The idea of investigating the relation of option and stock prices based just on the no-arbitrage assumption, but without assuming any model for the underlying price dynamics, has a long history in the financial economics literature. We introduce convex and, in particular semidefinite optimization methods, duality, and complexity theory to shed new light on this relation. For die single stock problem, given moments of the prices of the underlying assets, we show that we can find best-possible bounds on option prices with general payoff functions efficiently, either algorithmically (solving a semidefinite optimization problem) or in closed form. Conversely, given observable option prices, we provide best-possible bounds on moments of the prices of the underlying assets, as well as on the prices of other options on the same asset by solving linear optimization problems. For options that are affected by multiple stocks either directly (the payoff of the option depends on multiple stocks) or indirectly (we have information on correlations between stock prices), we find nonoptimal bounds using convex optimization methods. However, we show that it is NP-hard to find best possible bounds in multiple dimensions, We extend our results to incorporate transactions costs.
