Multiple Curves, One Price

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Pricing and Hedging Interest Rate Derivatives in the The Post Credit-Crunch Interest Rate Market

Multiple Curves, One Price

  1. 1. MULTIPLE CURVES, ONE PRICE The Post Credit-Crunch Interest Rate Market Global Derivatives Paris,17-21 May 2010 Marco Bianchetti Intesa Sanpaolo Bank, Risk Management, Market Risk, Pricing & Financial Modelling marco.bianchetti intesasanpaolo.com
  2. 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. 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 references “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 3
  4. 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 Banker’s Association (BBA, see http://www.bbalibor.com), o reference rate mentioned in ISDA standards for OTC transactions. Fixing mechanics: o each TARGET business day the BBA polls a panel of Banks for rate fixing on 15 maturities (1d-12M): “at what rate could you borrow funds, were you to do so by asking for and then accepting inter-bank offers in a reasonable market size just prior to 11 am (GMT)?”; o rate fixings are calculated, for each maturity, as the average of rates submissions after discarding highest and lowest quartiles (25%); o published around 11:45 a.m. (GMT), annualised rate, act/360 (Reuters page “LIBOR”); o calculation agent: Reuters. Currencies: GBP, USD, JPY, CHF, CAD, AUD, EUR, DKK, SEK, NZD. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 4
  5. 5. 1: Context & Market Practices: Libor interest rate [2] Libor definition amplified the rate at which each bank submits must be formed from that bank’s perception of its cost of funds in the interbank market; contributions must represent rates formed in London Market and not elsewhere; contributions must be for the currency concerned, not the cost of producing one currency by borrowing in another currency and accessing the required currency via the foreign exchange markets; the rates must be submitted by members of staff at a bank with primary responsibility for management of a bank’s cash, rather than a bank’s derivative book; the definition of “funds” is: unsecured interbank cash or cash raised through primary issuance of interbank Certificates of Deposit. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 5
  6. 6. 1: Context & Market Practices: Libor interest rate [3] Libor panels Composition 8-12-16 contributors per currency (a multiple of 4 because of the average calculation rule above); Selection criteria: o Guiding principle: “Banks chosen by the independent Foreign Exchange and Money Markets Committee to give the best representation of activity within the London money market for a particular currency”; o Criteria: Scale of market activity Reputation Perceived expertise in the currency concerned Review: annual review by BBA with FX & MM Committee; all panels and proposed banks are ranked according to their total money market and swaps activity over the previous year and selected according to the largest scale of activity with due concern given to the other 2 criteria. Sanctions: warning and successively exclusion from the panel. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 6
  7. 7. 1: Context & Market Practices: Libor interest rate [4] Libor panels per currency Banks AUD CAD CHF EUR GBP JPY USD DKK NZD SEK Panels Abbey National X 1 Bank of America X X X X 4 Bank of Montreal X 1 Bank of Nova Scotia X 1 Bank of Tokyo-Mitsubishi UFJ Ltd X X X X X 5 Barclays Banks plc X X X X X X X X X X 10 BNP Paribas X 1 Canadian Imperial Bank of Commerce X 1 Citibank NA X X X X X 5 Commonwealth Bank of Australia X X 2 Credit Suisse X X X 3 Deutsche Bank AG X X X X X X X X X X 10 HSBC X X X X X X X X X 9 JP Morgan Chase X X X X X X X X X 9 Lloyds Banking Group X X X X X X X X X X 10 Mizuho Corporate Bank X X X 3 National Australia Bank X X 2 National Bank of Canada X 1 Norinchukin Bank X X 2 Rabobank X X X X X X X X 8 Royal Bank of Canada X X X X 4 Royal Bank of Scotland Group X X X X X X X X X X 10 Société Générale X X X X 4 Sumitomo Mitsui X 1 UBS AG X X X X X X X X X 9 WestLB AG X X X X 4 Totals 8 12 12 16 16 16 16 8 8 8 Source: www.bbalibor.org, 31 Mar. 2010 “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 7
  8. 8. 1: Context & Market Practices: Libor interest rate [5] Libor questioned during the crisis The Bank for International Settlements reported that "available data do not support the hypothesis that contributor banks manipulated their quotes to profit from positions based on fixings“ (see J. Gyntelberg, P. Wooldridge, “Interbank rate fixings during the recent turmoil”, BIS Quarterly Review, Mar. 2008, ref. [III]). Risk Magazine reported rumors that “Libor rates are still not reflective of the true levels at which banks can borrow” (see P. Madigan, “Libor under attack”, Risk, Jun. 2008, ref. [VI]) The Wall Street Journal reported that some banks “have been reporting significantly lower borrowing costs for the Libor, than what another market measure suggests they should be” (see C. Mollenkamp, M. Whitehouse, The Wall Street Journal, 29 May 2008, ref. [V]). The British Banker’s Association commented that Libor continues to be reliable, and that other proxies are not necessarily more sound than Libor at times of financial crisis. The International Monetary Fund reported that "it appears that U.S. dollar LIBOR remains an accurate measure of a typical creditworthy bank’s marginal cost of unsecured U.S. dollar term funding“ (see Global Financial Stability Report, Oct. 2008, ch. 2, ref. [VII]). “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 8
