Multiple Curves, One Price

MULTIPLE CURVES, ONE PRICE
The Post Credit-Crunch Interest Rate Market

Global Derivatives
Paris,17-21 May 2010

Marco Bianchetti
Intesa Sanpaolo Bank, Risk Management, Market Risk, Pricing & Financial Modelling
marco.bianchetti intesasanpaolo.com

Acknowledgments and disclaimer

The author acknowledges fruitful discussions with M. De Prato, M. Henrard, M. Joshi,
C. Maffi, G. V. Mauri, F. Mercurio, N. Moreni, many colleagues in the Risk
Management. A particular mention goes to M. Morini and M. Pucci for their
encouragement and to F. M. Ametrano and the QuantLib community for the open-
source developments used here.

The views and the opinions expressed here are those of the author and do not
represent the opinions of his employer. They are not responsible for any use that
may be made of these contents.

“Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 2

Summary
1. Context and Market Practices
o Libor/Euribor/Eonia interest rates and derivatives
o Counterparty Risk and collateral agreements
o Eonia discounting or not ?
o Single-curve pricing & hedging interest-rate derivatives
o Multiple-curve pricing & hedging interest-rate derivatives
2. Multiple-Curve Framework
o General assumptions
o Double-curve pricing procedure
o No arbitrage and forward basis
3. Foreign-Currency Analogy approach
o Spot and forward exchange rates, quanto adjustment
o Pricing FRAs, swaps, caps/floors/swaptions
o Interpretation in terms of counterparty risk
4. Hedging in a Multiple-Curve Environment
5. Other Approaches
o Axiomatic approach
o Extended libor Market Model Approach
o Extended Short Rate Model Approach
o Counterparty Risk Approach
6. Conclusions
7. Selected references


1: Context & Market Practices:
Libor interest rate [1]
Libor definition and mechanics
(source: www.bbalibor.com, 31th March 2010)

Libor = London Interbank Offered rate,
o first published in 1986,
o sponsored by British Banker’s Association (BBA, see http://www.bbalibor.com),
o reference rate mentioned in ISDA standards for OTC transactions.
Fixing mechanics:
o each TARGET business day the BBA polls a panel of Banks for rate fixing on
15 maturities (1d-12M): “at what rate could you borrow funds, were you to do so
by asking for and then accepting inter-bank offers in a reasonable market size
just prior to 11 am (GMT)?”;
o rate fixings are calculated, for each maturity, as the average of rates
submissions after discarding highest and lowest quartiles (25%);
o published around 11:45 a.m. (GMT), annualised rate, act/360 (Reuters page
“LIBOR”);
o calculation agent: Reuters.
Currencies: GBP, USD, JPY, CHF, CAD, AUD, EUR, DKK, SEK, NZD.


Libor definition amplified

the rate at which each bank submits must be formed from that bank’s perception of
its cost of funds in the interbank market;
contributions must represent rates formed in London Market and not elsewhere;
contributions must be for the currency concerned, not the cost of producing one
currency by borrowing in another currency and accessing the required currency via
the foreign exchange markets;
the rates must be submitted by members of staff at a bank with primary
responsibility for management of a bank’s cash, rather than a bank’s derivative
book;
the definition of “funds” is: unsecured interbank cash or cash raised through primary
issuance of interbank Certificates of Deposit.


Libor panels

Composition 8-12-16 contributors per currency (a multiple of 4 because of the
average calculation rule above);
Selection criteria:
o Guiding principle: “Banks chosen by the independent Foreign Exchange and
Money Markets Committee to give the best representation of activity within the
London money market for a particular currency”;
o Criteria:
Scale of market activity
Reputation
Perceived expertise in the currency concerned
Review: annual review by BBA with FX & MM Committee; all panels and proposed
banks are ranked according to their total money market and swaps activity over the
previous year and selected according to the largest scale of activity with due
concern given to the other 2 criteria.
Sanctions: warning and successively exclusion from the panel.


Libor panels per currency
Banks AUD CAD CHF EUR GBP JPY USD DKK NZD SEK Panels
Abbey National X 1
Bank of America X X X X 4
Bank of Montreal X 1
Bank of Nova Scotia X 1
Bank of Tokyo-Mitsubishi UFJ Ltd X X X X X 5
Barclays Banks plc X X X X X X X X X X 10
BNP Paribas X 1
Canadian Imperial Bank of Commerce X 1
Citibank NA X X X X X 5
Commonwealth Bank of Australia X X 2
Credit Suisse X X X 3
Deutsche Bank AG X X X X X X X X X X 10
HSBC X X X X X X X X X 9
JP Morgan Chase X X X X X X X X X 9
Lloyds Banking Group X X X X X X X X X X 10
Mizuho Corporate Bank X X X 3
National Australia Bank X X 2
National Bank of Canada X 1
Norinchukin Bank X X 2
Rabobank X X X X X X X X 8
Royal Bank of Canada X X X X 4
Royal Bank of Scotland Group X X X X X X X X X X 10
Société Générale X X X X 4
Sumitomo Mitsui X 1
UBS AG X X X X X X X X X 9
WestLB AG X X X X 4
Totals 8 12 12 16 16 16 16 8 8 8
Source: www.bbalibor.org, 31 Mar. 2010


Libor questioned during the crisis

The Bank for International Settlements reported that "available data do not support the
hypothesis that contributor banks manipulated their quotes to profit from positions
based on fixings“ (see J. Gyntelberg, P. Wooldridge, “Interbank rate fixings during the recent turmoil”, BIS
Quarterly Review, Mar. 2008, ref. [III]).

Risk Magazine reported rumors that “Libor rates are still not reflective of the true levels
at which banks can borrow” (see P. Madigan, “Libor under attack”, Risk, Jun. 2008, ref. [VI])
The Wall Street Journal reported that some banks “have been reporting significantly
lower borrowing costs for the Libor, than what another market measure suggests they
should be” (see C. Mollenkamp, M. Whitehouse, The Wall Street Journal, 29 May 2008, ref. [V]).
The British Banker’s Association commented that Libor continues to be reliable, and
that other proxies are not necessarily more sound than Libor at times of financial crisis.

The International Monetary Fund reported that "it appears that U.S. dollar LIBOR
remains an accurate measure of a typical creditworthy bank’s marginal cost of
unsecured U.S. dollar term funding“ (see Global Financial Stability Report, Oct. 2008, ch. 2, ref. [VII]).


