INVESTMENT ACUMEN: AXA INVESTMENT MANAGERS’ RESEARCH REVIEW

Ethan Reiner, Head of Quantitative Research – Investment Solutions, AXA Investment Managers

AGE-OLD CUSTOMS IN THE NEW AGE

From Single to Multi-Curve Adapting to Change

A lively debate has been raging in the quantitative fi nance community since the burst of

the credit bubble in 2007. The burning issue involves what was, at least on the surface,

one of the simplest problems in the fi eld: how to price an interest rate swap. The question

is intimately linked to the construction of a risk-free yield curve, the cornerstone of

fi nance. Swaps had become standardised and liquid over the past 30 years and their

quotes are overwhelmingly used by banks and asset managers to track the interest

rate market and to build yield curves. Up until the crisis, simple no-arbitrage arguments

had allowed us to simply extract both discount rates and expected forward rates from

swap quotes. The hypotheses leading to the no-arbitrage derivation of swap prices have

proved faulty. As a result, academics and practitioners are faced with the challenge

of expanding interest rate theory to accommodate recent events. In a nutshell, the

working assumption in the past was that Libor (or Euribor), the underlying index used

for the fl oating payments of swaps, was a risk-free rate. Although this assumption was

known to be wrong, it was accepted as standard fare. The dramatic string of major bank

failures in 2007 culminating with the Lehman Brothers debacle in 2008, demonstrates

just how uncertain Libor really is. Some banks that were once Libor contributors are no

longer in the panel while others have disappeared. This article is a survey of the main

issues surrounding the construction of a theoretically consistent system of interest

rate curves in the post-crisis era. We will see that we can no longer model forward Libor

rates and risk-free discount rates with a single yield curve.

5

A new multiple curve theory has emerged

in the last four years that is meant to

provide separate estimates for risk-free

rates and forward rates. Pre-crisis academic

articles dealing with risky Libor (see Collin-

Dufresne and Solnik1) abound, however

their fi ndings were not embraced by most

practitioners as spreads were not signifi cant

enough to exploit. Although multiple-curve

solutions have been used for a number

of years in cross currency markets (see

Tuckman and Homé2), Bianchetti3 was

one of the fi rst public articles to discuss a

multi-curve solution in the single currency

context. Mercurio4 discusses the impact

of multi-curves on vanilla derivatives such

as swaptions and caps. In addition, he

considers a modifi cation of the Libor Market

Model that integrates a dynamic spread. An

original approach is proposed by Morini5,

where an attempt is made to explain recent

data by modeling the optionality underlying

the Libor index. The main objective of this

article is to highlight the reasons for the

emergence of a new theory and to present

the technical aspects of the multi-curve

algorithm. We will conclude with some

comments on dynamic extensions of the

approach as well as on the impacts on

trading and booking systems in fi nancial

institutions.

The End of the Libor Discounting

Paradigm

To get a clear understanding of the

profound changes that have affected

interest rate markets since August 2007,

it is worthwhile to step back and have a

look at pre-crisis working assumptions.

The risk-free curve was extracted from

deposits, futures and swap rates. A swap

is a bilateral agreement to exchange a

fl oating Libor rate against a known fi xed

rate over the life of the swap. Since the

perception of forward Libor rates is at the

heart of swap valuation, we take as our

point of departure the classic derivation of

the fair forward rate.

We review the textbook approach for

determining the forward rate and examine

the hypotheses which failed starting

August 2007. We assume a deposit

market in which banks may borrow and

lend without default risk or liquidity costs.

In our simple example we show how Bank

A can lock-in a borrowing (lending) rate

over a future period . Let be

the deposit rate for the period and the deposit rate for the period

.

Then we can show that the arbitrage free

forward rate for the future time interval

is

Indeed, if the bank wants to ensure it

can borrow at this rate for a notional of

Euros it can set up a simple strategy:

Borrow at the rate until

and immediately lend this amount at

until . Graph 1 illustrates the cash

fl ows of the strategy.

