We illustrate a problem in the self-financing condition used in the papers
"Funding beyond discounting: collateral agreements and derivatives pricing"
(Risk Magazine, February 2010) and "Partial Differential Equation
Representations of Derivatives with Counterparty Risk and Funding Costs" (The
Journal of Credit Risk, 2011). These papers state an erroneous self-financing
condition. In the first paper, this is equivalent to assuming that the equity
position is self-financing on its own and without including the cash position.
In the second paper, this is equivalent to assuming that a subportfolio is
self-financing on its own, rather than the whole portfolio. The error in the
first paper is avoided when clearly distinguishing between price processes,
dividend processes and gain processes. We present an outline of the derivation
that yields the correct statement of the self-financing condition, clarifying
the structure of the relevant funding accounts, and show that the final result
in "Funding beyond discounting" is correct, even if the self-financing
condition stated is not.Comment: added references, a footnote and updated abstrac

Brigo, Damiano

Buescu, Cristin

Liu, Qing

Pallavicini, Andrea

English

arXiv

Damiano Brigo

Cristin Buescu

Andrea Pallavicini

Qing Daphne Liu

Crossref

SSRN Electronic Journal

Illustrating a Problem in the Self-Financing Condition in Two 2010-2011 Papers on Funding, Collateral and Discounting

We illustrate a problem in the self-financing condition used in the papers "Funding beyond discounting: collateral agreements and derivatives pricing" (Risk Magazine, February 2010) and "Partial Differential Equation Representations of Derivatives with Counterparty Risk and Funding Costs" (The Journal of Credit Risk, 2011). These papers state an erroneous self-financing condition. In the first paper, this is equivalent to assuming that the equity position is self-financing on its own and without including the cash position. In the second paper, this is equivalent to assuming that a subportfolio is self-financing on its own, rather than the whole portfolio. The error in the first paper is avoided when clearly distinguishing between price processes, dividend processes and gain processes. We present an outline of the derivation that yields the correct statement of the self-financing condition, clarifying the structure of the relevant funding accounts, and show that the final result in "Funding beyond discounting" is correct, even if the self-financing condition stated is not

