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MULTIPLE CURVES, ONE PRICEMULTIPLE CURVES, ONE PRICEThe Post Credit-Crunch Interest Rate Market
Global DerivativesParis,17-21 May 2010
Marco BianchettiIntesa Sanpaolo Bank, Risk Management, Market Risk, Pricing & Financial Modellingmarco.bianchetti
intesasanpaolo.com
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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 RiskManagement. A particular mention goes to M. Morini and M. Pucci for theirencouragement 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 notrepresent the opinions of his employer. They are not responsible for any use that
may be made of these contents.
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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 Bankers 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 soby 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.
1: Context & Market Practices:Libor interest rate[1]
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Libor definition amplified
the rate at which each bank submits must be formed from that banks perception ofits 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 onecurrency by borrowing in another currency and accessing the required currency viathe foreign exchange markets;
the rates must be submitted by members of staff at a bank with primaryresponsibility for management of a banks cash, rather than a banks derivativebook;
the definition of funds is: unsecured interbank cash or cash raised through primaryissuance of interbank Certificates of Deposit.
1: Context & Market Practices:Libor interest rate[2]
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Libor panels
Composition 8-12-16 contributors per currency (a multiple of 4 because of theaverage 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 proposedbanks are ranked according to their total money market and swaps activity over theprevious year and selected according to the largest scale of activity with dueconcern given to the other 2 criteria.
Sanctions: warning and successively exclusion from the panel.
1: Context & Market Practices:Libor interest rate[3]
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1: Context & Market Practices:Libor interest rate[4]
Banks AUD CAD CHF EUR GBP JPY USD DKK NZD SEK PanelsAbbey National X 1
Bank of America X X X X 4Bank of Montreal X 1
Bank of Nova Scotia X 1Bank 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 1Canadian Imperial Bank of Commerce X 1
Citibank NA X X X X X 5Commonwealth Bank of Australia X X 2
Credit Suisse X X X 3Deutsche Bank AG X X X X X X X X X X 10
HSBC X X X X X X X X X 9JP 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 10Mizuho Corporate Bank X X X 3National Australia Bank X X 2
National Bank of Canada X 1Norinchukin Bank X X 2
Rabobank X X X X X X X X 8Royal Bank of Canada X X X X 4Royal Bank of Scotland Group X X X X X X X X X X 10
Socit Gnrale X X X X 4Sumitomo Mitsui X 1
UBS AG X X X X X X X X X 9WestLB AG X X X X 4
Totals 8 12 12 16 16 16 16 8 8 8
Libor panels per currency
Source: www.bbalibor.org, 31 Mar. 2010
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Libor questioned during the crisis
The Bank for International Settlements reported that "available data do not support thehypothesis 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, BISQuarterly Review, Mar. 2008, ref. [III]).
Risk Magazine reported rumors that Libor rates are still not reflective of the true levelsat 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 significantlylower 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 Bankers 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 LIBORremains an accurate measure of a typical creditworthy banks marginal cost of
unsecured U.S. dollar term funding (see Global Financial Stability Report, Oct. 2008, ch. 2, ref. [VII]).
1: Context & Market Practices:Libor interest rate[5]
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Euribor definition and mechanics(source: www.euribor.com, 31th March 2010)
Euribor = Euro Interbank Offered Rateo first published on 30 Dec. 1998;
o sponsored by the European Banking Federation (EBF) and by the FinancialMarkets 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
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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 relatedinstruments, 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 ofmaturities, on time, daily, accurately;
o must make the necessary organisational arrangements to ensure that deliveryof the rates is possible on a permanent basis without interruption due to human
or technical failure.
Sanctions: warning and successively exclusion from the panel.
1: Context & Market Practices:Euribor interest rate[2]
1 C & M k P i
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1: Context & Market Practices:Euribor interest rate[3]
Bank Country Bank Country
Erste Bank der sterreichischen Sparkassen RZB AIB GroupRaiffeisen Zentralbank sterreich AG Bank of Ireland
Dexia Bank Intesa SanpaoloFortis Bank Unicredit
KBC Monte dei Paschi di SienaNordea Finland Banque et Caisse d'pargne de l'tat Luxembourg
BNP - Paribas RBS N.V.Natixis Rabobank
Socit Gnrale ING BankCrdit Agricole s.a. Caixa Geral De Depsitos (CGD) Portugal
HSBC France Banco Bilbao Vizcaya Argentaria
Crdit Industriel et Commercial CIC Confederacion Espaola de Cajas de AhorrosLandesbank Berlin Banco Santander Central Hispano
WestLB AG La Caixa BarcelonaBayerische Landesbank Girozentrale Barclays Capital
Commerzbank Den Danske BankDeutsche Bank Svenska Handelsbanken
DZ Bank Deutsche Genossenschaftsbank Bank of Tokyo - MitsubishiLandesbank Baden-Wrttemberg Girozentrale J.P. Morgan Chase & Co.
