1.MULTIPLE CURVES, ONE PRICE The Post Credit-Crunch Interest Rate Market Global Derivatives Paris,17-21 May 2010Marco Bianchetti Intesa Sanpaolo Bank, Risk Management, Market Risk, Pricing & Financial Modelling marco.bianchetti intesasanpaolo.com

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 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 3. 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 referencesMultiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 3 4. 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 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 on15 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 sizejust prior to 11 am (GMT)?; o rate fixings are calculated, for each maturity, as the average of ratessubmissions after discarding highest and lowest quartiles (25%); o published around 11:45 a.m. (GMT), annualised rate, act/360 (Reuters pageLIBOR); o calculation agent: Reuters. Currencies: GBP, USD, JPY, CHF, CAD, AUD, EUR, DKK, SEK, NZD.Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 4 5. 1: Context & Market Practices: Libor interest rate [2]Libor definition amplifiedthe rate at which each bank submits must be formed from that banks 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 banks cash, rather than a banks derivative book; the definition of funds is: unsecured interbank cash or cash raised through primary issuance of interbank Certificates of Deposit. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 5 6. 1: Context & Market Practices: Libor interest rate [3]Libor panelsComposition 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 activityReputationPerceived 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.Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 6 7. 1: Context & Market Practices: Libor interest rate [4] Libor panels per currencyBanks AUD CAD CHF EUR GBP JPY USD DKK NZD SEK PanelsAbbey National X1 Bank of America X X X X4 Bank of MontrealX1 Bank of Nova Scotia X1 Bank of Tokyo-Mitsubishi UFJ LtdX X X X X5Barclays Banks plc X X X X X X X X X X10 BNP Paribas X1 Canadian Imperial Bank of CommerceX1 Citibank NA X X X X X5 Commonwealth Bank of AustraliaX X2 Credit Suisse X X X3 Deutsche Bank AGX X X X X X X X X X10HSBC X X X X X X X X X9JP Morgan ChaseX X X X X X X X X9Lloyds Banking Group X X X X X X X X X X10 Mizuho Corporate Bank X X X3 National Australia Bank X X2National Bank of CanadaX1 Norinchukin BankX X2Rabobank X X X X X X X X8Royal Bank of Canada X X X X4Royal Bank of Scotland Group X X X X X X X X X X10 Socit GnraleX X X X4 Sumitomo Mitsui X1 UBS AGX X X X X X X X X9 WestLB AG X X X X4Totals 8 1212161616 16 8 8 8 Source: www.bbalibor.org, 31 Mar. 2010 Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010p. 7 8. 1: Context & Market Practices: Libor interest rate [5]Libor questioned during the crisisThe 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 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 LIBOR remains 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]). Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010p. 8 9. 1: Context & Market Practices: Euribor interest rate [1] 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 apanel of European Banks for rate fixing on 15 maturities (1w-12M): what ratedo you believe one prime bank is quoting to another prime bank for interbankterm deposits within the euro zone?; o rate fixings are calculated, for each maturity, as the average of ratessubmissions 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: EURMultiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 9 10. 1: Context & Market Practices: Euribor interest rate [2]Euribor panelComposition 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.Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 10 11. 1: Context & Market Practices: Euribor interest rate [3]Euribor panelBank CountryBank CountryErste Bank der sterreichischen Sparkassen RZBAIB GroupAustria IrelandRaiffeisen Zentralbank sterreich AGBank of IrelandDexia BankIntesa SanpaoloFortis Bank Belgium Unicredit Italy KBCMonte dei Paschi di SienaNordeaFinland Banque et Caisse d'pargne de l'tat Luxembourg BNP - ParibasRBS N.V. NatixisRabobank NetherlandsSocit GnraleING BankFranceCrdit Agricole s.a.Caixa Geral De Depsitos (CGD) Portugal HSBC FranceBanco Bilbao Vizcaya ArgentariaCrdit Industriel et Commercial CIC Confederacion Espaola de Cajas de Ahorros Spain Landesbank BerlinBanco Santander Central Hispano WestLB AGLa Caixa BarcelonaBayerische Landesbank GirozentraleBarclays Capital Other EUCommerzbank Den Danske BankBanksDeutsche BankGermanySvenska HandelsbankenDZ Bank Deutsche GenossenschaftsbankBank of Tokyo - MitsubishiLandesbank Baden-Wrttemberg Girozentrale J.P. Morgan Chase & Co.International Norddeutsche Landesbank Girozentrale CitibankBanksLandesbank Hessen - Thringen GirozentraleUBS (Luxembourg) S.A. National Bank of GreeceGreece Source: www.euribor.org, 31 Mar. 2010Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 11 12. 