Innovations in Derivatives Markets pp 251-266 | Cite as

# Impact of Multiple-Curve Dynamics in Credit Valuation Adjustments

## Abstract

We present a detailed analysis of interest rate derivatives valuation under credit risk and collateral modeling. We show how the credit and collateral extended valuation framework in Pallavicini et al. (2011) can be helpful in defining the key market rates underlying the multiple interest rate curves that characterize current interest rate markets. We introduce the collateralized valuation measures and formulate a consistent realistic dynamics for the rates emerging from our analysis. We point out limitations of multiple curve models with deterministic basis considering valuation of particularly sensitive products such as basis swaps.

### Keywords

Multiple curves Evaluation adjustments Basis swaps Collateral HJM model## 1 Introduction

After the onset of the crisis in 2007, all market instruments are quoted by taking into account, more or less implicitly, credit- and collateral-related adjustments. As a consequence, when approaching modeling problems one has to carefully check standard theoretical assumptions which often ignore credit and liquidity issues. One has to go back to market processes and fundamental instruments by limiting oneself to use models based on products and quantities that are available on the market. Referring to market observables and processes is the only means we have to validate our theoretical assumptions, so as to drop them if in contrast with observations. This general recipe is what is guiding us in this paper, where we try to adapt interest rate models for valuation to the current landscape.

A detailed analysis of the updated valuation problem one faces when including credit risk and collateral modeling (and further funding costs) has been presented elsewhere in this volume, see for example [6, 7]. We refer to those papers and references therein for a detailed discussion. Here we focus our updated valuation framework to consider the following key points: (i) focus on interest rate derivatives; (ii) understand how the updated valuation framework can be helpful in defining the key market rates underlying the multiple interest rate curves that characterize current interest rate markets; (iii) define collateralized valuation measures; (iv) formulate a consistent realistic dynamics for the rates emerging from the above analysis; (v) show how the framework can be applied to valuation of particularly sensitive products such as basis swaps under credit risk and collateral posting;(vi) point out limitations in some current market practices such as explaining the multiple curves through deterministic fudge factors or shifts where the option embedded in the credit valuation adjustment (CVA) calculation would be priced without any volatility. For an extended version of this paper we remand to [3]. This paper is an extended and refined version of ideas originally appeared in [24].

## 2 Valuation Equation with Credit and Collateral

Classical interest-rate models were formulated to satisfy no-arbitrage relationships by construction, which allowed one to price and hedge forward-rate agreements in terms of risk-free zero-coupon bonds. Starting from summer 2007, with the spreading of the credit crunch, market quotes of forward rates and zero-coupon bonds began to violate usual no-arbitrage relationships. The main driver of such behavior was the liquidity crisis reducing the credit lines along with the fear of an imminent systemic break-down. As a result the impact of counterparty risk on market prices could not be considered negligible any more.

*t*for maturity

*T*,

*L*is the LIBOR rate and

*F*is the related LIBOR forward rate. A direct consequence is the impossibility to describe all LIBOR rates in terms of a unique zero-coupon yield curve. Indeed, since 2009 and even earlier, we had evidence that the money market for the Euro area was moving to a multi-curve setting. See [1, 19, 20, 27].

### 2.1 Valuation Framework

In order to value a financial product (for example a derivative contract), we have to discount all the cash flows occurring after the trading position is entered. We follow the approach of [25, 26] and we specialize it to the case of interest-rate derivatives, where collateralization usually happens on a daily basis, and where gap risk is not large. Hence we prefer to present such results when cash flows are modeled as happening in a continuous time-grid, since this simplifies notation and calculations. We refer to the two names involved in the financial contract and subject to default risk as investor (also called name “I”) and counterparty (also called name “C”). We denote by \(\tau _I\), and \(\tau _C\), respectively, the default times of the investor and counterparty. We fix the portfolio time horizon \(T>0\), and fix the risk-neutral valuation model \((\varOmega ,\mathscr {G},\mathbb {Q})\), with a filtration \((\mathscr {G}_t)_{t \in [0,T]}\) such that \(\tau _C\), \(\tau _I\) are \((\mathscr {G}_t)_{t \in [0,T]}\)-stopping times. We denote by \(\mathbb {E}_{t}\left[ \,\cdot \,\right] \) the conditional expectation under \(\mathbb {Q}\) given \(\mathscr {G}_t\), and by \(\mathbb {E}_{\tau _i}\left[ \,\cdot \,\right] \) the conditional expectation under \(\mathbb {Q}\) given the stopped filtration \(\mathscr {G}_{\tau _i}\). We exclude the possibility of simultaneous defaults, and define the first default event between the two parties as the stopping time \(\tau := \tau _C \wedge \tau _I \,\).

