Hello all, Below are two related questions on measure-numeraire relationships. The bank account variables below are my notation. Piterbarg writes the solution for the price of a collateralized derivative in two equivalent forms, in equations (3) and (5).With collateral C=0 in (3), the solution reduces to V(t)/B_rF(t) = E_t[ V(T)/B_rF(T) ] showing V(t)/B_rF is a martingale where the numeraire B_rF is a rolling bank account paying the funding rate rF.Similarly, with C=V in (5), the solution reduces to (7), i.e. V(t)/B_rC(t) = E_t[ V(T)/B_rC(T) ], showing that V(t)/B_rC is a martingale where the numeraire B_rC is the rolling bank account paying the collateral rate.1) Then it appears that choosing different values of C is equivalent to choosing different measures so that these different numeraires can continue to produce martingales. Is this a reasonable way to view what's happening when different values of C are chosen?2) The first line of (11) is in the "no-CSA" case, i.e. with C=0. It seems the numeraire is B_rF. The second line of (11) is a trivial identity. But in going to the third line expressed in the T-forward measure, is B_rC now considered to be the "old" numeraire -- i.e. the ratio of old to new numeraires being B_rC(t)/P_C(t,T) in order to transform to the T-forward measure relative to P_C(t,T)? I'm not clear on what is or isn't being treated as a numeraire in going from the first to third line.Many thanks in advance.

interesting point/question that you raise. indeed it would seem that C=0 and C=1 correspond to two different numeraires. however, the price of a derivative should be invariant under change of numeraire. this is clearly not the case. i think the crux is that a numeraire always uses / refers to one risk-free curve. here we have several discounting curves.

I have been writing somewhere else in this forum that the collateral bank account as a numeraire is wrong because it hinges on a shortcut to determine the discount rate when you write down the PDE for the derivatives contract. It can be shown quite easily that the bank account can be safely taken as the numeraire even when a collateral agreement is operating. So you do not need to introduce this new numeraire and thus you can avoid the inconsistencies you are pointing out.The proof is in a couple of my papers you can find in the net and whose links are also in the other threads of this forum.There is another reason why it is better to avoid using the collateral bank account as a numeraire: it can be justified only if you assume that the collateral is equal to the value of the contract that includes the effetc of the collateralisation. This is neglected at the moment because the risk-free rate and the collateral rate are basically the same (the EONIA/OIS). If for some reason they will be considered different in the market practice in the future, and the parties agree to set the collateral on a specific value of the contract, say the pure risk-free value (which will then differ from the value including the effects of the collateralisation), all the constructions by Piterbarg collapses, and the collateral account as a numeraire is impossible, whereas it is still possible to consistently use the bank account.

i never really understood why the collateral rate is per se the OIS rate and that this is the risk free rate.

as I understodd it this is just convention (anybody correct me if I'm wrong):- usually it is agreed on that the party receiving the collateral has to pay the ois-rate on the collateral amount (could also have been anything else)- consequently collateralized deals need to be discounted with ois (although in my opinion not completely correct. If I have to post collateral I need to found myself and my founding costs are definetively above ois)- as collateralized deals are assumed to be risk-free one takes the ois rate as the risk-free rateI am happy for any discussion ...cheers, bernd

yes, what i read in articles etc is that it is 'convention' as well. let's say i am the party receiving the cash collateral. i can lend the cash to another party for an overnight period, and the interest rate i receive on this is EONIA (unsecured lending/borrowing rate). the cash + eonia i receive the next day i can then return to the party that posted the cash collateral in the first place. makes sense. however, why did i lend the money unsecured? i could also have asked collateral (eg govies) on the cash collateral that i received and subsequently lend to another party. in that case the rate would be even lower than eonia since now it's secured. so basically my question is why is it always eonia that's assumed? and isn't collateralized overnight lending/borrowing closer to a risk free rate than the unsecured overnight lending/borrowing rate? maybe i am over-complicating stuff here / not familiar with the actual mechanics of lending/borrowing.

All, thanks for the replies. All are interesting in their own right and worthy of separate discussions but most perhaps are somewhat askance of the original question.Rather than focus on things like the collateral payment mechanism, or the virtues of a particular choice of numeraire, or why one can take the collateral rate as the OIS rate, I'm instead focusing on mathematical consequences based on equations in Piterbarg's paper.Given the PDE and its solution, the two different choices of collateral value (C=0 or C=V) produce two different numeraires each of which appears in its respective expression having the property of being a martingale. So it must be that the measure has changed based on the choice of collateral value -- and done so in a way so that dS still has the same drift. But (in line with frolloos) if C is fixed (i.e. if the discount curve is fixed), all standard changes of numeraire will leave the derivative value unchanged -- after all, the PDE is left unchanged if C remains fixed. However when C is changed, even though doing so has the apparent effect of choosing a different numeraire and creating a different martingale under a different measure, the value of the derivative must change since the PDE is changed and therefore it isn't equivalent to a standard measure change.Nevertheless, I am still brought back to my second question (in my original post) which concerns the measure changes in equation (11). If anyone sees the right way to look at the transition to the third line in that equation given the first, I'd be interested to hear your reply.Many thanks.

well, the measure is the same, as evidenced by the same SDE for the stock price. It is sometimes useful to think of the collateralized and uncolalteralized contract as paying different (continuous) "dividends" that arise from funding costs/benefits, and because those are different, they grow at different rates. Or, alternatively, should be discounted differently, even under the same measureHope this helpsVladimir

