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

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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4/66Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 4

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]



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.




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.



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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]).



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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

1: Context & Market Practices:Euribor interest rate[1]


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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 C & M k P i


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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

1 C t t & M k t P ti


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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

1 C t t & M k t P ti


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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:


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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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EUR 3M OIS rates vs 3M Depo rates

Quotations

Dec. 2005 -

May

2010 (source: Bloomberg)



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EUR 6M OIS rates vs 6M Depo rates

Quotations

Dec. 2005

Apr. 2010

(source: Bloomberg)



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EUR 3x6 FRA vs 3x6 fwd OIS rates

Quotations

Dec. 2005

Apr. 2010

(source: Bloomberg)



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EUR 6x12 FRA vs 6x12 fwd OIS rates

Quotations

Dec. 2005

Apr. 2010

(source: Bloomberg)



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EUR Basis Swap 5Y, 3M vs 6M

Quotations

May

2005

Apr. 2010

(source: Bloomberg)



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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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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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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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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.

1: Context & Market Practices:C i k d ll l


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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

1: Context & Market Practices:C t t i k d ll t l


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Euribor Rate (risk free)

Counterparty 1

Counterparty 2

Banks Funding Rate

Counterparty CDS Rate

CVA (Banks side)

CVA

1: Context & Market Practices:C t t i k d ll t l


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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

1: Context & Market Practices:Eonia Disco nting or Not ? [ ]


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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 ? -- -- --

1: Context & Market Practices:Eonia Discounting or Not ? [2]


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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


discounting = Euribor xM


discounting = Eonia

Forward premium ?

Eonia options ?

1: Context & Market Practices:Eonia Discounting or Not ? [3]


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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 =



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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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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

=

marcobianchettimultiplecurvesonepricev12-12745509596722-phpapp01

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