Robust Fundamental Theorem for Continuous Processes

Abstract

We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family P of possible physical measures. A robust notion NA 1 (P) of no-arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot be superhedged for free by using simple trading strategies. Our first main result is a version of the fundamental theorem of asset pricing: NA 1 (P) holds if and only if every P ∈ P admits a martingale measure which is equivalent up to a certain lifetime. The second main result provides the existence of optimal superhedging strategies for general contingent claims and a representation of the superhedging price in terms of martingale measures.

Valuation and Parities for Exchange Options

“…However, as stated in Theorem 2.1 below, the structure of the filtered probability space described in subsection 1.1 allows for such a construction under Assumption 1.2. A proof of Theorem 2.1 in this exact setting appears in [BBKN14]; of course, results of a similar nature have appeared previously-see, for example, [Föl72], [Mey72], [DS95], [PP10], [Ruf13], and [PR14].…”

On the Monotonicity Principle of Optimal Skorokhod Embedding Problem

“…We also refer to the subsequent literature on martingale optimal transport by [6,7,8,14,15,19,20,21,23,25], etc.…”

A Risk-Neutral Equilibrium Leading to Uncertain Volatility Pricing

“…The non-Markovian version of this pricing operator is known as the G-expectation [50,51]. More recently, a rich literature considering a variety of hedging instruments and underlying models has emerged; see, among many others, [1,3,9,10,23,45] for models in discrete time and [8,13,15,16,18,19,22,25,26,27,29,31,32,43,46] for continuous-time models. We refer to [30,48] for surveys.…”