INTEREST RATE

BOOTSTRAPPING EXPLAINED

German BernhartXAIA Investment GmbHSonnenstraße 19, 80331 München, Germanygerman.bernhart@xaia.com

Date: February 18, 2013

Abstract The aim of this document is to describe the basics of discountcurve bootstrapping and to provide some insights into the under-lying principles. It is meant to be a comprehensive explanationof the standard techniques used when extracting discount fac-tors from market data. In particular, we present the newly es-tablished standards necessary to consistently price swaps withlonger tenors or payments/swaps in different currencies.

1 Introduction The aim of a bootstrapping procedure is to extract discount fac-tors from market quotes of traded products. Stated differently,we want to derive "the value of a fixed payment in the future",respectively "the fixed future repayment of money borrowed to-day", from market prices. Therefore, we always have to restrictour focus to a specific investment universe consisting of the el-ements we want to consider. Here, we focus on bootstrappingfrom swap curves, whereas the procedure for bond curves isslightly different. It is shown what the underlying assumptionsare and how these procedures are based on static no-arbitrageconsiderations.In the following, we will focus on the extraction of discount fac-tors from European overnight index swaps (OIS), called EONIAswaps. There are several reasons for that:

• During the financial crisis, the spread between OIS and EU-RIBOR/LIBOR rates with longer tenor significantly increasedas the previously neglected risks related to unsecured banklending became apparent. Consequently, OIS rates can nowbe considered the best available proxy for risk-free rates (seee.g. Hull and White 2012). For academics such as Hull andWhite this is a strong argument why those rates have to beused when valuing derivatives.

• Furthermore, more and more derivative contracts are collat-eralized and this trend will be reinforced by current regulatorychanges (EMIR in Europe, Dodd-Frank in the US). Especiallyin the inter-dealer market (origin of the quoted prices pub-lished by the inter-dealer brokers) most trades are collateral-ized. There are several papers (see e.g. Piterbarg (2010) orFujii et al. (2010a)) arguing that for the valuation of thosetrades, discount factors from OIS swaps have to be used,since overnight rates can be earned/have to be paid on cashcollateral. For that reason, it is standard now in the inter-dealer market to value swaps based on OIS discount factors.

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Some clearing houses have changed their valuation proce-dures accordingly as well.

• The aim of the document is to give an introduction into thegeneral bootstrapping procedure. This procedure is indepen-dent from the chosen reference rate respectively the corre-sponding swap universe. Consequently, the chosen specialcase can serve as an introduction to the concept in general.

Our aim is to compute D(t, T ), the discount factor at time t forthe horizon T . In classical school mathematics with constant in-terest rate r and continuous compounding this quantity would begiven as D(t, T ) = exp(−r (T − t)). There are many other waysto describe a yield curve, e.g. zero rates or forward rates, butthey always depend on the underlying compounding and day-count conventions. Since most applications rely on discount fac-tors anyway, we directly consider those.

2 One-curve bootstrap

2.1 Simple setup Starting point of a bootstrap is the decision for a reference rate.As mentioned before, we will consider European overnight rates.Consequently, we are given:

• a "riskless" asset B with value process {B(t)}t≥0, B(0) = 1.This asset mirrors an investment in the reference rate rOIS ,i.e. continuously investing money into an asset paying the ref-erence rate. Consequently,

B(t) =

nt∏j=1

(1 +

rOISj dj

360

), t ≥ 0,

where nt is the number of business days in the consideredperiod from today 0 until t and rj the reference rate fixed onday j relevant for the next dj days (usually one day as it isan overnight rate, except for weekends and holidays). Theexistence of such an asset is always implicitly assumed butsometimes not stated explicitly. However, it is THE centralassumption of a bootstrap. We want to compute discount fac-tors based on the assumption that on every date, one is ableto borrow/invest at the given reference rate. It becomes clearhere why using EURIBOR rates with longer tenor is difficultas it does not seem reasonable to receive those rates on ariskless, respectively default free, asset.

• a set of EONIA swaps1 with maturities T1 < T2 < . . . <Tn, e.g. Ti ∈ {3m, 6m, 9m, 1y, 2y, . . . , 30y}, respectively thecorresponding par swap rates si. We assume that a swapwith maturity Ti exchanges payments at the dates T1 < T2 <. . . < Ti, which at Tj is a fixed payment of si (Tj − Tj−1)versus a floating payment2 of f(Tj−1, Tj) (Tj − Tj−1). The

1For more information regarding EONIA swaps, see e.g. http://www.euribor-ebf.eu/eoniaswap-org/about-eoniaswap.html.

