[Paper Review] On the value of being American

David Hobson, Anthony Neuberger|arXiv (Cornell University)|Apr 8, 2016

Stochastic processes and financial applications26 references5 citations

TL;DR

This paper establishes model-free upper bounds on the price of American options using only observed European call option prices, leveraging a semi-static hedging strategy that super-replicates the American claim under all consistent models. The key result is that the maximum model price equals the cost of the cheapest robust hedge, revealing that a significant portion of the American option's value stems from flexibility under model uncertainty.

ABSTRACT

The virtue of an American option is that it can be exercised at any time. This right is particularly valuable when there is model uncertainty. Yet almost all the extensive literature on American options assumes away model uncertainty. This paper quantifies the potential value of this flexibility by identifying the supremum on the price of an American option when no model is imposed on the data, but rather any model is required to be consistent with a family of European call prices. The bound is enforced by a hedging strategy involving these call options which is robust to model error.

Motivation & Objective

• To quantify the value of early exercise flexibility in American options when model uncertainty is present.
• To identify the supremum price of an American option without assuming a specific price process model.
• To construct a semi-static hedging strategy that super-replicates the American option payoff under all models consistent with observed European option prices.
• To demonstrate that standard model-based approaches underestimate the true value of American options due to restrictive assumptions on information flow and filtration.
• To show that the highest model price and the lowest hedging cost coincide, eliminating duality gap.

Proposed method

• The authors use a discrete-time, discrete-space framework to model the underlying asset price process with two regimes: constant before a jump time and stochastic after.
• They define a semi-static strategy that dynamically hedges in the underlying asset based on price level and exercise status, with static positions in European options.
• The strategy is constructed such that its cost equals the maximum possible model price of the American option across all consistent models.
• The approach relies on identifying a consistent model where the American option is exercised precisely at the time regime changes, maximizing model price.
• The solution is derived via linear programming, with the dual problem interpreted as a robust hedge, ensuring no duality gap.
• The method is extended to continuous time under weak assumptions, though the core insights are grounded in the discrete framework.

Experimental results

Research questions

• RQ1 What is the maximum possible price an American option can have when only European option prices are known and no model is imposed?
• RQ2 How can a robust hedging strategy be constructed that super-replicates the American option payoff under all models consistent with observed European option prices?
• RQ3 To what extent does model uncertainty contribute to the value of early exercise in American options?
• RQ4 Why do standard model-based approaches fail to capture the full value of American options?
• RQ5 Can the upper bound on the American option price be achieved by a semi-static strategy, and is there a duality between model price and hedging cost?

Key findings

• The upper bound on the American option price is equal to the cost of a semi-static hedging strategy that depends only on the underlying price level and whether the option has been exercised.
• The maximum model price is achieved in a two-regime model where the option is exercised at the moment the regime changes, reflecting the earliest resolution of model uncertainty.
• The duality gap between the model price and the hedging cost is zero, meaning the most expensive model price equals the cheapest super-replicating strategy cost.
• The value of the American option's early exercise feature is significantly underestimated in standard model-based approaches due to the assumption of a natural filtration and fixed information flow.
• The semi-static strategy is robust to model mis-specification and incurs lower transaction costs than dynamic hedging, as it requires trading only the underlying asset after time 0.
• The results remain robust under weak extensions to continuous time and state space, suggesting the discrete framework captures the essential economic intuition.