Interpolating Scattered European Swaption Volatility Cubes
Summary
The document describes the problem of filling gaps in a scattered cube of normal volatilities for European swaptions. The three dimensions are swap tenor, option expiry (term), and strike. The author seeks a practical interpolation approach and asks how volatility tends to vary along each dimension, including whether slices at a fixed tenor behave consistently across expiries.
It raises sequential slicing and flat extrapolation as possible pragmatic techniques, while noting that flat strike wings may create discontinuities. The author also recognizes that direct interpolation of quoted volatilities can introduce arbitrage opportunities and that calibrating a parametric model such as SABR is an alternative way to construct a cube. The document is a question rather than a proposed solution: it offers no market data, tested method, or conclusions about which dimensions are most consistent. Any interpolation choice therefore needs validation for smoothness and financial consistency.
Key ideas
- A swaption volatility cube varies across expiry, underlying swap tenor, and strike.
- Scattered quotes require interpolation or modeling to estimate volatility at unobserved points.
- Flat extrapolation on strike wings may create undesirable jumps.
- Direct interpolation of volatility can introduce arbitrage inconsistencies.
- Calibrating a parametric model such as SABR is presented as an alternative, not a demonstrated solution.
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Full text
# Interpolating the volatility cube of European Swaptions # Interpolating the volatility cube of European Swaptions I'm in a situation where I have a cube of European swaption volatilities (normal volatilities), which contains only scattered data. Since it is three dimensional (Tenor, Term, Strikes) I'm having a hard time wrapping my head around a nice way to interpolate this structure such that I get a "filled" cube, that doesn't contain any holes of data. For example, in two dimensions (e.g. a simple Cap/Floor Volatility surface) for my purposes sometimes it was fine even flat extrapolating on the wings, whereas in this case I fear that a flat extrapolation e.g. in the strike dimension would create weird jumps in the cube. I'm totally aware of the fact that there are not only much better ways of retreiving a nice cube (e.g. calibrating SABR parameters and retrieving it from those) but also a lot of other issues connected to the approach of directly interpolating on the vols (e.g. creating arbitrage opportunities). To make my question more obvious: - Do you have any hints for me on how to approach the interpolation of a Swaption Volatility cube with scattered data? - In order to apply simpler types of interpolation I considered slicing the cube. That brought me to the question: In which dimension (Tenor/Term/Strike) do I have which types of consistency? E.g. are Swaptions with the same tenor similar (i.e. 1Y Tenor Volatility will have little movement over the different expiry dates?) I can specify my question even more, if that's necessary please let me know. My goal is to gain some intuitive insights on the behaviour of the swaption vol in depending on the dimension tenor, term and strike and finally find a pragmatic approach to interpolate the cube. Thanks a lot!
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