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Stochastic PDEs for large portfolios with general mean-reverting volatility processes

Ben Hambly, Nikolaos Kolliopoulos

Abstract

We consider a structural stochastic volatility model for the loss from a large portfolio of credit risky assets. Both the asset value and the volatility processes are correlated through systemic Brownian motions, with default determined by the asset value reaching a lower boundary. We prove that if our volatility models are picked from a class of mean-reverting diffusions, the system converges as the portfolio becomes large and, when the vol-of-vol function satisfies certain regularity and boundedness conditions, the limit of the empirical measure process has a density given in terms of a solution to a stochastic initial-boundary value problem on a half-space. The problem is defined in a special weighted Sobolev space. Regularity results are established for solutions to this problem, and then we show that there exists a unique solution. In contrast to the CIR volatility setting covered by the existing literature, our results hold even when the systemic Brownian motions are taken to be correlated.

Stochastic PDEs for large portfolios with general mean-reverting volatility processes

Abstract

We consider a structural stochastic volatility model for the loss from a large portfolio of credit risky assets. Both the asset value and the volatility processes are correlated through systemic Brownian motions, with default determined by the asset value reaching a lower boundary. We prove that if our volatility models are picked from a class of mean-reverting diffusions, the system converges as the portfolio becomes large and, when the vol-of-vol function satisfies certain regularity and boundedness conditions, the limit of the empirical measure process has a density given in terms of a solution to a stochastic initial-boundary value problem on a half-space. The problem is defined in a special weighted Sobolev space. Regularity results are established for solutions to this problem, and then we show that there exists a unique solution. In contrast to the CIR volatility setting covered by the existing literature, our results hold even when the systemic Brownian motions are taken to be correlated.

Paper Structure

This paper contains 12 sections, 16 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Convergence of the empirical measure processes and the limiting SPDE
  3. Analysis of the volatility
  4. The two-dimensional density and the initial-boundary value problem
  5. Higher regularity of initial-boundary value problem solutions
  6. Uniqueness of solutions
  7. Proofs of certain important results
  8. Proof of Theorem \ref{['thm:2.6']}
  9. Proof of Theorem \ref{['thm:3.1']}
  10. Proof of Lemma \ref{['lem:5.3']}
  11. Proof of Lemma \ref{['lem:5.4']}
  12. Proof of Lemma \ref{['mainid']}

Key Result

Theorem 2.1

For each $N\in\mathbb{N}$ and any $t,\,s\geq0$, consider the random measure given by The sequence $v_{3,t,s}^{N}$ of three-dimensional empirical measures converges weakly to some measure $v_{3,t,s}$ for all $t,\,s\geq0$, $\mathbb{P}$-almost surely. Moreover, the measure-valued process $\{v_{3,t,s}:\,t,\,s\geq0\}$ is $\mathbb{P}$-almost surely continuous in both $t$ and $s$ under the

Theorems & Definitions (27)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 3.2
  • Theorem 3.3
  • Theorem 4.1
  • proof : Proof (outline)
  • ...and 17 more