post-shock short rate pricing

risk-magazine.net 83

liquidity and credit crises included large basis

spreads opening up between overnight collateralised instruments (for example, six-month Eonia) and non-collateralised fixings1 (for example, six-month Euribor). Deposits of the same maturity as fixings follow the same pattern. Yield curves built from instru-ments referencing these different rates demonstrate a significant basis (for example, building from Eonia swaps versus building from Euribor swaps). This means that new formulas are required for pricing previously standard instruments, for example, swaps, caps and swaptions. Recent literature (Bianchetti, 2010, Chibane & Sheldon, 2009, Fujii, Shimada & Takahashi, 2009, Kijima, Tanaka & Wong, 2009, Henrard, 2009, Mercurio, 2009 and 2010, Morini, 2009, Pallavicini & Tarenghi, 2010, and Ametrano & Bianchetti, 2010) offers different approaches using short-rate, Heath-Jarrow-Morton and Brace-Gatarek-Musiela settings with or without some foreign exchange analogy. However, within the

short-rate set-up, analytic swaption pricing formulas are lacking. Thus calibration to volatility data is awkward. While Kijima, Tanaka & Wong (2009) use a short-rate setting for multiple bond qualities, they do not provide direct analytic swaption pricing (they price via a Gram-Charlier expansion on the bond price dis-tribution). Short-rate settings can be, but are not necessarily, more appropriate when credit risk needs to be included, for exam-ple in credit value adjustment (CVA) (Brigo & Masetti, 2006, Brigo & Mercurio, 2006, and Brigo, Pallavicini & Papatheod-orou, 2009), because default can occur at any time.

We provide direct pseudo-analytic swaption pricing formulas within a short-rate set-up including the basis between overnight collateralised instruments (for example, six-month Eonia) and non-collateralised fixings (for example, six-month Euribor). We use a discounting curve to represent risk-free investment and a separate curve for market expectations of non-collateralised fix-ings. We call this second curve the fixing curve. Our approach differs from Kijima, Tanaka & Wong (2009) in that they use dis-counting and basis curves whereas we use discounting and fixing curves. We follow Bianchetti (2010) in using an explicit forex argument to deal with potential arbitrage considerations. Our approach can also be generalised as more data becomes available, especially with respect to Eonia-type swaptions (currently miss-ing from the market). Thus we propose a short-rate solution for the observed basis, enabling modelling of standard instruments and calibration to swaptions.

The forex analogy for Eonia/Euribor modelling, as in Kijima, Tanaka & Wong (2009) and Bianchetti (2010), is motivated by the fact that fixed-for-floating swaps have characteristics of quan-tos. They observe Euribor fixings, that is, fixings based on deposits of a specific tenor. Deposits are risky and funded. Then they pay exactly the number from the fixing in a collateralised and unfunded2 (that is, risk-free) setting. Furthermore, risky and risk-free assets are not interchangeable in the market. The same (uncol-lateralised) asset will have a different price from entities with dif-ferent risk levels, so there is some analogy to an exchange rate between risk levels (or qualities in the language of Kijima, Tanaka & Wong, 2009). The general approach of quoting a spread to price bonds from different issuers serves the same purpose. Kijima, Tan-aka & Wong (2009) use a pricing kernel approach to change units (from risky to risk-free) whereas Bianchetti (2010) uses a forex rate to change units. We regard risky and funded as analogous to one currency and collateralised and unfunded as analogous to another currency. Note that while individual risky entities do default, Euribor fixings are not strongly dependent on the default of any single entity. Market practice supports this view that different investment qualities exist in the separation of funded/collateral-ised (or risky/risk-free) worlds. We do not attempt to model the drivers of this separation, but start from the observable Eonia and Euribor instruments and derivatives.

Each different Euribor tenor fixing, for example, one month, three months or six months, represents a different level of risk (including liquidity risk). Thus we have a separate fixing curve for each tenor D that we label f

D to make the source tenor

explicit. (In forward-rate models, each different tenor forward is a different product.) We have a single discounting curve (Eonia).

