Innovations in Derivatives Markets pp 385403  Cite as
Negative Basis Measurement: Finding the Holy Scale
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Abstract
Investing into a bond and at the same time buying CDS protection on the same bond is known as buying a basis package. Loosely speaking, if the bond pays more than the CDS protection costs, the position has an allegedly riskfree positive payoff known as “negative basis”. However, several different mathematical definitions of the negative basis are present in the literature. The present article introduces an innovative measurement, which is demonstrated to fit better into arbitrage pricing theory than existing approaches. This topic is not only interesting for negative basis investors. It also affects derivative pricing in general, since the negative basis might act as a liquidity spread that contributes as a net funding cost to the value of a transaction; see Morini and Parampolini (Risk, 58–63, 2011, [23]).
Keywords
Negative basis measurement BondCDS basis Hidden yield1 Introduction
On first glimpse, it is surprising that investing into a bond and buying CDS protection on that underlying bond, henceforth called a basis package, can earn an attractive spread on top of the riskfree rate of return, as it appears to be free of default risk. This excess return over the riskfree rate is informally called negative basis ^{1}.; more formal definitions are given in the main body of this article. [8] has even devoted an entire book to the topic. If, conversely, the cost of CDS protection exceeds the bond earnings, one speaks of a positive basis. In this article, we only speak of negative bases, as fundamentally the concepts of positive and negative basis are simply inverse.
The appropriate measurement of negative basis plays an important role with regard to the cost of funding literature, which has become of paramount interest in the financial industry since the recent liquidity crisis. Generally speaking, this stream of literature reconsiders the pricing of derivatives under the new postcrisis fundamentals regarding funding, liquidity, and credit risk issues. Substantial contributions have been made, among others, by [5, 7, 12, 13, 23, 27, 29]. Loosely speaking, most references agree upon the fact that, at least under certain simplifying assumptions (full, bilateral, and continuous collateralization), derivative contracts can be evaluated in the traditional way, only the involved discount factors have to be adjusted by means of a spread accounting for funding and liquidity charges. In particular, [23] show in a simple, theoretical framework that the negative basis is a spread which plays an essential role in this regard. In order to set these theoretical findings into action in the industry’s pricing machinery, it is therefore an essential task to establish viable and reasonable measurements for the negative basis. The present article shows that this topic is not only important but also challenging, and contributes a careful comparison of three different measurement methods. In particular, we point out why the most common measurement approaches (denoted by (Z) and (PE) below) are not recommended, and propose a decent alternative.

Difference between Zspread of the bond and CDS running spread, as presented, e.g., in [8], and defined by Bloomberg on the screen YAS.

Parequivalent CDSmethodology, as described in the Appendix of [2], who apply this definition for an empirical study, see also [3].

A hidden yield approach that assumes the riskfree discounting curve to be a reference interest rate curve shifted by the (initially unknown) negative basis.
Important to note is that, according to all these definitions, a negative basis is assigned to a bond, not to an issuer. This means that two different bonds issued by the same company are allowed to have two different negative bases. This viewpoint stands in glaring contrast to some of the more macroeconomic considerations carried out in references cited in the next section. CDS protection typically refers to a whole battery of eligible bonds by a reference issuer, and normally the major driver for CDS spreads is considered to be the issuer’s default risk. However, some of the deliverable bonds might trade at diverse yields for reasons other than the issuer’s default risk—for instance legal issues, liquidity issues, or funding issues, cf. [21] and Sect. 2.
The rest of this article is organized as follows. Section 2 recalls reasons for the existence of negative basis. Section 3 introduces general notations, which are used throughout the remaining sections. Section 4 reviews the traditional methods (Z) and (PE), Sect. 5 discusses the innovative method (HY), and Sect. 6 concludes.
2 Why Does Negative Basis Exist?
There are a couple of intuitive explanations for the existence of negative basis, see, e.g., [1, 2, 4, 6, 10, 19, 24, 26, 30]. For the convenience of the reader, we briefly recall some of them in the sequel.