1: Context & Market Practices: 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 ofovernight unsecured lending transactions of that day and the weighted averagelending rate for these transactions; o rate fixing is calculated as the transaction volumes weighted average of ratessubmissions; 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 Rateo GBP: Sonia = Sterling Over Night Index Averageo JPY: Mutan rate Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 12 13. 1: Context & Market Practices: Xibor/Eonia interest rates discussion [1] Xibor discussionXibor 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 BasisSwaps, 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.Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010p. 13 14. 1: Context & Market Practices: Xibor/Eonia interest rates discussion [2] Eonia discussionEonia 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 banksin 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].Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 14 15. 1: Context & Market Practices:Xibor/Eonia interest rates discussion [3] LiborEuribor EoniaLondon InterBank Euro InterBankEuro OverNight DefinitionOffered RateOffered Rate Index Average Market London Interbank Euro InterbankEuro InterbankSide OfferOffer OfferEURLibor = Euribor, TARGET calendar, T+2, TARGET calendar, T+0, Rate quotation slight differences for otheract/360, three decimalact/360, three decimal specs currencies (e.g. act/365, T+0,places, tenor variable.places, tenor 1d.London calendar for GBPLibor). Maturities1d-12m 1w, 2w, 3w,1m,,12m1dPublication time 12.30 CET 11:00 am CET6:45-7:00 pm CET 39 banks from 15 EU8-16 banks (London based)Panel bankscountries + 4Same as Euribor per currencyinternational banksCalculation agentReutersReutersEuropean Central Bank Transactions based NoNoYesCounterparty riskYes Yes NegligibleLiquidity risk Yes Yes NegligibleTenor basisYes Yes No Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010p. 15 16. 1: Context & Market Practices: 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. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 16 17. 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 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. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 17 18. 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 by consecutive deposits.Explosion of basis swap rates (based on Xibor rates with different tenors)Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 18 19. 1: Context & Market Practices: Interest rate market segmentation [2]EUR 3M OIS rates vs 3M Depo ratesQuotations Dec. 2005 - May 2010 (source: Bloomberg)Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 19 20. 1: Context & Market Practices: Interest rate market segmentation [3]EUR 6M OIS rates vs 6M Depo ratesQuotations Dec. 2005 Apr. 2010 (source: Bloomberg)Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 20 21. 1: Context & Market Practices: Interest rate market segmentation [4] EUR 3x6 FRA vs 3x6 fwd OIS ratesQuotations Dec. 2005 Apr. 2010 (source: Bloomberg)Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 21 22. 1: Context & Market Practices: Interest rate market segmentation [5]EUR 6x12 FRA vs 6x12 fwd OIS ratesQuotations Dec. 2005 Apr. 2010 (source: Bloomberg)Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 22 23. 1: Context & Market Practices: Interest rate market segmentation [6] EUR Basis Swap 5Y, 3M vs 6MQuotations May 2005 Apr. 2010 (source: Bloomberg)Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 23 24. 1: Context & Market Practices: Interest rate market segmentation [7]65 Eonia vs Euribor (31.03.2010)Eonia vs 1MEonia vs 3M60Eonia vs 6MEonia vs 12M551M vs 3M1M vs 6M501M vs 12M3M vs 6M453M vs 12M6M vs 12M40 Basis spread (bps)35 30 25 20 15 1050 -5 1YR 2YR3YR4YR 5YR 6YR 7YR8YR 9YR10YR 11YR 12YR15YR 20YR 25YR30YRTerm EUR Basis Swaps Quotations as of 31 Mar. 2010 (source: Reuters, ICAP)Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 24 25. 1: Context & Market Practices: Interest rate market segmentation [8] 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. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010p. 25 26. 1: Context & Market Practices: 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)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 , f (T1,T2 ) Pd ( t,T2 ) R ( t; t,T2, Rf ) where:R ( t;T1,T2, Rf ) = Rf + ( 1 Rf ) EtQd [ qd (T1,T2 ) ]. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 50 51. 3: Foreign Currency Analogy: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 f (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 rate1Pd ( t,T1 ) R ( t; t,T1, Rf ) BAfd ( t;T1,T2 ) = 1 ,d (T1,T2 ) Pd ( t,T2 ) R ( t; t,T2, Rf )1Pd ( t,T1 ) 1R ( t; t,T1, Rf ) QAfd ( t;T1,T2 ) = . f (T1,T2 ) Pd ( t,T2 ) R ( t;T1,T2, Rf ) R ( t; t,T2, Rf ) Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 51 52. 