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                       King’s Research Portal  Link to publication record in King's Research PortalCitation for published version (APA):Brigo, D., Buescu, C., Pallavicini, A., & Liu, Q. (2012). Illustrating a problem in the self-financing condition in two2010-2011 papers on funding, collateral and discounting. http://arxiv.org/abs/1207.2316Citing this paperPlease note that where the full-text provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this maydiffer from the final Published version. If citing, it is advised that you check and use the publisher's definitive version for pagination,volume/issue, and date of publication details. 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Oct. 2023arXiv:1207.2316v2  [q-fin.PR]  16 Jul 2012Illustrating a problem in the self-financing condition in two2010-2011 papers on funding, collateral and discounting∗Damiano Brigo1, Cristin Buescu2, Andrea Pallavicini3 and Qing Liu11Department of Mathematics, King’s College London, on move to Imperial College2Department of Mathematics, King’s College London3Financial Engineering, Banca IMI, MilanFirst version: May 8, 2012. First posted on SSRN and arXiv on July 10, 2012.This version: July 17, 2012AbstractWe illustrate a problem in the self-financing condition used in the papers “Fund-ing beyond discounting: collateral agreements and derivatives pricing” (Risk Magazine,February 2010) and “Partial Differential Equation Representations of Derivatives withCounterparty Risk and Funding Costs” (The Journal of Credit Risk, 2011). These pa-pers state an erroneous self-financing condition. In the first paper, this is equivalent toassuming that the equity position is self-financing on its own and without including thecash position. In the second paper, this is equivalent to assuming that a subportfolio isself-financing on its own, rather than the whole portfolio. The error in the first paperis avoided when clearly distinguishing between price processes, dividend processes andgain processes. We present an outline of the derivation that yields the correct statementof the self-financing condition, clarifying the structure of the relevant funding accounts,and show that the final result in “Funding beyond discounting” is correct, even if theself-financing condition stated is not.AMS Classification Codes: 91B70, 91G10, 91G20, 91G40JEL Classification Codes: G12, G13Keywords: Funding cost, cost of funding, funding and discounting, self-financing strategy, tradingstrategies, hedging.∗We are grateful to Daniele Perini for interesting suggestions and discussion. Given the specific momentand the current turmoil affecting the top management of some important banks following the LIBOR fixingscandal, see for example [12], and given that the authors of the articles we are scrutinizing are all affiliated withone of said banks, we feel it is important to point out that this is not meant as a criticism to the quantitativeteams of said bank. This is simply a technical paper addressing a problem in the above published articlesand has nothing to do with the affiliation of such articles’ authors, or with the articles publisher. Finally, thisarticle reflects only the authors opinion and in no way represents the authors employers, present and past.1D. Brigo, C. Buescu, A. Pallavicini, Q. Liu. Illustrating a problem in the self-financing condition 21 IntroductionCost of funding has become a paramount topic in the industry. One just has to look at thenumber of presentations and streams at modeling conferences of 2010-2012 that deal with thistopic to realize how much research effort is being put into it. Despite this sustained effort,the literature is still in its infancy. One of the difficulties is that the funding problem ties innaturally with credit risk, collateral modeling and discounting.An initial analysis of the problem of replication of derivative transactions under collateral-ization in a Black and Scholes framework, but without the complicated and subtle intricaciesof default risk, has been considered in [19].The fundamental impact of collateralization on default risk and on CVA and DVA hasbeen analyzed in [8], and more recently in [3] and [2].The works [3] and [2] look at CVA and DVA gap risk under several collateralizationstrategies, with or without re-hypothecation, as a function of the margining frequency, withwrong-way risk and with possible instantaneous contagion. Minimum threshold amounts andminimum transfer amounts are also considered. We cite also [1] and [4] for a list of