  9. 9. 1: Context & Market Practices: Euribor interest rate [1] Euribor definition and mechanics (source: www.euribor.com, 31th March 2010) Euribor = Euro Interbank Offered Rate o first published on 30 Dec. 1998; o sponsored by the European Banking Federation (EBF) and by the Financial Markets Association (ACI). Fixing mechanics: o each TARGET business day the European Banking Federation (EBF) polls a panel of European Banks for rate fixing on 15 maturities (1w-12M): “what rate do you believe one prime bank is quoting to another prime bank for interbank term deposits within the euro zone?”; o rate fixings are calculated, for each maturity, as the average of rates submissions after discarding highest and lowest 15%; o published at 11:00 a.m. (CET) for spot value (T+2), annualised rate, act/360, three decimal places (Reuters page “EURIBOR=”); o calculation agent: Reuters Currencies: EUR “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 9
  10. 10. 1: Context & Market Practices: Euribor interest rate [2] Euribor panel Composition on Mar. 2010: 39 banks from 15 EU countries + 4 international banks; Selection criteria: o “…active players in the euro money markets in the euro-zone or worldwide and if they are able to handle good volumes in euro-interest rate related instruments, especially in the money market, even in turbulent market condition”; o “first class credit standing, high ethical standards and enjoying an excellent reputation”; Review: ”periodically reviewed by the Steering Committee to ensure that the selected panel always truly reflects money market activities within the euro zone”. Banks obligations: o must quote "the best price between the best banks“, “for the complete range of maturities”, “on time”, “daily”, “accurately”; o must make “the necessary organisational arrangements to ensure that delivery of the rates is possible on a permanent basis without interruption due to human or technical failure”. Sanctions: warning and successively exclusion from the panel. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 10
  11. 11. 1: Context & Market Practices: Euribor interest rate [3] Euribor panel Bank Country Bank Country Erste Bank der Österreichischen Sparkassen RZB AIB Group Austria Ireland Raiffeisen Zentralbank Österreich AG Bank of Ireland Dexia Bank Intesa Sanpaolo Fortis Bank Belgium Unicredit Italy KBC Monte dei Paschi di Siena Nordea Finland Banque et Caisse d'Épargne de l'État Luxembourg BNP - Paribas RBS N.V. Natixis Rabobank Netherlands Société Générale ING Bank France Crédit Agricole s.a. Caixa Geral De Depósitos (CGD) Portugal HSBC France Banco Bilbao Vizcaya Argentaria Crédit Industriel et Commercial CIC Confederacion Española de Cajas de Ahorros Spain Landesbank Berlin Banco Santander Central Hispano WestLB AG La Caixa Barcelona Bayerische Landesbank Girozentrale Barclays Capital Other EU Commerzbank Den Danske Bank Banks Deutsche Bank Germany Svenska Handelsbanken DZ Bank Deutsche Genossenschaftsbank Bank of Tokyo - Mitsubishi Landesbank Baden-Württemberg Girozentrale J.P. Morgan Chase & Co. International Norddeutsche Landesbank Girozentrale Citibank Banks Landesbank Hessen - Thüringen Girozentrale UBS (Luxembourg) S.A. National Bank of Greece Greece Source: www.euribor.org, 31 Mar. 2010 “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 11
  12. 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 of overnight unsecured lending transactions of that day and the weighted average lending rate for these transactions; o rate fixing is calculated as the transaction volumes weighted average of rates submissions; o published at 6:45-7:00 p.m. (CET) for today value (T+0), annualised rate, act/360, three decimal places (Reuters page “EONIA=“). o Calculation agent: European Central Bank Overnight rates in other currencies: o USD: Federal Funds Effective Rate o GBP: Sonia = Sterling Over Night Index Average o JPY: Mutan rate “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 12
  13. 13. 1: Context & Market Practices: Xibor/Eonia interest rates discussion [1] Xibor discussion Xibor is based on: o offered rates on unsecured funding; o expectations, views and beliefs of the panel banks about borrowing rates in the currency money market (see e.g. P. Madigan, “Libor under attack”, Risk, Jun. 2008, ref. [VI]). As any interest rate expectation, Xibor includes informations on: o the counterparty credit risk/premium, o the liquidity risk/premium and thus its not a risk free rate, as already well known before the crisis (see e.g. B. Tuckman, P. Porfirio, “Interest Rate Parity, Money Market Basis Swaps, and Cross-Currency Basis Swaps”, Lehman Brothers, Jun. 2003, ref. [1]). Lending/borrowing Xibor rates is tenor dependent: “The age of innocence – when banks lent to each other unsecured for three months or longer at only a small premium to expected policy rates – will not quickly, if ever, return” (M. King, Bank of England Governor, 21 Oct. 2008). The Xibor panel may change over time, panel banks may be replaced by other banks with higher credit standing. Borrowers and lenders will not be Xibor forever. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 13
  14. 14. 1: Context & Market Practices: Xibor/Eonia interest rates discussion [2] Eonia discussion Eonia is based on unsecured lending (offer side) transactions of the panel banks in the Euro money market; Eonia is used by ECB as a method of effecting and observing the transmission of the monetary policy actions; Eonia includes informations on: o the monetary policy effects, o the short term liquidity expectations of the panel banks in the Euro money market; Eonia holds the shortest rate tenor available (one day), carries negligible counterparty credit and liquidity risk and thus it is the best available market proxy to a risk free rate. See also Goldman Sachs, “Overview of EONIA and Update on EONIA Swap Market”, Mar. 2010, ref. [XV]. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 14
