Two Curves, One Price:Pricing & Hedging Interest Rate Derivatives Using

Different Yield Curves for Discounting and Forwarding

Marco Bianchetti∗

Risk Management, Market Risk, Pricing and Financial ModelingBanca Intesa Sanpaolo, piazza P. Ferrari 10, 20121 Milan, Italy

Version 1.4, Jan. 29th, 2009

Abstract

In this paper we revisit the problem of pricing and hedging plain vanilla single-currency interest rate derivatives using different yield curves for market coherentestimation of discount factors and forward rates with different underlying rate tenors(e.g. Euribor 3 months, 6 months,.etc.).Within such double-curve-single-currency framework, adopted by the market

after the liquidity crisis started in summer 2007, standard single-curve no arbitragerelations are no longer valid and can be formally recovered through the introductionof a basis adjustment. Numerical results show that the basis adjustment curvesmay display, in trouble market times, an oscillating micro-term structure, stronglydependent on the quality of the bootstrapping. Such shapes may induce appreciableeffects on the price of interest rate instruments, in particular when people switchesfrom the single-curve towards the double-curve framework.Recurring to the foreign-currency analogy we also derive the no arbitrage double-

curve market-like formulas for basic plain vanilla interest rate derivatives, FRA,swaps, cap/floors and swaptions in particular. These expressions include an extraquanto adjustment term typical of cross-currency derivatives, naturally originatedby the change between the numeraires associated to the two yield curves, thatcarries on a volatility and correlation dependence. Numerical scenarios confirm thatsuch correction can be non-negligible, thus making the market prices, in principle,

∗The author acknowledges fruitful discussions with M. De Prato, C. Maffi, F. Mercurio, N. Moreni,M. Morini, M. Pucci and with many colleagues in the Risk Management. A particular mention is dueto F. Ametrano and to the QuantLib community for the open-source developements used to computethe numerical results reported in the paper. The opinions expressed here are solely of the author anddo not represent in any way those of his employer.

not arbitrage free. In practice arbitrage opportunities are hidden by the marketincompleteness.JEL Classifications: E45, G13.

Keywords: liquidity, crisis, yield curve, forward curve, discount curve, pricing, hedg-ing, interest rate derivatives, FRAs, swaps, basis swaps, caps, floors, swaptions,basis adjustment, quanto adjustment, measure changes, no arbitrage.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Single-Curve Pricing & Hedging Interest-Rate Derivatives . . . . . . . . 31.2 From Single to Double-Curve Paradigm . . . . . . . . . . . . . . . . . . 4

2 Double-Curve Framework, No Arbitrage and Basis Adjustment . . . . . . . . 72.1 General Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Pricing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 No Arbitrage Revisited and Basis Adjustment . . . . . . . . . . . . . . . 10

3 Foreign-Currency Analogy and Quanto Adjustment . . . . . . . . . . . . . . . 123.1 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Swap Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Double-Curve Pricing & Hedging Interest Rate Derivatives . . . . . . . . . . . 194.1 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1. Introduction

One of the many consequences of the liquidity crisis started in the second half of 2007 hasbeen a strong increase of the basis spreads quoted on the market between single-currencyinterest rate instruments, swaps in particular, characterized by different underlyingrate tenors (e.g. Euribor3M1, Euribor6M, etc.), reflecting the increased liquidity riskand the corresponding preference of financial institutions for receiving payments withhigher frequency (quarterly instead of semi-annualy, for instance). Such asymmetryhas induced a sort of “segmentation” of the interest rate market into sub-areas, mainlycorresponding to instruments with 1M, 3M, 6M, 12M underlying rate tenors. Each areais characterized, in principle, by its own internal dynamic, reflecting the different viewsand interests of the market players.

In figs. 1.1, 1.2 we show a snapshot of the market quotations as of 30 Sep. 2008 (asensitive date for quarterly balance sheet results of financial institutions and industries)

1Euro Interbank Offered Rate, the rate at which euro interbank term deposits within the euro zoneare offered by one prime bank to another prime bank (see e.g. www.euribor.org).

2

for six basis swap curves corresponding to the four Euribor tenors 1M, 3M, 6M, 12M.As one can see, the basis spreads are monotonically decreasing from over 100 to around4 basis points. There is neither way nor any good reason to ignore such quotations in amarket-coherent pricing framework of interest rate derivatives.

We stress that the present market situation described above is nothing else thata new equilibrium configuration determined by the pressure of the increased illiquidityforce, that enlarges well known effects hystorically very small and traditionally neglectedbefore the crisis.

1.1. Single-Curve Pricing & Hedging Interest-Rate Derivatives

Such evolution of the financial markets has triggered a general reflection about themethodology used to price and hedge interest rate derivatives, namely those financialinstruments whose price depends on the present value of future interest rate-linkedcashflows. The pre-crisis standard market practice (which does not automatically meangood practice) can be summarized in the following procedure (see e.g. refs. [1]-[4]):

1. select one finite set of the most convenient (e.g. liquid) vanilla interest rate instru-ments traded in real time on the market with increasing maturities; for instance,a very common choice in the EUR market is a combination of short-term EURdeposit, medium-term Futures on Euribor3M and medium-long-term swaps onEuribor6M;

2. build one yield curve using the selected instruments plus a set of bootstrappingrules (e.g. pillars, priorities, interpolation, etc.);

3. compute on the same curve forward rates, cashflows2, discount factors and workout the prices by summing up the discounted cashflows;

4. compute the delta sensitivity and hedge the resulting delta risk using the suggestedamounts (hedge ratios) of the same set of vanillas.

For instance, a 5.5Y maturity EUR floating swap leg on Euribor1M (not directlyquoted on the market) is commonly priced using discount factors and forward ratescalculated on the same depo-Futures-swap curve cited above. The corresponding deltasensitivity is calculated by shoking one by one the curve pillars and the resulting deltarisk is hedged using the suggested amounts (hedge ratios) of 5Y and 6Y Euribor6Mswaps3.

2within the present context of interest rate derivatives we focus in particular on forward rate depen-dent cashflows. See also eq. 2.11.

3we refer here to the case of local yield curve bootstrapping methods, for which there are no sensitivitydelocalization effect (see refs. [1], [2]).

3

We stress that this is a single-currency-single-curve approach, in that a unique curveis built and used to price and hedge any interest rate derivative on a given currency.Thinking in terms of more fundamental variables, e.g. the short rate, this is equivalentto assume that there exist a unique fundamental underlying short rate process able tomodel and explain the whole term structure of interest rates of any tenor.

It is also a relative pricing approach, because both the price and the hedge of aderivative are calculated relatively to a set of vanillas quoted on the market. We noticealso that the procedure is not strictly guaranteed to be arbitrage-free, because discountfactors and forward rates obtained through interpolation are, in general, not necessarilyconsistent with the no arbitrage condition; in practice bid-ask spreads and transactioncosts virtually hide any arbitrage possibility.

Finally, we stress that the first key point in the procedure above is much more amatter of art than of science, because there is not an unique financially sound choice ofbootstrapping instruments and, in principle, none is better than the others.

The methodology described above can be extended, in principle, to more complicatedcases, in particular when a model of the underlying interest rate evolution is used tocalculate the future dynamic of the yield curve and the expected cashflows. The volatil-ity and (eventually) correlation dependence carried by the model implies, in principle,the bootstrapping of a variance/covariance matrix (two or even three dimensional) andhedging the corresponding sensitivities (vega and rho) using volatility and correlationdependent vanilla market instruments. In practice just a small subset of such quota-tions is available on the market, and thus only some portions of the variance/covariancematrix can be extracted from the market. In this note we will focus only on the basicmatter of yield curves and forget the volatility/correlation dimensions.

1.2. From Single to Double-Curve Paradigm

Unfortunately, the pre-crisis approach outlined above is no longer consistent, at least inthis simple formulation, with the present market configuration.

First, it does not take into account the market information carried by the basis swapspreads, now much larger than in the past and no longer negligible.

