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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

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 particular, let us mention the violation of standard nonarbitrage relation between forward rates and zero-coupon bonds. Even though we will not address the latter problem in the present paper, we would like to underline how it is very connected to the topic we will treat, as witnessed by the recent, wide, and growing literature linked to the so-called

In what follows, we will focus on a different issue which emerged after the last financial crisis, namely, the problem of pricing derivatives contracts, including the possibility of the

Recently, several works appeared that try to include counterparty risk, that is, the risk to each party of a contract that the counterparty will be unable to meet contractual obligations, as well as funding issues in pricing financial contracts, leading to a systematic treatment of both of them. In particular, to our knowledge, the first attempts in the direction of developing a concrete framework able to treat both the counterparty risk and the funding constraints can be found in [

In what follows, we also exploit the approach developed in [

The main contribution of the present work is thus to give rigorous and general mathematical foundation of the previously introduced setting. In particular, following [

Let us recall that the first rigorous treatment of delay differential equations dates back to the monograph [

Analogously, in [

Recently, the development of the theory of delayed stochastic differential equations has made one step further to include, besides the delay in the forward SDE, also a delay component in the backward equation. In particular, in [

To overcome the latter problem, some additional assumptions have to be taken into account, as in [

In the present paper, we exploit the aforementioned results obtained in [

Moreover, we underline that analogous approaches can be fruitfully exploited within frameworks characterized by stochastic optimal control problems, as has been made in, for example, [

The present paper is so structured: in Section

In what follows, we introduce the mathematical setting that allows us to derive the delayed pricing equation we are interested in. The aforementioned framework will be first introduced in its complete generality; for example, we will also consider path-dependent coefficients. Moreover, we underline the notion that the main novelty of the present work is represented by the fact that we can also consider a backward equation with delayed generator, together with the associated Feynman-Kac representation theorem. For the sake of completeness, we report here what has been essentially derived in [

Let us briefly introduce the setting the

In what follows, we define the mathematical framework needed to construct the solutions of the Kolmogorov equation we will treat later; see, for example, [

We will denote

Let

Let

Moreover, taking

As it is standard when dealing with delay equations, we will exploit the following notation: for a path

We say that

Let

Exploiting previous notations and following [

For a continuous function

The following theorem states a functional version of Itô’s formula; see, for example, [

Let

If

In what follows, mainly following [

Let us denote by

Let

Let us consider the process

Taking into account what we have introduced so far, we are in position to state the main object of analysis in the present work, namely, the following functional path-dependent PDE:

We now introduce the space of the test functions:

The definition of a viscosity solution to the functional PDE (

Let

For any

For any

One says that

One says that

The present section is devoted to the characterization of the delayed forward-backward system we are interested in. In particular, we will look for a triplet,

Let us first focus on the forward component

Let us consider two nonanticipative functionals

there exists

In light of Assumptions

Let the coefficients _{
1})-(A_{
2}). Let

In order to analyse the delayed backward SDE appearing in (

(i) Let

Moreover, we will equip the spaces

Further, in order to deal with the delayed backward SDE appearing in (

Let

There exist

The function

The function

Hence, we are in position to report the result which states both the existence and the uniqueness for the solution to the BSDE with delayed generator we are considering; see, for example, [

If Assumptions _{
1})–(B_{
3}) hold true and

The present subsection is mainly based on [_{1})–(B_{3}) hold true.

It is worth mentioning that typical generators

If

Let us define the function

Let Assumptions

See, for example, [

Let Assumptions

See, for example, [

In the present section, we are going to apply previously derived results to the pricing of financial derivatives under counterparty risk and funding issues. In order to derive the pricing equation, we closely follow [

Let us consider a standard filtered probability space

In order to work in a realistic and concrete financial framework, we include the risk of default. In particular, we denote by

Until now, we have worked under the strong assumption that there exists a risk-free rate

Given a rate

Following [

In particular, we have first to consider the payments due to the contract itself, which is a predictable process

We also have a random variable

We consider further a collateral account

Allowing also for rehypothecation, namely, allowing banks and brokers to use assets that have been posted as collateral by their clients, we end up with the following cash flow:

A similar convention holds for

Eventually, summing up all the aforementioned cash flows (

For a concrete example of how (

Switching to the default-free filtration, we can exploit the results stated in [

For any

In particular, we have that, for any

Let

So, by an application of Lemmas

By (

In the previous section, we have derived the BSDE that describes the evolution of the financial portfolio, while, in the present section, we are going to better specify the mathematical assumptions regarding (

We assume that the dividend process

In what follows, all the rates

We assume that

The hedging term

Last but not least, since this is the main novelty of the present approach, we assume that the collateral depends on portfolio past values. As said above, this implies that the BSDE is highly irregular, and even the existence and uniqueness of a solution are in general not granted under standard assumptions. However, as pointed out in [

We would like to underline the notion that previous choice is just one of the possible schemes admitted in our mathematical setting; see, for example, (

The aforementioned assumptions can be better formalized as follows.

One has the following:

The parameters

In light of Assumptions

It is worth mentioning that, in the setting represented by (

With respect to the forward-backward system (

Let one consider the forward-backward delayed system (

Because of Assumptions

We would like to underline the notion that the scheme described by (

Let one consider the forward-backward delayed system (

The proof is analogous to the one provided for Theorem

Inspired by the increasing attention to financial models which take into account credit risk factors, we have generalized results provided in [

Our BSDE approach generalizes the approach derived in [

We would like to underline the notion that the previous approach will be the base of our future works related to the fundamental topic of allowing for closeout rule with delay. Previous situation usually happens when one considers the time gap between the actual default of a party and the real closure of a contract. In such interval of time, it may happen that also the second party could default. Therefore, when one has to price a contract, such a time delay has to be taken into account; see, for example, [

With respect to the aforementioned setting, we believe that both BSDEs techniques and the

According to the recent literature, it is also important to note that the BSDEs techniques investigated in the present paper are also strongly connected to theoretical and, respectively, applied, questions (see, e.g., [

The authors declare that they have no competing interests regarding the publication of this paper.

The authors would like to thank the