A Risk-Sharing Framework of Bilateral Contracts
aa r X i v : . [ q -f i n . M F ] D ec A RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS ∗ JUNBEOM LEE † , STEPHAN STURM ‡ , AND
CHAO ZHOU § Abstract.
We introduce a two-agent problem which is inspired by price asymmetry arising from fundingdifference. When two parties have different funding rates, the two parties deduce different fair prices for derivativecontracts even under the same pricing methodology and parameters. Thus, the two parties should enter the derivativecontracts with a negotiated price, and we call the negotiation a risk-sharing problem. This framework defines thenegotiation as a problem that maximizes the sum of utilities of the two parties. By the derived optimal price, weprovide a theoretical analysis on how the price is determined between the two parties. As well as the price, the risk-sharing framework produces an optimal amount of collateral. The derived optimal collateral can be used forcontracts between financial firms and non-financial firms. However, inter-dealers markets are governed by regulations.As recommended in Basel III, it is a convention in inter-dealer contracts to pledge the full amount of a close-outprice as collateral. In this case, using the optimal collateral, we interpret conditions for the full margin requirementto be indeed optimal.
Key words.
Bilateral contracts, risk-sharing, piece-wise concave utility, collateral
AMS subject classifications.
1. Introduction.
In the aftermath of the financial crisis, it has become customary in recentyears for trading desks to make several adjustments in derivative transactions for counterpartydefault risk, funding spreads, collateral, etc. For pricing derivatives with the collective adjustments,many methodologies have been developed by, e.g., [47, 23, 19, 50, 29]. However, it is known thatthe fair values derived by the developed methodologies are not fully recouped from counterparties.This can possibly due to inclusion of funding spread. For traders, if the increased funding costsare not compensated from the counterparty, it will be losses on the trades. However, consideringthe choices of funding in derivative prices is a violation of Modigliani–Miller (MM) theorem. ForMM theorem to be valid, the absence of frictional financial distress costs is required; see [44, 49].Therefore, considering funding cost/benefit may be justified by market frictions.Even so, there still remain some puzzles related to the funding adjustment. First, when fundingcost/benefit is accepted, it gives rise to asymmetry of theoretical prices between two contractorseven under the same pricing methodology and parameters. The value fair to one party may not befair to the counterparty since funding rates of the counterparty is different from those of the otherparty. Second, as asked by [36, 37], if funding cost should be really considered, possibly due tomarket frictions, why do banks buy Treasury bonds that return less than their funding costs?Motivated by the related issues, we introduce a two-agent problem. Instead of using the indi-vidual fair values, two parties may enter a contract through negotiation by sharing costs, as brieflymentioned by [43]. We describe the negotiation problem as maximizing the sum of utilities of bothparties and we call this a risk-sharing problem. Then, the risk-sharing framework theoreticallyanalyzes how the price equilibrium is determined and we can interpret the questions on fundingadjustment by using the derived price. The optimal price from the risk-sharing framework willdepend on the risk aversion parameters and relative negotiation power between the two parties, butthey are not observable in markets. Therefore, the importance of this study is on providing soundtheoretical interpretations for the puzzles on funding difference.The other part of the solution in the risk-sharing framework is collateral. In recent times, mostOTC derivative contracts are collateralized. There are multiple procedures for the margin, but in ∗ Junbeom Lee and Chao Zhou received the support from the French Ministry of Foreign Affairs and the Merlionprogramme as well as the Singapore MOE AcRF grants R-146-000-255-114 and R-146-000-243-114. † Department of Sales and Trading, Yuanta Securities Korea ([email protected]). Most part of this workwas carried out when the author was in National University of Singapore. Opinions in this paper are those of theauthor, and do not represent the view of Yuanta Securities Korea. ‡ Department of Mathematical Sciences, Worcester Polytechnic Institute ([email protected]). § Department of Mathematics, National University of Singapore ([email protected])1
J. LEE, S. STURM, C. ZHOU bilateral contracts, it is general to post variation margin and initial margin . In our model, thefocus is on variation margin which traces mark-to-market exposures. As stated in [13, p.15], “forvariation margin, the full amount necessary to fully collateralise the mark-to-market exposure ofthe non-centrally cleared derivatives must be exchanged.” This full collateralization on variationmargin has been settled as a market convention in inter-dealer transactions. On the other hand,there is no such convention between banks and sovereign or corporate clients. Indeed, it is partlyor not collateralized for contracts between financial firms and non-financial firms. Therefore, the risk-sharing framework provides the optimal amount of variation margin for the contract betweena financial firm and non-financial firm. Since inter-dealer contracts are governed by regulation inpractice, we interpret the meaning behind the margin requirement.The optimal collateral in our model is represented by a certain stochastic process. Thus, full variation margin may not be optimal in general. However, we do not conclude that the conventionis unreasonable.
Variation margin is posted on a daily or intra-day basis. If the amount wascalculated by a complicated rule at each time, the amount would be unacceptable for some partiesand this can be a possible cause of conflict. Hence, rather than coming to a sensitive conclusion, weanalyze the situation for the margin requirement to be optimal. The market convention will turnout to be based on certain conditions on funding cost/benefit considered in derivative prices andhedging strategy taken by two parties. Especially, we will see later that the full margin requirementis related to the absence of market friction.One mathematical difficulty to deal with the risk-sharing problem is that the amount by breachof contract is given by piece-wise concave functions. Mathematically similar problems were solvedby [24, 15, 14, 51]. In [24], portfolio optimization problems were considered where the agent switchesutilities. They used duality method that cannot be applied to our problem as we cannot impose apositive constraint for the portfolio. In [15, 14], the piece-wise concave property arose from differentlending/borrowing rates and they solved the optimization problem by using HJB equations. Intheir problem, the associated HJB equations had a homothetic property. Moreover, with a mildassumption that the lending rate is smaller than the borrowing rate, the Hamiltonian becamecontinuous in their cases. However, in our problem, we deal with two state processes taken by twoutilities, so we cannot make use of a similar approach.We circumvent the above difficulties by imposing some conditions on funding spread dependingon choices of utilities. For the funding spread, we assume that the lending and borrowing rates arethe same for each party. To be more precise, the two parties fund themselves on their own fundingrate which may not be the same as OIS rate, but the lending and borrowing rates are the same.Moreover, the funding costs/benefits for delivering collateral of one or both parties will be ignoredfor characterization of the optimal solution. More precisely, we will examine two cases. First, wewill consider two risk-averse agents whose funding rates for delivering margin are OIS rate. Second,we will also consider one risk-averse agent and one risk-neutral agent, and in this case, the fundingrate of the risk-neutral agent does not need to be OIS rate. To streamline this paper, we mainlydeal with the two risk-averse agents in main sections and report the second case in Appendix C.This funding condition can be understood that the party is an entity which invests the capitalwithout or with a small leverage, or the party can post collateral with secured funding. Even thoughthe secured funding for variation margin is not so general, some realistic cases are discussed by [3].In addition, This setup on funding spread can be partly justified by the results in [42] which showedthat, in many classes of derivatives, hedgers do not need to switch funding state between lendingand borrowing positions. In particular, it is guaranteed that if a hedger does not enter borrowingstate and the lending rate is same as OIS rate, we can ignore the funding impacts.In our model, we include default risk, funding spread, and collateral. We consider incompletemarkets that hedgers cannot access to assets for hedging default risk such as bonds and CDSs. Thereference filtration is generated by a Brownian motion. The mark-to-market exposure is calculatedas clean price which is the classical risk-neutral price without default risks and funding spread.Moreover, for risk-averse agents, we consider exponential utilities. For simplicity, we consider non-
RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS risk-sharing problem is introduced. We start from defining a filtration andmaking an assumption on default intensities in subsection 2.1. Before giving the details, we explainour motivation with a simple model in subsection 2.2. Then we describe cash-flow which aredetermined by dividends, margins, and close-out amount. Both parties entering the contract willhave a portfolio given in subsection 2.3 depending on the cash-flow. The introduced risk-sharing problem is maximizing the sum of utilities of discounted portfolio values at termination of thecontract. In subsection 2.4, the original form of the risk-sharing problem is reduced so that it isrepresented on the reference filtration. Then we define admissible sets more precisely with thisreduced problem. We mainly deal with the reduced problem in this paper. In section 3, we definea dynamic version of the main problem, and optimal collateral is characterized. Then given theoptimal collateral, we derive a condition to find an optimal price in section 4 and examples aregiven in section 5.
2. Modeling.2.1. Mathematical Setup.
We consider two parties entering a bilaterally cleared contract.We call the two parties “Agent A” and “Agent B”, respectively. In what follows, an index A (resp. B ) is used to stand for the Agent A (resp. Agent B). We consider a probability space (Ω , G , P ) withphysical probability P and let E be the expectation under P . For i ∈ { A, B } , we define non-negativerandom variables τ i on (Ω , G , P ) such that P ( τ i = 0) = 0 and P ( τ i > t ) > , for any t ≥ , torepresent default times of the agents. We let τ := τ A ∧ τ B , ¯ τ := τ ∧ T, where T > is the maturity of a certain derivative contract. We denote by ( W t ) t ≥ a d -dimensionalstandard Brownian motion under P . The reference filtration F = ( F t ) t ≥ is the usual naturalfiltration of ( W t ) t ≥ , and the full filtration G is defined as G = ( G t ) t ≥ = (cid:16) F t ∨ σ (cid:0) { τ i ≤ u } : u ≤ t, i ∈ { A, B } (cid:1)(cid:17) t ≥ . Then, we consider a filtered probability space (Ω , G , G , P ) . Note that for any i ∈ { A, B } , τ i is a G -stopping time but may fail to be an F -stopping time. Unless stated, every process is a ( P , G ) -semimartingale. As a convention, for any G -progressively measurable process ( u t ) t ≥ and ( P , G ) -semimartingale ( U t ) t ≥ , R ts u s d U s = R ( s,t ] u s d U s , where the integral is well defined. In addition, forany G -stopping time θ and process ( ξ t ) t ≥ , we denote ξ θ · := ξ ·∧ θ , and when ξ θ − exists, denote ∆ ξ θ := ξ θ − ξ θ − . For i ∈ { A, B } , t ≥ , we also let G it := P ( τ i > t |F t ) and G t := P ( τ > t |F t ) . The following assumption stands throughout this paper.