Liquidity issues: Some bond issues are distributed only among a few investors. If one of these investors has to sell her bonds, for instance due to regulatory requirements or demand for liquidity, supply may exceed demand and thus the price of the bond must drop significantly in order for the bond to be sold. At the same time the CDS price might remain unaffected.

Funding costs: From a pure credit risk perspective, selling CDS protection economically is the same risk as buying the underlying bond. However, buying a bond requires an initial investment that must be funded, whereas selling CDS protection typically requires much less initial funding (unless the CDS upfront exceeds the bond price). Therefore, in times of high funding costs there is an incentive to sell CDS rather than to buy bonds, which might lead to an increase in supply of CDS protection, making it cheap relative to bond prices.

Market segmentation: Empirical observations suggest that bond trades sometimes have larger volumes and might be motivated much less by quantitative aspects than CDS trades. Arguing similarly, [6, p. 5, l. 5–7] conjecture that “marketimplied [risk] measures have a stronger impact on the CDS market, while the more easily available rating information affects the bond market more strongly”. Such instrumentspecific differences might contribute to the existence of negative basis.

Legal risk: The bond of the negative basis position might bear certain risks that cannot be protected against by means of a CDS. Examples are certain collective action clauses, debt restructuring events, or call rights for the bond issuer. Such “legal gaps” explain parts of the negative basis.

Counterparty credit risk: A joint default of both the CDS counterparty and the issuer of the bond could lead to a loss for the basis position.^{2} These potential losses imply that CDS protection is not \(100\,\%\) and consequently might contribute to the negative basis, see, e.g., [5, 22].

Marktomarket risk: The negative basis might further increase after one has entered into the position, due to one of the aforementioned reasons. In this case, one loses money due to marktomarket balancing. In theory, one gets this money back eventually, but it might occur that marktomarket losses exceed one’s personal tolerance level during the bond’s lifetime. In this case, one has to exit the position and realize the loss. This risk is especially significant if the negative basis position is levered (which has happened heavily during the financial crisis). Part of the negative basis might be viewed as a risk premium for taking this marktomarketrisk.
Basis “arbitrageurs” are investors that try to earn the negative basis by investing into basis packages. This means that they consider the negative basis an adequate compensation for taking the aforementioned risks. In classical arbitrage theory, their appearance improves trading liquidity. Counterintuitively, however, [9] argue that the advent of CDS was detrimental to bond markets and [20] find some evidence that basis arbitrageurs bring new risks into the corporate bond markets.
3 General Notations
All definitions to follow rely on the pricing of CDS and a plain vanilla coupon bond according to the most simple mathematical setup we can think of. This is in order to make the article as readerfriendly as possible; furthermore, we think the setup is already rich enough in order to convey the main ideas. The only randomness considered in the present article is the default time of the bond issuer, which is formally defined on a probability space \((\varOmega ,\mathscr {F},\mathbb {Q})\), with state space \(\varOmega \), \(\sigma \)algebra \(\mathscr {F}\), and probability measure \(\mathbb {Q}\). Expected values with respect to the pricing measure \(\mathbb {Q}\) are denoted by \(\mathbb {E}\). The default intensity \(\lambda (.)\) of the issuer’s default time \(\tau \) is assumed to be deterministic, i.e. \(\mathbb {Q}(\tau >t)=\exp (\int _{0}^{t}\lambda (s)\,\mathrm {d}s)\). Sometimes the function \(\lambda (.)\) is constant, sometimes piecewise constant, depending on our application. For example, the computation of a socalled Zspread requires \(\lambda (.)\) to be constant,^{3} whereas the joint consistent pricing of several CDS quotes with different maturities requires \(\lambda (.)\) to be piecewise constant.

