Valuation in the Absence of Dynamic Hedging
If we hold the point of view of a treasury desk, particularly one of a development or small financial institution that swaps the bond straight away with another counterpart, why do we care about correctly pricing an instrument? After all, the bond and the coupon leg of the asset swap cancel each other, so technically any value would do. The argument would seem even stronger for institutions with good credit ratings for which the discount factors of the bond and the coupon leg of the swap are fairly close (there is a small correction ).
At first the above theory seems sensible, however, there are a few cases and reasons why it is not. The first and most important has to do with the exit price of a trade. We have already mentioned this in Section 6.2.2 when we discussed the valuation of exotic structures.
For transactions that, like the majority of swaps and the totality of exotic swaps, are not quoted officially on an exchange but rely on internal valuations, the exit price is the last agreed price before the transaction is interrupted, usually, voluntarily.
A swap is made of a series of payments that each side owes the other. At payment date the amount is a known quantity and each side settles with the other. However, as we mentioned in Section 2.4.1, there needs to be an agreement on the net present value of both legs (the present value of one minus the present value of the other, which we called mark to market [MTM]) for collateral issues. The party who sees a negative MTM needs to post collateral in quantity equal (or at least proportional) to the MTM. This means that the suggestion in the opening paragraph that any value would do for the coupon leg of the asset swap does not hold. Unless we have a value that is reasonably in line with our counterpart there will be collateral disputes.
Collateral issues would be a strong enough reason for the importance of a correct valuation, however, at closer inspection, there might be reasons for grayness. Because of netting agreements (collateral is managed not at trade level but at portfolio level) one side might wish to accept a mistake on the other if it allows it to pay less collateral. Since the real payment on settlement date is not in dispute, the issue is only of collateral. There might be even more business-driven reasons, such as a client's line of credit.
Each institution, in order to disperse credit risk, sets itself limits on how much business it does with a specific counterpart and this limit is in terms of MTM. An investment bank eager to do business with the treasury of another institution might be willing to accept a lower MTM (from its own point of view) if this means it will remain within the limits of the treasury.
Let us imagine that investment bank ABC and treasury XYZ have two swaps between them, a plain vanilla with fixed coupon C, easy to value, and an exotic one. From ABC's point of view the vanilla one has MTM MTMV
large and positive. The exotic swap linked to some function f(Lm) has MTM MTME
which is large and negative. We have, however,
that is, the difference in mark-to-market results in an overall positive collateral position for ABC and the treasury XZY would need to post collateral of an amount .
Let us imagine that XYZ makes an error in valuing the exotic swaps resulting in an even greater MTM (i.e., more negative for ABC) such that
Although this on paper would technically mean ABC owes more to XYZ, ABC might let it pass because, overall (after netting the two swaps), it reduces the collateral it receives from XYZ to. This is a bit risky, but otherwise XYZ (treasuries usually have stricter limits than banks) might go somewhere else for business if net MTM had gone above the accepted limit for counterparty ABC. In general ABC sees this type of error as an involuntary way on XYZ's part to free the credit line and therefore allow ABC to do more business. ABC knows that on settlement date all will be solved by the fact that the payment becomes a certainty.
If collateral management offers reasons going beyond the strictly math- ematical/financial ones for accepting a misprice of a transaction, exit prices constitute a different matter. As the name suggests, an exit price is the last agreement between two counterparts: whereas a collateral dispute will be cleared, in part, by the actual settlement, there is nothing beyond the exit price, so it is in both parties' interest to value it correctly. We could envisage the need to calculate an exit price in two main situations: the bond terminates and we, as issuers of the debt, need to interrupt the asset swap accordingly or we need to transfer the asset swap to another counterpart.
In the following discussion let us lighten the notation a little. Let us define by Bt the value at time t of a bond with nominal value N paying a coupon linked to some function fj (which can be a fixed or floating coupon), that is,
and let us define the mark to market of an asset swap as , that is, the value at time t of a swap with principal N that pays LIBOR plus a fixed spread fixed at a time t in exchange for the coupons of the bond, as
Using the above the bond plus asset swap structure at inception (we choose inception to be time t1) will be given by
Assuming the bond was issued at par, both sides of the above at time t1 are equal to zero. Let us imagine now the situation in which the bond is terminated. The reasons for which a bond terminates can be multiple, however, the most common ones are either a trigger event such as the one shown for a TARN structure in Section 6.2.1 or the issuer deciding to call the bond, or a buyback, that is, an institution repurchasing its own debt. Let us consider the second situation as an example and consider thus the case in which the issuer at time returns the nominal value to the investor.