frequentlyasked questions on the subject.The fundamental funding implications in presence of default risk have been considered in[17], see also [7]. In order to highlight some essential features of funding costs these worksfocus on particularly simple products, such as zero coupon bonds or loans.The work in [14] analyzes implications of currency risk for collateral modeling.The above references constitute a beginning for the funding cost literature, but do not havethe level of generality needed to include all of the above features in a consistent frameworkthat can then be used to manage complex products. A general theory is still missing. Theonly exceptions so far are [6] and [5], which resort to a PDE approach and do not deal with thehidden complexities of collateral modeling and mark-to-market discontinuities at default, anddo not state explicitly the funding assumptions for all the assets in their replicating portfolio.The other general result is presented in the series [9], [10], [11], which resort to backwardsstochastic differential equations.The paper [18] follows the same level of generality of [9] and tries to deal with dif-ferent models of funding policy, while still accounting for CVA, DVA, collateral and re-hypothecation.A few available books have started including funding costs analysis, see for example [16]and [4].In this note we address an important problem with the self-financing condition used in[19] and [6]. The necessity for writing specifically a note is due to the fact that [19] is highlyquoted at industry conferences worldwide in the practitioners space1, while [6] has receivedextensive exposure, and we believe it is therefore important to highlight this issue.This erroneous self-financing condition stems from a common failure in applying thestochastic Leibnitz rule. This error is common even in mainstream textbooks such as [15]and has been pointed out explicity for example in [20] (Exercise 4.102).1 “For his work showing how to reconcile these complicating factors within a deriva-tives pricing framework [the Author] wins Risk’s quant of the year award for an unprece-dented second time. [The Author]’s paper “Funding beyond discounting” was the first toshow how funding rates and the posting of collateral can affect the pricing of derivatives.”http://www.risk.net/risk-magazine/feature/1934297/quant-vladimir-piterbarg-barclays-capital2We thank Chris Kenyon for reminding us of this exercise in [20], see also [16]D. Brigo, C. Buescu, A. Pallavicini, Q. Liu. Illustrating a problem in the self-financing condition 32 The self-financing condition and the problem in [19]The problem is immediate. We also notice that the same problems affects the proofs in [6].Formula (2) in [19] reads, for the portfolio Π that replicates the derivative V (we useidentical notation)V (t) = Π(t) = ∆(t)S(t) + γ(t) (2)where S is “price process” of the underlying asset, and γ is the “cash amount split among anumber of accounts [...]”.Then [19] goes on:“On the other hand, from (2), by the self-financing conditiondγ(t) = dV (t)−∆(t) dS(t) (∗)[...]”We argue that this formulation of the self-financing condition is wrong.It is enough to differentiate directly (2) to obtaindV (t) = d(∆(t)S(t)) + d γ(t), (∗∗)and combine this last equation with (*) to obtaind(∆(t)S(t)) = ∆(t) dS(t) (Wrong) .The boxed equation, immediately implied from the assumptions in [19], would imply thatthe position in the risky asset S is self-financing on its own, in that the change in the totalvalue of the position, namely d(∆(t)S(t)), is funded by the asset market movements alone:∆(t) dS(t).A further consequence of the above error follows immediately from the stochastic Leibnitzrule, leading tod∆t = 0 (wrong),and indeed if equity needs to be self-financing on its own the only possibility is that theamount of equity is constant (there is no re-balancing of the single position).3 The self-financing condition and the problem in [6]The same problem affects the self-financing condition stated in [6]. In that work it is statedexplicitly that the portfolio consisting of the stock S, the bond PB of party B, the bond PCof party C, and β(t) cash is self-financing. This portfolio value can be written as in the firstEquation of [6] following (3.2), namely (we use identical notation)−V̂ (t) = Π(t) = δ(t)S(t) + αB(t)PB(t) + αC(t)PC(t) + β(t). (#)The self-financing condition is stated in Equation (3.3) of [6] which reads−dV̂ (t) = δ(t)dS(t) + αB(t)dPB(t) + αC(t)dPC (t) + dβ(t). (##)D. Brigo, C. Buescu, A. Pallavicini, Q. Liu. Illustrating a problem in the