We will also consider the market sub-filtration \(({\mathscr {F}}_t)_{t \ge 0}\) that one obtains implicitly by assuming a separable structure for the complete market filtration \(({\mathscr {G}}_t)_{t \ge 0}\). \({\mathscr {G}}_t\) is then generated by the pure default-free market filtration \({\mathscr {F}}_t\) and by the filtration generated by all the relevant default times monitored up to *t* (see for example [2]).

*r*associated with the risk-neutral measure. We therefore need to define the related stochastic discount factor

*D*(

*t*,

*u*,

*r*) that in general will denote the risk-neutral default-free discount factor, given by the ratio

*B*is the bank account numeraire, driven by the risk-free instantaneous interest rate \(r_t\) and associated to the risk-neutral measure \(\mathbb {Q}\). This rate \(r_t\) is assumed to be \((\mathscr {F}_t)_{t \in [0,T]}\) adapted and is the key variable in all pre-crisis term structure modeling.

\(\pi _u\) is the coupon process of the product, without credit or debit risk and without collateral cash flows;

\(C_u\) is the collateral process, and we use the convention that \(C_u>0\) while

*I*is the collateral receiver and \(C_u<0\) when*I*is the collateral poster. \(( r_u - c_u ) C_u\) are the collateral margining costs and the collateral rate is defined as \(c_t := c^+_t 1_{\{C_t>0\}} + c^-_t 1_{\{C_t<0\}}\) with \(c^\pm \) defined in the CSA contract. In general we may assume the processes \(c^+,c^-\) to be adapted to the default-free filtration \({\mathscr {F}}_t\).\(\theta _{u}=\theta _{u}(C,\varepsilon )\) is the on-default cash flow process that depends on the collateral process \(C_u\) and the close-out value \(\varepsilon _u\).

^{1}It is primarily this term that originates the credit and debit valuation adjustments (CVA/DVA) terms, that may also embed collateral and gap risk due to the jump at default of the value of the considered deal (e.g. in a credit derivative), see for example [5].

Notice that the above valuation equation (2) is not suited for explicit numerical evaluations, since the right-hand side is still depending on the derivative price via the indicators within the collateral rates and possibly via the close-out term, leading to recursive/nonlinear features. We could resort to numerical solutions, as in [11], but, since our goal is valuing interest-rate derivatives, we prefer to further specialize the valuation equation for such deals.

### 2.2 The Master Equation Under Change of Filtration

*x*and \((x)^-=-(-x)^+\). For an extended discussion of the term \(\theta _\tau \) we refer to [3]. Moreover, to derive an explicit valuation formula we assume that gap risk is not present, namely \(\tilde{V}_{\tau -} = \tilde{V}_\tau \), and we consider a particular form for collateral and close-out prices, namely we model the close-out value as

^{2}

### Proposition 1

## 3 Valuing Collateralized Interest-Rate Derivatives

As we mentioned in the introduction, we will base our analysis on real market processes. All liquid market quotes on the money market (MM) correspond to instruments with daily collateralization at overnight rate (\(e_t\)), both for the investor and the counterparty, namely \(c_t \doteq e_t \,\).

Notice that the collateral accrual rate is symmetric, so that we no longer have a dependency of the accrual rates on the collateral price, as opposed to the general master equation case. Moreover, we further assume \(r_t \doteq e_t \,\).

