Innovations in Derivatives Markets pp 37-52 | Cite as

# Analysis of Nonlinear Valuation Equations Under Credit and Funding Effects

## Abstract

We study conditions for existence, uniqueness, and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al. (Funding valuation adjustment: a consistent framework including CVA, DVA, collateral, netting rules and re-hypothecation, 2011, [11]). These equations take the form of semi-linear 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 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 semi-linear PDE or FBSDEs and their classical 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 Brigo and Pallavicini (J. Financ. Eng. 1(1):1–60 (2014), [3]), Pallavicini and Brigo (Interest-rate modelling in collateralized markets: multiple curves, credit-liquidity effects, CCPs, 2011, [10]), Pallavicini et al. (Funding valuation adjustment: a consistent framework including cva, dva, collateral, netting rules and re-hypothecation, 2011, [11]), Pallavicini et al. (Funding, collateral and hedging: uncovering the mechanics and the subtleties of funding valuation adjustments, 2012, [12]), and Brigo et al. (Nonlinear valuation under collateral, credit risk and funding costs: a numerical case study extending Black–Scholes, [5]).

### Keywords

Counterparty credit risk Funding valuation adjustment Funding costs Collateralization Nonlinearity valuation adjustment Nonlinear valuation Derivatives valuation Semi-linear PDE FBSDE BSDE Existence and uniqueness of solutions## 1 Introduction

This is a technical paper where we analyze in detail invariance, existence, and uniqueness of solutions for nonlinear valuation equations inclusive of credit risk, collateral margining with possible re-hypothecation, and funding costs. In particular, we study conditions for existence, uniqueness, and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al. (2011) [11]. After briefly summarizing the cash flows definitions allowing us to extend valuation to default closeout, 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 straightforwardly to semi-linear PDEs or more generally to FBSDEs. We study conditions for existence and uniqueness of such solutions.

We formalize an invariance theorem showing 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 at all. In other words, we do not need to proxy the risk-free rate with any actual market rate, since it acts as an instrumental variable that does not manifest itself in our final valuation equations. Indeed, our final semi-linear PDEs or FBSDEs and their classical solutions depend only on contractual, market or treasury rates and contractual closeout specifications once we use a hedging strategy that is defined as a straightforward generalization of the natural delta hedging in the classical setting.

The equations’ derivations, their numerical solutions, and the invariance result had been analyzed numerically and extended to central clearing and multiple discount curves in a number of previous works, including [3, 5, 10, 11, 12], and the monograph [6], which further summarizes earlier credit and debit valuation adjustment (CVA and DVA) results. We refer to such works and references therein for a general introduction to comprehensive nonlinear valuation and to the related issues with valuation adjustments related to credit (CVA), collateral (LVA), and funding costs (FVA). In this paper, given the technical nature of our investigation and the emphasis on nonlinear valuation, we refrain from decomposing the nonlinear value into valuation adjustments or XVAs. Moreover, in practice such separation is possible only under very specific assumptions, while in general all terms depend on all risks due to nonlinearity. Forcing separation may lead to double counting, as initially analyzed through the Nonlinearity Valuation Adjustment (NVA) in [5]. Separation is discussed in the CCP setting in [3].

The paper is structured as follows.

Section 2 introduces the probabilistic setting, the cash flows analysis, and derives a first valuation equation based on conditional expectations. Section 3 derives an FBSDE under the default-free filtration from the initial valuation equation under assumptions of conditional independence of default times and of default-free initial portfolio cash flows. Section 4 specifies the FBSDE obtained earlier to a Markovian setting and studies conditions for existence and uniqueness of solutions for the nonlinear valuation FBSDE and classical solutions to the associated PDE. Finally, we present the invariance theorem: when adopting delta-hedging, the solution does not depend on the risk-free rate.

## 2 Cash Flows Analysis and First Valuation Equation

*T*, between two financial entities, the investor

*I*and the counterparty

*C*. Both

*I*and

*C*are supposed to be subject to default risk. In particular we model their default times with two \(\mathscr {G}\)-stopping times \(\tau _I,\tau _C\). We assume that the stopping times are generated by Cox processes of positive, stochastic intensities \(\lambda ^I\) and \(\lambda ^C\). Furthermore, we describe the

*default-free*information by means of a filtration \((\mathscr {F}_u)_{u\ge 0}\) generated by the price of the underlying \(S_t\) of our contract. This process has the following dynamic under the measure \(\mathbb {Q}\):

*risk-free*rate. We then suppose the existence of a risk-free account \(B_t\) following the dynamics

