Optimization based option pricing bounds via piecewise polynomial super- and sub-martingales

In this paper we first prove sufficient conditions for a continuous function of a diffusion process to be a super- or sub-martingale. This result is then used to create piecewise polynomial super- and sub-martingale bounds on option prices via a polynomial optimization problem. The polynomial optimization problem is solved under a sum-of-squares paradigm and thus uses semi-definite programming. The results are tested on a Black-Scholes example where a piecewise polynomial function of degree four in both the stock value and time is used to compute upper and lower bounds.