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THE IRONY IN THE DERIVATIVES DISCOUNTING PART II: THE CRISIS

MARC HENRARD

Abstract. Libor derivative pricing has changed with the crisis; Libor is not anymore one unam-biguous curve as a large basis has appeared between different Libor tenors. A previous approach

to derivative discounting is reviewed at the light of those changes. The valuation of so calledlinear derivatives, the yield curve construction and the valuation of vanilla options is analyzed.Copyright c 2009 by Marc Henrard.

1. Introduction

In the article Henrard (2007) we asked howthe cash-flows in Libor1 derivatives should bediscounted. The question was then viewed asa trivial and unquestionable matter. Moreoverasking such a question was non-politically cor-rect as the derivatives values enter in the market-to-market accounting of banks and the matterhad legal implication. Ironically the article, en-titled The irony in derivative discounting, waspublished in July 2007 just one month beforethe Libor/OIS spread (and other spreads) wentwild.

In mathematics, the art of pos-ing a question is more impor-tant than the art of solving one.

Georg Cantor, 1867

In the introduction of the above mentionedarticle we quoted Cantor, a quote repeated here,and expressed our belief that the question askedthen was important and our hope it was wellposed. After the crisis trigger, the importanceof correct curve selection appears even moreclearly. Unfortunately it appears also that,

against our hope, our question was not perfectlyposed. We did no imagine that Libor wouldbecome an ambiguous notion with large basisspreads between tenors. The present article,

written as the part II of derivatives discounting,proposes a solution to these new challenges.

The importance of the discounting ques-tion is attested by the numerous recent relatedliterature (e.g. Kijima et al. (2009), Ame-trano and Bianchetti(2009),Chibane and Shel-don(2009), Bianchetti(2009),Mercurio(2009),Morini(2009)) and the efforts in most banks toadjust their systems to the new reality. Some re-search explain the reason the derivatives basedon Libor with different tenors should be priceddifferently through credit risk analysis like in

Morini(2009). Here we will not try to explainthe reason behind the differences and take themas a starting point. We try to propose a coherentvaluation framework for the derivatives based ondifferent Libor tenors. Those frameworks havereceived several names: two curves, one price,derivative tenor curves, discounting-estimation,discounting-forecast or discounting-forward. Toour knowledge no theoretical fundamental sim-ple analysis of the framework has been proposedyet; this is what we try to achieve here.

In that new context all the standard ap-proaches to derivatives pricing and in particularthe valuation of so called linear products like in-terest rate swaps (IRS), forward rate agreements(FRA), OIS and futures need to be rethought

Date: First version: 4 July 2008; this version: 20 December 2009.Key words and phrases. Coherent pricing, interest rate derivative pricing, Libor, multi-curves, discounting,

forward, cost of funding, discounting, irony.JEL classification: G13, E43, C63.

AMS mathematics subject classification: 91B28, 91B24, 91B70, 60G15, 65C05, 65C30.In memoriam Jacques Henrard, who gave me a taste for questions, specially the non-politically correct ones.1In this note we use the term Libor as a generic term for all rates fixing with a similar rule and in particular

include under that term the Euribor(EUR) and Tibor (JPY).

1

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2 M. HENRARD

carefully. Some extra assumptions on the re-lation between the different curves need to beadded. Even the notions of yield curves and dis-count factors need to be handled carefully.

Our framework is based on a unique curveused to discount all the cash-flows related to thederivatives, whatever the tenor they are relatedto. Here we are interested mainly in derivatives,but obviously all cash-flows (from derivatives,bonds, deposits, etc.) should be treated in thesame way. In the first article we called that curvethefunding curve, in reference to the need for de-rivative desks to fund themselves to obtain cash.The most used name today, and probably a bet-ter choice, is thediscounting curve. We will also

use the latter name in the sequel.The choice of that discounting curve is byitself an open question. Different people willchoose different curves. One possibility is torelate it to the rate that can be achieved forthe remaining cash of the desk. For that rea-son the OIS curve is often selected, as the fund-ing/investment is often done on an overnight ba-sis. In time of liquidity shortage, this may be alow rate, not taking into account the real fund-ing cost. A curve based on Libor may be morereasonable; in that case one should select one(and only one) of the Libor tenors. The impact

of the discounting curve on IRS is not major(often less than 10% of the delta). What is cer-tainly important is to have a coherent curve withno jump from one choice to another.

In this article we suppose that the discount-ing curve is given. Our job will start with the

construction of the curves used to estimate thethe Libors (in a sense to be explained latter).Those curves are often called estimation, fore-cast or forward curves. Here the plural shouldbe used as there is a different curve for each Li-bor tenor. In practice the most frequently usedtenors are one, three, six and twelve months;in theory one could have a curve for each tenor(two, four, etc months). For those curves we willalso use the termsyield,discount factorandfor-ward rate, even if those terms may have a differ-ent meaning in this context. In particular theywill never be used for discounting2.

