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Mild to classical solutions for XVA equations under stochastic volatility

Damiano Brigo, Federico Graceffa, Alexander Kalinin

Abstract

We extend the valuation of contingent claims in presence of default, collateral and funding to a random functional setting and characterise pre-default value processes by martingales. Pre-default value semimartingales can also be described by BSDEs with random path-dependent coefficients and martingales as drivers. En route, we generalise previous settings by relaxing conditions on the available market information, allowing for an arbitrary default-free filtration and constructing a broad class of default times. Moreover, under stochastic volatility, we characterise pre-default value processes via mild solutions to parabolic semilinear PDEs and give sufficient conditions for mild solutions to exist uniquely and to be classical.

Mild to classical solutions for XVA equations under stochastic volatility

Abstract

We extend the valuation of contingent claims in presence of default, collateral and funding to a random functional setting and characterise pre-default value processes by martingales. Pre-default value semimartingales can also be described by BSDEs with random path-dependent coefficients and martingales as drivers. En route, we generalise previous settings by relaxing conditions on the available market information, allowing for an arbitrary default-free filtration and constructing a broad class of default times. Moreover, under stochastic volatility, we characterise pre-default value processes via mild solutions to parabolic semilinear PDEs and give sufficient conditions for mild solutions to exist uniquely and to be classical.
Paper Structure (19 sections, 20 theorems, 157 equations)

This paper contains 19 sections, 20 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and basic concepts
  4. Representations of conditional expectations
  5. Construction of conditionally independent hitting times
  6. A financial market model with default
  7. Model specifications
  8. Incorporation of all relevant cash flows, costs and benefits
  9. The pre-default valuation equation
  10. A parabolic equation for the pre-default valuation
  11. A general stochastic volatility model
  12. The logarithmised pre-default valuation PDE
  13. Proofs for the preliminary results and the market model
  14. Proofs for the representations of conditional expectations
  15. Proofs for the conditionally independent hitting times
  16. ...and 4 more sections

Key Result

Lemma 2.1

Any $[0,\infty]$-valued random variable $X$ satisfies for all $s,t\in [0,T]$ with $s\leq t$.

Theorems & Definitions (56)

  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Example 2.9
  • Proposition 2.10
  • ...and 46 more