Abstract
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.
Highlights
Financial markets provide option prices for a discrete set of strike prices and maturity dates.In order to price over-the-counter vanilla options with different strikes, or to hedge complex derivatives with vanilla options, it is useful to have a continuous arbitrage-free representation of the option prices, or equivalently of their implied volatilities
Breeden and Litzenberger (1978) have shown that any path-independent claim can be valued by integrating over the probability density implied by market option prices
The calibration consists firstly in computing an arbitrage-free set of call option prices from the market mid quotes according to Section 6.2, secondly in computing the B-spline initial guess following Section 4, and thirdly in minimizing the error measure represented by Equation (18) with a Levenberg–Marquardt minimizer
Summary
Financial markets provide option prices for a discrete set of strike prices and maturity dates. A rudimentary, but popular, representation is to interpolate market implied volatilities with a cubic spline across option strikes This may not be arbitrage-free as it does not preserve the convexity of option prices in general. The output of their technique is a discrete set of option prices, which, while relatively dense, must still be interpolated carefully to obtain the price of options whose strike falls in between nodes This paper considers another approach, based on the stochastic collocation technique of Grzelak and Oosterlee (2017). Instead of collocating on a polynomial as in Le Floc’h and Oosterlee (2019), we explore various ways to use a monotonic spline, including B-spline parametrizations This allows for a richer representation, with as many parameters as there are market option strikes.
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