Cross Currency Valuation and Hedging in the Multiple Curve Framework
aa r X i v : . [ q -f i n . P R ] J a n CROSS CURRENCY VALUATION AND HEDGING IN THE MULTIPLE CURVEFRAMEWORK
ALESSANDRO GNOATTO AND NICOLE SEIFFERTJanuary 30, 2020
Abstract.
We generalize the results of Bielecki and Rutkowski (2015) on funding and collateralizationto a multi-currency framework and link their results with those of Piterbarg (2012), Moreni and Pallavicini(2017), and Fujii et al. (2010b).In doing this, we provide a complete study of absence of arbitrage in a multi-currency market where,in each single monetary area, multiple interest rates coexist. We first characterize absence of arbitragein the case without collateral.After that we study collateralization schemes in a very general situation: the cash flows of thecontingent claim and those associated to the collateral agreement can be specified in any currency.We study both segregation and rehypothecation and allow for cash and risky collateral in arbitrarycurrency specifications. Absence of arbitrage and pricing in the presence of collateral are discussedunder all possible combinations of conventions.Our work provides a reference for the analysis of wealth dynamics, we also provide valuation formulasthat are a useful foundation for cross-currency curve construction techniques. Our framework providesalso a solid foundation for the construction of multi-currency simulation models for the generation ofexposure profiles in the context of xVA calculations. Introduction
Since the 2007-2009 financial crisis several assumptions underlying financial valuation have beenquestioned. Several spreads have emerged (more precisely widened) between certain interest rates(notably between overnight and unsecured Ibor rates) and these rates in turn differ from interestrates agreed in the context of repurchase agreements (repo rates). From a modeling perspective thisresulted in the development of multi curve interest rate models as in Henrard (2007), Bianchetti (2010),Moreni and Pallavicini (2014), Mercurio (2010), Henrard (2014), Grbac et al. (2015), Cr´epey et al.(2015) Cuchiero et al. (2016) and Cuchiero et al. (2019) among others.Even before the financial crisis, financial institutions employed many different funding strategies tosupport their trading activity. Borrowing cash from the internal treasury desk (as implicitly assumedin classical asset pricing theory) is only one among different possibilities to fund a transaction. Evenbefore the crisis repurchase agreements and collateralization constituted a possible and understoodway to finance cash flows, mainly aimed at managing counterparty risk.In a collateralization agreement, the agents participating in a transaction regularly exchange cashflows in order to reduce the outstanding exposure of a contract. A collateralized transaction is, ina nutshell, very similar in its nature to a transaction on a futures contract, where margin calls areregularly exchanged. One important difference is that in a collateralized transaction, the party whoreceives collateral typically pays an interest to the party who posts the collateral (either in the formof cash or shares of a risky asset with low volatility/high rating).
Mathematics Subject Classification.
JEL Classification
E43, G12.
Key words and phrases.
FX, cross-currency basis, multiple curves, FVA, CollVA, Basel III, Collateral.
The financial crisis implied a more widespread adoption of such alternative funding strategies.Collateralization agreements have now become a common aspects in the business relation betweenfinancial institutions.If we couple the increased importance of collateralization agreements with the emergence of spreadsbetween interest rates, we understand that some care is needed in the context of valuation and hedging.Since interest rates differ and since multiple sources of funding are possible, then one needs to carefullymodel the funding policy in order to obtain pricing formulas that are consistent with the contractualconditions of a certain transaction. If multiple sources of funding are employed, the spreads betweenthe interest rates linked to the different sources of funding must be taken into account.The problem described above has given rise to a large stream of literature aiming at reconcilingthe theory behind arbitrage free valuation with the current market setting. We cite, among others,Piterbarg (2010), Castagna (2011), Pallavicini et al. (2011), Pallavicini et al. (2012), Antonov et al.(2015), Cr´epey (2015a), Cr´epey (2015b), Brigo et al. (2015) and Brigo and Pallavicini (2014). Thecontribution of Bielecki and Rutkowski (2015) consists in presenting a sound martingale pricing frame-work that accounts for funding costs and collateralization. Such framework is then reconciled withthe results of Piterbarg (2010) in a pure diffusive setting. The contributions mentioned above restrictthemselves to a single currency framework.Funding strategies and collateralization agreements become even more involved as soon as we allowfor multiple currencies. Financial institutions may fund the trading activity in any currency. Alsocollateral might be posted in arbitrary currencies. From the perspective of the hedger, i.e. the partywho shorts a contingent claim, collateral might be posted or received either in domestic currency, orin the currency of the contractual cash flows, or even in a third foreign currency. Collateralizationagreements might grant the collateral provider the option to choose the currency he/she uses to postcollateral, thus providing the option to post collateral by using the currency with cheapest fundingcost. Such feature is often referred to as collateral choice option. The presence of collateral choiceoptions turns the valuation of even plain vanilla payoffs into a non-trivial problem.Collateralization in multiple currencies has been analyzed already in some contributions. Piterbarg(2012) studies funding strategies in multiple currencies by using FX swaps as basic collateralizedinstrument to create funding strategies in multiple currencies. He describes the cash flows of a col-lateralized FX swap contract (a combination of a spot and forward FX transaction) and from suchanalysis he obtains the dividend process of the collateralized FX swap, which depends on the collateralrate agreed between the two counterparties of the FX swap. According to Piterbarg, such collateralrate is unrelated to the domestic or the foreign collateral rate in the two economies involved in thetransaction.Fujii et al. (2010a) provide a valuation formula for contingent claims with currency dislocationsbetween contractual and collateral cashflows. Their choice of the Num´eraire is the unsecured fundingrate and the drift of the exchange rate they obtain is in line with the classical single curve theory:it is the difference between the domestic and the foreign unsecured funding rate. Concerning thecontribution of Fujii et al. (2010a) Piterbarg (2012) observes that the rate of the FX swap he obtainscorresponds to the difference of unsecured funding rates in Fujii et al. (2010a). The approach ofPiterbarg (2012) has been later expanded by Moreni and Pallavicini (2017). Anyway, even though theunderlying assumptions of Piterbarg (2012), Fujii et al. (2010a) and Moreni and Pallavicini (2017) areslightly different between each other, they reach similar conclusions in terms of pricing formulas andmodel dynamics.