Norddeutsche Landesbank Girozentrale CitibankLandesbank Hessen - Thringen Girozentrale UBS (Luxembourg) S.A.
National Bank of Greece Greece Source: www.euribor.org, 31 Mar. 2010
InternationalBanks
Other EUBanks
Euribor panel
Spain
Netherlands
Italy
Ireland
Germany
France
Belgium
Austria
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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 Averageo JPY: Mutan rate
1: Context & Market Practices:Eonia interest rate
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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/premiumand 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 smallpremium to expected policy rates will not quickly, if ever, return(M. King, Bank ofEngland 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.
1: Context & Market Practices:Xibor/Eonia interest rates discussion[1]
1 Conte t & Market Practices
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Eonia discussion
Eonia is based on unsecured lending (offer side) transactions of the panel banks inthe Euro money market;
Eonia is used by ECB as a method of effecting and observing the transmission ofthe monetary policy actions;
Eonia includes informations on:o the monetary policy effects,
o the short term liquidity expectations of the panel banksin the Euro money market;
Eonia holds the shortest rate tenor available (one day), carries negligiblecounterparty 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].
1: Context & Market Practices:Xibor/Eonia interest rates discussion[2]
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1: Context & Market Practices:Xibor/Eonia interest rates discussion[3]
Libor Euribor Eonia
DefinitionLondon InterBank
Offered RateEuro InterBankOffered Rate
Euro OverNightIndex Average
Market London Interbank Euro Interbank Euro Interbank
Side Offer Offer Offer
Rate quotationspecs
EURLibor = Euribor,slight differences for other
currencies (e.g. act/365, T+0,London calendar for GBPLibor).
TARGET calendar, T+2,act/360, three decimalplaces, tenor variable.
TARGET calendar, T+0,act/360, three decimal
places, tenor 1d.
Maturities 1d-12m 1w, 2w, 3w,1m,,12m 1d
Publication time 12.30 CET 11:00 am CET 6:45-7:00 pm CET
Panel banks8-16 banks (London based)
per currency
39 banks from 15 EUcountries + 4
international banksSame as Euribor
Calculation agent Reuters Reuters European Central BankTransactions based No No Yes
Counterparty risk Yes Yes Negligible
Liquidity risk Yes Yes Negligible
Tenor basis Yes Yes No
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1: Context & Market Practices:Xibor and counterparty risk
Suppose an investor interested to enter into a 6M deposit on Xibor rate. There are atleast two different alternatives:
choose Bank A, enter today into a 6M deposit, and get your money plus interest backin 6 months if Bank A has not defaulted;
choose Bank A, enter today into a 3M deposit, get your money plus interest back in 3months if Bank A has not defaulted, then rechoose a second Bank B (the same oranother), enter into a second 3M deposit and get your money plus interest back in 3months 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 nothingI can do). Hence a 6M loan is riskier than the two corresponding 3M+3M loans, and the6M 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 mustcompensate the party paying the higher 6M rate, hence the positive basis 3M-6M.
The same applies to any other rate pair with different tenors.
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1: Context & Market Practices: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 getscash 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 getscash plus interest back in 3M if the borrower has not defaulted, then it rechecks itsliquidity, loans the excess liquidity for the next 3M and gets cash plus interest back in 6Mif 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 loanis riskier than the two corresponding 3M+3M loans, and the 6M fixing must, all otherthings 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 ratemust 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.
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1: Context & Market Practices: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 byconsecutive deposits.