It easy to see that the net fl ows

correspond exactly to the fl ows of

borrowing over the period at

the rate : an infl ow of at and

an outfl ow of at . In fact, this replication strategy

shows that this is the unique fair rate as

seen at time 0: If Bank B were to offer

a lower lending rate then Bank A could

turn this into a profi t by borrowing at the

forward rate offered by Bank B and

selling the above replication strategy.

This ensures a profi t for Bank A of

at time with no initial cost at time 0.

If we relax default and liquidity

assumptions we immediately realise that

the above strategy is not risk-free. Default

of the deposit will cause Bank A to

suffer a loss; if there is a liquidity squeeze

then bid-ask spreads widen and the initial

zero cost assumption for setting up the

strategy breaks down. These potentially

catastrophic effects for the arbitrage

strategy are not new to market players;

however, they were all but ignored for the

construction of forward rates.

Strong evidence that something had

gone awry started to surface in August

2007, when practitioners remarked an

unusual divergence between short maturity

overnight index swap rates and same

Graph 1: The forward rate replication stategy

maturity deposit rates. Graph 2 displays

this phenomenon for three-month maturity

instruments.

To better understand what is at stake in

this graph we recall the defi nition of the

two rates under consideration:

■ The 3m Euribor index is a daily average

of bank offered rates for three-month

unsecured lending. The contributing panel

of banks is said to be a representative

rates, if we were to assume no default

risk, then we could prove that a large

divergence between the overnight index

swap and the Euribor rate should lead to an

arbitrage opportunity. This was probably the

assumption prior to the burst of the credit

bubble, on average the spread between

the two rates over the period 2000 to

2007 was 6 basis points with a standard

deviation of 2 basis points. Over the period

August 2007 to September 2011 the

average spread was 55 basis points with

a standard deviation of 35 basis points.

Similar results are found when we compare

these same instruments at different

maturities such as 1M, 6M or 12M.

Graph 3 shows the evolution of the spread

over the period 2000-2011 and highlights

the impressive increase over the period

starting 2007-2011. Graph 4 illustrates the

spread distribution and provides compelling

evidence that a true change of regime has

occurred.

This dramatic shift was not well understood

at fi rst. However, it is now largely admitted

that this spread represents a major risk

that is now being priced by markets. One

can no longer ignore the impact of credit

risk on interest rates... not even when we

consider the very fundamental object which

is the yield curve.

It is now clear that the upshot of recent

debates can be simply stated:

Libor no longer represents a risk-free rate.

Strictly speaking, Libor never was a risk-free

rate. However, prior to the credit crisis the

difference between Libor rates and risk-

free rates of comparable maturities was

negligible. This is no longer the case: we

now observe a persistent spread between

Overnight Index Swaps rates and Libor

rates of the same maturity. Markets have

sample of prime banks belonging to the

European Banking Federation.

■ The 3m Overnight Index Swap is the

fi xed rate that counterparties are willing

to pay in three months against the daily

compounded overnight rate over the

same period.

It is important to note that the overnight

index swap is a collateralised instrument

that bears essentially no credit risk. Going

back to our basic example of forward

Graph 2: Historic 3M Euribor and 3M overnight index swap quotes

for the period 2000-2011

Source: Bloomberg

Graph 3: Historic spread between 3M Euribor and 3M overnight index swap quotes

for the period 2000-2011

Source: Bloomberg

0%

1%

2%

3%

4%

5%

6%

08/05/00 08/05/01 08/05/02 08/05/03 08/05/04 08/05/05 08/05/06 08/05/07 08/05/08 08/05/09 08/05/10 08/05/11

3M Overnight Index Swap

3M Euribor

0,0%

0,5%

1,0%

1,5%

2,0%

08/05/00 08/05/01 08/05/02 08/05/03 08/05/04 08/05/05 08/05/06 08/05/07 08/05/08 08/05/09 08/05/10 08/05/11

Burst of the creditbubble

Lehman aftermath

Sovereign debt crisis

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7AGE-OLD CUSTOMS IN THE NEW AGE

From Single to Multi-Curve: Adapting to Change

had to rethink and adapt previous models

in order to account for this new reality.