Second, it does not take into account that the interest rate market is segmentatedinto sub-areas corresponding to instruments with different underlying rate tenors, char-acterized, in principle, by different dynamics (e.g. short rate processes). Thus, pricingand hedging an interest rate derivative on a single yield curve mixing different under-lying rate tenors can lead to “dirty” results, incorporating the different dynamics, andeventually the inconsistencies, of different market areas, making prices and hedge ratiosless stable and more difficult to interpret. On the other side, the more the vanillas andthe derivative share the same homogeneous underlying rate, the better should be therelative pricing and the hedging.

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Third, by no arbitrage, discounting must be univocal: two identical future cash-flows of whatever origin must display the same present value; hence we need an uniquediscounting curve.

The market practice has thus evolved to take into account the new market informa-tions cited above, that translate into the additional requirement of homogeneity : as faras possible, interest rate derivatives with a given underlying rate tenor should be pricedand hedged using vanilla interest rate market instruments with the same underlying.The corresponding pricing procedure will be formalized and justified in sec. 2.2; wesummarize here the following modified working procedure:

1. build one discounting curve using the preferred procedure;

2. selectmultiple separated sets of vanilla interest rate instruments traded in real timeon the market with increasing maturities, each set homogeneous in the underlyingrate (typically with 1M, 3M, 6M, 12M tenors);

3. build multiple separated forwarding curves using the selected instruments plustheir bootstrapping rules;

4. compute on each forwarding curve the forward rates and the corresponding cash-flows relevant for pricing derivatives on the same underlying;

5. compute the corresponding discount factors using the discounting curve and workout prices by summing up the discounted cashflows;

6. compute the delta sensitivity and hedge the resulting delta risk using the suggestedamounts (hedge ratios) of the corresponding set of vanillas.

For instance, the 5.5Y floating swap leg cited in the previous section should bepriced using Euribor1M forward rates calculated on an “pure” 1M forwarding curve,bootstrapped only on Euribor1M vanillas, plus discount factors calculated on the dis-counting curve. The corresponding delta sensitivity should be calculated by shokingone by one the pillars of both yield curves, and the resulting delta risk hedged using thesuggested amounts (hedge ratios) of 5Y and 6Y Euribor1M swaps plus the suggestedamounts of 5Y and 6Y instruments from the discounting curve (see sec. 4.2 for moredetails about the hedging procedure).

The improved approach described above is more consistent with the present marketsituation, but - there is no free lunch - it does demand much more additional efforts.First, the discounting curve clearly plays a special and fundamental role, and mustbe built with particular care. This “pre-crisis” obvious step has become, in the presentmarket situation, a very subtle and controversial point, that would require a whole paperin itself. In fact, while the forwarding curves construction is driven by the underlying

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rate tenor homogeneity principle, for which there is (now) a general market consensus,there is no longer general consensus for the discounting curve construction. At least twodifferent practices can be encountered on the market: a) the old “pre-crisis” approach(e.g. the depo, Futures and swap curve cited before), that can be justified with theprinciple of maximum liquidity (plus a little of inertia), and b) the Eonia4 curve, justifiedwith no risky or collateralized counterparties, and by increasing liquidity (see e.g. thediscussion in ref. [5]). Second, building multiple curves requires multiple quotations:much more interest rate bostrapping instruments must be considered (deposits, Futures,swaps, basis swaps, FRAs, etc.), which are available on the market with different degreesof liquidity and can display transitory inconsistencies. Third, non trivial interpolationalgorithms are crucial to produce smooth forward curves (see e.g. refs. [1]-[3]). Fourth,multiple bootstrapping instruments implies multiple sensitivities, so hedging becomesmore complicated. Last but not least, pricing libraries, platforms, reports, etc. mustbe extendend, configured, tested and released to manage multiple and separated yieldcurves for forwarding and discounting, not a trivial task for quants, developers and ITpeople.

In this paper we assume the existence of a convenient methodology (described indetail in ref. [6]) for bootstrapping a discounting curve plus multiple forwarding curvescharacterized by different underlying rate tenors (Euribor1M, 3M, 6M and 12M), and wefocus on the consequences for pricing and hedging interest rate derivatives. In particularin sec. 2 we fix the notation, we revisit some general concept of standard, no arbitragesingle-curve pricing and we formulate the principles for double-curve pricing, showinghow no arbitrage can be formally recovered with the introduction of a basis adjustment.In sec. 3 we use the foreign-currency analogy to derive a single-currency version ofthe quanto adjustment (typical of cross-currency derivatives) to be applied to forwardrates. In sec. 4 we derive the no arbitrage double-curve pricing expression for basicsingle-currency interest rate derivatives, zero coupon bonds, FRA, swaps and cap/flooroptions in particular. Finally, in sec. 5 we summarize the conclusions.

Some of these topics have been approached also in other papers (see e.g. refs. [7]-[12]). In particular W. Boenkost and W. Schmidt [9] discuss two methodologies forpricing cross-currency basis swaps, the first of which (the actual pre-crisis commonmarket practice), does coincide, once reduced to the single-currency case, with thedouble-curve pricing procedure described here. The concerns expressed by these authors,that such approach was not arbitrage free and not consistent with the pre-crisis single-curve market practice for pricing single-currency swaps, have now been overcome bythe market evolution towards a generalized double-curve pricing approach. Recently M.Kijima et al. [10] have extended the approach of ref. [9] to the case of three curves for

4Euro OverNight Index Average, the rate computed as a weighted average of all overnightrates corresponding to unsecured lending transactions in the euro-zone interbank market (see e.g.www.euribor.org).

6

discount factors, swap forward rates and bond forward rates, bootstrapped using swaps,cross-currency swaps and governative bonds, respectively. Finally, simultaneously to thedrafting of the present paper, M. Morini [11] is approaching the same problem in termsof counterparty risk, while F. Mercurio [12] is rebuilding the whole theory “from scratch”in terms of modified Libor Market Model.

The present work is complementary to those cited before in the sense that: a) weadopt a practitioner’s perspective, discussing in detail the current market practicesand keeping things as simple as possible; b) we explain why the market has evolvedfrom a single-curve towards a double-curve pricing framework in the case of single-currency interest rate instruments (to our knowledge, this is discussed in detail only inref. [12]); c) we proof how non-arbitrage is broken-up and can be recovered in terms ofbasis adjustment; d) we use the foreign-currency analogy to derive pricing formulas forbasic interest rate derivatives including the quanto adjustment arising from the changebetween the numeraires (or probability measures) naturally associated to forwardingand discounting curves; e) last but not least, we consistently keep together both pricingand hedging issues, whose intimate connection is at the heart of quantitative finance,trading and risk management.

2. Double-Curve Framework, No Arbitrage and Basis Adjustment

2.1. General Assumptions

We start by postulating the existence of two different interest rate markets, denoted byMx, x = d, f, characterized by the same currency and by two distinct bank accountsBx and yield curves x in the form of a continuous term structure of discount factors,

x = T −→ Px (t0, T ) , T ≥ t0 , (2.1)

where t0 is the (common) reference date of the curves (e.g. settlement date, or today)and Px (t, T ) denotes the price at time t ≥ t0 of the x -zero coupon bond for maturityT , such that Px (T, T ) = 1.