  15. 15. 1: Context & Market Practices: Xibor/Eonia interest rates discussion [3] Libor Euribor Eonia London InterBank Euro InterBank Euro OverNight Definition Offered Rate Offered Rate Index Average Market London Interbank Euro Interbank Euro Interbank Side Offer Offer Offer EURLibor = Euribor, TARGET calendar, T+2, TARGET calendar, T+0, Rate quotation slight differences for other act/360, three decimal act/360, three decimal specs currencies (e.g. act/365, T+0, places, tenor variable. places, tenor 1d. London calendar for GBPLibor). Maturities 1d-12m 1w, 2w, 3w,1m,…,12m 1d Publication time 12.30 CET 11:00 am CET 6:45-7:00 pm CET 39 banks from 15 EU 8-16 banks (London based) Panel banks countries + 4 Same as Euribor per currency international banks Calculation agent Reuters Reuters European Central Bank Transactions based No No Yes Counterparty risk Yes Yes Negligible Liquidity risk Yes Yes Negligible Tenor basis Yes Yes No “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 15
  16. 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. 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. 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. 19. 1: Context & Market Practices: Interest rate market segmentation [2] EUR 3M OIS rates vs 3M Depo rates Quotations Dec. 2005 - May 2010 (source: Bloomberg) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 19
  20. 20. 1: Context & Market Practices: Interest rate market segmentation [3] EUR 6M OIS rates vs 6M Depo rates Quotations Dec. 2005 – Apr. 2010 (source: Bloomberg) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 20
  21. 21. 1: Context & Market Practices: Interest rate market segmentation [4] EUR 3x6 FRA vs 3x6 fwd OIS rates Quotations Dec. 2005 – Apr. 2010 (source: Bloomberg) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 21
  22. 22. 1: Context & Market Practices: Interest rate market segmentation [5] EUR 6x12 FRA vs 6x12 fwd OIS rates Quotations Dec. 2005 – Apr. 2010 (source: Bloomberg) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 22
  23. 23. 1: Context & Market Practices: Interest rate market segmentation [6] EUR Basis Swap 5Y, 3M vs 6M Quotations May 2005 – Apr. 2010 (source: Bloomberg) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 23
  24. 24. 1: Context & Market Practices: Interest rate market segmentation [7] 65 Eonia vs Euribor (31.03.2010) Eonia vs 1M Eonia vs 3M 60 Eonia vs 6M Eonia vs 12M 55 1M vs 3M 1M vs 6M 50 1M vs 12M 3M vs 6M 45 3M vs 12M 6M vs 12M 40 Basis spread (bps) 35 30 25 20 15 10 5 0 -5 1YR 2YR 3YR 4YR 5YR 6YR 7YR 8YR 9YR 10YR 11YR 12YR 15YR 20YR 25YR 30YR Term EUR Basis Swaps Quotations as of 31 Mar. 2010 (source: Reuters, ICAP) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 24
  25. 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 2010 p. 25
  26. 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)<0 and expects to (globally) pay future cashflows to counterparty A (B has a debt with A). The reverse holds if NPV(A)<0 and NPV(B)>0. Such credit exposure can be mitigated through a guarantee, called collateral. In banking, collateral has two meanings: asset-based lending: the traditional secured lending, with unilateral obligations, secured in the form of property, surety, guarantee or other; capital market collateralization: used to secure trade transactions, with bilateral obligations, secured by more liquid assets such as cash or securities, also known as margin. Capital market collateralization: physical delivery from the debtor to the creditor (with or without transfer of property) of liquid financial instruments or cash with NPV corresponding to the NPV of the trade, as a guarantee for mutual obligations. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 26
  27. 27. 1: Context & Market Practices: Counterparty risk and collateral [2] Collateral mechanics Regulated markets Over the counter markets Not all trades are collateralised, it Collateralisation: All trades are collateralised depends on the agreements between the counterparties Financial highly standardised highly customised instruments: There is a Clearing House that acts There is no Clearing House, direct as counterparty for any trade and interaction between the Clearing House: establish settlement and counterparties, ad hoc contracts margination rules are used Daily settlement and margination, Settlement and Most used contracts are: collateral in cash of main margination ISDA Master Agreement currencies or highly rated bonds execution: Credit Support Annex (CSA) (govies) Collateral Overnight rate Depend on the agreements interest: “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 27
  28. 28. 1: Context & Market Practices: Counterparty risk and collateral [3] ISDA Master Agreement o Standardised contracts proposed and maintained by the International Swaps and Derivatives Association (ISDA). o Widely used by most financial operators to regulate OTC transactions. o Netting clause: counterparties are allowed to calculate the total net reciprocal credit exposure (total NPV = algebraic sum of the NPVs of all mutual transactions) Credit Support Annex: accessory document to the ISDA Master Agreement that establish the collateral and margination rules between the counterparties. There are two main CSA versions: o UK CSA: most used in Europe, with property transfer of the collateral (cash or assets) from the debtor to the creditor, that can freely use it; o US CSA: most used in the US, the collateral (cash or assets) is deposited by the debtor in a vincolated bank account of the creditor (there is no property transfer of the collateral). (source: F. Ametrano, M. Paltenghi, Risk Italia Nov. 2009, ref. [XII]) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 28