Explosion of basis swap rates (based on Xibor rates with different tenors)
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1: Context & Market Practices:Interest rate market segmentation[2]
EUR 3M OIS rates vs 3M Depo rates
Quotations
Dec. 2005 -
May
2010 (source: Bloomberg)
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1: Context & Market Practices:Interest rate market segmentation[3]
EUR 6M OIS rates vs 6M Depo rates
Quotations
Dec. 2005
Apr. 2010
(source: Bloomberg)
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1: Context & Market Practices:Interest rate market segmentation[4]
EUR 3x6 FRA vs 3x6 fwd OIS rates
Quotations
Dec. 2005
Apr. 2010
(source: Bloomberg)
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1: Context & Market Practices:Interest rate market segmentation[5]
EUR 6x12 FRA vs 6x12 fwd OIS rates
Quotations
Dec. 2005
Apr. 2010
(source: Bloomberg)
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1: Context & Market Practices:Interest rate market segmentation[6]
EUR Basis Swap 5Y, 3M vs 6M
Quotations
May
2005
Apr. 2010
(source: Bloomberg)
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1: Context & Market Practices:Interest rate market segmentation[7]
EUR Basis Swaps
Quotations as of 31 Mar. 2010 (source: Reuters, ICAP)
Eonia vs Euribor (31.03.2010)
-5
0
5
10
15
20
25
30
35
40
45
50
55
60
65
1YR
2YR
3YR
4YR
5YR
6YR
7YR
8YR
9YR
10YR
11YR
12YR
15YR
20YR
25YR
30YR
Term
Basisspread(bps)
Eonia vs 1M
Eonia vs 3M
Eonia vs 6MEonia vs 12M
1M vs 3M
1M vs 6M
1M vs 12M
3M vs 6M
3M vs 12M
6M vs 12M
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Interest rate market segmentation[8]
Apparently similar interest rate instruments with different underlying rate tenors arecharacterised, in practice, by different liquidity and credit risk premia, reflecting thedifferent 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 inwhich an unique short rate process was ableto model and explain the whole term structureof interest rates of all tenors, towards a sort
of market segmentation into sub-areascorresponding to instruments with differentunderlying rate tenors, characterised,in principle, by distinct dynamics,e.g. different short rate processes.
Notice that market segmentation was already present (and well understood) beforethe credit crunch (see e.g. B. Tuckman, P. Porfirio, Interest Rate Parity, Money Market Basis Swaps, andCross-Currency Basis Swaps, Lehman Brothers, Jun. 2003) but not effective due to negligible
basis spreads.
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Counterparty risk and collateral[1]
Typical financial transactions generate streams of future cashflows, whose total netpresent value (NPV = algebraic sum of all discounted expected cashflows) implies acredit exposure between the two counterparties.If, for counterparty A, NPV(A)>0=> counterparty A expects to (globally) receive futurecashflows from counterparty B (A has a credit with B), and, on the other side,
counterparty B has NPV(B)
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Counterparty risk and collateral[3]
ISDA Master Agreemento 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 mutualtransactions)
Credit Support Annex:
accessory document to the ISDA Master Agreement that establish the collateral andmargination 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 transferof the collateral).
(source: F. Ametrano, M. Paltenghi, Risk Italia Nov. 2009, ref. [XII])
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Counterparty risk and collateral[4]
CSA characteristics:
Exposure: potential loss that the creditor would suffer in case of default of thecounterparty before trade maturity. It is measured in terms of cost of replacement thecost 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 thecollateral 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.
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Counterparty risk and collateral[5]
CSA characteristics (contd):
Valuation agent: the counterparty that calculates the exposure and the collateral formargination; if not specified, the burden lies with the counterparty that calls theCollateral.
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 counterpartythe exposure and the collateral to be exchanged.
Interest rate: the rate of remuneration of the collateral; normally it is the flat overnightrate in the base currency.
Dispute resolution: how to redeem any disagreements on the exposure and collateralvaluation.
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Counterparty risk and collateral[6]
Collateral Cons
Funding volatility sensitivity
Possible liquidity squeeze
Structural and running costs
Operational risks: settlement andMTM mismatch
Legal risk
Collateral Pros
Counterparty risk reduction
Credit management optimization
Capital ratios reduction
(Basilea II) Increased business opportunities
Funding at overnight rate
Periodic check of credit exposure
and portfolio NPV
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Counterparty risk and collateral[7]
Euribor Rate (risk free)
Counterparty 1
Counterparty 2
Banks Funding Rate
Counterparty CDS Rate
CVA (Banks side)
CVA
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Counterparty risk and collateral[8]
Euribor Rate (includes both credit andliquidity issues among Euribor Banks)
Banks Funding Rate (no CSA)(includes Banks cost of liquidity over CDS)
Eonia Rate (CSA, risk free)
Counterparty 1
Counterparty 2
CVA (bilateral)
Liquidity
Value
Adjustment ?
Counterparty CDS Rate
Banks CDS Rate (includes theBanks default risk)
CVA (Ctp
side) CVA (Banks side)
RatesCVA
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Eonia Discounting or Not ?[1]
Is the market discounting at Eonia ?Interest
RateSwaps
Interest RateCMS andOptions
Inflation Credits Equity Commodities
Intra-day YES NO (?) ? NO (?) NO (?) NO (?)
End of day NO (?) NO (?) NO (?) NO (?) NO (?) NO (?)
