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We consider the problem of finding a consistent upper price bound for exoticoptions whose payoff depends on the stock price at two different predeterminedtime points (e.g. Asian option), given a finite number of observed call pricesfor these maturities. A model-free approach is used, only taking into accountthat the (discounted) stock price process is a martingale under theno-arbitrage condition. In case the payoff is directionally convex we obtainthe worst case marginal pricing measures. The speed of convergence of the upperprice bound is determined when the number of observed stock prices increases.We illustrate our findings with some numerical computations.

Consistent upper price bounds for exotic options given a finite number of call prices and their convergence