I'd appreciate if someone could shed light on the following two related questions regarding Fujii, Shimada and Takahashi's "A Note on Construction of Multiple Swap Curves with and without Collateral" http://ssrn.com/abstract=1440633.Section 3.1 of the paper discusses products with "perfect and continuous collateralization with zero threshold by cash".The process for the collateral account is given aswhere V(s) is the value of the collateral account at time s; y(s) is the difference between the risk-free rate r(s) and the rate c(s) paid on the collateral (the rates are assumed different), i.e. y(s)=r(s)-c(s); h(s) is the value of the derivative that is being collateralized; and a(s) is the "number of positions" of the derivative.Note the position size a(s) itself is allowed to change. The authors assume a(s) is time dependent because they state they wish to apply a self-financing strategy to the collateral. But the position size in many collateralized trades doesn't change. I'd like to better understand why this device of letting a(s) vary works and therefore how to interpret V(s). Any helpful comments would be greatly appreciated.Secondly:The general solution of the above equation is (T is final maturity):The authors choose the initial condition as V(t) = h(t). Then a(s) is set equal to which is chosen in order to collapse the general solution to the following:but which shows V(T) is not equal to the value h(T) of the derivative. What then is the meaning of the value of the account V(T) how is a(s) helping?Thanks in advance.

Hi Hattusa,I gave a different derivation to the valuation of a collataralized deal: it differs from from the one by Fujii et al. and by the most famus one by Piterbarg, which is not fully satisfying as well.There is much more to say in any case. Hidden assumptions of the collateral discounting are quite subtleThe links to my papers are http://iasonltd.com/FileUpload/files/pr ... %202.0.pdf and http://iasonltd.com/FileUpload/files/FX ... ateral.pdf"

Hi Antonio -- thanks very much for the references. I did in fact come across your papers earlier and I am currently going through them with interest. I'll let you know if I have any comments/questions. I hope to remove questions that come to mind on several points in various papers, like the one I referred to in starting this thread, by relating them to one another.Rick

Hi all,Ancast's papers are helpful generally with this question, but just wondered if anyone can shed light on the intuition of the specific Fujii et al derivation?Like Hattusa, I am struggling to understand the a(s) term, particulary:(1) why isn't it constant?, and(2) given that it isn't constant, why it is of the form assumed?Any help would be much appreciated.Cheers,Donal

Antonios paper (only worked through the single currency one up to now) is the most detailled and explicit one on the topic I read so far. Very good.Donal, I understood the Fuji Paper (A Note on Construction of Multiple Swap Curves with and without Collateral from March 2010) as follows:If you are e.g. long a derivative at time t which value is h(t) [ this is the value we seek to determine ] , then you get exactly this amount on your collateral account V, i.e. V(t) = h(t). (However at the same time you pay of course h(t) as the option premium to the cpty, so effectively no payment occurs at t)If the rate to be paid on the collateral is different from the rate you get on risk free investments, i.e. y(s) != 0, say always > 0 you earn money on the collateral account at the rate y(s). The trading strategy in formulas (26) say that this money is reinvested in the derivative contract. This is not what happens in reality where the reference amount of the derivative is usually constant, but an artificial strategy for valuation purposes only.One then sees that under this trading strategy at maturity the derivative payoff h(T) is replicated by the collateral account V(T) times e^{-\int_t^T y(s) ds} (, the latter factor accounting for the (in our example) growing position in the derivative.)Valuing the collateral account under the risk neutral measure then gives the desired pricing formula (28) for the derivative value at time t.cheersPeter

Hi all,I struggled to understand this particular paragraph initially also so I thought that I would add a few comments to Peter's explanation in case they might help.h(t) is the value at time t of the final payoff h(T) at time T and any intermediate cashflows arising from the collateral account of one unit of the derivative with price process h(s). The authors arrive at the pricing equation (28) for h(t) using two methods. Method 1, described by Peter, is in the body of the text and method 2 is in the footnote.Method 1a(s) is introduced in order to transfer all the intermediate flows on the collateral account to the terminal time T. In particular, all cash in the collateral account is reinvested in the derivative so that a(s)h(s) = V(s) for all s in [t,T]. This strategy is self-financing and you have no flows in the period [t,T) and a terminal flow of a(T) units of the derivative i.e. a terminal flow of a(T)h(T). The two set of flows i.e. {a(T)h(T) at time T} vs {h(T) at time T and the intermediate collateral account flows} are equivalent. Hence, you can take the risk neutral expectation of the discounted flow a(T)h(T) at time T to obtain the pricing equation (28) for h(t).Regarding the initial question of how to interpret V(s): I would interpret {V(s):s in [t,T]} as the value process of the collateral account for the trading strategy where you hold a(s) units of the derivative at each time s in [t,T]. It follows that X(s):=V(s)/a(s) is the value process of the collateral account for one unit of the derivative.Method 2I found the method in footnote 10 more intuitive. h(t) is the risk neutral expectation of the discounted terminal value h(T) plus the risk neutral expectation of all the intermediate flows on the collateral account discounted. It is then shown that dh(t) = c(t)h(t)dt + dM(t) where M(t) is a martingale wrt the risk neutral measure. Equation (28) then follows immediately.Hope this interpretation is reasonable and that it helps. My main problem with the formulae was the assumption that y(s) has the same value when V(s)>0 and V(s)<0. V(s)>0 is discussed in the paper i.e. invest the collateral at r(s) and pay c(s) on the collateral. When V(s)<0, this implies that you can borrow at r(s) to fund the collateral account and receive c(s) on the posted collateral. The only situation where I can see that the symmetric nature of y(s) will approximately hold is if the bank can borrow & lend ON at the rate r(s)=c(s) in which case the arguments above are trivial. Am I missing something here or is this the way that others look at r(s).Thanks,Francis.