Euribor interest rate [1]
Euribor definition and mechanics
(source: www.euribor.com, 31th March 2010)

Euribor = Euro Interbank Offered Rate
o first published on 30 Dec. 1998;
o sponsored by the European Banking Federation (EBF) and by the Financial
Markets Association (ACI).
Fixing mechanics:
o each TARGET business day the European Banking Federation (EBF) polls a
panel of European Banks for rate fixing on 15 maturities (1w-12M): “what rate
do you believe one prime bank is quoting to another prime bank for interbank
term deposits within the euro zone?”;
o rate fixings are calculated, for each maturity, as the average of rates
submissions after discarding highest and lowest 15%;
o published at 11:00 a.m. (CET) for spot value (T+2), annualised rate, act/360,
three decimal places (Reuters page “EURIBOR=”);
o calculation agent: Reuters
Currencies: EUR


Euribor panel

Composition on Mar. 2010: 39 banks from 15 EU countries + 4 international banks;
Selection criteria:
o “…active players in the euro money markets in the euro-zone or worldwide and
if they are able to handle good volumes in euro-interest rate related
instruments, especially in the money market, even in turbulent market
condition”;
o “first class credit standing, high ethical standards and enjoying an excellent
reputation”;
Review: ”periodically reviewed by the Steering Committee to ensure that the
selected panel always truly reflects money market activities within the euro zone”.
Banks obligations:
o must quote "the best price between the best banks“, “for the complete range of
maturities”, “on time”, “daily”, “accurately”;
o must make “the necessary organisational arrangements to ensure that delivery
of the rates is possible on a permanent basis without interruption due to human
or technical failure”.
Sanctions: warning and successively exclusion from the panel.


Euribor panel
Bank Country Bank Country
Erste Bank der Österreichischen Sparkassen RZB AIB Group
Austria Ireland
Raiffeisen Zentralbank Österreich AG Bank of Ireland
Dexia Bank Intesa Sanpaolo
Fortis Bank Belgium Unicredit Italy
KBC Monte dei Paschi di Siena
Nordea Finland Banque et Caisse d'Épargne de l'État Luxembourg
BNP - Paribas RBS N.V.
Natixis Rabobank Netherlands
Société Générale ING Bank
France
Crédit Agricole s.a. Caixa Geral De Depósitos (CGD) Portugal
HSBC France Banco Bilbao Vizcaya Argentaria
Crédit Industriel et Commercial CIC Confederacion Española de Cajas de Ahorros
Spain
Landesbank Berlin Banco Santander Central Hispano
WestLB AG La Caixa Barcelona
Bayerische Landesbank Girozentrale Barclays Capital
Other EU
Commerzbank Den Danske Bank
Banks
Deutsche Bank Germany Svenska Handelsbanken
DZ Bank Deutsche Genossenschaftsbank Bank of Tokyo - Mitsubishi
Landesbank Baden-Württemberg Girozentrale J.P. Morgan Chase & Co. International
Norddeutsche Landesbank Girozentrale Citibank Banks
Landesbank Hessen - Thüringen Girozentrale UBS (Luxembourg) S.A.
National Bank of Greece Greece Source: www.euribor.org, 31 Mar. 2010


Eonia interest rate
Eonia definition and mechanics
(source: www.euribor.com, 31th March 2010)

Eonia = Euro Over Night Index Average, first published and sponsored as Euribor.
Panel banks: same as Euribor
Fixing mechanics:
o each TARGET business day each panel bank submits the total volume of
overnight unsecured lending transactions of that day and the weighted average
lending rate for these transactions;
o rate fixing is calculated as the transaction volumes weighted average of rates
submissions;
o published at 6:45-7:00 p.m. (CET) for today value (T+0), annualised rate,
act/360, three decimal places (Reuters page “EONIA=“).
o Calculation agent: European Central Bank
Overnight rates in other currencies:
o USD: Federal Funds Effective Rate
o GBP: Sonia = Sterling Over Night Index Average
o JPY: Mutan rate

Xibor/Eonia interest rates discussion [1]
Xibor discussion

Xibor is based on:
o offered rates on unsecured funding;
o expectations, views and beliefs of the panel banks about borrowing rates in the
currency money market (see e.g. P. Madigan, “Libor under attack”, Risk, Jun. 2008, ref. [VI]).
As any interest rate expectation, Xibor includes informations on:
o the counterparty credit risk/premium,
o the liquidity risk/premium
and thus its not a risk free rate, as already well known before the crisis (see e.g. B.
Tuckman, P. Porfirio, “Interest Rate Parity, Money Market Basis Swaps, and Cross-Currency Basis
Swaps”, Lehman Brothers, Jun. 2003, ref. [1]).

Lending/borrowing Xibor rates is tenor dependent: “The age of innocence – when
banks lent to each other unsecured for three months or longer at only a small
premium to expected policy rates – will not quickly, if ever, return” (M. King, Bank of
England Governor, 21 Oct. 2008).

The Xibor panel may change over time, panel banks may be replaced by other
banks with higher credit standing. Borrowers and lenders will not be Xibor forever.


Eonia discussion

Eonia is based on unsecured lending (offer side) transactions of the panel banks in
the Euro money market;
Eonia is used by ECB as a method of effecting and observing the transmission of
the monetary policy actions;
Eonia includes informations on:
o the monetary policy effects,
o the short term liquidity expectations of the panel banks
in the Euro money market;
Eonia holds the shortest rate tenor available (one day), carries negligible
counterparty credit and liquidity risk and thus it is the best available market proxy to
a risk free rate.

See also Goldman Sachs, “Overview of EONIA and Update on EONIA Swap Market”, Mar. 2010, ref. [XV].


Libor Euribor Eonia
London InterBank Euro InterBank Euro OverNight
Definition
Offered Rate Offered Rate Index Average
Market London Interbank Euro Interbank Euro Interbank
Side Offer Offer Offer
EURLibor = Euribor,
TARGET calendar, T+2, TARGET calendar, T+0,
Rate quotation slight differences for other
act/360, three decimal act/360, three decimal
specs currencies (e.g. act/365, T+0,
places, tenor variable. places, tenor 1d.
London calendar for GBPLibor).
Maturities 1d-12m 1w, 2w, 3w,1m,…,12m 1d
Publication time 12.30 CET 11:00 am CET 6:45-7:00 pm CET
39 banks from 15 EU
8-16 banks (London based)
Panel banks countries + 4 Same as Euribor
per currency
international banks
Calculation agent Reuters Reuters European Central Bank
Transactions based No No Yes
Counterparty risk Yes Yes Negligible
Liquidity risk Yes Yes Negligible
Tenor basis Yes Yes No


Xibor and counterparty risk
Suppose an investor interested to enter into a 6M deposit on Xibor rate. There are at
least two different alternatives:
choose Bank A, enter today into a 6M deposit, and get your money plus interest back
in 6 months if Bank A has not defaulted;
choose Bank A, enter today into a 3M deposit, get your money plus interest back in 3
months if Bank A has not defaulted, then rechoose a second Bank B (the same or
another), enter into a second 3M deposit and get your money plus interest back in 3
months if Bank B has not defaulted.
Cleary the second 3M+3M strategy carries a credit risk lower than the first 6M strategy,
where I can only choose once (if Bank A is in bad waters after 3 months there is nothing
I can do). Hence a 6M loan is riskier than the two corresponding 3M+3M loans, and the
6M fixing must, all other things equal, be higher than the 3M fixing.
Basis swap 3M6M: if the counterparties are under CSA (with daily margination in
particular) the credit risk is negligible. Therefore the party paying the lower 3M rate must
compensate the party paying the higher 6M rate, hence the positive basis 3M-6M.
The same applies to any other rate pair with different tenors.