*t*we have \(\mathscr {G}_t=\mathscr {F}_t\vee \mathscr {H}^I_t\vee \mathscr {H}_t^C\) where

*conditionally independent*with respect to \(\mathscr {F}\), i.e. for any \(t>0\) and \(t_1,t_2\in [0,t]\), we assume \(\mathbb {Q}\{\tau _I>t_1,\tau _C>t_2 | \mathscr {F}_t\}=\mathbb {Q}\{\tau _I>t_1|\mathscr {F}_t\}\mathbb {Q}\{\tau _C>t_2 | \mathscr {F}_t\}\). Moreover, we indicate \(\tau =\tau _I\wedge \tau _C\) and with these assumptions we have that \(\tau \) has intensity \(\lambda _u=\lambda _u^I+\lambda _u^C\). For convenience of notation we use the symbol \(\bar{\tau }\) to indicate the minimum between \(\tau \) and

*T*.

### Remark 1

We suppose that the measure \(\mathbb {Q}\) is the so-called *risk-neutral* measure, i.e. a measure under which the prices of the traded non-dividend-paying assets discounted at the risk-free rate are martingales or, in equivalent terms, the measure associated with the numeraire \(B_t\).

### 2.1 The Cash Flows

To price this portfolio we take the conditional expectation of all the cash flows of the portfolio and discount them at the risk-free rate. An alternative to the explicit cash flows approach adopted here is discussed in [4].

- The payments due to the contract itself: modeled by an \(\mathscr {F}\)-predictable process \(\pi _t\) and a final cash flow \(\varPhi (S_T)\) payed at maturity modeled by a Lipschitz function \(\varPhi \). At time
*t*the cumulated discounted flows due to these components amount to$$ {1}_{\{ \tau >T\}}D(0,T)\varPhi (S_T)+\int _t^{\bar{\tau }}D(t,u)\pi _udu. $$ - The payments due to default: in particular we suppose that at time \(\tau \) we have a cash flow due to the default event (if it happened) modeled by a \(\mathscr {G}_\tau \)-measurable random variable \(\theta _\tau \). So the flows due to this component are$$ {1}_{\{ t<\tau<T\}}D(t,\tau )\theta _\tau ={1}_{\{ t<\tau <T\}}\int _t^TD(t,u)\theta _u d{1}_{\{ \tau \le u\}}. $$
- The payments due to the collateral account: more precisely we model this account by an \(\mathscr {F}\)-predictable process \(C_t\). We postulate that \(C_t>0\) if the investor is the collateral taker, and \(C_t<0\) if the investor is the collateral provider. Moreover, we assume that the collateral taker remunerates the account at a certain interest rate (written on the CSA); in particular we may have different rates depending on who the collateral taker is, so we introduce the ratewhere \(c_t^+,c_t^-\) are two \(\mathscr {F}\)-predictable processes. We also suppose that the collateral can be re-hypothecated, i.e. the collateral taker can use the collateral for funding purposes. Since the collateral taker has to remunerate the account at the rate \(c_t\), the discounted flows due to the collateral can be expressed as a cost of carry and sum up to$$\begin{aligned} c_t={1}_{\{ C_t>0\}}c_t^++{1}_{\{ C_t\le 0\}}c_t^- \ , \end{aligned}$$(1)$$ \int _t^{\bar{\tau }}D(t,u)(r_u-c_u)C_udu. $$
- We suppose that the deal we are considering is to be hedged by a position in cash and risky assets, represented respectively by the \(\mathscr {G}\)-adapted processes \(F_t\) and \(H_t\), with the convention that \(F_t>0\) means that the investor is borrowing money (from the bank’s treasury for example), while \(F<0\) means that
*I*is investing money. Also in this case to take into account different rates in the borrowing or lending case we introduce the rateThe flows due to the funding part are$$\begin{aligned} f_t={1}_{\{ V_t-C_t>0\}}f_t^++{1}_{\{ V_t-C_t\le 0\}}f_t^-. \end{aligned}$$(2)For the flows related to the risky assets account \(H_t\) we assume that we are hedging by means of repo contracts. We have that \(H_t>0\) means that we need some risky asset, so we borrow it, while if \(H<0\) we lend. So, for example, if we need to borrow the risky asset we need cash from the treasury, hence we borrow cash at a rate \(f_t\) and as soon as we have the asset we can repo lend it at a rate \(h_t\). In general \(h_t\) is defined as$$ \int _t^{\bar{\tau }}D(t,u)(r_u-f_u)F_udu. $$Thus we have that the total discounted cash flows for the risky part of the hedge are equal to$$\begin{aligned} h_t={1}_{\{ H_t>0\}}h^+_t+{1}_{\{ H_t\le 0\}}h^-_t. \end{aligned}$$(3)$$ \int _t^{\bar{\tau }}D(t,u)(h_u-f_u)H_udu. $$

*C*is to be omitted from the following equation) and the risky asset accounts, i.e.