We ignore recovery risk: Upon default, the bond holder receives the constant proportion \(R \in [0,1]\) of her nominal. Default is assumed to instantaneously trigger a credit event of the CDS. The bond is assumed to be a deliverable security in the auction following the CDS trigger event, and the auction process is assumed to yield the same recovery rate R. Although this is an unrealistic assumption in principle (see, e.g., [17]), a negative basis investor can always eliminate recovery risk by delivering his bonds into the auction (physical settlement), in which case he gets compensated by the (nominalmatched) CDS for the nominal loss of the bond.^{4} Consequently, our assumption is not severe for the present purpose.

We ignore interest rate risk: The discounting curve is deterministic and the discount factors are denoted by \(DF(t):=\exp (\int _{0}^{t}r(s)\,\mathrm {d}s)\) with some given deterministic short rate function r(.). All presented negative basis figures are measurements relative to the applied short rate function r(.).

\({t^{(B)}_j}\) denotes the coupon payment dates of the bond.

The bond’s lifetime is denoted by T, i.e. T denotes the last coupon payment date, which at the same time is the redemption date. Moreover, the bond is assumed to pay a constant coupon rate C at each coupon payment date.

\({t^{(C)}_i}\) denotes the payment dates of the considered CDS contracts, which typically are quarterly on the 20th of March, June, September, and December, respectively, according to the terms and conditions of ISDA standard contracts.^{5}

