Generalized Arbitrage-Free SVI Volatility Surfaces

Abstract

Abstract. In this paper we propose a generalization of the recent work by Gatheral and Jacquier [J. Gatheral and A. Jacquier, Quant. Finance, 14 (2014) 1. Introduction. European option prices are usually quoted in terms of the corresponding implied volatility, and over the last decade a large number of papers (both from practitioners and academics) has focused on understanding its behavior and characteristics. The most important directions have been toward (i) understanding the behavior of the implied volatility in a given model [1, 2, 7, 12] and (ii) deciphering its behavior in a model-independent way, as in [17,21,20]. These results have provided us with a set of tools and methods to check whether a given parameterization is free of arbitrage or not. 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.Recently, Gatheral and Jacquier [10] proposed a new class of implied volatility parameterization, based on the previous works by Gatheral [8]. In particular, they provide explicit sufficient and-in a certain sense-almost necessary conditions ensuring that such a surface is free of arbitrage. We shall recall the exact definition of arbitrage and see that it can be decomposed into two elements: butterfly arbitrage and calendar spread arbitrage. This new class depends on the maturity and can hence be used to model the whole volatility surface,

Perturbation analysis of sub/super hedging problems

“…Absence of arbitrage can equivalently be stated as conditions on the shape of the total implied variance as shown in Gatheral and Jacquier (2014), Guo et al. (2016a). In particular under proportional dividends, absence of Calendar Spread arbitrage is equivalent to for all and (Gatheral & Jacquier, 2014, Lemma 2.1).…”

Perturbation Analysis of Sub/Super Hedging Problems

“…is a positive distribution, where ∂ kk w(·, ·) is defined in the distributional sense. This condition in turn is equivalent to the Call price function being convex [37,Proposition 4.8]. Assumption 3.12 ensures that ∂ t w(k, t) is well-defined for all t > 0 and ∂ k w(k, t) can be taken to be right of left derivative at k if w is not differentiable there.…”

“…Absence of static arbitrage can equivalently be stated as conditions on the shape of the total implied variance as shown in [35,37]. In particular under proportional dividends, absence of Calendar Spread arbitrage is equivalent to ∂ t w(k, t) ≥ 0 for all k ∈ R and t > 0 [35,Lemma 2.1].…”

Explicit no arbitrage domain for sub-SVIs via reparametrization

“…Up to now and to the best of our knowledge, the only known volatility model (meaning: a formula for the implied volatility) with an explicit no arbitrage domain was the SSVI slice, with the (restrictive) conditions obtained by Gatheral and Jacquier ([3]), and extended to the characterization of the full no arbitrage domain in the decorrelated case in [4]. To this extent, the present work is a big leap forward, since we obtain 3 new families: the Vanishing Downward/Upward one, the Symmetric one, and the correlated SSVI, with explicit no arbitrage domains (a single-variable boundary function has to be computed numerically for SSVI).…”