3: Foreign Currency Analogy: Pros & ConsPROsCONsSimple and familiar framework, Plain vanilla prices acquire no additional effort, just analogy.volatility and correlationdependence (but this is commonto all approaches).Convexity adjustment emerges Parameters (exchange rate and naturally, as in more complexits volatility) presently not approaches.observable on the market,historical estimate only.Straightforward interpretation inUsual CONs of FX quanto terms of counterparty risk.adjustment applies (e.g. no smile)Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 52 53. 4: Hedging:Multiple Delta Hedging [1] 1. Given any portfolio of interest rate derivatives with price ( t,T , Rmkt ), compute thedelta risk with respect to all curves C = {Cd ,C f 1 ,...,C fn } as RNC ( t,T , Rk ) mktNC N k ( t,T , R ) = ( mkt) = mktt,T , Rk mkt ,k =1 k =1 j =1 Rk , jmktRwhere NC is the number of yield curves involved and Rk is the vector of N kbootstrapping market data (yield curve pillars) associated to yield curve Ck .2. Be careful to take properly into account all the delta components due to multiplecurve bootstrapping: the forwarding zero curves {C f 1 ,...,C fn } , in particular, dependdirectly on their corresponding input market instruments with tenor f, but alsoindirectly on the discounting curve, RZ One price, two curves, ( t,T , Rd ) Z Nd Nd mkt d ( t,T , Rd ) = mktd mkt,three deltas (see ref.[8])j =1 =1 Z Rd , jN fR N fZ Z ( t,T , Rf ) Z mkt f Nd N f ( t,T , Rf ) Z mktf ( t,T , Rf ) = mktf +,j =1 =1 Z Rfmkt,jj =1 =1 Z f mktRd , j Zwhere Z f is the vector of N f zero rates pillars in the zero rate curve Cf.Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 53 54. 4: Hedging: Multiple Delta Hedging [2] 3. eventually aggregate the delta sensitivity on the selected subset H of the most liquidmarket instruments used for hedging RH = { R1 ,..., RN H } (hedging instruments);HH NH ( t,T , RH) ( t,T , R )H RjH ,j =1 ( t ,T , R H) 4. calculate hedge ratios: h j ( t ,T , RH) = jH ( t ) , R jH H ( t , T j , R jH j) jH ( t , T j , R jH )=.H R jH 5. where j ( t ) is the market price (unit nominal) of the corresponding hedginginstrument, such that the hedged portfolio has zero delta NH tot ( t ,T , R ) =H ( t ,T , RH) h j ( t ) H ( t , T j , R jH ) , j j =1 ( t ,T , R H NH ) NH H ( t , T j , R jH ) h j (t ) j tot ( t ,T , R H ) Rk H Rk H k =1 j =1NH ( t ,T , R H )= h k ( t ) k ( t ) = 0,H k =1 Rk H Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 54 55. 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]). Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 55 56. 5: Other Approaches:Axiomatic ApproachSee ref. [8]: M. Henrard, The Irony in the Derivatives Discounting - Part II: TheCrisis, Jul. 2009, SSRN working paper http://ssrn.com/abstract=1433022.Individuate the minimal set of definitions and assumptions for coherent multiple yieldcurve pricing of plain vanillas (FRA, IRS).Find a (convexity) adjustment factor for FRA/Futures.Discuss delta hedging. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 56 57. 5: Other Approaches:Extended Libor Market Model Approach [1]See ref. [9]: F. Mercurio, "Post Credit Crunch Interest Rates: Formulas and MarketModels", Bloomberg, Jan. 2009, and ref. [17]: F. Mercurio, LIBOR Market Modelswith Stochastic Basis, Mar. 2010.Assume as the fundamental bricks the OIS forward rate, the FRA rate and theirspread, defined as, respectively, x ( )xxTQd kxx 1 Pd ( t,Tkx1 ) Fk t := Fd ( t;Tk 1,Tk ) = Et [ Ld (Tk 1,Tk ) ] = x 1 ,k Pd ( t,Tkx ) T Qd k Lx ( t )k := FRA ( t;Tkx1,Tkx) :=Et [ Lx (Tkx1,Tkx ) ], Sk ( t ) := Lx ( t ) Fkx ( t ) .xkIn ref. [17] assume, in the single FRA rate tenor case, general stochastic volatilitydynamics for each OIS forward and spread, uncorrelated, derive pricing expressionfor caplets and swaptions as integrals over the full smile structure, and provideexamples using various specific dynamics.Address the multi FRA rate tenor case finding the simplest stochastic-volatilitydynamics that preserve consistency across different tenors. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 57 58. 5: Other Approaches:Extended Libor Market Model Approach [2]See ref. [13]: M. Fujii, Y. Shimada, A. Takahashi, A Market Model of Interest Rateswith 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 theirspread.Assume model stochastic basis spreads in a HJM framework.Address both single- and multi-currency cases. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 58 59. 5: Other Approaches:Extended Short Rate Model ApproachSee 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 andgovernative short rates.Find approximated analytic pricing expression for european caps/floors/swaptionsand 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 shortrates, plus an uncorrelated lognormal process for the exchange rate.Find a pseudo-analytic european swaption pricing expression. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 59 60. 