Arbitrage-free smoothing of the implied volatility surface

Abstract

The pricing accuracy and pricing performance of local volatility models crucially depends on absence of arbitrage in the implied volatility surface: an input implied volatility surface that is not arbitrage-free invariably results in negative transition probabilities and/ or negative local volatilities, and ultimately, into mispricings. The common smoothing algorithms of the implied volatility surface cannot guarantee the absence arbitrage. Here, we propose an approach for smoothing the implied volatility smile in an arbitrage-free way. Our methodology is simple to implement, computationally cheap and builds on the well-founded theory of natural smoothing splines under suitable shape constraints. Unlike other methods, our approach also works when input data are scarce and not arbitrage-free. Thus, it can easily be integrated into standard local volatility pricers.2

Novel no-arbitrage conditions for options written on defaultable assets

“…Although these two conditions are not always required for the construction of an arbitrage-free option surface, they are commonly stated in the literature (see for example Aït-Sahalia and Duarte (2003), Fengler (2009), Monnier (2013) and Orosi (2011)). …”

“…Moreover, although interpolation performed in the implied volatility space has several advantages (see for example Figlewski (2009) andOrosi (2012)), the resulting call option surface is not arbitrage-free. To generate a suitable call option surface from an implied volatility surface, Fengler (2009) and Orosi (2014a, b) employ arbitrage-free interpolants.…”

Improved lower bounds of call options written on defaultable assets

“…Some important examples of financial engineering applications are the pricing illiquid exotic derivatives with arbitrary payoffs, copula-based pricing of multi-asset products, and reconstructing a local volatility surface. For example, Monteiro et al (2011) show that implied RND can be used to accurately price Europeanstyle binary options, Cherubini and Luciano (2002) use implied RND to price bivariate equity options and Fengler (2009) uses an interpolant to recover the local volatility surface. Figlewski (2009) points out that interpolation is typically performed in the implied volatility space, which involves fitting a spline or a loworder polynomial to the available data.…”

Generalized Arbitrage-Free SVI Volatility Surfaces

“…In particular, given a set of observed data (say, European Calls and Puts for different strikes and maturities), it is of fundamental importance to determine a methodology ensuring that both interpolation and extrapolation of this data are also arbitrage-free. Such approaches have been carried out, for instance, in [4,6,22]. Several parameterizations of the implied volatility surface have now become popular, in particular [8,13,15], albeit not ensuring absence of arbitrage.…”