ULTI-CURRENCY SETTING 3
Our aim in the present paper is to generalize the martingale pricing approach of Bielecki and Rutkowski(2015) and show that we can generalize martingale pricing to cover the results in the references above.Unlike Piterbarg (2012) and Moreni and Pallavicini (2017) we do not postulate that contracts arenatively collateralized. In our view collateralization is a feature of the relation between the hedgerand the counterparty and absence of arbitrage should be guaranteed irrespective of the presence or notof collateralization agreements, hence our first objective is to discuss absence of arbitrage in an uncol-lateralized multi currency market in the presence of multiple interest rates. In a world market withan arbitrary number of currencies L , in each currency area we allow for country specific submarketswith d k risky assets k = 1 . . . , L and each risky assets has a dedicated funding (repo) account. Eachcurrency area features an unsecured funding account and we employ strategies based on such unse-cured accounts to construct arbitrage free transactions on the spot foreign exchange rate. The benefitof such approach is twofold: we do not need to introduce derivatives to discuss absence of arbitrage(FX swaps involve a spot and a forward transaction) so that we can discuss absence of arbitrage ofthe market featuring only underlying securities, and we also disentangle the issue absence of arbitragefrom the description/modelization of collateralization agreements. In this sense we are following moreclosely the approach of Fujii et al. (2010a) which we fully map to the setting of Bielecki and Rutkowski(2015). This constitutes the topic of Section 2 and Section 3In Section 4, we introduce collateralization agreements. We deliberately choose to closely followthe presentation of Bielecki and Rutkowski (2015) and we present collateralization under segrega-tion/rehypothecation and we allow for collateral to be posted in the form of cash or units of a riskyasset. Out extension involves the possibility that the collateral is posted/received in an arbitrary cur-rency k ∈ { , . . . , L } . The findings of Section 4 allow us to discuss in Section 5 pricing of contingentclaims in the presence of collateral under any currency. We obtain first general formulas extendingBielecki and Rutkowski (2015). Later we specialize our valuation formulas in a pure diffusive settingin Section 6 which extends the literature in two ways: on the one side, we obtain pricing formulas con-sistent with Fujii et al. (2010a) and Moreni and Pallavicini (2017) extending Bielecki and Rutkowski(2015), on the other side, based on the findings of Section 3 we provide a sound construction of crosscurrency diffusion models in the presence of multiple interest rates in each single currency. Such crosscurrency models are of paramount importance in the context of xVA computations in the industry:as explained in e.g. Cesari et al. (2009), Green (2015), Lichters et al. (2015) and Sokol (2014), themarket standard for xVA involves the computation of valuation adjustment at the level of the fullportfolio as a way to capture the beneficial effect of netting agreements and this in turn implies theneed to construct high dimensional Monte Carlo simulation models simultaneously covering all riskfactors in all currencies relevant for the portfolio between the hedger and the counterparty. For adiscussion of the subtleties in the computation of valuation adjustments in the presence of nettingagreements we refer the reader to Biagini et al. (2019).2. Multi-currency trading under funding costs
We follow the notations of Bielecki and Rutkowski (2015). We fix a finite time horizon
T > , G , G , P ) be a filtered probability space where the filtration G = ( G ) t ∈ [0 ,T ] satisfies the usualconditions. We assume that G is trivial. All processes to be introduced in the sequel are assumed tobe G -adapted c´adl´ag semimartingales.Let k , k = 1 , . . . , L , L ∈ N be an index for different currency areas. For some k , k = e , which corre-sponds to the domestic currency. Let S i,k denote the ex-dividend price of the i -th risky asset traded ALESSANDRO GNOATTO AND NICOLE SEIFFERT in unit of currency k , i = 1 , . . . , d k , where d k is the number of risky assets traded in terms of the cur-rency with index k . Every asset has a cumulative dividend stream D i,k . As in Bielecki and Rutkowski(2015) we do not postulate that the processes S i,k , i = 1 , . . . , d k , k = 1 , . . . , L are positive.The trading desk can use different sources of funding, each being represented by a suitable familyof cash accounts. For unsecured funding, we assume that the trading desk can fund her activityby unsecured borrowing or lending in different currencies, hence we introduce the cash accounts B ,k = B k , k = 1 , . . . , L . If borrowing and lending rates differ, we write B k,b and B k,l for theborrowing and lending unsecured funding cash account of currency k .For every risky asset, we have an asset-specific funding account, which we call repo-account. Weintroduce B i,k as the funding account associated to the asset S i,k . In case borrowing and lending ratesdiffer, we write B i,k,b and B i,k,l for the borrowing and lending repo cash accounts associated to the i -th risky asset under currency k .We introduce a notation for foreign exchange rates. Let X k ,k , k , k = 1 , . . . , L the price of oneunit of currency k in terms of currency k . In terms of the usual FORDOM convention in currencymarkets we have e.g. for EURUSD X USD,EUR is the price in EUR of 1 USD.
Assumption 2.1.
We introduce the following processes:i) ex-dividend price process S i,k , i = 1 , . . . , d k , k = 1 , . . . , L are real-valued RCLL semimartin-gales.ii) cumulative dividend streams D i,k , i = 1 , . . . , d k , k = 1 , . . . , L are processes of finite variationwith D i,k = 0 .iii) exchange rate processes X k ,k , k , k = 1 , . . . , L are positive-valued RCLL semimartingales.iv) funding accounts B j,k i = 1 , . . . , d k are strictly positive and continuous processes of finitevariation with B j,k = 1 .v) positive or negative dividends from the i − k -th risky asset are invested in the correspondingfunding account B i,k . In line with Bielecki and Rutkowski (2015) we assume that prices are real-valued (for example, theprice of an interest rate swap might be negative), foreign exchange rates are however assumed to bepositive. Based on the last item of the above assumption, we introduce the following objects.