In the next section we set the main hypothesisused for the framework. Then we price the sim-

ple derivatives (IRS, FRA, futures). The termsimple has to be understood with a pinch of salt.In this framework a FRA is really a contingentclaim and a curve hypothesis is needed to priceit. Of course the futures require a so called con-vexity adjustment and thus a model.

The framework could be extended to valuecross currency products in a coherent multi-currency framework. In that case the discount-ing curves in the different currencies should belinked through cross-currency basis. This is notelaborated here.

Hopefully this time our questions are well

posed. To justify our attempt on asking simi-lar question again, we quote A. Einstein

The important thing is not tostop questioning.

Albert Einstein

2. Estimation hypothesis and linear products

As hinted by the title, we will consider thediscounting of cash-flows. Our first hypothesisrelates to it; its name D refers to it.

D: The instrument paying one unit in uis an asset for each u. Its value in t isdenotedPD(t, u).

With this curve we are able to price fixedcash-flows.

Our goal is to price Libor related derivativesand in particular IRS. Thus we need hypothe-sis saying at least that those instruments existin our framework. We call a (j-Libor) floating

coupon an instrument that pays in t2 the Li-bor fixing for the tenor j on the period [t1, t2]as fixed in t0 (0 t0 t1) multiplied by theconventional year fraction. The lag between t0

and t1 is called the spot lag and is usually twobusiness days3. The difference between t2 andt1 isj months (in the appropriate convention).

As the month addition t+ j months will beoften used we will use the notation abuse t+jfor that date.

L: The value of a (Libor) floating couponis an asset for each tenor and each fixingdate.

2Except for FRA where the discounting using the Libor fixing rate is part of the contract description.3This is the case in EUR and USD. In GBP the lag is 0 day.

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IRONY IN DISCOUNTING: THE CRISIS 3

This hypothesis is implicit in most of theabove quoted papers. We prefer to state it ex-plicitly as this is not the consequence of the ex-istence of the discounting curve.

In this article all asset prices are continuousin time.

Once we have assumed that the instrument isan asset, we can give a name to its price. We doit indirectly through a curve Pj.

Definition 1. Theforward curvePj is the con-tinuous function such that,Pj(t, t) = 1,Pj(t, T)is an arbitrary function fort < T < Spot(t) +jand, forT Spot(t),

(1) PD(t, T+j) Pj(t, T)

Pj

(t, T+j)

1is the price int of the floating coupon with startdateT and maturity dateT+j.

The reason behind the definition is to obtainthe usual formulas involving forward rate com-putation and discounting. The same terms dis-counting curveand forward ratewill still be usedeven if they represent different realities.

La mathematique est lart dedonner le meme nom a deschoses differentes.

Henri Poincare.

Note that the definition itself is arbitrary.One could fix any j month period (not onlythe first one) and deduce the rest of the curvefrom there. Or one could even take an arbi-trary decomposition of the j months interval insub-intervals and distribute those sub-intervalsarbitrarily on the real axis in such a way that,modulo j months, they recompose the initial jmonths period. One could also change the valueofPj(t, t).

We will come back to the curve constructionin the next section. Note that our definition ofestimation curve is similar to the one ofChibaneand Sheldon(2009) but different from the one ofKijima et al. (2009), Ametrano and Bianchetti(2009),Mercurio(2009) and Bianchetti(2009).

InMercurio(2009) the curve definition is dif-ferent from our but he barely uses those curves.He defines market models on quantities equiv-alent to our Fj defined below. In some sensethe quantities Lj defined in his Remark 1 arediscretized versions of our curves Pj .

We insist that the formula above on Pj is adefinition. We will relate those numbers to somefinancial intuition later, but for the moment Pj

is simply a function. If the floating coupon hasa price, such a function exists for each tenor j.This curve is not unique but once the first jmonths period is fixed, Pj(t, T) is well definedand unique. At this stage the only link betweenthe curve and a market rate is that the Liborrate fixing in t0 for the periodj is

(2) Ljt0 =1

Pj(t0, Spot(t0))

Pj(t0, Spot(t0) +j) 1

where is the year fraction. To obtain thisequality the price continuity was used.

2.1. Interest Rate Swap. With that hypothe-sis, the computation of vanilla interest rate swap(IRS) prices is straightforward. The hypothesiswas selected for that reason. An IRS is describedby a set of fixed coupons or cash-flows ciat datesti (1 i n). For those flows, the discount-ing is used. It also contains a set of floatingcoupons or cash-flows over the periods [ti1, ti]with ti =ti1+ j (1 i n)

4. The value of a(fixed rate) receiver IRS is(3)ni=1

ciPD

(t, ti)

ni=1

PD

(t, ti)Pj(t, ti1)

Pj(t, ti) 1

.

