mathematical modeling of interest rates: challenges … · mathematical modeling of interest rates:...

Mathematical Modeling of Interest Rates:Challenges and New Directions

Fabio Mercurio

Bloomberg L.P.

SIAM TutorialMinneapolis - Sunday, July 8, 2012

Before August 2007• Before the credit crunch of 2007, interest rates in the market

showed typical textbook behavior. For instance:

Example 1A floating rate bond where LIBOR is set in advance and paidin arrears is worth par (=100) at inception.

-0 T1 T2 TM

6

100

Ti−1 Ti

6

100 · τi · L(Ti−1,Ti)

where τi is the “length” of the interval (Ti−1,Ti ], andL(Ti−1,Ti) denotes the LIBOR at Ti−1 for maturity Ti .

LIBOR: London Inter-Bank Offer Rate, the interest rate thatbanks charge each other for loans over predefined timehorizons. It is officially fixed once a day by the BBA.

Before August 2007Example 2The forward rate implied by two deposits coincides with thecorresponding FRA rate: FDepo = FFRA.

The forward rate implied by the two deposits with maturity T1and T2 is defined by:

FDepo =1

T2 − T1

[P(0,T1)

P(0,T2)− 1

]The corresponding FRA rate is the (unique) value ofK = FFRA for which the following swap(let) has zero value attime t = 0.

-

t = 0 T1 T2

L(T1,T2)− K

Before August 2007

Example 3Compounding two consecutive 3m forward LIBOR ratesyields the corresponding 6m forward LIBOR rate:(

1 + 14F 3m

1)(

1 + 14F 3m

2)

= 1 + 12F 6m

whereF 3m

1 = F (0; 3m,6m)

F 3m2 = F (0; 6m,9m)

F 6m = F (0; 3m,9m)

-

t = 0 3m 6m 9m︷ ︸︸ ︷︷ ︸︸ ︷︸ ︷︷ ︸

F 3m1 F 3m

2

F 6m

Before and after August 2007

• These textbook relations are examples of the no-arbitragerules that held in the market until 3 years ago.

• In fact, differences between theoretically-equivalent rateswere present in the market, but generally regarded asnegligible.

• For instance, deposit rates and OIS rates for the samematurity used to track each other, but keeping a distance (thebasis) of a few basis points.

• Then August 2007 arrived, and our convictions began towaver: The liquidity crisis widened the basis betweenpreviously-near rates.

• Consider the following graphs ...

Before and after August 2007

• Since the credit crunch of 2007, the LIBOR-OIS basis hasbeen neither deterministic nor negligible:

USD 3m OIS rates vs 3m LIBOR rates

Before and after August 2007

USD 6m OIS rates vs 6m LIBOR rates

Before and after August 2007

EUR 3m EONIA rates vs 3m LIBOR rates

Before and after August 2007

EUR 6m EONIA rates vs 6m LIBOR rates

Before and after August 2007• Likewise, since August 2007 the basis between different

tenor LIBORs has been neither deterministic nor negligible:

USD 5y swaps: 1m vs 3m

Before and after August 2007

EUR 5y swaps: 3m vs 6m

Before and after August 2007What about arbitrage opportunities?

• Let us consider USD market quotes as of March 25 2009:• The 3x6 forward rate implied by the 3m and the 6m deposit

rates was FD = 2.31%.• The quoted FRA rate was FX = 1.26%.

• Again, an apparent inconsistency.• But ...• What happens if one tries to “arbitrage" the two markets?• In the good old times, we would exploit the misalignment

between deposit and FRA rates by implementing anarbitrage strategy, which has the following features:

• The initial value of the strategy is 0.• The strategy enables to lock in a positive gain in 6m.

• Let us set T1 = 3m and T2 = 6m.

Before and after August 2007What about arbitrage opportunities?

• Let us denote by D(t ,T ) the time-t deposit price for maturityT , and by L(T1,T2) the LIBOR rate

L(T1,T2) =1τ1,2

[1

D(T1,T2)− 1

]• The arbitrage strategy is as follows. At time t = 0:

a) Buy (1 + τ1,2FD) deposits with maturity T2, paying

(1 + τ1,2FD)D(0,T2) = D(0,T1) dollars

b) Sell 1 deposit with maturity T1, receiving D(0,T1) dollars;c) Enter a (payer) FRA, paying out at time T1

τ1,2(L(T1,T2)− FX )

1 + τ1,2L(T1,T2)

• The value of this strategy at time 0 is zero.

Before and after August 2007What about arbitrage opportunities?

• At time T1, b) plus c) yields

τ1,2(L(T1,T2)− FX )

1 + τ1,2L(T1,T2)− 1 = −

1 + τ1,2FX

1 + τ1,2L(T1,T2)< 0

• To pay this, we then sell the T2-deposits, remaining with

1 + τ1,2FD

1 + τ1,2L(T1,T2)−

1 + τ1,2FX

1 + τ1,2L(T1,T2)=

τ1,2(FD − FX )

1 + τ1,2L(T1,T2)> 0

at T1, which is equivalent to τ1,2(FD − FX ) received at T2.• This is clearly an arbitrage. (Too good to be true!)• In fact, there are issues we can no longer neglect:

i) Possibility of default before T2 of the counterparty we lentmoney to;

ii) Possibility of liquidity crunch at times 0 or T1;iii) Regulatory requirements, etc.

Explaining the divergence of “equivalent” ratesIntroducing a simple credit model

• Let us denote by τt the default time of the generic interbankcounterparty at time t .

• We assume independence between default and rates andzero recovery.

