Using Cubic Splines to Interpolate a Swaption Volatility Surface
Summary
The document considers how to estimate a missing point in a swaption volatility grid indexed by option maturity and swap tenor. The questioner has quoted Black-model volatilities at selected maturities and tenors, and is unsure whether simple linear interpolation along tenor is reliable. They also mention anomalous values at short maturities and tenors, and the effort involved in implementing a two-dimensional method.
The response recommends cubic spline interpolation for one-dimensional curves and bicubic interpolation for a two-dimensional grid. It describes these approaches as practical and commonly used, but supplies no swaption-specific validation, comparison against market conventions, or method for handling outliers. Its supporting claim about splines working well on macroeconomic time series does not establish that they preserve desirable properties of an interest-rate volatility surface. The recommendation is therefore a starting point; users still need to check the resulting surface for plausibility and suitability to their pricing task.
Key ideas
- A swaption volatility grid can be interpolated across maturity and tenor.
- Cubic splines are suggested for interpolation along one dimension.
- Bicubic interpolation is suggested for estimating values within a two-dimensional grid.
- The response does not demonstrate that these methods handle swaption-specific outliers or market constraints.
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Full text
# Interpolation of volatility curve for Swaption # Interpolation of volatility curve for Swaption I have found volatility in the black model for swaption for different maturity (1-2-3-6-9M, 1Y, 18M, 2-10Y, 15-20-25-30Y) and Tenor (1-10Y, 15-20-25-30Y). Now I need another values (Maturity: 2, Tenor: 12). I work with Excel without add-ins, I tried linear interpolation between (2,10) and (2,15), but I have some doubt on this method. I know some advanced inteprolation techniques (spline) in 2 dimension wich I could use for a given maturity, but it could take some time to implement a bicubic spline interpolation method. I could also use a (Maturity,Tenor) Interpolation, but I have some odd values for short maturity/short tenor. I would like to remove these " outliers". There is only discussion on advanced volatility interpolation for option. What would be a reliable/fast method to interpolate Volatility(Maturity,Tenor) ? I don't need a generic interpolation method but some suggestion on how to improve them for volatility interpolation, or a more complex interpolation method (not too complex) wich has given some good results. Here are my data so you could see what I am doing, the graph is a 1D cubic interpolation on maturity (step 1/12) then on tenor (step 1). ## Answer by Quantopik (score 2) https://quant.stackexchange.com/a/11446 One of the most used interpolation techniques is the cubic spline interpolation. Here you can find an overview of that, while, on Mathworks.com, you can find the tutorial to implement that in Matlab directly simply by using the spline(x,Y,xx) command function. It is not difficult to implement and, moreover, it gives pretty reliable results. I never tried to interpolate options data, but, anyway, it is proved it works pretty well on the most of macroeconomic time-series. In the case you need for interpolating in 2-d, you can use the bi-cubic interpolation technique; Here you can find an example in Matlab.
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