Techniques and approaches for pricing American option

This thesis is concerned with the theory of optimal stopping and martingale optimal transport, and its applications to the pricing and hedging of American-type contingent claims. In the first chapter we revisit the classical optimal stopping problem in continuous time and explore a delicate connection between semimartingale and Markovian formulations of the problem. More specifically, in the Markovian setting we are motivated by the question of whether the value function, corresponding to the optimal stopping problem, belongs to a certain class of functions (i.e. the domain of the extended or martingale generator) associated to the underlying Markov process. We show that the answer follows naturally from the fundamental property of the value process in a more general, semimartingale setting. We investigate applications of these results to the dual formulation of the optimal stopping problem and the classical smooth fit principle. The goal of the second chapter is to study the problem in martingale optimal transport, which is to move mass from a starting law (on R) to a terminal law (on R) in a way which respects the martingale property. One method is the `shadow embedding' of Beiglbock and Juillet [10]. Using the potential functions of the starting and terminal laws, we show how to explicitly construct the associated shadow measure. We also discuss the properties of the left-curtain martingale coupling, which is a coupling that arises (via shadow measure) from a certain parametrisation of the marginals. This coupling turns out to be optimal for the novel optimal martingale transport with stopping problem studied in the third chapter. The third chapter studies the problem of finding the highest robust or model independent price of the American put option given the prices of liquid European options, in a simple (but non-trivial) two time period setting. Combining ideas from the theory of optimal stopping and martingale optimal transport, we find, under some simplifying but still general conditions on the given data, the optimal model and the optimal stopping time. We also explicitly calculate the cheapest superhedging trading strategy. In the fourth chapter our goal is to find a specific geometric description of the left-curtain martingale coupling, which can be viewed as a martingale counterpart of the monotone Hoeffding-Frechet coupling in the classical optimal transport. While this is of independent interest, we also show that this generalised martingale coupling maximises the price of the American put option (studied in the third chapter under some simplifying assumptions).