• The value at time t of a deposit starting at t and with maturityT is

D(t ,T ) = E[e−

∫ Tt r(u) du1{τt>T}|Ft

]= P(t ,T )E

[1{τt>T}|Ft

]where

• r is the default-free instantaneous rate• P(t ,T ) is the price of a default-free zero-coupon bond

Explaining the divergence of “equivalent” rates

• SettingQ(t ,T ) := E

[1{τt>T}|Ft

],

the LIBOR rate L(T1,T2) is given by

L(T1,T2) =1τ1,2

[1

D(T1,T2)− 1

]=

1τ1,2

[1

P(T1,T2)

1Q(T1,T2)

− 1].

• Consider a T1xT2 FRA and denote by FX its FRA rate.• Assuming no counterparty risk, its time-0 value can be

written as

0 = E[e−

∫ T10 r(u) du τ1,2(L(T1,T2)− FX )

1 + τ1,2L(T1,T2)

]

Explaining the divergence of “equivalent” rates

• Carrying out the calculations, we have:

0 = E[e−

∫ T10 r(u) du τ1,2(L(T1,T2)− FX )

1 + τ1,2L(T1,T2)

]= E

[e−

∫ T10 r(u) du

(1−

1 + τ1,2FX

1 + τ1,2L(T1,T2)

)]= E

[e−

∫ T10 r(u) du

(1− (1 + τ1,2FX )P(T1,T2)Q(T1,T2)

)]= P(0,T1)− (1 + τ1,2FX )P(0,T2)E

[Q(T1,T2)

]• Therefore, the value of the FRA rate FX is:

FX = FFRA(0; T1,T2) =1τ1,2

[P(0,T1)

P(0,T2)

1E

[Q(T1,T2)

] − 1]

Explaining the divergence of “equivalent” rates

• Let us now assume that the OIS swap curve is a good proxyfor the risk-free curve.

• The OIS forward rates are then given by

FOIS(0; T1,T2) =1τ1,2

[P(0,T1)

P(0,T2)− 1

]• We can easily see that the FRA rate can be (arbitrarily)

higher than the corresponding forward OIS rate.• In fact,

1τ1,2

[P(0,T1)

P(0,T2)

1E

[Q(T1,T2)

] − 1]

︸ ︷︷ ︸FRA rate

>1τ1,2

[P(0,T1)

P(0,T2)− 1

]︸ ︷︷ ︸

OIS fwd

since 0 < Q(T1,T2) < 1.

Explaining the divergence of “equivalent” rates

• Similarly, the forward rate implied by the two depositsD(0,T1) and D(0,T2), i.e.

FDepo(0; T1,T2) =1τ1,2

[D(0,T1)

D(0,T2)− 1

]=

1τ1,2

[P(0,T1)

P(0,T2)

Q(0,T1)

Q(0,T2)− 1

]will be larger than the FRA rate FX if

Q(0,T1)

Q(0,T2)>

1E

[Q(T1,T2)

]• This happens when the market expectation for the credit

premium in L(T1,T2), inversely ∝ Q(T1,T2), is low comparedto the value implied by Q(0,T1) and Q(0,T2).

A recap of formulas under our simple credit model

Rate Classic formulas Model formulas

FFRA(0; T1,T2)1

τ1,2

[P(0,T1)P(0,T2)

− 1]

1τ1,2

[P(0,T1)P(0,T2)

1E [Q(T1,T2)]

− 1]

FOIS(0; T1,T2)1

τ1,2

[P(0,T1)P(0,T2)

− 1]

1τ1,2

[P(0,T1)P(0,T2)

− 1]

FDepo(0; T1,T2)1

τ1,2

[P(0,T1)P(0,T2)

− 1]

1τ1,2

[P(0,T1)P(0,T2)

Q(0,T1)Q(0,T2)

− 1]

General considerationsA practitioner’s approach

• The previous analysis provides a simple theoreticaljustification for the divergence of rates that used to beequivalent.

• Such rates, in fact, become “compatible" as soon as credit(and/or liquidity and/or regulatory) risks are taken intoaccount.

• However, instead of resorting to hybrid modeling,practitioners deal with the discrepancies above bysegmenting market rates.

• This results in the construction of different zero-couponcurves for different market rate tenors:1m, 3m, 6m, 1y, ...

• Besides these forward (or projection or growth) LIBORcurves, one also constructs one, or more, discount curves.

General considerationsA global look at market rates

• We no longer have a single zero-coupon curve for eachgiven currency.

EUR market rates (as of 17 October 2011)

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

Eonia

Deposits

FRAs (3m)

FRAs (6m)

Swaps (1m)

Swaps (3m)

Swaps (6m)

General considerationsConstruction of forward curves

• Banks construct different zero-coupon curves for eachmarket rate tenor: 1m, 3m, 6m, 1y, ...

Example: The EUR 6m curve

General considerationsThe practice of OIS discounting

• Under CSA, it is market practice to use OIS discounting:• Since June 2010, USD, Euro and GBP trades in SwapClear

(LCH.Clearnet) have been revalued using OIS discounting.• Since September 2010, swaption prices in the London

inter-dealer option market have been quoted on a forwardbasis for a number of European currencies.

Definitions in the multi-curve worldDiscount curve

• We assume OIS discounting.• Given a tenor x and an associated time structureT x = {0 < T x

0 , . . . ,TxMx}, with T x

i − T xi−1 = x , i = 1, . . . ,Mx ,

OIS forward rates are defined as in the classic single-curveparadigm:

F xi (t) := FD(t ; T x

i−1,Txi ) =

1τ x

i

[PD(t ,T x

i−1)

PD(t ,T xi )

− 1],

for i = 1, . . . ,Mx , where• τ x

i is the year fraction for the interval (T xi−1,T

xi ];

• PD(t ,T ) denotes the discount factor at time t for maturity Tfor the discount (OIS) curve.