Next we postulate the usual no arbitrage relation in each market Mx,

Px (t, T2) = Px (t, T1)× Px (t, T1, T2) , (2.2)

where t ≤ T1 < T2 and Px (t, T1, T2) denotes the forward discount factor from time T2to time T1, prevailing at time t. The financial meaning of expression 2.2 is that, in eachmarket Mx, given a cashflow of one unit of currency at time T2, its corresponding valueat time t < T2 must be the same both if we discount in one single step from T2 to t, usingthe discount factor Px (t, T2), and if we discount in two steps, first from T2 to T1, usingthe forward discount Px (t, T1, T2) and then from T1 to t, using Px (t, T1). Denoting with

7

Fx (t;T1, T2) the simple compounded forward rate associated to Px (t, T1, T2), resettingat time T1 and covering the time interval [T1;T2], we have

Px (t, T1, T2) =Px (t, T2)

Px (t, T1)=

1

1 + Fx (t;T1, T2) τx (T1, T2), (2.3)

where τx (T1, T2) is the year fraction between times T1 and T2 with daycount dcx, fromeq. 2.2 we obtain the familiar no arbitrage expression

Fx (t;T1, T2) =1

τx (T1, T2)

∙1

Px (t, T1, T2)− 1¸

=Px (t, T1)− Px (t, T2)

τx (T1, T2)Px (t, T2). (2.4)

Eq. 2.4 can be also derived (see e.g. ref. [13], par. 1.4) as the fair value condition attime t of the Forward Rate Agreement (FRA) contract with payoff at maturity T2 givenby

FRA (T2;T1, T2,K,N) = Nτx (T1, T2) [Lx (T1, T2)−K] , (2.5)

Lx (T1, T2) =1− Px (T1, T2)

τx (T1, T2)Px (T1, T2)(2.6)

where N is the nominal amount, Lx (T1, T2, dcx) is the T1-spot Euribor rate and K the(simply compounded) strike rate (sharing the same daycount convention for simplicity).Introducing expectations we have, ∀t ≤ T1 < T2,

FRA (t;T1, T2,K,N) = Px (t, T2)EQT2x

t [FRA (T2;T1, T2,K,N)]

= NPx (t, T2) τx (T1, T2)

½EQ

T2x

t [Lx (T1, T2)]−K

¾= NPx (t, T2) τx (T1, T2) [Fx (t;T1, T2)−K] , (2.7)

where QT2x denotes the Mx-T2-forward measure, EQt [.] denotes the expectation at time

t w.r.t. measure Q and filtration Ft (encoding the market information available up totime t), and we have used the standard martingale property of forward rates

Fx (t;T1, T2) = EQT2x

t [Fx (T1;T1, T2)] = EQT2x

t [Lx (T1, T2)] (2.8)

holding in each interest rate market Mx.

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2.2. Pricing Procedure

Now we consider any derivative written on a single underlying interest rate with n futurecoupons with payoffs π = π1, ..., πn, generating n cashflows c = c1, ..., cn at futuredates T = T1, ..., Tn, t < T1 < ... < Tn. Following the discussion in sec. 1.2, wepostulate the following generalized, double-curve-single-currency pricing procedure:

1. assume d as the discounting curve and f as the forwarding curve;

2. calculate any relevant spot/forward rate on the forwarding curve f as

Lf (t, Ti) =1− Pf (t, Ti)

τf (t, Ti)Pf (t, Ti), t < Ti, (2.9)

Ff (t;Ti−1, Ti) =Pf (t, Ti−1)− Pf (t, Ti)

τf (Ti−1, Ti)Pf (t, Ti), t ≤ Ti−1 < Ti, (2.10)

3. calculate cashflows ci, i = 1, ..., n, as expectations of the i-th coupon payoff πiwith respect to the discounting Ti-forward measure Q

Tid ,

ci = c (t, Ti, πi) = EQTid

t [πi] ; (2.11)

4. calculate the price at time t by discounting each cashflow c (t, Ti, πi) using thecorresponding discount factor Pd (t, Ti) obtained from the discounting curve dand summing up,

π (t;T) =nXi=1

Pd (t, Ti) c (t, Ti, πi)

=nXi=1

Pd (t, Ti)EQTid

t [πi] . (2.12)

Notice that steps 3 and 4 above have been formulated in terms of the particularpricing measure QTi

d associated to the numeraire Pd (t, Ti). This is convenient in ourcontext because it emphasizes that discounting must be associated to the discountingcurve. Obviously they can be reformulated in terms of any other equivalent measureassociated to different numeraires.

If we apply the paradigm above to the FRA case, we obtain that the single-curveFRA price in eq. 2.7 traslates into the following generalized, double-curve expression

FRA (t;T1, T2,K,N) = NPd (t, T2) τf (T1, T2)

½EQ

T2d

t [Ff (T1;T1, T2)]−K

¾. (2.13)

We will compute such expectation in par. 3.

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2.3. No Arbitrage Revisited and Basis Adjustment

The first consequence of the assumptions above is that, clearly, standard single-curveno arbitrage relations such as eq. 2.4 are broken up, being

Pd (t, T1, T2) =Pd (t, T2)

Pd (t, T1)=

1

1 + Fd (t;T1, T2) τd (T1, T2)

6= 1

1 + Ff (t;T1, T2) τf (T1, T2)=

Pf (t, T2)

Pf (t, T1)= Pf (t, T1, T2) . (2.14)

Clearly, no arbitrage is immediately recovered by postulating the following generalizeddouble-curve no arbitrage relation

Pd (t, T1, T2) =Pd (t, T2)

Pd (t, T1)

=1

1 + Ff (t;T1, T2)BAfd (t, T1, T2) τ f (T1, T2), (2.15)

or the equivalent simple transformation rule for forward rates

Fd (t;T1, T2) = Ff (t;T1, T2)BAfd (t, T1, T2) . (2.16)

We call the conversion factor BAfd (t, T1, T2) in eqs. 2.15-2.16 (forward) basis adjust-ment. From eq. 2.16 we can express it as a ratio between forward rates or, equivalently,in terms of discount factors from d and f curves as

BAfd (t, T1, T2) =Fd (t;T1, T2)

Ff (t;T1, T2)

=τf (T1, T2)

τd (T1, T2)

Pf (t, T2)

Pd (t, T2)

Pd (t, T1)− Pd (t, T2)

Pf (t, T1)− Pf (t, T2). (2.17)

Notice that if d = f we recover the single-curve case BAfd (t, T1, T2) = 1.In eq. 2.16 we have chosen a multiplicative definition of the basis adjustment.

Obviously the alternative additive definition is completely equivalent

BA0fd (t, T1, T2) = Fd (t;T1, T2)− Ff (t;T1, T2)

=Pd (t, T1)− Pd (t, T2)

τd (T1, T2)Pd (t, T2)− Pf (t, T1)− Pf (t, T2)

τf (T1, T2)Pf (t, T2)

= Ff (t;T1, T2) [BAfd (t, T1, T2)− 1] . (2.18)

The latter is more useful for comparisons with the market basis spreads of figs. 1.1-1.2.

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We stress that the basis adjustment in eqs. 2.17-2.18 is a straightforward conse-quence of the assumptions above, essentially the existence of two curves and no arbi-trage. In practice its value depends on the basis spread between the quotations of thetwo sets of vanilla instruments used in the bootstrapping of the two curves d and f .The advantage of expressions 2.17, 2.18 is that they allows for a direct computation ofthe basis adjustment between forward rates for any time interval [T1, T2], which is therelevant quantity for pricing and hedging interest rate derivatives. On the other side,the limit of expression 2.17-2.18 is that they reflects the statical5 differences betweenthe two interest rate markets Md, Mf carried by the two curves d, f , but they arecompletely independent of the interest rate dynamics in Md and Mf .

Notice also that, in principle, we can also use our approach to bootstrap a new yieldcurve from a given yield curve plus a given basis adjustment. Inverting eq. 2.17 weobtain the following recursive relations

Pd,i =τf,iPf,i

τd,i [Pf,i−1 − Pf,i]BAfd,i + τf,iPf,i−1Pd,i−1

=τf,iPf,i

τd,i [Pf,i−1 − Pf,i] + τd,iτf,iPf,iBA0fd,i + τf,iPf,i−1

Pd,i−1, (2.19)

Pf,i =τd,iPd,iBAfd,i

τf,i [Pd,i−1 − Pd,i] + τd,iPd,i−1BAfd,iPf,i−1

=τd,iPd,i

τd,iPd,i + τf,i [Pd,i−1 − Pd,i]− τd,iτf,iPd,iBA0fd,i

Pf,i−1, (2.20)

where we have shortened the notation by putting τx (Ti−1, Ti) = τx,i, Px (t, Ti) = Px,i,BAfd (t, Ti−1, Ti) = BAfd,i. Given the yield curve up to step Px,i−1 plus the basisadjustment for the step i− 1→ i, the equations above can be used to obtain the nextstep Px,i.