2For simplicity, we omit daycount conventions, assume both legs to haveequal payment dates, different swaps to have coinciding payment dates on

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floating payment is linked to the reference rate by

f(Tj−1, Tj) (Tj − Tj−1) =

nTj∏j=nTj−1

+1

(1 +

rOISj dj

360

)− 1.

The value of the sum of all fixed payments will be calledFixedLeg, the value of the sum of all floating payments willbe called FloatingLeg. Based on the definition of discountfactors, it obviously holds that

FixedLegi = si

i∑j=1

D(0, Tj)(Tj − Tj−1).

There is an obvious connection between both aforementionedproducts, which is that

f(Tj−1, Tj) (Tj − Tj−1) =B(Tj)

B(Tj−1)− 1

holds. That is, the swaps are structured in a way that the floatingpayments equal the interest earned on the reference rate. Fromthat, we can deduce that an investment of 1 in B at t = 0 canyield all the required floating payments plus an additional fixedpayment of 1 at the last payment date. Consequently, the valueof a floating leg with maturity Ti is given by

FloatingLegi = 1−D(0, Ti).

Requiring a swap to have legs with the same value, as the swapcan be entered with zero upfront costs, we end up with the follow-ing set of equations the discount factors with maturities Ti shouldsatisfy to match market prices:

si

i∑j=1

(Tj − Tj−1)D(0, Tj) = 1−D(0, Ti), 1 ≤ i ≤ n. (1)

Iteratively solving for D(0, Ti) by

D(0, Ti) =1− si

∑i−1j=1(Tj − Tj−1)D(0, Tj)

1 + si (Ti − Ti−1), 1 ≤ i ≤ n, (2)

is commonly known as bootstrapping the discount factors. Thisis also the formula we use, however, we would like to motivate itin a different way using direct replication arguments. The aim ofthis is to allow for a better understanding of the resulting prices.That is, we can motivate it by simple static replication arguments.

overlapping time intervals and furthermore, that longer-dated swaps extendthe set of fixed leg payment dates of the previous swaps by one additionalpayment date. The first assumptions can be easily relaxed, to relax the lastassumption we need additional methods which we will present in Section2.3.

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2.2 Intuitive interpretation A discount factor represents today’s (T0 = 0) value of a fixedpayment in the future. If we can replicate this payment structureby a portfolio consisting of the given instruments, the discountfactor is given as the respective portfolio’s market value today.For the maturities Ti of the given swaps, such an approach canbe applied. We will start from the smallest maturity and prove therest by induction.For T1, consider a portfolio consisting of an investment of 1 intothe bank account B and a receiver swap with maturity T1. Buy-ing a receiver swap corresponds to paying the floating leg (here:B(T1)/B(T0)− 1) in a swap contract and receiving the fixed leg(here: s1(T1−T0)). The payments of the respective portfolio canbe found in Table 1, using3 B(T0) = B(0) = 1. Consequently,

t = T0 t = T1

Swap 0 s1(T1 − T0)− (B(T1)/B(T0)− 1)

Bank account −B(T0) B(T1)

Portfolio −1 1 + s1(T1 − T0)

Table 1: Payments of the considered portfolio at the two relevantdates T0 = 0 and T1.

we were able to construct a portfolio where a fixed payment of 1at T0 = 0 yields a fixed payment of 1+s1(T1−T0) at T1, allowingfor the computation of D(0, T1) by

D(0, T1) =1

1 + s1 (T1 − T0).