• Consistently with OIS discounting, we assume that ourpricing measures are defined by the OIS discount curve.

Definitions in the multi-curve worldForward LIBOR rate

• The forward LIBOR rate at time t for the period [T xi−1,T

xi ] is

denoted by Lxi (t) and defined by

Lxi (t) = ET x

iD

[Lx(T x

i−1,Txi )|Ft

],

where• Lx(T x

i−1,Txi ) denotes the LIBOR set at T x

i−1 with maturity T xi ;

• ETD denotes expectation under the T -forward measure

associated with the discount (OIS) curve;• Ft denotes the “information” available at time t .

• As in single-curve modeling, Lxi (t) is the fixed rate to be

exchanged at time T xi for Lx(T x

i−1,Txi ) so that the swaplet

has zero value at time t :

-

t

0Value swaplet:

Time: T xi−1 T x

i

Lx(T xi−1,T

xi )− Lx

i (t)

Definitions in the multi-curve worldForward LIBOR rate

The previous definition of forward rate in a multi-curve setup is natural for the following reasons:

1. Lxi (t)=ET x

iD

[Lx(T x

i−1,Txi )|Ft

]coincides with the classically

defined forward rate in the limit case of a single curve:

ET xi

D

[Lx(T x

i−1,Txi )|Ft

]=ET x

iD

[FD(T x

i−1;Txi−1,T

xi )|Ft

]=FD(t ;T x

i−1,Txi )

2. The rate Lxi (T x

i−1) coincides with the LIBOR rate Lx(T xi−1,T

xi ):

Lxi (T x

i−1) = ET xi

D

[Lx(T x

i−1,Txi )|FT x

i−1

]= Lx(T x

i−1,Txi )

3. The time-0 value Lxi (0) can be stripped from market data.

4. The rate Lxi (t) is a martingale under the corresponding OIS

forward measure QT xi .

5. This definition allows for a natural extension of the marketformulas for swaps, caps and swaptions.

The valuation of interest rate swaps (IRSs)(under the assumption of distinct forward and discount curves)

• Given times T xa , . . . ,T x

b , consider an IRS whose floating legpays at each T x

k the LIBOR rate with tenor T xk − T x

k−1 = x ,which is set (in advance) at T x

k−1, i.e.

τ xk Lx(T x

k−1,Txk )

where τ xk denotes the year fraction.

• The time-t value of this payoff is:

FL(t ; T xk−1,T

xk ) = τ x

k PD(t ,T xk )ET x

kD

[Lx(T x

k−1,Txk )|Ft

]=: τ x

k PD(t ,T xk )Lx

k (t)

• The swap’s fixed leg is assumed to pay the fixed rate K ondates T S

c , . . . ,T Sd , with year fractions τS

j .

The valuation of interest rate swaps• The IRS value to the fixed-rate payer is given by

IRS(t ,K ) =b∑

k=a+1

τ xk PD(t ,T x

k )Lxk (t)− K

d∑j=c+1

τSj PD(t ,T S

j )

• We can then calculate the corresponding forward swap rateas the fixed rate K that makes the IRS value equal to zero attime t :

Sxa,b,c,d(t) =

∑bk=a+1 τ

xk PD(t ,T x

k )Lxk (t)∑d

j=c+1 τSj PD(t ,T S

j )

• In particular, at t = 0 we get:

Sx0,b,0,d(0) =

∑bk=1 τ

xk PD(0,T x

k )Lxk (0)∑d

j=1 τSj PD(0,T S

j )

where Lx1(0) is the first floating payment (known at time 0).

The valuation of interest rate swaps

• In practice, this swap rate formula can be used to bootstrapthe rates Lx

k (0).• The bootstrapped Lx

k (0) can then be used to price otherswaps based on the given tenor.

Swap rate Formulas

OLD∑b

k=1 τ xk P(0,T x

k )F xk (0)∑d

j=1 τSj P(0,T S

j )=

1−P(0,T Sd )∑d

j=1 τSj P(0,T S

j )

NEW∑b

k=1 τ xk PD(0,T x

k )Lxk (0)∑d

j=1 τSj PD(0,T S

j )

The valuation of caplets• Let us consider a caplet paying out at time T x

k

τ xk [Lx(T x

k−1,Txk )− K ]+.

• The caplet price at time t is given by:

Cplt(t ,K ; T xk−1,T

xk ) = τ x

k PD(t ,T xk )ET x

kD

{[Lx(T x

k−1,Txk )− K ]+|Ft

}= τ x

k PD(t ,T xk )ET x

kD

{[Lx

k (T xk−1)− K ]+|Ft

}• The rate Lx

k (t) = ETkD [Lx(T x

k−1,Txk )|Ft ] is, by definition, a

martingale under QT xk

D .• Assume that Lx

k follows a (driftless) geometric Brownian

motion under QT xk

D .• Straightforward calculations lead to a (modified) Black

formula for caplets.

The valuation of European swaptions• A payer swaption gives the right to enter at time T x

a = T Sc an

IRS with payment times for the floating and fixed legs givenby T x

a+1, . . . ,Txb and T S

c+1, . . . ,TSd , respectively.