We now discuss a numerical example of the basis adjustment in a realistic marketsituation. We consider the four interest rate underlyings I = I1M , I3M , I6M , I12M,where I = Euribor index, and we bootstrap from market data five separated yieldcurves = d, I1Mf , I2Mf , I6Mf , I12Mf , using the first one for discounting and theothers for forwarding. We follow the methodology described in ref. [6] and we use thecorresponding open-source development in the QuantLib framework [14]. The discount-ing curve d is built following a typical “pre-crisis” standard recipe from the most liquiddeposit, 3M Futures and 6M swap contracts; the other curves are built from mixings ofdepos, FRAs, Futures, swaps and basis swaps with homogeneous underlying rate tenors;a smooth algorithm (monotonic cubic spline on log discounts) is used for interpolations6.

5we remind that the discount factors in eqs. 2.17-2.17 are calculated on the curves d, f followingthe recipe described in sec. 2.2, not using any dynamical model for the evolution of the rates.

6 technicalities of the curves construction are not crucial in the present context.

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In fig. 2.1 we plot the 6M-tenor forward rates calculated on d and I6Mf as of 15Sept. 2008, end of day. This is an highly stressed market period, just at the Lehmandefault and after the Fannie Mae and Freddie Mac federal takeover (Sep. 8, 2008). Theeffects of such main market events are clearly visible in the crazy roller-coaster look-upof the curves. The small differences in the two short-term structures derive from theuse of different market instruments in the two bootstrappings, while the medium andlong-term similarity is due to the common use of 6M swap quotes. Similar patterns areobserved also in the other 1M, 3M, 12M curves (not reported here).

In fig. 2.2 (upper panels) we plot the term structure of the four correspondingmultiplicative basis adjustment curves calculated through eq. 2.17. In the lower panelswe also plot the additive basis ajustment given by eq. 2.18. The higher short-termbasis adjustments (left panels) are due to the higher short-term market basis spreads(see Figs. 1.1-1.2). We observe in particular that the medium-long-term I6Mf − dbasis (dash-dotted green lines in the right panels) are close to 1 and 0, respectively, asexpected from the common use of 6M swaps. A similar, but less evident, behaviouris found in the short-term I3Mf − d basis (continuous blue line in the left panels), asexpected from the common 3M Futures and the uncommon deposits. The two remainingbasis curves I1Mf − d and I12Mf − d are generally far from 1 or 0 because of differentbootstrapping instruments. Obviously such details depend on our arbitrary choice ofthe discounting curve.

Overall, we notice that all the basis curves xf − d reveal a complex micro-termstructure, not present in the smooth and monotonic basis swaps market quotes of figs.1.1-1.2, essentially due to an amplification effect of small local differences between thed and xf yield curves. In general, such richer term structure is a very sensitive testof the quality of the bootstrapping procedure (interpolation in particular), and also anindicator of the tiny, but observable, differences between different interest rate marketareas. Obviously these causes may have appreciable effects on the price of similarinterest rate instruments. We can appreciate in fig. 2.3 the perverse effects of a nonsmooth bootstrapping (linear interpolation on zero rates, a common market practice).The angular points in the lower panel clearly show the inadequacy of the bootstrap,but the very strong oscillations in the (additive) basis adjustment in the upper panel(notice the different scales w.r.t. fig. 2.2, lower panels) allows to further appreciate theunnatural differences induced in similar forward instruments priced on the two curves.

3. Foreign-Currency Analogy and Quanto Adjustment

We come now to the problem of calculating expectations as in eq. 2.12 in general and2.13 in particular. This will involve the dynamical properties of the two interest ratemarkets Md and Mf , or, in other words, will require to model the dynamics for theinterest rates in Md and Mf . This task is easily accomplished by recurring to the

12

natural analogy with cross-currency derivatives. Going back to the beginning of sec.2, we can identify Md and Mf with the domestic and foreign markets, d and f withthe corresponding curves, and the bank accounts Bd (t), Bf (t) with the correspondingcurrencies, respectively7. Within this framework, we can recognize on the r.h.s of eq.2.15 the forward discount factor from time T2 to time T1 expressed in domestic currency,and on the r.h.s. of eq. 2.13 the expectation of the foreign forward rate w.r.t the domesticforward measure. Hence, the computation of such expectation must involve the quantoadjustment commonly encountered in the pricing of cross-currency derivatives. Thederivation of such adjustment can be found in standard textbooks. Anyway, in orderto fully appreciate the parallel with the present double-curve-single-currency case, it isuseful to run through it once again. In particular, we will adapt to the present contextthe discussion found in ref. [13], chs. 2.9 and 14.4.

3.1. Forward Rates

In the double—curve-double-currency case, no arbitrage requires the existence at anytime t0 ≤ t ≤ T of a spot and a forward exchange rate between equivalent amounts ofmoney in the two currencies such that

cd (t) = xfd (t) cf (t) , (3.1)

Xfd (t, T )Pd (t, T ) = xfd (t)Pf (t, T ) , (3.2)

where the subscripts f and d stand for foreign and domestic, cd (t) is any cashflow(amount of money) at time t in units of domestic-currency and cf (t) is the correspondingcashflow at time t (the corresponding amount of money) in units of foreign currency.Obviously Xfd (t, T ) → xfd (t) for t → T . Expression 3.2 is a direct consequence ofno arbitrage. This can be understood with the aid of fig. 3.1: starting from top rightcorner in the time vs currency/yield curve plane with an unitary cashflow at time T > tin foreign currency, we can either move along path A by discounting at time t oncurve f using Pf (t, T ) and then by changing into domestic currency units using thespot exchange rate xfd (t), ending up with xfd (t)Pf (t, T ) units of domestic currency;or, alternatively, we can follow path B by changing at time T into domestic currencyunits using the forward exchange rate Xfd (t, T ) and then by discounting on d usingPd (t, T ), ending up with Xfd (t, T )Pd (t, T ) units of domestic currency. Both paths stopat bottom left corner, hence eq. 3.2 must hold by no arbitrage.

Our double-curve-single-currency case is immediately obtained from the discussionabove by thinking to the subscripts f and d as shorthands for forwarding and discountingand by recognizing that, having a single currency, the spot exchange rate must collapse

7notice the lucky notation: “d” stands either for “discounting” or“domestic” and “f ” for “forward-ing” or “foreign”, respectively.

13

to 1. We thus have

xfd (t) = 1, (3.3)

Xfd (t, T ) =Pf (t, T )

Pd (t, T ). (3.4)

Obviously for d = f we recover the single-currency, single-curve case Xfd (t, T ) = 1 ∀T . The interpretation of the forward exchange rate in eq. 3.4 within this framework isstraightforward: it is nothing else that the counterparty of the (forward) basis adjust-ment in eq. 2.16 for discount factors on the two yield curves d and f . They satisfythe following relation

BAfd (t, T1, T2) = Xfd (t, T2)τf (T1, T2)

τd (T1, T2)

× Pd (t, T1)− Pd (t, T2)

Pd (t, T1)Xfd (t, T1)− Pd (t, T2)Xfd (t, T1). (3.5)

Notice that we could forget the foreign currency analogy above and start by postulatingXfd (t, T ) as in eq. 3.4, name it (discount) basis adjustment and proceed with the nextstep.

We proceed by assuming, according to the standard market practice, the following(driftless) lognormal martingale dynamic for the f (foreign) forward rate

dFf (t;T1, T2)

Ff (t;T1, T2)= σf (t) dW

T2f (t) , t ≤ T1, (3.6)

where σf (t) is the volatility (positive deterministic function of time) of the process,

under the probability space³Ω,Ff , QT2

f

´with the filtration Ff

t generated by the brow-

nian motion WT2f under the forwarding (foreign) T2−forward measure QT2

f , associatedto the f (foreign) numeraire Pf (t, T2).

