We consider two inverse problems motivated by questions in mathematical finance. In the first two chapters (Part 1) we recover processes consistent with given perpetual American option prices. In the third and fourth chapters (Part 2) we construct model-independent bounds for prices of contracts based on the realized variance of an asset price process. The two parts are linked by the question of how to recover information about asset price dynamics from option prices: in part one we assume knowledge of perpetual American option prices while in the second part we will assume knowledge of European call and put option prices. Mathematically, the first part of the thesis presents a framework for constructing generalised diffusions consistent with optimal stopping values. The second part aims at constructing bounds for path-dependent functionals of martingales given their terminal distribution.