Definition 2.2.
The cumulative dividend price S i,cld,k in units of currency k is given as S i,cld,kt := S i,kt + B i,kt Z (0 ,t ] (cid:16) B i,ku (cid:17) − dD i,ku . (2.1) The cumulative dividend S i,cld,k ,k of the asset traded in units of currency k , expressed in units ofcurrency k is given as S i,cld,k ,k t := S i,k t X k ,k t + B i,k t Z (0 ,t ] (cid:16) B i,k u (cid:17) − X k ,k u dD i,k u . (2.2) the discounted cumulative dividend price ˆ S i,cld,k := ( B i,k ) − S i,cld,k in units of currency k satisfies ˆ S i,cld,kt := ˆ S i,kt + Z (0 ,t ] (cid:16) B i,ku (cid:17) − dD i,ku . (2.3) The discounted cumulative dividend ˆ S i,cld,k ,k := ( B i,k ) − S i,cld,k ,k t of the asset traded in units ofcurrency k , expressed in units of currency k satisfies ˆ S i,cld,k ,k = ˆ S i,k t X k ,k t + Z (0 ,t ] (cid:16) B i,k u (cid:17) − X k ,k u dD i,k u (2.4) ULTI-CURRENCY SETTING 5
Contracts and trading strategies.Definition 2.3.
A dynamic portfolio, denoted as ϕ = ( ξ, ψ ) with ϕ = ( ξ, ψ ) = (cid:16) ξ , , . . . , ξ d , , , ξ , , . . . , ξ d L ,L , ψ , , . . . , ψ d , , ψ , , . . . , ψ d L ,L (cid:17) , (2.5) consists of risky securities S i,k , i = 1 , . . . , d k , k = 1 , . . . , L , the cash accounts B ,k = B k , k = 1 , . . . , L ,for unsecured borrowing and lending, and funding/repo-accounts B i,k , i = 1 , . . . , d k , k = 1 , . . . , L usedfor funding of the i − k -th risky asset. We will use the shorthand ψ k = ψ ,k , k = 1 , . . . , L . In line with Bielecki and Rutkowski (2015) weconsider the following contracts. Definition 2.4.
By a bilateral financial contract, or simply a contract, we mean an arbitrary RCLLprocess of finite variation, denoted by A k to emphasize that the contract is denominated in terms ofcurrency k . The process A k is aimed to represent the cumulative cash flows of a given contract fromtime until its maturity date T . By convention, we set A k − = 0 . The process A k represents the flows from the perspective of the hedger and includes the initialflow A k taking place at the contract’s inception. As shown in Bielecki and Rutkowski (2015) it canbe used to describe contracts with multiple cash flows during the contract’s lifetime, with the cashflow at time 0 representing the (yet to be determined) price p k of the claim, in units of currency k . For example, in the case of a European call option written on the exchange rate X e,k , one has A et = p e [0 ,T ] ( t ) − (cid:16) X e,kT − K (cid:17) + [ T ] ( t ), K > A k ,k , to denote the flows of contracts natively denominated in units of currency k , whenexpressed in units of currency k . Assuming R (0 ,t ] X k ,k u dA k u is a square integrable random variable,for any choice of the indices k , k we write A k ,k t := p k X k ,k [0 ,T ] ( t ) − Z (0 ,t ] X k ,k u dA k u (2.6)and set p k := p k X k ,k . For example a call option written on a generic asset S i,k , has the followingstream of cash flows in units of domestic currency A e,k t = p e [0 ,T ] ( t ) − X e,k T (cid:16) S i,k T − K (cid:17) + [ T ] ( t ), K > Definition 2.5.
A trading strategy is a triplet ( x, ϕ, A k ) , where x is the initial endowment of thehedger, ϕ represents the hedging portfolio and A k are contractual cash flows in currency k . We denote by V k ( x, ϕ, A k ) the wealth process of the trading strategy ( x, ϕ, A k ) expressed undercurrency k . When k = e we simply omit the currency index and write V ( x, ϕ, A k ) = V e ( x, ϕ, A k ).We have V ( x, ϕ,
0) = x and V ( x, ϕ, A k ) = x + A e,k = x + p e . We introduce the following regularityassumption. Assumption 2.6.
We assume thati) ξ i,k i = 1 , , . . . , d k , k = 1 , . . . , L are arbitrary G -predictable processes.ii) ψ j,k j = 0 , , . . . , d k , k = 1 , . . . , L are arbitrary G -adapted processes.all processes above are such that the stochastic integrals used in what follows are well defined. Let us introduce the concept of self-financing trading strategy.
ALESSANDRO GNOATTO AND NICOLE SEIFFERT
Definition 2.7.
Let k be any fixed currency belonging to the set of currencies { , . . . , L } . Given thehedger’s initial endowment x , we say that a trading strategy ( x, ϕ, A k ) , associated with a contract A k is self financing, whenever the wealth process V ( x, ϕ, A k ) , which is given by the formula V t ( x, ϕ, A k ) = L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t , (2.7) satisfies V t ( x, ϕ, A k ) = x + L X k =1 d k X i =1 "Z (0 ,t ] X e,k u ξ i,k u (cid:16) dS i,k u + dD i,k u (cid:17) + Z (0 ,t ] ξ i,k u S i,k u d X e,k u + Z (0 ,t ] ξ i,k u d h S i,k , X e,k i u + d k X j =0 (cid:20)Z t X e,k u ψ j,k u dB j,k u + Z t ψ j,k u B j,k u d X e,k u (cid:21) + A e,k t . (2.8) Remark . In a single currency case Definition 2.7 corresponds to Definition 2.3 in Bielecki and Rutkowski(2015).2.2.
Basic multi-currency setting.