Post-shock short-rate pricingThe basis between swaps referencing funded fixings and swaps referencing overnight collateralised fixings has increased in importance with the 2007–09 liquidity and credit crises. This basis means that new pricing models for fixed-income staples such as caps, floors and swaptions are required. Recently, new formulas have been proposed using market models. Here, Chris Kenyon presents equivalent pricing in a short-rate framework, which is important for applications involving credit, such as credit value adjustment, because default can occur at any time

The2007–09

cuttingedge.inteReStRAtedeRiVAtiVeS

1 In fixed income, ‘fixing’ is a noun. It means the value at which a rate is fixed once a day for contract settlement, for example, ‘Euribor is the rate at which euro interbank term deposits are offered by one prime bank to another prime bank within the EMU zone, and is published at 11:00am (central European time) for spot value (T + 2)’2 Of course, when swaps go off-market they require funding inasmuch as they are out-of-the-money, and they are not part of an asset-swap package, etc

NOT FOR REPRODUCTION

84 RiskNovember2010


Unlike pre-crisis short-rate modelling, different levels of risk are now priced significantly differently and are not interchangeable. Thus each fixing curve is specific to the tenor that it was con-structed from. While we could calculate a one-month rate from a six-month tenor fixing curve, it will not be the same as the one-month fixing, because it represents a different level of risk. Thus a short-rate approach in the post-crisis world provides tenor fixings at different future times, but not the fixings from different tenors (at least not with the level of risk of the different tenor). This is inherent in the post-crisis market: fixing curves multiply whether in short-rate or forward-rate models. Thus the new post-crisis world causes fundamental changes to short-rate modelling. Previously, short-rate models could provide future floating rates for any tenor, but now a different fixing curve is required for each tenor.

A short-rate set-up, for the observed basis between discount-ing and fixing, is potentially useful for a variety of problems, for example, CVA, as it offers a complementary approach to for-ward-rate modelling (Brigo, Pallavicini & Papatheodorou, 2009, and Brigo & Masetti, 2006). In addition, the credit world in general has a strong connection to short-rate modelling because default can happen at any time, not just at tenor multi-ple maturities. Brigo & Mercurio (2006) expand on this point and conclude that the forward-rate approach will be less domi-nant in the credit space (see Chapter 23). As they state, a short-rate approach combines naturally with intensity-based hazard-rate modelling. Thus this article complements existing forward-rate multi-curve set-ups (Heath-Jarrow-Morton and Brace-Gatarek-Musiela) (Bianchetti, 2010, Henrard, 2009, Mercurio, 2009 and 2010, Ametrano & Bianchetti, 2010, and Pallavicini & Tarenghi, 2010) and extends Kijima, Tanaka & Wong, 2009).

ModelWe put ourselves in the European context by using Euribor and Eonia. The equations are general and this naming is only for convenience.n Discount curve. We use a risk-free curve as our discount curve. For a concrete example, we take Eonia as the risk-free discount curve. We aim to price standard fixed-income instru-ments (for example, swaps and swaptions) subject to collateral agreements. Thus the yield curve for similarly collateralised instruments is appropriate for discounting whatever fixings they reference.n Fixing curve. We calibrate a second curve, or ‘fixing curve’, to reproduce fixings and market expected fixings (when used together with the discount curve). Note that each different fix-ing tenor D leads to a different fixing curve f

D. For a concrete

example, consider the six-month Euribor fixing and market quoted standard euro swaps. This follows the concept in Ame-trano & Bianchetti (2010) and Kijikma, Tanaka & Wong (2009) of having a discount curve and a separate curve that, together with the discount curve, reproduces market expecta-tions of fixings and instruments based on them, for example, forwards and swaps.

Note that a forward rate agreement (FRA) payment is not the same as a swap payment for the same fixing because a FRA is paid in advance (discounted with the fixing by definition), whereas a standard swap fixing is paid in arrears.