• Therefore, the swaption payoff at time T xa = T S

c is

[Sx

a,b,c,d(T xa )− K

]+d∑

j=c+1

τSj PD(T S

c ,TSj ),

where K is the fixed rate and

Sxa,b,c,d(t) =

∑bk=a+1 τ

xk PD(t ,T x

k )Lxk (t)∑d

j=c+1 τSj PD(t ,T S

j )

• We setCc,d

D (t) =d∑

j=c+1

τSj PD(t ,T S

j )

The valuation of swaptions• The swaption payoff is conveniently priced under the swap

measure Qc,dD , whose associated numeraire is Cc,d

D (t):

PS(t ,K ; T xa , . . . ,T

xb ,T

Sc+1, . . . ,T

Sd ) =

d∑j=c+1

τSj PD(t ,T S

j )

· EQc,dD

{[Sx

a,b,c,d(T xa )− K

]+ ∑dj=c+1 τ

Sj PD(T S

c ,T Sj )

Cc,dD (T S

c )|Ft

}

=d∑

j=c+1

τSj PD(t ,T S

j ) EQc,dD

{[Sx

a,b,c,d(T xa )− K

]+|Ft}

• Hence, also in a multi-curve set up, pricing a swaption isequivalent to pricing an option on the underlying swap rate.

• Assuming that Sxa,b,c,d is a (lognormal) martingale under

Qc,dD , we obtain a (modified) Black formula for swaptions.

The new market formulas for caps and swaptions

Type Formulas

OLD Cplt τ xk P(t ,T x

k ) Bl(K ,F x

k (t), vk

√T x

k−1 − t)

NEW Cplt τ xk PD(t ,T x

k ) Bl(K ,Lx

k (t), vk

√T x

k−1 − t)

OLD PS∑d

j=c+1 τSj P(t ,T S

j ) Bl(K ,SOLD(t), ν

√T x

a − t)

NEW PS∑d

j=c+1 τSj PD(t ,T S

j ) Bl(K ,Sx

a,b,c,d(t), ν√

T xa − t

)

Pricing general interest rate derivatives

• We just showed how to value swaps, caps, and swaptionsunder the assumption of distinct discount (OIS) and forwardcurves.

• What about exotics?• The pricing of general interest rate derivatives should be

consistent with the practice of using OIS discounting. In fact:• A Bermudan swaption should be more expensive than the

underlying European swaptions. In addition, on the lastexercise date, a Bermudan swaption becomes a Europeanswaption.

• A one-period ratchet is equal to a caplet.• Etc ...

• We must forsake the traditional single-curve models andswitch to a multi-curve framework.

How do we build a multi-curve model?

• Interest-rate multi-curve modeling is based on modeling thejoint evolution of a discount (OIS) curve and multiple forward(LIBOR) curves.

• Most banks are currently using a deterministic basis set-up.They choose a model for the OIS curve (short-rate, HJM,LMM, ...), and then build the forward LIBOR curves at adeterministic spread over the OIS curve.

• In general, forward curves can be modeled either directly orindirectly by modeling (possibly stochastic) basis spreads.

• Modeling the OIS curve is necessary for two reasons:• Swap rates depend on OIS discount factors.• The pricing measures we consider are those defined by the

OIS curve.

• Calculating swaption prices in closed form may be hard ingeneral.

The multi-curve LIBOR Market Model (McLMM)

• In the classic (single-curve) LMM, one models the jointevolution of a set of consecutive forward LIBOR rates.

• What about the multi-curve case?• When pricing a payoff depending on same-tenor LIBOR

rates, it is convenient to model rates Lxk .

• This choice is also convenient in the case of a swap-ratedependent payoff. In fact, we can write:

Sxa,b,c,d(t) =

∑bk=a+1 τ

xk PD(t ,T x

k )Lxk (t)∑d

j=c+1 τSj PD(t ,T S

j )=

b∑k=a+1

ωxk (t)Lx

k (t)

ωxk (t) :=

τ xk PD(t ,T x

k )∑dj=c+1 τ

Sj PD(t ,T S

j )

• But, is the modeling of forward rates Lxk enough?

The McLMM: Alternative formulations• In fact, we also need to model the OIS forward rates F x

k ,k = 1, . . . ,Mx :

F xk (t) =

1τ x

k

[PD(t ,T x

k−1)

PD(t ,T xk )

− 1]

• Denote by Sxk (t) the additive basis spread

Sxk (t) := Lx

k (t)− F xk (t)

• By definition, both Lxk and F x

k are martingales under the

forward measure QT xk

D , and thus their difference Sxk is as well.

• The LMM can be extended to the multi-curve case in threedifferent ways by:

1. Modeling the joint evolution of rates Lxk and F x

k .2. Modeling the joint evolution of rates Lx

k and spreads Sxk .

3. Modeling the joint evolution of rates F xk and spreads Sx

k .

An example of McLMMModeling rates F x

k and Lxk

• Given the set of times T x = {0 < T x0 , . . . ,T

xMx}, which are

compatible with the given tenor x , we assume that each rateLx

k (t) evolves under QT xk

D according to

dLxk (t) = σk (t)Lx

k (t) dZk (t), t ≤ T xk−1

• Likewise, we assume that

dF xk (t) = σD

k (t)F xk (t) dZ D

k (t), t ≤ T xk−1

• The drift of X ∈ {Lxk ,F

xk } under Q

T xj

D is equal to

Drift(X ; QT x

jD ) = −

d〈X , ln(PD(·,T xk )/PD(·,T x

j ))〉tdt

where 〈·, ·〉t denotes instantaneous covariation at time t .