Next, since Xfd (t, T2) in eq. 3.4 is the ratio between the price at time t of a d(domestic) tradable asset (xfd (t)Pf (t, T2) in eq. 3.2, or Pf (t, T2) in eq. 3.4 withxfd (t) = 1) and the numeraire Pd (t, T2), it must evolve according to a (driftless) mar-tingale process under the associated discounting (domestic) T2−forward measure QT2

d ,

dXfd (t, T2)

Xfd (t, T2)= σX (t) dW

T2X (t) , t ≤ T2, (3.7)

where σX (t) is the volatility (positive deterministic function of time) of the process andWT2

X is a brownian motion under QT2d such that

dWT2f (t) dWT2

X (t) = ρfX (t) dt. (3.8)

14

Now, in order to calculate expectations such as in the r.h.s. of eq. 2.13, we mustswitch from the forwarding (foreign) measure QT2

f associated to the numeraire Pf (t, T2)

to the discounting (domestic) measure QT2d associated to the numeraire Pd (t, T2). In our

double-curve-single-currency language this amounts to transform a cashflow on curvef to the corresponding cashflow on curve d. Recurring to the change-of-numerairetecnique (cfr. refs. [13], [15], [16]) we obtain that the dynamic of Ff (t;T1, T2) underQT2d acquires a non-zero drift

dFf (t;T1, T2)

Ff (t;T1, T2)= µf (t) dt+ σf (t) dW

T2f (t) , t ≤ T1, (3.9)

µf (t) = −σf (t)σX (t) ρfX (t) , (3.10)

and that Ff (T1;T1, T2) is lognormally distributed under QT2d with mean and variance

given by

EQT2d

t

∙ln

Ff (T1;T1, T2)

Ff (t;T1, T2)

¸=

Z T1

t

∙µf (u)−

1

2σ2f (u)

¸du, (3.11)

VarQT2d

t

∙ln

Ff (T1;T1, T2)

Ff (t;T1, T2)

¸=

Z T1

tσ2f (u) du. (3.12)

We thus obtain the following expressions, for t0 ≤ t < T1,

EQT2d

t [Ff (T1;T1, T2)] = Ff (t;T1, T2)QAfd

¡t, T1, σf , σX , ρfX

¢, (3.13)

QAfd

¡t, T1, σf , σX , ρfX

¢= e

R T1t µf (u)du = e−

R T1t σf (u)σX(u)ρfX(u)du. (3.14)

We conclude that the foreign-currency analogy allows us to compute the expectationof a forward rate on curve f w.r.t. the discounting measure QT2

d associated to thediscounting curve d through the numeraire Pd (t, T2) in terms of a well-known quantoadjustment, typical of cross-currency derivatives. From the discussion above it is clearthat such adjustment essentially follows from a change between the probability measuresQT2f andQT2

d , or numeraires Pf (t, T2) and Pd (t, T2), naturally associated to the two yieldcurves, f and d, respectively.

Obviously we can also define an additive quanto adjustment as

EQT2d

t [Ff (T1;T1, T2)] = Ff (t;T1, T2) +QA0fd¡t, T1, σf , σX , ρfX

¢, (3.15)

QA0fd¡t, T1, σf , σX , ρfX

¢= Ff (t;T1, T2)

£QAfd

¡t, T1, σf , σX , ρfX

¢− 1¤, (3.16)

where the second relation comes from eq. 3.13. Notice that the expressions 3.14 dependson the average over the time interval [t, T1] of the product of the volatility σf of the

15

f (foreign) forward rates Ff , of the volatility σX of the forward exchange rate Xfd

between curves f and d, and of the correlation ρfX between Ff and Xfd. It doesnot depend either on the volatility σd of the d (domestic) forward rates Fd or on anystochastic quantity after time T1. The latter fact is actually quite natural, because thestochasticity of the forward rates involved ceases at their fixing time T1. The dependenceon the cashflow time T2 is actually implicit in eq 3.14, because the volatilities and thecorrelation involved are exactly those of the forward and exchange rates on the timeinterval [T1, T2]. Notice in particular that a non-trivial adjustment is obtained if andonly if the forward exchange rate Xfd is stochastic (σX 6= 0) and correlated to theforward rate Ff (ρfX 6= 0); otherwise expression 3.14 collapses to the single curve caseQAfd = 1.

The volatilities and the correlation in eq. 3.14 can be extracted from market data.In the present market situation, the volatility σf can be extracted from quoted cap/flooroptions on Euribor6M, while for other rate tenors and for σX and ρfX one must resortto historical estimates. Conversely, given a basis adjustment term structure, such thatin fig. 2.2, we could take σf from the market, assume for simplicity ρfX ' 1 (or anyother functional form), and bootstrap out a term structure for the volatility σX . Noticethat in this way we are also able to compare informations about the internal dynamicsof different market sub-areas. We will give some numerical estimate of the quantoadjustment in the next section.

Finally, we may derive a relation between the quanto and the basis adjustments.Combining eqs. 3.13, 3.15 with eqs. 2.17, 2.18 we obtain

BAfd (t, T1, T2)

QAfd

¡t, T1, σf , σX , ρfX

¢ = EQT2d

t [Ld (T1, T2)]

EQT2d

t [Lf (T1, T2)]

, (3.17)

BA0fd (t, T1, T2)−QA0fd¡t, T1, σf , σX , ρfX

¢= EQ

T2d

t [Ld (T1, T2)− Lf (T1, T2)] (3.18)

3.2. Swap Rates

The discussion above can be remapped, with some attention, to swap rates. Given twopayment dates vectors T = T0, ..., Tn, S = S0, ..., Sm, T0 = S0, for the floating andthe fixed leg of the swap, respectively, the corresponding fair swap rate on curve f isdefined as

Sf (t,T,S) =

nPi=1

Pf (t, Ti) τf (Ti−1, Ti)Ff (t;Ti−1, Ti)

Af (t,S), t ≤ T0, (3.19)

16

where

Af (t,S) =mXj=1

Pf (t, Sj) τf (Sj−1, Sj) (3.20)

is the annuity on curve f . Following the standard market practice, we observe that,assuming the annuity as the numeraire, the swap rate in eq. 3.19 is the ratio betweena tradable asset (the value of the swap floating leg on curve f ) and the numeraireAf (t,S), and thus it is a martingale under the associated forwarding (foreign) swapmeasure QSf . Hence we can assume, mimicking eq. 3.6, a driftless geometric brownianmotion for the swap rate under QSf ,

dSf (t,T,S)

Sf (t,T,S)= νf (t,T,S) dW

T,Sf (t) , t ≤ T0, (3.21)

where υf (t,T,S) is the volatility (positive deterministic function of time) of the processand WT,S

f is a brownian motion under QSf .Then, mimicking the discussion leading to eqs. 3.3-3.4, the following identity

mXj=1

Pd (t, Sj) τd (Sj−1, Sj)Xfd (t, Sj) = xfd (t)mXj=1

Pf (t, Sj) τf (Sj−1, Sj)

= Af (t,S) , (3.22)

must hold by no arbitrage between the two curves f and d. Defining a swap forwardexchange rate Yfd (t,S) such that

mXj=1

Pd (t, Sj) τd (Sj−1, Sj)Xfd (t, Sj) = Yfd (t,S)mXj=1

Pd (t, Sj) τd (Sj−1, Sj)

= Yfd (t,S)Ad (t,S) , (3.23)

we obtain the expression

Yfd (t,S) =Af (t,S)

Ad (t,S), (3.24)

equivalent to eq. 3.4. Hence, since Yfd (t,S) is the ratio between the price at time tof the d (domestic) tradable asset xfd (t)Af (t,S) and the numeraire Ad (t,S), it mustevolve according to a (driftless) martingale process under the associated discounting(domestic) swap measure QSd ,

dYfd (t,S)

Yfd (t,S)= νY (t,S) dW

SY (t) , t ≤ T0, (3.25)

17

where vY (t,S) is the volatility (positive deterministic function of time) of the processand WS