In the one curve pricing approach, the IRSare usually priced through either the Libor for-ward approach or the cash-flow equivalent ap-proach. The Libor forward approach consistsin estimating the forward Libor rate from thediscount factors and discounting the result frompayment date to today. To keep that intuition,we define the Libor forward rate in our frame-work as the figure we have to use to keep thesame formula.

Definition 2. TheLibor forward rate over theperiod [ti1, ti] is given at time t by

(4) Fjt(ti1, ti) = 1

i

Pj(t, ti1)

Pj(t, ti) 1

.

With that definition the IRS price is

ni=1

ciPD(t, ti)

ni=1

PD(t, ti)iFjt(ti1, ti).

4In practice, due to week-ends and holidays, the periods used for the fixings can be slightly different from thepayment dates. We will not make that distinction here.

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4 M. HENRARD

Note the fundamental difference betweenLjt and

Fjt. The object L is, by hypothesis L, a fun-

damental element of our economy; the secondone is defined by the above equality. Note alsothat our definitions of L and F are oppositeto Mercurio (2009); our Fjt(ti1, ti) is equiva-

lent to his Lji (t) and our Ljt0

corresponds to hisFj . The definitions of F and L coincide forti1= Spot(t0):

Ljt0 =Fjt0

(Spot(t0), Spot(t0) +j).

The cash-flow equivalentapproach consists inreplacing the (receiving) floating leg by receiv-ing the notional at the period start and payingthe notional at the period end. We would like

to have a similar result in our new framework.To this end we define

(5) jt (u, u +j) = Pj(t, u)

Pj(t, u +j)

PD(t, u +j)

PD(t, u) .

With that definition, a floating coupon price is

PD(t, ti)

Pj(t, ti1)

Pj(t, ti) 1

= PD(t, ti)

jt (ti1, ti)

PD(t, ti1)

PD(t, ti) 1

= jt (ti1, ti)P

D(t, ti1) PD(t, ti).

This last value is equal to the value of receiving

jt notional at the period start and paying thenotional at the period end.

If the forward discounting rateFDt (ti1, ti) isdefined in the standard way, the floating couponprice can also be written as

PD(t, ti)i

jtF

Dt (ti1, ti) +

1

i(jt 1)

which is the formula proposed in Henrard(2007).

A consequence of the hypothesis L and thedefinition of jt is that is a martingale inthe PD(., ti1) numeraire. The Libor coupon

value is jt (ti1, ti)PD(t, ti1) PD(t, ti). The

coupon is an asset due to L and so its valuedivided by the numeraire PD(t, ti1) is a mar-tingale. The second term is also an asset, henceits rebased value is also a martingale. The lastterm is thus also a martingale and its value isjt P

D(t, ti1)/PD(t, ti1) =

jt . This proves

that jt is a martingale under the PD(., ti1)-

measure.Like in the one curve framework, we can de-

fine a forward swap rate. This is the rate for

which the vanilla IRS price is 0:

Sj

t =

n

i=1 iFjt(ti1, ti)P

D(t, ti)ni=1 iPD(t, ti)

.

2.2. Forward Rate Agreement. A ForwardRate Agreement (FRA) is an instrument linkedto aj -month period, a fixing date t0and a fixedrate K. At the fixing date t0, the Libor rateLjt0 is recorded. The contractual payment int1= Spot(t0) (the start date) is

(Ljt0 K)

1 + Ljt0.

The origin of the formula is the difference be-tween the Libor fixing and the fixed rate dis-

counted at the Libor rate. The rate is not paidat the end of its period but at the start and isdiscounted by itself; it is not directly a float-ing coupon as we defined it above. In thatsense, our definition of a FRA is in line withreal FRA term sheet and different from the oneof Ametrano and Bianchetti (2009), Bianchetti(2009),Chibane and Sheldon(2009) andMercu-rio(2009). We will need a new hypothesis thatindicates that this instrument is an asset in oureconomy. In a general contingent claim formula,its price would be

N0E

N1t1 (Lj

t0 K)1 + Ljt0

.

Here we can not use the usual trick of postpon-ing the payment tot2by multiplying by 1+ L

jt0

and selecting Pj(., t2) as the numeraire to sim-plify the formula. The reason is that an invest-ment is not done at the libor rate but at thediscounting rate. Note that Fjt is the ratio ofan asset (floating coupon) and PD(., ti). In the

PD(., ti) numeraire measure,Fjt is a martingale.