  29. 29. 1: Context & Market Practices: Counterparty risk and collateral [4] CSA characteristics: Exposure: potential loss that the creditor would suffer in case of default of the counterparty before trade maturity. It is measured in terms of cost of replacement the cost for the creditor to enter in the same deal with another counterparty. Base currency: reference currency of the contract and of the collateral. Eligible currency: one or more currencies alternative to the base currency. Eligible credit support: the collateral assets agreed by the counterparties, generally cash or AAA bonds (mainly govies). Haircut: valuation percentage applied to the Eligible Credit Support to reduce the collateral asset volatility, proportional to the asset residual life. Independent amount: the amount transferred at CSA inception, indepentent on the NPV dynamics. Threshold: the maximum exposure allowed between two counterparties without CSA; it depends on the credit worthiness of the counterparties. Minimum transfer amount (MTA): the threshold for margination; it depends on the counterparties’ ratings. Rounding: the rounding to be applied to the MTA. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 29
  30. 30. 1: Context & Market Practices: Counterparty risk and collateral [5] CSA characteristics (cont’d): Valuation agent: the counterparty that calculates the exposure and the collateral for margination; if not specified, the burden lies with the counterparty that calls the Collateral. Valuation date: exposure calculation and margination frequency; it may be daily, weekly or monthly; daily margination allows for the best guarantee against credit risk. Notification time: when the Valuation Agent communicates to the other counterparty the exposure and the collateral to be exchanged. Interest rate: the rate of remuneration of the collateral; normally it is the flat overnight rate in the base currency. Dispute resolution: how to redeem any disagreements on the exposure and collateral valuation. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 30
  31. 31. 1: Context & Market Practices: Counterparty risk and collateral [6] Collateral Pros Collateral Cons Counterparty risk reduction Funding volatility sensitivity Credit management optimization Possible liquidity squeeze Capital ratios reduction Structural and running costs (Basilea II) Operational risks: settlement and Increased business opportunities MTM mismatch Funding at overnight rate Legal risk Periodic check of credit exposure and portfolio NPV “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 31
  32. 32. 1: Context & Market Practices: Counterparty risk and collateral [7] CVA Counterparty 2 Counterparty CDS Rate Counterparty 1 Bank’s Funding Rate CVA (Bank’s side) Euribor Rate (risk free) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 32
  33. 33. 1: Context & Market Practices: Counterparty risk and collateral [8] CVA Rates Counterparty 2 Counterparty CDS Rate Counterparty 1 CVA (bilateral) Bank’s Funding Rate (no CSA) (includes Bank’s cost of liquidity over CDS) Bank’s CDS Rate (includes the Bank’s default risk) CVA (Ctp side) CVA (Bank’s side) Euribor Rate (includes both credit and liquidity issues among Euribor Banks) Liquidity Value Adjustment ? Eonia Rate (CSA, risk free) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 33
  34. 34. 1: Context & Market Practices: Eonia Discounting or Not ? [1] Is the market discounting at Eonia ? Interest Interest Rate Rate CMS and Inflation Credits Equity Commodities Swaps Options Intra-day YES NO (?) ? NO (?) NO (?) NO (?) End of day NO (?) NO (?) NO (?) NO (?) NO (?) NO (?) Collateral NO NO NO NO NO NO Balance NO NO NO NO NO NO sheet Markit NO NO NO NO NO NO SwapClear NO -- -- -- -- -- ICAP YES NO ? -- -- -- “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 34
  35. 35. 1: Context & Market Practices: Eonia Discounting or Not ? [2] Main Broker Methodology Instrument Present methodology Future methodology Swap forwarding = Euribor xM forwarding = Euribor xM discounting = Euribor xM discounting = Eonia Basis Swap forwarding1 = Euribor xM1 forwarding1 = EuriborxM1 forwarding2 = Euribor xM2 forwarding2 = EuriborxM2 discounting = Min(EuriborxM1,EuriborxM2) discounting = Eonia CMS As Basis Swaps As Basis Swaps CMS S.O. As CMSs As CMSs CCS As Basis Swaps As Basis Swaps Caps/Floors/ forwarding = Euribor xM forwarding = Euribor xM Swaptions discounting = Euribor xM discounting = Eonia Forward premium ? Eonia options ? “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 35
  36. 36. 1: Context & Market Practices: Eonia Discounting or Not ? [3] Market Phase Transition SwapClear (London Clearing House) Possible Triggers Main Brokers Currencies EUR (Eonia), USD (Fed Fund rate) at the beginning Spot starting: NPV = 0, constant swap rate IR Swaps Forward starting: NPV = 0, variable forward swap rate Constant premiums, variable Black’s implied volatility, variable smile IR Options (variable ATM) CMS NPV = 0, constant swap spreads, variable beta SABR CMS Spread Options Constant premiums, variable (bilognormal) implied correlations Inflation Swaps NPV = 0, constant ZC, variable YoY Inflation Options Constant premiums, variable Black’s implied volatility Equity Options Constant premiums and dividends, variable Black’s implied volatility CDS NPV = 0, variable default probability Constant prices, variable credit spread absorbing the liquidity/credit Bonds risk inside Xibor “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 36
