Implied volatility in strict local martingale models

TL;DR

This paper analyzes how implied volatility patterns in asset models can reveal the presence of strict local martingales, which are associated with price bubbles, through asymptotic expansions of the volatility smile.

Contribution

It provides the first asymptotic expansion for implied volatility in strict local martingale models, enabling detection of price bubbles from market data.

Findings

Asymptotic expansion characterizes the right wing of implied volatility in strict local martingale models.

Strict local martingale property can be inferred from implied volatility asymptotics.

Duality methods relate right-wing expansions to left-wing behaviors in models with mass at zero.

Abstract

We consider implied volatilities in asset pricing models, where the discounted underlying is a strict local martingale under the pricing measure. Our main result gives an asymptotic expansion of the right wing of the implied volatility smile and shows that the strict local martingale property can be determined from this expansion. This result complements the well-known asymptotic results of Lee and Benaim-Friz, which apply only to true martingales. This also shows that `price bubbles' in the sense of strict local martingale behaviour can in principle be detected by an analysis of implied volatility. Finally we relate our results to left-wing expansions of implied volatilities in models with mass at zero by a duality method based on an absolutely continuous measure change.