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  • #6172
    rkapl
    Participant

    Dear All!

    We’ve discovered a Problem with the funding cost/benefit adjustment figure:
    Following the logic that expected positive exposures (after collateralisation) should lead to high(er) FCA and expected negative exposures to high(er) FBA, the results (and the source code, see OREAnalytics/postprocess.cpp, method PostProcess::updateStandAloneXVA(), line 646 ff.) suggest that that ORE calculates this reverted:

    `
    Real borrowingSpreadDcf = 0.0;
    if (!borrowingCurve.empty())
    borrowingSpreadDcf = borrowingCurve->discount(d0) / borrowingCurve->discount(d1) – oisCurve->discount(d0) / oisCurve->discount(d1);
    Real fbaIncrement = cvaS0 * dvaS0 * borrowingSpreadDcf * tradeEPE_[tradeId][j + 1];
    tradeFBA_[tradeId] += fbaIncrement;

    Real lendingSpreadDcf = 0.0;
    if (!lendingCurve.empty())
    lendingSpreadDcf = lendingCurve->discount(d0) / lendingCurve->discount(d1) – oisCurve->discount(d0) / oisCurve->discount(d1);
    Real fcaIncrement = cvaS0 * dvaS0 * lendingSpreadDcf * tradeENE_[tradeId][j + 1];
    tradeFCA_[tradeId] += fcaIncrement;

    Considering that funding costs depend on the borrowing spread and funding benefits on the lending spread, the above suggests even more an exchange of the two terms (see also Gregory, The xVA Challenge, page 343 on FVA calculation).

    -regards,
    Roland

    #6173
    rkapl
    Participant

    To add some data to the topic, below the results from Example 10, modified to express an almost “perfect” CSA (0 MTA, 0 Threshold, 1D MarginingFrequency (Call/Post) and 2W MarginPeriodOfRisk):

    CSADetails/Bilateral: Bilateral
    NettingSetId CVA DVA FBA FCA
    CPTY_A 14.844 24.696 -2.332 11.789

    CSADetails/Bilateral: PostOnly
    NettingSetId CVA DVA FBA FCA
    CPTY_A 153.061 11.094 -23.768 5.307

    CSADetails/Bilateral: CallOnly
    NettingSetId CVA DVA FBA FCA
    CPTY_A 8.033 217.001 -1.270 102.676

    ActiveCSAFlag:false
    NettingSetId CVA DVA FBA FCA
    CPTY_A 146.251 203.399 -22.706 96.195

    -regards,
    Roland

    #6188
    Anonymous
    Inactive

    Hi Roland,

    ORE is using the methodology outlined in “Modern Derivatives Pricing and Credit Exposure Analysis” by Lichters, Stamm and Gallagher. In this the FBA and FCA labels are different to Gregory (in effect they are swapped), this is reflected in the code.

    The assumption here is that FBA and FCA is applied to an already collateralised trade, in that case an increase in EPE would mean that the exposure to your counterparty has increased, thus they must post more margin to you and thus you get a benefit, so an EPE increase leads to an FBA increase. Equally an ENE increase leads to an FCA increase. This is covered in Appendix A.5 of the user guide.

    Gregory on the other hand is considering an un-collateralised trade and considering the FBA and FCA on a collateralised hedging trade, this is why it appears the two labels are reversed.

    I agree it appears a bit confusing and is probably not documented as well as it could be, what do you think would be good here? more comments in the code or a longer explanation in the user guide?

    Regards,
    Niall.

    #6189
    rkapl
    Participant

    Dear Niall!

    Sorry to be insisting on that, but after reading the chapter from “Modern Derivatives Pricing and Credit Exposure Analysis” by Lichters, Stamm and Gallagher and quoting them (hope this is allowed…):
    “If the uncollateralized derivative has a positive value for the bank, the offsetting trade will require collateral; if the value is negative, collateral
    will be received. Posted collateral only accrues interest at the collateral rate, while it is usually funded via unsecured borrowing in the money market, and so will create a cost of the funding spread.
    The opposite is true for any received collateral: if it is rehypothecable, it saves the bank the funding spread or can be invested, generating a funding benefit of the lending spread.”

    “Based on this argument, a simple definition of FVA can be given in a very similar fashion as the sum of unilateral CVA and DVA which we defined by (8.2), namely as an expectation of exposure times funding spreads, see for example Chapter 14 in Gregory [74]”

    “The interpretation of this formula is that if the current exposure of the uncollateralized derivative at some future point in time is positive, the bank will pay collateral on the offsetting trade, which costs an equal amount of funding at the bank’s borrowing rate.
    Likewise, a negative current exposure creates a funding benefit which can be invested at the lending rate.”

    Following these arguments, it seems to me that both the formula in “Modern Derivatives Pricing and Credit Exposure Analysis” and the formula in the userguide (and the implementation following both) seems to be mixed up. I can also see a potential source of confusion, as the offsetting (hedge) trade’s ENE (causing a collateral requirement on the bank) is actually the original trade’s EPE (as the cashflows need to be completely offsetting) and vice versa with the EPE.

    I have created an example case for your argument above (FBA and FCA is applied to an already collateralised trade, in that case an increase in EPE would mean that the exposure to your counterparty has increased, thus they must post more margin to you and thus you get a benefit, so an EPE increase leads to an FBA increase), in this the FBA/FCA is actually correct, however this seems to be not the point of FVA, which focuses on the uncollateralised parts (trades, imperfect CSA, etc.) of the business.

    Anyway, I’d rather make this a configurable calculation, as it is quite fundamental and only easily revertable in case of borrowing_spread = lending_spread.

    -regards,
    Roland

    #6190
    rkapl
    Participant

    Dear Niall!

    Sorry, I’ve just seen that it IS revertable also for borrowing_spread != lending_spread, it’s just the parts of the formula being incorrectly denoted.

    -regards,
    Roland

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