  37. 37. 1: Context & Market Practices: Single-Curve Pricing & Hedging IR Derivatives Pre credit-crunch single curve market practice: select a single set of the most convenient (e.g. liquid) vanilla interest rate instruments traded on the market with increasing maturities and build a single yield curve C using the preferred bootstrapping procedure (pillars, priorities, interpolation, etc.); for instance, a very common choice in the EUR market was a combination of short-term EUR deposit, medium-term FRA/Futures on Euribor3M and medium- long-term swaps on Euribor6M; compute, on the same curve C, forward rates, cashflows, discount factors and work out the prices by summing up the discounted cashflows; compute the delta sensitivity and hedge the resulting delta risk using the suggested amounts (hedge ratios) of the same set of vanillas. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 37
  38. 38. 1: Context & Market Practices: Multiple-Curve Pricing & Hedging IR Derivatives Post credit-crunch multiple curve market practice: build a single discounting curve Cd using the preferred bootstrapping procedure; build multiple distinct forwarding curves Cf1… Cfn using the preferred distinct selections of vanilla interest rate instruments, each homogeneous in the underlying rate tenor (typically 1M, 3M, 6M, 12M); compute the forward rates with tenor f on the corresponding forwarding curve Cf and calculate the corresponding cashflows; compute the corresponding discount factors using the discounting curve Cd and work out prices by summing the discounted cashflows; compute the delta sensitivity and hedge the resulting delta risk using the suggested amounts (hedge ratios) of the corresponding set of vanillas. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 38
  39. 39. 2: Multiple-Curve Framework: Basic Assumptions and notation [1] 1. There exist multiple different interest rate sub-markets Mx, x = {d,f1 ,…,fn} characterized by the same currency and by distinct bank accounts Bx and yield curves in the form of a continuous term structure of discount factors C x := {T → Px ( t0 ,T ) ,T ≥ t0 } , where t0 is the reference date of the curves (e.g. settlement date, or today) and Px(t,T) denotes the price at time t≥t0 of the Mx-zero coupon bond for maturity T, such that Px(T,T) = 1. 2. In each sub-market Mx we postulate the usual no arbitrage relation Px ( t,T2 ) = Px ( t,T1 ) × Px ( t,T1,T2 ) where Px(t,T1,T2) denotes the Mx-forward discount factor from time T2 to time T1, prevailing at time t. The financial meaning of the expression above is that in each market Mx, given a cashflow of one unit of currency at time T2, its corresponding value at time t < T2 must be the same, both if we discount in one single step from T2 to t, using the discount factor Px(t,T2), and if we discount in two steps, first from T2 to T1, using the forward discount Px(t,T1,T2) and then from T1 to t, using Px(t,T1). “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 39
  40. 40. 2: Multiple-Curve Framework: Basic Assumptions and notation [2] 3. We denote with Fx(t; T1; T2) the simple compounded forward rate associated, In each sub-market Mx to Px(t,T1,T2), resetting at time T1 and covering the time interval [T1; T2], such that Px ( t,T2 ) 1 Px ( t,T1,T2 ) = := , Px ( t,T1 ) 1 + Fx ( t;T1,T2 ) τx (T1,T2 ) where τx(T1,T2) is the year fraction between times T1 and T2 with daycount dcx. From the relations above we obtain the familiar no arbitrage expression 1 ⎡ 1 ⎤ Fx ( t;T1,T2 ) = ⎢ − 1⎥ τx (T1,T2 ) ⎢⎣ Px ( t,T1,T2 ) ⎥⎦ 1 Px ( t,T1 ) − Px ( t,T2 ) = τx (T1,T2 ) Px ( t,T2 ) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 40
  41. 41. 2: Multiple-Curve Framework: Basic Assumptions and notation [3] 4. The eq. above can be also derived (see e.g. ref. [A], sec. 1.4) as the fair value condition at time t of the Forward Rate Agreement (FRA) contract with payoff at maturity T2 given by FRAx (T1;T1,T2 , K ) = N τx (T1,T2 ) [ Lx (T1,T2 ) − K ], 1 − Px (T1,T2 ) Lx (T1,T2 ) = τx (T1,T2 ) Px (T1,T2 ) where N is the nominal amount, Lx(T1,T2) is the T1-spot Xibor rate for maturity T2 and K the (simply compounded) strike rate (sharing the same daycount convention for simplicity). Introducing expectations we have, t≤T1<T2, FRAx ( t;T1,T2 , K ) = NPx ( t,T2 ) τx (T1,T2 ) { } T Qx 2 Et [ Lx (T1,T2 ) ] − K = NPx ( t,T2 ) τx (T1,T2 ) [ Fx ( t;T1,T2 ) − K ], T where Qx 2 denotes the Mx-T2-forward measure associated to the numeraire Px(t,T2), EQ [ . ] denotes the expectation at time t w.r.t. measure Q and filtration Ft, encoding t the market information available up to time t, and we have assumed the standard martingale property of forward rates in the sub-market Mx. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 41
  42. 42. 2: Multiple-Curve Framework: Pricing Procedure 1. assume Cd as the discounting curve and Cf as the forwarding curve; 2. calculate any relevant spot/forward rate on the forwarding curve Cf as Pf ( t , Ti −1 ) − Pf ( t , Ti ) Ff ( t ;Ti −1 , Ti ) = , t ≤ Ti −1 < Ti , τ f (Ti −1 , Ti ) Pf ( t , Ti ) 3. calculate cashflows ci, i = 1,...,n, as expectations of the i-th coupon payoff πi with T respect to the discounting Ti - forward measure Q d i T Qd i ci := c ( t,Ti , πi ) = Et [ πi ]; 4. calculate the price π at time t by discounting each cashflow ci using the corresponding discount factor Pd ( t,Ti ) obtained from the discounting curve Cd and summing, n T π ( t,T ) = ∑ P ( t,Ti ) Et Qd i [ πi ]; i =1 5. Price FRAs as { [ Ff (T1 ;T1,T2 ) ] − K } T Qd 2 FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) τ f (T1,T2 ) Et “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 42
