Editing Interest rate swap (section)

==Valuation and pricing==
{{further information|Rational pricing#Swaps}}
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 -->IRSs are bespoke financial products whose customisation can include changes to payment dates, notional changes (such as those in amortised IRSs), accrual period adjustment and calculation convention changes (such as a [[day count convention]] of 30/360E to ACT/360 or ACT/365).

A vanilla IRS is the term used for standardised IRSs. Typically these will have none of the above customisations, and instead exhibit constant notional throughout, implied payment and accrual dates and benchmark calculation conventions by currency.<ref name=PTIRDs /> A vanilla IRS is also characterised by one leg being "fixed" and the second leg "floating" referencing an {{Nowrap|-IBOR}} index. The net [[present value]] (PV) of a vanilla IRS can be computed by determining the PV of each fixed leg and floating leg separately and summing. For pricing a mid-market IRS the underlying principle is that the two legs must have the same value initially; see further [[Rational pricing#Valuation at initiation|under Rational pricing]].

Calculating the fixed leg requires discounting all of the known cashflows by an appropriate discount factor:

:<math>P_\text{fixed} = N  R  \sum_{i=1}^{n_1} d_i v_i</math>
where <math>N</math> is the notional, <math>R</math> is the fixed rate, <math>n_1</math> is the number of payments, <math>d_i</math> is the decimalised day count fraction of the accrual in the i'th period, and <math>v_i</math> is the discount factor associated with the payment date of the i'th period.

Calculating the floating leg is a similar process replacing the fixed rate with forecast index rates:
:<math>P_\text{float} = N \sum_{j=1}^{n_2} r_j d_j v_j</math>
where <math>n_2</math> is the number of payments of the floating leg and <math>r_j</math> are the forecast {{Nowrap|-IBOR}} index rates of the appropriate currency.

The PV of the IRS from the perspective of receiving the fixed leg is then:
:<math>P_\text{IRS} = P_\text{fixed} - P_\text{float}</math>

Historically IRSs were valued using discount factors derived from the same curve used to forecast the {{Nowrap|-IBOR}} rates (i.e. the erstwhile [[reference rate]]s). This has been called "self-discounted". Some early literature described some incoherence introduced by that approach and multiple banks were using different techniques to reduce them. It became more apparent with the [[2008 financial crisis]] that the approach was not appropriate, and alignment towards discount factors associated with physical [[collateral (finance)|collateral]] of the IRSs was needed.

Post crisis, to accommodate credit risk, the now-standard pricing approach is the '''multi-curve framework''', applied where forecast discount factors and {{Nowrap|-IBOR}} (see below re MRRs) exhibit disparity. 
Note that the economic pricing principle is unchanged: leg values are still identical at initiation. See [[Financial economics#Derivative pricing|Financial economics § Derivative pricing]] for further context. 
Here, [[overnight index swap]] (OIS) rates are typically used to derive discount factors, since that index is the standard inclusion on [[Credit Support Annex]]es (CSAs) to determine the rate of interest payable on collateral for IRS contracts. As regards the rates forecast, since the [[LIBOR–OIS spread|basis spread]] between [[LIBOR]] rates of different maturities widened during the crisis, forecast curves are generally constructed for each [[Libor#Maturities|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>

Regarding the curve build, see: 
<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 />
Under the old framework a single self-discounted curve was [[Bootstrapping (finance)|"bootstrapped"]] for each tenor; 
i.e.: solved such that it exactly returned [[Yield curve#Construction of the full yield curve from market data|the observed prices of selected instruments]]—IRSs, with [[forward rate agreement|FRAs]] in the short end—with the build proceeding sequentially, date-wise, through these instruments. 
Under the new framework, the various curves are [[best fit]]ted 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 [[Basis swap|swapped for]]<ref name="CQF">[[Professional certification in financial services#Certificate in Quantitative Finance|CQF Institute]]. [https://www.cqfinstitute.org/sites/default/files/2021-02-fitch-multicurve-V-1-1_0.pdf  "Multi-curve and collateral framework"]</ref> 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 — very often [[Newton's method#Multidimensional formulations|a modification of Newton's method]]. 
The forecast-curves for other tenors can be solved in a "second stage", bootstrap-style, with discounting on the now-solved OIS curve.

Various approaches to solving curves are possible. 
Modern methods 
<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>
tend to employ [[global optimization|global optimizers]] with complete flexibility in the parameters that are solved relative to the calibrating instruments used to tune them. (Maturities corresponding to input instruments are referred to as "pillar points".) These optimizers will seek to minimize some [[objective function]] - here matching the observed instrument values - and this assumes that some [[interpolation]] mode has been configured for the curves.

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) [[Currency swap#Extended description|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 [[Covered interest arbitrage|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>

Starting in 2021, [[Libor#LIBOR cessation and alternatives available|LIBOR is being phased out]], with replacements including other "market reference rates" (MRRs) such as [[Secured Overnight Financing Rate|SOFR]] and [[Tokyo Overnight Average Rate|TONAR]]. (These MRRs are based on secured [[Overnight market|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>

The complexities of modern curvesets mean that there may not be discount factors available for a specific {{Nowrap|-IBOR}} index curve. These curves are known as 'forecast only' curves and only contain the information of a forecast {{Nowrap|-IBOR}} index rate for any future date. Some designs constructed with a discount based methodology mean forecast -IBOR index rates are implied by the discount factors inherent to that curve:
:<math>r_j = \frac{1}{d_j} \left ( \frac{x_{j-1}}{x_j} - 1 \right ) 
</math> where <math>x_{i-1}</math> and <math>x_{i}</math> are the start and end ''discount factors'' associated with the relevant forward curve of a particular {{Nowrap|-IBOR}} index in a given currency.

To price the mid-market or par rate, <math>S</math> of an IRS (defined by the value of fixed rate <math>R</math> that gives a net PV of zero), the above formula is re-arranged to:
:<math>S = \frac{\sum_{j=1}^{n_2}r_j d_j v_j}{ \sum_{i=1}^{n_1} d_i v_i}</math>

In the event old methodologies are applied the discount factors <math>v_k</math> can be replaced with the self discounted values <math>x_k</math> and the above reduces to:
:<math>S = \frac{x_0 - x_{n_2}}{ \sum_{i=1}^{n_1} d_i x_i}</math>

In both cases, the PV of a general swap can be expressed exactly with the following intuitive formula:<math>P_\text{IRS} = N(R-S)A</math> 
where <math>A</math> is the so-called [[Annuity]] factor <math display=inline>A = \sum_{i=1}^{n_1} d_i v_i</math> (or <math display=inline>A = \sum_{i=1}^{n_1} d_i x_i</math> for self-discounting). This shows that the PV of an IRS is roughly linear in the swap par rate (though small non-linearities arise from the co-dependency of the swap rate with the discount factors in the Annuity sum).