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

Miryana Grigorova, James Wheeldon

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.

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

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.
Paper Structure (21 sections, 12 theorems, 153 equations)

This paper contains 21 sections, 12 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. The Underlying Probability Setup
  3. BSDEs with Multiple Default Jumps
  4. BSDE with Multiple Default Jumps: Properties
  5. A Priori Estimates for BSDEs with Multiple Default Jumps
  6. Existence and Uniqueness for BSDEs with Multiple Default Jumps
  7. lambda-p-Linear BSDEs with Multiple Default Jumps
  8. Comparison Theorems for BSDEs with Multiple Default Jumps
  9. Pricing of European Options in Markets with Multiple Defaults
  10. Pricing in a Linear Financial Market with two Defaultable Risky Assets
  11. The Case Where $D$ is Predictable
  12. The Case Where $D$ is Optional
  13. Change of measure
  14. The Case Where $D$ is Predictable
  15. The Case Where $D$ is Optional
  16. ...and 6 more sections

Key Result

Theorem 2.1

For any $(\mathbb{P}, \mathbb{G})$-square-integrable martingale $(m_t)_{t\in[0,T]}$ there exist unique $\mathbb{G}$-predictable processes $z\in\mathcal{H}_T^2$ and $k^i\in\mathcal{H}_{\lambda^i,T}^2$ for all $i\in\{1,\ldots,p\}$, such that the following martingale representation property holds,

Theorems & Definitions (39)

  • Theorem 2.1: Martingale Representation Property
  • Definition 2.2: $\lambda^{(p)}$-Admissible Driver
  • Remark 2.3
  • Definition 2.4: BSDE with a $\lambda^{(p)}$-Admissible Driver
  • Definition 2.5: BSDE with a Generalized $\lambda^{(p)}$-Admissible Driver
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 29 more