Xibor and liquidity risk
Suppose a Bank with excess liquidity (cash) to lend today at Xibor rate for 6 month.
There are at least two different alternatives:
the Bank checks its liquidity today, it loans the excess liquidity today for 6M and gets
cash plus interest back in 6M if the borrower has not defaulted;
the Bank checks its liquidity today, it loans the excess liquidity today for 3M and gets
cash plus interest back in 3M if the borrower has not defaulted, then it rechecks its
liquidity, loans the excess liquidity for the next 3M and gets cash plus interest back in 6M
if the borrower has not defaulted;
Cleary the first 6M strategy carries a liquidity risk higher than the second 3M+3M
strategy: if in 3M the Bank needs liquidity it is allowed to stop lending. Hence a 6M loan
is riskier than the two corresponding 3M+3M loans, and the 6M fixing must, all other
things equal, be higher than the 3M fixing.
Basis swap 3M6M: if the counterparties are under CSA (with daily margination in
particular) the liquidity risk is negligible. Therefore the party paying the lower 3M rate
must compensate the party paying the higher 6M rate, hence the positive basis 3M-6M.
The same applies to any other rate pair with different tenors.


Interest rate market segmentation [1]
Stylized facts:

Divergence between deposit (Xibor based) and OIS (Overnight based) rates.

Divergence between FRA rates and the corresponding forward rates implied by
consecutive deposits.

Explosion of basis swap rates (based on Xibor rates with different tenors)



EUR 3M OIS rates vs 3M Depo rates
Quotations Dec. 2005 - May 2010 (source: Bloomberg)



EUR 6M OIS rates vs 6M Depo rates
Quotations Dec. 2005 – Apr. 2010 (source: Bloomberg)



EUR 3x6 FRA vs 3x6 fwd OIS rates



EUR 6x12 FRA vs 6x12 fwd OIS rates



EUR Basis Swap 5Y, 3M vs 6M
Quotations May 2005 – Apr. 2010 (source: Bloomberg)


65 Eonia vs Euribor (31.03.2010) Eonia vs 1M
Eonia vs 3M
60 Eonia vs 6M
Eonia vs 12M
55 1M vs 3M
1M vs 6M
50 1M vs 12M
3M vs 6M
45 3M vs 12M
6M vs 12M
40
Basis spread (bps)

35

30

25

20

15

10

5

0

-5
1YR

2YR

3YR

4YR

5YR

6YR

7YR

8YR

9YR

10YR

11YR

12YR

15YR

20YR

25YR

30YR
Term

EUR Basis Swaps
Quotations as of 31 Mar. 2010 (source: Reuters, ICAP)


Apparently similar interest rate instruments with different underlying rate tenors are
characterised, in practice, by different liquidity and credit risk premia, reflecting the
different views and interests of the market players.

Thinking in terms of more fundamental variables, e.g. a short rate, the credit crunch
has acted as a sort of symmetry breaking mechanism: from a (unstable) situation in
which an unique short rate process was able
to model and explain the whole term structure
of interest rates of all tenors, towards a sort
of market segmentation into sub-areas
corresponding to instruments with different
underlying rate tenors, characterised,
in principle, by distinct dynamics,
e.g. different short rate processes.

Notice that market segmentation was already present (and well understood) before
the credit crunch (see e.g. B. Tuckman, P. Porfirio, “Interest Rate Parity, Money Market Basis Swaps, and
Cross-Currency Basis Swaps”, Lehman Brothers, Jun. 2003) but not effective due to negligible
basis spreads.


Counterparty risk and collateral [1]
Typical financial transactions generate streams of future cashflows, whose total net
present value (NPV = algebraic sum of all discounted expected cashflows) implies a
credit exposure between the two counterparties.
If, for counterparty A, NPV(A)>0 => counterparty A expects to (globally) receive future
cashflows from counterparty B (A has a credit with B), and, on the other side,
counterparty B has NPV(B)<0 and expects to (globally) pay future cashflows to
counterparty A (B has a debt with A). The reverse holds if NPV(A)<0 and NPV(B)>0.
Such credit exposure can be mitigated through a guarantee, called collateral.

In banking, collateral has two meanings:
asset-based lending: the traditional secured lending, with unilateral obligations,
secured in the form of property, surety, guarantee or other;
capital market collateralization: used to secure trade transactions, with bilateral
obligations, secured by more liquid assets such as cash or securities, also
known as margin.

Capital market collateralization: physical delivery from the debtor to the creditor (with
or without transfer of property) of liquid financial instruments or cash with NPV
corresponding to the NPV of the trade, as a guarantee for mutual obligations.


Collateral mechanics
Regulated markets Over the counter markets
Not all trades are collateralised, it
Collateralisation: All trades are collateralised depends on the agreements
between the counterparties
Financial
highly standardised highly customised
instruments:
There is a Clearing House that acts There is no Clearing House, direct
as counterparty for any trade and interaction between the
Clearing House:
establish settlement and counterparties, ad hoc contracts
margination rules are used
Daily settlement and margination,
Settlement and Most used contracts are:
collateral in cash of main
margination ISDA Master Agreement
currencies or highly rated bonds
execution: Credit Support Annex (CSA)
(govies)
Collateral
Overnight rate Depend on the agreements
interest:


ISDA Master Agreement
o Standardised contracts proposed and maintained by the International Swaps
and Derivatives Association (ISDA).
o Widely used by most financial operators to regulate OTC transactions.
o Netting clause: counterparties are allowed to calculate the total net reciprocal
credit exposure (total NPV = algebraic sum of the NPVs of all mutual
transactions)

Credit Support Annex:
accessory document to the ISDA Master Agreement that establish the collateral and
margination rules between the counterparties. There are two main CSA versions:
o UK CSA: most used in Europe, with property transfer of the collateral (cash or
assets) from the debtor to the creditor, that can freely use it;
o US CSA: most used in the US, the collateral (cash or assets) is deposited by the
debtor in a vincolated bank account of the creditor (there is no property transfer
of the collateral).