Assumption (i) ( G t ) t ≥ is non-increasing and absolutely continuous with respect toLebesgue measure.(ii) For any i ∈ { A, B } , there exists a process h i , defined as h it := lim u ↓ u P ( t < τ i ≤ t + u, τ > t |F t ) P ( τ > t |F t ) , and ( M it ) t ≥ := (cid:0) τ i ≤ t ∧ τ − R t ∧ τ h is d s (cid:1) t ≥ is a ( P , G ) -martingale. J. LEE, S. STURM, C. ZHOU
We denote h := h A + h B . By (i) in Assumption 2.1, there exists an F -progressively measurableprocess ( h t ) t ≥ such that h t = lim u ↓ u P ( t < τ ≤ t + u |F t ) P ( τ > t |F t ) , and ( M t ) t ≥ := (cid:0) τ ≤ t − R t ∧ τ h s d s (cid:1) t ≥ is also a ( P , G ) -martingale. When τ A and τ B are independenton F h ∆ := h − h = 0 . In general, it is not the case. Moreover, by (i) in Assumption 2.1 and [27, Corollary 3.4]), τ avoidsany F -stopping time. In other words, for any F -stopping time τ F , P ( τ = τ F ) = 0 . (2.1) Remark G is only ( P , G ) -supermartingale,thus, by the Doob-Meyer decomposition, there exist a ( P , G ) -martingale ν and non-increasingprocess υ such that G = ν + υ . Therefore, assuming G is non-increasing is equivalent to setting ν = 0 , i.e., we ignore some parts of random effects in default times for mathematical simplic-ity. In addition, the condition (i) is equivalent to the statement that for any F -martingale ξ , thestopped process ξ τ is a G -martingale, see Proposition 3.4 in [33]. Thus, the condition is close to( H )-hypothesis.On this setup, we can reduce the full filtration using Lemma A.1 reported in Appendix A. Fordenoting the spaces of random variables and processes, we use standard notations which are givenin Appendix B. Before delving into the details, we will explain a motivation of our risk-sharing problem with a simple model. Let us consider two agents, A and B with constant fundingrates R A and R B . We moreover, consider a situation that the Agent A buys from the Agent Ban uncollateralized bond of unit notional amount and maturity T . If the two agents were able tofund by the (so-called) risk-free rate r , the fair value of the two parties would be e − rT . However,when R i , i ∈ { A, B } , are not equal to r , the fair values of two parties are different to the parties,and the two parties would want to recoup their individual adjustments: ( e − R i T − e − rt ) . Given theasymmetry by funding difference, we want to model how the price is determined.To this end, let p be the adjustment price “given to A ” on top of the (clean) risk-neutral price e − rT , e.g., for the bond contract, A pays − e − rT + p to B at initiation of the contract. This moneyis invested in their funding accounts up to T , and at the maturity, the Agent A will receive dollaramount from the Agent B. Therefore, the respective profit and loss of the two parties at T will be V A,pT = ( − e − rT + p ) e R A T + 1 , V B,pT = ( e − rT − p ) e R B T − . For the time being, we assume that the two parties both have the same preference of exponentialutility as U ( x ) = − e − x . Then, we find an optimal adjustment price p ∗ to maximize the two parties’aggregated utility of discounted P/L by their own funding rates, namely, p ∗ = arg max p ∈ R (cid:20) U (cid:0) − e − rT + p + e − R A T (cid:1) + λU (cid:0) e − rT − p − e − R B T (cid:1)(cid:21) , (2.2)for some λ > . The parameter λ can be interpreted as a relative bargaining power of B . Bystraightforward calculation, (2.2) becomes p ∗ = − e − R A T + e − R B T e − rT − ln ( λ )2 . (2.3) RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS λ = 1 , the optimal adjustmentprice p ∗ is determined as the middle of the individual adjustments, i.e., p ∗ = − h ( e − R A T − e − rT ) + ( e − R B T − e − rT ) i . (2.4)In addition, as λ increases, the contract becomes more advantageous to the Agent B. In the casethat the Agent B is a government, λ can be large possibly due to tax benefits in buying treasurysecurities. However, it should be mentioned that λ is generally not observable in markets, so theimportance of our model mainly remains in theoretical analysis. In the following sections, wedescribe the agents’ P/L with more details in terms of hedging portfolios for entering a derivativecontract. Remark funding transfer policy (FTP) that is beneficial to both parties wasalso discussed by [4]. It was shown that (2.4) is one of the choices satisfying their condition; see[4, Proposition 5.1]. However, there may be many choices of the FTP satisfying the condition, soinstead, we investigate the prices which are the best to the parties.
In this section, under CVA, DVA,funding spread, and collateral, we define the two parties’ hedging portfolios for entering a contract.We mostly depict the hedging portfolio in view of A . Then the portfolio of B can be derived by asimilar way. We begin this section by explainingthe cash-flow in bilateral contracts. Consider two agents who want to enter a bilateral contract whichexchanges promised dividends. We denote the cumulative dividend process by D . We assume that D is an F -adapted càdlàg process and is independent of defaults. The value of D is determined byan n -dimensional F -adapted (i.e., non-defaultable) underlying asset S = ( S , . . . , S n ) that satisfiesthe following stochastic differential equation (SDE): d S it = µ it S it d t + ( σ it ) ⊤ S it d W t , ≤ i ≤ n, (2.5)where σ i ∈ R d and µ i ∈ R are F -predictable. Moreover, we denote µ ∈ R n and σ ∈ R n × d such that ( µ ) i = µ i and row( σ ) i = σ i .It is not assumed that n = d . In other words, the considered market may or may not becomplete regardless of whether assets to hedge default risk, such as CDSs and bonds, are traded.In this paper, we consider markets with the absence of assets to hedge the default risk. We onlyassume that for all t , σ t is of full rank so that we can define the risk premium Λ as a solution of σ Λ = ( µ − r ) , (2.6)where := (1 , . . . , ∈ R d and r is an F -adapted process which represents overnight indexed swap(OIS) rate. We will later use Λ for a pricing measure to define close-out amount. Recall that theexistence of Λ guarantees arbitrage-free condition in classical context. However, since the classicaldefinition of arbitrage opportunity does not reflect adequately the hedger-specific nature of bilateralcontracts, there have been many studies to redefine arbitrage opportunity properly in the contextof bilateral contracts. The condition being developed is slightly different from paper to paper, butoften absence of arbitrage opportunity is obtained with similar conditions to (2.6). See, for example,[12, Proposition 3.3]. For definitions of hedger-specific arbitrage opportunities , readers can refer to[12, 10, 6, 7, 8, 41, 40].We set, as a convention, a positive value (resp. negative) of dividend process at a certainmoment to mean that the Agent A pays to (resp. is paid by) the Agent B. For example, if A sells aput option on S with the exercise price κ and maturity T , then for any t ≥ , D t = t ≥ T ( κ − S T ) + .Note that the initial price exchanged at initiation of the contract is not a part of D . We will include J. LEE, S. STURM, C. ZHOU the initial price in the hedging portfolios as their initial value. Because jumps of an F -adapted càdlàgprocess are exhausted by F -stopping times by [35, Theorem 4.21], and τ avoids any F -stopping time(recall (2.1)), D does not jump at default, i.e., ∆ D τ = 0 , almost surely.(2.7)Let us turn to explain close-out amount and margin process. The obligation on dividend D maynot be fully honored at one party’s default. For the default risk, covenants of the close-out amountand collateral are documented in a Credit Support Annex before initiation of the contract. At theevent of default, the dividend stream stops and CSA close-out amount should be settled. However,because of the default, the defaulting party would not be able to pay the full close-out amount. Tomitigate the risk of losses at default, collateral is exchanged between the two parties. In bilateralcontracts, variation margin and initial margin are posted in general, and the close-out amount isoften determined as mark-to-market exposure.In our model, only variation margin is a part of the control variables of our stochastic controlproblem that will be introduced later. We exclude initial margin for simplicity. As stated in[13, p.12], “the amount of variation margin reflects the size of this current exposure,” and it isrecommended that “the full amount necessary to fully collateralise the mark-to-market exposureof the non-centrally cleared derivatives must be exchanged” [13, p.15]. The meaning behind thisregulation will be discussed later based on our model. Remark initial margin , readers can refer to [45, 22]. In addition, initial margin causes associated BSDEs anticipated. For the numerical simulation of anticipatedBSDEs, readers can refer to [1].Now, we depict the close-out amount and variation margin mathematically. One of the popularchoices to calculate the market exposure is clean price which is basically the classical risk-neutralprice. As used in the classical pricing, we use the “so-called” risk-free rate. Note that it is a littleout of context to call it the risk-free rate since arguments under bilateral contracts are from thereality that dealers cannot access to the risk-free rate, yet it is acknowledged that OIS rate is thebest proxy for the so-called risk-free rate. Thus, in what follows, we use OIS rate for evaluating the clean price , and denote it by ( r t ) t ≥ . We assume ( r t ) t ≥ is F -adapted and denote by B the moneymarket account on ( r t ) t ≥ , namely B t := exp (cid:18) Z t r s d s (cid:19) for any t ≥ . We can find the pricing measure Q such that B − S i , ≤ i ≤ n , are ( Q , F ) -local martingales because σ is of full rank as in (2.6). Let e t denote the new mark-to-market exposure at t ≤ T . We assumethat the market exposure is calculated as clean price : Assumption
For any t ∈ [0 , T ] , e t = B t E Q (cid:16) R Tt B − s d D s (cid:12)(cid:12)(cid:12) F t (cid:17) . By Assumption 2.5, we can derive some properties of ( e t ) t ≥ . We report the proof in Appen-dix D. Lemma (i) e T = 0 .(ii) e ¯ τ = τ ≤ T e τ .(iii) d e t = (cid:0) r t e t + B t Z t Λ t (cid:1) d t + B t Z t d W t − d D t , for t ∈ [0 , T ] , for some Z ∈ H ,dT,loc .(iv) e τ − = e τ almost surely.Remark Z is closely related with the delta risk of ˜ e t = B − t e t . Indeed, if e is Malliavin differentiable and is a smooth function of S t , i.e., ˜ e t = ˜ e ( t, S t ) , then by Clark-Ocone A part of ISDA Master Agreement. RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS Z t = D t ˜ e t = [diag( S t ) σ t ] ⊤ ∇ S ˜ e ( t, S t ) , where D t is the Malliavin derivative operator at t and ∇ S is the classical gradient. We shall see later that Z has a special role to interpret themeaning behind full collateralization. In addition, if D ∈ S T , we can choose Z in H ,dT .Let m t denote the amount of variation margin posted at t ≤ T . Similarly, m t ≥ (resp. m t < )means that A posts (resp. receives) the margin to B at time t ≤ T . We assume ( m t ) t ≥ is an F -adapted càdlàg process. Note that m is chosen to be F -adapted for consistency in financial modeling.The collateral is required because we do not know the full information of default. Therefore, theamount of collateral is calculated only by available information F . The admissible set will be definedmore precisely when the risk-sharing problem is introduced.Once one party announces bankruptcy, the margin process stops. Therefore, at the default τ ≤ T , the amount of collateral is m τ − , but wealth which amounts to e τ + ∆ D τ (= e τ a.s ) shouldbe transferred from A to B . In addition, the loss by breach of the contract will be inflicted to theAgent B (resp. A) only when τ = τ A and e τ ≥ m τ − (resp. τ = τ B and e τ < m τ − ). Denoting theloss rate of the Agent A (resp. B) by L A (resp. L B ) , the amount by breach is τ = τ A L A ( e τ − m τ − ) + − τ = τ B L B ( e τ − m τ − ) − . We assume that L i , i ∈ { A, B } , are positive constant. Finally, we can define the full cash-flow C as C t := τ>t D t + τ ≤ t (cid:0) D τ + e τ (cid:1) − τ = τ A ≤ t L A ( e τ + ∆ D τ − m τ − ) + + τ = τ B ≤ t L B ( e τ + ∆ D τ − m τ − ) − . By (2.7) and the last item in Lemma 2.6, for any t ≤ T , almost surely C t = τ>t D t + τ ≤ t (cid:0) D τ + e τ (cid:1) − τ = τ A ≤ t L A ( e τ − m τ − ) + + τ = τ B ≤ t L B ( e τ − m τ − ) − . (2.8)In the next section, we define a self-financing portfolios to hedge against C with more details.We construct the portfolio in view of the Agent A since the portfolio of the Agent B is in most wayssimilar. Before proceeding, we provide some remarks related to possible extensions of our model. Remark clean price is not an appropriate close-out amountsince the Agent B’s default is not considered. However, taking default risk of the AgentB into the exposure may heavily penalize the surviving party, because the default event ofone party can negatively affect the creditworthiness of the surviving party, especially whenthe defaulting member has an impact on systemic risk. For such discussion, readers canrefer to [21].(ii) In practice, variation margin is called on intra-day basis (say, two or three times per day).In this paper, we assume a continuous margin process for simplicity. One may want tomodel variation margin as a càdlàg step process to describe reality more precisely, cf. [20].(iii) Underlying assets subject to defaults are beyond the scope of this paper. For modelingwith emphasis on contagion risk, readers may want to refer to [38, 17, 18, 16].(iv) In reality, it is hard to estimate exact default intensities. For example, dependence be-tween the Agent B’s exposure and default probability is not negligible, which is sometimescalled right/wrong way risk, but it is challenging to estimate the dependence from marketquotation. Thus, such issues lead to robust pricing arguments. See [34, 9].