Y is a brownian motion under QSd such that

dWT,Sf (t) dWS

Y (t) = ρfY (t,T,S) dt. (3.26)

Now, applying again the change-of-numeraire tecnique of sec. 3.1, we obtain that thedynamic of the swap rate Sf (t,T,S) under the discounting (domestic) swap measureQSd acquires a non-zero drift

dSf (t,T,S)

Sf (t,T,S)= λf (t,T,S) dt+ νf (t,T,S) dW

T,Sf (t) , t ≤ T0, (3.27)

λf (t,T,S) = −νf (t,T,S) νY (t,S) ρfY (t,T,S) , (3.28)

and that Sf (t,T,S) is lognormally distributed under QSd with mean and variance givenby

EQSd

t

∙ln

Sf (T0,T,S)

Sf (t,T,S)

¸=

Z T0

t

∙λf (u,T,S)−

1

2ν2f (u,T,S)

¸du, (3.29)

VarQSdt

∙ln

Sf (T0,T,S)

Sf (t,T,S)

¸=

Z T0

tfν2f (u,T,S) du. (3.30)

We thus obtain the following expressions, for t0 ≤ t < T0,

EQSd

t [Sf (T0,T,S)] = Sf (t,T,S)QAfd

¡t,T,S, νf , νY , ρfY

¢, (3.31)

QAfd

¡t,T,S, νf , νY , ρfY

¢= e

R T0t λf (u,T,S)du = e−

R T0t νf (u,T,S)νY (u,S)ρfY (u,T,S)du.

(3.32)

The same considerations as in sec. 3.1 apply. In particular, we observe that theadjustment in eqs. 3.31, 3.33 follows from a change between the probability measuresQSf and QSd , or numeraires Af (t,S) and Ad (t,S), naturally associated to the two yieldcurves, f and d, respectively, once swap rates are considered. In the present marketsituation, the volatility νf (u,T,S) in eq. 3.32 can be extracted from quoted swaptionson Euribor6M, while for other rate tenors and for νY (u,S) and ρfY (u,T,S) one mustresort to historical estimates.

An additive quanto adjustment can also be defined as before

EQSd

t [Sf (T0,T,S)] = Sf (t,T,S) +QA0fd¡t,T,S, νf , νY , ρfY

¢, (3.33)

QA0fd¡t,T,S, νf , νY , ρfY

¢= Sf (t,T,S)

£QAfd

¡t,T,S, νf , νY , ρfY

¢− 1¤. (3.34)

18

4. Double-Curve Pricing & Hedging Interest Rate Derivatives

4.1. Pricing

The discussion above allows us to derive the no arbitrage, double-curve-single-currencypricing formulas for interest rate derivatives. The recipes are, basically, eqs. 3.13-3.14or 3.31-3.32.

The simplest interest rate derivative is a floating zero coupon bond paying at timeT a single cashflow depending on a single spot rate (e.g. the Euribor) fixed at timet < T ,

ZCB (T ;T,N) = Nτf (t, T )Lf (t, T ) . (4.1)

Being

Lf (t, T ) =1− Pf (t, T )

τf (t, T )Pf (t, T )= Ff (t; t, T ) , (4.2)

the price at time t ≤ T is given by

ZCB (t;T,N) = NPd (t, T ) τf (t, T )EQTd

t [Ff (t; t, T )]

= NPd (t, T ) τf (t, T )Lf (t, T ) . (4.3)

Notice that the basis adjustment in eq. 4.3 disappears and we are left with the standardpricing formula, modified according to the double-curve paradigm.

Next we have the FRA, whose payoff is given in eq. 2.5 and whose price at timet ≤ T1 is given by

FRA (t;T1, T2,K,N) = NPd (t, T2) τf (T1, T2)

½EQ

T2d

t [Ff (T1;T1, T2)]−K

¾= NPd (t, T2) τf (T1, T2)

£Ff (t;T1, T2)QAfd

¡t, T1, ρfX , σf , σX

¢−K

¤. (4.4)

Notice that in eq. 4.4 for K = 0 and T1 = t we recover the zero coupon bond price ineq. 4.3.

For a (payer) floating vs fixed swap with payment dates vectors T,S as in sec. 3.2we have the price at time t ≤ T0

Swap (t;T,S,K,N)

=nXi=1

NiPd (t, Ti) τf (Ti−1, Ti)Ff (t;Ti−1, Ti)QAfd

¡t, Ti−1, ρfX,i, σf,i, σX,i

¢−

mXj=1

NjPd (t, Sj) τd (Sj−1, Sj)Kj . (4.5)

19

For constant nominal and fixed rate the fair (equilibrium) swap rate is given by

Sf (t,T,S) =

nPi=1

Pd (t, Ti) τf (Ti−1, Ti)Ff (t;Ti−1, Ti)QAfd

¡t, Ti−1, ρfX,i, σf,i, σX,i

¢Ad (t,S)

,

(4.6)where

Ad (t,S) =mXj=1

Pd (t, Sj) τd (Sj−1, Sj) (4.7)

is the annuity on curve d.For caplet/floorlet options on a T1-spot rate with payoff at maturity T2 given by

cf (T2;T1, T2,K,ω,N) = NMax ω [Lf (T1, T2)−K] τf (T1, T2) , (4.8)

the standard market-like pricing expression at time t ≤ T1 ≤ T2 is modified as follows

cf (t;T1, T2,K,ω,N) = NEQT2d

t [Max ω [Lf (T1, T2)−K] τf (T1, T2)]= NPd (t, T2) τf (T1, T2)Bl

£Ff (t;T1, T2)QAfd

¡t, T1, ρfX , σf , σX

¢,K, µf , σf , ω

¤,

(4.9)

where ω = +/− 1 for caplets/floorlets, respectively, and

Bl [F,K,µ, σ, ω] = ω£FΦ

¡ωd+

¢−KΦ

¡ωd−

¢¤, (4.10)

d± =ln F

K + µ (t, T )± 12σ2 (t, T )

σ (t, T ), (4.11)

µ (t, T ) =

Z T

tµ (u) du, σ2 (t, T ) =

Z T

tσ2 (u) du, (4.12)

is the standard Black-Scholes formula. Hence cap/floor options prices are given at t ≤ T0by

CF (t;T,K,ω,N) =nXi=1

cf (Ti;Ti−1, Ti,Ki,ωi,Ni)

=nXi=1

NiPd (t, Ti) τf (Ti−1, Ti)

×Bl£Ff (t;Ti−1, Ti)QAfd

¡t, Ti−1, ρfX,i, σf,i, σX,i

¢,Ki, µf,i, σf,i, ωi

¤, (4.13)

Finally, for swaptions on a T0-spot swap rate with payoff at maturity T0 given by

Swaption (T0;T,S,K,N) = NMax [ω (Sf (T0,T,S)−K)]Ad (T0,S) , (4.14)

20

the standard market-like pricing expression at time t ≤ T0, using the discounting swapmeasure QSd associated to the numeraire Ad (t,S) on curve d, is modified as follows

Swaption (t;T,S,K,N) = NAd (t,S)EQSdt Max [ω (Sf (T0,T,S)−K)]

= NAd (t,S)Bl£Sf (t,T,S)QAfd

¡t,T,S, νf , νY , ρfY

¢,K, λf , νf , ω

¤. (4.15)

where we have used eq. 3.31 and the quanto adjustment termQAfd

¡t,T,S, νf , νY , ρfY

¢is given by eq. 3.32.

When two or more different underlying interest-rates are present, pricing expressionsmay become more involved. An example is the spread option, for which the reader canrefer to, e.g., ch. 14.5.1 in ref. [13]).

The calculations above show that also basic interest rate derivatives prices includea quanto adjustment and are thus volatility and correlation dependent. In fig. 4.1 weshow some numerical scenario for the quanto adjustment in eqs. 3.14, 3.16. We seethat, for realistic values of volatilities and correlation, the magnitudo of the additiveadjustment may be non negligible, ranging from a few basis points up to over 10 basispoints. Time intervals longer than the 6M period used in fig. 4.1 further increase theeffect. Notice that positive correlation implies negative adjustment, thus lowering theforward rates that enters the pricing formulas above.