(source: F. Ametrano, M. Paltenghi, Risk Italia Nov. 2009, ref. [XII])


CSA characteristics:

Exposure: potential loss that the creditor would suffer in case of default of the
counterparty before trade maturity. It is measured in terms of cost of replacement the
cost for the creditor to enter in the same deal with another counterparty.
Base currency: reference currency of the contract and of the collateral.
Eligible currency: one or more currencies alternative to the base currency.
Eligible credit support: the collateral assets agreed by the counterparties, generally
cash or AAA bonds (mainly govies).
Haircut: valuation percentage applied to the Eligible Credit Support to reduce the
collateral asset volatility, proportional to the asset residual life.
Independent amount: the amount transferred at CSA inception, indepentent on the
NPV dynamics.
Threshold: the maximum exposure allowed between two counterparties without CSA;
it depends on the credit worthiness of the counterparties.
Minimum transfer amount (MTA): the threshold for margination; it depends on the
counterparties’ ratings.
Rounding: the rounding to be applied to the MTA.


CSA characteristics (cont’d):

Valuation agent: the counterparty that calculates the exposure and the collateral for
margination; if not specified, the burden lies with the counterparty that calls the
Collateral.
Valuation date: exposure calculation and margination frequency; it may be daily,
weekly or monthly; daily margination allows for the best guarantee against credit risk.
Notification time: when the Valuation Agent communicates to the other counterparty
the exposure and the collateral to be exchanged.
Interest rate: the rate of remuneration of the collateral; normally it is the flat overnight
rate in the base currency.
Dispute resolution: how to redeem any disagreements on the exposure and collateral
valuation.



Collateral Pros Collateral Cons

Counterparty risk reduction Funding volatility sensitivity
Credit management optimization Possible liquidity squeeze
Capital ratios reduction Structural and running costs
(Basilea II) Operational risks: settlement and
Increased business opportunities MTM mismatch
Funding at overnight rate Legal risk
Periodic check of credit exposure
and portfolio NPV



CVA

Counterparty 2
Counterparty CDS Rate
Counterparty 1

Bank’s Funding Rate

CVA (Bank’s side)

Euribor Rate (risk free)



CVA Rates

Counterparty 2
Counterparty CDS Rate
Counterparty 1

CVA (bilateral)
Bank’s Funding Rate (no CSA)
(includes Bank’s cost of liquidity over CDS)

Bank’s CDS Rate (includes the
Bank’s default risk)

CVA (Ctp side) CVA (Bank’s side)
Euribor Rate (includes both credit and
liquidity issues among Euribor Banks)

Liquidity Value Adjustment ?
Eonia Rate (CSA, risk free)


Eonia Discounting or Not ? [1]
Is the market discounting at Eonia ?
Interest Interest Rate
Rate CMS and Inflation Credits Equity Commodities
Swaps Options

Intra-day YES NO (?) ? NO (?) NO (?) NO (?)

End of day NO (?) NO (?) NO (?) NO (?) NO (?) NO (?)

Collateral NO NO NO NO NO NO

Balance
NO NO NO NO NO NO
sheet

Markit NO NO NO NO NO NO

SwapClear NO -- -- -- -- --

ICAP YES NO ? -- -- --



Main Broker Methodology
Instrument Present methodology Future methodology
Swap forwarding = Euribor xM forwarding = Euribor xM
discounting = Euribor xM discounting = Eonia
Basis Swap forwarding1 = Euribor xM1 forwarding1 = EuriborxM1
forwarding2 = Euribor xM2 forwarding2 = EuriborxM2
discounting = Min(EuriborxM1,EuriborxM2) discounting = Eonia
CMS As Basis Swaps As Basis Swaps
CMS S.O. As CMSs As CMSs
CCS As Basis Swaps As Basis Swaps
Caps/Floors/ forwarding = Euribor xM forwarding = Euribor xM
Swaptions discounting = Euribor xM discounting = Eonia
Forward premium ?
Eonia options ?


Market Phase Transition
SwapClear (London Clearing House)
Possible Triggers
Main Brokers
Currencies EUR (Eonia), USD (Fed Fund rate) at the beginning
Spot starting: NPV = 0, constant swap rate
IR Swaps
Forward starting: NPV = 0, variable forward swap rate
Constant premiums, variable Black’s implied volatility, variable smile
IR Options
(variable ATM)
CMS NPV = 0, constant swap spreads, variable beta SABR
CMS Spread Options Constant premiums, variable (bilognormal) implied correlations
Inflation Swaps NPV = 0, constant ZC, variable YoY
Inflation Options Constant premiums, variable Black’s implied volatility
Equity Options Constant premiums and dividends, variable Black’s implied volatility
CDS NPV = 0, variable default probability
Constant prices, variable credit spread absorbing the liquidity/credit
Bonds
risk inside Xibor


Single-Curve Pricing & Hedging IR Derivatives

Pre credit-crunch single curve market practice:

select a single set of the most convenient (e.g. liquid) vanilla interest rate
instruments traded on the market with increasing maturities and build a single yield
curve C using the preferred bootstrapping procedure (pillars, priorities, interpolation,
etc.); for instance, a very common choice in the EUR market was a combination of
short-term EUR deposit, medium-term FRA/Futures on Euribor3M and medium-
long-term swaps on Euribor6M;
compute, on the same curve C, forward rates, cashflows, discount factors and work
out the prices by summing up the discounted cashflows;
compute the delta sensitivity and hedge the resulting delta risk using the suggested
amounts (hedge ratios) of the same set of vanillas.


Multiple-Curve Pricing & Hedging IR Derivatives

Post credit-crunch multiple curve market practice:

build a single discounting curve Cd using the preferred bootstrapping procedure;
build multiple distinct forwarding curves Cf1… Cfn using the preferred distinct
selections of vanilla interest rate instruments, each homogeneous in the underlying
rate tenor (typically 1M, 3M, 6M, 12M);
compute the forward rates with tenor f on the corresponding forwarding curve Cf and
calculate the corresponding cashflows;
compute the corresponding discount factors using the discounting curve Cd and work
out prices by summing the discounted cashflows;
compute the delta sensitivity and hedge the resulting delta risk using the suggested
amounts (hedge ratios) of the corresponding set of vanillas.


2: Multiple-Curve Framework:
Basic Assumptions and notation [1]
1. There exist multiple different interest rate sub-markets Mx, x = {d,f1 ,…,fn}
characterized by the same currency and by distinct bank accounts Bx and yield
curves in the form of a continuous term structure of discount factors
C x := {T → Px ( t0 ,T ) ,T ≥ t0 } ,
where t0 is the reference date of the curves (e.g. settlement date, or today) and
Px(t,T) denotes the price at time t≥t0 of the Mx-zero coupon bond for maturity T,
such that Px(T,T) = 1.

2. In each sub-market Mx we postulate the usual no arbitrage relation
Px ( t,T2 ) = Px ( t,T1 ) × Px ( t,T1,T2 )
where Px(t,T1,T2) denotes the Mx-forward discount factor from time T2 to time T1,
prevailing at time t. The financial meaning of the expression above is that in each
market Mx, given a cashflow of one unit of currency at time T2, its corresponding
value at time t < T2 must be the same, both if we discount in one single step from
T2 to t, using the discount factor Px(t,T2), and if we discount in two steps, first from
T2 to T1, using the forward discount Px(t,T1,T2) and then from T1 to t, using Px(t,T1).