  43. 43. 2: Multiple-Curve Framework: No Arbitrage and Forward Basis Classic single-curve no arbitrage relations are broken: for instance, by specifying the subscripts d and f as prescribed above we obtain the two eqs. Pd ( t,T2 ) = Pd ( t,T1 ) Pf ( t,T1,T2 ) , 1 Pf ( t,T2 ) Pf ( t,T1,T2 ) = = , 1 + Ff ( t ;T1,T2 ) τ f (T1,T2 ) Pf ( t,T1 ) that clearly cannot hold at the same time. No arbitrage is recovered by taking into account the forward basis as follows 1 1 Pf ( t,T1,T2 ) = := , 1 + Ff ( t;T1,T2 ) τf (T1,T2 ) 1 + [ Fd ( t;T1,T2 ) + BAfd ( t;T1,T2 ) ] τd (T1,T2 ) for which we obtain the following static expression in terms of discount factors 1 ⎡ Pf ( t,T1 ) Pd ( t,T1 ) ⎤ BAfd ( t;T1,T2 ) = ⎢ − ⎥. τd (T1,T2 ) ⎢⎣ Pf ( t,T2 ) Pd ( t,T2 ) ⎥⎦ “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 43
  44. 44. 2: Multiple-Curve Framework: Forward Basis Yield Curves 80 Forward Basis 0Y-3Y 10 Forward Basis 3Y-30Y 60 8 6 40 4 20 basis points basis points 2 0 0 -20 -2 -4 -40 1M vs Disc 1M vs Disc 3M vs Disc -6 3M vs Disc -60 6M vs Disc 6M vs Disc 12M vs Disc -8 12M vs Disc -80 -10 May-09 May-10 May-11 Nov-09 Nov-10 Nov-11 Aug-09 Aug-10 Aug-11 Feb-09 Feb-10 Feb-11 Feb-12 Feb-12 Feb-15 Feb-18 Feb-21 Feb-24 Feb-27 Feb-30 Feb-33 Feb-36 Feb-39 Forward basis (bps) as of end of day 16 Feb. 2009, daily sampled 3M tenor forward rates calculated on C1M, C3M , C6M , C12M curves against Cd taken as reference curve. Bootstrapping as described in ref. [2]. The richer term structure of the forward basis curves provides a sensitive indicator of the tiny, but observable, statical differences between different interest rate market sub- areas in the post-credit crunch interest rate world, and a tool to assess the degree of liquidity and credit issues in interest rate derivatives' prices. Provided that… “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 44
  45. 45. 2: Multiple-Curve Framework: Bad Yield Curves 6% 3M curves 0Y-30Y 4 Forward Basis 3Y-30Y 2 5% 0 basis points 4% -2 3% -4 -6 1M vs Disc 2% 3M vs Disc zero rates -8 6M vs Disc forward rates 12M vs Disc 1% -10 Feb-09 Feb-11 Feb-13 Feb-15 Feb-17 Feb-19 Feb-21 Feb-23 Feb-25 Feb-27 Feb-29 Feb-31 Feb-33 Feb-35 Feb-37 Feb-39 Feb-12 Feb-15 Feb-18 Feb-21 Feb-24 Feb-27 Feb-30 Feb-33 Feb-36 Feb-39 Left: 3M zero rates (red dashed line) and forward rates (blue continuous line). Right: forward basis. Linear interpolation on zero rates has been used. Numerical results from QuantLib (www.quantlib.org). …smooth forward yield curves are used…Non-smooth bootstrapping techniques, e.g. linear interpolation on zero rates (still a diffused market practice), produce zero curves with no apparent problems, but ugly forward curves with a sagsaw shape inducing, in turn, strong and unnatural oscillations in the forward basis (see ref. [3]). “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 45
  46. 46. 3: Foreign Currency Analogy: Spot and Forward Exchange Rates A second issue regarding no arbitrage arises in the double-curve framework: { [ Ff (T1 ;T1,T2 ) ] − K } T Qd 2 FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) τ f (T1,T2 ) Et ≠ Pd ( t,T2 ) τ f (T1,T2 ) [ Ff (T1 ;T1,T2 ) − K ] 1. Double-curve-double-currency: ⇒ d = domestic, f = foreign c d ( t ) = x fd ( t ) c f ( t ) , X fd ( t , T ) Pd ( t , T ) = x fd ( t ) P f ( t , T ), x fd ( t 0 ) = x fd , 0. 2. Double-curve-single-currency: ⇒ d = discounting, f=forwarding cd ( t ) = x fd ( t )c f ( t ) , X fd ( t , T ) Pd ( t , T ) = x fd ( t ) Pf ( t ,T ) , x fd ( t 0 ) = 1. Picture of no arbitrage definition of the forward exchange rate. Circuitation (round trip) ⇒ no money is created or destructed. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 46
  47. 47. 3: Foreign Currency Analogy: Quanto Adjustment 1. Assume a lognormal martingale dynamic for the Cf (foreign) forward rate dFf ( t ;T1,T2 ) = σ f ( t ) dW fT2 ( t ) , Q T2 ↔ Pf ( t ,T2 ) ↔ C f ; f Ff ( t ;T1,T2 ) 2. since x fd ( t ) Pf ( t,T ) is the price at time t of a Cd (domestic) tradable asset, the forward exchange rate must be a martingale process dX fd ( t ,T2 ) T T = σ X ( t ) dWX 2 ( t ) , Qd 2 ↔ Pd ( t , T2 ) ↔ C d , X fd ( t ,T2 ) dW fT2 ( t ) dWX 2 ( t ) = ρ fX ( t ) dt ; T with 3. by changing numeraire from Cf to Cd we obtain the modified dynamic dFf ( t ;T1,T2 ) = μ f ( t ) dt + σ f ( t ) dW fT2 ( t ) , Qd 2 ↔ Pd ( t ,T2 ) ↔ C d , T Ff ( t ;T1,T2 ) μ f ( t ) = −σ f ( t ) σ X ( t ) ρ fX ( t ) ; 4. and the modified expectation including the (additive) quanto-adjustment T Qd 2 E t [ L f (T1,T2 ) ] = Ff ( t ;T1 ,T2 ) + QAfd ( t ;T1 , σ f , σ X , ρ fX ), ⎡ T1 ⎤ QAfd ( t ;T1, σ f , σ X , ρ fX ) = Ff ( t ;T1,T2 ) ⎢ exp ⎣ ∫t μ f ( s ) ds − 1 ⎥ . ⎦ “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 47
  48. 48. 3: Foreign Currency Analogy: Pricing Plain Vanillas [1] 1. FRA: FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) τ f (T1,T2 ) × [ Ff ( t;T1,T2 ) + QAfd ( t,T1, σ f , σX , ρfX ) − K ] m 2. Swaps: Swap ( t;T , S, K ) = −∑ Pd ( t, S j ) τd ( S j −1, S j )K j n j =1 +∑ Pd ( t, ST ) τ f (Tj −1,Tj )[ Ff ( t;Ti −1,Ti ) + QAfd ( t,Ti −1, σ f ,i , σX ,i , ρfX ,i ) ]. i =1 n 3. Caps/Floors: CF ( t;T , K , ω ) = ∑ Pd ( t,Ti ) τd (Ti −1,Ti ) i =1 ×Black [ Ff ( t;Ti −1,Ti ) + QAfd ( t,Ti −1, σ f ,i , σX ,i , ρfX ,i ), Ki , μf ,i , v f ,i , ωi ], 4. Swaptions: Swaption ( t;T , S, K , ω ) = Ad ( t, S ) ×Black [ S f ( t;T , S ) + QAfd ( t,T , S, ν f , νY , ρfY ), K , λf , v f , ω ]. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 48
