Innovations in Derivatives Markets pp 3752  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 rehypothecation, 2011, [11]). 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 rehypothecation, and treasury funding costs, we show how such cash flows, when presentvalued in an arbitragefree setting, lead to semilinear 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 riskneutral valuation approach based on a locally riskfree bank account growing at a riskfree rate, our final valuation equations do not depend on the riskfree rate. Indeed, our final semilinear PDE or FBSDEs and their classical solutions depend only on contractual, market or treasury rates and we do not need to proxy the riskfree 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 (Interestrate modelling in collateralized markets: multiple curves, creditliquidity effects, CCPs, 2011, [10]), Pallavicini et al. (Funding valuation adjustment: a consistent framework including cva, dva, collateral, netting rules and rehypothecation, 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 Semilinear PDE FBSDE BSDE Existence and uniqueness of solutions1 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 rehypothecation, 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 rehypothecation and treasury funding costs, we show how such cash flows, when presentvalued in an arbitragefree setting, lead straightforwardly to semilinear 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 riskneutral valuation approach based on a locally riskfree bank account growing at a riskfree rate, our final valuation equations do not depend on the riskfree rate at all. In other words, we do not need to proxy the riskfree 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 semilinear 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 defaultfree filtration from the initial valuation equation under assumptions of conditional independence of default times and of defaultfree 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 deltahedging, the solution does not depend on the riskfree rate.
2 Cash Flows Analysis and First Valuation Equation
Remark 1
We suppose that the measure \(\mathbb {Q}\) is the socalled riskneutral measure, i.e. a measure under which the prices of the traded nondividendpaying assets discounted at the riskfree 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 riskfree 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 rehypothecated, 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)$$ \int _t^{\bar{\tau }}D(t,u)(r_uc_u)C_udu. $$
 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$$\begin{aligned} f_t={1}_{\{ V_tC_t>0\}}f_t^++{1}_{\{ V_tC_t\le 0\}}f_t^. \end{aligned}$$(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$$ \int _t^{\bar{\tau }}D(t,u)(r_uf_u)F_udu. $$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)$$ \int _t^{\bar{\tau }}D(t,u)(h_uf_u)H_udu. $$
Remark 2
In the classic noarbitrage 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 deltahedging interpretation emerges from the combined effect of working under the defaultfree 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.
 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 repoborrow \(\varDelta _t\) stock on the repomarket.
 4.
To do this, I borrow \(H_t = \varDelta _t S_t\) cash at time t from the treasury.
 5.
I repoborrow 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 \(V_tC_t\).
 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 V in a classic delta hedging setting.$$\begin{aligned} H_t(1+h_t \, dt )  \varDelta _t S_{t+dt} =  \varDelta _t \, dS_t + h_t H_t \, dt \end{aligned}$$
 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 \(V_tC_t+C_t(1+c_t\, dt)+(V_tC_t)f_t\, dt=V_t(1+f_t\, dt)+C_t(c_tf_t\, dt)\).
I then give to the treasury the cash \(V_{t+dt}\) I just obtained, the net effect being$$\begin{aligned} V_{t+dt}  V_t(1+f_t\, dt)C_t(c_tf_t)\, dt = dV_t  f_t V_t \, dtC_t(c_tf_t) \, dt \end{aligned}$$  16.I now have that the total amount of flows is:$$\begin{aligned}  \varDelta _t \, dS_t + h_t H_t \, dt+dV_t  f_t V_t \, dtC_t(c_tf_t) \, dt \end{aligned}$$
 17.Now I presentvalue the above flows in t in a riskneutral 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 V in \([t,t+dt)\) expressing the funding costs. Setting the above expression to zero we obtain$$\begin{aligned}&{\mathbb {E}}_t [  \varDelta _t \, dS_t + h_t H_t \, dt+dV_t  f_t V_t \, dtC_t(c_tf_t) \, dt] \\&\quad = \varDelta _t (r_t  h_t) S_t\, dt + (r_t  f_t) V_t \, dtC_t(c_tf_t) \, dtd \varphi (t) \\&\quad = H_t(r_t  h_t)\, dt + (r_t  f_t) (H_t+F_t+C_t) \, dtC_t(c_tf_t) \, dtd \varphi (t) \\&\quad = (h_tf_t)H_t\, dt+(r_t  f_t)F_t\, dt+(r_tc_t)C_t\, dt d \varphi (t) \end{aligned}$$which coincides with the definition given earlier in (6).$$\begin{aligned} d \varphi (t) = (h_tf_t)H_t\, dt+(r_t  f_t)F_t\, dt+(r_tc_t)C_t\, dt \end{aligned}$$
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
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

at first to default time \(\tau \) we compute the closeout 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 closeout 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 closeout value \(\epsilon _t\) is equal to \(\widetilde{V}_t\) (this adds a source of nonlinearity with respect to choosing a riskfree 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 deltahedging 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

