 jzw123
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Joined: February 23rd, 2011, 1:14 pm

### ois discounting effect on par swaps

Hi there,I am trying to figure out the impact of ois discounting on par swaps and its consequent impact on libor curve constructions. As par swaps are usually used in combination with overight rates, EDfuture rates, etc as part of a boot strapping process to get the libor discount factors. Par swap rates under ois discounting are going to effectively yield ois discount factors not libor discount factors. Am I right? And if I am, how the market is coping with this? For off par swap valuation, do we use forward rates projected from the libor curve for floating payment and use OIS to discount the cashflow to get PV or do we use fwd rates projected from the OIS discount curve for floating payment then PV the cashflows with OIS curve?Many thanks in advance mathmarc
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Joined: March 18th, 2003, 6:50 am

### ois discounting effect on par swaps

QuoteOriginally posted by: jzw123 I am trying to figure out the impact of ois discounting on par swaps and its consequent impact on libor curve constructions. As par swaps are usually used in combination with overight rates, EDfuture rates, etc as part of a boot strapping process to get the libor discount factors. Par swap rates under ois discounting are going to effectively yield ois discount factors not libor discount factors. Am I right? And if I am, how the market is coping with this? For off par swap valuation, do we use forward rates projected from the libor curve for floating payment and use OIS to discount the cashflow to get PV or do we use fwd rates projected from the OIS discount curve for floating payment then PV the cashflows with OIS curve?In a (now standard) multi-curve framework there is one discounting curve and several forward (Ibor) functions. I use the term "function" instead of "curve" as it is not a real yield curve; it is the function that when used in the appropriate formula provides the correct price for Ibor coupons. The term "Ibor discounting factor" is also a misuse of language as the Ibor function is never used for discounting.The base of the framework is coherence. The function is used everywhere in the same way, in curve construction and in off-par swap valuation. The "forward Ibor rate" is estimated with the Ibor function and discounted with the discounting curve. Again the term "forward rate" is a misuse of language, I should use the term "number to be used in the formula", but this is not a very practical term. This "forward Ibor rate" is never estimated with the discounting/OIS curve.A small detail, in the multi-curve framework as described above you would not use overnight rate to construct the Ibor curve (at least not in EUR and USD, due to the 2 day spot lag). The curve below the spot date is never used (no need to invent something there) and the curve below the first period (spot+3m for a 3m Ibor, spot+6m for 6m Ibor, etc.) is arbitrary. jzw123
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### ois discounting effect on par swaps

Do you have some concrete examples I can look at? Or any good web sites? Thanks mathmarc
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### ois discounting effect on par swaps

QuoteOriginally posted by: jzw123Do you have some concrete examples I can look at? Or any good web sites?There are certainly a lot of articles you can look at. An early list isAmetrano, Ferdinando and Bianchetti, Marco, "Bootstrapping the illiquidity: multiple yield curves construction for market coherent forward rates estimation", Working paper, Banca IMI/Banca IntesaSanpaolo, 2009. (Also MODELING INTEREST RATES, Fabio Mercurio, ed., Risk Books, Incisive Media) http://ssrn.com/abstract=1371311Bianchetti, Marco, "Two curves, one price: pricing and hedging interest rate derivatives decoupling forwarding and discounting yield curves, Banca Intesa San Paolo, 2009. http://ssrn.com/abstract=1334356Chibane, Messaoud and Sheldon, Guy, "Building curves on a good basis", Shinsei Bank, 2009. http://ssrn.com/abstract=1394267Henrard, Marc, "The Irony in the Derivatives Discounting Part II: The Crisis ", SSRN 2009. http://ssrn.com/abstract=1433022 and Wilmott Journal, Vol. 2, pp. 301-316, 2010Kijima, Masaaki and Tanaka, Keiichi and Wong, Tony, A multi-quality model of interest rates, Quantitative Finance, March 2009, 133--145.Mercurio, Fabio, "Interest Rates and The Credit Crunch: New Formulas and Market Models", Bloomberg Financial Markets (BFM), 2009. http://ssrn.com/abstract=1332205Morini, M., "Credit modelling after the subprime crisis, Working paper, 2009There are also some good open source libraries that implement those approaches. I don't know if you just want to build one example or need to develop a full system for a bank/hedge fund/fund manager. Cuchulainn
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### ois discounting effect on par swaps

Mathmarc,Do you have links to these open source project? Thanks. BTW we have implemented them in C#. It would be interesting to compare the deisgns and and output.
Last edited by Cuchulainn on April 22nd, 2012, 10:00 pm, edited 1 time in total.
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### ois discounting effect on par swaps

HelloI have a further sub-question on this topic, regarding the fact that USD OIS outright swaps do not extend beyond the 10y point, and it is therefore necessary to substitute in FF vs LIBOR basis swaps to extend the OIS and LIBOR curves simultaneously beyond the 10y point.In my view, this should be absolutely trivial through the following argument:For the FF vs LIBOR basis swap:PV of (FF + Spread) leg = PV of LIBOR floating leg (1)For the outright LIBOR IR Swap:PV of LIBOR fixed leg = PV of LIBOR floating leg (2)Therefore by combining (1) and (2):PV of (FF+Spread) Leg = PV of LIBOR fixed legThis removes the need to solve for LIBOR discount factors, because on the RHS of the equation we have a PV equivalent to SUM[(Tau(i) * Coupon * DF(OIS, i)]So we can solve for OIS discount factors without having to solve for LIBOR discount factors simultaneously using this framework (given that LIBOR discount factors don't appear as unknowns in the equations above).Is there a flaw in my economic or mathematical reasoning? Are banks and institutions employing this "shortcut" (which is possible if the maturities of the FF vs LIBOR basis swaps match the maturity of the LIBOR outright IRS)? If not, why not?Thanks kelang
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Joined: November 14th, 2011, 4:53 pm

### ois discounting effect on par swaps

many places do this from 2Y aboveQuoteOriginally posted by: PvalAnal85HelloI have a further sub-question on this topic, regarding the fact that USD OIS outright swaps do not extend beyond the 10y point, and it is therefore necessary to substitute in FF vs LIBOR basis swaps to extend the OIS and LIBOR curves simultaneously beyond the 10y point.In my view, this should be absolutely trivial through the following argument:For the FF vs LIBOR basis swap:PV of (FF + Spread) leg = PV of LIBOR floating leg (1)For the outright LIBOR IR Swap:PV of LIBOR fixed leg = PV of LIBOR floating leg (2)Therefore by combining (1) and (2):PV of (FF+Spread) Leg = PV of LIBOR fixed legThis removes the need to solve for LIBOR discount factors, because on the RHS of the equation we have a PV equivalent to SUM[(Tau(i) * Coupon * DF(OIS, i)]So we can solve for OIS discount factors without having to solve for LIBOR discount factors simultaneously using this framework (given that LIBOR discount factors don't appear as unknowns in the equations above).Is there a flaw in my economic or mathematical reasoning? Are banks and institutions employing this "shortcut" (which is possible if the maturities of the FF vs LIBOR basis swaps match the maturity of the LIBOR outright IRS)? If not, why not?Thanks  