This of course is in accordance with Equation (2).Now, we can iteratively extend this procedure for the remainingmaturities. Assume that we are able to replicate fixed paymentsat Ti for i < n and thus know D(0, Ti), i < n. We are tryingto replicate a fixed payment at Tn. As before, invest 1 into thebank account and additionally buy a receiver swap with maturityTn. As mentioned earlier, the investment into the bank accountcan replicate the floating payments of the swap. Additionally, wewill replicate the payments of the fixed leg up to Tn−1 using thereplicating portfolios we have by our "induction hypothesis". Thepayments of the respective portfolio can be found in Table 2 (onthe last page). In total, we found a portfolio that replicates a fixedpayment of 1+ sn(Tn−Tn−1) at Tn, where today one has to pay1−sn

∑n−1j=1 (Tj−Tj−1)D(0, Tj). Solving this for D(0, Tn) yields

D(0, Tn) =1− sn

∑n−1j=1 (Tj − Tj−1)D(0, Tj)

1 + sn (Tn − Tn−1),

again being in accordance with Equation (2). To summarize, wehave shown by an iterative argument that Equation (2) can bemotivated by simple replication arguments. Unfortunately, this

3Note that in practice, when constructing this replicating portfolio, there willtake place an exchange of collateral during the lifetime of the transactionto account for a change of market value of the swap contract. However,assuming that we will have to pay EONIA on received collateral and receiveEONIA on payed collateral, this has no impact on the final payments as weassumed we can invest/finance this collateral at EONIA (using B) as well.

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approach as well as Equation (2) in general only hold for thediscount factors with maturities Ti and the case that additionalswaps only add one extra payment date. However, it hopefullyallows for a better intuitive understanding of bootstrapping pro-cedures in general.

2.3 Implementation In practice, the available data will in most cases not fulfill theideal assumptions described in the previous paragraphs. For ex-ample, one might use other contracts than swaps on the shortend or one might have "overlapping" contracts. However, we donot want to elaborate on the process of selecting the contractsdeemed the most adequate or liquid. Another aspect consistsof the fact that in the iterative procedure of solving Equation (2),there might be more than one unknown variable. This happenswhen payment dates of longer contracts do not coincide withthose of shorter contracts or when a longer contract has severalpayment dates after the maturity of the previous contract. This isan important aspect we want to illustrate.Assume that we have already bootstrapped the values D(0, Ti), 1 ≤i ≤ n, and now, we consider a swap with maturity T , swap rates and payment dates t1 < t2 < . . . < tk = T . To be con-sistent with market quotes, we are looking for discount factorsD(0, tj), 1 ≤ j ≤ k, such that

s

k∑j=1

(tj − tj−1)D(0, tj) = 1−D(0, T ). (3)

If tj ∈ {T1, . . . , Tn}, the corresponding discount factor D(0, tj)is already given. Here, we consider the fact that more than oneof those discount factors is unknown. Consequently, there is nounique solution to the problem at hand.Consider a tj such that there exists Ti < tj ≤ Ti+1. This isa "known" problem, as after a completed bootstrap we will alsoneed discount factors for time points in between the others. Oneusually employs some kind of interpolation, not based on thediscount factors directly but instead based on forward or zerorates describing those discount factors. As the discount fac-tors used in the bootstrap should be the same as the discountfactors used later for pricing purposes, it is clear that alreadyin the bootstrapping procedure the desired interpolation schemehas to be used. Hagan, West (2006) argue similarly and com-pare existing interpolation approaches. The interested reader isreferred to their paper for a full description of the related prob-lems/considerations.Consider now the remaining tj such that tj > Tn. If there isonly one of them, Equation (3) obviously has a unique solutionas given in Equation (2). If there are several of them, the pre-vious considerations regarding interpolation have to be appliedagain. That is, we only have to consider D(0, T ) = D(0, tk) asan unknown variable whereas D(0, tj) with Tn < tj < T haveto be computed via interpolation, all depending on the unknownvariable D(0, T ). In that case, a solution to Equation (3), whichis now an equation with one unknown variable D(0, T ), can notbe found in closed form but instead must be found via numerical

555

methods.There are several criteria an interpolation scheme has to fulfillfrom the perspective of a market maker, e.g. the resulting hedg-ing strategies should be reasonable. From a buyside perspec-tive, the "best" interpolation scheme probably is the one used bymarket makers as this results in realistic/tradable swap rates fornon-standard maturities. Keeping in mind the intuitive interpreta-tion of a bootstrap, one could actually use those rates/swaps tobuild a replicating portfolio which in turn determines the value ofa fixed payment in the future.

