Interpolating Option Surfaces While Preserving No-Arbitrage
Summary
The document considers filling gaps in observed option premiums using tenor and moneyness as the two surface coordinates. The question proposes inverse distance weighting of nearby contracts and artificial zero-premium points at extreme moneyness as a simple, model-light approach. It focuses on out-of-the-money options and seeks a smooth fit that remains close to market quotes, while setting aside volatility history, rates, and in-the-money contracts.
The included response clarifies that ordinary equity and foreign exchange options are typically represented with expiry and strike, moneyness, or delta as the two coordinates; the option premium or implied volatility is the resulting value. Instruments with an additional underlying tenor, such as swaptions, may require a cube. It cautions that simple linear interpolation can violate no-arbitrage conditions and suggests fitting SABR by section. The answer is brief and does not provide a full interpolation procedure, parameter choices, or validation evidence; the proposed artificial boundary points are not assessed.
Key ideas
- Standard equity and foreign exchange option surfaces use expiry and strike-related coordinates.
- Swaptions may require an additional underlying tenor dimension.
- Simple linear interpolation may violate no-arbitrage conditions.
- The response suggests fitting SABR parameters to a section before obtaining interpolated volatility.
- The document does not evaluate inverse distance weighting or its proposed boundary constraints.
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# Simple approach to interpolate option surface
# Simple approach to interpolate option surface
Let's set the spot price as 1 (spot price of underlying security) and express each option contract as a point in 3D space $$ \{ x, y, z \} = \{ tenor, moneyness, premium \} $$ where the premium is also relative, like moneyness.
Past volatility, interest rates and in-the-money options are ignored, we are interested only in out-of-the money options.
Additionally we can add imaginary asymptotical points, contracts with $moneyness = 0$ and $moneyness = 10$ (10x spot price) and set its premium as zero. Just to introduce asymptotical constraints for our 3D surface.
All this would give us a set of points in 3D space. What would be the simple approach to interpolate it? Find points in-between? Find premium for arbitrary values of moneyness and tenor?
The simplest approach would be the Inverse distance weighting, find N nearest contracts to the given point $ \{ x, y \} = \{ tenor, moneyness \} $ and then compute the weighted average of its premiums.
I wonder if there are better simple options? I don't want to use BlackSholes model, because I don't intuitively understand how it works and I don't care about the past volatility and don't want to make too much assumptions about the surface. I want to stay close to the surface defined by the real prices, whatever shape it has and just interpolate it to fill the gaps. Just any universal and relatively simple method with the single assumption that surface is more or less smooth.
P.S.
Just to be sure we are talking about the same things.
tenor - how much days remained till the expiration date, moneyness - how far strike price is from the spot price (relative, to spot price), premium - how much money you get if you sell option (relative, to spot price).
## Answer by David Duarte (score 2)
https://quant.stackexchange.com/a/55365
As has been said in the comments, unless you are working with an asset class that has a second dimension, i.e, swaptions where you have not only the option expiry but also the underlying tenor, a surface would suffice. In the swaptions case you can either have a surface for ATM options or a given strike, or you will need a cube (3 dimensions): option expiry, swap tenor and strike (or moneyness).
For Equities and FX, for example, you will have 2 dimensions: (1) option expiry and (2) strike or moneyness or delta.
The option premium or vol (Black or Normal) would be the result value for each combination of the two dimensions.
As for interpolation, you won't be able to do a simple linear interpolation because that might violate nonarbitrage conditions.
SABR interpolation is simple enough and you just need to fit the parameters to a given section and get your vol from the model.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.