# Nonlinear Valuation with XVAs: Two Converging Approaches

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## Abstract

**:**

## 1. Introduction

**nonlinear valuation paradigms**that are based on more advanced mathematical tools, such as semi-linear PDEs or nonlinear BSDEs (see, e.g., [1,2,3,4,5,6,7,8,9]).

## 2. Risk-Neutral Valuation with Adjusted Cash Flows

#### 2.1. Clean Price of an OTC Contract

**Definition**

**1**

**(Non-defaultable, uncollateralized derivative).**

**Definition**

**2**

**(Clean price).**

#### 2.2. Adjusted Price of a Defaultable Collateralized Contract

**Cost of collateralization.**When default risk is present, it is customary to mitigate it by implementing a collateralization procedure, which is also known among practitioners as "margining". Let ${C}_{t}$ denote the level of the collateral account at time t, as specified by the credit support annex (CSA), which defines the terms for the provision of collateral by the parties in the contract. For conciseness, we do not differentiate between the initial and variation margins so ${C}_{t}$ represents the total collateralization. By convention, ${C}_{t}>0$ means that the collateral has been overall posted by the counterparty to protect the investor, and the investor has to pay instantaneous interest ${c}_{t}^{b}$ on the related amount. If ${C}_{t}<0$, then the investor posts collateral for the counterparty and is remunerated at interest ${c}_{t}^{l}$. Therefore, the effective collateral accrual rate $\overline{c}$ is given by ${\overline{c}}_{t}:={c}_{t}^{b}{\mathrm{\U0001d7d9}}_{\{{C}_{t}>0\}}+{c}_{t}^{l}{\mathrm{\U0001d7d9}}_{\{{C}_{t}<0\}}$. Here, we assume the collateral account can be rehypothecated (see [30] for a discussion on how collateralization impacts the close-out specification). We denote the discounted net cash flows due to the variation margining procedure over $[t,s]$ by $\gamma (t,s,C)$, where

**Close-out cash flows.**The most significant manifestation of the credit default risk is the actual default itself, and we consider the cash flow to and from the investor at the first default. Once a default event occurs, the contract is terminated (we have close-out) and all payments that are accelerated and become due are calculated under the close out payoff. Contractual cash flows have been exchanged up until the arrival of the first default time, either that of the investor I or that of the counterparty C. We define the first-to-default time $\tau $ as $\tau :={\tau}_{C}\wedge {\tau}_{I}$, resulting in an effective maturity $\widehat{\tau}$ of the contract given by $\widehat{\tau}:=\tau \wedge T$. An important feature of a defaultable contract is the credit support annex (CSA) close-out payoff, which occurs when one of the parties defaults either before or at the maturity of the contract. To define the CSA close-out payoff ${\vartheta}_{\tau}(R,C)$ on the event $\{\tau \le T\}$, we first define the random variable $\mathrm{Y}={Q}_{\tau}-{C}_{\tau -}$, where Q is the CSA close-out valuation process of the contract inclusive of the increment $\Delta {A}_{\tau}={A}_{\tau}-{A}_{\tau -}$, representing a (possibly null) promised bullet dividend at $\tau $ and ${C}_{\tau -}$ is the value of the collateral process C at the moment of the first default. Since the margin account is not updated at the moment of the first default, it can be represented as ${\tilde{C}}_{t}={\mathrm{\U0001d7d9}}_{\{t<\tau \}}{C}_{t}+{\mathrm{\U0001d7d9}}_{\{t\ge \tau \}}{C}_{\tau -}$, so that ${\tilde{C}}_{\tau}={\tilde{C}}_{\tau -}$.