This makes sense because \(e_t\) being an overnight rate, it embeds a low counterparty risk and can be considered a good proxy for the risk-free rate \(r_t\). We will describe some of these MM instruments, such as OIS and Interest Rate Swaps (IRS), along with their underlying market rates, in the following sections. For the remaining of this section we adopt the perfect collateralization approximation of Eq. (1) to derive the valuation equations for OIS and IRS products, hence assuming no gap-risk, while in the numeric experiments of Sect. 4 we will consider also uncollateralized deals. Furthermore, we assume that daily collateralization can be considered as a continuous-dividend perfect collateralization. See [4] for a discussion on the impact of discrete-time collateralization on interest-rate derivatives.

### 3.1 Overnight Rates and OIS

*x*and maturity

*T*is given by

*K*is the fixed rate payed by the OIS. Furthermore, we can introduce the (par) fix rates \(K=E_t(T,x;e)\) that make the one-period OIS contract fair, namely priced 0 at time

*t*. They are implicitly defined via

^{3}as

### 3.2 LIBOR Rates, IRS and Basis Swaps

*T*and tenor

*x*is given by

*K*is the fix rate payed by the IRS. Furthermore, we can introduce the (par) fix rates \(K=F_t(T,x;e)\) that render the one-period IRS contract fair, i.e. priced at zero. They are implicitly defined via

*T*-forward measure \(\mathbb {Q}^{T;e}\),

*T*-forward measure and in particular, we can write LIBOR forward rates as

### 3.3 Modeling Constraints

^{4}Brownian motions with correlation matrix \(\rho \), and the volatility vector processes \(\sigma ^P\) and \(\sigma ^F\) may depend on bonds and forward LIBOR rates themselves.

## 4 Interest-Rate Modeling

We can now specialize our modeling assumptions to define a model for interest-rate derivatives which is on one hand flexible enough to calibrate the quotes of the MM, and on the other hand robust. Our aim is to use an HJM framework using a single family of Markov processes to describe all the term structures and interest rate curves we are interested in.

In the literature many authors proposed generalizations of the HJM framework to include multiple yield curves. In particular, we cite the works of [12, 13, 14, 16, 20, 21, 22, 23]. A survey of the literature can be found in [17].

In such works the problem is faced in a pragmatic way by considering each forward rate as a single asset without investigating the microscopical dynamics implied by liquidity and credit risks. However, the hypothesis of introducing different underlying assets may lead to over-parametrization issues that affect the calibration procedure. Indeed, the presence of swap and basis-swap quotes on many different yield curves is not sufficient, as the market quotes swaption premia only on few yield curves. For instance, even if the Euro market quotes one-, three-, six- and twelve-month swap contracts, liquidly traded swaptions are only those indexed to the three-month (maturity one-year) and the six-month (maturities from two to thirty years) Euribor rates. Swaptions referring to other Euribor tenors or to overnight rates are not actively quoted.

In order to solve such problem [23] introduces a parsimonious model to describe a multi-curve setting by starting from a limited number of (Markov) processes, so as to extend the logic of the HJM framework to describe with a unique family of Markov processes all the curves we are interested in.

### 4.1 Multiple-Curve Collateralized HJM Framework

- (i)
existence of OIS rates, which we can describe in terms of instantaneous forward rates \(f_t(T;e)\);

- (ii)
existence of LIBOR rates assigned by the market, typical underlyings of traded derivatives, with associated forwards \(F_t(T,x;e)\);

- (iii)
no arbitrage dynamics of the \(f_t(T;e)\) and the \(F_t(T,x;e)\) (both being (

*T*,*e*)-forward measure martingales); - (iv)
possibility of writing both \(f_t(T;e)\) and \(F_t(T,x;e)\) as functions of a common family of Markov processes, so that we are able to build parsimonious yet flexible models.

*N*-dimensional) volatility processes \(\sigma _t(T)\) and \(\varSigma _t(T,x),\) the vector of

*N*independent \(\mathbb {Q}^{T;e}\)-Brownian motions \(W_t^{T;e},\) and the set of deterministic shifts

*k*(

*T*,

*x*), such that \(\lim _{x \rightarrow 0} x k(T,x) = 1\). This limit condition ensures that the model approaches a standard default- and liquidity-free HJM model when the tenor goes to zero. We bootstrap \(f_0(T;e)\) and \(F_0(T,x;e)\) from market quotes.