From Libor to Overnight

Index-based Discounting

The important observations brought up

above have led practitioners to seek a

robust framework that can account for

the following elements when pricing a

collateralised interest rate swaps:

■ An interest rate swap involves payments

linked to Libor which is a risky unsecured

rate;

■ Quoted swap rates refer to collateralised

agreements that bear no counterparty risk.

Two possible approaches come to mind:

■ The credit approach: Use a risk-free

interest rate for discounting default free

cash fl ows and add an appropriate credit

spread to account for risky rates such as

future Libor rates;

■ The segmentation approach: Model the

interest rate market as a segmented

market in which each partition contains

only instruments which reference the

same underlying Libor tenor.

The credit approach is presented by

Mercurio4 and is extended by Morini5

to investigate the dynamics as well as

the embedded optionality present in the

Libor index. From a theoretical standpoint

the credit approach is quite satisfying

since it should facilitate the inclusion

of previously ignored concepts such as

funding and liquidity into rates modeling.

This approach is quite involved, however,

since Libor does not behave like a classical

defaultable counterparty: Libor contains

default risk but can never default. Currently,

practitioners seem to be leaning towards

the segmentation approach in implementing

pricing systems as evidenced in a large

number of recent articles (see Traven7 for

example). Since overnight index swaps

play a central role in this model, it is also

known as the OIS discounting model. The

rest of this article focuses on the technical

aspects of this new approach.

Bootstrapping before

and after the credit crunch

The popular term bootstrap calibration refers

to an iterative algorithm that extracts a yield

curve from a set of interest rate instruments

including swap quotes. It enables us

to start with an ordered set of quotes

for swaps maturing at times

and successively obtain

discount factors

which constitute the key points of the yield

curve. In this section we will describe what

has changed in the bootstrap procedure in

the context of the segmented or multi-curve

interest rate model.

Strictly speaking Libor

never was a risk-free rate.

However, prior to the credit

crisis the difference between

Libor rates and risk-free rates

of comparable maturities was

negligible. This is no longer

the case

We begin with a general description of the

approach and then give a detailed account

of how the algorithm works for most major

currencies. We will fi rst assume that we are

dealing with a market in which a full set of

Overnight Indexed Swaps are quoted. This

is the case for EUR, GBP, JPY and CHF. Next

we will look at some of the concrete details

that apply to the EUR market. Discussion

of tenor swaps - swaps in which fl oating

rates of different tenors are exchanged - is

deferred until the next section, where the

specifi cs of the USD market are considered.

To make things clear we require some notation:

0

100

200

300

400

500

600

700

800

900

-0,020,01

0,050,08

0,110,15

0,180,21

0,250,28

0,310,35

0,380,41

0,440,48

0,510,54

Freq

uenc

y

Graph 4: Euribor - OIS Distributions 2000-2010

Overview of the methodology

Our goal is to build an arbitrage-free

framework that is adapted to the

collateralised interest rate swap market. The

approach can be viewed as a straightforward

extension of techniques that have been

used for years in cross currency markets

(see Tuckman for a detailed presentation).

The main idea is to split the procedure into

two parts: in the fi rst we use a subset of

instruments to determine discount factors;

we then reuse these discount factors to

extract forward rates from other sets of

instruments. Graph 5 summarises the

multi-curve method. For a given currency we

consider a large universe of interest rate

instruments covering all the underlying Libor

tenors available.

The calibration procedure is separated into

three broad steps:

1. Partition the instruments according to

tenor.

2. Obtain a discount curve using

the overnight index segment of the

market .3. For each segment corresponding to a

Libor tenor we obtain projection curve

via a bootstrap procedure that

uses the previously obtained curve for

discounting fl ows.

can be expressed as

This conclusion is reached by showing that

the stream of Libor rates on the right hand

side of the fi rst equation can be replicated

by a rolling Libor deposit strategy whose

value at time 0 is given by .

This is known as the Libor discounting

paradigm.

When taking into account the possibility

of default, the replication strategy is not

risk-free.

The key to understanding the multiple

curve approach is the observation that the

replication argument fails and that the fair

swap rate formula is not valid.

The risk-neutral value of the Libor leg must

be computed as the present value of risky

future rates. The expression

can no longer be simplifi ed: represents risk-free discount factor to be

applied to certain payoffs; represent

the future risky rates for unsecured loans.