An example of McLMM• Let us assume j < k . The case j > k is analogous.• The log of the ratio of the two numeraires can be written as

lnPD(t ,T x

k )

PD(t ,T xj )

= ln1∏k

h=j+1(1 + τ xh F x

h (t))= −

k∑h=j+1

ln(1 + τ xh F x

h (t))

• We thus get

Drift(Lxk ; Q

T xj

D ) = −d〈Lx

k , ln(PD(·,T xk )/PD(·,T x

j ))〉tdt

=k∑

h=j+1

d〈Lxk , ln

(1 + τ x

h F xh)〉t

dt

=k∑

h=j+1

τ xh

1 + τ xh F x

h (t)d〈Lx

k ,Fxh 〉t

dt

An example of McLMMDynamics under a general forward measure

Proposition. The dynamics of Lxk and F x

k under QT x

jD are:

j < k :

dLx

k (t) = σk (t)Lxk (t)

[k∑

h=j+1

ρL,Fk,h τ

xh σ

Dh (t)F x

h (t)1 + τ x

h F xh (t)

dt + dZ jk (t)

]

dF xk (t) = σD

k (t)F xk (t)

[k∑

h=j+1

ρD,Dk,h τ

xh σ

Dh (t)F x

h (t)1 + τ x

h F xh (t)

dt + dZ j,Dk (t)

]

j = k :

{dLx

k (t) = σk (t)Lxk (t) dZ j

k (t)dF x

k (t) = σDk (t)F x

k (t) dZ j,Dk (t)

j > k :

dLx

k (t) = σk (t)Lxk (t)

[−

j∑h=k+1

ρL,Fk,h τ

xh σ

Dh (t)F x

h (t)1 + τ x

h F xh (t)

dt + dZ jk (t)

]

dF xk (t) = σD

k (t)F xk (t)

[−

j∑h=k+1

ρD,Dk,h τ

xh σ

Dh (t)F x

h (t)1 + τ x

h F xh (t)

dt + dZ j,Dk (t)

]

An example of McLMMDynamics under the spot LIBOR measure

• The spot LIBOR measure QT x

D associated with timesT x = {T x

0 , . . . ,TxMx} is the measure whose numeraire is

BT x

D (t) =PD(t ,T x

β(t)−1)∏β(t)−1j=0 PD(T x

j−1,Txj ),

where β(t) = m if T xm−2 < t ≤ T x

m−1, m ≥ 1, and T x−1 := 0.

• Application of the change-of numeraire technique leads to:

dLxk (t) = σk (t)Lx

k (t)[ k∑

h=β(t)

ρL,Fk ,h τ

xh σ

Dh (t)F x

h (t)

1 + τ xh F x

h (t)dt + dZ d

k (t)]

dF xk (t) = σD

k (t)F xk (t)

[ k∑h=β(t)

ρD,Dk ,h τ

xh σ

Dh (t)F x

h (t)

1 + τ xh F x

h (t)dt + dZ d ,D

k (t)]

An example of McLMMThe pricing of caplets

• The pricing of caplets in this multi-curve lognormal LMM isstraightforward. We get:

Cplt(t ,K ; T xk−1,T

xk ) = τ x

k PD(t ,T xk ) Bl(K ,Lx

k (t), vk (t))

where

vk (t) :=

√∫ T xk−1

tσk (u)2 du

• As expected, this formula is analogous to that obtained in thesingle-curve lognormal LMM.

• Here, we just have to replace the “old” forward rates with thecorresponding “new” ones, and use the discount factors ofthe OIS curve.

An example of McLMMThe pricing of swaptions

• Our objective is to derive an analytical approximation for theimplied volatility of swaptions.

• To this end, we recall that

Sxa,b,c,d(t) =

b∑k=a+1

ωxk (t)Lx

k (t), ωxk (t) =

τ xk PD(t ,T x

k )∑dj=c+1 τ

Sj PD(t ,T S

j )

• Contrary to the single-curve case, the weights ωxk (t) are not

functions of forward LIBOR rates only, since they alsodepend on discount factors calculated on the OIS curve.

• Therefore we can not write, under Qc,dD ,

dSxa,b,c,d(t) =

b∑k=a+1

∂Sxa,b,c,d(t)∂Lx

k (t)σk (t)Lx

k (t) dZ c,dk (t)

An example of McLMMThe pricing of swaptions

• However, we can resort to a standard approximationtechnique and freeze the weights ωx

k at their time-0 value:

Sxa,b,c,d(t) ≈

b∑k=a+1

ωxk (0)Lx

k (t),

thus also freezing the dependence of Sxa,b,c,d on rates F x

h .• Hence, we can write:

dSxa,b,c,d(t) ≈

b∑k=a+1

ωxk (0)σk (t)Lx

k (t) dZ c,dk (t).

• Like in the classic single-curve LMM, we then:• Match instantaneous quadratic variations• Freeze forward and swap rates at their time-0 value

An example of McLMMThe pricing of swaptions

• This immediately leads to the following (payer) swaptionprice at time 0:

PS(0,K ; T xa , . . . ,T

xb ,T

Sc+1, . . . ,T

Sd )

=d∑

j=c+1

τSj PD(0,T S

j ) Bl(K ,Sx

a,b,c,d(0),Va,b,c,d),

where the swaption volatility (multiplied by√

T xa ) is given by

Va,b,c,d =

√√√√ b∑h,k=a+1

ωxh(0)ωx

k (0)Lxh(0)Lx

k (0)ρh,k

(Sxa,b,c,d(0))2

∫ T xa

0σh(t)σk (t) dt

• Again, this formula is analogous in structure to that obtainedin the single-curve lognormal LMM.

A different class of McLMMsA framework for the single-tenor case

• In the previous example, we modeled the joint evolution ofrates F x

k and Lxk .