Note that Pj can not be an asset. If it wasthe case we would have, for any numeraire N,

PD(0, t) = N0E

N1t PD(t, t)

= N0E

N1t 1

= N0E

N1t Pj(t, t)

= Pj(0, t).

The two curves PD and Pj would be identical,which is a contradiction to our stated goal ofdecoupling them.

We need extra assumptions. The goal of thisarticle is to obtain a relatively simple coherentand practical approach to Libor derivatives pric-ing.

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IRONY IN DISCOUNTING: THE CRISIS 5

Our next hypothesis is that the spread be-tween the curves, as defined through the quan-tityj

t, is independent of the curves.

SI: The multiplicative coefficient betweendiscount factor ratios, jt (u, u+ j), de-fined in Equation (5) is independent ofthe ratioPD(t, u)/PD(t, u +j).

For the pricing of options, we will impose astricter hypopthesis.

S0: The multiplicative coefficient betweendiscount factor ratios, jt (u, u+ j), de-fined in Equation (5) constant through

time: jt (u, v) = j0(u, v) for all t and u.

With the independence hypothesis, the ad-

justment t

j is a martingale for any PD

(., u)-measure or for the cash account measure. Thechange of numeraire on the independent variabledoes not affect the martingale property ofjt .

The hypothesis S0 can be viewed as theequivalent of the constant continuously com-pounded spread used in Henrard (2007). Herethe spread is not constant across maturities butdeterministic and given by its initial values. Thespread is ln(j(u, v))/(u v). In that sense thisframework is a direct extension of the one devel-oped in the previous work adapted to the currentmarket situation where the spreads are not equal

for all maturities.The hypothesis S0 is equivalent5 to the hy-

pothesis that jt is deterministic. The equiva-lence can be obtained easily through a martin-gale argument.

We have

1+Ljt0(t1, t2) =Pj(t0, t1)

Pj(t0, t2)=j(t1, t2)

PD(t0, t1)

PD(t0, t2).

Theorem 1. Under hypothesisD, L andSItheprice in t of thej months FRA with fixing datet0, and rateK is

PD(t, t1)(Fj

t

K)

1 + Fjt=PD(t, t2)

(Fj

t

K)

j(t1, t2) .

where t1= Spot(t0) andt2= Spot(t0) +j.

The result can be obtained trough simple ma-nipulation of the above coefficients and the in-dependent hypothesis on .

The above result relies on hypothesis SI.Even if the floating coupon value is given in L

this is not enough to price the FRA; an extrahypotheses is necessary. The FRA discountingwith the Libor rate between start date and enddate creates an adjustment represented by thecoefficientj .

Note that, as already mentioned in Henrard(2007), the IRS is not anymore a portfolio ofFRA with same notional.

The above formula is used by some systemsdecoupling discounting and forward curves. Thehypothesis SIjustifying such a formula is seldomprovided.

A FRA is at-the-money when the rate K issuch that the instrument value is 0. Using theabove result, it is the case when

K=Fj

0 =1

j(t1, t2) P

D(0, t1)PD(0, t2)

1

.

This is essentially the FRA fair rate obtainedinMercurio(2009) with

j = 1

R+ (1 R) E[Q(t1, t2)].

The adjustment factors j can be linked tocredit related parameters as recovery rate anddefault probability. It is also linked to the quan-tity KL(t, t1, t2) defined in (Kijima et al., 2009,Equation (4.7)). In their article the spread islinked to some model parameters, in our case itis fitted to the market curves.

2.3. Libor futures. A general pricing formulafor eurodollar futures in the Gaussian HJMmodel was proposed in Henrard (2005a). Theformula extended a previous result proposed inKirikos and Novak (1997). The formula wasbriefly extended to the discounting framework inHenrard(2007). We now study the instrumentunder our new hypothesis.

The futures are liquid only for the threemonth Libor up to two or three year. To a lesserextend some one month futures are available on

the shorter part of the curve.The future fixing date is denoted t0. The fix-

ing is on the Libor rate between t1 = Spot(t0)and t2 = t1+ j. The accrual factor for the pe-riod [t1, t2] is , the fixing is linked to the yieldcurve by

1 + Ljt0 =Pj(t0, t1)

Pj(t0, t2).

5Another possible hypothesis proposed byTanaka(2009) would be that the ratio Pj(t, u)/PD(t, u) is determin-

istic. This hypothesis implies the one we propose. That ratio can not be constant (it tends to one as t tends to u)We prefer to deal with constant quantities.

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6 M. HENRARD

The futures price is jt . On the fixing date, therelation between the price and the rate is

j

t0 = 1 Lj

t0 .The futures margining is done on the futuresprice (multiplied by the notional and divided by4).

The exact notation for the HJM one-factormodel used here is the one ofHenrard(2005a).