Collateral NO NO NO NO NO NO
Balancesheet
NO NO NO NO NO NO
Markit NO NO NO NO NO NO
SwapClear NO -- -- -- -- --
ICAP YES NO ? -- -- --
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Eonia Discounting or Not ?[2]
Main Broker Methodology
Instrument Present methodology Future methodology
Swap forwarding = Euribor xM
discounting = Euribor xM
forwarding = Euribor xM
discounting = Eonia
Basis Swap forwarding1 = Euribor xM1
forwarding2 = Euribor xM2
discounting = Min(EuriborxM1,EuriborxM2)
forwarding1 = EuriborxM1
forwarding2 = 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/Swaptions
forwarding = Euribor xM
discounting = Euribor xM
forwarding = Euribor xM
discounting = Eonia
Forward premium ?
Eonia options ?
1: Context & Market Practices:Eonia Discounting or Not ? [3]
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Eonia Discounting or Not ?[3]
Market Phase Transition
Possible TriggersSwapClear (London Clearing House)
Main Brokers
Currencies EUR (Eonia), USD (Fed Fund rate) at the beginning
IR Swaps Spot starting: NPV = 0, constant swap rateForward starting: NPV = 0, variable forward swap rate
IR OptionsConstant premiums, variable Blacks implied volatility, variable smile(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 Blacks implied volatility
Equity Options Constant premiums and dividends, variable Blacks implied volatility
CDS NPV = 0, variable default probability
BondsConstant prices, variable credit spread absorbing the liquidity/credit
risk inside Xibor
1: Context & Market Practices:Single Curve Pricing & Hedging IR Derivatives
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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 yieldcurve C using the preferred bootstrapping procedure (pillars, priorities, interpolation,
etc.); for instance, a very common choice in the EUR market was a combination ofshort-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 workout the prices by summing up the discounted cashflows;
compute the delta sensitivity and hedge the resulting delta risk using the suggestedamounts (hedge ratios) of the same set of vanillas.
1: Context & Market Practices:Multiple-Curve Pricing & Hedging IR Derivatives
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Post credit-crunch multiple curve market practice:
build a single discounting curve Cdusing the preferred bootstrapping procedure; build multiple distinct forwarding curves Cf1 Cfnusing the preferred distinct
selections of vanilla interest rate instruments, each homogeneous in the underlyingrate tenor (typically 1M, 3M, 6M, 12M);
compute the forward rates with tenor fon the corresponding forwarding curve Cfandcalculate the corresponding cashflows;
compute the corresponding discount factors using the discounting curve Cdand workout prices by summing the discounted cashflows;
compute the delta sensitivity and hedge the resulting delta risk using the suggestedamounts (hedge ratios) of the corresponding set of vanillas.
Multiple-Curve Pricing & Hedging IR Derivatives
2: Multiple-Curve Framework:Basic Assumptions and notation [1]
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Basic Assumptions and notation[1]
1. There exist multiple different interest rate sub-markets Mx
, x = {d,f
1
,,fn
}
characterized by the same currency and by distinct bank accounts Bx
and yieldcurves in the form of a continuous term structure of discount factors
where t0
is the reference date of the curves (e.g. settlement date, or today) and
Px
(t,T)
denotes the price at time tt0
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
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
).
( ) ( ) ( )1 12 2, , , ,x x xP t T P t T P t T T =
( ){ }0 0: , , ,x xC T P t T T t =
2: Multiple-Curve Framework:Basic Assumptions and notation [2]
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Basic Assumptions and notation[2]
3. We denote with Fx
(t; T
1
; T
2
)
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
wherex
(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
( )
( )
( ) ( ) ( )
2
1 21 1 2 1 2
, 1
, , : ,, 1 ; , ,
x
x x x x
P t T
P t T T P t T F t T T T T = =
+
( )( ) ( )
( )
( ) ( )
( )
1 21 2 1 2
1 2
1 2 2
1 1; , 1
, , ,
1 , ,
, ,
xx x
x x
x x
F t T T T T P t T T
P t T P t T
T T P t T
=
=
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Basic Assumptions and notation[3]
4. The eq. above can be also derived (see e.g. ref. [A], sec. 1.4) as the fair valuecondition at time t
of the Forward Rate Agreement (FRA) contract with payoff at
maturity T2
given by
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 conventionfor simplicity). Introducing expectations we have, tT1
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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 Cfas
3. calculate cashflows ci
, i = 1,...,n, as expectations of the i-th coupon payoff i
withrespect to the discounting Ti-
forward measure
4. calculate the price
at time t by discounting each cashflow ci using thecorresponding discount factor obtained from the discounting curve Cdandsumming,
5. Price FRAs as
( ),d iP t T
TidQ
( )( ) ( )
( ) ( )
11 1
1
, ,; , , ,
, ,
f i f i f i i i i
f i i f i
P t T P t T F t T T t T T
T T P t T
=