3 Additional curves in the same

currency (dual curve stripping)

Until the beginning of the recent financial crisis, bootstrappingdiscount factors from EURIBOR/LIBOR rates was sufficient asthere was only a negligible spread between swaps with differenttenor. However, this has changed considerably. The risk relatedto unsecured bank lending became apparent and consequently,the longer the tenor of EURIBOR/LIBOR rates, the higher therisk investors associate with those rates and thus, the higher therates. It is not possible to value a swap with a different tenorbased on the previously bootstrapped discount factors. For that,another procedure, the so called dual curve stripping or curvecooking is needed, which we will present in the following.

• Now, we additionally consider a set of swaps with floatingpayments linked to EURIBOR rates with a given tenor. Notethat we do not assume that it is possible to invest at this rate,especially not riskless! One might include an asset replicat-ing an investment at this rate later on to explain the results orto model the dynamics of such a system, but then one has totake account of the related default risk as well.

• Those swaps are contracts exchanging fixed payments ver-sus floating payments linked to an exogenous index, the pay-ments themselves are not exposed to credit risk as one usu-ally considers collateralized swap contracts. In particular,they are not exposed to the credit risk causing the basis spread.Furthermore, based on the previously mentioned literaturediscount factors from OIS discounting have to be used whenprizing those collateralized contracts.

• The value of fixed riskless payments is given based on thebootstrapping procedure in the previous section. Thus, whatwe can extract from the set of swaps considered here is theimplied (expected) value of the floating payments.

• Stated differently, in this part of the bootstrap we are notconcerned with the "value of a fixed payment in the future"anymore. Now, we try to extract a "description" of the addi-tional swap curve/the related EURIBOR rate which allows usto value arbitrary products linked to those floating paymentsaccordingly. In other words, we have to extract the "value ofa floating payment linked to that rate", i.e. the value of thefloating payments payed in those contracts. As curves areoften described using a discount curve, we will also describethis curve in terms of discount factors later on, even though

666

we are merely interested in the forward rates determining thefloating payments.

Let L3M (Ti, Ti+1) denote the EURIBOR rate for the time periodbetween Ti and Ti+1 (in our example, the tenor is assumed to be3 month), which is fixed at Ti and payed at Ti+1 in a swap. As inSection 2.2, the idea would be to iteratively construct portfoliosusing the given instruments, which exchange a fixed paymenttoday, PV (0), against a payment of L3M (Ti, Ti+1)(Ti+1 − Ti)at Ti+1. Using the language of financial mathematics, we canrestate

PV (0) = ”Today's value of a future payment of

L3M (Ti, Ti+1)(Ti+1 − Ti)”

= D(0, Ti+1)EQTi+1 [L3M (Ti, Ti+1)](Ti+1 − Ti)

= D(0, Ti+1)F3M (0, Ti, Ti+1)(Ti+1 − Ti), (4)

where QTi+1 denotes the Ti+1-forward measure using risk-neutralvaluation. PV (0) can also be translated into a related forwardrate F 3M (0, Ti, Ti+1), i.e. the rate such that a contract exchang-ing F 3M (0, Ti, Ti+1) against L3M (Ti, Ti+1) at Ti+1 has valuezero today4.Those forward rates, respectively the curve of those forward ratesTi 7→ F 3M (0, Ti, Ti+1), are the objects we are interested in.When valuing a specific (collateralized) derivative consistent withmarket prices, we can replace the unknown floating rate in thecalculation by the respective forward rate (see Equation (4)). Asbefore, we end up with the following set of equations the forwardsshould satisfy to match market prices:

s3Mi

ni∑j=1

(Tj − Tj−1)D(0, Tj)

=

ni∑j=1

D(0, Tj)F3M (0, Tj−1, Tj)(Tj − Tj−1), (5)

for 1 ≤ i ≤ n, where ni corresponds to the number of paymentdates5 for the i-th swap and s3Mi the corresponding par spread.As in the previous section, we might need some interpolationtechniques during this bootstrapping procedure, as in every stepof the bootstrap mostly either two or four forward values haveto be determined. To state the problem in a similar manner asbefore, where we were concerned with discount factors, it mightbe favorable to state the problem in terms of a discount factorcurve. Therefore, we will encode this information in a "discountfactor curve"6 P 3M (0, T ) using the relation

F 3M (0, Tj−1, Tj) =1

Tj − Tj−1

(P 3M (0, Tj−1)

P 3M (0, Tj)− 1

).