3. We denote with Fx(t; T1; T2) the simple compounded forward rate associated, In
each sub-market Mx to Px(t,T1,T2), resetting at time T1 and covering the time
interval [T1; T2], such that

Px ( t,T2 ) 1
Px ( t,T1,T2 ) = := ,
Px ( t,T1 ) 1 + Fx ( t;T1,T2 ) τx (T1,T2 )
where τx(T1,T2) is the year fraction between times T1 and T2 with daycount dcx.
From the relations above we obtain the familiar no arbitrage expression

1 ⎡ 1 ⎤
Fx ( t;T1,T2 ) = ⎢ − 1⎥
τx (T1,T2 ) ⎢⎣ Px ( t,T1,T2 ) ⎥⎦
1 Px ( t,T1 ) − Px ( t,T2 )
=
τx (T1,T2 ) Px ( t,T2 )


4. The eq. above can be also derived (see e.g. ref. [A], sec. 1.4) as the fair value
condition at time t of the Forward Rate Agreement (FRA) contract with payoff at
maturity T2 given by

FRAx (T1;T1,T2 , K ) = N τx (T1,T2 ) [ Lx (T1,T2 ) − K ],
1 − Px (T1,T2 )
Lx (T1,T2 ) = τx (T1,T2 )
Px (T1,T2 )
where N is the nominal amount, Lx(T1,T2) is the T1-spot Xibor rate for maturity T2
and K the (simply compounded) strike rate (sharing the same daycount convention
for simplicity). Introducing expectations we have, t≤T1<T2,

FRAx ( t;T1,T2 , K ) = NPx ( t,T2 ) τx (T1,T2 ) { }
T
Qx 2
Et [ Lx (T1,T2 ) ] − K
= NPx ( t,T2 ) τx (T1,T2 ) [ Fx ( t;T1,T2 ) − K ],
T
where Qx 2 denotes the Mx-T2-forward measure associated to the numeraire Px(t,T2),
EQ [ . ] denotes the expectation at time t w.r.t. measure Q and filtration Ft, encoding
t
the market information available up to time t, and we have assumed the standard
martingale property of forward rates in the sub-market Mx.


Pricing Procedure
1. assume Cd as the discounting curve and Cf as the forwarding curve;
2. calculate any relevant spot/forward rate on the forwarding curve Cf as
Pf ( t , Ti −1 ) − Pf ( t , Ti )
Ff ( t ;Ti −1 , Ti ) = , t ≤ Ti −1 < Ti ,
τ f (Ti −1 , Ti ) Pf ( t , Ti )
3. calculate cashflows ci, i = 1,...,n, as expectations of the i-th coupon payoff πi with
T
respect to the discounting Ti - forward measure Q d i
T
Qd i
ci := c ( t,Ti , πi ) = Et [ πi ];
4. calculate the price π at time t by discounting each cashflow ci using the
corresponding discount factor Pd ( t,Ti ) obtained from the discounting curve Cd and
summing, n
T
π ( t,T ) = ∑ P ( t,Ti ) Et Qd i
[ πi ];
i =1
5. Price FRAs as

{ [ Ff (T1 ;T1,T2 ) ] − K }
T
Qd 2
FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) τ f (T1,T2 ) Et


No Arbitrage and Forward Basis
Classic single-curve no arbitrage relations are broken: for instance, by
specifying the subscripts d and f as prescribed above we obtain the two eqs.
Pd ( t,T2 ) = Pd ( t,T1 ) Pf ( t,T1,T2 ) ,
1 Pf ( t,T2 )
Pf ( t,T1,T2 ) = = ,
1 + Ff ( t ;T1,T2 ) τ f (T1,T2 ) Pf ( t,T1 )

that clearly cannot hold at the same time. No arbitrage is recovered by taking
into account the forward basis as follows

1 1
Pf ( t,T1,T2 ) = := ,
1 + Ff ( t;T1,T2 ) τf (T1,T2 ) 1 + [ Fd ( t;T1,T2 ) + BAfd ( t;T1,T2 ) ] τd (T1,T2 )

for which we obtain the following static expression in terms of discount factors

1 ⎡ Pf ( t,T1 ) Pd ( t,T1 ) ⎤
BAfd ( t;T1,T2 ) = ⎢ − ⎥.
τd (T1,T2 ) ⎢⎣ Pf ( t,T2 ) Pd ( t,T2 ) ⎥⎦


Forward Basis Yield Curves
80 Forward Basis 0Y-3Y 10 Forward Basis 3Y-30Y
60 8
6
40
4
20
basis points

basis points
2
0 0

-20 -2
-4
-40 1M vs Disc 1M vs Disc
3M vs Disc -6 3M vs Disc
-60 6M vs Disc 6M vs Disc
12M vs Disc -8
12M vs Disc
-80 -10
May-09

May-10

May-11
Nov-09

Nov-10

Nov-11
Aug-09

Aug-10

Aug-11
Feb-09

Feb-10

Feb-11

Feb-12

Feb-12

Feb-15

Feb-18

Feb-21

Feb-24

Feb-27

Feb-30

Feb-33

Feb-36

Feb-39
Forward basis (bps) as of end of day 16 Feb. 2009, daily sampled 3M tenor forward rates calculated on
C1M, C3M , C6M , C12M curves against Cd taken as reference curve. Bootstrapping as described in ref. [2].

The richer term structure of the forward basis curves provides a sensitive indicator of
the tiny, but observable, statical differences between different interest rate market sub-
areas in the post-credit crunch interest rate world, and a tool to assess the degree of
liquidity and credit issues in interest rate derivatives' prices. Provided that…


Bad Yield Curves
6% 3M curves 0Y-30Y 4 Forward Basis 3Y-30Y

2
5%
0

basis points
4% -2

3% -4

-6
1M vs Disc
2% 3M vs Disc
zero rates -8 6M vs Disc
forward rates 12M vs Disc
1% -10
Feb-09
Feb-11
Feb-13
Feb-15
Feb-17
Feb-19
Feb-21
Feb-23
Feb-25
Feb-27
Feb-29
Feb-31
Feb-33
Feb-35
Feb-37
Feb-39

Feb-12

Feb-15

Feb-18

Feb-21

Feb-24

Feb-27

Feb-30

Feb-33

Feb-36

Feb-39
Left: 3M zero rates (red dashed line) and forward rates (blue continuous line). Right: forward basis. Linear
interpolation on zero rates has been used. Numerical results from QuantLib (www.quantlib.org).