### Remark 2

In the classic no-arbitrage theory and in a complete market setting, without credit risk, the hedging process *H* would correspond to a delta hedging strategy account. Here we do not enforce this interpretation yet. However, we will see that a delta-hedging interpretation emerges from the combined effect of working under the default-free filtration \(\mathscr {F}\) (valuation under partial information) and of identifying part of the solution of the resulting BSDE, under reasonable regularity assumptions, as a sensitivity of the value to the underlying asset price *S*.

### 2.2 Adjusted Cash Flows Under a Simple Trading Model

We now show how the adjusted cash flows originate assuming we buy a call option on an equity asset \(S_T\) with strike *K*. We analyze the operations a trader would enact with the treasury and the repo market in order to fund the trade, and we map these operations to the related cash flows. We go through the following steps in each small interval \([t,t+dt]\), seen from the point of view of the trader/investor buying the option. This is written in first person for clarity and is based on conversations with traders working with their bank treasuries.

*t*:

- 1.
I wish to buy a call option with maturity

*T*whose current price is \(V_t = V(t,S_t)\). I need \(V_t\) cash to do that. So I borrow \(V_t\) cash from my bank treasury and buy the call. - 2.
I receive the collateral amount \(C_t\) for the call, that I give to the treasury.

- 3.
Now I wish to hedge the call option I bought. To do this, I plan to repo-borrow \(\varDelta _t\) stock on the repo-market.

- 4.
To do this, I borrow \(H_t = \varDelta _t S_t\) cash at time

*t*from the treasury. - 5.
I repo-borrow an amount \(\varDelta _t\) of stock, posting cash \(H_t\) as a guarantee.

- 6.
I sell the stock I just obtained from the repo to the market, getting back the price \(H_t\) in cash.

- 7.
I give \(H_t\) back to treasury.

- 8.
My outstanding debt to the treasury is \(V_t-C_t\).

- 9.
I need to close the repo. To do that I need to give back \(\varDelta _t\) stock. I need to buy this stock from the market. To do that I need \(\varDelta _t S_{t+dt}\) cash.

- 10.
I thus borrow \(\varDelta _t S_{t+dt}\) cash from the bank treasury.

- 11.
I buy \(\varDelta _t\) stock and I give it back to close the repo and I get back the cash \(H_t\) deposited at time

*t*plus interest \(h_tH_t\). - 12.I give back to the treasury the cash \(H_t\) I just obtained, so that the net value of the repo operation has beenNotice that this \(- \varDelta _t dS_t\) is the right amount I needed to hedge$$\begin{aligned} H_t(1+h_t \, dt ) - \varDelta _t S_{t+dt} = - \varDelta _t \, dS_t + h_t H_t \, dt \end{aligned}$$
*V*in a classic delta hedging setting. - 13.
I close the derivative position, the call option, and get \(V_{t+dt}\) cash.

- 14.
I have to pay back the collateral plus interest, so I ask the treasury the amount \(C_t(1+c_t\, dt)\) that I give back to the counterparty.