The funding sources of an agent can be an ex-ternal funding provider, treasury department, repo markets, etc. After the financial crisis, suchfunding rates do not represent the risk-free rate (approximately OIS rate in recent times). Weconsider F -adapted processes ( R i,mt ) t ≥ , i ∈ { A, B } , to represent the margin funding rates offeredfrom margin lenders, and denote that for t ≥ , B i,mt := exp (cid:16) Z t R i,ms d s (cid:17) . (2.9) J. LEE, S. STURM, C. ZHOU
A Br m R A,m R B,m
Margin providerFunding desk Margin providerFunding desk R A R B Fig. 1: Interest rates among parties. When a party is funded from its internal funding provider fordelivering collateral, R i,m = R i . In practice, r m is often chosen as federal funds rate or EONIArate, i.e., r m = r .Another cash-flow stream associated with margin process is remuneration from margin receivers.When variation margin is pledged by (resp. received by) the Agent A, the Agent B should remu-nerate (resp. should be remunerated by) the Agent A with respect to an interest rate. We let r m and B m denote the remuneration rate and account of the Agent A. Therefore, for each party, thenet cost/benefit involved in posting the margin is determined by R i,m − r m , i ∈ { A, B } , and wedenote this spread by s i,m , i.e., s i,m := R i,m − r m , i ∈ { A, B } . (2.10)In general, rehypothecation is allowed for variation margin , in other words, the margin account canbe used to maintain the hedging portfolio. We moreover, assume that the two parties should financetheir operations by interest rates R i , i ∈ { A, B } , for constructing the rest of the portfolios. Wedenote the associated funding account and spread by B i and s i , i ∈ { A, C } , respectively, i.e., B it := exp (cid:16) Z t R is d s (cid:17) , (2.11) s it := R it − r t . (2.12)Then, each party constructs their hedging portfolio using the above accounts and risky as-sets ( B i , B i,m , B m , S ) , i ∈ { A, B } . Let ϕ i := ( η i , η i,m , η m , η i,S ) denote the respective units of ( B i , B i,m , B m , S ) , i ∈ { A, B } , in the hedging portfolios and we call ϕ i the trading strategy of theAgent i . We assume that ϕ i , i ∈ { A, B } , are G -predictable and use the convention that a positiveunit of trading strategy means long position.If the Agent A posts collateral which amounts to m t at t , she needs to deliver η A,mt shares ofthe account B A,mt from the margin lender. Then, the Agent A will have η mt shares of the marginaccount B mt to which the remuneration from the Agent B is accrued. Thus, η m B m = m, (2.13) η A,m B A,m + η m B m = 0 , (2.14) η B,m B B,m − η m B m = 0 . (2.15) Remark rehypothecated . Sometimes itis possible that risky assets can be posted as collateral and the margin account is segregated , which
RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS self-financing portfolio . Definition If V At = V At ( ϕ A , C ) , t ∈ [0 , T ] , defined as V At = η At B At + η A,mt B A,mt + η mt B mt + η A,St S t , (2.16) satisfies V At = V A + Z t ∧ ¯ τ η As d B As + Z t ∧ ¯ τ η A,ms d B A,ms + Z t ∧ ¯ τ η ms d B ms + Z t ∧ ¯ τ η A,Ss d S s − C t ∧ ¯ τ , (2.17) for any t ∈ [0 , T ] , then V A is called the self-financing portfolio of the Agent A.Remark t > ¯ τ , V At = V A ¯ τ . In (2.17), C t ∧ ¯ τ = C t , for any t ≥ , by thedefinition (2.8).The self-financing portfolio of the Agent B is defined similarly. The difference is the direction of variation margin and C . Then, by (2.10), (2.13)-(2.17), we can see that self-financing portfolioprocesses of the Agent A and, similarly, the Agent B follow d V At = (cid:16) R At V At − s A,mt m t + η A,St ⊙ S t [ µ t − R At ] (cid:17) d t + η A,St ⊙ S t σ t d W t − d C t , (2.18) d V Bt = (cid:16) R Bt V Bt + s B,m m t + η B,St ⊙ S t [ µ t − R Bt ] (cid:17) d t + η B,St ⊙ S t σ t d W t + d C t , (2.19)where ⊙ is component-wise product. If we consider an agent who has a naked position againstmarket risk, we set η i,S = 0 . Before examining whether (2.18) and (2.19) are well defined, we firstwant to introduce our target problem.We find the best initial price and amount of variation margin to optimize the aggregatedutilities of both parties. If there were no adjustment in pricing, the classical risk-neutral price e should be exchanged at initiation of the contract. Let p denote the amount paid to the Agent A ontop of e . Therefore, initial price paid to the Agent A is e + p . More precisely, denoting the initialendowment of each party by ν A and ν B , V A = ν A + e + p, and V B = ν B − e − p. Thus, V i depends on the choice of ( p, m ) . For simplicity of notations, we often suppress ( p, m ) ,e.g., V i = V i,p,m , i ∈ { A, B } . Then, with an admissible set A , utilities U i : R → R , and λ > , wedefine the risk-sharing problem as follows: ( p ∗ , m ∗ ) = arg max ( p,m ) ∈A E h U A (cid:0) ( B A ¯ τ ) − V A,p,m ¯ τ (cid:1) + λU B (cid:0) ( B B ¯ τ ) − V B,p,m ¯ τ (cid:1)i . (2.20)We will define A more precisely in the following section. Note that hedging strategies are notcontrol variables. In other words, we assume that two parties choose their strategies by their ownmethodologies not by the risk-sharing framework. It can be said that λ is the relative bargainingpower of the Agent B, or how much the Agent A wants to enter the contract. One can also thinkof λ as the belief of how much funding spread should be acknowledged in derivative transactions.0 J. LEE, S. STURM, C. ZHOU
As we set the conventions in subsection 2.3.2, p ∗ is the amount paid to the Agent A on top of e . This additional payment is necessary because of default risk and funding spread. If two partiesprice the contracts individually, the calculated prices may be different to each party because ofdifferent funding spread on this model. Therefore, when the contract is made with the initial price e + p ∗ , some parties should accept a cost. Thus, p ∗ can be seen as the cost that is agreed by thetwo parties to enter the contract, so we call p ∗ agreement-cost . We also call m ∗ optimal collateral(or margin), and ( p ∗ , m ∗ ) risk-sharing contract .Before giving the detail, we first provide motivation about the discounting factors behind thechoice of our model (2.20). In (2.20), the values of the portfolios were adjusted by discountingfactors. The discounting factors are necessary for a fairness since the two agents have differentfunding rates. In general, the higher default risk is, the higher funding rate is. However, a hedgingportfolio grows with respect to its funding rate (recall (2.18) and (2.19)). Therefore, without thediscounting factors, we penalize a party under a healthier credit condition.One may want to put the discounting factors outside of utilities as it is a typical choice inportfolio optimization literature. In this case, when the portfolio processes evolve forwardly, theeffect of funding rates is mixed with risk aversion parameters in the utilities. However, the futurevalue is purely discounted without consideration of risk aversions, so we would again end up withpunishing or rewarding a certain party depending on risk aversions. An argument in the samecontext was discussed in [48].For the utilities, we will investigate two cases: U A ( x ) = x, U B ( x ) = − e − γ B x , (2.21) U A ( x ) = − e − γ A x , U B ( x ) = − e − γ B x , (2.22)for some γ i > . We choose the exponential utilities mainly for simplicity. To use a power utility,we need for V i , i ∈ { A, B } , to be lower bounded. To this end, boundedness condition should beimposed to ( p, m ) , but this makes the exposition more complicated. Moreover, an explicit form ofoptimal collateral is not generally obtained under power utilities.To solve the risk-sharing problem, we need different restrictions to funding spread dependingon the choice of utilities for characterizing the optimal collateral. The restrictions are requiredmainly because the value functions w.r.t variation margin is not concave. More precisely, we willneed that s B,m = 0 in (2.21), and s A,m = s B,m = 0 in (2.22). The conditions on funding spreadcan be assumed not only when the capital structure of a party has small leverage but also whenthe party achieves secured funding for variation margin . This situation is not common, but someexamples for the secured funding were discussed by [3].There is another interpretation to keep the funding condition without loss of much generality,which is partly justified by a complete market argument. It was shown in complete market modelsthat an agent can guarantee that they do not switch their position of funding state between lendingand borrowing position, depending on the structure of the payoff. This binary nature of fundingstate is related to whether payoff functions are non-increasing or non-decreasing with respect tounderlying assets. For the details, see Proposition 5.8 in [32] and refer to [42]. To streamlinethis paper, we deal with cases of (2.22), in the main sections. For the cases of risk-neutral agent,we report the analysis in Appendix C. Therefore, the following assumption stands throughout thefollowing sections except Appendix C.
Assumption (i) s A,m = s B,m = 0 ,(ii) U A ( x ) = − e − γ A x and U B ( x ) = − e − γ B x . In the next section, we represent (2.20) in a reduced form with a more precise definition of theadmissible set. (2.20) is one type of principal-agent problems. This problem is often called the firstbest case in typical principal-agent context. In general, it is challenging to solve principal-agentproblems because the solvability of involved equations, e.g., coupled FBSDEs, is not easy to obtain.
RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS
We start this section with introducing a long list of notations.The following notations are often used in this paper. For i ∈ { A, B } , t ∈ [0 , T ] , ¯ V At := ( B At ) − ( V At − e t ∧ ¯ τ ) , ¯ V Bt := ( B Bt ) − ( V Bt + e t ∧ ¯ τ ) , (2.23) v t := ( B At ) − e t , c t := ( B At ) − m t , (2.24) δ t := v t − c t , K t := B At ( B Bt ) − , (2.25) π it := ( B it ) − η i,St ⊙ S t σ t , ∆ it := B t ( B it ) − Z t , (2.26) ¯ φ At := π At − ∆ At , ¯ φ Bt := π Bt + ∆ Bt , (2.27) σ t Λ it := ( µ t − R it ) , b it := Λ it − Λ t , (2.28) Θ t ( δ ) := τ A = t L A δ + − τ B = t L B δ − . (2.29)We give some remarks on the above notations. Remark ¯ V i , i ∈ { A, B } are (discounted) adjustment processes. By (2.24), v isthe discounted market exposure, and c is the discounted collateral, and by (2.25), δ is the differencebetween the two processes. Note that δ = v − c = 0 means full collateralization. By (2.26), ∆ i , i ∈ { A, B } , are the delta-risk of the market exposure adjusted by the funding rate of each party.By (2.27), ¯ φ i , i ∈ { A, B } , are the difference between the amount invested in the risky assets anddelta risk of the clean price ,i.e., ¯ φ i can be seen as the hedging error. If the Agent B does not hedgethe market risk, we have ¯ φ B = ∆ B . Notice that if b i = 0 , then R i = r , by (2.28).We will find the projections of ¯ V i onto F , then we will deal with the risk-sharing problem mainlywith the reduced processes. For any i ∈ { A, B } , we let φ i denote the F -predictable reduction of ¯ φ i until ¯ τ . Namely, φ i , i ∈ { A, B } , are F -predictable and t ≤ ¯ τ ¯ φ it = t ≤ ¯ τ φ it . By Itô’s formula and (2.24), v satisfies, for t ∈ [0 , T ] , d v t = (cid:0) − s At v t + ∆ At Λ t (cid:1) d t + ∆ At d W t − ( B At ) − d D t . Note that v is exogenously given. Thus, if R i , i ∈ { A, B } , are independent with V i and ¯ V i arewell-defined, then V i are also well defined by (2.23). Theorem
Assume s i , s i,m , i ∈ { A, B } , are bounded and X i ∈{ A,B } Z T (cid:16) | δ t | + | Λ it | + | φ it | + | b it | + | ∆ it | (cid:17) d t < ∞ , a.s.Then, the following processes v A and v B , are well-defined: d v At = (cid:0) φ At Λ At + ∆ At b At + s At v t (cid:1) d t + φ At d W t , (2.30) d v Bt = (cid:0) φ Bt Λ Bt − ∆ Bt b Bt − s Bt K t v t (cid:1) d t + φ Bt d W t . (2.31) Moreover, assume that R i , i ∈ { A, B } , are independent with V i , and v A = ν A + p,v B = ν B − p. Then v i , i ∈ { A, B } , are F -optional reductions of ¯ V i until ¯ τ , i.e., v i , i ∈ { H, C } , are F -optionaland t< ¯ τ ¯ V it = t< ¯ τ v it , for any t ≥ . J. LEE, S. STURM, C. ZHOU
Proof.
It is easy to check the first assertion. To check the second part, we apply Itô’s formulato ( B At ) − V At and this yields d (cid:0) ( B At ) − V At (cid:1) = − R At ( B At ) − V At d t + ( B At ) − d V At = t ≤ ¯ τ π At Λ At d t + t ≤ ¯ τ π At d W t − ( B At ) − d C t . In addition, by (2.8) together with v τ = v τ − , a.s, ( B At ) − d C t = t ≤ τ ( B At ) − d D t + d( τ ≤ t ) v τ − − d( τ ≤ t )Θ τ ( δ τ − ) . (2.32)Then, by combining (2.32) and (2.33), we have d ¯ V At = d (cid:0) ( B At ) − ( V At ) − v t ∧ ¯ τ (cid:1) = t ≤ ¯ τ (cid:0) s At v t + ¯ φ At Λ At + ∆ At b At (cid:1) d t + t ≤ ¯ τ ¯ φ At d W t − t>τ ( B At ) − d D t − d( τ ≤ t ) (cid:0) v τ − − Θ τ ( δ τ − ) (cid:1) . (2.33)It follows that d (cid:0) t< ¯ τ ¯ V At (cid:1) = ¯ V At − d( t< ¯ τ ) + t ≤ ¯ τ d ¯ V At − δ ¯ τ (d t )∆ ¯ V A ¯ τ = ¯ V A ¯ τ − d( t< ¯ τ ) + t ≤ ¯ τ (cid:0) s At v t + ¯ φ At Λ At + ∆ At b At (cid:1) d t + t ≤ ¯ τ ¯ φ At d W t − d( t ≥ ¯ τ ) (cid:0) v τ − − Θ τ ( δ τ − ) (cid:1) − δ ¯ τ (d t )∆ ¯ V A ¯ τ = t ≤ ¯ τ d v At − d( t ≥ ¯ τ ) ¯ V A ¯ τ − d( t ≥ ¯ τ ) (cid:0) v τ − − Θ τ ( δ τ − ) (cid:1) . Let Y t := t< ¯ τ v At + t ≥ ¯ τ ( v Aτ − − v τ − + Θ τ ( δ τ − )) . Again, by Itô’s formula together with v τ = v τ − ,a.s, d Y t = t ≤ ¯ τ d v At − d( ¯ τ ≤ t ) v At − + d( ¯ τ ≤ t ) (cid:0) v Aτ − − v τ − + Θ τ ( δ τ − (cid:1) = t ≤ ¯ τ d v At − d( ¯ τ ≤ t ) (cid:0) v τ − − Θ τ ( δ τ − ) (cid:1) . Thus, if Y = ¯ V A , we obtain Y t = ¯ V At , for any t ∈ [0 , T ] . Moreover, v A is the F -optional reductionof ¯ V A and more precisely, ¯ V At = t< ¯ τ v At + t ≥ ¯ τ ( v Aτ − − v τ − + Θ τ ( δ τ − )) . (2.34)Similarly, we can attain that ¯ V Bt = t< ¯ τ v Bt + t ≥ ¯ τ ( v Bτ − + K τ − v τ − − K τ − Θ τ ( δ τ − )) . (2.35)Notice that control of m is equivalent to that of δ since e is given exogenously. Thus, we solve (2.20)with respect to the two state processes depending on δ : V i,p,m = V i,p,δ . Moreover, we denote that c ∗ := ( B A ) − m ∗ and δ ∗ := v − c ∗ .Now, we are ready reduce the risk-sharing problem. Recall from (2.20) that our goal is tomaximize the sum of utilities of discounted portfolios over all ( p, δ ) ∈ A : E h U A (cid:0) ( B A ¯ τ ) − V A,p,δ ¯ τ (cid:1) + λU B (cid:0) ( B B ¯ τ ) − V B,p,δ ¯ τ (cid:1)i . (2.36) RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS E h U A (cid:0) ( B A ¯ τ ) − V A,p,δ ¯ τ (cid:1)i = E h U A (cid:0) T <τ v A,pT + τ ≤ T ( v A,pτ − + Θ δτ ) (cid:1)i = E (cid:20) G T U A ( v A,pT ) + Z T G t h h At U A (cid:0) v A,pt + L A δ + t (cid:1) + h Bt U A (cid:0) v A,pt − L B δ − t (cid:1)i d t (cid:21) , and E h U B (cid:0) ( B B ¯ τ ) − V B,p,δ ¯ τ (cid:1)i = E h U B (cid:0) T <τ v B,pT + τ ≤ T ( v B,pτ − − K τ Θ δτ ) (cid:1)i = E (cid:20) G T U B ( v B,pT ) + Z T G t h h At U B (cid:0) v B,pt − L A K t δ + (cid:1) + h Bt U B (cid:0) v B,pt + L B K t δ − t (cid:1)i d t i . We define g t := δ ≥ g + t + δ< g − t , where g + t ( v A , v B , δ ) := G t h h At (cid:0) U A ( v A + L A δ ) + λU B ( v B − L A K t δ ) (cid:1) + h Bt (cid:0) U A ( v A ) + λU B ( v B ) (cid:1)i g − t ( v A , v B , δ ) := G t h h Bt (cid:0) U A ( v A + L B δ ) + λU B ( v B − L B K t δ ) (cid:1) + h At (cid:0) U A ( v A ) + λU B ( v B ) (cid:1)i . For the above reduction to be valid, we assume the following integrbility condition: X i ∈{ A,B } (cid:20) | U i ( v iT ) | + Z T | U i ( v it ) | d t (cid:21) < ∞ , (2.37)and we define the admissible set of collateral D for a given Borel set A ⊆ R as follows: Definition δ ∈ D , if δ ∈ H T and(i) δ ∈ A , d P ⊗ d t − a.s,(ii) E (cid:2) R T (cid:12)(cid:12) g t ( v A,pt , v
B,pt , δ t ) (cid:12)(cid:12) d t (cid:3) < ∞ . Then, the risk-sharing problem can be rewritten as max ( p,δ ) ∈A E (cid:20) G T U A ( v A,pT ) + λG T U B ( v B,pT ) + Z T g t ( v A,pt , v
B,pt , δ t ) d t (cid:21) , (2.38)where A = R × D and v i , i ∈ { A, B } , are defined in (2.23)-(2.28), (2.30), and (2.31). Note that v i does not depend on δ since we assume s A,m = s B,m = 0 . When the Agent A is risk-neutral, weonly need that s B,m = 0 , and in this case D should be defined in a slightly different way. We willdiscuss this with more details in Appendix C. Remark risk-sharing framework can be thought of as a two-agent problem.Since there is no party who can solely decide the contract, the mathematical structure ofour risk-sharing problem is different from that of typical principal-agent problems. Forexample, for the Agent A, in both perspectives of funding impacts and loss given defaults,posting collateral to the Agent B is not beneficial. Say, we consider A as an agent, subjectto B as a principal. Then, if we solve the agent problem first, e.g., as in [30], it alwaysgives the trivial solution δ ∗ = v − c ∗ = ∞ .4 J. LEE, S. STURM, C. ZHOU (ii) One may want to take stochastic calculus of variation as in [31]. However, note that g ispiece-wise concave in δ . Even when g is concave, it may not be differentiable. Therefore,if we take stochastic calculus of variation, we will face a very challenging FBSDE witha discontinuous coefficient in the drift of (forward) SDE. A similar case was dealt withby [26]. However, in our case, we encounter multi-dimensional FBSDE with a degeneratevolatility and unbounded coefficients. The solvability of such FBSDE is beyond the scopeof this paper and we leave it as future research. Instead, in this paper, we impose someconditions on funding cost/benefit in delivering the collateral, s i,m , i ∈ { A, B } , dependingon the utilities, and use verification argument.
