{{short description|Mathematical finance concept}} <!-- #REDIRECT Interest rate swap #Multi-curve framework --> In mathematical finance, the '''multi-curve framework''' refers to <ref>Staff (2023). [https://www.rebellionresearch.com/multi-curve-framework "Multi-curve Framework"], ''rebellionresearch.com''</ref> the use of multiple interest rate curves to price the various fixed income securities and derivatives, based on their characteristics, particularly tenor, but also currency.
==Context== {{further|Financial economics#Derivative pricing|Interest rate swap#Valuation and pricing}} Historically interest rate swaps, IRSs, were valued using discount factors derived from the same curve used to forecast the LIBOR ({{Nowrap|-IBOR}}) rates for payment (the erstwhile reference rates; see below re MRRs). This has been called "self-discounted". Following the 2008 financial crisis, <ref name="Everything">Ametrano, Ferdinando M. and Bianchetti, Marco. (2013). [https://ssrn.com/abstract=2219548 "Everything You Always Wanted to Know About Multiple Interest Rate Curve Bootstrapping but Were Afraid to Ask".]</ref> however, it became apparent that the approach was not appropriate, and alignment towards discount factors associated with physical collateral of the IRSs was needed. <ref name="Henrard I">Henrard M. (2007). ''The Irony in the Derivatives Discounting'', Wilmott Magazine, pp. 92–98, July 2007. [http://ssrn.com/abstract=1349024 SSRN preprint.]</ref> <ref name="Henrard II">Henrard M. (2010). ''The Irony in the Derivatives Discounting Part II: The Crisis'', Wilmott Journal, Vol. 2, pp. 301–316, 2010. [http://ssrn.com/abstract=1433022 SSRN preprint.]</ref> <ref>Bianchetti M. (2010). ''Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives Decoupling Forwarding and Discounting Yield Curves'', Risk Magazine, August 2010. [http://ssrn.com/abstract=1334356 SSRN preprint.]</ref>
Thus, the now-standard pricing approach is the "multi-curve framework" where separate ''discount curves'' and ''forecast curves'' are built; here, respectively: *Overnight index swap (OIS) rates are typically used to derive discount factors, since that index is the standard inclusion on Credit Support Annexes (CSAs) to determine the rate of interest payable on collateral for IRS contracts. *As regards the payment / rates forecast, since the basis spread between LIBOR rates of different maturities widened during the crisis, forecast curves are generally constructed for each LIBOR tenor used in floating rate derivative legs.<ref>[https://www.kpmg.com/Global/en/IssuesAndInsights/ArticlesPublications/Documents/multi-curve-valuation-approaches-part-1.pdf Multi-Curve Valuation Approaches and their Application to Hedge Accounting according to IAS 39], Dr. Dirk Schubert, KPMG</ref>
==Curve construction== {{see also|Interest rate swap#Valuation and pricing}} Although the Multi-curve framework modifies the overall approach, there is no change to the economic pricing principle: swap leg values are still identical at initiation (see {{section link|Rational pricing#Swaps}}). What differs is that, as above, separate curves are constructed for payments and for discounting.
Thus, regarding the curve build, the following emerges. <ref>M. Henrard (2014). [https://link.springer.com/book/10.1057/9781137374660 ''Interest Rate Modelling in the Multi-Curve Framework: Foundations, Evolution and Implementation.''] Palgrave Macmillan {{ISBN|978-1137374653}}</ref> <ref>See section 3 of Marco Bianchetti and Mattia Carlicchi (2012). [https://arxiv.org/ftp/arxiv/papers/1103/1103.2567.pdf ''Interest Rates after The Credit Crunch: Multiple-Curve Vanilla Derivatives and SABR'']</ref> <ref name=PTIRDs>[http://www.tradinginterestrates.com Pricing and Trading Interest Rate Derivatives: A Practical Guide to Swaps], J H M Darbyshire, 2017, {{ISBN|978-0995455528}}</ref> <ref name="Everything"/> Under the old framework a single self-discounted curve was "bootstrapped" for each tenor; i.e.: solved such that it exactly returned the observed prices of selected instruments—IRSs, with FRAs in the short end—with the build proceeding sequentially, date-wise, through these instruments. Under the new framework, the various curves are best fitted to observed market prices as a "curve set": one curve for discounting, and one for each IBOR-tenor "forecast curve"; the build is then based on quotes for IRSs ''and'' OISs, with FRAs included as before. Here, since the observed average overnight rate plus a spread is swapped for<ref name="CQF"/> the {{Nowrap|-IBOR}} rate over the same period (the most liquid tenor in that market), and the {{Nowrap|-IBOR}} IRSs are in turn discounted on the OIS curve, the problem entails a nonlinear system, where all curve points are solved at once, and specialized iterative methods are usually employed (see further following). The forecast-curves for other tenors can be solved in a "second stage", bootstrap-style, with discounting on the now-solved OIS curve.
