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

Abstract: We propose a method to determine the expectation of the supremum of the price process in stochastic volatility models. It can be applied 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.