#Fengler arbitrage smoothing

[genius-mathematics-consultants]

Volatility smoothing algorithms to remove arbitrage from volatility surfaces

Implied volatility surfaces are usually constructed from the market prices of a number of options of different strikes and expiries. Because of poor data quality, a trader valuing options from such a surface will typically generate arbitrage opportunities. In fact, even if the pillar observations are free of arbitrage, cubic spline interpolation can introduce arbitrage.

Volatility smoothing is the process of finding the cubic spline surface that fits the input data as closely as possible, under the restriction that it be arbitrage free. This turns out to be a quadratic optimization problem, as discussed in the very useful paper "Arbitrate Free Smoothing" of M.R Fengler.

We've implemented Fengler's algorithm in python, and in this short article we illustrate how it can be used to dramatically improve the quality of volatility data. The algorithm is fast enough to be run across millions of rows of volatility data.

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