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Nonlinear valuation under credit, funding, and margins: Existence, uniqueness, invariance, and disentanglement,☆☆

https://doi.org/10.1016/j.ejor.2018.10.046Get rights and content

Highlights

  • We evaluate derivatives in presence of default risk, collateral, and funding costs.

  • We demonstrate the well posedness of the valuation equations.

  • We show that under delta hedging the valuation is independent from the risk-free rate.

  • We analyse the overlap of funding costs and treasury Debt Valuation Adjustment.

  • We illustrate how to consistently evaluate netting sets and portfolios.

Abstract

Since the 2008 global financial crisis, the banking industry has been using valuation adjustments to account for default risk and funding costs. These adjustments are computed separately and added together by practitioners as if the valuation equations were linear. This assumption is too strong and does not allow to model market features such as different borrowing and lending rates and replacement default closeout. Hence we argue that the full valuation equations are nonlinear, and this paper is devoted to studying the nonlinear valuation equations introduced in Pallavicini et al (2011).

We illustrate all the cash flows exchanged by the parties involved in a derivative contract, in presence of default risk, collateralisation with re-hypothecation and funding costs. Then we show how to obtain semi-linear PDEs or Forward Backward Stochastic Differential Equations (FBSDEs) from present-valuing said cash flows in an arbitrage-free setup, and we study the well-posedness of these PDEs and FBSDEs in a viscosity and classical sense.

Moreover, from a financial perspective, we discuss cases where classical valuation adjustments (XVA) can be disentangled. We show how funding costs are offset by treasury valuation adjustments when one takes a whole-bank perspective in the valuation, while the same costs are not offset by such adjustments when taking a shareholder perspective. We show that although we use a risk-neutral valuation framework based on a locally risk-free bank account, our final valuation equations do not depend on the risk-free rate. Finally, we show how to consistently derive a netting set valuation from a portfolio level one.

Introduction

Before the financial crisis of 2007–2008, the credit risk of counterparties with a high rating was not rigorously accounted for in the valuation of over-the-counter (OTC) derivatives. The credit events on Fannie Mae, Freddie Mac, Lehman Brothers, Washington Mutual, Landsbanki, Glitnir, and Kaupthing were definitely a hint that things would have needed to change. Another event that modified the financial landscape was the abrupt rise in the difference between the rate implied by overnight indexed swaps (OISs) and the LIBOR rate. The increment of this basis was a sign of the sudden realisation by market participants of the credit and liquidity risk present in the interbank market. These risks lead the dealers to reconsider the valuation of OTC claims and amend the book value of their deals by various adjustments. After the financial crisis the size of OTC markets has shrank but still remains important. For example the market value of outstanding OTC derivative contracts was reported to be $24.7 trillion end of 2012, with $632.6 trillion in notional value (BIS2013).

In the classical theory of derivative pricing, one usually evaluates a financial contract (for example an option or an interest rate swap) without taking into account the default risk of the contracting parties. Hence, the value of a contract can be in general determined by taking a conditional expectation, under a risk-neutral measure, of the discounted promised cash flows. As we have illustrated above, after the financial crisis of 2007–2008 the exposition towards counterparty risk for OTC contracts became of first importance. This is not to say that credit or counterparty risk were ignored before (see for example the reviews of the field Bielecki, Rutkowski, 2002, Duffie, Singleton, 2012, or the early work Brigo & Masetti, 2005), but after the financial crisis it became definitely a major topic in both practitioners and academic literature. More specifically, a particular way to account for default and credit risk gained a lot of traction, leading to the so called Valuation Adjustments. In particular banks already had a big infrastructure of pricing libraries to compute the value of contracts without accounting for default risks (the so called default-free value). Hence, it made a lot of sense to define adjustments to be added to the default-free price in order to recover the actual value of the contract being examined. The literature on valuation adjustments is very vast and has many contributions both from practitioners and academics. For a broad introduction to the topic, we suggest the monographs Brigo, Morini, and Pallavicini (2013) and Gregory (2010). Here we just highlight the aspects of the problem which are closer to this work.

The first valuation adjustment to consider is the so called credit valuation adjustment (CVA), which adjusts the value of a deal to take into account the possible losses due to the default of the counterparty in the considered contract. More precisely, CVA is a positive quantity, subtracted from the default-free price that is increasing with respect to the counterparty default probability. Clearly, in order to suffer a loss due to the default of a counterparty, the deal should generate a positive exposure towards said counterparty. For example if a bank sells a call option to a client, the bank is not exposed to the client default and hence this position does not need a CVA. On the other hand if the bank buys a call option from a client, then if the client defaults the bank might suffer losses due to the inability of the client of delivering the option’s payoff. CVA is very intuitive, in fact we are simply saying that the value of an option bought from a risky party is lower than one traded with a default free one. CVA is a well known adjustment, and market players account for it since a while and is well established in the literature (see for example Brigo & Masetti, 2005).

