Normalization for Implied Volatility

Abstract

We study specific nonlinear transformations of the Black-Scholes implied volatility to show remarkable properties of the volatility surface. Model-free bounds on the implied volatility skew are given. Pricing formulas for the European options which are written in terms of the implied volatility are given. In particular, we prove elegant formulas for the fair strikes of the variance swap and the gamma swap.

Log-Modulated Rough Stochastic Volatility Models

“…Hyper-rough volatility, in a sense analogous to a H < 0 model, has also been considered [24,22], but also in this case spot volatility is not dened. The extreme T −1/2 speed of explosion for the skew expected in the H → 0 limit has been shown to be a model-free bound [26,12], and is reached under local volatility through a volatility function with a singularity ATM [30], but this poses the problem of time-consistency (see also [11]). To the best of our knowledge, this extreme behavior of the skew has not been shown for any (time-consistent) stochastic volatility model, where the volatility is a proper process.…”

Smiles in delta

“…This follows from the fact that d 2 (−∞) = −N −1 (P (S T = 0)) as shown in proposition 2.4 of [1]. Now, it holds…”

“…Since l is increasing, the latter condition is equivalent to l 1 2 < 0. When δ > 1 2 , then N −1 (δ) is positive and both the solutions could be valid. Under the requirement l(δ) ≤ N −1 (δ) 2…”

Volatility has to be rough

“…is given in [14] and shown to be sharp in [29]. This extreme skew corresponds to the H-power law with H = 0.…”