thank you Francis for this nice elaboration.Concerning your last point (different funding and deposit rates) :- provided that the collateral amount can be hypothecated, it can be seen as a source of funding. Therefore it may be a valid assumption that the derivative desk pays _and_ receives the same rate for the collateral account amount to / from the funding desk. Don't you think ? Then, the same rate r can be used in pricing, which also may well be different from the collateral rate c.- if you want r to be different for V > 0 and V < 0 : This case is in fact covered in Antonios paper (the first one, chapter 4, in particula formula 34). However seting \gamma = 1 in this formula (full collateralization) one gets the same pricing as in the previous case, namely discounting at the collateral rate c if I understand it correctly. This is quite counterintuitive, isn't it ?

Hi Peter,Thanks for your feedback and other helpful posts on this topic.I am struggling to understand your first comment. Would it be possible to expand on it a little for the case of a USD swap fully collateralised in USD cash where the collateral account earns/incurs Fed Funds i.e. where c(s) is a continuous representation of Fed Funds? Suppose the swap has an initial value of zero i.e. V(0) = h(0) = 0. If the swap value moves positive, we get posted collateral and deposit it in a risk-free bank account earning r(s). As a proxy for this, I think of lending the posted collateral overnight to a highly rated institution, bank A say, and earning approximately Fed Funds. So for the case of V(s) > 0, I could see r(s) approximately equal to c(s). As you state, they are not necessarily equal. If the swap value moves negative, we are asked to post collateral that we need to borrow. Assuming we are not perceived to be as creditworthy as bank A above, I would expect the borrowing to cost more than r(s). How does the hypothecation argument work here to attract r(s) on your borrowing?On your second comment, I haven't read Anotonio's paper(s) properly yet. I will probably be back with questions/comments when I do!Thanks again for the help,Francis.

Hi Francis, QuoteOriginally posted by: frank82 I am struggling to understand your first comment. [...] If the swap value moves positive, we get posted collateral and deposit it in a risk-free bank account earning r(s).what I meant was that in this case we do not deposit the posted collateral in a risk-free bank account but rather see it as a source of funding, therefore earning our own funding spread on it (= r = the same rate in the case when the swap value is negative and we need to fund the collateral amount), at least in the sense of an internal flow between derivative and funding desk. In other words the funding desk pays the funding rate r, no matter if it gets the money from outside or from the derivative desk via the collateral account. However this model assumes that the funding desk always needs / accepts funding at rate r, or as Fries puts it, the bank can always "buy back its own bonds".(The rate r being the funding rate may be significantly higher than a "risk free" rate, but in my opinion this is ok to price within the usual framework, since we just need a rate r at which we can lend and borrow money at the same time, right ?)When we want to differentiate between funding and deposit rate things get much more complicated, and in particular I would expect that even a fully collaterized deal will then have non zero net funding costs, i.e. the valuation should not be independent of the own funding spread, should it ? That's why I do not quite get what formula (34) says in Antonios paper.Best regardsPeter

Hi Peter,Thanks a lot, that explains your comment. I agree that I think r can take any value as long as you can use it for both lending and borrowing.Regarding your comment on Antonio's paper, it does appear counterintuitive that the funding spread does not enter the pricing equation (34) when \gamma=1. It may become clear when I read the paper more carefully. If so, I will post a comment.Thanks,Francis.

Hi Peter and Francis,Careful in Antonio's first paper (the one Peter refers to). While C is defined to be the collateral account (bottom of page 3), Eq.(12) for dC actually measures the change in something other than the funds paid to the counterparty. To see this, take gamma=1 for simplicity (full collateralization), then dC in Eq.(12) cannot equal the sum of dV and the collateral interest, for then the derivative would be overcollateralized. The end result Eq. (26) is still correct and can be obtained without making use of Eq. (12), e.g. through a cashflow analysis, or via Piterbarg. A slight modification to Antonio's derivation goes as follows: Eq.(16) and its differential must both hold over dt. In the differential, C increases by only its own interest over dt. This does not include readjustment of C at the beginning of the next dt, but the hedge generates the rest so that Eq.(16) still holds. Eliminating the hedge constants leads to Eq.(26).