…smooth forward yield curves are used…Non-smooth bootstrapping techniques, e.g.
linear interpolation on zero rates (still a diffused market practice), produce zero curves
with no apparent problems, but ugly forward curves with a sagsaw shape inducing, in
turn, strong and unnatural oscillations in the forward basis (see ref. [3]).


3: Foreign Currency Analogy:
Spot and Forward Exchange Rates
A second issue regarding no arbitrage arises in the double-curve framework:

{ [ Ff (T1 ;T1,T2 ) ] − K }
T
Qd 2
FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) τ f (T1,T2 ) Et
≠ Pd ( t,T2 ) τ f (T1,T2 ) [ Ff (T1 ;T1,T2 ) − K ]
1. Double-curve-double-currency: ⇒
d = domestic, f = foreign
c d ( t ) = x fd ( t ) c f ( t ) ,
X fd ( t , T ) Pd ( t , T ) = x fd ( t ) P f ( t , T ),
x fd ( t 0 ) = x fd , 0.

2. Double-curve-single-currency: ⇒ d
= discounting, f=forwarding
cd ( t ) = x fd ( t )c f ( t ) ,
X fd ( t , T ) Pd ( t , T ) = x fd ( t ) Pf ( t ,T ) ,
x fd ( t 0 ) = 1.
Picture of no arbitrage definition of the forward exchange rate.
Circuitation (round trip) ⇒ no money is created or destructed.

Quanto Adjustment
1. Assume a lognormal martingale dynamic for the Cf (foreign) forward rate
dFf ( t ;T1,T2 )
= σ f ( t ) dW fT2 ( t ) , Q T2 ↔ Pf ( t ,T2 ) ↔ C f ;
f
Ff ( t ;T1,T2 )
2. since x fd ( t ) Pf ( t,T ) is the price at time t of a Cd (domestic) tradable asset, the
forward exchange rate must be a martingale process
dX fd ( t ,T2 ) T T
= σ X ( t ) dWX 2 ( t ) , Qd 2 ↔ Pd ( t , T2 ) ↔ C d ,
X fd ( t ,T2 )
dW fT2 ( t ) dWX 2 ( t ) = ρ fX ( t ) dt ;
T
with
3. by changing numeraire from Cf to Cd we obtain the modified dynamic
dFf ( t ;T1,T2 )
= μ f ( t ) dt + σ f ( t ) dW fT2 ( t ) , Qd 2 ↔ Pd ( t ,T2 ) ↔ C d ,
T
Ff ( t ;T1,T2 )
μ f ( t ) = −σ f ( t ) σ X ( t ) ρ fX ( t ) ;
4. and the modified expectation including the (additive) quanto-adjustment
T
Qd 2
E t [ L f (T1,T2 ) ] = Ff ( t ;T1 ,T2 ) + QAfd ( t ;T1 , σ f , σ X , ρ fX ),
⎡ T1 ⎤
QAfd ( t ;T1, σ f , σ X , ρ fX ) = Ff ( t ;T1,T2 ) ⎢ exp
⎣ ∫t μ f ( s ) ds − 1 ⎥ .
⎦

Pricing Plain Vanillas [1]
1. FRA: FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) τ f (T1,T2 )
× [ Ff ( t;T1,T2 ) + QAfd ( t,T1, σ f , σX , ρfX ) − K ]
m
2. Swaps:
Swap ( t;T , S, K ) = −∑ Pd ( t, S j ) τd ( S j −1, S j )K j
n j =1
+∑ Pd ( t, ST ) τ f (Tj −1,Tj )[ Ff ( t;Ti −1,Ti ) + QAfd ( t,Ti −1, σ f ,i , σX ,i , ρfX ,i ) ].
i =1
n
3. Caps/Floors:
CF ( t;T , K , ω ) = ∑ Pd ( t,Ti ) τd (Ti −1,Ti )
i =1
×Black [ Ff ( t;Ti −1,Ti ) + QAfd ( t,Ti −1, σ f ,i , σX ,i , ρfX ,i ), Ki , μf ,i , v f ,i , ωi ],

4. Swaptions: Swaption ( t;T , S, K , ω ) = Ad ( t, S )
×Black [ S f ( t;T , S ) + QAfd ( t,T , S, ν f , νY , ρfY ), K , λf , v f , ω ].


Pricing Plain Vanillas [2]
100 Quanto Adjustment (additive)
80
60
Quanto adj. (bps)

40
Numerical scenarios for the (additive)
20
quanto adjustment, corresponding to
0 three different combinations of (flat)
-20 volatility values as a function of the
-40 correlation. The time interval is fixed
-60
Sigma_f = 10%, Sigma_X = 2.5% to T1-t=10 years and the forward rate
Sigma_f = 20%, Sigma_X = 5% to 3%.
-80 Sigma_f = 30%, Sigma_X = 10%
-100
-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0
Correlation

We notice that the adjustment may be not negligible. Positive correlation implies
negative adjustment, thus lowering the forward rates. The standard market practice,
with no quanto adjustment, is thus not arbitrage free. In practice the adjustment
depends on market variables not directly quoted on the market, making virtually
impossible to set up arbitrage positions and locking today positive gains in the future.


A Simple Credit Model
Both the forward basis and the quanto adjustment discussed above find a simple
financial explanation in terms of counterparty risk. We adapt the simple credit model
from Mercurio (ref. [9]) to the present context and notation.
If we identify:
Pd(t,T) = default free zero coupon bond,
Pf(t,T) = risky zero coupon bond emitted by a generic risky (Xibor) counterparty for
maturity T and with recovery rate Rf ,
τ(t)>t = (random) counterparty default time observed at time t,
qd ( t,T ) = EtQd {1[ τ ( t )>T ] } = default probability after time T expected at time t,
we obtain the following expressions
Pf ( t,T ) = Pd ( t,T ) R ( t; t,T, Rf ),
1 ⎡ Pd ( t,T1 ) R ( t; t,T1, Rf ) ⎤
Ff ( t;T1,T2 ) = ⎢ − 1⎥ ,
τf (T1,T2 ) ⎢⎣ Pd ( t,T2 ) R ( t; t,T2, Rf ) ⎥⎦
where: R ( t;T1,T2, Rf ) = Rf + ( 1 − Rf ) EtQd [ qd (T1,T2 ) ].