For a CDS with maturity T, the (usually standardized) running coupon is denoted by s(T) and the upfront payment to be made at CDS settlement by \(\text {upf}(T)\).
 The expected discounted value of the sum of all premium payments to be made by the CDS protection buyer (the premium leg) is denoted by^{6}$$\begin{aligned}&EDPL(\lambda (.),r(.),s(T),\text {upf}(T),T) \\&\qquad :=\text {upf}(T)+s(T)\,\!\!\!\sum _{0< t_i^{(C)} \le T}\!\!\!\big (t^{(C)}_it^{(C)}_{i1}\big )\,DF\big (t_i^{(C)}\big )\,\mathbb {Q}\big (\tau >t_i^{(C)}\big ) \\&\qquad =\text {upf}(T)+s(T)\,\!\!\!\sum _{0 < t_i^{(C)} \le T}\!\!\!\big (t^{(C)}_it^{(C)}_{i1}\big )\,DF\big (t_i^{(C)}\big )\,e^{\int _{0}^{t_i^{(C)}}\!\!\lambda (s)\,\mathrm {d}s}. \end{aligned}$$
 The expected discounted value of the sum of all default compensation payments to be made by the CDS protection seller (the default/protection leg) is denoted by$$\begin{aligned} EDDL(\lambda (.),r(.),R,T):&=(1R)\,\mathbb {E}[1_{\{\tau \le T\}}\,DF(\tau )] \\&= (1R)\,\int _{0}^{T}DF(y)\,\lambda (y)\,e^{\int _{0}^{y}\lambda (s)\,\mathrm {d}s}\,\mathrm {d}y. \end{aligned}$$
 The model price of the bond is given by$$\begin{aligned} \text {Bond}(\lambda (.),r(.),R,C,T)&:=C\,\!\!\!\sum _{0<t_j^{(B)}\le T}\!\!\! \big (t_j^{(B)}t_{j1}^{(B)}\big )\,DF\big (t_j^{(B)}\big )\,\mathbb {Q}\big (\tau>t_j^{(B)}\big ) \\&\quad \,\, +DF(T)\,\mathbb {Q}(\tau >T)+ R \,\mathbb {E}[1_{\{\tau \le T\}}\,DF(\tau )]\\&=C\,\!\!\!\sum _{0<t_j^{(B)}\le T} \!\!\!\big (t_j^{(B)}t_{j1}^{(B)}\big )\,DF\big (t_j^{(B)}\big )\,e^{\int _{0}^{t_j^{(B)}}\!\!\lambda (s)\,\mathrm {d}s}\\&\quad +DF(T)\,e^{\int _{0}^{T}\lambda (s)\,ds}+ R \,\int _{0}^{T}DF(y)\,\lambda (y)\,e^{\int _{0}^{y}\lambda (s)\,\mathrm {d}s}\,\mathrm {d}y. \end{aligned}$$
4 Traditional Measurements
4.1 The ZSpread Methodology
The main idea of the Zspread methodology is to define the negative basis as the difference between (expected) annualized bond earnings and annualized protection costs. This method is described, e.g., in [8]. The negative basis \(NB^{(Z)}\) is computed by the following algorithm.
Definition 1
 1.
A reference discounting curve, resp. the associated short rate r(.), is chosen and used in all subsequent steps, e.g. bootstrapped from quoted prices for interest rate derivatives according to one of the methods described in [15, 16].
 2.
From a term structure of quoted CDS with different maturities, piecewise constant intensities \(\lambda (.)\) are bootstrapped, as described, e.g., in [25]. For this, a recovery assumption is made, i.e. R is model input.^{7}
 3.Denoting by B the quoted market price of the bond, the bond’s Zspread z is defined as the root of the function^{8}if existent. In words, the Zspread is the amount by which the reference short rate r(.) needs to be shifted parallelly in order for the discounted bond cash flows to match the market quote. The root, whenever existing at all, is unique.$$\begin{aligned} x \mapsto \text {Bond}(x,r(.),0,C,T)  B, \end{aligned}$$(1)
 4.The (zeroupfront) running CDS spread s(T) for a CDS contract, whose maturity matches the bond’s maturity, is defined asi.e. the fair running spread when no upfront payment is present.$$\begin{aligned} s(T):=\frac{EDDL(\lambda (.),r(.),R,T)}{EDPL(\lambda (.),r(.),1,0,T)}, \end{aligned}$$
 5.
\(NB^{(Z)}:=zs(T)\).
Intuitively, the Zspread z is a measure of the annualized excess return of the bond on top of the “riskfree” rate r(.), whereas s(T) is the annualized CDS protection cost. Hence, \(NB^{(Z)}\) equals the difference between earnings and costs (expected in case of survival). If the function (1) does not have a root in \((0,\infty )\), this means that the bond is less risky than the default risk intrinsic in the chosen discounting curve r(.). Especially since the liquidity crisis, when the interbank money transfer ran dry, significant spreads between discounting curves obtained from overnight rates and LIBORbased swap rates are observed. Consequently, one could recognize, e.g., German government bonds with a “negative Zspread” with respect to the interest rate curve r(.), which was obtained from 6month EURIBOR swap rates. For such reasons it has become market standard to extract the “riskfree” discounting curve from overnight rates rather than from LIBORbased swap rates. Moreover, [19] point out that the difference between bond yields and CDS spreads can depend on whether treasury rates or swap rates are used for discounting. Since negative basis investors are typically trading in the high yield sector, the function (1) normally does have a root in \((0,\infty )\) for several canonical choices of r(.), be it extracted from swap rates with overnight tenor, 3month tenor, or 6month tenor. But it is important to stress that all presented negative basis measurements are always relative measures depending on the applied interest rate curve r(.).

Imprecision: Earnings and costs are not measured accurately, but only approximately. The Zspread is only a rough estimate for the expected annualized earnings, and the zeroupfront running CDS spread is also not really tradable, but only a fictitious quantity. Furthermore, the Zspread is earned on the bond value, whereas the CDS spread is paid on the (bond and) CDS nominal, which may result in a nonsense measurement for bonds trading away from par, see Example 1 below. To this end, [10] proposes to replace the Zspread by an asset swap spread. It is possible to define more accurate measurements of earnings and costs taking into account actual cash flows. However, in the present article we do not elaborate on these finetunings, since the “earnings and costs”perspective in general suffers from the following second difficulty.