- 15.
My outstanding debt plus interest (at rate

*f*) to the treasury is \(V_t-C_t+C_t(1+c_t\, dt)+(V_t-C_t)f_t\, dt=V_t(1+f_t\, dt)+C_t(c_t-f_t\, dt)\).I then give to the treasury the cash \(V_{t+dt}\) I just obtained, the net effect being$$\begin{aligned} V_{t+dt} - V_t(1+f_t\, dt)-C_t(c_t-f_t)\, dt = dV_t - f_t V_t \, dt-C_t(c_t-f_t) \, dt \end{aligned}$$ - 16.I now have that the total amount of flows is:$$\begin{aligned} - \varDelta _t \, dS_t + h_t H_t \, dt+dV_t - f_t V_t \, dt-C_t(c_t-f_t) \, dt \end{aligned}$$
- 17.Now I present-value the above flows in
*t*in a risk-neutral setting.This derivation holds assuming that \({\mathbb {E}}_t [ dS_t] = r_t S_t \, dt\) and \({\mathbb {E}}_t [ dV_t] = r_t V_t \, dt - d \varphi (t)\), where \(d \varphi \) is a dividend of$$\begin{aligned}&{\mathbb {E}}_t [ - \varDelta _t \, dS_t + h_t H_t \, dt+dV_t - f_t V_t \, dt-C_t(c_t-f_t) \, dt] \\&\quad =- \varDelta _t (r_t - h_t) S_t\, dt + (r_t - f_t) V_t \, dt-C_t(c_t-f_t) \, dt-d \varphi (t) \\&\quad = -H_t(r_t - h_t)\, dt + (r_t - f_t) (H_t+F_t+C_t) \, dt-C_t(c_t-f_t) \, dt-d \varphi (t) \\&\quad = (h_t-f_t)H_t\, dt+(r_t - f_t)F_t\, dt+(r_t-c_t)C_t\, dt -d \varphi (t) \end{aligned}$$*V*in \([t,t+dt)\) expressing the funding costs. Setting the above expression to zero we obtainwhich coincides with the definition given earlier in (6).$$\begin{aligned} d \varphi (t) = (h_t-f_t)H_t\, dt+(r_t - f_t)F_t\, dt+(r_t-c_t)C_t\, dt \end{aligned}$$

## 3 An FBSDE Under \(\mathscr {F}\)

We aim to switch to the default free filtration \(\mathscr {F}=(\mathscr {F}_t)_{t\ge 0}\), and the following lemma (taken from Bielecki and Rutkowski [1] Sect. 5.1) is the key in understanding how the information expressed by \(\mathscr {G}\) relates to the one expressed by \(\mathscr {F}\).

### Lemma 1

*X*and any \(t\in \mathbb {R}_+\), we have:

*Y*there exists an \(\mathscr {F}_t\)-measurable random variable

*Z*such that

What follows is an application of the previous lemma exploiting the fact that we have to deal with a stochastic process structure and not only a simple random variable. Similar results are illustrated in [2].

### Lemma 2

### Proof

where the penultimate equality comes from the fact that the default times are conditionally independent and if we define \(\varLambda _X(u)=\int _0^u\lambda ^X_sds\) with \(X\in \{I,C\}\) we have that \(\tau _X=\varLambda _X^{-1}(\xi _X)\) with \(\xi _X\) mutually independent exponential random variables independent from \(\lambda ^X\).^{1} A similar result will enable us to deal with the default cash flow term. In fact we have the following (Lemma 3.8.1 in [2])

### Lemma 3

*LGD*indicates the loss given default, typically defined as \(1-REC\), where

*REC*is the corresponding recovery rate and \((x)^+\) indicates the positive part of

*x*and \((x)^-=-(-x)^+\). The meaning of these flows is the following, consider \(\theta _\tau \):

at first to default time \(\tau \) we compute the close-out value \(\epsilon _{\tau }\);

if the counterparty defaults and we are net debtor, i.e. \(\epsilon _{\tau }-C_\tau \le 0\) then we have to pay the whole close-out value \(\varepsilon _\tau \) to the counterparty;

if the counterparty defaults and we are net creditor, i.e. \(\epsilon _{\tau }-C_\tau >0\) then we are able to recover just a fraction of our credits, namely \(C_\tau +REC_C(\varepsilon _\tau -C_\tau )=REC_C\varepsilon _\tau +LGD_CC_\tau =\varepsilon _\tau -LGD_C(\varepsilon _\tau -C_\tau )\) where \(LGD_C\) indicates the loss given default and is equal to one minus the recovery rate \(REC_C\).

A similar reasoning applies to the case when the Investor defaults.

### Remark 3

## 4 Markovian FBSDE and PDE for \(\widetilde{V}_t\) and the Invariance Theorem

the dividend process \(\pi _u\) is a deterministic function \(\pi (u,S_u)\) of

*u*and \(S_u\), Lipschitz continuous in \(S_u\);the rates \(r,f^\pm ,c^\pm ,\lambda ^I,\lambda ^C\) are deterministic bounded functions of time;

the rate \(h_t\) is a deterministic function of time, and does not depend on the sign of

*H*, namely \(h^+=h^-\), hence there is only one rate relative to the repo market of assets;the collateral process is a fraction of the process \(\widetilde{V}_{u}\), namely \(C_u=\alpha _u\widetilde{V}_{u}\), where \(0\le \alpha _u\le 1\) is a function of time;

the close-out value \(\epsilon _t\) is equal to \(\widetilde{V}_t\) (this adds a source of nonlinearity with respect to choosing a risk-free closeout, see for example [6] and [5]);

the diffusion coefficient \(\sigma (t,S_t)\) of the underlying dynamic is Lipschitz continuous, uniformly in time, in \(S_t\);

we consider a delta-hedging strategy, and to this extent we choose \(\widetilde{H}_t=S_t\frac{Z_t}{\sigma (t,S_t)}\); this reasoning derives from the fact that if we suppose \(\widetilde{V}_t=V(t,S_t)\) with \(V(\cdot ,\cdot )\in C^{1,2}\) applying Ito’s formula and comparing it with Eq. (12), we have that \(\sigma (t,S_t)\partial _SV(t,S_t)=Z_t\).

