Unbounded Volatility in the Uncertain Volatililty Model

Abstract

We work in the Uncertain Volatility Model setting of Avellaneda, Levy, Paras [1] and Lyons [10] (cf. also [11]). We first look at European options in a market with no interest rate and focus on the extreme case where the volatility has a lower bound but no upper bound. We show that the smallest riskless selling price of the claim is the Black-Scholes price (at volatility given by the lower bound) of an option with payoff the smallest concave function above the initial payoff. We next extend our results to the case with interest rate.

Keywords: European options, Hamilton-Jacobi-Bellman Equation, Stochastic Control, Superstrategies

JEL Classification: G10, G12, G13