Inaccurate hedge: The measurement assumes that bond and CDS have the same maturity and nominals and furthermore implicitly assumes a survival until maturity. Upon a default event the PnL of the position might be considerably different, depending on the timing of the default, see Fig. 1 in Example 1 below. Hence, the assumed CDS hedge cannot really be considered to be defaultrisk eliminating (it might either profit from or lose on a default event), and consequently the number \(NB^{(Z)}\) does not deserve to be called a return figure after elimination of default risk, which the negative basis should be in our opinion.
4.2 The ParEquivalent CDS Methodology
The parequivalent CDS methodology is described in the Appendix of [2]. A similar idea is also outlined in [8, p. 101 ff] and [3]. The negative basis \(NB^{(PE)}\) is computed along the steps of the following algorithm.
Definition 2
 1.
A reference discounting curve, resp. the associated short rate r(.), is chosen and used in all subsequent steps, e.g. bootstrapped from quoted prices for interest rate derivatives according to one of the methods described in [15, 16].
 2.
From a term structure of CDS contracts on the reference entity, piecewise constant intensities \(\lambda (.)\) are bootstrapped, as described, e.g., in [25]. For this a recovery assumption is made, i.e. R is model input.
 3.The (zeroupfront) running CDS spread s(T) for a CDS contract, whose maturity matches the bond’s maturity, is defined asi.e. the fair running spread when no upfront payment is present.$$\begin{aligned} s(T):=\frac{EDDL(\lambda (.),r(.),R,T)}{EDPL(\lambda (.),r(.),1,0,T)}, \end{aligned}$$
 4.Denoting by B the quoted market price of the bond, a shift \(\tilde{z}\) is defined as the root of the functionif existent. In words, the bond is priced with the default intensities \(\lambda (.)\) that are consistent with CDS quotes, which are then shifted parallelly until the bond’s market quote is matched.$$\begin{aligned} x \mapsto \text {Bond}(\lambda (.)+x,r(.),R,C,T)  B, \end{aligned}$$
 5.A second (zeroupfront) running CDS spread \(\tilde{s}(T)\) for a CDS contract, whose maturity matches the bond’s maturity, is defined asi.e. the fair spread when no upfront payment is present, but now with the shifted intensity rates \(\lambda (.)+\tilde{z}\), which are required in order to price the bond correctly.$$\begin{aligned} \tilde{s}(T):=\frac{EDDL(\lambda (.)+\tilde{z},r(.),R,T)}{EDPL(\lambda (.)+\tilde{z},r(.),1,0,T)}, \end{aligned}$$
 6.
\(NB^{(PE)}:=\tilde{s}(T)s(T)\).
The main idea of (PE) is to question the default probabilities bootstrapped from the given CDS quotes, and to adjust them in order to match the bond quote. On a high level, this negative basis measurement is based on the difference between default probabilities that are required in order to match the bond price and default probabilities that are required in order to fit the CDS quotes.

No link to arbitrage pricing theory: In our view, there is no convincing economic argument as to why two different survival functions for the same default time should be used. In particular, the method provides no joint pricing model for bond and CDS that explains the negative basis as one of its parameters. The method is “decoupled” from arbitrage pricing theory.