3. Optimal Collateral.
In this section, we characterize the optimal collateral in the risk-sharing problem by using martingale optimality principle. Then we argue by verification that thecharacterized collateral is indeed an optimal solution. First, we solve the problem with respect toonly variation margin δ , with a fixed initial price p ∈ R : max δ ∈D E (cid:20) G T U A ( v A,p,δT ) + λG T U B ( v B,p,δT ) + Z T g t ( v A,p,δt , v
B,p,δt , δ t ) d t (cid:21) . (3.1)Then, the agreement-cost p ∗ will be found with the given optimal variation margin δ ∗ . However,mainly because of the non-concave property of our problem addressed in Remark 2.16, we needto impose some restrictions to the funding spread in delivering collateral for the characterization.Recall that we consider two cases: a risk-neutral Agent A and risk-averse the Agent B investingtheir capital with a small leverage so that s B,m = 0 , and two risk averse parties with s B,m = s A,m = 0 . As we mentioned, the mathematical analysis for a risk-neutral agent is deferred toAppendix C. In what follows, we first derive an optimal collateral for both cases, then we givefinancial interpretations later. The two most notable features are the weak dependence with defaultintensities and relationship with the full margin requirement. The discussion about the relationshipbetween the optimal collateral and margin requirement is an important part of this paper.Thisfunding condition is not necessary for finding p ∗ if δ is not a control variable, e.g. A = { δ } forsome δ ∈ R .We define a dynamic version of (2.38) and use martingale optimality principle (MOP) as in[39]. To this end, we define a set of controls which coincide with a given ε ∈ D up to a certain time t ≤ T . We denote the set by D ( t, ε ) , i.e., for ε ∈ D , D ( t, ε ) := (cid:8) δ ∈ D (cid:12)(cid:12) δ . ∧ t = ε ·∧ t (cid:9) . Now, define thedynamic version of (3.1) as J εt ( p ) := ess sup δ ∈D ( t,ε ) E (cid:20) G T U A ( v A,pT ) + λG T U B ( v B,pT ) + Z Tt g s ( v A,ps , v
B,ps , ε s ) d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . (3.2)Then we characterize the optimal collateral by using martingale optimality principle. By MOP, ( J εt ) ≤ t ≤ T is chosen as a càdlàg version such that for any ε ∈ D , n J εt + Z t g s ( v As , v Bs , ε s ) d s o ≤ t ≤ T is a ( P , F ) -supermartingale. Moreover, for the optimal collateral δ ∗ for J , n J δ ∗ t + Z t g s ( v As , v Bs , δ ∗ s ) d s d s o ≤ t ≤ T is a ( P , F ) -martingale. When the admissibility is guaranteed, a solution to (3.1) can be found byverification.Before moving on, to represent the optimal collateral by one stochastic process, we define aprocess ( X t ) t ≥ such that U A ( X t ) := U A ( v At ) − U B (cid:0) v Bt − v B (cid:1) = − exp (cid:20) − γ A (cid:16) v At − γ B γ A ( v Bt − v B ) (cid:17)(cid:21) . (3.3) RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS X t = v At − ( γ B /γ A )( v Bt − v B ) . More precisely, X is given by X t = v A + Z t h s t v t + φ At Λ At − γ B γ A φ Bt Λ Bt + ∆ At b At + γ B γ A ∆ Bt b Bt i d s + Z t φ s d W s , (3.4)Then, (3.1) will be represented w.r.t X , and where φ t := φ At − ( γ B /γ A ) φ Bt and s t := s At +( γ B /γ A ) s Bt K t . Theorem
Assume that the integrability condition (2.37) hold. Define δ ∗ t ( p, x ) := arg max δ ∈ A (cid:8) U A ( x ) ψ At ( δ ) + λU B ( v B,p ) ψ Bt ( δ ) (cid:9) , (3.5) ψ At ( δ ) := − h At U A ( L A δ + ) − h Bt U A ( − L B δ − ) , (3.6) ψ Bt ( δ ) := − h At U B ( − L A K t δ + ) − h Bt U B ( L B K t δ − ) . (3.7) If δ ∗ ( p, X ) ∈ D , then δ ∗ ( p, X ) is a solution of (3.1). Moreover, J = E (cid:20) β T (cid:2) U A ( X T ) + λU B ( ν B − p ) (cid:3) + Z T ˆ f t ( p, X t ) d t (cid:21) , (3.8) where β t := − G t U B (cid:0) v Bt − v B (cid:1) and ˆ f t ( p, x ) := β t (cid:2) U A ( x ) ψ At ( δ ∗ t ( p, x )) + λU B ( ν B − p ) ψ Bt ( δ ∗ t ( p, x )) (cid:3) . (3.9) Proof.
For ε ∈ D , define ξ εt := J t + R t g s ( v As , v Bs , ε s ) d s . Notice that J t is independent of ε ∈ D and by (3.6) and (3.7), − (cid:2) G t U B ( v Bt − v B ) (cid:3) − g t ( v At , v Bt , ε t ) = U A ( X t ) ψ At ( ε t ) + λU B ( v B ) ψ Bt ( ε t ) . Therefore, for any ε ∈ D , ξ ε − ξ δ ∗ ( p,X ) is a ( P , F ) -supermartingale. Moreover, for any ǫ ∈ DE (cid:2) ξ ǫT − ξ δ ∗ ( p,X ) T (cid:3) ≤ E (cid:2) ξ ǫ − ξ δ ∗ ( p,X )0 (cid:3) = 0 . (3.10)Thus, (3.8) is obtained where the admissibility of δ ∗ ( p, X ) is guaranteed.To find the explicit form of δ ∗ ( p, X ) , we consider A = R and represent (3.5) as δ ∗ t ( p, x ) := arg max δ ∈ A (cid:0) δ< f − ( t, p, x, δ ) + δ ≥ f + ( t, p, x, δ ) (cid:1) , for some functions f − , f + . Then, f i , i ∈ {− , + } are continuously differentiable in δ and for any ( t, p, x ) , there exist I it ( p, x ) such that ∂ δ f i ( t, p, x, I it ( p, x )) = 0 . (3.11)Then, δ ∗ ( p, X ) can be attained at I − , I + , and zero. We can easily see that f − ( t, p, x, δ ) := h Bt (cid:2) U A ( x + L B δ ) + λU B ( ν B − p − L B K t δ ) (cid:3) + h At (cid:2) U A ( x ) + λU B ( ν B − p ) (cid:3) , (3.12) f + ( t, p, x, δ ) := h At (cid:2) U A ( x + L A δ ) + λU B ( ν B − p − L A K t δ ) (cid:3) + h Bt (cid:2) U A ( x ) + λU B ( ν B − p − L B K t ) (cid:3) . (3.13)6 J. LEE, S. STURM, C. ZHOU
Therefore, we obtain that I − t ( p, x ) := γ B ν B − γ B p − γ A x − ln (cid:0) λK t γ B γ A (cid:1) L B ( γ B K t + γ A ) , (3.14) I + t ( p, x ) := γ B ν B − γ B p − γ A x − ln (cid:0) λK t γ B γ A (cid:1) L A ( γ B K t + γ A ) . (3.15)The exact form of δ ∗ can be obtained by characterizing the region n max R f − > max R f + o . The calculation of the region is a straightforward but tedious; see, e.g., [24]. We only obtain theexact form for a simple case which will be seen later. We complete Theorem 3.1 by the next lemma.The proof is reported in Appendix D.
Lemma
Let A = R , and assume ( e t ) t ≥ , ( Z t ) t ≥ , ( π it ) t ≥ , i ∈ { A, B } , are bounded. Then δ ∗ ( p, X ) ∈ D and (2.37) hold.Example π A = ∆ A , π B = 0 . They enter a bond contract that is paid by the Agent A, namely, D = J T, ∞ K .We assume that OIS rate ( r t ) t ≥ follows the next SDE: d r t = k ( θ − r t ) d t + ρ √ r t d W Q t , for some k, θ, ρ ∈ R and a risk-neutral measure Q . Moreover, we assume that h i , i ∈ { A, B } , arebounded and s B = s B,m = 0 . Then, by Clark-Ocone formula, for t ≥ , Z t = − ρ √ r t A ( t, T ) B − t e t , where e t = A ( t, T ) e − r t A ( t,T ) ,A ( t, T ) := (cid:16) ae ( a + k )( T − t ) / a + ( a + k )( e a ( T − t ) − (cid:17) kθ/ρ ,A ( t, T ) := 2( e a ( T − t ) − a + ( a + k )( e a ( T − t ) − ,a := p k + 2 ρ . Since r > , Z is bounded. Hence, all conditions in Lemma 3.2 are satisfied.Now, we are ready to discuss the financial interpretation of the optimal collateral. In the nextsection, the financial meanings of (3.14)-(3.15) and the relationship with the margin requirementwill be discussed. In this section, we provide financial interpretations of the opti-mal collateral derived in the previous sections. Collateral is posted for default risk. In our model,there are two main components in default risk: intensities and loss rates. We first discuss a weakdependence between the optimal collateral and default intensities. We begin with giving the explicitform of δ ∗ in the following lemma. We report the proof in Appendix D. RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS Lemma
Assume A = R . Then, δ ∗ t ( p, x ) is given by δ ∗ t ( p, x ) =(0 ∨ I + t ( p, x )) + (0 ∧ I − t ( p, x )) , where (3.16) I − t ( p, x ) := − γ A x − γ B pL B ( γ B K t + γ A ) + γ B ν B − ln (cid:0) λK t γ B γ A (cid:1) L B ( γ B K t + γ A ) , (3.17) I + t ( p, x ) := − γ A x − γ B pL A ( γ B K t + γ A ) + γ B ν B − ln (cid:0) λK t γ B γ A (cid:1) L A ( γ B K t + γ A ) . (3.18)Note from (3.16)-(3.18) that the optimal collateral depends only on the loss rates L i not on defaultintensities h i , which is a rather natural consequence. Collateral is required for loss given defaultnot for the default itself. Put differently, collateral is about how much loss would be inflicted atdefault and not about how likely default occurs. Recalling δ ∗ = v − c ∗ and observing (3.17) and(3.18), the magnitude of the optimal variation margin c ∗ increases as L i , i ∈ { A, B } , increase.We discuss the effect of loss rates with more details. By (3.16), when δ ∗ ( p, X ) ≤ , I − ( p, X ) = δ ∗ ( p, X ) . In this case, as L B increases, c ∗ = v − δ ∗ ( p, X ) decrease sbecause of the increased averageloss of collateral posted to the Agent B. On the other hand, when δ ∗ ( p, X ) ≥ , the optimalcollateral c ∗ = v − δ ∗ ( p, X ) , is independent of L B and increases w.r.t L A . Again, this is becausethe high loss rate makes it risky for the Agent B to post collateral to the Agent A.The relationships with p and λ are self-explanatory. If the contract starts from giving a highprice p , to the Agent A at initiation of the contract, the Agent A needs to post more collateral inreturn. Moreover, the higher λ is, i.e., the strong bargaining power the Agent B has, the more theAgent A should post more collateral.In addition, recalling X t = v At − ( γ B /γ A )( v Bt − v B ) , it seems that (3.16) suggests that optimalcollateral ratio should be decided by the relative performance of each party. It is not obviouslyapplicable in practice. However, we can use X to derive an interesting interpretation from the fullmargin requirement of Basel III, which will be discussed in the next section. Remark . regardless of entities. Our model together with the practice on lossrates partly explains the margin requirement applied to all banks. In this section, we interpret the meaningbehind the inter-dealer market convention that is required by Basel III. It can be understood thatthe inter-dealer convention is δ ∗ ( p, X ) = 0 . By (3.14) and (3.15), the full margin convention requiresthat X t + 1 γ A ( s A − s B ) t = γ B γ A ( ν B − p ) − γ A ln (cid:18) λγ B γ A (cid:19) , d P ⊗ d t − a.s . (3.19)Therefore, since { X t + ( γ A ) − R t ( s As − s Bs ) d s } t ≥ should be constant, it is necessary that φ = 0 . Iftwo parties’ hedging strategies are chosen independently of each other, by (3.4), φ = 0 may mean φ A = φ B = 0 . It follows that π A = ∆ A and π B = − ∆ B . In other words, the two parties shouldhedge the delta-risk of market exposure. In addition, together with (3.4), this constant conditionimplies that (cid:18) s A + γ B γ A s B K t (cid:19) v + ∆ A b A + γ B γ A ∆ B b B + s A − s B γ A = 0 , d P ⊗ d t − a.s . (3.20)For (3.20) to hold with arbitrary ∆ A and ∆ B , we should have s A = s B = 0 . Therefore, for themarket convention to be optimal, the following two conditions are necessary: • both agents hedge the delta-risk of clean price , • funding spreads are not transferred to each party.8 J. LEE, S. STURM, C. ZHOU
The second item seems like an expected result because the condition δ ∗ = 0 inherently considerstwo parties whose earnings from the margin is symmetric. If one can make a profit or suffer a lossby margin process, δ ∗ = 0 may not be optimal. A debate is still underway whether funding spreadshould be recouped from counterparties and how to handle the accounting; see, e.g., [36, 37, 25, 5,2]. Indeed, in frictionless markets, choices of funding are separated with pricing as MM theoremproperly applies. However, with frictional distress costs, shareholders’ decision can depend on thechoices of funding. In such cases, the margin requirement is not optimal anymore. Therefore, thesecond condition on funding transfer can be understood that the margin requirement of Basel IIIinherently considers frictionless financial markets.In the next section, we will derive a maximum principle of p ∗ for (2.38). Mainly because of anissue from non-concavity, for finding the optimal pair ( p ∗ , δ ∗ ) , we need either s B,m = s A,m = 0 or A = { δ } , for some δ ∈ R . The second condition means that the variation margin c is fixed as agiven process.