The new model must therefore be capable

of distinguishing between estimates of

risk-free rates used for discounting and

unsecured forward rates. Assuming that we

have already obtained the risk-free discount

factor , the N-year swap rate

with underlying Libor tenor K

satisfi es

The only unknowns in the above equations

are the . Equipped with any

The model makes the following

assumptions:

■ The collateral underlying all swaps

under consideration is cash in the same

currency as the swaps themselves;

■ Swap quotes used in the model refer to

bilateral agreements with zero threshold

and continuous posting of collateral.

Although common credit support annex

agreements (CSA) provide more fl exibility in

terms of eligible collateral, our assumption is

consistent with broker quotes. Our universe

thus consists of futures, forward rate

agreements, tenor basis swaps and swaps.

Strictly speaking, money market deposits

should not be included in the model since

they represents uncollateralised instruments.

As we explain below, with some care, a

subset of deposit rates may be used in the

multi-curve system.

Swap Pricing

In the traditional swap valuation approach

one shows that by no arbitrage the fair

swap rate defi ned by

Graph 5: Overview of the multi-curve method algorithm

Market Segments Sequential Bootstrap

ON Deposit 1D Deposit OIS Swaps OIS Discount Curve

3M Deposit 3M Futures 3M FRAS 3M Swaps 3M Libor Projection Curve

6M Deposit 3M FRAS 6M Swaps 6M Libor Projection Curve

Interest Rate Market

OIS Swaps3M Swaps6M Swaps3M FRASDepositsFutures

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9AGE-OLD CUSTOMS IN THE NEW AGE

From Single to Multi-Curve: Adapting to Change

convenient interpolation assumption

we can sequentially extract the

. from available swap quotes

. It should now be

clear that before we can extract Libor

rates estimates from swap rates,

we must have a risk-free discount curve

in hand. We look at this issue in the next

section.

The discount curve generated

by overnight index swaps

Although there may be different methods

for obtaining risk-free discount factors,

in our approach we will focus on

information contained in overnight index

(OI) swaps. An OI swap is an agreement

to exchange a fi xed rate against a fl oating

rate determined by compounding the

overnight rate over the reference period.

In the overnight case, the rolling strategy

still works as long as we make the

reasonable the default risk of an overnight

loan is negligible. Therefore,

if S is a quote for an OIS we can write

The conclusion is that in the presence

of an OIS market we can construct

a risk-free discounting curve which may

be used for discounting cash fl ows

in all cash-collateralised derivatives.

Futures, FRAS and deposits

Futures and FRAS are collateralised

instruments that are priced in the

new framework. Indeed, the quoting

mechanisms are such that quoted forward

rates correspond exactly

to the projected Libor rates

Similarly, for futures prices we ignore

convexity adjustments

The output of the multi-curve model

consists of a discount curve and a set

of forward curves. Each forward curve

corresponds to market projections

concerning a particular tenor: the

instruments that contribute to a given

forward curve all contain information about

the risk of Libor rates of one specifi c tenor

1M, 3M, 6M, or 12M.

Money market deposits seem more diffi cult

to included in our setting since

■ They are not collateralised;

■ There is a possible tenor mismatch

(each quoted deposit rate refers to a

different tenor).

We remark, however, that for each forward

rate curve corresponding to a Libor tenor K

we can view the fi rst deposit rate of

appropriate tenor K as directly contributing

the fi rst projection point .

The Euro Curve

To make our treatment as concrete

as possible we will now consider

the details of the multi-curve algorithm

for the Euro market. The formulas

provided here can be employed in all

markets that have a similar structure

(GBP, JPY and CHF). The Euro market

can be partitioned in the following

manner:

Overnight 3M 6M

ON DepositTN Deposit1D Deposit

3M Deposit 6M Deposit

3M FRAs3M Futures

6M FRAs

OIS Swaps 3M Swaps 6M Swaps

The bootstrap calibration proceeds as

follows:

■ Use OIS quotes to

bootstrap the discount curve according the standard equation

■ Bootstrap the 6M curve:

- Start the curve with the 6M deposits

and 6M FRAS

- For the swaps use the formula

■ Bootstrap the 3M curve:

- Start the curve with the 3M deposits,

3M FRAS, and 3M Futures

- For the swaps use the formula

Details of the USD market

The USD swap market is substantially

different in its quotation convention.