No link to “earnings and costs”perspective: Unlike the method (Z), the method (PE) does not have a clear link to an earnings measure above a reference rate, which is what the negative basis is informally thought of.
5 An Innovative Methodology
In our opinion, the negative basis should be a spread on top of a reference discounting curve which can be earned without exposure to default risk. This means we question the usual assumption that the applied discounting curve r(.) is the appropriate riskfree rate to be used, because there is actually a higher rate that can be earned “riskfree” (recalling that default risk is the only risk within our tiny model). This motivates what we call the hidden yield approach. The negative basis \(NB^{(HY)}\) is computed along the steps of the following algorithm.
Definition 3
 1.
A reference discounting curve, resp. the associated short rate r(.), is chosen and used in all subsequent steps, e.g. bootstrapped from quoted prices for interest rate derivatives according to one of the methods described in [15, 16].
 2.
Denote by \(\lambda _x(.)\) the piecewise constant intensity rates that are bootstrapped from CDS market quotes, when the assumed discounting curve is \(r(.)+x\), as described, e.g., in [25]. The recovery rate R is fixed and chosen as model input.
 3.The negative basis \(NB^{(HY)}\) is defined as the root^{9} of the functionIn words, \(NB^{(HY)}\) is precisely the parallel shift of the reference short rate r(.) which allows for a calibration such that the model prices of bond and CDS match the observed market quotes for bond and CDS.$$\begin{aligned} x \mapsto \text {Bond}(\lambda _x(.),r(.)+x,R,C,T)  B. \end{aligned}$$
The idea of method (HY) can also be summarized as follows: If the riskfree interest rate curve is assumed to be \(r(.)+NB^{(HY)}\), then the market quotes for bond and CDS are arbitragefree (as we have found a corresponding pricing measure). It allows for the intuitive interpretation of the negative basis as a spread earned on top of a reference discounting rate after elimination of default risk. Abstractly speaking, assuming no transaction costs and availability of CDS protection at all maturities \(T > 0\) (= perfect market conditions), arbitrage pricing theory suggests the existence of a trading strategy which buys the bond and hedges it via CDS, and which earns^{10} precisely the rate \(r(.)+NB^{(HY)}\) until the minimum of default time \(\tau \) and bond maturity T. Since this way of thinking about \(NB^{(HY)}\) is its distinctive property and highlights its intrinsic coherence with arbitrage pricing theory, the following lemma demonstrates by a heuristic argument how the rate \(r(.)+NB^{(HY)}\) can be earned in a riskfree way.
Lemma 1
(The rate \(r(.)+NB^{(HY)}\) can be earned without default risk) Assuming perfect market conditions, there exists a (static) portfolio, which is long the bond and invested in several CDS, which earns the rate \(r(.)+NB^{(HY)}\) until \(\min \{\tau ,T\}\).
Proof
We present an example that demonstrates how different the three presented measurements of negative basis can be in practice. The specifications are inspired by a realworld case.
Example 1
We consider a bond with maturity \(T=3.5\) years paying a semiannual coupon rate of \(C=8.25\,\%\). It trades far below par value, namely at \(B=46.5\,\%\). An almost maturitymatched CDS contract is available at an upfront value of \(\text {upf}(T)=53\,\%\) with a running coupon of \(s(T)=5\,\%\), payed quarterly. This means a nominalmatched negative basis investment comes at a package price of \(46.5+53=99.5\,\%\), and pays a coupon rate of \(8.25 5=3.25\,\%\) until default (however, the bond and CDS coupon payments have different frequencies and payment dates). In the sequel we assume a recovery rate of \(R=20\,\%\), and the reference rate r(.) is bootstrapped from 3month tenorbased interest rate swaps according to the raw interpolation method described in [15, 16]. Because the bond trades far below par, the measurement (Z) is highly questionable and returns \(NB^{(Z)}=0.42\,\%\), which is clearly not an appropriate measurement. As indicated earlier, improved versions of earnings and costsmeasurements must be used in