This paper explores the stochastic collocation technique, applied on a monotonic spline, as an arbitrage-free and model-free interpolation of implied volatilities. We explore various spline formulations, including B-spline representations. We explain how to calibrate the different representations against market option prices, detail how to smooth out the market quotes, and choose a proper initial guess. The technique is then applied to concrete market options and the stability of the different approaches is analyzed. Finally, we consider a challenging example where convex spline interpolations lead to oscillations in the implied volatility and compare the spline collocation results with those obtained through arbitrage-free interpolation technique of Andreasen and Huge.

Arbitrage-free, B-spline, Implied volatility, Quantitative finance, Risk neutral density, Stochastic collocation
dx.doi.org/10.3390/risks7010030
Risks
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Le Floc’h, F.L, & Oosterlee, C.W. (2019). Model-free stochastic collocation for an arbitrage-free implied volatility, part II. Risks, 7(1), 1–23. doi:10.3390/risks7010030