free interpolation












After constructing an arbitrage-free surface, when analyzing a slice of the surface, do practitioners use a slice generated from the calibrated surface, or would they re-calibrate along the slice that they care about? I'm assuming it would be contextual, but what types of instances would you choose to re-fit on the slice or just use slice generated from the calibrated surface.

## Answer by Jesper Tidblom (score 2)

https://quant.stackexchange.com/a/83653

I am not a practitioner, but have some experience in volatility modelling. As you say, this is very contextual and it also depends on what model you are using to calibrate the surface.

If you are using some simple model with only a few parameters, like SABR or Heston, and try to calibrate the parameters in those to fit all given market volatilities as good as possible, you will only be able to match those very roughly. If you are doing some rough risk calculations this might be good enough. But if you actually want to price options to buy/sell on the market you cannot use that surface directly. This would lead to an arbitrage opportunity. However, you can fit a surface to to all given market volatilities perfectly by using the local volatility model. However, while this can match all market volatilities, it is not really used in practice due to its drawbacks when it comes to the model assumptions on the stochastic behavior of the volatility. In practice one typically fits a so called stochastic local volatility model which combines the local volatility model with a stochastic volatility model, like Heston, to get a perfect match to market volatilities as well as a more realistic stochastic behavior of the volatility. This model can then be used to price options in a consistent way.

## Answer by Arnoldik (score 1)

https://quant.stackexchange.com/a/83662

When working with implied volatility surfaces, interpolating directly on a slice (say, fixed expiry across strikes) can lead to arbitrage unless you're careful. Most practitioners I’ve seen recalibrate the whole surface if they’re analyzing a new slice, especially if the slice is used for pricing.

That said, there are "slice-aware" interpolation techniques like the SVI model or arbitrage-free cubic splines that preserve no-arbitrage conditions across both strikes and maturities. If you're using a pre-calibrated surface, slicing is fine for visualization or rough estimates — but if you're using that slice for hedging or pricing, it's safer to re-calibrate locally.

A good practice is to validate convexity and monotonicity after slicing, especially if you plan to interpolate further.