[Paper Review] Model-Independent Pricing of Asian Options via Optimal Martingale Transport

Florian Stebegg|arXiv (Cornell University)|Jan 1, 2014

Stochastic processes and financial applications27 references14 citations

TL;DR

This paper develops model-independent bounds for Asian option prices using optimal martingale transport theory. It provides geometric characterizations of extremal pricing models for discrete and continuous-time Asian options with one or two marginals, reducing the problem to optimal transport for cost functions like |x + y|, and establishes robust price ranges without assuming a specific stochastic model for the underlying asset.

ABSTRACT

Diese Arbeit beschäftigt sich mit der Berechnung von optimalen modellunabhängigen (robusten) Grenzen für den Preis einer Asiatischen Option mit diskreter beziehungsweise stetiger Durchschnittsbildung. Es werden geometrische Charakterisierungen der maximierenden und minimierenden Modelle zur Bepreisung von verschiedenen Typen von Asiatischen Optionen auf Prozessen in diskreter und stetiger Zeit hergeleitet. In diskreter Zeit wird das Problem auf die Lösung des Problems des optimalen Martingaltransports für die Kostenfunktion $|x+y|$ zurückgeführt. Im zeitstetigen Fall betrachten wir die Fälle mit einem bzw. zwei vorgegebenen Randverteilungen. Wir beschreiben die Pfadformen des maximierenden Modells für beide dieser Fälle, als auch die Pfadform des minimierenden Modells für den Fall mit einer gegebenen Randverteilung. Der Maximierer für das Problem mit zwei Randverteilungen wird auf das zeitdiskrete Problem zurückgeführt.

Motivation & Objective

• To address model risk in Asian option pricing by deriving robust, model-independent price bounds.
• To characterize the extremal martingale measures that yield the highest and lowest possible option prices under given marginal distributions.
• To extend optimal transport techniques from discrete to continuous-time settings for path-dependent options.
• To provide geometric solutions for optimal transport plans in the context of Asian options with one or two fixed marginals.
• To relate continuous-time problems to discrete-time counterparts through approximation and limit arguments.

Proposed method

• Formulates the model-independent pricing problem as a martingale optimal transport problem on continuous paths.
• Reduces the discrete-time problem to finding optimal martingale couplings for the cost function |x + y|.
• Uses Skorokhod embedding and duality in optimal transport to characterize extremal measures.
• Applies results from Galichon, Henry-Labordère, and Touzi on martingale optimal transport to exotic options.
• Employs pathwise stochastic integration and Lebesgue-Stieltjes integrals to define self-financing strategies in continuous time.
• Derives necessary and sufficient conditions for optimality via case analysis on the relative ordering of asset prices and transport plans.

Experimental results

Research questions

• RQ1 What are the model-independent upper and lower bounds for the price of an Asian option with discrete averaging?
• RQ2 How can the extremal martingale measures be geometrically characterized in discrete time for Asian options?
• RQ3 What is the structure of the optimal transport plan in continuous time with one or two given marginals?
• RQ4 How do the discrete-time results relate to the continuous-time case of Asian options with continuous averaging?
• RQ5 Can the optimal pricing model be explicitly described using transport theory for the cost function |x + y|?

Key findings

• The optimal martingale transport plan for discrete-time Asian options with two averaging points corresponds to minimizing or maximizing the expected value of |x + y| under the martingale constraint.
• In the continuous-time case with one marginal, the maximising model is characterised by a specific coupling that preserves the martingale property and extremalizes the average payoff.
• For two marginals in continuous time, the extremal models are related to the discrete-time problem with two marginals, via a limiting argument.
• The minimising model in the one-marginal case is explicitly characterised using optimal transport duality and geometric constraints on the joint distribution.
• The paper proves that the optimal transport plans are uniquely determined by the relative ordering of the underlying asset prices and the signs of the deviations from the mean.
• A complete classification of extremal transport plans (BTPs) is provided through case analysis, showing that optimality holds under specific inequalities involving price levels and transport weights.