4. Optimal Initial Prices.
Throughout this section, the conditions in Lemma 3.2 are as-sumed so that the admissibility is obtained. The next maximum principle for p ∗ is basically a firstorder condition. First, we consider the case that δ is not a control variable, i.e. A = { δ } , forsome δ ∈ R . In previous sections, we have assumed that s A,m = s B,m = 0 . However, when A issingleton, we do not need the condition on margin rate. Theorem
Assume A = { δ } , for some δ ∈ R . Therefore, δ ∗ = δ . Let X ∗ := X p ∗ ,δ ∗ , f ∗· := ˆ f · ( p ∗ , X ∗· ) , and for given t ≤ T , define Q t ∈ R as the set that f ∗ t ( · ) is not differentiable.Assume ( p ∗ ,X ∗ ) ∈ Q = 0 , d P ⊗ d t − a.s , (4.1) i.e., ( p ∗ , X ∗ ) does not fall in the non-differentiable set of ˆ f , d P ⊗ d t − a.s. Moreover, assume E " β T U ′ A ( X ∗ T ) − β T λU ′ B ( ν B − p ∗ )+ Z T ( p ∗ ,X ∗ t ) / ∈ Q t (cid:0) ∂ p ˆ f t ( p ∗ , X ∗ t ) + ∂ x ˆ f t ( p ∗ , X ∗ t ) (cid:1) d t = 0 . (4.2) Then p ∗ is the optimal initial price.Proof. Notice that f ∗ t ( · ) is concave for any t ∈ [0 , T ] , so is differentiable a.e. The maximumprinciple (4.2), is basically a first order condition. We only need to check whether ( p ∗ , X ∗ ) is notabsorbed in Q . Since we assume A = { δ } , for some δ ∈ R , δ ∗ does not depend on ( p, X ) . We let,for any process ϕ , ϕ ∗ := ϕ p ∗ and, for arbitrary p ∈ R , ∆ ϕ ∗ := ϕ p − ϕ ∗ . Then, E (cid:2) ∆ Y ∗ (cid:3) = E (cid:2) β T (cid:0) U A ( X pT ) − U A ( X ∗ T ) (cid:1) + β T λ (cid:0) U B ( ν B − p ) − U B ( ν B − p ∗ ) (cid:1) + Z T ∆ f ∗ t d t (cid:3) ≤ E (cid:2) β T U ′ A ( X ∗ T )∆ X ∗ T − β T λU ′ B ( ν B − p ∗ )∆ p ∗ + Z T ∆ f ∗ t d t (cid:3) ≤ E (cid:2) β T U ′ A ( X ∗ T )∆ X ∗ T − β T λU ′ B ( ν B − p ∗ )∆ p ∗ (cid:3) + E (cid:20) Z T ( p ∗ ,X ∗ t ) / ∈ Q t (cid:0) ∂ x ˆ f t ( p ∗ , X ∗ t )∆ X ∗ t + ∂ p ˆ f ( p ∗ , X ∗ t )∆ p ∗ (cid:1) d t (cid:21) . The last inequality is obtained by concavity of f ∗ and (4.1). Notice that ∆ X ∗ t = ∆ p ∗ , for any RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS t ∈ [0 , T ] . Therefore, by (4.2), we have E (cid:2) Y p − Y ∗ (cid:3) ≤ ∆ p ∗ E (cid:2) β T U ′ A ( X ∗ T ) − β T λU ′ B ( ν B − p ∗ ) (cid:3) + ∆ p ∗ E (cid:20) Z T ( p ∗ ,X ∗ t ) / ∈ Q t (cid:0) ∂ p ˆ f t ( p ∗ , X ∗ t ) + ∂ x ˆ f ( p ∗ , X ∗ t ) (cid:1) d t (cid:21) =0 . When we control ( p, δ ) together, two conditions on the funding spread, s A,m = s B,m = 0 , arerequired. The proof is analogous to that of Theorem 4.1.
Proposition
Assume s A,m = s B,m = 0 . Let X ∗ := X p ∗ , f ∗· := ˆ f · ( p ∗ , X ∗· ) , and for given t ≤ T , define Q t ∈ R as the set that f ∗ t ( · ) is not differentiable. Assume ( p ∗ ,X ∗ ) ∈ Q = 0 , d P ⊗ d t − a.s , i.e., ( p ∗ , X ∗ ) does not fall in the non-differentiable set of ˆ f , d P ⊗ d t − a.s. Moreover, assume E " β T U ′ A ( X ∗ T ) − β T λU ′ B ( ν B − p ∗ ) + Z T ( p ∗ ,X ∗ t ) / ∈ Q t (cid:0) ∂ p ˆ f t ( p ∗ , X ∗ t ) + ∂ x ˆ f t ( p ∗ , X ∗ t ) (cid:1) d t = 0 . Then ( p ∗ , δ ∗ ) is the risk-sharing contract. We deal with examples in the next section.
5. Examples.
In subsection 3.2, it was shown that delta-hedge of clean price and the absenceof market frictions are necessary for the full margin requirement to be optimal. We first derive the risk-sharing contract given the conditions.
Example s i,m = s i = φ i = 0 , i ∈ { A, B } , A = R . Therefore, X p = ν A + p , and β = G . We will check that ( p ∗ , δ ∗ ) = (ˆ p, where ˆ p := γ B ν B − γ A ν A γ B + γ A − γ B + γ A ln (cid:18) λγ B γ A (cid:19) . (5.1)By (3.17)-(3.18), we have I + t ( X pt , p ) = ( L A ) − (ˆ p − p ) , I − t ( X pt , p ) = ( L B ) − (ˆ p − p ) , where ˆ p is definedas (5.1). Thus, by taking p = ˆ p , we recover the full margin convention: δ ∗ = 0 . In addition, X ˆ p + L A I + = ν A + ˆ p,ν B − p + L B I − = ν B − ˆ p. Therefore, by (3.9), ˆ f t is differentiable at (ˆ p, X ˆ p ) , i.e., for t ∈ [0 , T ] , Q t = ∅ . Moreover, by straightforward calculation, ∂ p ˆ f t ( p, x ) = − ∂ x ˜ f t ( p, x ) , and U ′ A ( ν A + ˆ p ) = λU ′ B ( ν B − ˆ p ) . Therefore, we obtain ( p ∗ , δ ∗ ) = (ˆ p, . Note that when γ B = γ A , ν A = ν B = 0 , λ = 1 , then ˆ p = 0 .Thus, in this case, ( p ∗ , δ ∗ ) = (0 , .If we take γ B = γ A = 1 and ν B = ν A = 0 , (5.1) is reduced to ˆ p = − ln ( λ ) / . In particular,when the two parties have the same negotiation power, i.e., λ = 1 , we have ˆ p = 0 . It can be saidthat ˆ p represents the amount of adjustment by agents’ preference and negotiation power, whichare non-observable information in markets. Since it is hard for both parties to agree on suchparameters. In addition, it is notable that ˆ p does not depend on h i , L i , i ∈ { A, B } , because theprice is mathematically derived from fully collateralized contracts. If one party does not hedge thedelta-risk, we cannot have an explicit solution for ˆ p , so we discuss only the existence of p ∗ satisfying(4.2).0 J. LEE, S. STURM, C. ZHOU
Example s i,m = 0 , and π A = ∆ A , π B = 0 , i.e., φ A = 0 , φ B = ∆ B . We considerconstant default intensities and, without loss of generality, assume that γ A = γ B , L A = L B = 0 . , λ = 1 , and ν A = ν B = 0 . Therefore, for t ∈ [0 , T ] , X pt = p − Z t h(cid:16) s As + γ B γ A s Bs (cid:17) v s + ∆ Bs ( b Bs − Λ Bs ) + ∆ As b As i d s − Z t ∆ Bs d W s ,I it ( p, X pt ) = − X pt − p, i ∈ {− , + } . Since γ A = γ B , we denote U := U A = U B . By straightforward calculation, we can check that Q t = ∅ , for t ≤ T , and ∂ p ˆ f t ( p, X pt ) + ∂ x ˆ f t ( p, X pt ) = (cid:26) − γh B β t (cid:0) U ( X pt ) − U ( − p ) (cid:1) , − X pt − p ≥ , − γh A β t (cid:0) U ( X pt ) − U ( − p ) (cid:1) , − X pt − p < . Recall that X pT increases as p increases. Therefore, both ∂ p ˆ f + ∂ x ˆ f and [ U ′ ( X T ) − U ′ ( − p )] decreasew.r.t p . Moreover, both terms tend to ∞ (resp. −∞ ) as p → −∞ (resp. p → ∞ ) . Thus, thereexists p ∈ R satisfying (4.2). Once p ∗ is obtained, δ ∗ can be found as well, but in this case, ( p ∗ , δ ∗ ) may not be (ˆ p, , i.e., full collateralization may not be optimal.
6. Conclusion.
In this paper, we introduced a new risk-sharing framework to understandhow two parties enter bilateral contracts with the presence of entity-specific information such asdefault risk and funding spread. Based on our model, we can explain why banks buy Treasury bondsthat return less than their funding rate. The analysis of the optimal collateral in the risk-sharing framework interprets the meaning behind the margin requirement in Basel III: two parties hedgedelta risk of clean price and funding spread is not considered in derivative prices. Note that the fullcollateralization is really optimal in frictionless financial markets, which is an inherent assumptionin Basel III. It is possible that this conclusion can change if we include gap risk , KVA, and hedgingstrategies are also control variables. We leave such analysis as a further research topic.
Appendix A. An Auxiliary Lemma.
The next lemma is borrowed from [11] and oftenused in this paper.
Lemma
A.1.