^{2}

### Theorem 1

*T*]:

*K*such that

\(\sigma (t,x)^2\ge \frac{1}{K}\);

\(|f(t,x,y,z)-f(t,x',y',z')|+|g(x)-g(x')|\le K(|x-x'|+|y-y'|+|z-z'|)\);

\(|f(t,0,0,0)|+|g(0)|\le K\);

*classical*(i.e. \(C^{1,2}\)) solution to the following semilinear PDE

*B*(

*t*,

*s*,

*v*,

*z*) is not Lipschitz continuous in

*s*because of the hedging term. But, since the hedging term is linear in \(Z_t\) we can move it from the drift of the backward equation to the drift of the forward one. More precisely consider the following:

### Theorem 2

### Proof

*s*by assumption. The \(\theta \) term and the \((f_t(\alpha _t-1)-\lambda _t-c_t\alpha _t)v\) term are continuous and piece-wise linear, hence Lipschitz continuous and this concludes the proof.

Note that the *S*-dynamics in (16) has the repo rate *h* as drift. Since in general *h* will depend on the future values of the deal, this is a source of nonlinearity and is at times represented informally with an expected value \(\mathbb {E}^h\) or a pricing measure \(\mathbb {Q}^h\), see for example [5] and the related discussion on operational implications for the case \(h=f\).

### Theorem 3

(Solution to the Valuation Equation) Let \(S_t\) be the solution to Eq. (18) and *u*(*t*, *s*) the classical solution to Eq. (17). Then the process \((S_t,u(t,S_t),\sigma (t,S_t)\partial _su(t,S_t))\) is the unique solution to Eq. (13).

### Proof

\(|\mu (t,x)-\mu (t,x')|+|\sigma (t,x)-\sigma (t,x')|\le K|x-x'|\)

\(|\mu (t,x)|+|\sigma (t,x)|\le K(1+|x|)\)

\(|f(t,x,y,z)-f(t,x,y',z')|\le K(|y-y'|+|z-z'|)\)

\(|g(x)|+|f(t,x,0,0)|\le K(1+|x|^p)\)

*y*and

*z*we can verify that Eq. (13) satisfies the above-mentioned assumptions and hence has a unique solution.

### Remark 4

Since we proved that \(V_t=u(t,S_t)\) with \(u(t,s)\in C^{1,2}\), the reasoning we used, when saying that \(\widetilde{H}_t=S_t\frac{Z_t}{\sigma (t,S_t)}\) represented choosing a delta-hedge, it is actually more than a heuristic argument.

Moreover, since (17) does not depend on the risk-free rate \(r_t\) so we can state the following:

### Theorem 4

(Invariance Theorem) If we are under the assumptions at the beginning of Sect. 4 and we assume that we are backing our deal with a delta hedging strategy, then the price \(V_t\) can be calculated via the semilinear PDE (17) and does *not* depend on the risk-free rate *r*(*t*).

This invariance result shows that even when starting from a risk-neutral valuation theory, the risk-free rate disappears from the nonlinear valuation equations. A discussion on consequences of nonlinearity and invariance on valuation in general, on the operational procedures of a bank, on the legitimacy of fully charging the nonlinear value to a client, and on the related dangers of overlapping valuation adjustments is presented elsewhere, see for example [3, 5] and references therein.

## Footnotes

## Notes

### Acknowledgements

The opinions here expressed are solely those of the authors and do not represent in any way those of their employers. We are grateful to Cristin Buescu, Jean-François Chassagneux, François Delarue, and Marek Rutkowski for helpful discussion and suggestions that helped us improve the paper. Marek Rutkowski and Andrea Pallavicini visits were funded via the EPSRC Mathematics Platform grant EP/I019111/1.

The KPMG Center of Excellence in Risk Management is acknowledged for organizing the conference “Challenges in Derivatives Markets - Fixed Income Modeling, Valuation Adjustments, Risk Management, and Regulation”.

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