I am interested in the answer to hattusa's second question too. Basically, in Piterbarg's original paper, line (2) to line (3) of formula (11) is based on that the original risk-neutral numeraire is the collateralized bank account B_c(t). Then, the question remains open: will the risk-neutral bank account remains uniquely to be the bank account B_c(t) with and without collaterization, or it will be different as B_c(t) and B_f(t) with and without collateralization? If it is the latter, how can the derivation hold from line(2) to line (3)? =====Here is my current understanding of this. Instead of assuming to have 4 different measures, risk-free risk-neutral measure, risk-free T-forward measure, risky risk-neutral measure, and risky T-forward measure, it is assumed in this paper that there is only one risk-neutral measure, i.e., the risk-free risk-neutral measure, as implicitly stated right below Formula (4), and in this measure the numeraire is B_c(t) and the stock is drift at the rate of r_R-r_D. The introduction of risk-free T-fwd measure and risky T-fwd measure are from the same risk-neutral measure. With this assumption, Formula (11) holds. To me, this looks like that it has an implicit assumption that the spread between risky and risk-free short rate is deterministic so that the risk-free risk-neutral measure and risky risk-neutral measure are the same. (Strictly speaking, the measures are the same when that spread is not just deterministic but constant.) Any feedback? Appreciated! Danny

Hope it helps. Any feedback more than welcome.paperAbstract:In the classical quantitative finance literature it is assumed that there is a risk free rate at which hedgers can borrow and lend in the dynamic replication process of financial derivatives. In such a framework, under complete market conditions and absence of arbitrage opportunities, for a given numeraire whose price cannot vanish, prices of self financing portfolios divided by the numeraire behave like a martingales under a unique martingale measure associated with the numeraire. Nevertheless, in the current market environment a high percentage of deals are collateralized due to counperparty credit risk concerns. Depending on the collateral agreement, collateral can be in the form of cash in different currencies, but also in the form of assets (bonds, shares,...). In this paper we explore how the fundamental valuation theorem and the change of numeraire tollkit is reformulated under this new framework.Thanks in advance.Luis

Hi,I realize this is a relatively old post but I thought that I would contribute this in case it helps. Let . To move from line 2 to line 3 in equation 11, start with the numerator in the second line of equation (11) of the paper.whereThen defining the measure by gives the third line of equation (11) since:I hope this is correct and helps.Thanks,Francis.

As Frank mentioned, the third line from (11) follows from the second line in (11) directly from the definition of E^T_t: for any random variable X, E^T_t[X] := E_t[\exp(-\int_t^T r_C(u)du)X]/P_C(t,T). In the case of moving from the second to the third line of (11), X=\exp(-\int_t^T s_F(u)du)S(T). I have some issues with this paper. The first is less important: there's a potential problem using the assumption that \Delta(t)=(dV/dS)(t) is a self-financing trading strategy. As far as I can tell, any self-financing trading strategy must be of bounded variation, and constant when S is not 0; this follows from the Ito formula applied to the product \Delta(t)*S(t): d(\Delta*S)(t)=S(t)d\Delta(t)+\Delta(t)dS(t)+d<\Delta,S>(t).Self-financing is the equation (as I understand): d(\Delta*S)(t)=\Delta(t)dS(t), which implies from the above that S(t)d\Delta(t)=-d<\Delta,S>(t), so that \Delta(t) has bounded variation, which then implies that d<\Delta,S>(t)=0, which then implies that S(t)d\Delta(t)=0, so that \Delta can only increase when S is 0, i.e., it must be piecewise constant. (dV/dS)(t) will not satisfy this unless V is linear in S. Am I wrong about this? Of course, the zero-strike call and forward examples in the paper do satisfy this, but more general derivative payoffs do not.Why is delta hedging so often accepted as a self-financing strategy? Proofs justifying the Black-Scholes differential equation using this argument seem flawed to me.The more serious issue that I have with this paper is that I was unable to see how "rearranging terms in (3)" yields (5). How do you change the terms in the exponent from r_F to r_C? Integration by parts doesn't seem to work. Trying to change the numeraire also didn't work for me - anyways, that would imply that the expectation operator E_t is changing, which I'm pretty sure is not the case. Does the derivation of (5) use a Feynman-Kac line of reasoning similar to that used in deriving (3)? Am I missing something obvious?

After looking at it again, I see (5) follows from rearranging the equation in the line preceding (3) and then applying the same logic as that in deriving (3). I believe this paper now.

I have one more question about the measure topic:If the stock is repoable than it has grow risk-neutrally with its repo-rate (because this is the cheapest way of funding the stock - i.e. my opportunity costs). But what if the stock is not repoable?? The cheapest way of funding (for a bank A) is now to borrow the money unsecured from the market. However, the funding rates obviously differ across the market participants. As I see it, this market will not be arbitrage free as the law of one price does not even hold (bank B with a lower funding rate should be willing to pay more for a stock than bank A - assuming both have a same level of risk-aversion). Consequently, there is no risk-neutral measure and the whole theory breaks down.Is this correct?Cheers, bernd

QuoteOriginally posted by: piterbargwell, the measure is the same, as evidenced by the same SDE for the stock price. It is sometimes useful to think of the collateralized and uncolalteralized contract as paying different (continuous) "dividends" that arise from funding costs/benefits, and because those are different, they grow at different rates. Or, alternatively, should be discounted differently, even under the same measureHope this helpsVladimirI like this view. Looking at the collateralized contract as paying a dividend. In my view this leads directly to modelling this contract as a "market atom". That means there is no delta hedge between the collateralized and the uncollaterlized contract. These are 2 liquid traded assets that appear both in modelling a market. Is my view shared?