Theorem 2. Let0 t t0 t1 t2. In theHJM one-factor model on the discount curve un-

der the hypothesesD, L andSI, the price of the

futures fixing ont0 for the period[t1, t2] with ac-crual factor is given by

j

t = 1

1

Pj(t, t1)

Pj(t, t2) (t)

1

(6)

= 1 (t)Fjt +1

(1 (t))

where

(t) = exp

t0t

(s, t2)((s, t2) (s, t1))ds

.

Proof. Using the generic pricing future priceprocess theorem (Hunt and Kennedy,2004, The-orem 12.6),

jt = EN

1 Ljt0

Ft

.

InLjt0 , the only non-constant part is the ratio of

j-discount factors which is, up to jt0 the ratio

of D-discount factors. Using (Henrard, 2005a,Lemma 1) twice, we obtain

PD(t0, t1)

PD(t0, t2)

= PD(t, t1)

PD(t, t2)exp

1

2

t0t

2(s, t1)2(s, t2)ds

+

t0t

(s, t1) (s, t2)dWs

.

Only the second integral contains a stochasticpart. This integral is normally distributed with

variancet0

t ((s, t1) (s, t2))2ds. So the ex-

pected discount factors ratio value is reduced to

PD(t, t1)

PD(t, t2) exp

1

2

t0t

2(s, t1) 2(s, t2)ds

+

t0t

((s, t1) (s, t2))2ds.

The coefficientjt0 is independent and a martin-gale, hence we have the announced result.

3. Curve construction

Our goal in this section is to construct thecurves Pj(0, .). The three instruments detailedabove are the most used to construct the yieldcurves Pj(0, .). As in the previous section, wesuppose that the discounting curve PD(0, .) isgiven.

Note that in Mercurio (2009) the curveconstruction is not discussed and in Kijimaet al. (2009) the model used imposes a spe-cific parametrized shape to the spread betweencurves.

As described in the definition ofPj , we takePj(0, 0) = 1 and for Pj(0, t) an arbitrary con-tinuous function for t

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IRONY IN DISCOUNTING: THE CRISIS 7

value is known. Using the result on the valua-tion of such an instrument, the value is 0 whenKi= Fj

0

(si, e

i). The equality gives the value of

Pj(0, ei) based on the known value Pj(0, si).

Note that for the curve construction purpose,the FRA can be consider as a floating coupon.This is only true because the FRA value is 0 andthe discounting has no impact.

3.2. Futures. The futures are used only forj = 3. They could be used also for j = 1 wheresome short futures are quoted. There is usuallyless liquidity on those instruments and the exis-tence of short swaps make them less useful.

The mechanism is similar to the one for

FRAs. A set of futures with increasing maturityis selected. The HJM model parameters are sup-posed to be obtained (calibrated) already. The

market price of the future j0 gives the value of

Fj0 through Equation (6), this allows to obtain

the curve Pj up to the future maturity like inthe FRA case.

3.3. IRS. For longer maturities IRS are used.The IRS are usually quoted directly with onlyone tenor. To obtain the IRS with other tenorsone has to use basis swaps, swap that exchange

floating coupons in one tenor against floatingcoupon with another tenor. In EUR the basisswap are often quoted for a pair of IRS fixed vsfloating with different fixed rates. The spreadbeing the difference between the fixed rates. TheIRSs are sorted in increasing maturities and thecurve is obtained up to their maturity in sucha way that the value of the swap with marketcoupon as given by Equation (3) is 0. Interpo-lation is used when necessary.

3.4. The arbitrary curve. As mentionedabove, the curve up to the first tenor is arbitrary.

The intuition behind it is that the forward curveis used only to compute forwards and thus onlythe ratio between two values ofPj is important,

never a single value. This is true only to a cer-tain extend; not all the points of the curve areconstructed directly, some are interpolated.

Consider as an example the curve with j = 3.Suppose you construct the curve with an arbi-trary curve up to 3 months and then the curveup to two years with futures (this is an approachoften used). Here are two cases that you couldencounter if you are too arbitraryon the selec-tion. Take the case were there is one month tothe start of the next future. Take an arbitrarylow spot rate and a one month forward rate veryhigh. In that case, the curve is constructed withthe 3 month futures from that high rate and allthe rates are also high. This is not a problem for

your futures, they are all well priced as the ratiosof discount factors on those points are correct.How is a one year versus 3 month IRS priced inthat context? The one year rate is interpolatedbetween the 10 months and 13 month rates com-ing from the future, both of them too high. Sothe 12 month rate used for the forward is toohigh, the spot rate is too low and your one yearrate is too high. Even if all futures are correctlypriced, the swap which is interpolated is not.Even if one adds the one year swap in the curveconstruction one could create a similar problem.Take now the same initial arbitrary curve with

a very low two months rate. Use the futuresand the swap to construct the curve and nowtry to price a forward swap starting in 2 monthsand with one year maturity; this swap could beused as underlying of a swaption. It is not toodifficult to see that the forward rate would alsobe overestimated. For that reason we suggestto use financially meaningful numbers for thearbitrary values, for example the rate obtainedfrom shorter fixing. In our case, the three monthcurve would be constructed with one and twomonth Libor deposits to establish the arbitrarypart.