order to deal with such extreme situations of highly distressed bonds, but this lies outside the scope of the present article. The parequivalent CDS methodology returns the measurement \(NB^{(PE)}=2.29\,\%\), whereas the hidden yield methodology returns the significantly lower number \(NB^{(HY)}=1.18\,\%\). While the authors are not aware of a strategy how to monetize the (PE)measurement \(2.29\,\%\), Lemma 1 provides a clear interpretation for the (HY)measurement \(1.18\,\%\) in terms of an internal rate of return that can be earned on top of the riskfree rate, when the negative basis investment is structured as indicated in the proof of Lemma 1.
6 Conclusion
We proposed an innovative measurement for the negative basis, denoted \(NB^{(HY)}\). Compared to traditional approaches, it is based on an arbitragefree pricing model for the simultaneous pricing of the bond and the CDS, which provides a sound economic interpretation. Within a simple model with only default risk being present, the negative basis is perfectly explained as the spread on top of a reference interest rate curve r(.). It was pointed out how the rate \(r(.)+NB^{(HY)}\) can be earned without exposure to default risk.
Footnotes
 1.
sometimes also called bondCDS basis
 2.
However, counterparty credit risk can be reduced significantly by a negative basis investor when the CDS is collateralized, which is the usual case.
 3.
See below in Step 3 of Definition 1.
 4.
Interestingly, a mismatch between bond and CDS recovery is often favorable for the negative basis investor, since the CDS recovery rate tends to be lower than the bond recovery, see, e.g., [14]. Thus, it might make sense for a negative basis investor to opt for cash settlement of the CDS and sell his bonds in the marketplace, speculating on a favorable recovery mismatch.
 5.
 6.
For the sake of notational convenience we ignore accrued interest upon default, which can, of course, be incorporated easily.
 7.
If CDS prices are quoted in running spreads with zero upfronts, then these quotes typically come naturally equipped with a recovery assumption that is required in order to convert the running spreads into actually tradable standardized coupon and upfront payments. However, after this conversion the recovery rate is a free model parameter.
 8.
For a readerfriendly explanation of the Zspread see [28]. In particular, it is useful to observe that \(\text {Bond}(x,r(.),R,C,T)=\text {Bond}(0,r(.)+x,R,C,T)\) for \(R=0\), implying that the Zspread equals a constant default intensity under a zero recovery assumption.
 9.
Lemma A.1 in the Appendix guarantees that this root typically exists and is unique.
 10.
By “earning” \(r(.)+NB^{(HY)}\) we mean that the internal rate of return of the position is the reference rate r(.) plus a spread \(NB^{(HY)}\).
 11.
One says that \(\tau _1\) is less than \(\tau _2\) in the usual stochastic order, and the following computation is standard in the respective theory.
 12.
Here, we have used that the function \(\tau \mapsto \exp (\int _{0}^{\tau }r(s)+x+\varepsilon \,\mathrm {d}s)\,1_{\{\tau \le T_1\}}\) is nonincreasing if \(x \ge \inf \{r(t)\,:\,t \ge 0\}\).
 13.
Similar as in the induction start, we denote by \(\mathbb {E}_y[f(\tau )]\) the expectation over \(f(\tau )\) when the default time has piecewise constant intensity with the level y on the piece \((T_{k1},T_k]\).
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”.
References
 1.Andritzky, J., Singh, M.: The pricing of credit default swaps during distress. IMF Working Paper 06/254 (2006)Google Scholar
 2.Bai, J., CollinDufresne, P.: The determinants of the CDSBond basis during the financial crises of 2007–2009. Working Paper (2011)Google Scholar
 3.Beinstein, E., Scott, A., Graves, B., Sbityakov, A., Le, K., Goulden, J., Muench, D., Doctor, S., Granger, A., Saltuk, Y., Allen, P.: Credit Derivatives Handbook. J.P. Morgan Corporate Quantitative Research (2006)Google Scholar
 4.Blanco, R., Brennan, S., Marsh, I.W.: An empirical analysis of the dynamic relation between investmentgrade bonds and credit default swaps. J. Financ. 60(5), 2255–2281 (2005)CrossRefGoogle Scholar