Absence of arbitrage is a feature of the market that must holdirrespective of the particular funding strategy adopted: Absence of arbitrage should hold irrespectiveof the presence or absence of a collateralization agreement. Absence of arbitrage should hold first in abasic setting without any collateralization agreement. The introduction of collateralization agreementsshould be done in such a way as to preserve absence of arbitrage. In this section we start our discussionof absence of arbitrage. With this aim in mind, following Bielecki and Rutkowski (2015), we introducethe multi-currency basic model.
Definition 2.9.
We call basic multi-currency model with funding costs a market model in whichlending and borrowing accounts coincide: we have B ,k = B ,k,b = B ,k,l and B j,k = B j,k,b = B j,k,l .for all j = 1 , . . . , d k , k = 1 , . . . , L . and trading in funding accounts and risky assets is unconstrained. This simple setting is instrumental in analyzing more realistic models with further trading covenants.Following Bielecki and Rutkowski (2015), we introduce the concept of netted wealth , which will beinstrumental in characterizing absence of arbitrage in the multi-currency market: In fact, the conceptof martingale measure will be that of a measure such that the discounted netted wealth is a (local)martingale. As explained in Bielecki and Rutkowski (2015), this is necessary because the wealthprocess includes A k , and one needs to compensate such position with holdings on − A k . Definition 2.10.
The netted wealth V net ( x, ϕ, A k ) of a trading strategy is defined by the equality V net ( x, ϕ, A k ) := V ( x, ϕ, A k ) + V (0 , e ϕ, − A k ) , (2.9) where (0 , e ϕ, − A k ) is the unique self-financing trading strategy that uses holdings in B e to finance aposition A k : the trader borrows money from treasury (i.e. borrows units of B e ), purchases units of thecurrency k (i.e. buys units of B ,k ) and uses them to enter an unhedged position in the claim withdividend process A k , and leaves the position unhedged, meaning that for e ϕ we have ξ i,k = ψ j,k = 0 for any i = 1 , . . . , d k and j = 1 , . . . , d k . Notice that the net effect in e ϕ is that of a short position in the domestic unsecured account B e = B ,e and a long position on A k with two opposite positions in B ,k compensating each other. ULTI-CURRENCY SETTING 7
Lemma 2.11.
Assume B e = B e,b = B e,l , then the following holds, for all t ∈ [0 , T ] . V nett ( x, e ϕ, A k ) = V t ( x, ϕ, A k ) − B et Z [0 ,t ] dA e,k u B eu . (2.10) Proof.
This corresponds to Lemma 2.1 in Bielecki and Rutkowski (2015). We provide the details inwhat follows. We have V t (0 , e ϕ, − A k ) = ˜ ψ et B et We also have d (cid:18) V (0 , e ϕ, − A k ) B e (cid:19) t = dV t (0 , e ϕ, − A k ) B et − V t (0 , e ϕ, − A k )( B et ) dB et = ˜ ψ et dB et − dA e,k t B et − V t (0 , e ϕ, − A k )( B et ) dB et = − ( B et ) dA e,k t , where we used ˜ ψ et = V t (0 , e ϕ, − A k ) B et . Now, since we know that V (0 , e ϕ, − A k ) = − A e,k , we can integrateboth sides to conclude that V (0 , e ϕ, − A k ) = + B et Z [0 ,t ] dA e,k u B eu , and the conclusion immediately follows from the definition of netted wealth. (cid:3) Preliminary computation in the basic model.
Following Bielecki and Rutkowski (2015) we in-troduce, for i = 1 , . . . d k , k = 1 , . . . , L , the processes K i,k ,k t := Z (0 ,t ] B i,k u d ˆ S i,cld,k ,k u . (2.11)This process represents the wealth, denominated in units of currency k , discounted by the fundingaccount B i,k of a self-financing trading strategy that invests in the asset S i,k , the associated repo-account B i,k . For the sake of simplicity, the next process is only considered in terms of units of thedomestic currency e : K ϕ,k t := Z (0 ,t ] B eu d ˜ V u ( x, ϕ, A k ) − ( A e,k t − A e,k ) = Z (0 ,t ] B eu d ˜ V netu ( x, ϕ, A k ) , (2.12)where ˜ V net ( x, ϕ, A k ) := ( B e ) − V net ( x, ϕ, A k ) and ˜ V ( x, ϕ, A k ) := ( B e ) − V ( x, ϕ, A k ) and the lastequality follows from (2.10).The following proposition is instrumental for the analysis of absence of arbitrage in the basic modeland more advanced settings. We remark again that we are adopting the point of view of the domesticcurrency e , but analogous computations make it possible to obtain the same claims with respect toany currency denomination. Proposition 2.12.
For any self-financing strategy ϕ we have that, for every t ∈ [0 , T ] K ϕ,k t = L X k =1 d k X i =1 Z (0 ,t ] ξ i,k u dK i,e,k u + L X k =1 d k X i =1 Z t B eu B i,k u (cid:16) ψ i,k u B i,k u + ξ i,k u S i,k u (cid:17) X e,k u d (cid:18) B i,k B e (cid:19) u + L X k =1 d k X i =1 Z t B i,k u ψ i,k u d X e,k u + L X k =1 Z t B eu ψ k u d (cid:18) X e,k B k B e (cid:19) u . (2.13) ALESSANDRO GNOATTO AND NICOLE SEIFFERT
Assume also that the repo constraint holds, i.e. for all i = 1 , . . . , d k , k = 1 , . . . , L we have ζ i,k t := ψ i,k t B i,k t + ξ i,k t S i,k t = 0 , t ∈ [0 , T ] , (2.14) then we have K ϕ,k t = L X k =1 d k X i =1 Z (0 ,t ] ξ i,k u B i,k u X e,k u d (cid:18) S i,k B i,k (cid:19) u + X e,k u B i,k u dD i,k u + d (cid:20) S i,k B i,k , X e,k (cid:21) u ! + L X k =1 Z t B eu ψ k u d (cid:18) X e,k B k B e (cid:19) u . (2.15) Proof.