**Definition**

**3**

**(Close-out payoff).**

**Example**

**1.**

**Funding costs and benefits.**In this step, we focus on the funding costs for implementing the trading/hedging strategy and we add the relevant cash flows by adopting the procedure proposed in Pallavicini et al. [23]. Let ${F}_{t}$ be the cash account for the hedging of the trade and let ${H}_{t}$ stand for the value of the investor’s positions in the underlying risky asset, S. We assume that S can be traded through a repo (repurchase agreement) market, meaning that the risky asset, S, is funded using a cash account, ${F}_{t}^{S}$, and the equality ${F}_{t}^{S}=-{H}_{t}$ holds for every $t\in [0,T]$. The case of collateralized risky assets can be treated in the same way by interpreting the cash account as the collateral account for such assets. We assume there are two funding rates—${f}^{b}$ for borrowing money and ${f}^{l}$ for lending money—and similarly two repo rates, ${h}^{b}$ and ${h}^{l}$. The funding policy of the bank’s treasury is determined by funding rates for cost, ${f}^{b}$, and benefit, ${f}^{l}$, of carry of hedge accounts, which both depend on the funding policy of the bank.

**Definition**

**4**

**(Defaultable, collateralized contract).**

**Proposition**

**1**

**“Adjusted cash flows” pricing formula**). The risk-neutral price of the defaultable, collateralized contract $(A,R,C,\tau )$ inclusive of funding costs is obtained as, on the event $\{t<\tau \}$ for every $t\in [0,T]$,

- (${\overline{c}}_{u}-{r}_{u}$) for the cost-of-carry of the collateral account;
- (${r}_{u}-{\overline{f}}_{u}$) for the costs due to the funding account;
- (${r}_{u}-{\overline{h}}_{u}$) for the costs due to hedging in the repo market;

#### 2.3. Risk-Neutral Approach to Nonlinear Valuation Adjustments

## 3. Valuation by Replication in Linear Multi-Curve Markets

**Repo markets.**We denote by $({S}^{1},{S}^{2},\dots ,{S}^{d})$ the collection of prices of d non-defaultable risky assets, which do not pay dividends. Let ${B}^{i,l}$ (respectively, ${B}^{i,b}$) stand for the lending (respectively, borrowing) repo account corresponding to the ith risky non-defaultable asset. In the special case when ${B}^{i,l}={B}^{i,b}$, the single repo account for the asset ${S}^{i}$ is denoted by ${B}^{i}$. We assume that $d{B}_{t}^{i,l}={h}_{t}^{i,l}{B}_{t}^{i,l}\phantom{\rule{0.166667em}{0ex}}dt,\phantom{\rule{0.166667em}{0ex}}d{B}_{t}^{i,b}={h}_{t}^{i,b}{B}_{t}^{i,b}\phantom{\rule{0.166667em}{0ex}}dt$ and $d{B}_{t}^{i}={h}_{t}^{i}{B}_{t}^{i}\phantom{\rule{0.166667em}{0ex}}dt$ and the processes ${S}^{1},{S}^{2},\dots ,{S}^{d}$ are $\mathbb{G}$-semimartingales. Repo markets are nowadays well established for bonds and equities, although the lower quality as collateral means that the equity repo rate is invariably higher than investment-grade bond repo.

**Unsecured trading.**Let $({S}^{d+1},{S}^{d+2},\dots ,{S}^{m})$ be the collection of prices of m non-defaultable risky assets, which do not pay dividends and are traded through unsecured funding from the bank’s treasury. We assume that the processes ${S}^{d+1},{S}^{d+2},\dots ,{S}^{d+m}$ are $\mathbb{G}$-semimartingales. The lending (respectively, borrowing) cash account${B}^{l}$ (respectively, ${B}^{b}$) can be used by the investor for unsecured lending (respectively, borrowing) of cash from the bank’s treasury. When the borrowing and lending treasury rates coincide, the single treasury account is denoted by ${B}^{f}$. It is assumed that $d{B}_{t}^{l}={f}_{t}^{l}{B}_{t}^{l}\phantom{\rule{0.166667em}{0ex}}dt,\phantom{\rule{0.166667em}{0ex}}d{B}_{t}^{b}={f}_{t}^{b}{B}_{t}^{b}\phantom{\rule{0.166667em}{0ex}}dt$ and $d{B}_{t}^{f}={f}_{t}{B}_{t}^{f}\phantom{\rule{0.166667em}{0ex}}dt$ where the treasury funding rates ${f}^{l},{f}^{b}$ and f are $\mathbb{G}$-adapted processes.