• A different class of McLMMs can be constructed by definingthe joint evolution of rates F x

k and spreads Sxk under the spot

LIBOR measure QT x

D .• A single-tenor framework is based on assuming that, under

QT x

D , OIS rates follow general SLV processes:

dF xk (t) = φF

k (t ,F xk (t))ψF

k (V F (t))

·

[k∑

h=β(t)

τ xh ρh,kφ

Fh (t ,F x

h (t))ψFh (V F (t))

1 + τ xh F x

h (t)dt + dZ T

k (t)

]dV F (t) = aF (t ,V F (t)) dt + bF (t ,V F (t)) dW T (t)

A different class of McLMMsA general framework for the single-tenor case

where

• φFk , ψF

k , aF and bF are deterministic functions of theirrespective arguments

• Z T = {Z T1 , . . . ,Z

TMx} is an Mx -dimensional QT

D -Brownianmotion with instantaneous correlation matrix (ρk ,j)k ,j=1,...,Mx

• W T is a QTD -Brownian motion whose instantaneous

correlation with Z Tk is denoted by ρx

k for each k .• The stochastic volatility V F is assumed to be a process

common to all OIS forward rates.• We assume that V F (0) = 1.

Generalizations can be considered where each rate F xk has a

different volatility process.

A different class of McLMMsA general framework for the single-tenor case

• We then assume that also the spreads Sxk follow SLV

processes.• For computational convenience, we assume that spreads

and their volatilities are independent of OIS rates.• This implies that each Sx

k is a QTD -martingale as well.

• Finally, the global correlation matrix that includes all crosscorrelations is assumed to be positive semidefinite.

Remark. Several are the examples of dynamics that can beconsidered. Obvious choices include combinations (andpermutations) of geometric Brownian motions and of thestochastic-volatility models of Hagan et al. (2002) and Heston(1993). However, the discussion that follows is rather general andrequires no dynamics specification.

A different class of McLMMsCaplet pricing

• Let us consider the x-tenor caplet paying out at time T xk

τ xk [Lx

k (T xk−1)− K ]+

• Our assumptions on the discount curve imply that the capletprice at time t is given by

Cplt(t ,K ; T xk−1,T

xk )

= τ xk PD(t ,T x

k )ET xk

D

{[Lx

k (T xk−1)− K ]+|Ft

}= τ x

k PD(t ,T xk )ET x

kD

{[F x

k (T xk−1) + Sx

k (T xk−1)− K ]+|Ft

}• Assume we explicitly know the QT x

kD -densities fSx

k (T xk−1)

andfF x

k (T xk−1)

(conditional on Ft ) of Sxk (T x

k−1) and F xk (T x

k−1),respectively, and/or the associated caplet prices.

A different class of McLMMsCaplet pricing

• Thanks to the independence of the random variablesF x

k (T xk−1) and Sx

k (T xk−1) we equivalently have:

Cplt(t ,K ; T xk−1,T

xk )

τ xk PD(t ,T x

k )

=

∫ +∞

−∞ET x

kD

{[F x

k (T xk−1)− (K − z)]+|Ft

}fSx

k (T xk−1)

(z) dz

=

∫ +∞

−∞ET x

kD

{[Sx

k (T xk−1)− (K − z)]+|Ft

}fF x

k (T xk−1)

(z) dz

• One may use the first or the second formula depending onthe chosen dynamics for F x

k and Sxk :

• Lognormal• Heston• SABR• ...

A different class of McLMMsCaplet pricing

• To calculate the caplet price one needs to derive thedynamics of F x

k and V F under the forward measure QT xk

D .

• Notice that the QT xk

D -dynamics of Sxk and its volatility are the

same as those under QTD thanks to our independence

assumption.• To this end, we apply the standard change-of-numeraire

result, which in this case reads as:

Drift(X ; QT xk

D ) = Drift(X ; QTD ) +

d〈X , ln PD(·,T xk )

PD(·,T xβ(t)−1)

〉t

dt

where X is a continuous process, and 〈·, ·〉t denotes againinstantaneous covariation at time t .

A different class of McLMMsCaplet pricing

• The dynamics of F xk and V F under QT x

kD are thus given by:

dF xk (t) = φF

k (t ,F xk (t))ψF

k (V F (t)) dZ kk (t)

dV F (t) = aF (t ,V F (t)) dt + bF (t ,V F (t))

·

[−

k∑h=β(t)

τ xh φ

Fh (t ,F x

h (t))ψFh (V F (t))ρx

h1 + τ x

h F xh (t)

dt + dW k (t)

]

where Z kk and W k are QT x

kD -Brownian motions.

• By resorting to standard drift-freezing techniques, one canfind tractable approximations of V F for typical choices of aF

and bF , which will lead either to an explicit density fF xk (T x

k−1)or

to an explicit option pricing formula (on F xk ).

• This, along with the assumed tractability of Sxk , will finally

allow the calculation of the caplet price.

A different class of McLMMsSwaption pricing

• Let us consider a (payer) swaption, which gives the right toenter at time T x

a = T Sc an interest-rate swap with payment

times for the floating and fixed legs given by T xa+1, . . . ,T

xb

and T Sc+1, . . . ,T

Sd , respectively, with T x

b = T Sd and where the

fixed rate is K .• The swaption payoff at time T x

a = T Sc is given by

[Sx

a,b,c,d(T xa )− K

]+d∑

j=c+1

τSj PD(T S

c ,TSj ),

where the forward swap rate Sxa,b,c,d(t) is given by

Sxa,b,c,d(t) =

∑bk=a+1 τ

xk PD(t ,T x

k )Lxk (t)∑d

j=c+1 τSj PD(t ,T S

j ).

A different class of McLMMsSwaption pricing

• The swaption payoff is conveniently priced under Qc,dD :

PS(t ,K ; T xa , . . . ,T

xb ,T

Sc+1, . . . ,T

Sd )

=d∑

j=c+1

τSj PD(t ,T S

j ) Ec,dD

{ [Sx

a,b,c,d(T xa )− K

]+ |Ft}

• To calculate the last expectation, we set

ωxk (t) :=

τ xk PD(t ,T x

k )∑dj=c+1 τ

Sj PD(t ,T S

j )and write:

Sxa,b,c,d(t) =

b∑k=a+1

ωxk (t)Lx

k (t)

=b∑

k=a+1

ωxk (t)F x

k (t) +b∑

k=a+1

ωxk (t)Sx

k (t) =: F (t) + S(t)

A different class of McLMMsSwaption pricing

• The processes Sxa,b,c,d , F and S are all Qc,d

D -martingales.

