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### OIS discounting collateralized swaps proof

Hello,I wonder how can be proved/argued that if the collateral rate is OIS, then the collateralized swap cash flows should be discounted using the OIS curve. Thank you, cfaleontan
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Joined: July 21st, 2011, 6:26 pm

### OIS discounting collateralized swaps proof

you have to factor in your funding rate, too.www.fsa.go.jp/frtc/nenpou/2009/07-1.pdf
Last edited by cfaleontan on May 22nd, 2012, 10:00 pm, edited 1 time in total. miscelania
Topic Author
Posts: 187
Joined: April 1st, 2009, 6:05 am

### OIS discounting collateralized swaps proof

QuoteOriginally posted by: cfaleontanyou have to factor in your funding rate, too.www.fsa.go.jp/frtc/nenpou/2009/07-1.pdfThank you.I think that in case of collateralized swaps the discounting rate is computed considering the OIS curve, not the funding rate, right?I am just focusing here on collateralized instruments, but I do not know why they are discounted using OIS curve when the collateral is capitalized with the overnight rate. Martinghoul
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### OIS discounting collateralized swaps proof

You can't "prove" anything. However, if you just follow the cashflows, it becomes reasonably clear. ancast
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Joined: July 14th, 2002, 3:00 am

### OIS discounting collateralized swaps proof

There is no such discounting as an OIS discounting. It just derives from a mathematical fact in the derivation of the PDE when you establish a replicating strategy: "OIS discounting" is a shortcut to the more correct statement "risk-free discounting + adjustment due to the CSA collateral (assumed it is OIS remunerated ) clashes with the OIS disocunting". Besides a big assumption behind that is that the bank pays OIS on the collateral it receives and pays OIS to fund the collateral it needs to post. See here for a deeper analysis. For interest rates it is trikier, since in my opinion now swaps and FRAs are primary, not derivative secuirties. In the paper above also this issue is discussed. miscelania
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### OIS discounting collateralized swaps proof

OK. Thanks so much for your help. tula
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### OIS discounting collateralized swaps proof

You can derive it within the standard risk-neutral pricing framework, under the assumptions of continuous bilateral cash collateralization and zero thresholds / minimum transfer amounts. The standard pricing formula includes cash flows due to collateral posting as well, but this simplifies to a formula where you discount the terminal payoff at the (instantaneous) collateral rate c(t), ie.Btw this equality is already present in Johannes and Sundaresan 2003. There are multiple ways to derive it, eg via PDEs or using a martingale argument. The name OIS discounting probably comes from the fact that in most cases the collateral rate is the overnight rate. list
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Joined: October 26th, 2005, 2:08 pm

### OIS discounting collateralized swaps proof

QuoteOriginally posted by: ancastThere is no such discounting as an OIS discounting. It just derives from a mathematical fact in the derivation of the PDE when you establish a replicating strategy: "OIS discounting" is a shortcut to the more correct statement "risk-free discounting + adjustment due to the CSA collateral (assumed it is OIS remunerated ) clashes with the OIS disocunting". Besides a big assumption behind that is that the bank pays OIS on the collateral it receives and pays OIS to fund the collateral it needs to post. See here for a deeper analysis. For interest rates it is trikier, since in my opinion now swaps and FRAs are primary, not derivative secuirties. In the paper above also this issue is discussed.ancast, could you please to help with simple example to understand collateralization. In your paper, the 1-step binomial scheme let consider call option collaterized pricing. We have t = 0 and t = 1 = T; S ( 0 ) = S , S ( 1 ) is either Sd or Su ; call value C = C ( 0 ) Cd and Cu. From formulas (1) and(2) it looks that collateral is posted at t = 0 but its value 'V' does not specified in this case? list
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Joined: October 26th, 2005, 2:08 pm