The arbitrariness ofPj(0, t) over the first pe-riod is not broached in Chibane and Sheldon(2009) andMercurio(2009).

4. Delta

Once a curve is constructed and financial in-struments are priced with it, the next step is tocompute the risks associated, the first of which isthedeltas. By this we mean the change of valueof a financial instrument when the rate used to

construct the curve is moved by a small amount(usually one basis point).

In the discounting context, to parody the ti-tles ofHenrard(2005b) and Bianchetti (2009),the title of this section could be: One price, two

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8 M. HENRARD

curves and three deltas. Due to the two curves,there are two sets of financial instruments play-ing a role in the curve construction and as suchtwo deltas. But when one looks more in detailsthere are three ways the rates act. The first oneis the direct influence of the discounting rate onthe price, the second one is the market instru-ments rates used in the forward curve and thethird one is the indirect impact of the change ofthe discounting curve on the forward rate con-struction. The forward curve is constructed us-ing as input different instruments and the dis-counting curve. The three impacts can be eval-uated separately.

For the simple instruments mentioned above,

the impact of the discounting curve is relativelysmall. The total impact is null for at-the-moneyswaps.

In the standard one curve approach the curveis usually only constructed with deposits andswaps. When it is constructed also with fu-tures, the information is often translated intostandard tenor deposits and swaps rates (by in-terpolation) to obtain a standardized curve con-struction from which a standardized delta canbe obtained. Here the situation is different; bydefinition, there is no deposit forward rate. Oneis using FRA, but 3M FRA and 6M FRA rate

sensitivities have not the same meaning and cannot be added. A standardized delta can not beobtained through FRAs and futures. A wayaround the problem is to create artificial deposits

for the forward curve. Those terms are clearlyantinomic. What we mean by that is to createartificial market-like rates that applied to for-mulas similar to deposit formulas create a curvewith the same discount factors as the forwardcurve. From the output curvePj one creates ar-tificial standardized input instruments. In thisway the deltas computed are somehow compat-ible.

Even inside one of those curves, the compat-ibility is not certain. Take for example the 3months curve. The 6 months point could be rep-resented by an (artificial) 6 months deposit or byan (artificial) 6 months swap versus 3 month Li-bor. If the market rates of the two instrumentsare well selected, the curve constructed will bethe same. Nevertheless the delta computed withthose two approaches will not be the same.

Even if the curve construction and deltas def-initions are somehow arbitrary, one can obtain

some meaningful deltas. The next (non trivial)question is how to use them. In the traditionalone curve approach one would use swaps anddeposits used to construct the curve in decreas-ing maturity order to hedge the deltas. An at-the-money instrument of correct notional is en-tered into in such a way that the exposure of thelongest maturity is cancelled. Once this is doneone works inductively down to the overnightpoint to cancel all exposures.

In the two curves approach (or multi-curves ifone trades more than one tenor) one has not onebut two deltas for each maturity. One can useswaps to cancel the forward curve exposure atthe expense of leaving a (small) discounting ex-

posure. The discounting exposure could be can-celled by lending or borrowing some cash to acounterpart. Often derivative desks dont havethe authority (nor the willingness) to lend orborrow long term, so the exposures would re-main unhedged or compensated by exposure onother curves.

In some cases an hybridapproach is used forpricing and delta computation. The (precise)two curves approach described above is used forsimple instruments and a less precise one curveapproach is used for more complex instruments.The unique curve (that can be one of the forward

curves) is used in the traditional way. The levelat which the split operates can vary from vanillaswaption to exotic instruments. That hybrid ap-proach is dangerous. The one curve exposure socomputed is added to some two curve exposurein such a way that one does not know what thefigures actually represent. Such an hybrid ap-proach could lead to cases where the apparentexposure is 0 for all tenors while the actual ex-posure is globally 0 but there is a large basisposition.

Table 1 gives some examples of deltas forswaps and swaptions. The swaptions are dis-

cussed in the next section. The delta is splitin two for each instrument: the discount curve(both direct and indirect effects) and the for-ward curve. The swaps studied are annualversus three month Libor 6Mx5Y swaps withcoupon between 1 and 5%. As can be seen thesplit is largely dependent on the deal money-ness. The discounting curve delta is usually be-tween -7% and +7% of the total delta. Larger orsmaller coupons would give more extreme splits.Note that the at-the-money swaps (2.91% in our

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IRONY IN DISCOUNTING: THE CRISIS 9

example) have a 0 delta with respect to the dis-counting curve.