4Consequently, if one wanted to hedge against the uncertainty in future float-ing rates, one could lock in the forward rate by entering into such a forwardcontract at no cost.

5We again assume floating and fixed leg to have equal payment dates forsimplicity. Here, this does not make a difference at all.

6Note that this is of course not a real discount curve as the existence of sev-eral different discount curves induces arbitrage. Instead, it is just a conve-nient way to describing curves of interest rates.

777

This transforms Equation (5) to

s3Mi

ni∑j=1

(Tj − Tj−1)D(0, Tj)

=

ni∑j=1

D(0, Tj)

(P 3M (0, Tj−1)

P 3M (0, Tj)− 1

), 1 ≤ i ≤ n.

Note that here, as the two discount curves are different, theright hand side of the equation does not reduce to a telescopingsum. Replacing D by P 3M in the previous equation yields theformula commonly used before the crisis in a one curve frame-work, where the discounting curve also corresponds directly tothe considered EURIBOR rate. Obviously, both approaches dif-fer significantly. What is surprising on the first glance is that theresulting forward rates usually do not. However, there is an easyexplanation for this. In both cases, the following equations haveto be fulfilled:

s3Mi =

ni∑j=1

wij F

3M (0, Tj−1, Tj), 1 ≤ i ≤ n,

where wij are weights summing to one, either

wij :=

D(0, Tj) (Tj − Tj−1)∑nil=1(Tl − Tl−1)D(0, Tl)

"one-curve",

or

wij :=

P 3M (0, Tj) (Tj − Tj−1)∑nil=1(Tl − Tl−1)P 3M (0, Tl)

"dual-curve".

As long as the form of the considered discount curves is similar,the weights should be very similar7. Thus, the resulting forwardscan not differ too much as the weighted sums with very similarweights have to yield the original input swap rates again. Forthat reason, one might employ the techniques common beforethe crisis to get a good approximation of the forward rates. How-ever, when valuing swaps, especially forward starting swaps oron the run swaps, the right discount factors, i.e. the discount fac-tors from Section 2, have to be used.To demonstrate the procedure, the presented techniques wereimplemented using market data of February 11, 2013. Regard-ing the interpolation, we employed a raw interpolation scheme,which is linear on the logarithm of discount factors and corre-sponds to piecewise constant forward rates (see Hagan, West(2006)). Further details are omitted as they do not contribute toa better understanding of the principles presented here. Figure1 illustrates the EONIA interest rate curve in terms of the cor-responding zero rates z(0, T ) := − log(D(0, T ))/T . This is theresult of the techniques presented in the second section. Ad-ditionally, a 3M EURIBOR curve is depicted which results from"wrongly" applying the techniques of Section 2 to a set of swaps

7For example in the fictitious case that one discount curve equals the otherdiscount curve multiplied by a constant, the weights would be equal.

888

referencing the 3M EURIBOR rates directly, as it used to be com-mon. Indeed, the second curve is uniformly higher because ofthe higher default risk associated with the longer tenor. Wronglyusing the corresponding discount factors consequently would re-sult in wrong prices for most interest rate products. Furthermore,it is obvious that the structure of the involved discount curves issimilar, which according to our previous observation results invery similar forward rates for both approaches. The related for-ward rates for both approaches can be found in Figure 2. To rec-ognize any deviation, the difference between the forward ratesis shown in Figure 3. Similar figures for different curves can befound in Bloomberg starting from the <ICVS> screen.

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

years

zero

rat

e in

per

cent

EONIA3M EURIBOR

Fig. 1: The EONIA curve computed with the techniques of thesecond section based on market data of February 04,2013. Additionally, the 3M EURIBOR curve is plotted.

4 Including foreign currencies In Section 2, we computed the discount factors corresponding toa specific reference rate, assuming we were able to borrow/investat that rate. This means we now know today’s value of a fixedpayment in the future in our currency, i.e. the currency of the ref-erence rate. The aim of this section is to translate those discountfactors for our local currency into consistent discount factors fora foreign currency, which without loss of generality will be de-noted by $ and D$(t, T ). In other words, we are interested inthe discount factors for a different currency which result from ourlocal discount factors by borrowing/investing in the foreign cur-rency through a set of different products such as FX forwards orcross currency swaps. This is also the curve used when valuingpayments in a foreign currency in a contract collateralized in thelocal currency.Of course one could directly start from a comparable (in our caseovernight) rate in the foreign currency and proceed as in Section2. However, this need not necessarily be consistent to the dis-count factors in the currency one starts from (at least not any-more, see e.g. the significantly negative basis spreads for some

999

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

years

forw

ard

rate

in p

erce

nt

forward rates (true)forward rates (approx)

Fig. 2: The forward rates for the 3M EURIBOR computed with thetechniques of Section 3 is shown and compared with theapproximation based on a straight-forward application ofolder techniques.