### OIS discounting collateralized swaps proof

QuoteOriginally posted by: tulaYou can derive it within the standard risk-neutral pricing framework, under the assumptions of continuous bilateral cash collateralization and zero thresholds / minimum transfer amounts. The standard pricing formula includes cash flows due to collateral posting as well, but this simplifies to a formula where you discount the terminal payoff at the (instantaneous) collateral rate c(t), ie.Btw this equality is already present in Johannes and Sundaresan 2003. There are multiple ways to derive it, eg via PDEs or using a martingale argument. The name OIS discounting probably comes from the fact that in most cases the collateral rate is the overnight rate.What is V ( ) in this formula? ancast
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Joined: July 14th, 2002, 3:00 am

### OIS discounting collateralized swaps proof

QuoteOriginally posted by: listQuoteOriginally posted by: ancastThere is no such discounting as an OIS discounting. It just derives from a mathematical fact in the derivation of the PDE when you establish a replicating strategy: "OIS discounting" is a shortcut to the more correct statement "risk-free discounting + adjustment due to the CSA collateral (assumed it is OIS remunerated ) clashes with the OIS disocunting". Besides a big assumption behind that is that the bank pays OIS on the collateral it receives and pays OIS to fund the collateral it needs to post. See here for a deeper analysis. For interest rates it is trikier, since in my opinion now swaps and FRAs are primary, not derivative secuirties. In the paper above also this issue is discussed.ancast, could you please to help with simple example to understand collateralization. In your paper, the 1-step binomial scheme let consider call option collaterized pricing. We have t = 0 and t = 1 = T; S ( 0 ) = S , S ( 1 ) is either Sd or Su ; call value C = C ( 0 ) Cd and Cu. From formulas (1) and(2) it looks that collateral is posted at t = 0 but its value 'V' does not specified in this case?List, I think that there is a bit of misunderstading when you read the paper.C is not the value of the call, but it is the collateral posted at time 0. The value of the derivative is V (possibly a call option if you wish). So, assuming a given fraction \gamma of the value of the contract is collateralized, you can write that C = \gamma V.The example I present in the paper is with 3 steps, but is ideally the same as in the one-step case.On a more general note, something important often overlooked when people refers to OIS discounting, is that it is implicitly assumed that parties agree to revaluate the contract, and hence post collateral, by using not a risk-free value, but the value that includes the effect of the collateral (the LVA in my paper). Now, since in practice collateral is remunerated with OIS, which is also what is generally considered the risk-free rate, it happens that the LVA is nil and OIS discounting is equivalent to risk-free discounting. More generally, if the collateral is remunerated at a different rate, then the LVA is not zero, and in this case there may be some divergence on how to revaluate the deal (e.g.: one party may affirm that the collateral has to be computed on the risk-free value of the contract, not the value obtained with OIS discounting), and the amount of collateral to post. This also shows that OIS discounting is just a shortcut working only under specific assumptions. list
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### OIS discounting collateralized swaps proof

thanks ancast for respond and i indeed changed notations. My question was whether V is say BS price with C = 0, if we talk about about call option. if counterparties agree with collateral 100% , ie gamma = 1. Whether initial collateral will post at next date t = 1 or it might be at t = T = 1 in our example? How this initial value of collateral C is defined at initial moment at t = 0 , or t = 1 which one is correct ? or this initial value C is an agreed value.
Last edited by list on May 25th, 2012, 10:00 pm, edited 1 time in total. ancast
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### OIS discounting collateralized swaps proof

List,I am not sure I fully get your question: if you mean which is the value the parties consider to determine the amount to post as collateral, then this is exactly what I tried to say in my reply above. More sepcifically, when you evaluate a contract, you can assume that collateral (at any time starting from t=0) is the value that considers also the effects of the collateral itself (so that value + LVA is what is posted as collateral): in this case it is possible to show that the LVA can be included in the valuation simply by switching from risk-free discounting to collateral-rate discounting.You can alteratively assume that the collateral is computed on the base of the risk-free value of the contract (without including the LVA effect), so that it is not possible to use anymore the colltaeral-rate disocunting as a shortcut, but you have to separately evaluate the risk-free and the LVA components of the total value.In the discrete time setting at the beginning of my paper, only the first case is considered (also in the continuous time setting to be honest). The generalzation is quite simple once you et C = \gamma V*, where V* is not the value V computed recursively in the PDE or on the tree, but it is the risk-fee value evaluated for example by B&S formula (with risk-free disocunting).At the moment I am studying the CSA agreeement to understang which is the most reasonable choice complying with it. deepdish7
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Joined: October 16th, 2010, 6:57 pm