The swaptions used are receiver swaptions;they are priced with an extended Vasicek (Hull-White) model. Their deltas represents a frac-tion of the swap deltas. As we selected extremestrikes, the fraction is almost 0% on one sideand almost 100% on the other. The ratio is notequal between the discounting and the forwardparts. A swaption can not be perfectly (delta)

hedged with swaps only; some fixed cash-flows(deposits) may be needed. This is the usual du-ality between in-the-modeland out-of-the-modelhedging. As we have a model with one stochas-tic curve and a deterministic spread, it shouldbe possible to hedge all instruments based onthat curve. In the delta computation the curvesare moved separately, out of the model. This ishow different instruments (swap and swaptions)have non-similar deltas.

Cpn 1% Cpn 2% Cpn 3% Cpn 4% Cpn 5%Dsc Fwd Dsc Fwd Dsc Fwd Dsc Fwd Dsc Fwd

Swap (EUR) 0.28 -4.57 0.13 -4.57 -0.02 -4.57 -0.18 -4.57 -0.33 -4.57

Swap rel. -6.63 1 06.63 - 2.93 1 02.93 0.52 99.48 3.75 96.25 6.78 9 3.22Swaption (EUR) -0.00 -0.01 -0.00 -0.43 -0.05 -2.51 -0.18 -4.29 -0.33 -4.56Tot swpt/swap 0.29 9.67 55.66 94.19 99.86Swpt/swap -0.01 0.27 -2.84 9.31 212.05 54.84 100.41 93.95 99.99 99.85Figures in %, except when indicated otherwise. Figures in EUR are for a 10,000 EUR notional and a

one basis point shift.

Table 1. Delta for swaps and swaptions.

5. Libor contingent claims - Cash-flow equivalent

In this section a general approach to price Li-

bor related contingent claims is proposed underthe hypothesis S0 of a deterministic spread. Theapproach was already used for FRA and futuresin the previous sections.

To price contingent claims in the discountingforward framework, we propose to use any stan-dard model on the discount curve. The instru-ments are based on fixed cash-flows and Liborbased cash-flows. For the Libor based cash-flows, the cash-flow equivalent technique de-scribed at the end of Section 2.1 will be used.

Lets apply this approach to a swaption. Theunderlying swap value is given by Equation (3).Using the cash-flow equivalent description, the

equation can be replaced by an equation involv-

ing only PD:

IRS =

ni=1

ciPD(t, ti)

ni=1

j(ti1, ti)P

D(t, ti1) PD(t, ti)

.

Regrouping the terms with same discount factortogether and defining n,ti and di appropriately,one has

IRS =

ni=0

diPD(t, ti).

In that sense, the swaptions are equivalent tobond options with adjusted coupon and strikeprice in the discounting only framework. Thisresult is adapted to any model which models thewhole term structure andPD(., ti) in particular.

6. Other spread hypothesis

Our hypothesis S0 is one way to link thecurves. Other possibilities are presented below.

6.1. Market spread. Under this hypothesis,the spread between the market standard tenorswap rate and other forward is known for every

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10 M. HENRARD

fixing. LetMbe thestandard market frequency,the one for which we have market data.

MR: The spreads or basis

(7) Bj(t0, t1) = Sjt0

(t0, t1) SMt0

(t0, t1)

are know for every fixing date t0 an ev-ery tenor j .

Using the martingale property of the swap rate itis easy to verify that a deterministic hypothesisis equivalent to the proposed constant (forward)spread.

If one has a model for SM, the swap rateswith other tenors are obtained by a deterministicshift. This is particularly adapted to a Bache-

lier type model where the volatility would be thesame for rates with different tenors. This canbe used in a Black-like model under the under-standing that if the market rate is log-normal,the others are shifted log-normal.

This model is in practice very close to the onewe proposed in the previous sections which canbe viewed as an approach with known continu-ously compounded spreads.

6.2. Black spread. The simplest model usedto model interest rates derivatives is the Blackmodel on forward rates. The base equation is

(8) dSMt =SMt dWt.

The volatility is the one given by the mar-ket for the standard tenor and specific expiry,tenor and strike. The similar rate for the for-ward (non-market convention) curve is supposedto follow a similar equation

dSjt =Sjt dWt.

B1: The forward (swap) rate follows aBlack equation (between 0 and expiry)with the same Brownian motion than

the rate in the market convention.

Under that approach the spread between therates SM Sj is not constant nor determinis-tic. It is a constant proportion of the rate. Thespread grows (and reduces) with the rate:

Sjt = Sj

0

SM0

SMt .