Let V := V ( x, ϕ, A k ) and hence ˜ V := ( B e ) − V . Then we have d ˜ V t = − V t ( B et ) dB et + 1 B et L X k =1 d k X i =1 h X e,k t ξ i,k t (cid:16) dS i,k t + dD i,k t (cid:17) + ξ i,k t S i,k t d X e,k t + ξ i,k t d h S i,k , X e,k i t i + d k X j =0 h X e,k t ψ j,k t dB j,k t + ψ j,k t B j,k t d X e,k t i + dA e,k t = − B et ) L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t dB et + 1 B et L X k =1 d k X i =1 h X e,k t ξ i,k t (cid:16) dS i,k t + dD i,k t (cid:17) + ξ i,k t S i,k t d X e,k t + ξ i,k t d h S i,k , X e,k i t i + d k X j =0 h X e,k t ψ j,k t dB j,k t + ψ j,k t B j,k t d X e,k t i + dA e,k t . By regrouping terms we obtain d ˜ V t = L X k =1 d k X i =1 ξ i,k t d (cid:18) X e,k S i,k B et (cid:19) t + L X k =1 d k X i =1 ξ i,k t X e,k t B et dD i,k t + L X k =1 d k X i =1 ψ i,k t d (cid:18) X e,k B i,k B e (cid:19) t + L X k =1 ψ k t d (cid:18) X e,k B k B e (cid:19) t + ( B et ) − dA e,k t . We set ˜ S i,cld,e,k t := X e,k t S i,k t B et + Z (0 ,t ] X e,k u B eu dD i,k u . (2.16)We can then focus on K ϕ,k . We have dK ϕ,k t = B et d ˜ V t ( x, ϕ, A k ) − dA e,k t = L X k =1 d k X i =1 B et ξ i,k t d ˜ S i,cld,e,k t + L X k =1 d k X i =1 B et ψ i,k t d (cid:18) X e,k B i,k B e (cid:19) t + L X k =1 B et ψ k t d (cid:18) X e,k B k B e (cid:19) t = L X k =1 d k X i =1 B et ξ i,k t d (cid:18) S i,k X e,k B i,k B i,k B e (cid:19) t + L X k =1 d k X i =1 ξ i,k t X e,k t dD i,k t ULTI-CURRENCY SETTING 9 + L X k =1 d k X i =1 B et ψ i,k t d (cid:18) X e,k B i,k B e (cid:19) t + L X k =1 B et ψ k t d (cid:18) X e,k B k B e (cid:19) t = L X k =1 d k X i =1 B et ξ i,k t S i,k t X e,k t B i,k t d (cid:18) B i,k B e (cid:19) t + L X k =1 d k X i =1 ξ i,k t B i,k t d (cid:18) S i,k X e,k B i,k (cid:19) t + L X k =1 d k X i =1 B i, t ξ i,k t X e,k t B i,k t dD i,k t + L X k =1 d k X i =1 B et ψ i,k t d (cid:18) X e,k B i,k B e (cid:19) t + L X k =1 B et ψ k t d (cid:18) X e,k B k B e (cid:19) t . Since dK i,e,k t = B i,k t d ˆ S i,cld,e,k t , we have dK ϕ,k t = L X k =1 d k X i =1 ξ i,k t dK i,e,k t + L X k =1 d k X i =1 B et B i,k t ! B i,k t ψ i,k t X e,k t d (cid:18) B i,k B e (cid:19) t + B i,k t ψ i,k t B i,k t B et d X e,k t + ξ e,k t S i,k t X e,k t d (cid:18) B i,k B e (cid:19) t (cid:19) + L X k =1 B et ψ k t d (cid:18) X e,k B k B e (cid:19) t and (2.13) part is proven. Now, under the repo constraint (2.14), we have Z t B i,k u ψ i,k u B i,k u B eu d X e,k u = − Z t S i,k u ξ i,k u B i,k u B eu d X e,k u and so we obtain (2.15). (cid:3) Wealth dynamics in the basic model.
In Bielecki and Rutkowski (2015), the single currencyanalogue of the increment dK i,e,k t , i.e. dK i,e,et represents the change in price of the i -th asset, net offunding costs. This becomes clearer thanks to the following result, that follows from an applicationof the Ito product rule. Lemma 2.13.
The following equality holds true for t ∈ [0 , T ] . K i,e,k t = Z (0 ,t ] S i,k u d X e,k u − S i,k u X e,k u B i,k u dB i,k u + X e,k u dS i,k u + d h X e,k , S i,k i u + X e,k u dD i,k u (cid:17) . (2.17)Let us specialize (2.17) in a single currency setting, under the additional assumption that thefunding account B i,e is absolutely continuous with respect to the Lebesgue measure, so that we write dB i,et = r i,et B i,et dt , for some G -progressively measurable process r i,e = ( r i,et ) t ∈ [0 ,T ] . Since we obviouslyhave X e,et ≡ , t ∈ [0 , T ] we obtain K i,e,et = Z (0 ,t ] (cid:16) dS i,eu − S i,eu r i,eu du + dD i,k u (cid:17) . (2.18)Such expression is often referred to in the literature as the gain process from the i -th risky asset. Ina multi currency setting, however, this no longer holds, since we have a further term involving thecurrency risk related to the foreign repo cash account: In Proposition 2.12 we also have the term L X k =1 d k X i =1 Z t B i,k u ψ i,k u d X e,k u , which captures the impact of fluctuations of foreign exchange rates on the funding costs related toforeign repo positions. The definition of the martingale property for the gain process of risky assetsshould account also for this last source of funding costs, which is identically zero in the single-currencycase. Corollary 2.14.