The theory presented in the previous

sections still applies but some extra

work must be done in setting up the

bootstrapping procedure. Indeed, the

US market has few OIS quotes and

beyond 10Y one must use Fed funds

swaps in order to extract long term

information about overnight rates. We

begin by describing the set of available

instruments.

OIS Swaps

The USD market quotes OIS swaps based

on Fed Funds up to 10Y.

Leg 1: Fixed Payments - Annual Act/360

Leg 2: Compounded overnight Fed Funds -

Rate Annual Act/360.

Fed Fund Swaps

The USD market does not quote OIS

swaps directly for a number of maturities.

The closest liquid proxies are Fed

Fund swaps. These have the following

description:

Variable Leg 1: USD Libor 3M, Act/360

plus spread - Paid quarterly

Variable Leg 2: Daily average of the

overnight Fed Fund Rate plus spread,

Act/360 - Paid quarterly

Standard USD Swaps

The standard swaps in the US market are

3M Libor versus fi xed rate

Variable Leg; USD Libor 3M, Act/360 - Pay

Quarterly

Fixed Leg: Fixed Rate, 30/360 - Pay Semi-

Annually

Tenor Swaps

Semi-Annual 6M Libor swaps are not

quoted directly. The quoted market

consists of tenor swaps.

Variable Leg 1: USD Libor 3M, Act/360 -

Reset Quarterly, Pay Quarterly.

Variable Leg 2: USD Libor 6M, Act/360 -

Reset Semi-Annually, Pay Semi-Annually.

Determining discount rates

from USD Fed Fund swaps

If we wish to determine OIS from Fed Fund

versus 3M swaps we need to make some

approximation concerning the average Fed

Fund (FF) rate. One possible approach is

to assume that the daily compounded rate

over the same period gives the same rate:

where is the number of business

day over the relevant period. Under this

assumption, we can replicate the fl oating

Fed Fund payments by daily rolling of a

eliminate all the fl oating rates and obtain

equations in which only discount factors

need to be determined.

We remark in passing that for actual

applications an exact modeling is required

in order to properly reconstruct all market

instruments. We must therefore be careful

in the above equation to distinguish the

quarterly year fractions which use

the ACT/360 convention and the Semi-

Annual year fractions which use the

30/360 conventions.

The birth of the multi-

curve system has been a

diffi cult one. It is not the

mere complexity of the

model that is troublesome

but rather the fact that the

yield curve is such a basic

object in fi nance that it

appears everywhere

Bootstrapping 6M USD swaps

The approach is similar to the above

except that we now assume that

discount factors are known and we seek

the expected rates. We use the

equilibrium equations:

For Standard 3M Swaps:

For 3M vs 6M tenor Swaps:

where represents semi-annual

ACT/360 year fractions.

We eliminate the 3M rates to get:

Fed Funds investment over each of the

reference fl oating rate periods. What

distinguishes the USD market from

what we have seen so far is the need to

combine swap quotes of different types

to build synthetic swaps that will enable

us to determine risk-free rate. Specifi cally,

we use equilibrium equations from the 3M

versus fi xed market and the FF versus 3M

market:

For standard semi-annual vs quarterly:

Quarterly vs quarterly tenor swap:

where is the realized average

Fed Fund rate over the period starting at

and represents the quoted spread

of the Fed Funds over 3M Libor.

By subtracting one equation from the

other we get

Thus,

In the last equation we have used the

assumption that the Fed Fund leg can

be replicated by rolling a Fed Fund

investment over each fl oating period. The

important conclusion here is that we can

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11AGE-OLD CUSTOMS IN THE NEW AGEFrom Single to Multi-Curve: Adapting to Change

which defi nes the swaps we can use

to solve for the 6M Libor projection curves.

Bootstrapping 3M USD Swaps

The 3M segment in the US market quotes

in a conventional way. Therefore,

once we have obtained the discount

factors from the fi rst step of the algorithm.