We use a short-rate model for the discounting curve D, and for the fixing curve f

D, with short rates rD(t) and rfD

(t) respectively.3

Note that the level of risk that the fixing curve represents is explicitly referenced in D. Although the short rates are both denominated in the same currency, say, the euro, they represent theoretically different investment qualities (risks), as in Kijikma, Tanaka & Wong (2009). Thus there is an exchange rate between them. This set-up also mirrors that of Bianchetti (2010) to pre-serve no-arbitrage between alternative investment opportunities.

As in Kijikma, Tanaka & Wong (2009), the second (fixing) curve f

D represents an investment opportunity inasmuch as mar-

ket D fixing tenor risk curve bonds are available. While it may be difficult in practice to find a provider with the appropriate level of risk for a given tenor this does not affect the consistency of the derivation. All modelling abstracts from reality – for example, no risk-free (or overnight risk level) bonds exist because even previ-ously safe sovereign government bonds (for example, those of the US, the UK and Germany) have recently exhibited significant yield volatility.n Standard instruments. The set-up for standard instruments (swaps and swaptions) is very similar to that of Kijima, Tanaka & Wong (2009), Biancheti (2010), Ametrano & Bianchetti (2010) and Mercurio (2009). Our innovations are in the next section, where we show how to price them within a specific short-rate set-up.n Definitions. We define each of our base items here: short rates rD, rfD

; bank accounts Bl(t) (l = D, fD); and discount factors Pl(t, T).

Ft is the usual filtration at t. rD is the instantaneous rate of return of a risk-free investment. rfD

is the instantaneous rate of return of a risky investment with risk level corresponding to Libor with tenor D. Bl(t) = exp(∫ t

0rl(s)ds) is the bank account with level of risk l = D, f

D. Pl(t, T) = El[Bl(t)/Bl(T)|Ft] is the zero-coupon bond

with level of risk l = D, fD.

n Swaps. Swaps in this dual curve set-up are similar to differen-tial floating for fixed swaps (also known as floating for fixed quanto swaps) in Brigo & Mercurio (2006) (see page 623). To price swaps, we need to define the simply compounded (fixing curve) interest rate that the swaps fix on, Lk(t), at Tk−1 for the inter-est rate from Tk−1 to Tk, that is, tenor τ fix. Note that f

D refers to

tenor D or τ fix (either can be clearer in context). We define this rate in terms of the fixing curve with tenor D:

Lk t( ) :=1τ fix

PfΔ t,Tk−1( )PfΔ t,Tk( )

−1⎛

⎝⎜⎜

⎞

⎠⎟⎟

Using the terminology of Mercurio (2009) and the equations above, we have for the floating leg of a swap:

FL t :Ta ,...,Tb( ) = FL t :Tk−1,Tk( )k=a+1

b∑ = τkPD t,Tk( )Lk t( )

k=a+1

b∑

Note that since separate curves are used for discounting and fix-ing, there is no reason for a floating leg together with repayment of par at the end to price at par. A risk-free floating-rate bond should price to par, but that bond would not get Euribor coupons but Eonia coupons. This has previously been pointed out by Mer-curio (2009) and Kijima, Tanaka & Wong (2009).

From these definitions, we can bootstrap both discounting and fixing curves from an Eonia discount curve (built conventionally) and Euribor deposits and swaps. Note that we will need the dynamics of r*(t), where * is either D or f

D, under the T−forward

measure, that is, using the zero-coupon D-quality (risk-free) bond with maturity T, for swaption pricing.3 We need d later for an index, hence the use of D for the discounting curve



Figure 1 shows the discount curve (Eonia) and the fixing curve bootstrapped from Euribor swaps and six-month deposits. Note that a conventional Euribor curve (not shown) would be very close to the fixing curve. By conventional, we mean one boot-strapped from swaps alone, that is, done without reference to a risk-free discount curve.n Swaptions. For a European swaption with strike K and matu-rity T, directly following on from the formulas for a swap above, we have:

ES ω,T ,K,a,b,c,d[ ]

= PD 0,T( )EDT ω τi

fixPD T ,Ti( )Li T( )i=a+1

b∑

⎛

⎝⎜

⎡

⎣⎢⎢

⎧⎨⎪

⎩⎪

− τiKPD T ,Ti( )i=c+1

d∑

⎞

⎠⎟⎤

⎦⎥⎥

+⎫⎬⎪

⎭⎪

where ω = 1 for a payer swaption (and –1 for a receiver swaption), and the a, b, c, d, τ i

fix, τi take care of the different tenors and pay-ment frequencies of the two sides. Note that because the floating leg does not price to par we must include the specification of the floating coupons. These floating coupons are the market expecta-tion of the relevant fixings Li(T).n Short-rate pricing. We present semi-analytic pricing (that is, simulation and formulas) for swaptions under general affine short-rate models and pseudo-analytic pricing (that is, integration and formulas) in the Gaussian case.

For full analytic tractability, we use a one-factor Hull-White type model for the discount curve (for example, Eonia), expressed as a G1++ type model (analogous to the terminology in Brigo & Mercurio, 2006) for the discounting short rate rD(t) under the risk-free bank account numeraire for D:

rD t( ) = x t( )+ϕD t( ), rD 0( ) = rD,0dx t( ) = −aDx t( )dt +σDdW1 t( )

(1)

We also use a one-factor model for the fixing curve, that is, to fit the market-expected fixings, and we label this short rate rfD

(t) under the D tenor-level risk bank account numeraire for f

D:

rfΔ t( ) = y t( )+ϕ fΔ t( ), rfΔ 0( ) = rfΔ , 0dy t( ) = −a fΔ y t( )dt +σ fΔdW2 t( )

(2)

where (W1(t), W2(t)) is a two-dimensional Brownian motion with instantaneous correlation r. We also have a quality exchange rate (also known as the risk exchange rate) X with process under QfD

for the quantity of f

D-quality (or risk-level) investment required to

obtain one unit of D-quality investment:dX t( ) = rfΔ t( )− rD t( )( )X t( )dt + νX t( )dWX t( )

where we assume WX has zero correlation with W1(t) and with W2(t). This assumption implies that equation (2) is unchanged under the measure change from QfD

to QD.This is a very parsimonious representation relative to two

G2++ models. It is, however, sufficient since we only wish to calibrate to Eonia discount rates and Euribor swaptions. Note that neither Eonia caps/floors nor swaptions are liquid at present. It is tempting to use three-month swaptions and six-month swaptions to get implied Euribor swaption volatility. However, there is a significant difference in their liquidity so this is problematic.

n Swaption pricing. For a European swaption with strike K and maturity T, we have for any one-factor affine short-rate model (that is, bond prices available as A()eB()):

ES ω,T ,K,a,b,c,d[ ]

= PD 0,T( )ET ω τifixPD T ,Ti( )Li T( )

i=a+1

b∑

⎛

⎝⎜

⎡

⎣⎢⎢

⎧⎨⎪

⎩⎪


d∑

⎞

⎠⎟⎤

⎦⎥⎥

+⎫⎬⎪

⎭⎪

= PD 0,T( )ET ω PD T ,Ti( )i=a+1

b∑

⎛

⎝⎜

⎡

⎣⎢⎢

⎧⎨⎪

⎩⎪

PfΔ T ,Ti−1( )PfΔ T ,Ti( )