Let i ∈ { A, B } .(i) Let U be an F s -measurable, integrable random variable for some s ≥ . Then, for any t ≤ s , E ( s<τ U |G t ) = t<τ G − t E ( G s U |F t ) . (ii) Let ( U t ) t ≥ be a real-valued, F -predictable process and E | U ¯ τ | < ∞ . Then, E ( τ = τ i ≤ T U τ |G t ) = t<τ G − t E (cid:16) Z Tt h is G s U s d s (cid:12)(cid:12)(cid:12) F t (cid:17) . Appendix B. Spaces of Random Variables and Stochastic Processes.
In this paper,we denote spaces of random variables and stochastic processes as follows.
Definition
B.1.
Let m ∈ N and p ≥ . • L pT : the set of all F T -measurable random variables ξ , such that k ξ k p := E [ | ξ | p ] p < ∞ . • S pT : the set of all real valued, F -adapted, càdlàg processes ( U t ) t ≥ , such that k U k S pT := E (cid:0) sup t ≤ T | U t | p (cid:1) p < ∞ . Right continuous and left limit. RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS • H p,mT : the set of all R m -valued, F -predictable processes ( U t ) t ≥ , such that k U k H pT := E (cid:16) Z T (cid:12)(cid:12) U t (cid:12)(cid:12) p d t (cid:17) p < ∞ . • H p,mT,loc : the set of all R m -valued, F -predictable processes ( U t ) t ≥ , such that Z T (cid:12)(cid:12) U t (cid:12)(cid:12) p d t < ∞ , a.s. When d = 1 , we denote H pT := H p, T and H pT,loc := H p, T,loc . Appendix C. A Risk-Neutral Agent under Incremental Cash-flow.
In this section, wewill derive an optimal collateral with a risk-neutral Agent A: U A ( x ) = x . As in previous sections,the Agent B is risk-averse as U B ( x ) = − e − γ B x . In this case, we can relax the assumption onmargin funding rate of A . Then, we intend to derive similar arguments as in subsection 3.1 andsubsection 3.2 with assuming s A,m > . (C.1). Now, to model the incremental cash-flow, assume that the bank has had contracts given by someendowed càdlàg F -adapted processes ( D E , e E , m E ) before initiation of the new contract. If the twoparties do not enter the new contract, the cash-flow remains as C Et = τ>t D Et + τ ≤ t (cid:0) D Eτ + e Eτ (cid:1) − τ = τ A ≤ t L A ( e Eτ − m Eτ − ) + + τ = τ B ≤ t L B ( e Eτ − m Eτ − ) − . On the other hand, with the new contract, the exposure and margin become ( e E + e ) and ( m E + m ) ,respectively. Therefore, with the new contract, the summed cash-flows are C St := τ>t ( D Et + D t ) + τ ≤ t (cid:0) D Eτ + D τ + e Eτ + e τ (cid:1) − τ = τ A ≤ t L A ( e Eτ + e τ − m Eτ − − m τ − ) + + τ = τ B ≤ t L B ( e Eτ + e τ − m Eτ − − m τ − ) − . Thus, the amount that should be dealt with by the Agent A is the increment from C E to C S , namelyfor t ≤ T , C t := C St − C Et = D t ∧ ( τ − ) + τ ≤ t e τ − − τ = τ A ≤ t L A (cid:0) ( e τ − − m τ − + e Eτ − m Eτ − ) + − ( e Eτ − m Eτ − ) + (cid:1) + τ = τ B ≤ t L B (cid:0) ( e τ − − m τ − + e Eτ − m Eτ − ) − − ( e Eτ − m Eτ − ) − (cid:1) . (C.2)Thus, we denote the amount of breach of the contract as Θ t ( δ ) = τ A = t L A (cid:0) ( δ + δ Et ) + − ( δ Et ) + (cid:1) − τ B = t L B (cid:0) ( δ + δ Et ) − − ( δ Et ) − (cid:1) . (C.3)Moreover, by (C.1), the F -reduction of ¯ V A , which is derived in Theorem 2.14, becomes slightlydifferent as d v A,δt =( s A,mt δ t + α At ) d t + φ At d W t α At := s A, ∆ t v t + φ At Λ At + ∆ At b At s A, ∆ t := s At − s A,mt . Notice that v A depends on δ since we assumed that s A,m > , which is the main mathematicaldifference from the main sections. We still assume that the Agent B can deliver the collateral2 J. LEE, S. STURM, C. ZHOU without any excessive cost/benefit, i.e., s B,m = 0 . In this case, v B does not depend on δ . Becauseof the dependence between v A and δ , we impose a slightly stronger condition for the admissible setof collateral D : E (cid:20)(cid:12)(cid:12) G T U A ( v A,δT ) (cid:12)(cid:12) + (cid:12)(cid:12) λG T U B ( v BT ) (cid:12)(cid:12) + Z T (cid:12)(cid:12) g t ( v A,δt , v Bt , δ t ) (cid:12)(cid:12) d t (cid:21) < ∞ , (C.4)where we now denote g t := δ + δ E ≥ g + t + δ + δ E < g − t and g + t ( v A , v B , δ ) := G t (cid:2) h At U A (cid:0) v A + L A ( δ − ( δ Et ) − ) (cid:1) + λh At U B (cid:0) v B − L A K t ( δ − ( δ Et ) − ) (cid:1) + h Bt (cid:0) U A ( v A + L B ( δ Et ) − ) + λU B ( v B − L B K t ( δ Et ) − ) (cid:1)(cid:3) ,g − t ( v A , v B , δ ) := G t (cid:2) h Bt U A (cid:0) v A + L B ( δ + ( δ Et ) + ) (cid:1) + λh Bt U B (cid:0) v B − L B K t ( δ + ( δ Et ) + ) (cid:1) + h At (cid:0) U A ( v A − L A ( δ Et ) + ) + λU B ( v B + L A K t ( δ Et ) + ) (cid:1)(cid:3) . As in section 3, the first task is to characterize the optimal collateral by MOP. To this end, weslightly modify (3.1) by merging the one terminal condition G T v A,δT into d t -integral term. Observethat Itô’s formula yields d (cid:0) G t v A,δt (cid:1) = G t (cid:2) s A,mt δ t + α At − h t v At (cid:3) d t + G t v A,δt φ At d W t . (C.5)If Gv A,δ φ A ∈ H T , the Itô’s integral term is an ( P , F ) -local martingale. Thus, E (cid:2) G T v A,δT − ( ν A + p ) (cid:3) = E (cid:20) Z T G t (cid:2) s A,mt δ t + α At − h t v At (cid:3) d t (cid:21) . Thus, (3.1) can be written as max δ ∈D E (cid:20) ( ν A + p ) + λG T U B ( v BT )+ Z T (cid:2) g t ( v A,δt , v Bt , δ t ) + G t ( s A,mt δ t + α At − h t v A,δt ) (cid:3) d t (cid:21) . (C.6)Then, we define a dynamic version of (C.6) as: J εt ( p ) := ess sup δ ∈D ( t,ε ) E (cid:20) ( ν A + p ) + λG T U B ( v BT )+ Z Tt h g s ( v A,δs , v Bs , δ s ) + G s ( s A,ms δ s + α As − h s v A,δs ) i d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . (C.7)Then ( J εt ) ≤ t ≤ T is chosen as a càdlàg version such that for any ε ∈ D , n J εt + Z t (cid:2) g s ( v A,εs , v Bs , ε s ) + G s ( s A,ms ε s + α As − h s v A,εs ) (cid:3) d s o ≤ t ≤ T is an ( P , F ) -supermartingale. Moreover, for the optimal collateral δ ∗ for J , n J δ ∗ t + Z t (cid:2) g s ( v A,δ ∗ s , v Bs , δ ∗ s ) + G s ( s A,ms δ ∗ s + α As − h s v A,δ ∗ s ) (cid:3) d s o ≤ t ≤ T is an ( P , F ) -martingale. The detail is summarized in the following theorem. Before giving thetheorem, we introduce two notations. We separate δ from v A by denoting d˜ v At = d v A,δt − s A,mt δ t d t ,more precisely, ˜ v At = Z t α As d s + Z t φ As d W s . (C.8) RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS ˜ v A = 0 . In addition, we denote I t := Z Tt G s h ∆ s d s. (C.9)The optimal collateral will later be represented by ( I t ) t ≥ . This term appears in this section since s A,m can be positive. If we consider the cost of delivering collateral, then when to default becomesalso important. However, the effect of default time can be still marginal. Recall the definitionthat h ∆ = h − h . Therefore, ( I t ) t ≥ can be understood as a correcting term of collateral for thedependence of default times. When τ A and τ B are independent, we have h = h and I = 0 . Theorem
C.1.