As we said CVA is the adjustment due to the default risk of our counterparty. On the other hand there are cases (e.g. we sell an option) where our counterparty is at risk because of our default. From our counterparty’s perspective, the price should be adjusted by a CVA. This adjustment, due to our default risk is commonly referred to as debit valuation adjustment (DVA). It can be also seen as the CVA seen from the eyes of the other party. Building on the previous call option example, a very natural thing to do is to consider contracts that have a bilateral exposure, for example forward rate agreements (FRAs). In this case the contract exposes both parties to default risk.

This kind of contracts present both a CVA and a DVA. While DVA has the merit of keeping valuation bilateral (our CVA is the counterparty DVA, and our DVA is the counterparty CVA), this adjustment has some counterintuitive consequences: for example, a dealer could book a profit as a consequence of a decrease of her credit quality. This is not just a theoretical possibility: for example Citigroup reported in a press release on the first quarter revenues of 2009 that “Revenues also included [...] a net 2.5$ billion positive CVA on derivative positions, excluding monolines, mainly due to the widening of Citi’s CDS spreads”. Accounting standards by the FASB accept DVA while the Basel Committee does not recognise it in the context of regulatory capital accounting. Another aspect of DVA is the difficulty of hedging it. In fact, one cannot sell protection on itself, and hence banks are left just with proxy hedging in order to cover changes in their own default intensity, leaving jump to default uncovered. See again Brigo et al. (2013) for a discussion. Valuation adjustments such as CVA and DVA aim at pricing default risk, on the other hand, one could adopt a more conservative approach and compute the economic capital relative to default risk. Clearly, this has to be done taking into account the dependence structure of market and credit risk, with the intrinsic computational burden that this implies, see for example Grundke (2009). Since default risk became a prominent risk factor, the market and regulators have tried to implement ways to reduce it. In particular, the use of collateralisation is now widespread in order to mitigate the exposure at default. In practice, the parties involved in a contract are usually required to post some assets every day as a guarantee in case they default (see Section 2.1 for a more detailed explanation). Collateralisation is definitely effective in reducing counterparty risk, and regulators such as the European Market Infrastructure Regulation (EMIR) are pushing for a high level of collateralisation in the market, enforcing stricter collateral rules for over the counter contracts. Another way to reduce the impact of default risk are netting agreements. Such an agreement between two parties allows them to net the exposures resulting from trades in the netting agreement. This is beneficial because it reduces outstanding exposures and avoids double counting. The last tools against default risk that we will mention are Central Counterparties (CCP). A central counterparty is a well capitalised financial institution that provides clearing and settlement services for derivatives. In particular CCPs act as intermediaries between two trading parties taking the risk of the counterparty default and ensuring that the payments are performed even in case of a default. To achieve this CCPs use both netting across clients and high margining standards. For more details on CCPs see Cont and Kokholm (2014); Duffie and Zhu (2011) while for valuation of CCP cleared contracts we refer the reader to Brigo and Pallavicini (2014). Clearly collateralisation plays a role in reducing default risk and consequently CVA and DVA charges. On the other hand, contagion and gap risk effects may lead to residual CVA and DVA, as was shown for the case of credit default swap trades in Brigo, Capponi, and Pallavicini (2014a). More general contagion at a more systemic level has been investigated for example in Barro and Basso (2010). Another important aspect, investigated in the literature, is the analysis of the so called wrong-way risk, i.e. when the exposure to a counterparty is negatively correlated with the credit quality of the counterparty itself. Analyses of the dependence structure between exposure and default risk are treated for example in Baviera, Bua, and Pellicioli (2016), while Biffis, Blake, Pitotti, and Sun (2015); Brigo et al. (2014a); Brigo and Chourdakis (2009) specialise the problem to particular products: Longevity Swaps and Credit Default Swaps respectively.

