Abstract
We consider a nonlinear pricing problem that takes into account credit risk and funding issues. The aforementioned problem is formulated as a stochastic forward-backward system with delay, both in the forward and in the backward component, whose solution is characterized in terms of viscosity solution to a suitable type of path-dependent PDE.
Highlights
Starting from the spreading of the credit crunch in 2007, empirical lines of evidence have shown how some aspects of financial markets, neglected up to that point by theoretical models, are instead fundamental in concrete economical frameworks
In [37], the authors proposed a new type of stochastic delay equation, whose generator may depend on the past values of the backward stochastic differential equations (BSDEs) itself
The present paper is so structured: in Section 2, we introduce the mathematical setting; in particular, Section 2.1 is devoted to the introduction to the formalism used, whereas in Section 2.2 we state the main theoretical results that will be applied in Section 3 to the problem of obtaining a portfolio under credit risk and funding issues
Summary
Starting from the spreading of the credit crunch in 2007, empirical lines of evidence have shown how some aspects of financial markets, neglected up to that point by theoretical models, are instead fundamental in concrete economical frameworks. We exploit the approach developed in [18, 21], where the authors firstly consider the present value of a contract as the discounted present value of future payoffs, and include margin variations and counterparty risk in their valuation BSDE, which turns out to be risk-free rate independent The latter depends only on different funding rates which are directly observable on the markets. In [37], the authors proposed a new type of stochastic delay equation, whose generator may depend on the past values of the BSDE itself As mentioned above, this peculiarity is highly nontrivial, as witnessed by several examples reported in [37], where the authors show how the uniqueness property for the solution fails to be true.
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