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

\(f(t,x,y,z)f(t,x',y',z')+g(x)g(x')\le K(xx'+yy'+zz')\);

\(f(t,0,0,0)+g(0)\le K\);
Theorem 2
Proof
Note that the Sdynamics 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

\(\mu (t,x)\mu (t,x')+\sigma (t,x)\sigma (t,x')\le Kxx'\)

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

\(f(t,x,y,z)f(t,x,y',z')\le K(yy'+zz')\)

\(g(x)+f(t,x,0,0)\le K(1+x^p)\)
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 deltahedge, it is actually more than a heuristic argument.
Moreover, since (17) does not depend on the riskfree 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 riskfree rate r(t).
This invariance result shows that even when starting from a riskneutral valuation theory, the riskfree 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, JeanFrançois 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”.
References
 1.Bielecki, T.R., Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging. Springer, Heidelberg (2002)MATHGoogle Scholar
 2.Bielecki, T.R., JeanblancPicqué, M., Rutkowski, M.: Credit Risk Modeling. Osaka University Press, Osaka (2009)MATHGoogle Scholar
 3.Brigo, D., Pallavicini, A.: Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrongway risks. J. Financ. Eng. 1(1), 1–60 (2014)CrossRefGoogle Scholar
 4.Bielecki, T.R., Rutkowski, M.: Valuation and hedging of contracts with funding costs and collateralization. arXiv preprint arXiv:1405.4079 (2014)
 5.Brigo, D., Liu, Q., Pallavicini, A., Sloth, D.: Nonlinear valuation under collateral, credit risk and funding costs: a numerical case study extending Black–Scholes. arXiv preprint at arXiv:1404.7314. A refined version of this report by the same authors is being published in this same volume
 6.Brigo, D., Morini, M., Pallavicini, A.: Counterparty Credit Risk, Collateral and Funding with Pricing Cases for all Asset Classes. Wiley, Chichester (2013)CrossRefMATHGoogle Scholar
 7.Delarue, F.: On the existence and uniqueness of solutions to FBSDEs in a nondegenerate case. Stoch. Process. Appl. 99(2), 209–286 (2002)MathSciNetCrossRefMATHGoogle Scholar
 8.Duffie, D., Huang, M.: Swap rates and credit quality. J. Financ. 51(3), 921–949 (1996)CrossRefGoogle Scholar
 9.Nie, T., Rutkowski, M.: A bsde approach to fair bilateral pricing under endogenous collateralization. arXiv preprint arXiv:1412.2453 (2014)
 10.Pallavicini, A., Brigo, D.: Interestrate modelling in collateralized markets: multiple curves, creditliquidity effects, CCPs. arXiv preprint arXiv:1304.1397 (2013)
 11.Pallavicini, A., Perini, D., Brigo, D.: Funding valuation adjustment: a consistent framework including CVA, DVA, collateral, netting rules and rehypothecation. arXiv preprint arXiv:1112.1521 (2011)
 12.Pallavicini, A., Perini, D., Brigo, D.: Funding, collateral and hedging: uncovering the mechanics and the subtleties of funding valuation adjustments. arXiv preprint arXiv:1210.3811 (2012)
 13.Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)MathSciNetCrossRefMATHGoogle Scholar
 14.Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Rozovskii, B., Sowers, R. (eds.) Stochastic Differential Equations and their Applications. Lecture Notes in Control and Information Sciences, vol. 176, pp. 200–217. Springer, Berlin (1992)CrossRefGoogle Scholar
 15.Zhang, J.: Some fine properties of backward stochastic differential equations, with applications. Ph.D. thesis, Purdue University. http://wwwbcf.usc.edu/~jianfenz/Papers/thesis.pdf (2001)
Copyright information
Open Access This chapter is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link is provided to the Creative Commons license and any changes made are indicated.
The images or other third party material in this chapter are included in the work’s Creative Commons license, unless indicated otherwise in the credit line; if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation, users will need to obtain permission from the license holder to duplicate, adapt or reproduce the material.