A Simple Credit Model [2]
If Ld(T1,T2), Lf(T1,T2) are the risk free and the risky Xibor rates underlying the
corresponding derivatives, respectively, we obtain the familiar FRA pricing expression
modified with a credit term:
Pd ( t,T1 )
FRAf ( t;T1,T2, K ) = − [ 1 + K τf (T1,T2 ) ] Pd ( t,T2 ) ,
R ( t;T1,T2, Rf )
and we are able to express the forward basis and the quanto adjustment described
before in terms of risk free zero coupon bonds Pd(t,T) and of the expected recovery rate

1 Pd ( t,T1 ) ⎡ R ( t; t,T1, Rf ) ⎤
BAfd ( t;T1,T2 ) = ⎢ −1 ⎥,
τd (T1,T2 ) Pd ( t,T2 ) ⎢⎣ R ( t; t,T2, Rf ) ⎥⎦
1 Pd ( t,T1 ) ⎡ 1 R ( t; t,T1, Rf ) ⎤
QAfd ( t;T1,T2 ) = ⎢ − ⎥.
τ f (T1,T2 ) Pd ( t,T2 ) ⎢⎣ R ( t;T1,T2, Rf ) R ( t; t,T2, Rf ) ⎥⎦


Pros & Cons

PROs CONs

Simple and familiar framework, Plain vanilla prices acquire
no additional effort, just analogy. volatility and correlation
dependence (but this is common
to all approaches).

Convexity adjustment emerges Parameters (exchange rate and
naturally, as in more complex its volatility) presently not
approaches. observable on the market,
historical estimate only.

Straightforward interpretation in Usual CONs of FX quanto
terms of counterparty risk. adjustment applies (e.g. no smile)


4: Hedging:
Multiple Delta Hedging [1]
1. Given any portfolio of interest rate derivatives with price Π ( t,T , Rmkt ), compute the
delta risk with respect to all curves C = {Cd ,C f 1 ,...,C fn } as
R
NC
∂Π ( t,T , Rk )
mktNC N k
Δ ( t,T , R ) = ∑ Δ (
π mkt
) = ∑∑
π mkt
t,T , Rk mkt
,
k =1 k =1 j =1 ∂Rk , j
mkt R
where NC is the number of yield curves involved and Rk is the vector of N k
bootstrapping market data (yield curve pillars) associated to yield curve Ck .

2. Be careful to take properly into account all the delta components due to multiple
curve bootstrapping: the forwarding zero curves {C f 1 ,...,C fn } , in particular, depend
directly on their corresponding input market instruments with tenor f, but also
indirectly on the discounting curve,
R Z “One price, two curves,
∂Π ( t,T , Rd ) ∂Z α
Nd Nd mkt d
Δ ( t,T , Rd ) = ∑ ∑
π mkt
d mkt
, three deltas” (see ref.[8])
j =1 α =1 ∂Z α ∂Rd , j
N fR N fZ Z
∂Π ( t,T , Rf ) ∂Z α
mkt f Nd N f
∂Π ( t,T , Rf ) ∂Z α
mkt f
Δ ( t,T , Rf ) = ∑ ∑
π mkt
f
+∑∑ ,
j =1 α =1 ∂Z α ∂Rfmkt
,j j =1 α =1 ∂Z αf mkt
∂Rd , j
Z
where Z f is the vector of N f zero rates pillars in the zero rate curve Cf.


4: Hedging:
Multiple Delta Hedging [2]
3. eventually aggregate the delta sensitivity on the selected subset H of the most liquid
market instruments used for hedging RH = { R1 ,..., RN H } (hedging instruments);
H H

NH
∂Π ( t,T , RH )
Δ π
( t,T , R )H
∑ ∂RjH
,
j =1

∂Π ( t ,T , R H )
4. calculate hedge ratios: h j ( t ,T , R H
) = δ jH ( t ) ,
∂ R jH
∂ π H ( t , T j , R jH
j )
δ jH ( t , T j , R jH )=
.
H ∂ R jH
5. where π j ( t ) is the market price (unit nominal) of the corresponding hedging
instrument, such that the hedged portfolio has zero delta
NH
Π tot
( t ,T , R ) =
H
Π ( t ,T , R H
) − ∑ h j ( t )π H ( t , T j , R jH ) ,
j
j =1
⎡ ∂Π ( t ,T , R H
NH
) NH
∂ π H ( t , T j , R jH ) ⎤
∑ ⎢⎢ − ∑ h j (t )
j
Δ tot ( t ,T , R H ) ⎥
∂ Rk H
∂ Rk H ⎥
k =1 ⎣ j =1 ⎦
NH ⎡
∂Π ( t ,T , R H ) ⎤
= ∑⎢ − h k ( t ) δk ( t ) ⎥ = 0,
H
⎢
k =1 ⎣ ∂ Rk H ⎥
⎦

5: Other Approaches

Axiomatic approach: M. Henrard (ref. [8]).

Extended Libor Market Model approaches: F. Mercurio (ref. [9], [17]); M. Fujii, Y.
Shimada, A. Takahashi (ref. [13]).

Extended Short Rate Model approaches: F. Kijima et al (ref. [5]); C. Kenyon (ref. [18]).

Counterparty Risk approach: M. Morini (ref. [11]).


5: Other Approaches:
Axiomatic Approach
See ref. [8]: M. Henrard, ”The Irony in the Derivatives Discounting - Part II: The
Crisis”, Jul. 2009, SSRN working paper http://ssrn.com/abstract=1433022.
Individuate the minimal set of definitions and assumptions for coherent multiple yield
curve pricing of plain vanillas (FRA, IRS).
Find a (convexity) adjustment factor for FRA/Futures.
Discuss delta hedging.


Extended Libor Market Model Approach [1]
See ref. [9]: F. Mercurio, "Post Credit Crunch Interest Rates: Formulas and Market
Models", Bloomberg, Jan. 2009, and ref. [17]: F. Mercurio, “LIBOR Market Models
with Stochastic Basis”, Mar. 2010.
Assume as the fundamental bricks the OIS forward rate, the FRA rate and their
spread, defined as, respectively,
x ( ) x x
T
Qd k x x 1 ⎡ Pd ( t,Tkx−1 ) ⎤
Fk t := Fd ( t;Tk −1,Tk ) = Et [ Ld (Tk −1,Tk ) ] = x ⎢ − 1⎥ ,
τk ⎢⎣ Pd ( t,Tkx ) ⎥⎦
T
Qd k
Lx ( t )
k := FRA ( t;Tkx−1,Tkx ) := Et [ Lx (Tkx−1,Tkx ) ],
Sk ( t ) := Lx ( t ) − Fkx ( t ) .
x
k
In ref. [17] assume, in the single FRA rate tenor case, general stochastic volatility
dynamics for each OIS forward and spread, uncorrelated, derive pricing expression
for caplets and swaptions as integrals over the full smile structure, and provide
examples using various specific dynamics.
Address the multi FRA rate tenor case finding the simplest stochastic-volatility
dynamics that preserve consistency across different tenors.


Extended Libor Market Model Approach [2]
See ref. [13]: M. Fujii, Y. Shimada, A. Takahashi, “A Market Model of Interest Rates
with Dynamic Basis Spreads in the presence of Collateral and Multiple Currencies”,
Nov. 2009.

Assume as the fundamental bricks the OIS forward rate, the FRA rate and their
spread.
Assume model stochastic basis spreads in a HJM framework.
Address both single- and multi-currency cases.


