• convex order
• exotic options
• left-curtain transport plan
• martingale
• optimal transport
• potential functions
• shadow embedding

In this thesis we discuss results obtained from the martingale transportation problem as a tool to solve financial problems.
We know that there exist some financial models that allow us to price vanilla options: for instance, the Black-Scholes' model give formulas to calculate the value of call and put options. When we consider options as simple as vanilla options, the model give us a unique value for these prices. In the last years several models which allow to capture the risk of exotic options have emerged. These models depend on various parameters which can be calibrated more or less accurately to market price of liquid options (such as call/put options). This calibration procedure does not uniquely set the dynamics of forward prices which are only required to be (local) martingales according to the no-arbitrage framework. This could lead to a wide range of various prices of a given exotic option when evaluated using different models calibrated on the same market data.
What we do in this thesis is to review some results from the mathematical literature that determinate lower and upper bounds for prices of exotic options produced by models calibrated on the same market data, and therefore with similar marginals. \\
More precisely, we fix an exotic option depending only on the value of a single asset $S$ at discrete times $t_1<t_2$ and we denote by $\Phi(S_1,S_2)$ its payoff, where $\Phi$ is supposed to be some measurable funtion. In the no-arbitrage framework, the standard approach is to postulate a model, that is, a probability measure $\mathbb{Q}$ on $\R^2$ under which the process $(S_i)_{i=1}^2$
$$\mappa{S_i}{\R^2}{\R}, \quad S_i(s_1, s_2)=s_i, \quad i=1,2,$$
is required to be a discrete martingale on its own filtration.
By $S_0=s_0$ we denote the current spot price. \\
The fair value of $\Phi$ is then given as the expetation of the payoff $\mathbb{E}_{\mathbb{Q}}[\Phi].$ \\
Additionally we impose that our model is calibrated to a continuum of call options, which is equivalent to prescribe probability measures $\mu_1,\mu_2$ on the real line such that the one dimensional marginals of $\mathbb{Q}$ satisfy
$$\mathbb{Q}^i=Law_{S_i}=\mu_i \qquad i=1,2.$$
Let $\Pi_M(\mu_1,\mu_2)$ be the set of all martingale measures $\mathbb{Q}$ on $\R^2$ having marginals $\mathbb{Q}^1=\mu_1,\mathbb{Q}^2=\mu_2$. \\
We concentrate on the lower bound of the price and consider the problem
\begin{equation}\label{P}
P=\inf \set{\mathbb{E}_{\mathbb{Q}}[\Phi(s_1,s_2)] \, :\, \mathbb{Q}\in \Pi_M(\mu_1,\mu_2)}.
\end{equation}
Instead, the upper bound can be found analyzing the same problem but with $-\Phi$ instead of $\Phi$. \\
We show that, in the same way of the classical optimal transport, the set $\Pi_M(\mu_1,\mu_2)$ is a (weakly) compact set, and then, if we require that the "cost" function $\Phi(s_1,s_2)$ is lower semi-continuous, this problem has a minimum, provided the set $\Pi_M(\mu_1,\mu_2)$ is nonempty.
Then we review a necessary and sufficient condition (due to Strassen) for this set to be nonempty: the measure $\mu_1$ and $\mu_2$ must be in convex order.
We then consider the dual problem of \eqref{P}, which has an interesting financial interpretation in terms of hedging strategies. Since the dual problem does not exhibit nice compactness properties and hence the dual extremizer is not necessarily attained, we review conditions on $\Phi(s_1,s_2)$ for the supremum to be attained, following the approach in the article "Model-independent bounds for option prices: a mass transport approach" by M. Beiglbock, P. Henry-Labordère, F. Penkner.
Finally we construct an optimizer for the problem \eqref{P} for some types of $\Phi$, called the left-curtain transport plan, introducing the shadow embedding of a measure, following the approach in the article "On a problem of optimal transport under marginal martingale constraints" by M. Beiglbock, N. Juillet.