A CSA could allow for collateral, and hence interest payments on that collateral, in any currency.<ref name="UTokyoPaper">{{cite journal|last=Fujii|first=Masaaki Fujii|author2=Yasufumi Shimada |author3=Akihiko Takahashi |title=A Note on Construction of Multiple Swap Curves with and without Collateral|journal=CARF Working Paper Series No. CARF-F-154|date=26 January 2010|ssrn=1440633}}</ref> To accommodate this, banks include in their curve-set a USD discount-curve to be used for discounting {{Nowrap|local-IBOR}} trades which have USD collateral; this curve is sometimes called the (Dollar) "basis-curve". It is built by solving for observed (mark-to-market) cross-currency swap rates, where the local {{Nowrap|-IBOR}} is swapped for USD LIBOR with USD collateral as underpin. The latest, pre-solved USD-LIBOR-curve is therefore an (external) element of the curve-set, and the basis-curve is then solved in the "third stage". Each currency's curve-set will thus include a local-currency discount-curve and its USD discounting basis-curve. As required, a third-currency discount curve — i.e. for local trades collateralized in a currency other than local or USD (or any other combination) — can then be constructed from the local-currency basis-curve and third-currency basis-curve, combined via an arbitrage relationship known here as "FX Forward Invariance".<ref>Burgess, Nicholas (2017). [https://doi.org/10.2139/ssrn.3009281 ''FX Forward Invariance & Discounting with CSA Collateral'']</ref>
Various approaches to solving curves are possible. Modern methods tend to employ global optimizers with complete flexibility in the parameters that are solved relative to the calibrating instruments used to tune them. These optimizers will seek to minimize some objective function - here matching the observed instrument values - and this assumes that some interpolation mode <ref>P. Hagan and G. West (2006). [https://www.deriscope.com/docs/Hagan_West_curves_AMF.pdf Interpolation methods for curve construction]. ''Applied Mathematical Finance'', 13 (2):89—129, 2006.</ref> <ref>P. Hagan and G. West (2008). [http://web.math.ku.dk/~rolf/HaganWest.pdf Methods for Constructing a Yield Curve]. ''Wilmott Magazine'', May, 70-81.</ref> <ref>P du Preez and E Maré (2013). [http://www.scielo.org.za/pdf/sajems/v16n4/03.pdf Interpolating Yield Curve Data in a Manner That Ensures Positive and Continuous Forward Curves]. ''SAJEMS'' 16 (2013) No 4:395-406</ref> has been configured for the curves; the approach ultimately employed may be a modification of Newton's method. Maturities corresponding to input instruments are referred to as "pillar points"; often, these are solved directly, while other spot rates are interpolated. (Then, once solved, all that need be stored are the pillar point rates and the interpolation rule.)