• F is equal to the classic single-curve forward swap rate thatis defined by OIS discount factors, and whose reset andpayment times are given by T S

c , . . . ,T Sd .

• If the dynamics of rates F xk are sufficiently tractable, we can

approximate F (t) by a driftless stochastic-volatility process,F (t), of the same type as that of F x

k .• The process S is more complex, since it explicitly depends

both on OIS discount factors and on basis spreads.• However, we can resort to a standard approximation and

freeze the weights ωxk at their time-0 value, thus removing the

dependence of S on OIS discount factors.

A different class of McLMMsSwaption pricing

• We then assume we can further approximate S with adynamics S similar to that of Sx

k , for instance by matchinginstantaneous variations.

• After the approximations just described, the swaption pricebecomes

PS(t ,K ; T xa , . . . ,T

xb ,T

Sc+1, . . . ,T

Sd )

=d∑

j=c+1

τSj PD(t ,T S

j ) Ec,dD

{[F (T x

a ) + S(T xa )− K

]+|Ft}

which can be calculated exactly in the same fashion as theprevious caplet price.

A general recipe for multi-curve modelingA simple stochastic-basis framework

• In general, is there any easy-to-use recipe for multi-curvemodeling, i.e. for jointly modeling discount and forwardcurves at a stochastic spread over one another?

• The answer is yes. We can introduce a generalstochastic-basis framework that can be plugged in on top ofany term structure model for the OIS curve.

• Let us assume that the OIS curve evolution is described bysome classic single-curve model.

• We model additive LIBOR-OIS basis spreads

Sxi (t) = Lx

i (t)− F xi (t)

• Lxi and F x

i are both martingales under the OIS T xi -forward

measure QT xi

D . Hence Sxi is a QT x

iD -martingale as well.

A simple stochastic basis framework

• Our stochastic-basis framework is based on assuming that

Sxi (t) = ρx

i F xi (t) + αx

i[(1− λx

i )X xi (t) + λx

i]

where• αx

i := Sxi (0)− ρx

i F xi (0);

• ρxi ’s and λx

i ’s are constant parameters;• The basis factors X x

i are independent of OIS rates;• X x

i (0) = 1.

• LIBORs are then explicitly given by

Lxi (t) = F x

i (t) + ρxi F x

i (t) + αxi[(1− λx

i )X xi (t) + λx

i]

= (1 + ρxi )F x

i (t) + αxi[(1− λx

i )X xi (t) + λx

i]

• Swap rates can be calculated by using the new marketformula.

A simple stochastic basis frameworkTerminal correlations

• Using the independence of X xi (t) and F x

i (t), it is trivial tocalculate the following (terminal) T x

i -forward correlations:

Corr(F x

i (t),Sxi (t)

)= 1{ρx

i 6=0}sign(ρx

i )√1 +

(αxi )2(1−λx

i )2

(ρxi )2

Var[X xi (t)]

Var[F xi (t)]

Corr(F x

i (t),Lxi (t)

)= 1{ρx

i 6=−1}sign(1 + ρx

i )√1 +

(αxi )2(1−λx

i )2

(1+ρxi )2

Var[X xi (t)]

Var[F xi (t)]

• The terminal correlations between OIS forward rates,LIBORs and spreads corresponding to different time intervalsand/or tenors may also be calculated explicitly, depending onthe chosen OIS curve model.

Example I: a multi-curve LIBOR Market Model• Let us consider tenors x1 < x2 < · · · < xn with associated

time structures T xi = {0 < T xi0 , . . . ,T

xiMxi}, where

T xn ⊂ T xn−1 ⊂ · · · ⊂ T x1 =: T .• We assume that, under the OIS spot LIBOR measure QT

D ,the OIS forward rates F x1

k , k = 1, . . . ,M1, follow:

dF x1k (t) = σx1

k V F (t)[ 1τ x1

k+ F x1

k (t)][

V F (t)k∑

h=β(t)

ρh,kσx1h dt + dZ T

k (t)

]dV F (t) = εV F (t) dW T (t)

where:• σx1

k is a positive constant, for each k ;• dZTk (t)dZTj (t) = ρk,j dt , k , j = 1, . . . ,M1;• dW T (t)dZTk (t) = ψx

k dt ;• V F (0) = 1;• β(t) = m if T x

m−2 < t ≤ T xm−1, m ≥ 1, and T x

−1 := 0.

A multi-curve LIBOR Market Model

• The dynamics of forward rates F xk , for tenors x ∈ {x2, . . . , xn},

can be obtained by Ito’s lemma, noting that F xk can be written

in terms of shorter tenor rates F x1k as follows:

ik∏h=ik−1+1

[1 + τ x1h F x1

h (t)] = 1 + τ xk F x

k (t),

for some indices ik−1 and ik .• These dynamics of F x

k are consistent across different tenorsx . For instance, both 3m-rates and 6m-rates followshifted-lognormal processes with common SABR stochasticvolatility.

• This allows us to simultaneously price, with the same type offormula, caps and swaptions with different tenors x .