In more recent years, the funding valuation adjustment (FVA) was introduced. This adjustment takes into account the cost of funding the deal. Traders hedge the deal with a client on the market, maintaining a portfolio of positions in the underlying asset and cash. To maintain these positions, and to feed the collateral account that is usually attached to an OTC trade, the trader needs funds from the treasury of the bank, which in turn needs to raise this money from external funders. All the interest charges on these borrowing and lending activities contribute to the valuation of the deal, and this is precisely what the FVA accounts for. This adjustment can be quite sizable, for example Michael Rapoport reported in the Wall Street Journal, on Jan 14, 2014, that funding valuation adjustments costed J.P. Morgan Chase $1.5 billion in the fourth quarter results. The rise of this valuation adjustment is probably due to the fact that the spread between secured rates and unsecured ones has widened due to the perceived increase in the riskiness of the banking sector. This has risen the costs associated with the carrying of a derivative position. The literature that analyses the valuation of derivatives, accounting for funding and default risk is vast: Pallavicini, Perini, & Brigo, Pallavicini, Perini, Brigo, 2012 have a derivation of the valuation equations, Brigo, Liu, Pallavicini, and Sloth (2016) analyses the nonlinearity in said equations, while Bormetti, Brigo, Francischello, and Pallavicini (2017) deals with multicurve modeling (for a credit risk model in a multi-curve setup see for example Fanelli & crunch, 2016). From the practitioners side we have Burgard and Kjaer (2011); Piterbarg (2010) that frame the valuation problem into a PDE setting. For a backward stochastic differential equation approach, see instead Crépey and Song (2016); Nie and Rutkowski (2016). Another interesting aspect treated in the literature is how to reconcile the perspective of different agents related to the bank for what concerns funding and default costs. In particular see Andersen, Duffie, and Song (2018); Hull and White (2012) for an analysis of the shareholder/bondholder problem in assessing the costs of financing (this aspect is treated in Section 5.1) and Albanese and Andersen (2014); Burgard and Kjaer (2017) for a preliminary analysis of how to deal with netting sets (this subject will be treated in Section 5.2). Finally, for what concerns the numerical aspects of the problem see for example Kim and Leung (2016).

More recently, a capital valuation adjustment (KVA) is being discussed to account for the regulatory capital absorbed by the considered trade, see for example Green, Kenyon, and Dennis (2014). We will not treat KVA here since the industry didn’t reach a consensus on its definition yet.

Valuation adjustments are a useful instrument to partially assess the risk of a position but they might lead to think that the above mentioned risks and costs of a trading position are linear. The design of a framework for valuing trades in presence of all these market imperfections can easily become quite challenging. For example, even the simple fact that borrowing and lending do not happen at the same rate, or the use of replacement closeout upon first default, leads to nonlinear equations and hence to the sophisticated mathematical tools needed to handle them (the market features leading to these nonlinearities will be analysed in Section 2.1).

This paper builds on the valuation equations first introduced in Pallavicini et al. (2011) extending their scope and showing for the first time well posedness results for said equations. The valuation equations of Pallavicini et al. (2011) are the first to highlight the fixed point nature of the valuation in presence of funding costs and default risk. They represent also the first approach that allows to use both a risk-free or replacement closeout model (compare with Piterbarg, 2010 and Burgard & Kjaer, 2011). More specifically, we illustrate all the cash flows exchanged by the parties involved in a derivative contract, in presence of default risk, collateralisation with re-hypothecation and funding costs. Then, we show how to obtain a partial differential equaiton (PDE) and a forward backward differential equation (FBSDE) from present-valuing said cash flows in an arbitrage-free setup, and we study the well-posedness of these equations in a viscosity and classical sense. We highlight that our paper is the first work to treat both the PDE and the FBSDE of the valuation problem at a great detail level. The PDE perspective is useful since it highlights the links between our valuation problem and standard PDE based valuation techniques. On the other hand, the FBSDE formalism provides a connection with valuation via risk neutral expectations, powerful theorems for the regularity of the price (important for hedging), and alternative computational methods that can be used in high dimensions. We also show an interesting invariance result: while our starting point to derive our pricing equations is a classical risk-neutral valuation based on a locally risk-free rate, said rate does not appear in the final pricing equations that we obtain. This result frees us from having to infer the risk-free rate from market rates. In fact, if we choose as our hedging strategy a natural generalisation of the Black–Scholes delta hedging, our final PDEs and FBSDEs depend only on market quantities and contractual quantities. From a financial point of view we analyse the funding component of the value of a contract and deduce that it is offset by the funding benefits the bank faces in servicing trades when taking a whole-bank perspective. When these are negated, as in the case where one takes a shareholder perspective, funding costs remain. The whole bank versus shareholders debate was first brought to the general attention of the public in a basic setup in Hull and White (2012), and here we rigorously show under which assumptions it can be actually proven. Lastly, we also tackle the problem of how our pricing framework can be used in presence of multiple netting sets. This is a very realistic case of practical importance, and it is often neglected in the literature. In this work we show that, if we suppose that the bank is always net borrower on the market (banks usually are very leveraged companies), we can attribute a funding cost to each netting set and we can then carry on the valuation of the whole bank’s portfolio. This result is a rigorous improvement of the ones in Albanese and Andersen (2014); Burgard and Kjaer (2017).

