Fallback, cash flows and OIS discounting
In the IBOR fallback issue, there are different criteria we would like to impose in order to obtain a clean fallback. Among those criteria are the absence of value transfer and the coherence (IBOR is IBOR).
The fallback procedures proposed in ISDA consultation rely on the replacement of each IBOR fixing by an RFR-linked rate plus an adjustment spread.
The existence of CCP basis (and bilateral to CCP basis) which is potentially different between LIBOR products and OIS products leads to the following paradox: if we want to achieve the absence of value transfer, we need to select an adjustment spread that is different for the same IBOR and RFR depending on the clearing location, which is a violation of the coherence criterion. If we keep coherence, we have the same spread and consequently there is a value transfer.
Where is the paradox coming from? I would say that it is coming from a misunderstanding of the OIS discounting formula. The formula is written as (see Formula 8.6, based on Formula 8.3, Theorem 8.1 in my multi-curve book)
The formula looks like the valuation is obtained by discounting a cash flow. This is also what the popular name of the formula seems to imply; and this is also why I don't like the name. It gives the wrong intuition. The formula is correct but should not be interpreted as discounting a single cash flow. The reality is many cash flows (daily variation margin); the actual theoretical IBOR fixing is never paid as a single amount. The summary of all those daily cash flows into a single expectation is a very simple formula for a complex process. The formula is based on hypothesis, including perfect replication hypothesis by IBOR related instruments and OIS (hence the OIS discounting formula). But that replication is dynamic, with the hedging changing on a continuous basis, depending on the VM paid and the evolution of the market.
This approach is valid only where there is availability of the hedging instruments. In practice, it means that there is one formula coming from one pool of liquidity and one set of rules at each CCP (and at each bilateral agreement). Each CCP is its own world in term of hedging and price. Of course in practice it is possible to transfer risk (and thus price) from one CCP to another, but at a cost (the posting of IM, summarised in MVA figures). Between CCP prices there is not unique transformation of price but a price range, with the range depending of the IM rules and the cost of funding of potential arbitrageurs.
But this does not answer to the question: where is the paradox coming from. Based on the above analysis, my answer would be that there is no paradox in our criteria of asking for an absence of value transfer and a coherence on IBOR fixing. The problem/paradox/error comes from trying to solve the dynamic hedging problem with a static solution. The ISDA proposed solution (adjusted RFR + adjustment spread) is decided on the announcement date and is fixed from then on. Imposing a static solution to a dynamic problem leads to impossibility to obtain the price based on a dynamic world with segregated pools of liquidity.
The problem is not coming from our criteria, those are, to my opinion, fair requests, but from the constraints imposed on the solution which are not in line with the valuation framework where we want to apply them.
The fallback procedures proposed in ISDA consultation rely on the replacement of each IBOR fixing by an RFR-linked rate plus an adjustment spread.
The existence of CCP basis (and bilateral to CCP basis) which is potentially different between LIBOR products and OIS products leads to the following paradox: if we want to achieve the absence of value transfer, we need to select an adjustment spread that is different for the same IBOR and RFR depending on the clearing location, which is a violation of the coherence criterion. If we keep coherence, we have the same spread and consequently there is a value transfer.
Where is the paradox coming from? I would say that it is coming from a misunderstanding of the OIS discounting formula. The formula is written as (see Formula 8.6, based on Formula 8.3, Theorem 8.1 in my multi-curve book)
The formula looks like the valuation is obtained by discounting a cash flow. This is also what the popular name of the formula seems to imply; and this is also why I don't like the name. It gives the wrong intuition. The formula is correct but should not be interpreted as discounting a single cash flow. The reality is many cash flows (daily variation margin); the actual theoretical IBOR fixing is never paid as a single amount. The summary of all those daily cash flows into a single expectation is a very simple formula for a complex process. The formula is based on hypothesis, including perfect replication hypothesis by IBOR related instruments and OIS (hence the OIS discounting formula). But that replication is dynamic, with the hedging changing on a continuous basis, depending on the VM paid and the evolution of the market.
This approach is valid only where there is availability of the hedging instruments. In practice, it means that there is one formula coming from one pool of liquidity and one set of rules at each CCP (and at each bilateral agreement). Each CCP is its own world in term of hedging and price. Of course in practice it is possible to transfer risk (and thus price) from one CCP to another, but at a cost (the posting of IM, summarised in MVA figures). Between CCP prices there is not unique transformation of price but a price range, with the range depending of the IM rules and the cost of funding of potential arbitrageurs.
But this does not answer to the question: where is the paradox coming from. Based on the above analysis, my answer would be that there is no paradox in our criteria of asking for an absence of value transfer and a coherence on IBOR fixing. The problem/paradox/error comes from trying to solve the dynamic hedging problem with a static solution. The ISDA proposed solution (adjusted RFR + adjustment spread) is decided on the announcement date and is fixed from then on. Imposing a static solution to a dynamic problem leads to impossibility to obtain the price based on a dynamic world with segregated pools of liquidity.
The problem is not coming from our criteria, those are, to my opinion, fair requests, but from the constraints imposed on the solution which are not in line with the valuation framework where we want to apply them.
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