Formula (2.13) in Proposition 2.12 is equivalent to the following expressions. d ˜ V nett ( x, ϕ, A k ) = L X k =1 d k X i =1 ξ i,k t B i,k t B et d ˆ S i,cld,e,k t + L X k =1 d k X i =1 B i,k (cid:16) ψ i,k t B i,k t + ξ i,k t S i,k t (cid:17) X e,k t d (cid:18) B i,k B e (cid:19) t + L X k =1 d k X i =1 B i,k t B et ψ i,k t d X e,k t + L X k =1 ψ k t d (cid:18) X e,k B k B e (cid:19) t , (2.19) d ˜ V t ( x, ϕ, A k ) = L X k =1 d k X i =1 ξ i,k t B i,k t B et d ˆ S i,cld,e,k t + L X k =1 d k X i =1 B i,k (cid:16) ψ i,k t B i,k t + ξ i,k t S i,k t (cid:17) X e,k t d (cid:18) B i,k B e (cid:19) t + L X k =1 d k X i =1 B i,k t B et ψ i,k t d X e,k t + L X k =1 ψ k t d (cid:18) X e,k B k B e (cid:19) t + ( B et ) − dA e,k t , (2.20) dV t ( x, ϕ, A k ) = ˜ V t ( x, ϕ, A k ) dB et + L X k =1 d k X i =1 ξ i,k t dK i,e,k t + L X k =1 d k X i =1 B et B i,k (cid:16) ψ i,k t B i,k t + ξ i,k t S i,k t (cid:17) X e,k t d (cid:18) B i,k B e (cid:19) t + L X k =1 d k X i =1 B i,k t ψ i,k t d X e,k t + L X k =1 B et ψ k t d (cid:18) X e,k B k B e (cid:19) t + dA e,k t . (2.21) Proof.
We deduce (2.19) from dK ϕ,e,k t = B et d ˜ V nett ( x, ϕ, A k ) and (2.11). From (2.19) we deduce (2.20)due to d ˜ V t ( x, ϕ, A k ) = d ˜ V nett ( x, ϕ, A k ) + ( B et ) − dA e,k t . Finally, using dV t ( x, ϕ, A k ) = d ˜ V t ( x, ϕ, A k ) dB et + B et d ˜ V t ( x, ϕ, A k ) , we deduce (2.21). (cid:3) Pricing and hedging in an unsecured multi-currency market
In this section, we discuss the problem of pricing and hedging in a multi-currency market withfunding costs (i.e. multiple curves for different assets) but in the absence of collateralization. Thisprovides a sound foundation for a martingale pricing approach that we extend in subsequent sections
ULTI-CURRENCY SETTING 11 to include collateral in different currencies. As usual, our discussion is based on and generalizes thework of Bielecki and Rutkowski (2015) in a single currency setting. For the sake of simplicity, we workin the setting of the basic multi-currency model with funding costs, i.e. we exclude the possibility ofbid-offer spreads in the rates.3.1.
Arbitrage for hedger.
Let x be the initial endowment in units of the local currency e . Wedenote by V ( x ) the wealth process of a self-financing strategy ( x, ϕ , ϕ is the portfolio withall components set to zero except ψ ,e = ψ e . The wealth process of the strategy is simply V t ( x ) = xB et .Given a contract A k , an arbitrage opportunity is present in the market if the hedger can create ahigher netted wealth (i.e. by going long and short the contract, while leaving one position unhedged)at time T than the future value of the initial endowment. We restrict ourselves to admissible tradingstrategies as defined by the following. Definition 3.1.
A self-financing trading strategy ( x, ϕ, A k ) is admissible for the hedger whenever thediscounted netted wealth V net ( x, ϕ, A k ) is bounded from below by a constant. Definition 3.2.
An admissible trading strategy ( x, ϕ, A k ) is an arbitrage opportunity for the hedgerwith respect to the contract A k whenever the following conditions are satisfiedi) P (cid:0) V netT ( x, ϕ, A k ) ≥ V T ( x ) (cid:1) = 1 ii) P (cid:0) V netT ( x, ϕ, A k ) > V T ( x ) (cid:1) > Remark . From V t ( x ) = xB et we can rewrite the conditions in Definition 3.2 asi) P (cid:0) V netT ( x, ϕ, A k ) ≥ xB eT (cid:1) = 1 = ⇒ P (cid:16) ˜ V netT ( x, ϕ, A k ) ≥ x (cid:17) = 1ii) P (cid:0) V netT ( x, ϕ, A k ) > xB et (cid:1) > ⇒ P (cid:16) ˜ V netT ( x, ϕ, A k ) > x (cid:17) > V t ( x ) = x + B ,e,lt − x − B ,e,bt A classical textbook arbitrage strategy can be constructed in a market with two locally risk-freeassets growing at different rates. To preclude such trivial arbitrage opportunities, the repo constraint(2.14) becomes crucial. The financial meaning of the repo constraint is that the holdings on everyrisky asset are financed by a position on an-asset specific cash accounts and it is not possible to createlong-short positions on different cash accounts that produce riskless profits. Intuitively speaking, thismeans that, for every fixed currency k , the corresponding market consists of a combination of d k sub-markets, each consisting of a single risky asset with an associated financing account. Proposition 3.4.
Assume that all strategies available to the hedger are admissible in the sense ofDefinition 3.1 and satisfy the repo constraint (2.14) . If there exists a probability measure Q e on (Ω , G T ) , such that Q e ∼ P and such that the processes Z (0 ,t ] X e,k u d (cid:18) S i,k B i,k (cid:19) u + X e,k u B i,k u dD i,k u + d (cid:20) S i,k B i,k , X e,k (cid:21) u !! ≤ t ≤ T , (3.1) X e,k t B k t B et ! ≤ t ≤ T (3.2) i = 1 , . . . , d k , k = 1 . . . , L are local martingales, then the basic multi-currency model with fundingcosts is arbitrage free for the hedger for any contract. Proof.
