Rationale for OIS discounting for collateralized derivatives?

Can someone explain to me the rationale for why the market may be moving towards OIS discounting for fully collateralized derivatives?

Most counterparty agreements specify some sort of ois rate for the interest paid/received on posted collateral. So the OIS rate is the appropriate one to use for discounting future cash flows.

Prior to 2008 the OIS/Libor spread was small and stable, so you didn't really need to worry about this, but now it's much larger, so people are taking it into account. The reason it's "big news" now is that properly switching pricing systems over to use OIS discounting is a large change, so most places are only now getting this online.

If you assume that you do not have any market risk (a strange assumption, but it would hold for example if you are fully hedged), then a (correctly) collaterlized derivative does not have any net future cash flow. Clearly: if the derivative contract has a cash flow of -X, its value will go down by X and the collateral account will have a cash flow of +X (the corresponding collateral will be returned).

If there is no future cash flow, there is no discounting (in the sense of funding costs). However there is a new question now: what is the correct amount of collateral C we should post in t=0 to collaterlize the cash flow in t=T?

Since collateral is accrued according to the collateral contract by the OIS rate we would like to have that the accrued collateral account matches the cash-flow, that is

C * (1 + r * T) = X

where r is the OIS rate. That is we determine the collateral by OIS discounting

C = X / (1 + r * T).

OIS discounting is the way to determine the amount of collateral we have to post.

(You can make this argument mathematically correct (under some general assumptions) and show that collateralization is like having a different currency which has its own interest rate, I have some stuff on this (paper, spreadsheet for OIS bootstrapping, source code) here: http://www.finmath.net/spreadsheets/curvecalibration/ )

• Could you please give an example of a "derivative contract with a cash flow of -X"? I am not sure I understand that part, my understanding is that the value moves in- or out-of-the-money. How is there a cash flow? – user1157 May 27 '14 at 15:01
• It appears you are thinking of a (call) option only. In that case: being that option short would be an example. A more natural example is that of a swap exchanging fixed rate C versus floating rate L, i.e. X = C-L (negative for high L). The swap could be a structured one with a complex coupon C, but the above applies for plain products as well. – Christian Fries May 28 '14 at 17:28

The OIS rate is more stable than Libor, right? And according to this article from Risk Magazine:

The party that is owed money at the end of the swap will have been paying an OIS rate on the collateral it has been holding, and so the ultimate value of the cash it will receive will be the sum it is owed minus the overnight interest rate it has had to pay on this collateral.

There are three interest rates in play here:

1. Discounting market valuation
2. Discounting for collateral call valuation
3. Interest paid upon collateral (determined by the CSA terms)

In order for the collateralisation process not to generate surplus value to either participant, considering a single cashflow received today and then repaid a day later will give the answer that we need 1 = 2 = 3. Otherwise there will be a net flow of cash on today or tomorrow (or both).

First, consider net flows today. The person paying for the cash flow is paying out market value & receiving the collateral call value. We need 1 = 2 for the two to balance.

Second, take net flows tomorrow. That same person will be receiving the cash flow and repaying collateral with interest. For the values of these to balance then we need 2 = 3.

So then 1 = 2 = 3 in our example. Discounting for market value must follow discounting at the rate determined in the collateral agreement.

And since you can essentially synethesize any derivative by pasting together 1-day cashflows of this nature then this also estabishes that 1 = 2 = 3 for all derivatives.