5: Other Approaches:Counterparty Risk ApproachSee 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 Liborreplication can be explained by using the quoted Basis Swap spreads.Explain the market patterns of the Basis Swap spreads by modelling them as optionson the credit worthiness of the counterparty.Formalize the mathematical representation of the post-credit crunch interest ratemarket, introducing credit risk, allowing for no-fault standard rule andcollateralization, and using subfiltrations to model risky Libor rates.Compute change of numeraire and convexity adjustments for collateralizedderivatives tied to risky Libor, thus explaining the quanto-adjustment found byBianchetti in ref. [7] in a simpler framework. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 60 61. 6: Conclusions1. 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-curveno arbitrage conditions are broken and can be recovered taking into account theforward basis; once a smooth bootstrapping technique is used, the richer termstructure of the calculated forward basis curves provides a sensitive indicator of thetiny, but observable, statical differences between different interest rate market sub-areas. 3. Using the foreign-currency analogy we have computed the no arbitrage generaliseddouble-curve-single-currency market-like pricing expressions for basic interest ratederivatives, including a quanto adjustment arising from the change of numerairesnaturally associated to the two yield curves. Numerical scenarios show that thequanto adjustment can be non negligible. 4. Both the forward basis and the quanto adjustment have a simple interpretation interms of counterparty risk, using a simple credit model with a risk-free and a riskyzero coupond bonds. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 61 62. 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 derivato dalla crisi, Risk Italia, 26 Nov. 2009. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 62 63. 7: Main references [2]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). Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 63 64. 7: Main references[3]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--BusinessSchool of Finance & Management, May 2005. 3. M. Henrard, The Irony in the Derivatives Discounting, Mar. 2007, SSRN working paper,http://ssrn.com/abstract=970509. 4. M. Johannes, S. Sundaresan, The Impact of Collateralization on Swap Rates, Journal ofFinance 62, pages 383410, 2007. 5. M. Kijima, K. Tanaka, T. Wong, A Multi-Quality Model of Interest Rates, QuantitativeFinance, vol. 9, issue 2, pages 133-145, 2008. 6. F. Ametrano, M. Bianchetti, Bootstrapping the Illiquidity: Multiple Yield Curves ConstructionFor Market Coherent Forward Rates Estimation, in Modeling Interest Rates: LatestAdvances for Derivatives Pricing, edited by F. Mercurio, Risk Books, 2009. 7. M. Bianchetti, Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives UsingDifferent Yield Curves for Discounting and Forwarding, Jan. 2009, SSRN working paper,http://ssrn.com/abstract=1334356. 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 workingpaper, http://ssrn.com/abstract=1394267. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 64 65. 7: Main references [4]10. M. Henrard, The Irony in the Derivatives Discounting - Part II: The Crisis, Jul. 2009, SSRN working paper http://ssrn.com/abstract=1433022. 11. M. Morini, Solving the Puzzle in the Interest Rate Market, Oct. 2009, SSRN working paper, http://ssrn.com/abstract=1506046. 12. R. Preusser, Euribor vs Eonia: Clash of the Titans, Deutsche Bank, Fixed Income Special Report, Nov. 2009. 13. M. Fujii, Y. Shimada, A. Takahashi, A Survey on Modeling and Analysis of Basis Spreads, Nov. 2009, SSRN working paper, http://ssrn.com/abstract=1520619. 14. 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, SSRN working paper, http://ssrn.com/abstract=1520618. 15. M. Fujii, Y. Shimada, A. Takahashi, A Note on Construction of Multiple Swap Curves with and without Collateral, Jan. 2010, SSRN working paper, http://ssrn.com/abstract=1440633. 16. M. Fujii, Y. Shimada, A. Takahashi, On the Term Structure of Interest Rates with Basis Spreads, Collateral and Multiple Currencies, Jan. 2010, SSRN working paper, http://ssrn.com/abstract=1556487. 17. V. V. Piterbarg, Funding beyond discounting: collateral agreements and derivatives pricing, Risk, Feb. 2010, http://www.risk.net/digital_assets/735/piterbarg.pdf. 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. 19. F. Mercurio, LIBOR Market Models with Stochastic Basis, Mar. 2010, SSRN working paper, http://ssrn.com/abstract=1563685. Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010 p. 65 66. 7: Main references[5]20. A. Pallavicini, M. Tarenghi, Interest Rate Modelling with Multiple Yield Curves, forthcoming. papers per semester 6 Technical papers5 432 1 01 20072 20071 20082 20091 2009 2 2009 1 2010semester Multiple Curves, One Price - Marco Bianchetti Global Derivatives 2010p. 66