The rest of the paper is structured as follows. In Section 2 we introduce the financial problem that we aim to solve: the pricing of a derivative contract subject to counterparty risk, collateralisation and funding. We describe the mathematical setup, and we derive a first valuation equation based on conditional expectations. In Section 3, we start by showing how, with a change of filtration technique, we derive a pricing equation for the pre-default value process of our derivative. Furthermore we show how this pricing equation leads naturally to a Markovian FBSDE. Finally we derive a semi-linear PDE by assuming regularity of the BSDE solution. In Section 4, we study conditions for well-posedness of the nonlinear valuation FBSDE and the associated PDE. We then present our invariance result: under a delta-hedging assumption, the solution does not depend on the risk-free rate. In Section 5, we analyse the Funding Valuation Adjustment, disentangling the different contribution and highlighting the cancellation between the funding term and the DVA on the funding strategy. Furthermore we reconcile the whole bank view with the netting sets one. In Section 6 we show how our theoretical contributions can be applied in a numerical example to concrete derivatives such as options or forwards. Lastly in Section 7 we summarise our contributions and illustrate future research directions. A number of longer proofs, an example of how funding cash flows originate, and a brief introduction to BSDEs are found in appendices.

Section snippets

Cash flows analysis and first valuation equation

In this section we explain in detail what is the financial problem that we will solve in the next sections. We take the perspective of a bank that enters an OTC derivative contract with a counterparty, for example an interest rate swap or an option.

Pre-default value process

In this section we want to obtain a BSDE for the pre-default value of the contract we are considering. Financially, this means that we want to obtain a formula for the price up to default, since we are not really interested in what happens after the default event. Mathematically, this means that we can simplify our problem. In fact saying that we are interested in the pre-default price means that we are interested in the quantity 1{t < τ}Vt, and while Vt will in general be G-adapted, we know

FBSDE well-posedness and invariance results

We now analyse the well-posedness of both the FBSDE and the PDE of the previous section. In particular we will show existence and uniqueness results in a strong sense, while for weak existence see for example Antonelli and Ma (2003). Moreover we will analyse the dependence of the solution from the risk-free rate rt, and show that under a delta hedging hypothesis, the value process Vt does not depend on the risk-free rate.

Disentanglement of valuation adjustments

In this section we analyse the financial implications of what we have shown so far. In particular we deal with the interplay between funding terms and DVA ones, and we tackle the problem of evaluating many netting sets. As we will show in the numerical section, funding terms are the same order of magnitude as CVA terms and hence their overlap with other adjustments should be properly accounted for. Netting set evaluation, is a problem of big practical relevance that, to the best of our

A numerical example

In this section we give an illustrative example of the size of the different valuation adjustments in a simple setting. Even if, generally, banks use more sophisticated models, we believe that our stylised model is able to capture the relative size of valuation adjustments, and make us understand impact of netting without the clutter of a full-blown model. For this purpose we consider a six months European call option and a six months equity forward. The underlying is a stock modeled by the

Conclusions

In summary, we proposed a framework for the evaluation of derivatives subject to counterparty risk, collateralisation ad funding costs. Via a thorough analysis of all the cash flows happening during the life span of such a derivative, we have shown how the evaluation problem is intrinsically nonlinear, since many of these cash flows depend on the value of the derivative itself. To properly account for those nonlinearities, the valuation of a derivative in our framework requires the solution of

Acknowledgments

We are grateful to Cristin Buescu, Jean-François Chassagneux, François Delarue, Federico Graceffa and Marek Rutkowski for helfpul 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.

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    An unpublished preprint version of this paper is avaiable at arXiv:1506.00686 and SSRN: 2613010. A related proceedings version Brigo, Francischello, & Pallavicini, Brigo, Francischello, Pallavicini, 2016, greatly extended and improved in the current manuscript, has appeared in Innovations in Derivatives Markets, Springer International Publishing, 2016.

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    The opinions here expressed are solely those of the authors and do not represent in any way those of their employers.

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