*q*(

*u*,

*T*,

*x*) is a deterministic \(N\times N\) diagonal matrix function, and

*a*(

*s*) is a deterministic

*N*-dimensional vector function. The condition on

*q*(

*u*;

*T*,

*x*) being the identity matrix, when \(T=u\) ensures that a standard HJM framework holds for collateralized zero-coupon bonds.

*a*is a constant vector, and

*R*is the Cholesky decomposition of the correlation matrix that we want our \(X_t\) vector to have. In this case we obtain \(\sigma _t(u;T,x)= R \cdot e^{ - a (u-t) }\), where the exponential is intended to be component-wise. Then we note that \(X_t\) is a mean reverting Gaussian process while the \(Y_t\) process is deterministic.

*h*, as in [29]. More precisely we replace

*h*(

*t*) in (11) with \(h(t)\doteq \sqrt{v_t} R\). With

*a*and

*R*as before and \(v_t\) being a process with the following dynamic:

*k*(

*T*,

*x*), and introduces a dependence on the tenor in the volatility process.

*a*and

*R*as before and \(v_t\) being defined by (12). Here we have for the volatility \(\sigma _t(u;T,x) =\sqrt{v_t} R \cdot e^{\eta x - a (u-t) }\).

### 4.2 Numerical Results

We apply our framework to simple but relevant products: an IRS and a basis swap. We analyze the impact of the choice of an interest rate model on the portfolio valuation, in particular we measure the dependency of the price on the correlations between interest-rates and credit spreads, the so-called wrong-way risk. We model the market risks by simulating the following processes in a multiple-curve HJM model under the pricing measure \(\mathbb {Q}\). The overnight rate \(e_t\) and the LIBOR forward rates \(F_t(T;e)\) are simulated according to the dynamics given in Sect. 4.1. Maintaining the same notation of the aforementioned section, we choose \(N=2\), and for our numerical experiments we use a HW model, a Ch model and an MP model, all calibrated to swaption at-the-money volatilities listed on the European market.

We now analyze the impact of wrong-way risk on the bilateral adjustment, namely CVA plus DVA, of IRS and basis swaps when collateralization is switched off, namely we want to evaluate Eq. (1) when \(\alpha _t\doteq 0\). For an extended analysis see [3]. Wrong-way risk is expressed with respect to the correlation between the default intensities and a proxy of market risk, namely the short rate \(e_t\).

In Fig. 1 we show the variation of the bilateral adjustment for a ten years IRS receiving a fix rate yearly and paying 6 m Libor twice a year and for a ten years basis swap receiving 3 m Libor plus spread and paying 6 m Libor. It is clear that for a product like the IRS, not subject to the basis dynamic, we have that the big difference among the models is the presence of a stochastic volatility. In fact we can see that the Ch model and the MP model are almost indistinguishable while the results of the HW model are different from the stochastic volatility ones. Moreover we can observe that all the models have the same trend, i.e. the bilateral adjustment grows as correlation increase. In fact this can be explained by the fact that a higher correlation means that the deal will be more profitable when it will be more risky (since we are receiving the fixed rate and paying the floating one), hence the bilateral adjustment will be bigger.

In the case of a basis swap instead, we see that, as said before, the HW model and the Ch model do not have a basis dynamic and hence the curve represented is almost flat. On the other hand the MP model is able to capture the dynamics of the basis and hence we can see that the more the overnight rate is correlated with the credit risk the smaller the bilateral adjustment becomes.

We conclude by pointing out that our analysis will be extended to partially collateralized deals in future work. In such a context funding costs enter the picture in a more comprehensive way. Some initial suggestions in this respect were given in [24].

## Footnotes

- 1.
The closeout value is the residual value of the contract at default time and the CSA specifies the way it should be computed.

- 2.
- 3.
Notice that we are only defining a price process for hypothetical collateralized zero-coupon bond. We are not assuming that collateralized bonds are assets traded on the market.

- 4.
In the following we will consider

*N*-dimensional vectors as \(N\times 1\) matrices. Moreover, given a matrix*A*, we will indicate \(A^*\) its transpose, and if*B*is another conformable matrix we indicate*AB*the usual matrix product.

## Notes

### Acknowledgements

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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