Hi,In order to justify (to myself) using OIS discounting, I am looking at the simple case of a commitment by A to make a payment of |x| Euro to party B in one week from the spot date. I assume that the trade is marked to market daily, fully collateralised in Euro cash and that collateral earns EONIA. I then attempt to replicate the cashflows on the trade, including the collateral cashflows, using the ON market and the 1W EONIA swap. In the short note attached, I find that to replicate the cashflows, A must be paid |x| / (1 + r d) on the spot date, where r is the 1W EONIA rate and d is the associated day count fraction. Is this the way that others justify OIS discounting or is there a better argument (or a correct argument in the case that this one is wrong)? I have read papers describing the continous time justification but I have never seen a simple discrete time justification using actual market instruments to hedge the flows.If I extend the example by saying that the payment is denominated in EUR but that the collateral is USD, I get the same answer for the amount that A must be paid to replicate the cashflows (assuming that A can lend EUR ON and borrow USD ON). I must be missing something here because from what I have read, I think that the EUR/USD OIS basis spread should influence the discounting here?I would appreciate any comments or insights.Thanks,Frank.

For your first example, Bianchetti ("Two curves, one price:Pricing and hedging interest rate derivatives decoupling forwarding and discounting yield curves"), Mercurio ("Interest rates and credit crunch: New formulas and market models"), Piterbag ("Funding beyond discounting") show that the discounting for a contract that is collateralized continuously should be operated by the collateral rate...?For the second, especially by marking to market daily, you are going to run in cross currency basis spreads who's going to mess with the interest rates.

Hi APlus,Thanks for your feedback. I have read the papers that you referenced and from those papers it is clear that you use the discount factor derived from OIS swaps to discount the payment due in one week. However, I was trying to derive a simple hedging strategy, using market instruments, that exactly matches the cashflows associated with the payment i.e. the payment itself and all interim daily collateral flows.I agree with you on your second point also. Again, in this case, I am trying to come up with a simple hedging strategy that matches the EUR payment itself and all interim daily USD collateral flows. If I assume that A can lend EUR ON and borrow USD ON, then applying essentially the same strategy as above I get that EUR/USD OIS basis swaps do not enter the hedge. I was wondering where I was making a mistake/missing something.Thanks,Frank.

Not sure if this is what you are looking for regarding the EUR/USD.. (what is your base currency when you lend the EUR and borrow the USD?)Anyway, at some point, you are going to do a cross currency swap beteen the EUR and the USD, which I guess it's the thing that interest you.The problem comes from the fact that the OIS is not the risk free interest rate, but the rate with which you are "investing". When you are doing the EUR/USD swap, in fact you are doing 3 swaps instead of one: OIS (EUR) against risk-free(EUR), risk-free(EUR) against risk-free(USD), risk-free(USD) against OIS(USD). And that is how the cross currency basis spread enters the game, so you should also consider the cross currency basis swaps. The opinions are devided where this spread comes from, what is the percentage of its liquidity related part and of its credit related part, as before the cerdit crunch it has been practically zero, but the end point is that the CCBSpread is a market price. You can well get its values for the different tenors, but the problem is that it is not a "normal" interest rate curve, as you can not do extrapolation on it (the 1M(t), 1Y(t) and 5Y(t) spreads seem to behave indipendently). So you end up extrapolating a surface (value, Tenor, time) when doing evaluation.

If we stick with the simple example of the commitment to make a EUR payment, collateralised in EUR (daily posting, full MtM amount in cash with no thresholds or minimum margin amounts), in one week from the spot date.In the past this trade was said (incorrectly) to be worth x / (1 + L d) on the spot date, where L is 1W Euribor and d is the associated day count fraction. The simple argument (ignoring the collateral and assuming Libor was risk-free) was that if I was given |x| / (1 + L d) on spot, I could deposit it at rate L and have |x| in one week to make the payment due.I was wondering if there is a similar argument that shows that this trade is actually worth x / (1 + r d') on the spot date, where r is 1W EONIA and d' is the associated day count fraction. The attached note attempts to form such an argument but I am not sure if 1) it is a valid argument, 2) there is a better argument or 3) there is no simple hedge for this case. If collateral posting was weekly in this example, then the answer is simple since you just enter a 1W received fixed EONIA swap - the floating leg is covered by the interest you earn on the collateral account & the fixed leg grows to |x|.Thanks again,Frank.

On your first question about 'justifying OIS discounting', the best argument I have found is Piterbarg's 'Cooking with Collateral' in Risk, where he explicitly constructs an arbitrage that exists if you don't discount at the rate paid on the collateral. It seems to me that a lot of the other justifications start on shaky ground. Often the existence of risk free rates is assumed, without being very clear on exactly what this means. Also given there is no need to fund hedges there will never be any need to borrow money; neither is there any left over cash to deposit after hedging in a fully collateralised world. Finally there tends to be a lot of numeraires floating around, which I don't like either (a collateralised portfolio, including the collateral posting, is not worth anything, so what does it mean to measure the value of something worth nothing in terms of a numeraire?).

Thanks Joet for the helpful reference - I hadn't read this before.What I was trying to hedge in my note was the sale of [$]|x|[$] units of a domestic collateralised domestic zero coupon bond i.e. the instrument with price process [$]P_{d, d}(t,1W)[$] in the language of the paper. I realise now that I was making it harder than it needed it to be. I was thinking that I needed to be able to borrow and lend at the ON rate but as you say in your note, there is no need to borrow/lend in a fully collateralised setting. The hedge is just to enter an OIS swap with notional [$]|x| / (1 + r \delta)[$] where I receive fixed [$]r[$] and pay the ON rate (and assume for simplicity that there is a final notional exchange on both legs). The receive fixed leg matches the sold collateralized zero coupon bond (i.e. daily collateral flows & final amount). The pay floating leg (i.e. daily collateral flows & final amount) is worth the (negative of) the notional of the swap i.e. [$]x / (1 + r \delta)[$].

One should take care which type of cash flows is discounting: out- or in-.Because for the cash outflows you should consider your own spread (as funding cost/DVA) when you do the discounting (this makes your liabilities to decrease as your credit worthiness lowers/spread increases - which is the observed fact), so you discount at your own interest rate (risk free + own spread).