 5.Brigo, D., Capponi, A., Pallavicini, A.: Arbitragefree bilateral counterparty risk valuation under collateralization and application to Credit Default Swaps. Math. Financ. 24(1), 125–146 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 6.Bühler, W., Trapp, M.: Explaining the bondCDS basisthe role of credit risk and liquidity. In: Risikomanagement und kapitalmarktorientierte Finanzierung: Festschrift für Bernd Rudolph zum 65. Geburtstag, KnappVerlag, Frankfurt a. Main, pp. 375–397 (2009)Google Scholar
 7.Burgard, C., Kjaer, M.: In the balance. Risk pp. 72–75 (2011)Google Scholar
 8.Choudhry, M.: The Credit Default Swap Basis. Bloomberg Press, New York (2006)Google Scholar
 9.Das, S., Kalimipalli, M.: Did CDS trading improve the market for corporate bonds? J. Financ. Econ. 111(2), 495–525 (2014)CrossRefGoogle Scholar
 10.De Wit, J.: Exploring the CDSBond basis. National Bank of Belgium working paper No. 104 (2006)Google Scholar
 11.Doctor, S., White, D., Elizalde, A., Goulden, J., Toublan, D.D.: Differential discounting for CDS: J.P. Morgan Europe Credit Research (2012)Google Scholar
 12.Fries, C.: Discounting Revisited: Valuation Under Funding, Counterparty Risk and Collateralization. Working paper (2010), available at SSRN: http://ssrn.com/abstract=1609587
 13.Fujii, M., Shimada, Y., Takahashi, A.: Collateral Posting and Choice of Collateral Currency. CIRJE Discussion Papers (2010)Google Scholar
 14.Gupta, S., Sundaram, R.K.: Mispricing and arbitrage in CDS auctions. J. Deriv. 22(4), 79–91 (2015)CrossRefGoogle Scholar
 15.Hagan, P.S., West, G.: Interpolation methods for curve construction. Appl. Math. Financ. 13(2), 89–129 (2006)CrossRefzbMATHGoogle Scholar
 16.Hagan, P.S., West, G.: Methods for constructing a yield curve. Wilmott magazine pp.70–81 (2008)Google Scholar
 17.Höcht, S., Kunze, M., Scherer, M.: Implied recovery ratesauction and models. In: Innovations in Quantitative Risk Management. Springer, Berlin pp. 147–162 (2015)Google Scholar
 18.Hull, J., White, A.: Valuing credit default swaps I: no counterparty default risk. J. Deriv. 8(1), 29–40 (2000)CrossRefGoogle Scholar
 19.Hull, J., Predescu, M., White, A.: The relationship between credit default swap spreads, bond yields, and credit rating announcements. J. Bank. Financ. 28, 2789–2811 (2004)CrossRefGoogle Scholar
 20.Li, H., Zhang, W., Kim, G.H.: The CDSBond basis and the cross section of corporate bond returns. Working paper (2011)Google Scholar
 21.Longstaff, F.A., Neis, E., Mithal, S.: Corporate yield spreads: default risk or liquidity? New evidence from the creditdefault swap market. J. Financ. 60(5), 2213–2253 (2005)CrossRefGoogle Scholar
 22.Mai, J.F., Scherer, M.: Simulating from the copula that generates the maximal probability for a joint default under given (inhomogeneous) marginals. In: Melas, V.B. et al.: Topics in Statistical Simulation: Springer Proceedings in Mathematics and Statistics, vol. 114, pp. 333–341, Springer, Heidelberg (2014)Google Scholar
 23.Morini, M., Prampolini, A.: Risky funding with counterparty and liquidity charges. Risk pp. 58–63 (2011)Google Scholar
 24.O’Kane, D.: The link between Eurozone sovereign debt and CDS prices. EDHECRisk Institute Working Paper (2012)Google Scholar
 25.O’Kane, D., Turnbull, S.: Valuation of credit default swaps. Fixed Income Quantitative Research Lehman Brothers (2003)Google Scholar
 26.Palladini, G., Portes, R.: Sovereign CDS and bond pricing dynamics in the Euroarea. Centre for Economic Policy Research Discussion Paper No. 8651 (2011)Google Scholar
 27.Pallavicini, A., Perini, D., Brigo, D.: Funding Valuation Adjustment: A Consistent Framework Including CVA, DVA, Collateral, Netting Rules and ReHypothecation. Working paper (2011), available at SSRN: http://ssrn.com/abstract=1969114
 28.Pedersen, C.M.: Explaining the Lehman Brothers option adjusted spread of a corporate bond. Fixed Income Quantitative Credit Research, Lehman Brothers (2006)Google Scholar
 29.Piterbarg, V.: Funding beyond discounting: collateral agreements and derivatives pricing. Risk, pp. 97–102 (2010)Google Scholar
 30.Zhu, H.: An empirical comparison of credit spreads between the bond market and the credit default swap market. BIS Working Paper No. 160 (2004)Google Scholar
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