Convergence of Heston to SVI Proposed Extensions: Rational & Conjecture for the Convergence of Extended Heston to the Implied Volatility Surface Parametrization

Abstract

A mathematical and a market argument on the sub-linearity of the wings for the implied variance is given. Gatheral stochastic volatility inspired (SVI) parameterization claim to have two key properties that have led to its subsequent popularity with practitioners is exposed. Namely the linearity in the log-strike k as |k| → ∞ consistent with Roger Lees moment formula as well as its connection to the Heston model are examined more in details. Though correct, the former point led to the model subsequent decommission in the industry. We explain this apparent contradiction by pointing to a mathematically convenient chosen factor in the Heston model which we expose and consequently introduce couple candidates: the p-Heston and the Inferred Correlation models instead. The link between the latter and the SVI being broken, we propose a connection to the Implied Volatility surface Parametrisation (IVP) recently introduced and propose a conjecture between a mirror convergence towards these models using the parallel between the traditional Heston to SVI convergence.