Narrow Results

For $\mu$ and $\nu$ two probability measures on the real line such that $\mu$ is smaller than $\nu$ in the convex order, this property is in general not preserved at the level of the empirical measures ${\mu }_I=\frac{1}{I}{\sum }_{i=1}^I{\delta }_{X_i}$ and ${\nu }_J=\frac{1}{J}{\sum }_{j=1}^J{\delta }_{Y_j}$, where ${(X_i)}_{1\le i\le I}$ (resp., ${(Y_j)}_{1\le j\le J}$) are independent and identically distributed according to $\mu$ (resp., $\nu$). We investigate modifications of ${\mu }_I$ (resp., ${\nu }_J$) smaller than ${\nu }_J$ (resp., greater than ${\mu }_I$) in the convex order and weakly converging to $\mu$ (resp., $\nu$) as $I,J\to \infty$. According to Kertz & Rösler(1992), the set of probability measures on the real line with a finite first order moment is a complete lattice for the increasing and the decreasing convex orders. For $\mu$ and $\nu$ in this set, this enables us to define a probability measure $\mu \vee \nu$ (resp., $\mu \wedge \nu$) greater than $\mu$ (resp., smaller than $\nu$) in the convex order. We give efficient algorithms permitting to compute $\mu \vee \nu$ and $\mu \wedge \nu$ (and therefore ${\mu }_I\vee {\nu }_J$ and ${\mu }_I\wedge {\nu }_J$) when $\mu$ and $\nu$ have finite supports. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate martingale optimal transport problems and in particular to calculate robust option price bounds.

We consider the problem of finding a consistent upper price bound for exotic options whose payoff depends on the stock price at two different predetermined time points (e.g. Asian option), given a finite number of observed call prices for these maturities. A model-free approach is used, only taking into account that the (discounted) stock price process is a martingale under the no-arbitrage condition. In case the payoff is directionally convex we obtain the worst case marginal pricing measures. The speed of convergence of the upper price bound is determined when the number of observed stock prices increases. We illustrate our findings with some numerical computations.