We can use swap quotes directly to strip

out the 3M projection rates. The full

procedure for USD is clearly more involved

than for other currencies. Graph 6 gives a

schematic overview of the approach.

Interpolation and Extrapolation

Interpolation and extrapolation of the

discount curve in the multi-curve context

is extremely important. In practice,

undesirable effects may occur if we naively

extrapolate the bootstrapped rates. A

concrete example of what can go wrong

can be seen on the USD market where Fed

Fund swaps are not liquid beyond 30Y and

where at the same time we wish to include

long dated 3M versus fi x swaps (quoting

up to 60Y). We are thus confronted

with the diffi culty of determining forward

Libor rates without any direct market

information on risk-free rates beyond

30Y. Extrapolating the risk-free rate in

a naive way leads to discounting with a

rate higher than the projected 3M Libor

rate, which contradicts the assumption

that Fed Funds rates are inherently less

risky. A fi rst simple solution to such

anomalies consists in fi rst extrapolating

Fed Fund spreads over the missing

maturities. In this way we maintain a safe

distance between 3M forwards and the

risk-free rate. This leads to the creation

of a set of virtual swaps consistent with

quoted spreads up to the last known

maturity. These swaps enter the bootstrap

mechanism so that the fi nal interpolated/

extrapolated curves are the result of

interpolated/extrapolated instruments.

The above discussion also pertains to the

3M/6M spread: we suggest interpolating/

extrapolating the spread and then creating

the appropriate swaps that are fed into

the bootstrap procedure. In Graph 7, we

exhibit the interpolation method for the

USD case. The left hand panel shows

that the 3M projection curve falls below

Graph 6: Global view of the multi-curve curve building algorithm for USD

Market Segments Replace Tenor Swaps

ON Deposit 1D Deposit OIS Swaps 3M vs FF Swaps Virtual OIS Swaps

Sequential Bootstrap3M Deposit 3M Futures 3M FRAs 3M Swaps

6M Deposit 6M FRAs 3M vs 6 Swaps Virtual 6M Swaps

USD Interest Rate Market

OIS Swaps3M vs FF Swaps3M Swaps3M vs 6M Swaps

6M FRA3M FRADepositsFutures

Graph 7: The impact of spread interpolation / extrapolation on the discount curve in the case of USD

USD ZC Curves: Naive extrapolation

0,0%

0,5%

1,0%

1,5%

2,0%

2,5%

3,0%

3,5%

4,0%

4,5%

5,0%

0 10 20 30 40 50 60

Discount Curve

3M Fwd Curve

USD ZC Curves: Spread extrapolation

0,0%

0,5%

1,0%

1,5%

2,0%

2,5%

3,0%

3,5%

4,0%

4,5%

5,0%

0 10 20 30 40 50 60

Discount Curve

3M Fwd Curve

Sources: AXA IM, 2011

the discounting curve if we extrapolate

the risk-free curve in a naive manner. We

recommend the approach used in the

right panel where we extrapolate spreads

instead.

Extensions

The multi-curve system described here

is but the fi rst step in a new and exciting

direction. The subprime crisis has shaken

the world and the foundations of fi nancial

economic theory.

Derivative pricing theory is in some

sense being rewritten as regulation.

Collateral agreements and liquidity must

be integrated into our working hypotheses

in order to obtain more realistic models.

At the same time, practitioners require

models that are easy to understand

and easy to implement. It is yet to be

seen whether the new models that

will emerge will fi t the bill! Focusing on

interest rates, there is now a clear need

to take the spread between Libor rates

and discount rates into account. The

graphs shown earlier should convince

even the novice that this spread is

anything but a constant. It can be shown

that in the multi-curve context vanilla

products such as caps and swaptions

can be priced using Black-Scholes style

formulas without directly modeling

the spread volatility (see Mercurio4).

