European Options in Market Models with Multiple Defaults: the BSDE approach

TL;DR

This paper develops a BSDE framework for pricing and hedging European options in complex market models with multiple defaults, providing existence, uniqueness, and explicit solutions under various conditions.

Contribution

It introduces a generalized BSDE approach for multiple defaults, deriving explicit formulas and applying them to complex market scenarios with defaultable assets.

Findings

Established existence and uniqueness of solutions.

Derived explicit formulas for linear BSDEs.

Applied methods to real-world defaultable asset markets.

Abstract

We study non-linear Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and p default martingales. The driver of the BSDE with multiple default jumps can take a generalized form involving an optional finite variation process. We first show existence and uniqueness. We then establish comparison and strict comparison results for these BSDEs, under a suitable assumption on the driver. In the case of a linear driver, we derive an explicit formula for the first component of the BSDE using an adjoint exponential semimartingale. The representation depends on whether the finite variation process is predictable or only optional. We apply our results to the problem of pricing and hedging a European option in a linear complete market with two defaultable assets and in a non-linear complete market with p defaultable assets. Two examples of the latter market model are provided: an example where the seller of the option is a large investor influencing the probability of default of a single asset and an example where the large seller's strategy affects the default probabilities of all p assets.