  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.