Extended Short Rate Model Approach
See ref. [5]: M. Kijima, K. Tanaka, T. Wong, “A Multi-Quality Model of Interest Rates”,
Quantitative Finance, vol. 9, issue 2, pages 133-145, 2008.
Assume three yield curves: discounting, forwarding and governative (D-L-G).
Assume three correlated Hull-White processes for the discounting, forwarding and
governative short rates.
Find approximated analytic pricing expression for european caps/floors/swaptions
and bond options.

See ref. [18]: C. Kenyon, “Short-Rate Pricing after the Liquidity and Credit Shocks:
Including the Basis”, Feb. 2010, SSRN working paper,
http://ssrn.com/abstract=1558429.
Assume two correlated Hull-White processes for the discounting and forwarding short
rates, plus an uncorrelated lognormal process for the exchange rate.
Find a pseudo-analytic european swaption pricing expression.


Counterparty Risk Approach
See ref. [11]: M. Morini, “Solving the Puzzle in the Interest Rate Market”, Oct. 2009.

Discuss Libor rate and Libor-based derivatives before and after the crisis.
Show that the post-credit crunch basis between FRA rates and their spot Libor
replication can be explained by using the quoted Basis Swap spreads.
Explain the market patterns of the Basis Swap spreads by modelling them as options
on the credit worthiness of the counterparty.
Formalize the mathematical representation of the post-credit crunch interest rate
market, introducing credit risk, allowing for no-fault standard rule and
collateralization, and using subfiltrations to model risky Libor rates.
Compute change of numeraire and convexity adjustments for collateralized
derivatives tied to risky Libor, thus explaining the quanto-adjustment found by
Bianchetti in ref. [7] in a simpler framework.


6: Conclusions

1. We have reviewed the pre and post credit crunch market practices for pricing &
hedging interest rate derivatives.
2. We have shown that in the present double-curve framework standard single-curve
no arbitrage conditions are broken and can be recovered taking into account the
forward basis; once a smooth bootstrapping technique is used, the richer term
structure of the calculated forward basis curves provides a sensitive indicator of the
tiny, but observable, statical differences between different interest rate market sub-
areas.
3. Using the foreign-currency analogy we have computed the no arbitrage generalised
double-curve-single-currency market-like pricing expressions for basic interest rate
derivatives, including a quanto adjustment arising from the change of numeraires
naturally associated to the two yield curves. Numerical scenarios show that the
quanto adjustment can be non negligible.
4. Both the forward basis and the quanto adjustment have a simple interpretation in
terms of counterparty risk, using a simple credit model with a risk-free and a risky
zero coupond bonds.


7: Main references [1]

Recent references on interest rate markets:
I. Euribor and Eonia official website: http://www.euribor.org
II. Libor official website: http://www.bbalibor.com
III. Bank for International Settlements, “International banking and financial market
developments”, Mar. 2008 Quarterly Review, http://www.bis.org/publ/qtrpdf/r_qt0803.htm.
IV. Financial Stability Forum, “Enhancing Market and Institutional Resilience”, 7 Apr. 2008,
http://www.financialstabilityboard.org/publications/r_0804.pdf.
V. C. Mollenkamp, M. Whitehouse, "Study Casts Doubt on Key Rate: WSJ Analysis Suggests
Banks May Have Reported Flawed Interest Data for Libor", The Wall Street Journal, May
29th, 2008, http://online.wsj.com/article/SB121200703762027135.html?mod=MKTW.
VI. P. Madigan, “Libor under attack”; Risk, Jun. 2008, http://www.risk.net/risk-
magazine/feature/1497684/libor-attack.
VII. International Monetary Fund, Global Financial Stability Report, Oct. 2008, ch. 2,
http://www.imf.org/external/pubs/ft/gfsr/2008/02/index.htm.
VIII. F. Allen, E. Carletti, “Should Financial Institutions Mark To Market ?”, Financial Stability
Review, Oct. 2008.
IX. F. Allen, E. Carletti, “Mark To Market Accounting and Liquidity Pricing”, J. of Accounting
and Economics, 45, 2008.
X. D. Wood, “The Reality of Risk Free”, Risk, Jul. 2009.
XI. L. Bini Smaghi, ECB Conference on Global Financial Linkages, Transmission of Shocks
and Asset Prices, Frankfurt, 1 Dec. 2009.
XII. F. Ametrano, M. Paltenghi, “Che cosa è derivato dalla crisi”, Risk Italia, 26 Nov. 2009.


XIII. D. Wood, “Scaling the peaks on 3M6M basis”; Risk, Dec. 2009.
XIV. H. Lipman, F. Mercurio, “The New Swap Math”, Bloomberg Markets, Feb. 2010.
XV. C. Whittall, “The Price is Wrong”, Risk, March 2010.
XVI. Goldman Sachs, “Overview of EONIA and Update on EONIA Swap Market”, Mar. 2010.

Main reference textbooks on Interest Rate Modelling:
A. D. Brigo, F. Mercurio, "Interest Rate Models - Theory and Practice", 2nd ed., Springer,
2006.
B. L. B. G. Andersen, V. V. Piterbarg, “Interest Rate Modeling”, Atlantic Financial Press, 2010
(forthcoming).



Technical papers:
1. B. Tuckman, P. Porfirio, “Interest Rate Parity, Money Market Basis Swaps, and Cross-
Currency Basis Swaps”, Lehman Brothers, Jun. 2003.
2. W. Boenkost, W. Schmidt, “Cross currency swap valuation”, working paper, HfB--Business
School of Finance & Management, May 2005.
3. M. Henrard, ”The Irony in the Derivatives Discounting”, Mar. 2007, SSRN working paper,
4. M. Johannes, S. Sundaresan, “The Impact of Collateralization on Swap Rates”, Journal of
Finance 62, pages 383–410, 2007.
5. M. Kijima, K. Tanaka, T. Wong, “A Multi-Quality Model of Interest Rates”, Quantitative
Finance, vol. 9, issue 2, pages 133-145, 2008.
6. F. Ametrano, M. Bianchetti, “Bootstrapping the Illiquidity: Multiple Yield Curves Construction
For Market Coherent Forward Rates Estimation”, in “Modeling Interest Rates: Latest
Advances for Derivatives Pricing”, edited by F. Mercurio, Risk Books, 2009.
7. M. Bianchetti, “Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives Using
Different Yield Curves for Discounting and Forwarding”, Jan. 2009, SSRN working paper,
8. F. Mercurio, "Post Credit Crunch Interest Rates: Formulas and Market Models", Bloomberg,
Jan. 2009, SSRN working paper, http://ssrn.com/abstract=1332205.
9. M. Chibane, G. Sheldon, “Building curves on a good basis”, Apr. 2009, SSRN working
paper, http://ssrn.com/abstract=1394267.


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