  49. 49. 3: Foreign Currency Analogy: Pricing Plain Vanillas [2] 100 Quanto Adjustment (additive) 80 60 Quanto adj. (bps) 40 Numerical scenarios for the (additive) 20 quanto adjustment, corresponding to 0 three different combinations of (flat) -20 volatility values as a function of the -40 correlation. The time interval is fixed -60 Sigma_f = 10%, Sigma_X = 2.5% to T1-t=10 years and the forward rate Sigma_f = 20%, Sigma_X = 5% to 3%. -80 Sigma_f = 30%, Sigma_X = 10% -100 -1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0 Correlation We notice that the adjustment may be not negligible. Positive correlation implies negative adjustment, thus lowering the forward rates. The standard market practice, with no quanto adjustment, is thus not arbitrage free. In practice the adjustment depends on market variables not directly quoted on the market, making virtually impossible to set up arbitrage positions and locking today positive gains in the future. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 49
  50. 50. 3: Foreign Currency Analogy: A Simple Credit Model Both the forward basis and the quanto adjustment discussed above find a simple financial explanation in terms of counterparty risk. We adapt the simple credit model from Mercurio (ref. [9]) to the present context and notation. If we identify: Pd(t,T) = default free zero coupon bond, Pf(t,T) = risky zero coupon bond emitted by a generic risky (Xibor) counterparty for maturity T and with recovery rate Rf , τ(t)>t = (random) counterparty default time observed at time t, qd ( t,T ) = EtQd {1[ τ ( t )>T ] } = default probability after time T expected at time t, we obtain the following expressions Pf ( t,T ) = Pd ( t,T ) R ( t; t,T, Rf ), 1 ⎡ Pd ( t,T1 ) R ( t; t,T1, Rf ) ⎤ Ff ( t;T1,T2 ) = ⎢ − 1⎥ , τf (T1,T2 ) ⎢⎣ Pd ( t,T2 ) R ( t; t,T2, Rf ) ⎥⎦ where: R ( t;T1,T2, Rf ) = Rf + ( 1 − Rf ) EtQd [ qd (T1,T2 ) ]. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 50
  51. 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 τf (T1,T2 ) ] Pd ( t,T2 ) , R ( t;T1,T2, Rf ) and we are able to express the forward basis and the quanto adjustment described before in terms of risk free zero coupon bonds Pd(t,T) and of the expected recovery rate 1 Pd ( t,T1 ) ⎡ R ( t; t,T1, Rf ) ⎤ BAfd ( t;T1,T2 ) = ⎢ −1 ⎥, τd (T1,T2 ) Pd ( t,T2 ) ⎢⎣ R ( t; t,T2, Rf ) ⎥⎦ 1 Pd ( t,T1 ) ⎡ 1 R ( t; t,T1, Rf ) ⎤ QAfd ( t;T1,T2 ) = ⎢ − ⎥. τ f (T1,T2 ) Pd ( t,T2 ) ⎢⎣ R ( t;T1,T2, Rf ) R ( t; t,T2, Rf ) ⎥⎦ “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 51
  52. 52. 3: Foreign Currency Analogy: Pros & Cons PROs CONs Simple and familiar framework, Plain vanilla prices acquire no additional effort, just analogy. volatility and correlation dependence (but this is common to all approaches). Convexity adjustment emerges Parameters (exchange rate and naturally, as in more complex its volatility) presently not approaches. observable on the market, historical estimate only. Straightforward interpretation in Usual CONs of FX quanto terms of counterparty risk. adjustment applies (e.g. no smile) “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 52
  53. 53. 4: Hedging: Multiple Delta Hedging [1] 1. Given any portfolio of interest rate derivatives with price Π ( t,T , Rmkt ), compute the delta risk with respect to all curves C = {Cd ,C f 1 ,...,C fn } as R NC ∂Π ( t,T , Rk ) mktNC N k Δ ( t,T , R ) = ∑ Δ ( π mkt ) = ∑∑ π mkt t,T , Rk mkt , k =1 k =1 j =1 ∂Rk , j mkt R where NC is the number of yield curves involved and Rk is the vector of N k bootstrapping market data (yield curve pillars) associated to yield curve Ck . 2. Be careful to take properly into account all the delta components due to multiple curve bootstrapping: the forwarding zero curves {C f 1 ,...,C fn } , in particular, depend directly on their corresponding input market instruments with tenor f, but also indirectly on the discounting curve, R Z “One price, two curves, ∂Π ( t,T , Rd ) ∂Z α Nd Nd mkt d Δ ( t,T , Rd ) = ∑ ∑ π mkt d mkt , three deltas” (see ref.[8]) j =1 α =1 ∂Z α ∂Rd , j N fR N fZ Z ∂Π ( t,T , Rf ) ∂Z α mkt f Nd N f ∂Π ( t,T , Rf ) ∂Z α mkt f Δ ( t,T , Rf ) = ∑ ∑ π mkt f +∑∑ , j =1 α =1 ∂Z α ∂Rfmkt ,j j =1 α =1 ∂Z αf mkt ∂Rd , j Z where Z f is the vector of N f zero rates pillars in the zero rate curve Cf. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 53
  54. 54. 4: Hedging: Multiple Delta Hedging [2] 3. eventually aggregate the delta sensitivity on the selected subset H of the most liquid market instruments used for hedging RH = { R1 ,..., RN H } (hedging instruments); H H NH ∂Π ( t,T , RH ) Δ π ( t,T , R )H ∑ ∂RjH , j =1 ∂Π ( t ,T , R H ) 4. calculate hedge ratios: h j ( t ,T , R H ) = δ jH ( t ) , ∂ R jH ∂ π H ( t , T j , R jH j ) δ jH ( t , T j , R jH )= . H ∂ R jH 5. where π j ( t ) is the market price (unit nominal) of the corresponding hedging instrument, such that the hedged portfolio has zero delta NH Π tot ( t ,T , R ) = H Π ( t ,T , R H ) − ∑ h j ( t )π H ( t , T j , R jH ) , j j =1 ⎡ ∂Π ( t ,T , R H NH ) NH ∂ π H ( t , T j , R jH ) ⎤ ∑ ⎢⎢ − ∑ h j (t ) j Δ tot ( t ,T , R H ) ⎥ ∂ Rk H ∂ Rk H ⎥ k =1 ⎣ j =1 ⎦ NH ⎡ ∂Π ( t ,T , R H ) ⎤ = ∑⎢ − h k ( t ) δk ( t ) ⎥ = 0, H ⎢ k =1 ⎣ ∂ Rk H ⎥ ⎦ “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 54
  55. 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. 56. 5: Other Approaches: Axiomatic Approach See ref. [8]: M. Henrard, ”The Irony in the Derivatives Discounting - Part II: The Crisis”, Jul. 2009, SSRN working paper http://ssrn.com/abstract=1433022. Individuate the minimal set of definitions and assumptions for coherent multiple yield curve pricing of plain vanillas (FRA, IRS). Find a (convexity) adjustment factor for FRA/Futures. Discuss delta hedging. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 56