**Defaultable securities.**In order to guarantee that hedging of default risk is feasible, we also postulate that some defaultable securities are available for trade. Specifically, let ${Z}^{1}(t,T)$ and ${Z}^{2}(t,T)$ be the prices of T-maturity unit zero-coupon bonds issued by the investor’s bank and the counterparty’s entity. Of course, it is also possible to introduce credit default swaps (CDSs) in the present market model (see, e.g., Brigo et al. [32]). Let $\tau ={\tau}_{1}\wedge {\tau}_{2}={\tau}_{I}\wedge {\tau}_{C}$ where ${\tau}_{1}={\tau}_{I}$ and ${\tau}_{2}={\tau}_{C}$ are $\mathbb{G}$-stopping times, representing the default times of the investor and the counterparty, respectively. As before, we denote by $\widehat{\tau}:=\tau \wedge T$ the effective maturity of the contract.

#### 3.1. Linear Markets with Funding Costs and Default Risk

#### 3.1.1. Clean Price of a Financial Contract

**Definition**

**5.**

#### 3.1.2. Replication of a Non-Defaultable Collateralized Contract

**Definition**

**6.**

#### 3.1.3. Replication of a Defaultable Collateralized Contract

**Definition**

**7.**

**Definition**

**8**

**(Wealth process).**

**Definition**

**9**

**(Replication).**

**Remark**

**1.**

#### 3.2. Valuation in a Linear Multi-Curve Market

**Definition**

**10**

**(Ex-dividend price).**

**Lemma**

**1.**

**Proof.**

**Example**

**2.**

#### 3.2.1. Auxiliary Lemma

**Definition**

**11.**

**Lemma**

**2.**

**Proof.**

#### 3.2.2. Linear Probabilistic Valuation Formula

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

#### 3.2.3. Linear Risk-Neutral Valuation with Funding, Defaults, and Collateralization

**Corollary**

**2.**

#### 3.3. Linear Valuation with XVAs under Risk-Free Close-Out

**Definition**

**12.**

**Proposition**

**2.**

**Proof.**

## 4. Nonlinear Markets with Funding Costs and Default Risk

#### 4.1. Nonlinear Dynamics of the Value Process of a Trading Strategy

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

#### 4.2. Nonlinear Probabilistic Valuation Formula

**Theorem**

**2.**

**Proof.**

#### 4.3. Nonlinear Valuation with XVAs under Risk-Free Close-Out

**Corollary**

**3.**

**Proposition**

**3.**

#### 4.4. Nonlinear Pricing BSDE

**Proposition**

**4.**

## 5. Incomplete Market with Funding Benefit at Default

**Definition**

**13.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Brigo, D.; Buescu, C.; Francischello, M.; Pallavicini, A.; Rutkowski, M.
Nonlinear Valuation with XVAs: Two Converging Approaches. *Mathematics* **2022**, *10*, 791.
https://doi.org/10.3390/math10050791

**AMA Style**

Brigo D, Buescu C, Francischello M, Pallavicini A, Rutkowski M.
Nonlinear Valuation with XVAs: Two Converging Approaches. *Mathematics*. 2022; 10(5):791.
https://doi.org/10.3390/math10050791

**Chicago/Turabian Style**

Brigo, Damiano, Cristin Buescu, Marco Francischello, Andrea Pallavicini, and Marek Rutkowski.
2022. "Nonlinear Valuation with XVAs: Two Converging Approaches" *Mathematics* 10, no. 5: 791.
https://doi.org/10.3390/math10050791