On the integrability of the supremum of stochastic volatility models and other martingales

TL;DR

This paper introduces a method to bound the expectation of the supremum in stochastic volatility models, facilitating American option pricing and exploring martingales with non-integrable supremum.

Contribution

It provides a novel bounding technique for the supremum in stochastic volatility models and constructs martingales with non-integrable supremum.

Findings

Bounded the supremum expectation in rough Bergomi model

Established the existence of optimal stopping times for bounded payoffs

Constructed martingales with non-integrable supremum

Abstract

We propose a method to bound the expectation of the supremum of the price process in stochastic volatility models. It can be applied, for example, to the rough Bergomi model, avoiding the need to discuss finiteness of higher moments. Our motivation stems from the theory of American option pricing, as an integrable supremum implies the existence of an optimal stopping time for any linearly bounded payoff. Moreover, we survey the literature on martingales with non-integrable supremum, and give a new construction that yields uniformly integrable martingales with this property.