A multi-curve LIBOR Market Model• The dynamics for the x-tenor rate F x

k under QT xk

D are given by:

dF xk (t) = σx

k V F (t)[ 1τ x

k+ F x

k (t)]dZ k ,x

k (t)

dV F (t) = −ε[V F (t)]2ik∑

h=β(t)

σx1h ψ

x1h dt + εV F (t) dW k ,x(t)

• Regarding basis spreads dynamics in our framework, we setρx

k = λxk = 0, for all k and x , and assume that basis factors

X xk follow a one-factor geometric Brownian motion:

Sxk (t) = Sx

k (0)M(t), k = 1, . . . ,Mx ,

dM(t) = σM(t) dZ (t)

where Z is a Brownian motion independent of Z k ,xk and W k ,x

and σ is a positive constant. Clearly, Mx(0) = 1.

A multi-curve LIBOR Market ModelA numerical example

• By the independence of spreads and rates, a caplet pricecan be calculated as an integral of SABR prices times anormal density (we approximate linearly the drift term of V F ).

• The caplet pricing formula we obtain can be used to pricecaps on any tenor x .

• We consider a numerical example based on EUR data as ofSeptember 15th, 2010.

• We calibrate the market prices of standard x = 6m-tenorcaps using the model formula.

• We then price caps on a non-standard tenor x = 3m byusing the same formula, and assuming a specific correlationstructure ρi,j .

A multi-curve LIBOR Market ModelA numerical example

Figure: Absolute differences (in%) between market and model capvolatilities.

A multi-curve LIBOR Market ModelA numerical example

Figure: Absolute differences (in bp) between model-implied 3m-LIBORcap volatilities and model 6m-LIBOR ones.

Example II: a two-factor multi-curve model• Let us now assume that the OIS curve follows the one-factor

Hull-White (1990) model:

dr(t) = [ϑ(t)− ar(t)] dt + σ(t) dW (t)

where a is a positive constant parameter, and ϑ and σ aredeterministic functions of time.

• Hence, forward OIS rates are explicitly given by:

F xi (t) =

[A(t ,T x

i−1,Txi ) e−B(t ,T x

i−1,T xi )r(t) − 1

]/τ x

i

where A and B are deterministic functions of time.• Then we assume that basis factors X x

i follow a commongeometric Brownian motion X x :

X xi (t) = X x(t), for each i

dX x(t) = ηx(t)X x(t) dZ x(t), X x(0) = 1,

where ηx is a deterministic function of time, and theBrownian motion Z x is independent of OIS rates.

A two-factor multi-curve modelCaplet and swaption pricing

• Caplet and swaption prices can be expressed in terms of atwo-dimensional integral of the respective payoff functionstimes a bivariate normal density.

• Since forward LIBOR and swap rates are affine functions ofX x conditional on OIS rates, one integration can be carriedout explicitly.

• Therefore, caplets and swaptions can be pricedsemi-analytically by a one-dimensional integration of aclosed-form function of Black-Scholes type.

• Let us consider a (payer) swaption paying out at time T xa :[

Sxa,b,c,d(T x

a )− K]+

d∑j=c+1

τSj PD(T x

a ,TSj )

• The swaption payoff can be expressed as follows:[C(y(T x

a ))X x(T xa )− D(y(T x

a ))]+

A two-factor multi-curve modelCaplet and swaption pricing

• The payer swaption price at time 0 can be calculated bytaking expectation under the T x

a -forward measure:

PS(0,K ) = PD(0,T xa )ET x

a

{[C(y(T x

a ))X x(T xa )− D(y(T x

a ))]+

}= PD(0,T x

a )

∫ ∞

−∞h(C(y),D(y),VX (T x

a ))·[

normal densityof y(T x

a )

](y) dy

h(A,B,V ) :=

Bl

(A,B,V ,1

)if A,B > 0

Bl(A,B,V ,−1

)if A,B < 0

A− B if A ≥ 0,B ≤ 00 if A < 0,B ≥ 0

Bl(F ,K , v ,w) := wFΦ

(w

ln FK + v2

2v

)− wKΦ

(w

ln FK − v2

2v

),

A two-factor multi-curve modelExtension to a finer time structure

• For pricing purposes one may want to consider a finer timestructure than the initial T x .

• Let us then assume we are given a time structureT := {x < T1, . . . ,TM} that contains T x , with the possibleexception of T x

0 .• OIS forward rates at time t for the interval [Ti − x ,Ti ] can be

defined by the classic formula:

FD(t ; Ti − x ,Ti) =1x

[PD(t ,Ti − x)

PD(t ,Ti)− 1

], i = 1, . . . ,M

• The x-tenor forward LIBOR rate at time t for the interval[Ti − x ,Ti ] can then be defined as follows:

Lx(t ; Ti − x ,Ti)

= (1 + ρxi )FD(t ; Ti − x ,Ti) + αx

i[(1− λx

i )X x(t) + λxi]

A two-factor multi-curve modelBermudan swaptions pricing

K 2% 2.50% 3% 3.50% 4%Stochastic Basis 8.14% 6.28% 4.81% 3.75% 2.98%

Deterministic Basis 9.42% 7.47% 5.84% 4.50% 3.41%

Table: Prices of 10y-non-call-1y Bermudan (payer) swaptions.

A two-factor multi-curve modelSingle looks

K 25 30 35 40 45 50Stochastic Basis 35.93 31.75 27.67 23.74 20.00 16.50

Deterministic Basis 26.48 21.74 16.99 12.24 7.49 2.75

Table: Prices of single looks paying out max(CMS10y-CMS2y-K ,0) in5y. Strikes and prices are in bp.

A two-factor multi-curve modelRange accruals

Range 0%-1% 1%-2% 2%-3% 3%-4% 4%-5%Stochastic Basis 46.01% 65.64% 62.38% 42.39% 22.44%

Deterministic Basis 39.52% 51.52% 52.71% 43.31% 29.36%

Table: Prices of 6y-RACLs. Underlying index: 3m LIBOR. Window: 3y-6y.