Assuming the issuer follows financial logic, the bond is called when it is worth more than par by an amount 8, that is,. From our
perspective, immediately before we call the bond
at the same time the liability we have on the bond is compensated by our gain on the swap since, always from our own perspective,
The above is not entirely true. We mentioned in Section 7.2.2 that a change in the issuer's own credit rating has an effect on the relation between the bond and the swap price: one could imagine a strange scenario in which all market inputs are frozen except for a progressive deteriora- tion/improvement of the the credit rating of the issuer, in which case the bond would decrease/increase in value without being matched by a respective move in the asset swap mark to market. In this scenario we could not have the same variable 8 in both Equations 7.8 and 7.9. We are going to ignore this scenario and imagine instead a situation in which the credit rating of the issuer, our credit rating, is fairly constant and good, meaning that the present value of the coupon payments are the main drivers of bond and swap prices.
As far as the value of our assets is concerned, everything is fine only if we agree on the value of 8 with our swap counterpart. Otherwise, should the 8 on the swap side be smaller than the one we see on the bond side, we would be registering a loss.
If the example of early termination might seem a bit hard to grasp, after all, one might say the loss is simply between what we thought we had, and as far as the transaction itself is concerned there is no real loss. Let us not forget that in the modern economy the records of a company, in the form of financial statements, are crucial instruments and the belief a company has of the value of its own assets is a fundamental pillar of the economy because it is central to the way others see the health of the institution.
An example even more clear, however, is the case where we need to change, for some reason, swap counterpart. Let us imagine ourselves again at a point where Equation 7.8 is true. In order to change counterpart we have two options: either we enter into the same swap we had with the first counterpart, that is, a swap with mark to market
or we enter into a market asset swap (see Section 5.3.2) where the swap has principal equal to the value of the bond and a new asset swap spread such that the swap prices at par, that is, with
equal to zero. Let us consider the first case, which is probably easier and more common: we thus maintain constant the characteristics of the function fj appearing in the bond and coupon leg of the swap and we maintain constant the asset swap spread. In the simplest scenario we terminate the swap with the first counterparty, receive 8 (the value of the MTM of the swap, which is positive for us) and give it to the second counterpart to cover the fact they start the swap at a disadvantage, that is, with a negative MTM from their point of view.
This simple scenario relies on the assumption that the second counterpart also sees the value of the MTM of the swap as being equal to 8 and this situation is an argument in favor of the importance of valuing an instrument correctly even when carrying out static hedging. It is unlikely that all three parties–us, the first counterpart, A, and second counterpart, B–will agree on the value of δ. There could even be a situation in which B sees the MTM value as larger than 8, meaning that the lump sum we receive from A (assuming that at least the two of us have agreed) will not be enough to give to B and, assuming we proceed with B, the extra amount will be a realized loss. This loss, as opposed to the disagreements we were mentioning in the context of collateral, is real.
Another way of changing counterpart is through the process known as novation. This process is usually more common since the payment of a lump sum, as in the example above, while possible, rarely happens due to the likelihood of a disagreement in the value of the MTM. A novation is a legal process in which one of the parties in an agreement is exchanged for another. In our swap, we were facing A and now we are facing B. Should we all agree on the value of δ, the process then becomes a purely legal one. In most cases, however, B will want to be paid the difference between what they see and what we see as the value of δ.
From the examples above one obtains the impression that when exiting a trade there is always a loss for the debt issuer, after all, why can't it be the opposite, and why can't we receive something from B? There are many reasons for it, let us try to name a few.
A debt issuer, particularly one that is swapping the trade with an external institution, prizes static hedging exactly because it is static, a transaction that takes all hedging worries away until the maturity of the debt. If it needs (as opposed to wants as in the case of a buyback) to exit a trade, it is usually because something serious has happened, possibly a grave deterioration or even default of the swap counterpart. Because these types of events are never isolated, there is a great chance that the market itself is in turmoil with liquidity and optimism in short supply. In this environment the second counterpart will need an incentive to take on the trade.
A further reason, closely linked to the first though, is that a treasury desk swapping a bond might not be aware of the hedging costs incurred by a swap counterpart, and these hedging costs tend to increase in the type of environment when a change of counterpart is carried out.
All these considerations should be enough to realize that static hedging is not an alternative to a proper valuation of a financial instrument. These considerations are the reasons for which we have discussed treasury business and risk management only after having introduced topics such as curve construction, credit, and emerging markets' liquidity, which at first might have seemed unrelated. Discussing these topics was a way of showing how theoretical and practical elements can add complexity to the valuation of financial derivatives.