With such an hypothesis or a similar on Li-bor forward rate the pricing of FRA required toconstruct the curves PJ would be difficult.

Mercurio (2009) proposes such a model foreach rateSj but does not propose a relation be-tween them. This type of model is difficult tocalibrate as few options on non market standardtenor are available. If basis spread were to stay

large and volatile, such a market may appear.6.3. SABR/CEV spread. The base equa-tions are

dSMt = t(SMt )

dW1t.(9)

dt = tdW2t(10)

S1: The forward swap rate follows a SABRequation (between 0 and expiry) withthe same paremeters and Brownian mo-tion than the rate in the market conven-tion.

The result will depend on the parameter,like the computation of the delta in the SABRframework. Aclose to 1 will give a result closeto Black spread and aclose to 0 will give a re-sult close to market spread.

In the Black and SABR approaches, thespreads would increase when the rates increase.In the recent crisis, the spreads have increasedwhile the rates were decreasing. This approachwould be counterintuitive.

The three approaches mentioned above areconvenient only if the objective is only to modelone (swap) rate and not the whole term struc-ture. These approaches would not be the most

convenient for exotics (Bermuda, callable, etc.).

7. Conclusion

A previous article dealt with the concept ofdiscounting for Libor derivatives in presence ofconstant spread. It is here extended to priceLibor derivatives to the current market situa-tion where different Libor tenors imply differentswap rates (basis spread) and those spreads arematurity dependent and changing.

Simple hypotheses are introduced to proposea coherent framework. In that framework thevalue of simple Libor derivatives used to con-struct yield curves is detailed. The instrumentsare IRS, FRA and futures.

In a simplified framework a technique to priceany Libor contingent claim is sketched. With

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IRONY IN DISCOUNTING: THE CRISIS 11

that technique, the pricing in the multi-curves framework is not technically more difficult thanin the one curve approach.

Up to our knowledge this is the first proposalfor a coherent and simple method to price Liborderivatives with different reference tenors.

Acknowledgment: The author wishes to thanks his colleagues for their detailed feedback.

References

Ametrano, F. and Bianchetti, M. (2009). Bootstrapping the illiquidity: multiple yield curvesconstruction for market coherent forward rates estimation. Working paper, Banca IMI/BancaIntesaSanpaolo. Available at SSRN: http://ssrn.com/abstract=1371311. 1,3,4

Bianchetti, M. (2009). Two curves, one price: pricing and hedging interest rate derivatives de-coupling forwarding and discounting yield curves. Technical report, Banca Intesa San Paolo.Available at SSRN: http://ssrn.com/abstract=1334356. 1,3,4,7

Chibane, M. and Sheldon, G. (2009). Building curves on a good basis. Technical report, ShinseiBank. Available at SSRN: http://ssrn.com/abstract=1394267. 1,3,4,7

Henrard, M. (2005a). Eurodollar futures and options: Convexity adjustment in HJM one-factormodel. Working paper 682343, SSRN. Available at SSRN: http://ssrn.com/abstract=682343.5,6

Henrard, M. (2005b). Swaptions: 1 price, 10 deltas, and . . . 6 1/2 gammas. Wilmott Magazine,pages 4857. 7

Henrard, M. (2007). The irony in the derivatives discounting. Wilmott Magazine, pages 9298. 1,

4,5Hunt, P. J. and Kennedy, J. E. (2004). Financial Derivatives in Theory and Practice. Wiley seriesin probability and statistics. Wiley, second edition. 6

Kijima, M., Tanaka, K., and Wong, T. (2009). A multi-quality model of interest rates. QuantitativeFinance, pages 133145. 1,3,5,6

Kirikos, G. and Novak, D. (1997). Convexity conundrums. Risk, pages 6061. 5Mercurio, F. (2009). Interest rates and the credit crunch: new formulas and market models.

Technical report, QFR, Bloomberg. 1,3,4,5,6,7,10Morini, M. (2009). Solving the puzzle in the interest rate market. Working paper, IMI Bank Intesa

San Paolo. Available at SSRN: http://ssrn.com/abstract=1506046.Tanaka, K. (2009). Personal communication. 5

Contents

1. Introduction 12. Estimation hypothesis and linear products 22.1. Interest Rate Swap 32.2. Forward Rate Agreement 42.3. Libor futures 53. Curve construction 63.1. FRA 63.2. Futures 73.3. IRS 73.4. The arbitrary curve 74. Delta 7

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12 M. HENRARD

5. Libor contingent claims - Cash-flow equivalent 96. Other spread hypothesis 96.1. Market spread 96.2. Black spread 106.3. SABR/CEV spread 107. Conclusion 10References 11

Marc Henrard is Head of Interest Rate Modelling, Dexia Bank Belgium