Invariance, existence and uniqueness of solutions of nonlinear valuation PDEs and FBSDEs inclusive of credit risk, collateral and funding costs

TL;DR

This paper investigates the mathematical properties of nonlinear valuation equations incorporating credit risk, collateral, and funding costs, establishing conditions for their solutions and demonstrating their independence from the risk-free rate.

Contribution

It extends existing valuation models by providing existence, uniqueness, and invariance results for semilinear PDEs and FBSDEs in complex credit and funding contexts.

Findings

Solutions depend only on market, contractual, and treasury rates.

Established conditions for existence and uniqueness of solutions.

Proved invariance of valuation equations to the risk-free rate.

Abstract

We study conditions for existence, uniqueness and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al (2011). These equations take the form of semilinear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions allowing us to extend valuation to credit risk and default closeout, including collateral margining with possible re-hypothecation, and treasury funding costs, we show how such cash flows, when present-valued in an arbitrage free setting, lead to semi-linear PDEs or more generally to FBSDEs. We provide conditions for existence and uniqueness of such solutions in a viscosity and classical sense, discussing the role of the hedging strategy. We show an invariance theorem stating that even though we start from a risk-neutral valuation approach based on a locally risk-free bank account growing at a risk-free rate, our final valuation equations do not depend on the risk free rate. Indeed, our final semilinear PDE or FBSDEs and their classical or viscosity solutions depend only on contractual, market or treasury rates and we do not need to proxy the risk free rate with a real market rate, since it acts as an instrumental variable. The equations derivations, their numerical solutions, the related XVA valuation adjustments with their overlap, and the invariance result had been analyzed numerically and extended to central clearing and multiple discount curves in a number of previous works, including Pallavicini et al (2011), Pallavicini et al (2012), Brigo et al (2013), Brigo and Pallavicini (2014), and Brigo et al (2014).