0 5 10 15 20 25 30−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

years

diffe

renc

e in

bp

Fig. 3: The difference between the two forward rate curves in bp.

101010

cross-currency swaps). Hence, we start from payments in thelocal currency and "generate" payments in the foreign currencyby swapping payments using FX forwards or cross-currency ba-sis swaps (CCSs). If there existed FX forwards for the wholerange of considered maturities, we would be done at this pointas we could easily solve for the required discount factors usingthe relation

D$(0, T ) =D(0, T )

FXTFX0,

where FX0 denotes today’s value of one unit of the local cur-rency in the foreign currency and FXT the comparable amountfor the future date T determined by the corresponding FX for-ward. The intuitive meaning of this equation is that the value of 1in the foreign currency at T in the local currency is 1/FXT . To-day, this has the value D(0, T )/FXT in the local currency whichcorresponds to FX0D(0, T )/FXT in the foreign currency.Unfortunately, FX forwards are not liquidly traded for the wholerange of required maturities. For that reason, we have to usecross-currency swaps which makes the procedure considerablymore difficult. We thus also need the procedure described inSection 3 since the usually considered (most liquid) CCSs ex-change floating payments as EURIBOR in the local currencyversus floating payments in the foreign currency. For a moredetailed, academic description of the procedure and the relatedproblems the interested reader is referred to Fujii et al. (2010b)8.One has to consider the following structure. By entering into aCCS, one exchanges floating payments plus a basis spread inthe local currency versus floating payments in the foreign cur-rency. The value of floating payments in the local currency isknown from Section 3. However, both, the expected value of thefloating payments in the foreign currency, and the respective dis-count factors, are unknown. Consequently, one can not solvedirectly for the discount factors.Instead, one has to enter a second swap, this time a standardinterest rate swap in the foreign currency exchanging exactly thefloating payments in the foreign currency versus fixed paymentsin the foreign currency. This "removes" the floating $ paymentsfrom the structure. The complete payment structure is illustrated9

in Figure 4. Note that CCSs also involve an initial exchange of thenominal in the two related currencies and the initial amounts arepayed back at maturity. This offers a way to "fund" in the foreign

8Actually, the approach presented here slightly differs from the approach usedin Fujii et al. (2010b). Their idea is the following. As we have seen before,the resulting forward rates mainly depend on the form of the discount fac-tor curve. Thus, they compute the forward rates in the foreign currencyby simply applying the procedures presented in Sections 2 and 3, startingfrom a USD overnight reference rate. They then replace the unknown for-ward rates in our setting by those forward rates. Based on those forwardrates, they solve the resulting system of equations for the unknown discountfactors. However, it suffers from the same small mistake as our approachdoes, see Remark 4.1.

9Note that for some currency combinations, the structure might be even morecomplex. Since it is desirable to use the most liquid contracts, one might notbe able to swap the floating foreign payments directly into fixed payments.Instead, one swaps them into floating payments of a different tenor, whichthen can be swapped into fixed payments.

111111

currency. As for example demand to fund in US-$ is higher thansupply, this resulted in the considerably negative basis spreadsfor USD/EUR CCS. The result of the described bootstrappingprocedure basically yields the interest rate one has to pay on$-amounts borrowed by entering into a CCS.

Party B Party A Party C

CCS

at inception, t = 0

$ Notional

Notional

during lifetime

Floating payments & basis spreadFloating $ payments

at maturity

Notional

$ Notional

$-IRS

during lifetime

Fixed $ paymentsFloating $ payments

Fig. 4: Illustration of payment streams in the considered combi-nation of a CCS and a $-IRS.

