Option Pricing without Price Dynamics: A Probabilistic Approach
  Employing probabilistic techniques we compute best possible upper and lower bounds on the price of an option on one or two assets with continuous piecewise linear payoff function based on prices of simple call options of possibly distinct maturities and the no-arbitrage condition, but without any assumption on the price dynamics of underlying assets. We show that the problem reduces to solving linear optimization problems that we explicitly characterize. We report numerical results that illustrate the effectiveness of the algorithms we develop. 
Dimitris Bertsimas, Natasha Bushueva