[Paper Review] Second-order BSDEs with general reflection and game options under uncertainty

Anis Matoussi, Lambert Piozin+1 more|arXiv (Cornell University)|Dec 3, 2012

Stochastic processes and financial applicationsEconomics, Econometrics and Finance50 references22 citations

TL;DR

This paper establishes the existence and uniqueness of doubly reflected second-order backward stochastic differential equations (2BSDEs) with two general reflecting barriers under regularity and separation conditions on the barriers. It links these 2BSDEs to uncertain Dynkin games and provides a new framework for pricing American game options under volatility uncertainty, extending classical DRBSDE theory to model uncertainty in financial markets.

ABSTRACT

The aim of this paper is twofold. First, we extend the results of [33] concerning the existence and uniqueness of second-order reflected 2BSDEs to the case of two obstacles. Under some regularity assumptions on one of the barriers, similar to the ones in [10], and when the two barriers are completely separated, we provide a complete wellposedness theory for doubly reflected second-order BSDEs. We also show that these objects are related to non-standard optimal stopping games, thus generalizing the connection between DRBSDEs and Dynkin games first proved by Cvitanic and Karatzas [11]. More precisely, we show under a technical assumption that the second order DRBSDEs provide solutions of what we call uncertain Dynkin games and that they also allow us to obtain super and subhedging prices for American game options (also called Israeli options) in financial markets with volatility uncertainty

Motivation & Objective

• To extend the wellposedness theory of second-order reflected BSDEs from one to two reflecting barriers.
• To establish a connection between doubly reflected second-order BSDEs and optimal stopping games under model uncertainty.
• To provide a new approach for pricing American game options (Israeli options) in financial markets with volatility uncertainty.
• To generalize the classical connection between DRBSDEs and Dynkin games to the second-order setting under uncertainty.
• To develop a theoretical framework for superhedging and subhedging in incomplete markets with volatility ambiguity.

Proposed method

• Introduces doubly reflected second-order BSDEs with two barriers L and U, where the solution Y is constrained to stay between them.
• Imposes regularity conditions on one barrier (similar to those in [10]) and assumes complete separation between the two barriers.
• Uses a penalization method and a priori estimates to prove existence and uniqueness of solutions under the given assumptions.
• Applies a comparison theorem for g-supermartingales and constructs a weak limit for the penalized sequence of solutions.
• Establishes a link between the 2BSDE solution and the value process of an uncertain Dynkin game via a technical assumption on the generator.
• Applies the theory to derive superhedging and subhedging prices for American game options under volatility uncertainty.

Experimental results

Research questions

• RQ1 Under what conditions does a doubly reflected second-order BSDE with two barriers admit a unique solution?
• RQ2 How can second-order reflected BSDEs be connected to optimal stopping games under model uncertainty?
• RQ3 Can the theory of second-order BSDEs be used to price American game options when volatility is uncertain?
• RQ4 What is the role of the Mokobodski-type condition in the context of second-order reflected BSDEs?
• RQ5 How do the solutions of these 2BSDEs relate to the value functions of uncertain Dynkin games?

Key findings

• Under regularity and separation assumptions on the two barriers, the paper establishes a complete wellposedness theory for doubly reflected second-order BSDEs.
• The solution of the 2BSDE is shown to represent the value process of an uncertain Dynkin game under a technical assumption on the generator.
• The 2BSDE framework allows for the derivation of superhedging and subhedging prices for American game options in markets with volatility uncertainty.
• The existence of a solution is proven via a penalization scheme and a priori estimates, with convergence established through weak limits.
• The time regularity of the solution is analyzed via a downcrossing inequality for g-supermartingales, extending classical results to the second-order setting.
• The comparison theorem for g-supermartingales is used to establish the minimal action of the local time processes K+ and K−, ensuring the solution stays within the barriers.