reader to a recent Risk Magazine

article8 where two major banks make a

public statement to this effect. Despite

the urgency of the matter, however,

until recently discrepancies persisted

between front offi ce and middle offi ce

pricing systems. Starting mid-2011 most

major banks seemed to have fi nally

migrated their reporting from Libor to

OIS discounting. Banks are now moving

to the next challenge in this context

which involves accounting for the actual

collateral posted for each OTC deal: when

the collateral is not cash, the discounting

rate has to be appropriately adjusted. On

the buy side, integrating these changes

is also of great importance. With the

number of collateralised transactions on

the rise, asset managers are required

to provide independent valuations of

over-the-counter transactions. Buy side

companies tend to be more dependent on

third party fi nancial software for fi nancial

computations and in particular for yield

curve construction. At the time of writing,

many of the leading software providers

are yet to develop fully integrated multi-

curve solutions. In the short term, this

may lead asset managers dependent on

external software to use approximations

or other time-consuming short cuts until

their usual providers catch up with the

changes.

For more sophisticated products and

strategies it is necessary to simulate

the full set of curves. Progress in this

direction is provided by Mercurio who

shows how the standard Libor Market

Model can be enriched with a stochastic

spread. Pallavicini and Moreni6 present

a generalization of HJM framework and

in particular an extension of the popular

G2++ short rate model. Despite the slew

of recent academic articles attempting to

extend the theory, it seems far too early

to say which models will be embraced as

the most effective and usable. Clearly,

much experimentation and research lie

ahead before practitioners adopt a new

standard model.

Impact on Systems

The birth of the multi-curve system

has been a diffi cult one. It is not the

mere complexity of the model that is

troublesome but rather the fact that

the yield curve is such a ubiquitous

and fundamental concept in fi nance.

As a result, when a fi nancial institution

changes its assumptions and the way it

computes discount rates, it can expect

side effects and changes in all its

accounting and trading books. Banks

were the fi rst to be impacted by these

changes and as a result they have had to

adapt their pricing and booking quickly.

On this subject we refer the interested

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13

References

1 Collin-Dufresne, P., and Solnik, B. On the term structure of default premia in the swap and libor markets. The Journal of Finance LVI, 3 (June 2001).

2 Tuckman, B., and Hom´e, J.-B. Consistent pricing of FX forwards, cross currency basis swaps and interest rate swaps in several currencies. Lehman Brothers Fixed Income Research (December 2003).

3 Bianchetti, M. Two curves, one price: Pricing & hedging interest rate derivatives using different yield curves for dicounting and forwarding. Working Paper (2008).

4 Mercurio, F. Interest rates and the credit crunch: New formulas and market models. Working Paper (February 2009).

5 Morini, M. Solving the Puzzle in the Interest Rate Market. Working Paper (October 2009).

6 Pallavicini, A., and Moreni, N. Parsimonous HJM Modelling for Multiple Yield Curves.Working Paper (2010).

7 Traven, S. Pricing and hedging linear derivatives with an arbitrage-free set of interest rate curve.Barclays Capital Quantitative Credit Quarterly (2010-Q3/Q4).

8 Hallet, N., and Wilson, S. Funding valuation-a clear and present future. Risk Magazine (June2010). RBS and Barclay’s Capital joint sponsored statement.

AGE-OLD CUSTOMS IN THE NEW AGEFrom Single to Multi-Curve: Adapting to Change

ConclusionIn the wake of the economic crisis sparked by the collapse of the sub-prime housing market, many profound changes have impacted derivative trading and pricing. Perhaps the most striking changes involve the contamination of interest rate theory by credit risk. We have reviewed recent events and described the fi nancial phenomena that have led to a new theory for interest rate curve construction. Although most market participants now agree that the credit spread present in Libor is now a fundamental part of interest rate theory, there is still no clear consensus on how to deal with it in practice. Nevertheless, many agree that the multi-curve approach is a consistent and tractable extension of the methods we applied before the credit bubble. Whereas previously, fi nancial institutions had more time to adjust to major theoretical innovations, the shock to interest rate hypotheses seems to have taken everyone by surprise. Due to the pressing demand for collateralised transactions, many practitioners feel rushed to adapt to these changes before the industry as a whole agrees on a new standard for curve construction. In the coming years we expect that these innovations will be absorbed by academia and industry and that some agreed standard models will emerge. Given the fundamental and complex nature of the change we conjecture that this transition will take more time than previous ones.

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