Assume that the repo constraint (2.14) holds. This implies that we can write ψ i,k t = − ξ i,k t S i,k t B i,k t .Then looking at (2.19) we have L X k =1 d k X i =1 Z (0 ,t ] B i,k u B eu ψ i,k u d X e,k u = − L X k =1 d k X i =1 Z (0 ,t ] B i,k u B eu ξ i,k u S i,k u B i,k u d X e,k u , and also L X k =1 d k X i =1 Z (0 ,t ] ξ i,k u B i,k u B eu d ˆ S i,cld,e,k u = L X k =1 d k X i =1 Z (0 ,t ] ξ i,k u B i,k u B eu d (cid:18) S i,k X e,k B i,k (cid:19) u + X e,k u B i,k u dD i,k u ! . Using the above expressions we can write˜ V nett ( x, ϕ, A k ) = x + L X k =1 d k X i =1 Z (0 ,t ] ξ i,k u B i,k u B eu (cid:18) X e,k u d (cid:18) S i,k B i,k (cid:19) u . + X e,k u B i,k u dD i,k u + d (cid:20) S i,k B i,k , X e,k (cid:21) u ! + L X k =1 Z t ψ k u d (cid:18) X e,k B k B e (cid:19) u . (3.3)Using assumptions (3.1) and (3.2), we observe that (3.3) is a local martingale bounded from below bya constant, hence by Fatou Lemma it is a supermartingale. (cid:3) Fair valuation in the presence of funding costs.
We work under the assumption that themodel is arbitrage free for the hedger for any contract. We wish to describe the fair price of a contractat time zero from the perspective of the hedger, (i.e. from the perspective of the seller of the contract).Recall the notation p e ∈ R for the price of the claim. Recall also that p e = A e,k . We use the followingstandard convention: if p e > p e from the counterparty,whereas p e < p e to the counterparty. The following is along thelines of Bielecki and Rutkowski (2015), Definition 3.5. Definition 3.5.
We say that ¯ p e = A e,k ∈ R is a hedger’s fair price if, for any self-financing tradingstrategy (cid:0) x, ϕ, A k (cid:1) such that the discounted wealth ˜ V (cid:0) x, ϕ, A k (cid:1) is bounded from below by a constantwe have either P (cid:16) V T (cid:16) x, ϕ, A k (cid:17) = V T ( x ) (cid:17) = 1 or P (cid:16) V T (cid:16) x, ϕ, A k (cid:17) < V T ( x ) (cid:17) > . If the price ¯ p e is too high, then we have an arbitrage as defined in the following, which is theanalogue of Bielecki and Rutkowski (2015) Definition 3.6. Definition 3.6.
We say that a quadruplet (cid:0) p e , x, ϕ, A k (cid:1) , where p e ∈ R and (cid:0) x, ϕ, A k (cid:1) is an admis-sible trading strategy such that the discounted wealth process ˜ V (cid:0) x, ϕ, A k (cid:1) is bounded from below by aconstant is a hedger’s arbitrage opportunity for A k at price p e if P (cid:16) V T (cid:16) x, ϕ, A k (cid:17) ≥ V T ( x ) (cid:17) = 1 ULTI-CURRENCY SETTING 13 or P (cid:16) V T (cid:16) x, ϕ, A k (cid:17) > V T ( x ) (cid:17) > . The following result characterizes the hedger’s fair price and generalizes Proposition 3.2 in Bielecki and Rutkowski(2015).
Proposition 3.7.
Under the assumptions of Proposition 3.4, ¯ p e ∈ R is a hedger’s fair price, whenever,for any admissible trading strategy (cid:0) x, ϕ, A k (cid:1) satisfying the repo constraint (2.14) we have either P ¯ p e + L X k =1 d k X i =1 Z (0 ,T ] B i,k u B eu (cid:16) ξ i,k u d ˆ S i,cld,e,k u + ψ i,k u d X e,k u (cid:17) + L X k =1 Z (0 ,T ] ψ k u d (cid:18) X e,k B k B e (cid:19) u + Z (0 .T ] ( B eu ) − dA e,k u = 0 = 1 or P ¯ p e + L X k =1 d k X i =1 Z (0 ,T ] B i,k u B eu (cid:16) ξ i,k u d ˆ S i,cld,e,k u + ψ i,k u d X e,k u (cid:17) + L X k =1 Z (0 ,T ] ψ k u d (cid:18) X e,k B k B e (cid:19) u + Z (0 .T ] ( B eu ) − dA e,k u < > Proof.
We recall Lemma 2.11 and make use of (2.19). We have1 = P (cid:16) V T (cid:16) x, ϕ, A k (cid:17) = V T ( x ) (cid:17) = P (cid:18) ˜ V T (cid:16) x, ϕ, A k (cid:17) = V T ( x ) B eT (cid:19) = P p e + x + L X k =1 d k X i =1 Z (0 ,T ] B i,k u B eu (cid:16) ξ i,k u d ˆ S i,cld,e,k u + ψ i,k u d X e,k u (cid:17) + L X k =1 Z (0 ,T ] ψ k u d (cid:18) X e,k B k B e (cid:19) u + Z (0 .T ] ( B eu ) − dA e,k u = x From which we obtain the first relation. The second is proven analogously. (cid:3) Multi Currency trading under funding costs and collateralization
We consider the situation where the hedger posts or receives collateral in a currency k ∈ { , . . . , L } ,represented by a process C k , which is right continuous and G -adapted. In the literature on coun-terparty credit risk the symbol C is often used to denote the so-called variation margin . Nowadaysfinancial institutions also exchange another collateral called initial margin which is meant to providea form of over-collateralization. Our discussion in the present section aims at covering most collat-eral conventions, so that the formulas we derive can be suitably combined in order to describe eithervariation margin or initial margin or even a situation where multiple types of collateral are present.As any random variable, C t can be split in its positive and negative part. In particular, we adopt thefollowing convention • C k , + t is the value of collateral received by the hedger to the counterparty in currency k . • C k , − t is the value of collateral posted by the hedger from the counterparty in currency k . We will allow for collateral to be posted in any currency and either in the form of cash (whichconstitutes the most common form) or risky assets. For this, following Bielecki and Rutkowski (2015)we introduce two dedicated assets, denominated in the currency k . This means that for one of thecurrencies in the set { , . . . , L } we will have d k + 2 assets. For simplicity we assume that, whencollateral is posted in terms of a risky asset, the currency of the posted and received collateral arethe same. The situation where the two currencies differ is a rather uncommon situation which canhowever be accommodated in principle in our set-up. We make the following assumptions: • The risky asset S d k +1 is delivered by the hedger as collateral. • The risky asset S d k +2 is received by the hedger as collateral. • Unsecured funding borrowing and lending rates coincide for all currencies, i.e. B k ,b = B k ,l = B k , k = 1 , . . . , L . • Repo funding accounts have identical borrowing and lending rates B i,k ,b = B i,k ,l = B i,k forevery i = 1 , . . . , d k , k = 1 , . . . , L . • We also assume that the collateral account satisfies C k T = 0, meaning that, when trading isover, the collateral is returned to its legal owner. • Depending on the underlying collateral convention, the hedger receives or pays interest con-tingent on being the poster or receiver of collateral: the hedger receives interest paymentsbased on B c,k ,l or pays interests based on B c,k ,b . More precisely, the amount of interest isdetermined by η k ,l := ( B c,k ,l ) − ( C k ) − and respectively η k ,b := − ( B c,k ,b ) − ( C k ) + . (4.1)4.1. Collateral conventions.
