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

## TL;DR

This paper establishes a model-free method to compute consistent upper bounds for exotic options based on finite call prices, analyzing convergence and providing numerical illustrations.

## Contribution

It introduces a novel approach to determine upper price bounds for exotic options using a finite set of call prices without relying on specific models.

## Key findings

- Derived worst-case marginal pricing measures for directionally convex payoffs
- Analyzed the convergence rate of upper price bounds as the number of observed prices increases
- Provided numerical examples demonstrating the theoretical results

## Abstract

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.