Imposing no‐arbitrage conditions in implied volatilities using constrained smoothing splines

Abstract

We apply constrained smoothing B-splines to the construction of arbitrage-free implied volatilities and derived measures. The constrained smoothing B-splines allows the imposition of the constraints of monotonicity and convexity given by the no-arbitrage conditions in the pricing function. We illustrate the methodology in the construction of implied volatilities and also in the construction of derived measures such as risk-neutral densities, showing that it can be used as an effective tool for general treatment of option prices.

Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints

“…Alternative kernel regression estimators are proposed by Birke and Pilz (2009) and Fan and Mancini (2009). As regards SNP techniques, a number of polynomial spline methods have been suggested to date: B-splines in Wang et al (2004), Laurini (2011), andCorlay (2013); smoothing splines in Yatchew and Härdle (2006), Monteiro et al (2008), and Fengler (2009); linear splines in Härdle and Hlávka (2009). Other SNP-type estimators are based on the Edgeworth expansion as in Jarrow and Rudd (1982), on Hermite polynomials as in Madan and Milne (1994) and Jondeau and Rockinger (2001), or on approximation methods, such as the positive convolution approximation as in Bondarenko (2003) and the nonparametric density mixtures as in Yuan (2009).…”

Foreign exchange rate volatility smiles and smirks

“…We leave a study of the no‐arbitrage version of the implied volatility for future work with reference to the related study by Hagan et al 27 In fact, many studies address arbitrage‐free volatility models. Among such studies, several studies concentrate on the arbitrage‐free interpolation of implied volatilities or equivalently of option prices 28‐32 . Grzelak and Oosterlee 33 present arbitrage‐free option prices based on the SABR model using stochastic collocation.…”

Arbitrage‐free call option surface construction using regression splines

“…Laurini [10] proposes a similar approach using constrained smoothing B-splines. Furthermore, Fengler and Hin [11] suggest a tensor-product B-spline estimator for the call price surface.…”