  57. 57. 5: Other Approaches: Extended Libor Market Model Approach [1] See ref. [9]: F. Mercurio, "Post Credit Crunch Interest Rates: Formulas and Market Models", Bloomberg, Jan. 2009, and ref. [17]: F. Mercurio, “LIBOR Market Models with Stochastic Basis”, Mar. 2010. Assume as the fundamental bricks the OIS forward rate, the FRA rate and their spread, defined as, respectively, x ( ) x x T Qd k x x 1 ⎡ Pd ( t,Tkx−1 ) ⎤ Fk t := Fd ( t;Tk −1,Tk ) = Et [ Ld (Tk −1,Tk ) ] = x ⎢ − 1⎥ , τk ⎢⎣ Pd ( t,Tkx ) ⎥⎦ T Qd k Lx ( t ) k := FRA ( t;Tkx−1,Tkx ) := Et [ Lx (Tkx−1,Tkx ) ], Sk ( t ) := Lx ( t ) − Fkx ( t ) . x k In ref. [17] assume, in the single FRA rate tenor case, general stochastic volatility dynamics for each OIS forward and spread, uncorrelated, derive pricing expression for caplets and swaptions as integrals over the full smile structure, and provide examples using various specific dynamics. Address the multi FRA rate tenor case finding the simplest stochastic-volatility dynamics that preserve consistency across different tenors. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 57
  58. 58. 5: Other Approaches: Extended Libor Market Model Approach [2] See ref. [13]: M. Fujii, Y. Shimada, A. Takahashi, “A Market Model of Interest Rates with Dynamic Basis Spreads in the presence of Collateral and Multiple Currencies”, Nov. 2009. Assume as the fundamental bricks the OIS forward rate, the FRA rate and their spread. Assume model stochastic basis spreads in a HJM framework. Address both single- and multi-currency cases. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 58
  59. 59. 5: Other Approaches: Extended Short Rate Model Approach See ref. [5]: M. Kijima, K. Tanaka, T. Wong, “A Multi-Quality Model of Interest Rates”, Quantitative Finance, vol. 9, issue 2, pages 133-145, 2008. Assume three yield curves: discounting, forwarding and governative (D-L-G). Assume three correlated Hull-White processes for the discounting, forwarding and governative short rates. Find approximated analytic pricing expression for european caps/floors/swaptions and bond options. See ref. [18]: C. Kenyon, “Short-Rate Pricing after the Liquidity and Credit Shocks: Including the Basis”, Feb. 2010, SSRN working paper, http://ssrn.com/abstract=1558429. Assume two correlated Hull-White processes for the discounting and forwarding short rates, plus an uncorrelated lognormal process for the exchange rate. Find a pseudo-analytic european swaption pricing expression. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 59
  60. 60. 5: Other Approaches: Counterparty Risk Approach See ref. [11]: M. Morini, “Solving the Puzzle in the Interest Rate Market”, Oct. 2009. Discuss Libor rate and Libor-based derivatives before and after the crisis. Show that the post-credit crunch basis between FRA rates and their spot Libor replication can be explained by using the quoted Basis Swap spreads. Explain the market patterns of the Basis Swap spreads by modelling them as options on the credit worthiness of the counterparty. Formalize the mathematical representation of the post-credit crunch interest rate market, introducing credit risk, allowing for no-fault standard rule and collateralization, and using subfiltrations to model risky Libor rates. Compute change of numeraire and convexity adjustments for collateralized derivatives tied to risky Libor, thus explaining the quanto-adjustment found by Bianchetti in ref. [7] in a simpler framework. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 60
  61. 61. 6: Conclusions 1. We have reviewed the pre and post credit crunch market practices for pricing & hedging interest rate derivatives. 2. We have shown that in the present double-curve framework standard single-curve no arbitrage conditions are broken and can be recovered taking into account the forward basis; once a smooth bootstrapping technique is used, the richer term structure of the calculated forward basis curves provides a sensitive indicator of the tiny, but observable, statical differences between different interest rate market sub- areas. 3. Using the foreign-currency analogy we have computed the no arbitrage generalised double-curve-single-currency market-like pricing expressions for basic interest rate derivatives, including a quanto adjustment arising from the change of numeraires naturally associated to the two yield curves. Numerical scenarios show that the quanto adjustment can be non negligible. 4. Both the forward basis and the quanto adjustment have a simple interpretation in terms of counterparty risk, using a simple credit model with a risk-free and a risky zero coupond bonds. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 61
  62. 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 è derivato dalla crisi”, Risk Italia, 26 Nov. 2009. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 62
  63. 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. 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--Business School 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 of Finance 62, pages 383–410, 2007. 5. M. Kijima, K. Tanaka, T. Wong, “A Multi-Quality Model of Interest Rates”, Quantitative Finance, vol. 9, issue 2, pages 133-145, 2008. 6. F. Ametrano, M. Bianchetti, “Bootstrapping the Illiquidity: Multiple Yield Curves Construction For Market Coherent Forward Rates Estimation”, in “Modeling Interest Rates: Latest Advances for Derivatives Pricing”, edited by F. Mercurio, Risk Books, 2009. 7. M. Bianchetti, “Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives Using Different Yield Curves for Discounting and Forwarding”, Jan. 2009, SSRN working paper, 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 working paper, http://ssrn.com/abstract=1394267. “Multiple Curves, One Price” - Marco Bianchetti – Global Derivatives 2010 p. 64

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