The standard market practice for pricing interest rate derivatives does not considerthe quanto adjustment, thus leaving, in principle, the door open to arbitrage oppor-tunities. In practice the adjustment depends on variables presently not quoted on themarket, making virtually impossible to set up arbitrage positions to lock today positivegains in the future. Obviously, it is always possible to bet on a view of the futurerealizations of the volatilties and correlation.

4.2. Hedging

We come now to the problem of hedging. We assume to have a trading portfolio filledwith a variety of interest rate derivatives with different underlying rate tenors. The firstissue is how to calculate the delta sensitivity. In principle, the answer is straightforward:having recognized interest-rates with different tenors as different underlyings, and hav-ing constructed multiple yield curves =

nd, I1f , ...,

INf

ousing homogeneous market

instruments, we must coherently calculate the delta with respect to the market rate ofeach bootstrapping instrument8. In practice this can be computationally cumbersome,given the higher number of market instruments involved.

Once the delta sensitivity is known for each pillar of each relevant curve, the nextissues of hedging are the choice of the set of hedging instruments and the calculation

8with the obvious caveat of avoiding double counting of those instruments eventually appearing inmore than one curve (3M Futures for instance could appear both in d and in I3Mf curves).

21

of the corresponding hedge ratios. In principle, there are two alternatives: a) the setof hedging instruments overlaps exactly the set of bootstrapping instruments; or, b) itis a subset restricted to the most liquid bootstrapping instruments. The first choiceallows for a straightforward calculation of hedge ratios and representation of the deltarisk distribution of the portfolio. But, in practice, people prefer to hedge using the mostliquid instruments, both for better confidence in their market prices and for reducingthe cost of hedging. Hence the second strategy generally prevails. In this case thecalculation of hedge ratios requires a three-step procedure: first, the delta is calculatedon the basis of all bootstrapping instruments; second, it is re-aggregated, pillar by pillar,on the basis of hedging instruments, using the appropriate mapping rules; then, hedgeratios are calculated. The disadvantage of this second choice is, clearly, that some risk -the basis risk in particular - is only partially hedged: hence, a particular care is requiredin the choice of the hedging instruments.

A final issue regards portfolio management. In principle one could keep all the inter-est rate derivatives together in a single portfolio, pricing each one with its appropriateforwarding curve, discounting all cashflows with the same discounting curve, and hedg-ing using the preferred choice described above. A possible alternative is the segregationof homogeneous contracts (with the same underlying interest rate index) into dedicatedsub-portfolios, each managed with its appropriate curves and hedging tecniques. Theeventually remaining non-homogeneous instruments (those not separable in pieces de-pending on a single underlying) can be redistributed in the portfolios above accordingto their prevailing underlying (if any), or put in other isolated portfolios, to be handledwith special care. The main advantage of this second approach is to “clean up” thetrading books, “cornering” the more complex deals in a controlled way, and to allowa clearer and self-consistent representation of the sensitivities to the different underly-ings, and in particular of the basis risk of each sub-portfolio, thus allowing for a cleanerhedging.

5. Conclusions

We have discussed how the liquidity crisis and the resulting changes in the market quota-tions, in particular the very high basis swap spreads, have forced the market practice toevolve the standard procedure adopted for pricing and hedging single-currency interestrate derivatives. The new double-curve paradigm involves the bootstrapping of multi-ple yield curves using separated sets of vanilla interest rate instruments homogeneous inthe underlying rate (typically with 1M, 3M, 6M, 12M tenors). Prices, sensitivities andhedge ratios of interest rate derivatives on a given underlying rate tenor are calculatedusing the corresponding forward curve with the same tenor, plus a second distinct curvefor discount factors.

We have shown that the old, well-known, standard single-curve no arbitrage relations

22

are no longer valid and can be recovered with the introduction of a (forward) basisadjustment, for which simple statical expressions are given in eqs. 2.17-2.18 in terms ofdiscount factors from the two curves. Our numerical results have shown that the basisadjustment curves, in particular in a realistic stressed market situation, may display anoscillating term structure, not present in the smooth and monotonic basis swaps marketquotes and more complex than that of the discount and forward curves. Such richermicro-term structure is caused by amplification effects of small local differences betweenthe discount and forwarding curves and constitutes both a very sensitive test of thequality of the bootstrapping procedure (interpolation in particular), and an indicator ofthe tiny, but observable, differences between different interest rate market areas. Bothof these causes may have appreciable effects on the price of interest rate instruments, inparticular when one switches from the single-curve towards the double-curve framework.

Recurring to the foreign-currency analogy we have also been able to recompute theno arbitrage double-curve-single-currency market-like pricing formulas for basic interestrate derivatives, zero coupon bonds, FRA, swaps caps/floors and swaptions in particular.Such prices depend on forward or swap rates on curve f corrected with the well-known quanto adjustment typical of cross-currency derivatives, naturally arising fromthe change between the numeraires, or probability measures, naturally associated to thetwo yield curves. The quanto adjustment depends on the volatility σf of the forwardrates Ff on f , of the volatility σX of the forward exchange rate Xfd between f and d,and of the correlation ρfX between Ff and Xfd. In particular, a non-trivial adjustmentis obtained if and only if the forward exchange rates Xfd are stochastic (σX 6= 0) andcorrelated to the forward rate Ff (ρfX 6= 0). Analogous considerations hold for theswap rate quanto adjustment. Numerical scenarios show that the quanto adjustmentcan be non negligible for realistic values of volatilities and correlation. The standardmarket practice does not take into account the adjustment, thus being, in principle, notarbitrage free, but, in practice the market does not trade enough instruments to set uparbitrage positions. Hence only bets are possible on the future realizations of voloatilityand correlation.

References

[1] P. S. Hagan, G. West, “Methods for Constructing a Yield Curve”, Wilmott Maga-zine, p 70-81, May 2008.

[2] P. S. Hagan, G. West, “Interpolation Methods for Curve Construction”, AppliedMathematical Finance, Vol. 13, No. 2, 89—129, June 2006.

[3] L. Andersen, “Discount Curve Construction with Tension Splines”, Review ofDerivatives Research, Springer, vol. 10, issue 3, pages 227-267, Dec. 2007.

23

[4] U. Ron, “A Practical Guide to Swap Curve Construction”, Working Paper 2000-17,Bank of Canada, Aug. 2000.

[5] P. Madigan, “Libor Under Attack”, Risk Magazine, Jun. 2008,http://www.risk.net/public/showPage.html?page=797090.

[6] F. Ametrano, M. Bianchetti, “Smooth Yield Curves Bootstrapping For ForwardLibor Rate Estimation and Pricing Interest Rate Derivatives”, to be published in“Modelling Interest Rates: Latest Advances for Derivatives Pricing”, edited by F.Mercurio, Risk Books.

[7] E. Fruchard, C. Zammouri, E. Willems, “Basis for change”, Risk, 8(10), 70—75,1995.

[8] B. Tuckman, P. Porfirio, “Interest Rate Parity, Money Market Basis Swaps, andCross-Currency Basis Swaps”, Fixed Income Liquid Markets Research, LehmanBrothers, Jun 2003.

[9] W. Boenkost, W. Schmidt, “Cross currency swap valuation”, Working Paper, HfB—Business School of Finance & Management, May 6, 2005, http://www.frankfurt-school.de/dms/publications-cqf/CPQF_Arbeits2.pdf.

[10] M. Kijima, K. Tanaka, T. Wong, “A Multi-Quality Model of Interest Rates”, Quan-titative Finance, 2008.

[11] M. Morini, “Credit Modelling After the Subprime Crisis”, Marcus Evans course,2008.

[12] F. Mercurio, “Post Credit Crunch Interest Rates: Formulas and Market Models”,working paper, Bloomberg, 2008, http://ssrn.com/abstract=1332205.

[13] D. Brigo, F. Mercurio, “Interest Rate Models - Theory and Practice”, 2nd edition,Springer 2006.