self-financing condition 4The reader has clearly seen where the problem is: if we now differentiate Equation (#) andequate the resulting equation with (##) we immediately obtain:d(δ(t)S(t) + αB(t)PB(t) + αC(t)PC(t)) = δ(t)dS(t) + αB(t)dPB(t) + αC(t)dPC (t),which clearly implies that the portfolio of the three assetsS, PB , PCis self-financing. This is clearly at odds with [6] stating instead that the entire portfolioS, PB , PC , cashis self-financing.This is the same problem that afflicts the self-financing condition in [19], except that hereit is distributed across more than one asset.4 A sketch of the correct formulation in the framework of [19]Since the derivation of the result is important, as it provides a description of the fundingaccount and of the funding strategy, we believe it is appropriate at this point to illustrate theproper formulation for [19]. We point out that we do not discuss the assumptions in [19], aswe rather concentrate on the correct formulation of the self-financing condition under theirassumptions. For a more comprehensive framework that includes explicit default modeling,collateral modeling, re-hypothecation and debit valuation adjustments we refer the readerelsewhere, for example [18] or [9].One of the problems in the above derivation is that it does not distinguish between gainprocesses, price processes and dividend processes, and in confusing them brings about aproblem. We present below an informal and yet correct account of the theory. The fulltheory can be found in Duffie [13].We define an asset Ai to be described by a price process PAiand a dividend process DAi.The gain process is defined asGAi := PAi +DAi .Now we assume that i belongs to a finite set of indices {1, 2, . . . , n}, and call A the vector ofassets with components Ai.A trading strategy in A is defined as a pairΠ := (θA, A)where θ is a vector of portfolio positions (number of units) in each asset.The value of the trading strategy is defined asV Π := θA1 PA1 + . . .+ θAnPAnwhile the gain process of the strategy isdGΠ := θ1dGA1 + . . .+ θndGAn .D. Brigo, C. Buescu, A. Pallavicini, Q. Liu. Illustrating a problem in the self-financing condition 5A trading strategy is termed self-financing if its dividend is zero (there is no dividendoutflow or inflow from the portfolio), namely ifGΠ − V Π = 0.This implies in particular thatdGΠ = dV Πord(θA1 PA1 + . . .+ θAnPAn ) = θ1dGA1 + . . .+ θndGAnbut notice that in generald(θA1 PA1 + . . .+ θAnPAn ) 6= θ1dPA1 + . . . + θndPAn . (∗ ∗ ∗)Indeed, the self-financing condition for the strategy does not imply thatdGAi = dPAifor the single asset i. Clearly, if all single assets have null dividend process, it follows thatthe self-financing condition for the strategy implies an equality in (***), but more generallythis does not hold.Finally, a claim Y is replicated by the strategy Π ifV Y = V Π, GY = GΠfor all t ≥ 0 and up to the claim maturity.We now apply the above framework to the setup in [19].For a derivative with value V Y we seek a self-financing strategy that replicates it. We trythe strategy (θA, A) =: Π.We define the vector A of assets: A1 to be a repo contract for the risky asset, the riskyasset price being denoted by St; A2 is a collateral position, the collateral price being C, andA3 is the funding position, whose price is denoted by α.For such positions we have the price processesPA1 = 0, PA2 = C, PA3 = α (1)and the gain processesdGA1 = dS + (rD − rR)Sdt, dGA2 = rCCdt, dGA3 = rFαdt, (2)where rD is the rate at which stock dividends are paid, rR is the short rate on funding securedvia repo, rC is the short rate of cash/collateral, and rF is the short rate for unsecured funding.We omit the superscript “A” in the following to lighten notation.One can immediately write the dividend processes for the three accounts as follows:dD1 = dG1 − dP1 = dS + (rD − rR)Sdt, D1(0) = 0, (3)dD2 = dG2 − dP2 = rCCdt− dC, D2(0) = 0,dD3 = dG3 − dP3 = rFαdt− dα, D3(0) = 0.The components of θ areD. Brigo, C. Buescu, A. Pallavicini, Q. Liu. Illustrating a problem in the self-financing condition 6θ1 = ∆, θ2 = 1, θ3 = 1.Notice that here the single asset dividend processes are not null. Hence we have (***).The self-financing condition requiresGΠ = V Πwhereas the replication condition requiresV Π = V Y .It follows thatGΠ = V Y .Let us now take a look at V Π and GΠ:V Π = ∆0 + 1C + 1α ⇒ α = V Π − C = V Y − C.We notice that, by plugging α = V Y − C into Equations (1), (2) and (3), we obtainall the dynamics of the price, gain and dividend processes expressed in terms of the marketobservable