−1⎛

⎝⎜⎜

⎞

⎠⎟⎟


d∑

⎞

⎠⎟⎤

⎦⎥⎥

+⎫⎬⎪

⎭⎪

= PD 0,T( ) ω ADi=a+1

b∑

⎛

⎝⎜

⎡

⎣⎢⎢

⎧⎨⎪

⎩⎪R2

⌠

⌡

⎮⎮ T ,Ti( )e−BD T ,Ti( )x

×AfΔ T ,Ti−1( )AfΔ T ,Ti( )

e −BfΔ T ,Ti−1( )+BfΔ T ,Ti( )( )y −1⎛

⎝⎜⎜

⎞

⎠⎟⎟

− τiKAD T ,Ti( )e−BD T ,Ti( )x

i=c+1

d∑

⎞

⎠⎟⎤

⎦⎥⎥

+⎫⎬⎪

⎭⎪f x, y( )dxdy

(3)

where A*, B*, * = D, fD are the affine factors for the risk-free and f

D

quality bond prices in their respective units of account. Note that up to this point the equations apply to any affine short-rate model. However, to go further analytically we require the joint distribu-tion of x and y under the T-forward measure, that is, f(x, y) above. This is available for Gaussian models but not for Cox-Ingersoll-Ross-type specification. This formula could be applied in a Cox-Ingersoll-Ross-type specification by combining simulation up to T with the affine bond formulas used above.

We require equations (1) and (2) in the TD−forward measure (that is, using the zero-coupon bond from the discounting curve D as the numeraire). Using standard change-of-numeraire machinery, we obtain:

5 10 15 20

1

2

3

4

00

Note: a conventional Euribor curve (not shown) would be veryclose to the fixing curve

Fixing curveDiscounting curve

1eoniaandeuribordiscountingandfixingcurves



86 RiskNovember2010


dx t( ) = −aDx t( )−σD2

aD1− e−aD T−t( )( )

⎡

⎣⎢

⎤

⎦⎥dt +σDdW1

T

dy t( ) = −a fΔ y t( )−ρσ fΔσD

aD1− e−aD T−t( )( )⎡

⎣⎢

⎤

⎦⎥dt +σ f dW2

T

This is because the Radon-Nikodym derivative dQTD/dQD uses the

zero-coupon discounting bond, hence only x is involved in the measure change (not y as well).

These have explicit solutions:

x t( ) = x s( )e−aD t−s( ) −MxT s,t( ) +σD e−aD t−u( ) dW1

T u( )st∫

y t( ) = y s( )e−a fΔ t−s( ) −MyT s,t( )+σ fΔ e−a fΔ t−u( )

st∫ dW2

T u( )

where:

MxT s,t( ) = σD

2

aD1− e−aD T−t( )( )

⎡

⎣⎢

⎤

⎦⎥e−aD t−u( )

s

t⌠

⌡

⎮⎮ du

MyT s,t( ) = ρ

σ fΔσD

aD1− e−aD T−t( )( )⎡

⎣⎢

⎤

⎦⎥e

−a fΔ t−u( )

s

t⌠

⌡⎮⎮ du

Now we can express a European swaption price as follows.n Theorem 1. The arbitrage-free price at time t = 0 for the above European unit-notional swaption is given by numerically calcu-lating the following one-dimensional integral:

ES ω,T ,K,a,b,c,d( ) = −ωPD 0,T( ) δ x( ) e−12

x−µxσx( )

2

σ x 2π−∞

∞⌠

⌡

⎮⎮⎮

Φ −ωh1 x( )( )− λieκi x( )Φ −ωh2 x( )( )

i=a+1

b∑

⎡

⎣⎢

⎤

⎦⎥dx

where:

ω =1 for a payer swaption(and −1 for a receiver swaption)

δ x( ) := AD T ,Ti( )e−BD T ,Ti( )x

i=a+1

b∑

+ τii=c+1

d∑ KAD T ,Ti( )e−BD T ,Ti( )x

h1 x( ) :=y x( )−µyσ y 1−ρxy

2−ρxy x −µx( )σ x 1−ρxy

2

h2 x( ) := h1 x( )+αiσ y 1−ρxy2

αi := − −BfΔ T ,Ti−1( ) + BfΔ T ,Ti( )( )

λi x( ) := 1δ x( )

AD T ,Ti( )e−BD T ,Ti( )x AfΔ T ,Ti−1( )AfΔ T ,Ti( )