==Transition== Starting in 2021, LIBOR is being phased out, with replacements including other "market reference rates" (MRRs) such as SOFR and TONAR. (These MRRs are based on secured overnight funding transactions). With the coexistence of "old" and "new" rates in the market, multi-curve and OIS curve "management" is necessary, with changes required to incorporate new discounting and compounding conventions, while the underlying logic is unaffected; see.<ref>Fabio Mercurio (2018). [https://www.ieor.columbia.edu/files/seas/content/docs/columbia2018.pdf SOFR So Far: Modeling the LIBOR Replacement]</ref><ref>FINCAD (2020). [https://fincad.com/sites/default/files/2020-08/New_Datasheet_Curve_Building_End_of_Libor_A4.pdf Future-Proof Curve-Building for the End of Libor]</ref><ref>Finastra (2020). [https://www.finastra.com/sites/default/files/2020-05/brochure_transitioning-from-libor-fusion-sophis-factsheet.pdf Transitioning from LIBOR to alternative reference rates]</ref>
==Other curves== The reference to "multi-curves" may sometimes include the various curves relating to credit quality. Thus, post-crisis, investment banks will value their ''bonds'' using CSA-linked discount curves, while adjusting the expected cashflows - coupons and "face" - for default risk via the use of an issuer "credit curve". More broadly, <ref>Lorenzo Silotto, Marco Scaringi, and Marco Bianchetti (2023). [https://arxiv.org/pdf/2107.10377v3 "Everything You Always Wanted to Know About XVA Model Risk but Were Afraid to Ask"]. arxiv.org</ref> under XVA — a notable post-crisis development — CSA discount curves are combined with the counterparty's survival curve, and also with the relevant funding curves, so as to model the various "valuation adjustments". Where the underlying-instrument exhibits optionality — caps and floors, swaptions, embedded derivatives — so a volatility "cube" will be further required.
==Risk management== {{see|Interest rate swap#Risks}} The separation of discounting and forward curves, the explicit modeling of basis risk, and the incorporation of credit and volatility considerations all add layers of complexity as compared to the simpler self-discounting regime. Thus, financial risk management in a multi-curve environment involves a more nuanced approach to hedging, economic / risk capital, and stress testing. <ref name="Henrard I"/><ref name="Henrard II"/> <ref name="CQF">CQF Institute. [https://web.archive.org/web/20211102180254/https://cqfinstitute.org/sites/default/files/2021-02-fitch-multicurve-V-1-1_0.pdf "Multi-curve and collateral framework"]</ref> Since (individual) positions are (potentially) affected by numerous instruments not obviously related, a "real challenge" <ref name="CQF"/> is to then hedge portfolios and describe risks coherently; hedge accounting is similarly complicated. <ref>Dr. Dirk Schubert (2012). {{cite web| title=Multi-Curve Valuation Approaches and their Application to Hedge Accounting according to IAS 39 |url=https://assets.kpmg.com/content/dam/kpmg/pdf/2013/01/Multi-Curve-Hedge-Accounting-2012-KPMG.pdf|publisher=KPMG}}</ref> For stress testing, the bank must model the impact of the (macro-economic) scenario in question on the various (major) risk factors, incorporating, as far as possible, their interrelationships. Relatedly, to return an estimate of value at risk (VaR) the bank must now simulate changes in the OIS curve, forward curves, basis spreads, and credit spreads, while also ensuring that correlations between these risk factors are properly modeled. Allocation of the resultant economic capital — required, e.g., for RAROC calculations at the operational dealing "desk" level — is similarly complicated due to the multiple risk factors (and if required, partial sensitivities <ref>Dóra Balog (2011). {{cite web |title = Capital allocation in financial institutions: the Euler method |url=https://www.econstor.eu/bitstream/10419/108225/1/MTDP1126.pdf}} IEHAS Discussion Papers, No. MT-DP - 2011/26</ref>) involved.
==See also== *{{section link|Yield curve|Construction of the full yield curve from market data}} *{{section link|Fixed-income attribution |Modeling the yield curve}}
==References== <references/>
==Further reading == * Henrard M. (2014) [https://link.springer.com/book/10.1057/9781137374660 ''Interest Rate Modelling in the Multi-curve Framework: Foundations, Evolution, and Implementation.''] Palgrave Macmillan. Applied Quantitative Finance series. June 2014. {{ISBN|978-1-137-37465-3}}.
Category:Financial economics Category:Mathematical finance Category:Fixed income analysis Category:Interest rates Category:Bonds (finance) Category:Swaps (finance) Category:Financial models