[14] QuantLib is an open-source object oriented C++ financial library(http://www.quantilib.org).

[15] F. Jamshidian, “An Exact Bond Option Formula”, Journal of Finance, 44, pp.205-209, 1989.

[16] H. Geman, N. El Karoui, J.C.Rochet, “Changes of Numeraire, Changes of Prob-ability Measure and Option Pricing”, J. of Applied Probability, Vol. 32, n. 2, pp.443-458, 1995.

24

Figure 1.1: Quotations as of 30 Sep. 2008 for five basis swap curves corresponding tothe four Euribor swap curves 1M, 3M, 6M, 12M (source: Reuters, contributor: ICAP).

25

EUR Basis swaps

0

10

20

30

40

50

60

70

80

90

100

110

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

11Y

12Y

15Y

20Y

25Y

30Y

basi

s sp

read

(bps

)

3M vs 6M1M vs 3M1M vs 6M6M vs 12M3M vs 12M1M vs 12M

Figure 1.2: Euribor basis swap spreads (basis points) versus swap maturity (years) fromFig. 1.1. The curve 1M-12M has been deduced from the quoted 3M-12M and 1M-3Mcurves.

26

6M curve

3.00%

3.25%

3.50%

3.75%

4.00%

4.25%

4.50%

4.75%

5.00%

09/2

008

09/2

011

09/2

014

09/2

017

09/2

020

09/2

023

09/2

026

09/2

029

09/2

032

09/2

035

09/2

038

09/2

041

09/2

044

09/2

047

09/2

050

09/2

053

09/2

056

09/2

059

09/2

062

09/2

065

forw

ard

rate

(%)

Discount curve

3.00%

3.25%

3.50%

3.75%

4.00%

4.25%

4.50%

4.75%

5.00%

09/2

008

09/2

011

09/2

014

09/2

017

09/2

020

09/2

023

09/2

026

09/2

029

09/2

032

09/2

035

09/2

038

09/2

041

09/2

044

09/2

047

09/2

050

09/2

053

09/2

056

09/2

059

09/2

062

09/2

065

forw

ard

rate

(%)

Figure 2.1: Forward curves, plotted with 6M-tenor forward ratesF (t0; t, t+ 6M, act/360 ), t daily sampled and t0 = 15 Sep. 2008. Upper panel:forward curve from 6M curve I6Mf ; lower panel: forward curve from discount curve d(see description in the text). The effects of the market stress are clearly visible in thecrazy roller-coaster look-up of the curves. The same pattern is observed also in the1M, 3M, 12M curves (not reported here).

27

Basis adjustment (multiplicative)

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

Sep-

08

Dec

-08

Mar

-09

Jun-

09

Sep-

09

Dec

-09

Mar

-10

Jun-

10

Sep-

10

Dec

-10

Mar

-11

Jun-

11

Sep-

11

basi

s ad

just

men

t

1M vs Disc3M vs Disc6M vs Disc12M vs Disc

Basis adjustment (additive)

-60

-40

-20

0

20

40

60

80

100

120

Sep-

08

Dec

-08

Mar

-09

Jun-

09

Sep-

09

Dec

-09

Mar

-10

Jun-

10

Sep-

10

Dec

-10

Mar

-11

Jun-

11

Sep-

11

basi

s sp

read

(bps

)

1M vs Disc3M vs Disc6M vs Disc12M vs Disc

Basis adjustment (multiplicative)

0.97

0.98

0.99

1.00

1.01

1.02

1.03

Sep-

11

Sep-

14

Sep-

17

Sep-

20

Sep-

23

Sep-

26

Sep-

29

Sep-

32

Sep-

35

Sep-

38

Sep-

41

Sep-

44

Sep-

47

basi

s ad

just

men

t

1M vs Disc3M vs Disc6M vs Disc12M vs Disc

Basis adjustment (additive)

-15

-10

-5

0

5

10

15Se

p-11

Sep-

14

Sep-

17

Sep-

20

Sep-

23

Sep-

26

Sep-

29

Sep-

32

Sep-

35

Sep-

38

Sep-

41

Sep-

44

Sep-

47

basi

s sp

read

(bps

)

1M vs Disc3M vs Disc6M vs Disc12M vs Disc

Figure 2.2: Upper panels: multiplicative basis adjustments from eq. 2.17 as of 15 Sep.2008 (end of day), for daily sampled 6M-tenor forward rates as in fig. 2.1, calculatedon I1Mf , I3Mf , I6Mf and I12Mf curves against d taken as reference curve. Lower panels:equivalent plots of the additive basis adjustment of eq. 2.18 between the same forwardrates (basis points). Left panels: 0Y-3Y data; Right panels: 3Y-40Y data on magnifiedscales. The higher short-term adjustments seen in the left panels are due to the highershort-term market basis spread (see Figs. 1.1-1.2). The oscillating term structureobserved is due to the amplification of small differences in the term structures of thecurves.

28

Basis adjustment (additive)

-25

-20

-15

-10

-5

0

5

10

15

20

25

Sep

-08

Sep

-10

Sep

-12

Sep

-14

Sep

-16

Sep

-18

Sep

-20

Sep

-22

Sep

-24

Sep

-26

Sep

-28

Sep

-30

Sep

-32

Sep

-34

Sep

-36

Sep

-38

Sep

-40

Sep

-42

Sep

-44

Sep

-46

Sep

-48

basi

s sp

read

(bps

)

1M vs Disc3M vs Disc6M vs Disc12M vs Disc

Forward curve 6M

3.00%

3.25%

3.50%

3.75%

4.00%

4.25%

4.50%

4.75%

5.00%

09/2

008

09/2

011

09/2

014

09/2

017

09/2

020

09/2

023

09/2

026

09/2

029

09/2

032

09/2

035

09/2

038

09/2

041

09/2

044

09/2

047

09/2

050

09/2

053

09/2

056

09/2

059

09/2

062

09/2

065

forw

ard

rate

(%)

Figure 2.3: Same as in figs. 2.1 and 2.2, but with linear interpolation on zero rates (acommon market practice). The angular points in the lower panel clearly show the inade-quacy of the boostrap, but the very strong oscillations in the (additive) basis adjustmentin the upper panel (notice the different scales w.r.t. fig. 2.2, lower panels) allows to fur-ther appreciate the innatural differences induced in similar forward instruments pricedon the two curves.

29

Figure 3.1: Picture of no-arbitrage interpretation for the forward exchange rate in eq.3.2. Moving, in the yield curve vs time plane, from top right to bottom left cornerthrough path A or path B must be equivalent. Alternatively, we may think to no-arbitrage as a sort of zero “circuitation”, sum of all trading events following a closedpath starting and stopping at the same point in the plane. This description is equivalentto the traditional “table of transaction” picture, as found e.g. in fig. 1 of ref. [8].

30

Quanto Adjustment (multiplicative)

0.96

0.97

0.98

0.99

1.00

1.01

1.02

1.03

1.04

-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0Correlation

Qua

nto

adj.

Sigma_f = 20%, Sigma_X = 5%Sigma_f = 30%, Sigma_X = 10%Sigma_f = 40%, Sigma_X = 20%

Quanto Adjustment (additive)

-20

-15

-10

-5

0

5

10

15

20

-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0Correlation

Qua

nto

adj.

(bps

)

Sigma_f = 20%, Sigma_X = 5%Sigma_f = 30%, Sigma_X = 10%Sigma_f = 40%, Sigma_X = 20%

Figure 4.1: Numerical scenarios for the quanto adjustment. Upper panel: multiplicative(from eq. 3.14); lower panel: additive (from eq. 3.16). In each figure we show the quantoadjustment corresponding to three different combinations of (flat) volatility values as afunction of the correlation. The time interval is fixed to T1 − t = 0.5 and the forwardrate entering eq. 3.16 to 4%, a typical value in fig. 2.1. We see that, for realisticvalues of volatilities and correlation, the magnitudo of the additive adjustment may beimportant.

31