quantitiesS, C, V.We have immediately by definition of gain process thatdGΠ = θ1dG1 + θ2dG2 + θ3dG3,namelydGΠ = ∆[dS + (rD − rR)Sdt] + 1rCCdt+ rF (VY − C)dt.At the same time since we are imposing GΠ = V Y , we have dV Y = dGΠ given by theequation above. On the other hand, through Ito’s formula,dV Y =∂V Y∂tdt+∂V Y∂SdS +12∂2V Y∂S2d〈S〉2and by equating the right hand sides of the last two equations we obtain∆[dS + (rD − rR)Sdt] + 1rCCdt+ rF (VY − C)dt =∂V Y∂tdt+∂V Y∂SdS +12∂2V Y∂S2d〈S〉2,from which∆ =∂V Y∂S, ∆(rD − rR)S + rCC + rF (VY − C) =∂V Y∂t+12∂2V Y∂S2σ2SS2.From now on the derivation may continue as in [19] with their equation (3).D. Brigo, C. Buescu, A. Pallavicini, Q. Liu. Illustrating a problem in the self-financing condition 75 ConclusionWe illustrated a problem with the self-financing condition used in [19] and [6]. This wrongcondition implies that the portfolio without cash is self-financing on its own. Since a soundfunding theory depends crucially on financing costs, the violation of the self-financing condi-tion is an important problem. Fortunately, the error can be corrected and the same resultobtained in a rigorous way for [19], and we briefly sketched the correct setup, casting light onthe necessary distinction between dividend processes, price processes and gain processes.References[1] D. Brigo. Counterparty risk FAQ: Credit VaR, PFE, CVA, DVA, closeout, netting,collateral, re-hypothecation, WWR, Basel, funding, CCDS and margin lending. WorkingPaper, 2011.[2] D. Brigo, A. Capponi, and A. Pallavicini. Arbitrage-free bilateral counterparty risk valua-tion under collateralization and re-hypothecation with application to CDS. MathematicalFinance, 2011. Accepted for publication.[3] D. Brigo, A. Capponi, A. Pallavicini, and V. Papatheodorou. Collateral margining inarbitrage-free counterparty valuation adjustment including re-hypotecation and netting.Working Paper, 2011.[4] D. Brigo, M. Morini, and A. Pallavicini. Counterparty Credit Risk, Collateral and Fund-ing with pricing cases for all asset classes. Wiley, Forthcoming, 2012.[5] C. Burgard and M. Kjaer. In the balance. Risk Magazine, October, 2011.[6] C. Burgard and M. Kjaer. Partial differential equation representations of derivativeswith counterparty risk and funding costs. The Journal of Credit Risk, 7 (3):1–19, 2011.[7] A. Castagna. Funding, liquidity, credit and counterparty risk: Links and implications.Working Paper, 2011.[8] U. Cherubini. Counterparty risk in derivatives and collateral policies: the replicatingportfolio approach. In L. Tilman, editor, ALM of Financial Institutions. InstitutionalInvestor Books, 2005.[9] S. Crépey. A BSDE approach to counterparty risk under funding constraints. WorkingPaper, 2011.[10] S. Crépey. Bilateral counterparty risk under funding constraints Part I: Pricing. Forth-coming in Mathematical Finance, 2012.[11] S. Crépey. Bilateral counterparty risk under funding constraints Part II: CVA. Forth-coming in Mathematical Finance, 2012.[12] K. Crowley and A. Choudhury. Bloomberg. http://www.bloomberg.com/news/2012-07-05/made-in-london-scandals-risk-city-s-reputation-as-finance-center.html, 6 July 2012.[13] D. Duffie. Dynamic Asset Pricing Theory. Princeton University Press, 3rd edition, 2001.D. Brigo, C. Buescu, A. Pallavicini, Q. Liu. Illustrating a problem in the self-financing condition 8[14] M. Fujii, Y. Shimada, and A. Takahashi. Collateral posting and choice of collateralcurrency. Working Paper, 2010.[15] J. Hull. Options, Futures and other Derivatives. 8th Edition. Prentice Hall, 2011.[16] C. Kenyon, and R. Stamm. Discounting, Libor, CVA and Funding. Palgrave MacMillan,2012.[17] M. Morini and A. Prampolini. Risky funding: A unified framework for counterparty andliquidity charges. Risk Magazine, March, 2011.[18] A. Pallavicini, D. Perini, and D. Brigo. Funding Valuation Adjustment: FVA consistentwith CVA, DVA, WWR, Collateral, Netting and re-hyphotecation. ssrn.com, 2011.[19] V. Piterbarg. Funding beyond discounting: collateral agreements and derivatives pricing.Risk Magazine, 2:97–102, 2010.[20] S. Shreve. Stochastic Calculus for Finance II: Continuous-Time Models. Springer Verlag,Heidelberg, 2004.

Illustrating a problem in the self-financing condition in two 2010-2011 papers on funding, collateral and discounting

Illustrating a problem in the self-financing condition in two 2010-2011
  papers on funding, collateral and discounting

Illustrating a problem in the self-financing condition in two 2010-2011 papers on funding, collateral and discounting

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