Black-Scholes in a CEV random environment

TL;DR

This paper introduces a novel approach to modeling small-maturity implied volatility smiles by randomizing the Black-Scholes variance with a CEV-generated distribution, capturing a range of explosion behaviors.

Contribution

It proposes a new method of randomizing Black-Scholes volatility using CEV distributions to better model small-maturity implied volatility smiles.

Findings

Captures a range of explosion rates similar to exponential Lévy and fractional stochastic volatility models.

Provides a flexible framework to modulate implied volatility smile steepness.

Offers an alternative to jump and rough volatility models for small-maturity options.

Abstract

Classical (It\^o diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential L\'evy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see \cite{Tank} for an overview), and more recently rough volatility models \cite{AlosLeon, Fukasawa}. We suggest here a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential L\'evy models and fractional stochastic volatility models.