κi x( ) := −αi µy −121−ρxy

2( )σ y2αi +ρxyσ y

x −µxσ x

⎡

⎣⎢

⎤

⎦⎥

where y_ = y

_(x) is the unique solution of:

ADi=a+1

b∑ T ,Ti( )e−BD T ,Ti( )x AfΔ T ,Ti−1( )

AfΔ T ,Ti( )e −BfΔ T ,Ti−1( )+BfΔ T ,Ti( )( )y −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= τiKADi=c+1

d∑ T ,Ti( )e−BD T , Ti( )x

and:

µx := −MxT 0,T( )

µy := −MyT 0,T( )

σ x := σD1− e−2aDT

2aD

σ y := σ fΔ1− e−2a fΔT

2a fΔ

ρxy :=ρσDσ fΔ

aD + a f( )σ xσ y1− e− aD+a fΔ( )T⎡⎣⎢

⎤⎦⎥

and:

A* T ,Ti( ) =P* 0,Ti( )P* 0,T( )

exp 12V T ,Ti ,a*,σ*( )(

⎡

⎣⎢

−V 0,Ti ,a*,σ*( )+V 0,T ,a*,σ*( ))⎤⎦

B* T ,Ti( ) = 1− ea* Ti−T( )

a*V T ,Ti ,a*,σ*( )

=σ*2

a*2 Ti −T( )+ 2

a*e−a* Ti−T( ) 1

a*e−2a* Ti−T( ) −

32a*

⎛

⎝⎜

⎞

⎠⎟

where * is either fD or D.

n Proof. We manipulate equation (3) of ES above into the same form as for Theorem 4.2.3 in Brigo & Mercurio (2006) and pro-vide an appropriate new definition of y

_ to finish the proof.

The []+ part of equation (3) of ES is of the form:

Uie−Vix Fie

−Giy −1( )i=1

n∑ − M j

j=1

m∑ eN jx

Expanding:

Uie−VixFie

−Giy

i=1

n∑ − Uie

−Vix

i=1

n∑ − M j

i= j

m∑ eN jx

Now if we multiply and divide by the two negative terms we obtain:

− Uke−Vkx + M je

N jx

j=1

m∑

k=1

n∑⎛

⎝⎜⎜

⎞

⎠⎟⎟

1− Uie−Vix

Σk=1n Uke

−Vkx + Σ j=1m M je

N jxFie

−Giy

i=1

n∑

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Now define y_ = y

_(x) as the unique solution of:

Uie−Vix

i∑ Fie

−Giy −1( ) = M jj∑ eN jx

So if we freeze x we have the same form as Theorem 4.2.3 in Brigo & Mercurio (2006), and the rest of the proof is immediate.



As an example, we calibrated to Eonia, Euribor swaps and Euribor swaptions using data as of February 8, 2010. We use the following calibration parameters: aD = 1.0, sD = 0.4, afD

= 0.1, sfD

= 0.0105 and r = –0.95.Swaption calibration with the G1++/G1++ model is similar to

that in Brigo, Pallavicini & Papatheodorou (2009) using a G2++ model, which also uses post-crisis data (see figure 2).

ConclusionWe provide pseudo-analytic pricing for swaptions in a dis-counting/fixing multi-curve Gaussian short-rate set-up. Each different tenor fixing D gives rise to a different fixing curve f

D.

Semi-analytic swaption pricing for general affine short-rate models is possible by combining simulation up to maturity with analytic bond formulas at swaption maturity. Pricing of other standard instruments, for example, caps, are simplifica-tions of the swaption formulas provided. Structurally, the pseudo-analytic calculations are very similar to G2++ pseudo-analytic swaption pricing in Brigo & Mercurio (2006). This means that the current calculations could be directly extended to a G1++/G2++ set-up or G2++/G2++ set-up once Eonia swap-tions, which are currently not available, become liquid. This work extends Kijima, Tanaka & Wong (2009), which also worked in the short-rate setting but did not provide pseudo-analytic swaption pricing.