Assume that Gh ∆ is deterministic. Define b δ t (˜ v At , v Bt ) := arg max δ ∈ A (cid:2) ˜ g t (˜ v At , v Bt , δ ) + I t s A,mt δ (cid:3) , (C.10) ˜ g t ( v A , v B , δ ) := δ + δ E ≥ ˜ g + t ( v A , v B , δ ) + δ + δ E ≥ ˜ g + t ( v A , v B , δ ) , ˜ g it ( v A , v B , δ ) := g it ( v A , v B , δ ) + G t (cid:0) s A,mt δ + α At − h t v A (cid:1) , i ∈ {− , + } . If b δ (˜ v A , v B ) ∈ D , then ( b δ t (˜ v At , ˜ v Bt )) ≤ t ≤ T is a solution of (3.1).Proof. For ε ∈ D , we define an ( P , F ) -semimartingale ( Y t ) t ≥ as Y t := J εt − E (cid:20) Z Tt G u h ∆ u (cid:18) Z t s A,ms ε s d s (cid:19) d u (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = J εt − E (cid:20) Z t (cid:18) Z Tt G u h ∆ u d u (cid:19) s A,ms ε s d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = J εt − Z t I t s A,ms ε s d s. Notice that Y does not depend on ε . We also define ξ εt := Y t + Z t I t s A,ms ε s d s + Z t (cid:2) g s ( v A,εs , v Bs , ε s ) + G s ( s A,ms ε s + α As − h s v A,εs ) (cid:3) d s. (C.11)To simplify (C.11), note that g t ( v A,εt , v Bt ,ε t ) + G t ( s A,mt ε t + α At − h t v A,εt )= (cid:2) g t ( v A,εt , v Bt , ε t ) − G t h t v A,εt (cid:3) + G t h t v A,εt − h t v A,εt + G t ( s A,mt ε t + α At )= (cid:2) g t ( v A,εt , v Bt , ε t ) − G t h t v A,εt (cid:3) + G t h ∆ t v A,εt + G t ( s A,mt ε t + α At ) . In addition, by Fubini’s theorem, Z t G u h ∆ u h Z u s A,ms ε s d s i d u = Z t s A,ms ε s h Z ts G u h ∆ u d u i d s. Moreover, Z t G u h ∆ u v A,εu d u = Z t G u h ∆ u h v A,εu − Z u s A,ms ε s d s i d u + Z t G u h ∆ u h Z u s A,ms ε s d s i d u = Z t G u h ∆ u ˜ v Au d u + Z t s A,ms ε s h Z ts G u h ∆ u d u i d s, (C.12)4 J. LEE, S. STURM, C. ZHOU and for t ≤ T , g t ( v A,εt , v Bt , ε t ) − G t h t v A,εt + Gh t ˜ v At = g (˜ v At , v Bt , ε t ) . (C.13)Therefore, (C.11) can be rewritten as Y t + Z t I t s A,ms ε s d s + Z t (cid:2) g s ( v A,εs , v Bs , ε s ) + G s ( s A,ms ε s + α As − h s v A,εs ) (cid:3) d s = Y t + Z t I t s A,ms ε s d s + Z t s A,ms ε s h Z ts G u h ∆ u d u i d s + Z t (cid:16) g s ( v A,εs , v Bs , ε s ) − G s h s v A,εs + Gh ∆ s v As + G s ( s A,ms ε s + α As ) d s = Y t + Z t I s s A,ms ε s d s + Z t (cid:16) g s (˜ v As , v Bs , ε s ) + G s ( s A,ms ε s + α As − h ˜ v As ) (cid:17) d s = Y t + Z t I s s A,ms ε s d s + Z t ˜ g s (˜ v As , v Bs , ε s ) d s. Then, since Y is independent of ε ∈ D , by the assumption of admissibility of b δ (˜ v A , v B ) , for any ε ∈ D , we have that ξ ε − ξ b δ (˜ v A ,v B ) is an ( P , F ) -supermartingale. It follows that for any ε ∈ D , E (cid:2) ξ εT − ξ b δ (˜ v A ,v B ) T (cid:3) ≤ E (cid:2) ξ ε − ξ b δ (˜ v A ,v B )0 (cid:3) = 0 . Then, by the admissibility, b δ (˜ v A , v B ) is a solution of(3.1).The last step is to show b δ (˜ v A , v B ) is admissible given some conditions. We consider A = R andfind the explicit form of b δ (˜ v A , v B ) for the case. Then, the integrability condition is easy to check.First, notice that ˜ g + , ˜ g − are continuously differentiable and strictly concave in δ . Thus, for any ( t, v A , v B ) , there exists b I it ( v A , v B ) , i ∈ {− , + } such that ∂ δ ˜ g it ( v A , v B , b I it ( v A , v B )) + s A,mt I t = 0 (C.14)Then, it is easy to check that b δ (˜ v A , v B ) is attained at I − , I + , and − δ E . Observe the precise formsof ˜ g i , i ∈ {− , + } , are ˜ g − t ( v A , v B , δ ) := G t (cid:2) h ∆ t v A + ( h Bt L B + s A,mt ) δ + α At + ( h Bt L B − h At L A )( δ Et ) + + λh Bt U B (cid:0) v B − L B K t ( δ + ( δ Et ) + ) (cid:1) + λU B ( v B + L A K t ( δ Et ) + ) (cid:1)(cid:3) , ˜ g + t ( v A , v B , δ ) := G t (cid:2) h ∆ t v A + ( h At L A + s A,mt ) δ + α At + ( h Bt L B − h At L A )( δ Et ) − + λh At U B (cid:0) v B − L A K t ( δ − ( δ Et ) − ) (cid:1) + λh Bt U B ( v B − L B K t ( δ Et ) − ) (cid:1)(cid:3) . Therefore, assuming h i L i > , i ∈ { A, B } , I i , i ∈ {− , + } , can be explicitly represented as b I − t ( v A , v B ) = − ( δ Et ) + + v B K t L B + 1 γ B K t L B ln (cid:18) G t [ h Bt L B + s A,mt ] + s A,mt I t G t λγ B K t h Bt L B (cid:19) , (C.15) b I + t ( v A , v B ) =( δ Et ) − + v B K t L A + 1 γ B K t L A ln (cid:18) G t [ h At L A + s A,mt ] + s A,mt I t G t λγ B K t h At L A (cid:19) . (C.16)Then Theorem C.1 is completed by the next lemma. The proof is similar to that of Lemma 3.2, sowe omit it. Lemma
C.2.
Let A = R . Assume that δ E , e, Z, π i , i ∈ { A, B } , are bounded. Moreover, assume h i L i ∈ H T and ( G + I ) s A,m Gh i L i , i ∈ { A, B } , (C.17) are bounded. Then b δ (˜ v A , v B ) ∈ D . RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS b δ (˜ v A , v B ) depends on h A and h B . As mentioned, this dependencearises from the funding impact of s A,m , and by setting s A,m = 0 , (C.15)-(C.15) are reduced to b I − t ( v A , v B ) = − ( δ Et ) + + v B K t L B − ln ( λγ B K t ) γ B K t L B , (C.18) b I + t ( v A , v B ) =( δ Et ) − + v B K t L A − ln ( λγ B K t ) γ B K t L A . (C.19)In addition, we can derive similar interpretations as in subsection 3.1. In what follows, we moreover,assume that all parameter are constant and the default times are independent on F , i.e., I = 0 .The full collateral convention can be said that δ ∗ = v − c ∗ = 0 and δ E = 0 , d P ⊗ d t -a.s. Therefore, I i = 0 , for any t ≤ T . By (C.15) and (C.16), full collateralization requires that v Bt − γ B ( s A − s B ) t = 1 γ B ln (cid:16) λγ B h i L i h i L i + s A,m (cid:17) , i ∈ {
A, B } , d P ⊗ d t − a.s . (C.20)In particular, ( v Bt − ( s A − s B ) t/γ B ) t ≥ should be constant. Thus, (C.20) implies that φ B = π B + ∆ B = 0 , i.e., delta-hedge, and − s B Kv + φ B Λ Bt − ∆ B b B − s A − s B γ B = 0 , d P ⊗ d t − a.s . (C.21)Consider a contract such that Z = 0 , so necessarily v = 0 and ∆ B = 0 . Since (C.21) should holdfor all contracts such that Z = 0 , (C.21) implies that s B = b B = s A − s B = 0 . Equivalently, by(2.12), (2.28), s B = s A = s A,m = 0 . Therefore, the margin requirement hinges on the assumption ofabsence of funding impacts and delta-hedge of the Agent B. No property of the Agent A’s hedgingstrategy was derived since we assumed the Agent A is risk-neutral.
Appendix D. Proofs of Lemmas.
Proof of Lemma . (i) is from the definition and (ii) is a directly obtained from (i). For(iii), notice that B − t e t + R t B − s d D s is an ( Q , F ) -local martingale. Thus, by (local) martingalerepresentation property, there exists Z ∈ H ,dT,loc such that for any t ≥ , B − t e t + Z t B − s d D s = Z t Z s d W Q s , where W Q is the Brownian motion under Q , i.e., W Q t = W t + R t Λ s d s . Therefore, ( e t ) t ≥ followsthe SDE: d e t = r t e t d t + B t Z t d W Q t − d D t = (cid:0) r t e t + B t Z t Λ t (cid:1) d t + B t Z t d W t − d D t . (D.1)By (iii), ( e t ) t ≥ is an F -adapted càdlàg process, but τ avoids F -stopping times. Thus, ∆ e τ = 0 almost surely, equivalently e τ − = e τ a.s. Proof of Lemma . Let i ∈ { A, B } and Ψ( x ) := e Cx for C ∈ R . It suffices to show that forany C ∈ R , Ψ( X ) , Ψ( v i ) , are in S T . Note that v = ( B A ) − e is bounded and, by (2.26) and (2.27), ∆ i and φ i are also bounded. Denoting α A := φ A Λ A + ∆ A b A + s A vα B := φ B Λ B − ∆ B b B − Ks B v, J. LEE, S. STURM, C. ZHOU we can write d v it = α it d t + φ it d W t . Applying Itô’s formula to Ψ( v i ) , dΨ( v it ) = (cid:0) Cα it + ( Cφ it ) / (cid:1) Ψ( v it ) d t + Cφ it Ψ( v it ) d W t . (D.2)By the assumptions, the coefficients in (D.2) are uniformly Lipsitch continuous. Thus, there existsa unique solution of (D.2) such that E h sup t ≤ T | Ψ( v it ) | i < ∞ . In particular, U i ( v i ) ∈ S T , so we obtain the integrability condition (2.37). It is similarly obtainedthat Ψ( X ) ∈ S T . Let ( C t ) t ≥ be an arbitrary bounded deterministic process. Then, we also have exp ( CX ) ∈ S T and it follow that U A (cid:18) − γ A γ B K + γ A X (cid:19) ∈ S T U B (cid:18) Kγ A γ B K + γ A X (cid:19) ∈ S T . Thus, by (3.5)-(3.7), (3.14), and (3.15), we obtain δ ∗ ( p, X ) ∈ D . Proof of Lemma . Recall from (3.12) and (3.13) that ∂ δ f + ( t, p, x, δ ) := h At L A (cid:2) − γ A U A ( x + L A δ ) + λγ B K t U B ( ν B − p − L A K t δ ) (cid:3) ,∂ δ f − ( t, p, x, δ ) := h Bt L B (cid:2) − γ A U A ( x + L B δ ) + λγ B K t U B ( ν B − p − L B K t δ ) (cid:3) . Let I it ( p, x ) denote the function such that ∂ δ f i ( t, p, x, I it ( p, x )) = 0 . Since f i , i ∈ {− , + } , areconcave w.r.t δ , I i uniquely exists. Let us denote ˜ f + ( t, p, x ) := max ≤ δ f + ( t, p, x, δ ) , ˜ f − ( t, p, x ) := max δ ≤ f − ( t, p, x, δ ) , ˜ f ( t, p, x ) := max δ ∈ R f ( t, p, x, δ ) . Then it follows that ˜ f ( t, p, x ) := f ( t, p, x, δ ∗ ( t, p, x )) , = ˜ f + ( t, p, x ) ˜ f + ( t,p,x ) ≥ ˜ f − ( t,p,x ) + ˜ f − ( t, p, x ) ˜ f + ( t,p,x ) ≤ ˜ f − ( t,p,x ) . Thus, for finding δ ∗ , we should characterize the region that ˜ f + ( t, p, x ) ≥ ˜ f − ( t, p, x ) . The solutions δ i , i ∈ {− , + } , of ˜ f i can be found explicitly: δ + t ( p, x ) = (cid:26) , > ∂ δ f + ( t, p, x, ,I + t ( p, x ) , ≤ ∂ δ f + ( t, p, x, ,δ − t ( p, x ) = (cid:26) I − t ( p, x ) , ∂ δ f − ( t, p, x, ≤ , , < ∂ δ f − ( t, p, x, . Moreover, notice that since h i ≥ and L i ≥ , ∂ δ f + (0) ∗ ∂ δ f − (0) ≥ . The rest of the proof ismerely a straightforward comparison of ˜ f i in each region. In what follows, we suppress t, x, p .(I) Let ≤ ∂ δ f + (0) ∧ ∂ δ f − (0) . In other words, γ A x + γ B p ≤ γ B ν B − ln (cid:0) λK t γ B γ A (cid:1) . RISK-SHARING FRAMEWORK OF BILATERAL CONTRACTS I i ≥ , i ∈ {− , + } , and δ − = 0 . Moreover, ˜ f − − ˜ f + = f − (0) − f + ( I + ) = f + (0) − f + ( I + ) ≤ . Hence, δ ∗ = δ + = I + ≥ .(II) Let > ∂f + (0) ∨ ∂f − (0) . Then, δ + = 0 and I − ≤ . Therefore, by similar calculation, δ ∗ = δ − = I − ≤ . Acknowledgments.
We thank Stéphane Crépey for spending much time to help us improvethis paper.
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