Functional Co-monotony of Processes with Applications to Peacocks and Barrier Options

We show that several general classes of stochastic processes satisfy a functional co-monotony principle, including processes with independent increments, Brownian bridge, Brownian diffusions, Liouville processes, fractional Brownian motion. As a first application, we recover and extend some recent results about peacock processes obtained by Hirsch et al. in (Peacocks and Associated Martingales, with Explicit Constructions, Bocconi & Springer, 2011, 430p) [see also (Peacocks sous l’hypothèse de monotonie conditionnelle et caractérisation des 2-martingales en termes de peacoks, thèse de l’Université de Lorraine, 2012, 169p)] which were themselves motivated by a former work of Carr et al. in (Finance Res. Lett. 5:162–171, 2008) about the sensitivities of Asian options with respect to their volatility and residual maturity (seniority). We also derive semi-universal bounds for various barrier options.