The post-crisis world, with significant basis spreads between tenors, produces a fundamental change in short-rate models apart from having separate discounting and fixing f

D curves.

Each fixing curve can only provide fixings (floating coupons) for the tenor D from which it was constructed. Creating a cou-pon for a different tenor Dother will not reproduce market-expected fixings for the other tenor. This is because the other tenor represents a different level of risk. This is inherent in the post-crisis market: fixing curves multiply whether in short-rate or forward-rate models. Thus the new post-crisis world causes fundamental changes to short-rate modelling: fixings curves can produce floating coupons for any date but only for the tenor D associated with the fixing curve f

D.

There is a significant lack of data in the market at present. Eonia/overnight indexed swap swaptions are missing, for exam-ple. We calibrate our model jointly to give Eonia and Euribor volatility. However, it is not possible to explicitly test our identi-fication with current market data. Also, six-month volatility data does not give full information about three-month volatil-ity. Although futures options are liquid and can help, the long end of the volatility curve is simply missing for non-standard tenors. Without this data pricing, options on non-standard tenor fixings (for example, three-month Euribor, 12-month dol-lar Libor) is problematic.

The Gaussian framework is convenient analytically but, as is well known, permits negative rates. In this context, it also per-mits negative basis spreads dependent on the strength of the correlation between the driving processes, their relative volatili-ties and the strength of their respective mean reversions. It is possible to obtain analytic results with factor-correlated Cox-Ingersoll-Ross processes where the correlation is done at the process level (not at the level of the driving Brownian motions, which remain independent). However, this does not give rise to constant instantaneous correlations and is outside the scope of this article.

A short-rate set-up for multiple yield curves, that is, discount-

ing and fixing, is potentially useful for a variety of problems, for example, CVA, where forward-rate set-ups are less natural, as in Brigo, Pallavicini & Papatheodorou (2009). Thus this article complements existing market model approaches (Henrard, 2009, Mercurio, 2009, and Ametrano & Bianchetti, 2010). n

chrisKenyonisassociatedirectorandheadofstructuredcreditvaluationatdepfaBankindublin.HewouldliketothankdonalgallagherandRolandStammforusefuldiscussions.theviewsexpressedherearehisonly,andnootherrepresentationshouldbeattributed.email:[email protected]

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Brigo D and F Mercurio, 2006Interest rate models: theory and practiceSpringer,secondedition

Brigo D, A Pallavicini and V Papatheodorou, 2009Bilateral counterparty risk valuation for interest-rate products: impact of volatilities and correlationsSSRNeLibrary

Chibane M and G Sheldon, 2009Building curves on a good basisSSRNeLibrary

Fujii M, Y Shimada and A Takahashi, 2009A market model of interest rates with dynamic basis spreads in the presence of collateral and multiple currenciesUniversityofTokyoandShinseiBank,availableatwww.e.u-tokyo.ac.jp/cirje/research/dp/2009/2009cf698.pdf

Henrard M, 2009The irony in the derivatives discounting part II: the crisisSSRNeLibrary

Kijima M, K Tanaka and T Wong, 2009A multi-quality model of interest ratesQuantitativeFinance9(2),pages133–145

Mercurio F, 2009Interest rates and the credit crunch: new formulas and market modelsBloombergPortfolioResearchPaperno2010-01-FRONTIERS

Mercurio F, 2010A Libor market model with stochastic basisSSRNeLibrary

Morini M, 2010Solving the puzzle in the interest rate marketSSRNeLibrary

Pallavicini A and M Tarenghi, 2010Interest-rate modeling with multiple yield curvesSSRNeLibrary

References

1

1 2 3 4 5 7 10 2 3 4 5 6 78 9 10

Maturity

Tenor

–400

–200

0

200

400

bp

2Swaptioncalibrationforg1++/g1++model(errorsinbasispoints)


post-shock short rate pricing

Documents

shortrate setup

shortrate settings

noncollateralised fixings

shortrate solution

eonia swaps

sixmonth eonia

basis curves

euribor swaps