Under some simplifying assumptions (assuming coinciding pay-ment dates and daycount conventions for all involved swaps, etc.)we end up with the following system of equations that has to befulfilled in order to match market quotes:

D(0, Tni) +

ni∑j=1

D(0, Tj) (F (0, Tj−1, Tj) + bsi) (Tj − Tj−1)

= D$(0, Tni) + ci

ni∑j=1

D$(0, Tj) (Tj − Tj−1),

for 1 ≤ i ≤ n, where ci denotes the par spread of the $-IRS andbsi the basis spread of the involved CCS, which is payed on topof the floating payments in the local currency (depending on theconvention of the CCS). The left hand side of these equationsrepresents the value of all payments received by party A and theright hand side the value of all payments made by party A (ex-cluding the value of the initial payments offsetting each other).Note that the floating $ payments do not show up in these equa-tions. When iteratively solving the system of equations one willhave to employ the previously mentioned bootstrapping proce-dures again.In Figure 5, the result for the previously used exemplary day isshown. We present the USD curve consistent with the EONIAcurve which is derived using CCSs and FX forwards. For com-parison, a directly computed USD curve is added. It is obviousthat it makes a huge difference which of the following curves is

121212

used. Interestingly, the resulting curve resembles more a shiftedUSD OIS curve than the EONIA curve we start from, though noUSD OIS market quotes are used.

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

years

zero

rat

e in

per

cent

EONIAUSD O/N (CCS−adjusted)USD O/N (un−adjusted)

Fig. 5: The EONIA curve plus the consistent corresponding USDcurve derived by the presented procedure using CCSsand FX forwards. The USD OIS curve which is com-puted by applying the procedure presented in Section 2directly to swap data in the foreign currency is added forcomparison.

Remark 4.1Actually, there occurs a small mistake in this computation. How-ever, it seems to be unavoidable. By setting those paymentsagainst each other and in particular, by using the results fromSection 2 and 3, we implicitly assume that the collateral underly-ing the CCS and the foreign currency interest rate swap is alsoposted in EUR, the local currency. However, this is usually notthe case and the market quotes available and used in the boot-strap normally correspond to USD-collateralized contracts.

References P. S. Hagan, G. West, Interpolation Methods for Curve Construc-tion, Applied Mathematical Finance, Vol. 12, No. 2, (2006) pp.89–129.

M. Fujii, Y. Shimada, A. Takahashi, Collateral posting and choiceof collateral currency - implications for derivatives pricing andrisk management, CIRJE working paper (2010).

M. Fujii, Y. Shimada, A. Takahashi, A note on construction of mul-tiple swap curves with and without collateral, FSA ResearchReview, Vol. 6, (March 2010).

V. Piterbarg, Funding beyond discounting: collateral agreementsand derivatives pricing, Risk (February 2010), pp. 97–102.

131313

t=

T0

t=

T1

t=

T2

...

t=

Tn−1

t=

Tn

Rep

licat

ion

ofs n(T

1−

T0)

D(0,T

1)·

s n(T

1−T0)

−s n(T

1−T0)

0..

.0

0

Rep

licat

ion

ofs n(T

2−

T1)

D(0,T

2)·

s n(T

2−T1)

0−s n(T

2−T1)

...

00

. . .. . .

. . .. . .

. ..

. . .. . .

Rep

licat

ion

ofs n(T

n−1−Tn−2)

D(0,T

n−1)·

s n(T

n−1−Tn−2)

00

...−s n(T

n−1−Tn−2)

0

Sw

apw

ithm

atur

ityTn

01−B(T

1)/B(T

0)

+s n(T

1−T0)

1−B(T

2)/B(T

1)

+s n(T

2−T1)

...

1−B(T

n−1)/B(T

n−2)

+s n(T

n−1−Tn−2)

1−B(T

n)/B(T

n−1)

+s n(T

n−Tn−1)

Ban

kac

coun

t−B(T

0)

B(T

1)/B(T

0)−1

B(T

2)/B(T

1)−1

...

B(T

n−1)/B(T

n−2)−

1B(T

n)/B(T

n−1)−1

+1

Port

folio

s n∑ n−

1j=

1(T

j−

Tj−

1)D

(0,T

j)−

1

00

...

01+

s n(T

n−Tn−1)

Tabl

e2:

Pay

men

tsof

the

cons

ider

edpo

rtfo

lioat

ever

yre

leva

ntda

te.

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