### OIS discounting collateralized swaps proof

i'd rather ask, why OIS are now used for projection curves. This does not make just any sense to me (whereas using OIS for discounting makes full sense)projection curves - I mean in an interest-rate swap with 3M LIBOR quarterly floating payments. why the hell is the new general approach today to project forward cashflows using OIS curve, not the LIBOR zero curve, as it always used to be. when the cashflows will actually depend on future LIBOR rate, not on future OIS rate. this seems just very strange to me
Last edited by deepdish7 on May 26th, 2012, 10:00 pm, edited 1 time in total. ancast
Posts: 144
Joined: July 14th, 2002, 3:00 am

### OIS discounting collateralized swaps proof

QuoteOriginally posted by: deepdish7i'd rather ask, why OIS are now used for projection curves. This does not make just any sense to me (whereas using OIS for discounting makes full sense)projection curves - I mean in an interest-rate swap with 3M LIBOR quarterly floating payments. why the hell is the new general approach today to project forward cashflows using OIS curve, not the LIBOR zero curve, as it always used to be. when the cashflows will actually depend on future LIBOR rate, not on future OIS rate. this seems just very strange to me????. Who told you that, deepdish7? This is exactly what it is NOT happening now in practice, where you use 2 curves, one to disocunt and the other to project forwards.
Last edited by ancast on May 26th, 2012, 10:00 pm, edited 1 time in total. list
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Joined: October 26th, 2005, 2:08 pm

### OIS discounting collateralized swaps proof

QuoteOriginally posted by: ancastList,I am not sure I fully get your question: if you mean which is the value the parties consider to determine the amount to post as collateral, then this is exactly what I tried to say in my reply above. More sepcifically, when you evaluate a contract, you can assume that collateral (at any time starting from t=0) is the value that considers also the effects of the collateral itself (so that value + LVA is what is posted as collateral): in this case it is possible to show that the LVA can be included in the valuation simply by switching from risk-free discounting to collateral-rate discounting.You can alteratively assume that the collateral is computed on the base of the risk-free value of the contract (without including the LVA effect), so that it is not possible to use anymore the colltaeral-rate disocunting as a shortcut, but you have to separately evaluate the risk-free and the LVA components of the total value.In the discrete time setting at the beginning of my paper, only the first case is considered (also in the continuous time setting to be honest). The generalzation is quite simple once you et C = \gamma V*, where V* is not the value V computed recursively in the PDE or on the tree, but it is the risk-fee value evaluated for example by B&S formula (with risk-free disocunting).Thanks for reply. I am trying to undestand mechanics of the CSAAt the moment I am studying the CSA agreeement to understang which is the most reasonable choice complying with it.Thanks for detail explanation. I am trying understand CSA rules: what are fixed rules and what are flexible agreements. In binomial scheme with CSA = 0 everything is known: S, Su, Sd, payoff for scenario 'up' and 'down', and call value at t=0 that is V. Let for simplicity gamma is 100%. In such setting whether one can define a unique value of collateral at t = 0 and at what time the collateral is posted? and actually in general setting it is not clear who have to pay collateral as two parties could default? ------------Subjectively thinking, though it can be wrong that this setting does not complete. If we deal with binomial scheme in its classical setting for say call option we do not specify credit risk, ie we need something to add. Talking for the option on stock credit risk is something different than to talk about bonds. Thus if underlying is a corporate bond we need something to complete the standard binomial scheme to say reduced form model of the credit risk?
Last edited by list on May 26th, 2012, 10:00 pm, edited 1 time in total.  