Hi everybody,just to stress the fact that in the last version of the paper (you can find at www.iasonltd.com/resources ) there is an explicit distinction between collateral C and cash collateral account {\cal C}.Since what matters in the end to replicate the contract is the sum of all cash-flows, it is the variation of the collateral account (that includes the interests earned/paid) that enters the replication portfolio, even though the collateral evolves like dC = dV. I have to thank Hattusa for pointing that to me in a private conversation, since in the earlier version of the paper I did not properly separated the two concepts, although the results were not affected since I still worked with the relevant quantity, that is the collateral account (and not simply the collateral).The problem with buying back own bonds was investigated by myself elsewhere (the article has been recently published). I am not denying that this is possible, but I think I show quite irrefutably that there is no benefit originated by that, so assigning to the desk generating cash, the funding rate according to an internal transfer pricing since the bank can buy back its own bond, would produce an internal transfer of P&L from one desk to another, but the result is always nil for the bank as a whole. The problem now is to decide if the contract has to be valued from the desk's point of view or from the bank's point of view. I believe that the bank's perspective is to be preferred: I see a choral participation to the replication of the contract. I would agree with Fries and the others supporting the funding benefit argument only if the cash generated could actually be lent outside the bank at the bank's funding rate to a risk-free counterparty. For more on this, see my Dynamic replication of the DVA.

Hi Antonio,I get the points you mention. Thank you. But is there really a connection to my problem below concerning formula (34) ? Still I understand that you would discount a fully collaterized deal (\gamma=1) on the collateral curve, the result thus being independent of the spread between deposit and funding rate. Is that corect ?

QuoteOriginally posted by: pcaspersHi Antonio,Still I understand that you would discount a fully collaterized deal (\gamma=1) on the collateral curve, the result thus being independent of the spread between deposit and funding rate. Is that corect ?The effective discounting would be indipendent from the funding rate, the decomposition of the value in FVA, LVA and risk-free value would depend on the funding rate in any case, even with \gamma = 1.Plus, the value can still depend on the funding rate that enters in the drift of the underlying asset (see the 5 case below formula 34). This is only when Delta hedge cannot be operated via a repo contract, in which case you have always the repo rate in the drift of the underlying asset.

QuoteOriginally posted by: ancastPlus, the value can still depend on the funding rate that enters in the drift of the underlying asset (see the 5 case below formula 34). This is only when Delta hedge cannot be operated via a repo contract, in which case you have always the repo rate in the drift of the underlying asset.I am thinking about the following test case: The contract is just one cashflow -C(T) that I have to pay at time T. When entering the deal I receive the pv C(0) from the counterparty and have to put it on the collateral account. If the collateral rate is deterministic there will be no intermediate margin payments, and formula 34 is correct. However when the collateral rate is stochastic I will have to pay amounts to the collateral account or receive amounts from the account depending on the intermediate valutaions C(t) of the contract. If I have to fund the payed amounts at a different (say higher) rate than I earn on the received amounts, I would expect to have positive net funding costs. Therefore I would have expected the funding spread to enter formula 34. [ assume here the dynamics of the underlying independent of the collateral rate dynamics - the drift term of the underlying will have no influence on the pricing, since the payoff does not depend on any underlying ].This is related to the comment in the Fuji paper, but there funding and deposit rate are the same, hence no net effect:QuoteFuiji et. al... It is also useful to interpret the results in terms ofthe funding cost for the possessed positions. First, let us consider the case where there is a receiptof cash at a future time (hence, positive present value) from the underlying contract. In this case,we are immediately posted an equivalent amount of cash as its collateral, on which we need to paythe collateral rate and return its whole amount in the end. We consider it as a loan where we fundthe position at the expense of the collateral rate. On the other hand, if there is a payment of cashat future time (negative present value), the required collateral posting can be interpreted as a loanprovided to the counter party with the same rate. Therefore, compared to the non-collateralized trade(and hence, Libor funding), we get more in the case of positive present value since we can fund theloan cheaply, but lose more in the case of negative value due to the lower return from the loan lent tothe client.Maybe of course there is also a flaw in my way of seeing this.The other point:QuoteOriginally posted by: ancastThe problem now is to decide if the contract has to be valued from the desk's point of view or from the bank's point of view. I believe that the bank's perspective is to be preferred: I see a choral participation to the replication of the contract. I would agree with Fries and the others supporting the funding benefit argument only if the cash generated could actually be lent outside the bank at the bank's funding rate to a risk-free counterparty.I think there shouldn't be "unfair" internal flows between the desks, i.e. if the derivative desk earns the banks funding rate from the funding desk on the collateral this should imply that effectively the bank can earn that rate externally using the collateral. But lending to an external counterparty at the funding rate is not the only possibility, reducing or replacing funding (if operable) that else would have to be done using other sources is also enough to justify this model, or don't you think so ?