Arbitrage-free pricing of American options in nonlinear markets

TL;DR

This paper extends the nonlinear arbitrage-free pricing framework to American options, providing detailed valuation analysis and a BSDE approach for explicit pricing and hedging in nonlinear markets.

Contribution

It advances the understanding of American option pricing in nonlinear markets by extending European option results and employing reflected BSDEs for explicit solutions.

Findings

Extended valuation results for American options in nonlinear markets

Developed a BSDE-based approach for explicit pricing and hedging

Provided conditions for desirable properties of solutions

Abstract

We re-examine and extend the findings from the recent paper by Dumitrescu, Quenez and Sulem (2018) who studied American and game options in a particular market model using the nonlinear arbitrage-free pricing approach developed in El Karoui and Quenez (1997). In the first part, we provide a detailed study of unilateral valuation problems for the two counterparties in an American-style contract within the framework of a general nonlinear market. We extend results from Bielecki and Rutkowski (2015) and Bielecki, Cialenco and Rutkowski (2018) who examined the case of a European-style contract. In the second part, we present a BSDE approach, which is used to establish more explicit pricing, hedging and exercising results when solutions to reflected BSDEs have additional desirable properties.