To be self-contained, let us recall the standard conventions for collat-eral. • Segregation
Under segregation, the collateral amount must be kept in a separate accountand is not available as a source of funding for the trading activity. The hedger, when he/shereceives collateral, can not use it for trading: he/she is only allowed to receive possibly zerointerest based on B d k +2 ,k . The dynamics of the wealth of the hedger do not depend on theamount of cash or shares of the asset S d k +2 ,k he/she receives. On the contrary the amountof cash or shares of the asset S d k +1 he/she posts to the counterparty has an effect on thedynamics of the portfolio. Segregation is the standard for the exchange of initial margin. • Rehypothecation
Under rehypothecation, the hedger is allowed to use the cash or the sharesof securities he/she receives to fund his/her trading activity. Rehypothecation constitutes themost adopted convention for the variation margin.The dynamics of the hedger’s wealth differ under segregation or rehypothecation. There is also animpact of whether collateral is posted in form of cash or risky assets We also need to address theaction that is undertaken by the hedger when he/she receives collateral: we need to discuss how suchamount of wealth is reinvested. The reinvestment activity will be modelled by means of an additionalcash account, that we wish to make specific on the fact that we have rehypothecation or segregation:to this aim we introduce the cash accounts B d k +2 ,k ,s in case of segregation and B d k +2 ,k ,h in case ofrehypothecation. The following notation helps to distinguish between those cases. In a single currencycase this reflects Definition 4.2 and Definition 4.3 of Bielecki and Rutkowski (2015). Definition 4.1 (Cash collateral) . Cash collateral is specified as follows:
ULTI-CURRENCY SETTING 15
Repo MarketCounterparty Hedger η d k +2 ,k t B d k +2 ,k t ξ d k +2 ,k t S d k +2 ,k t ξ d k +2 ,k t S d k +2 ,k t Figure 1.
Collateral reinvestment process in case a risky asset is used.(i)
If the hedger receives cash collateral denoted by ( C k ) + t , he/she pays interest determined by theborrowing account B c,k ,bt and ( C k ) + t . In case of segregation, he/she receives interest based onthe amount ( C k ) + t and the cash account B d k +2 ,k ,st . Under rehypothecation, the hedger mayuse the amount ( C k ) + t to fund the trading activity before maturity: in particular he/she usesunits of B d k +2 ,k ,ht for his/her own trading purposes. Recall that { B d k +2 ,k ,s , B d k +2 ,k ,h } isa specification of B d k +2 ,k , monitoring the underlying collateral convention, as well for thecorresponding strategy { η d k +2 ,k ,s , η d k +2 ,k ,h } . (ii) If the hedger posts cash collateral, he/she delivers the amount ( C k ) − t , borrowed from the cashaccount B d k +3 ,k t and receives interest determined by B c,k ,lt in return. As collateral is postedin form of cash, the following equalities hold for any t ∈ [0 , T ] : ξ d k +1 ,k t = ψ d k +1 ,k t = 0 , η d k +3 ,k t B d k +3 ,k t = − ( C k t ) − (4.2)We assume the hedger receives or delivers shares of the risky asset S d k +1 ,k , which are supposed tohave low credit risk and should be uncorrelated with the underlying trading portfolio. We stress thefollowing fact: due to the assumption that the asset is uncorrelated with the underlying portfolio, inthe case where the collateral is received there is no reason to include the holdings of such asset in theportfolio. Instead, if the asset is posted, the hedger needs to fund and create a position in such assetin order to fulfil the margin call. Definition 4.2 (Risky asset collateral) . Risky collateral is specified as follows: (i)
If the hedger receives ξ d k +2 ,k t shares of the risky asset S d k +2 ,k t , used as collateral, he/she iscommitted to pay interest to the counterparty determined by B c,k ,bt and ( C k ) + t = ξ d k +2 ,k t S d k +2 ,k t .In case of segregation where reinvesting collateral is not allowed, the hedger receives intereston a basic bank deposit determined by B d k +2 ,k ,st similar to the cash collateral case. (ii) If the hedger delivers a number of shares ξ d k +1 ,k t of the risky asset S d k +1 ,k t to the coun-terparty, funded by the account B d k +1 ,k t he/she receives interest determined by the collateralaccount B c,k ,lt in return. Hence, the following setting can be defined for t ∈ [0 , T ] : ( C k t ) − = ξ d k +1 ,k t S d k +1 ,k t , η d k +3 ,k t = 0 ,ξ d k +1 ,k t S d k +1 ,k t + ψ d k +1 ,k t B d k +1 ,k t = 0(4.3) which implies ψ d k +1 ,k t B d k +1 ,k t = − ( C k t ) − for all t ∈ [0 , T ] . At this point, it is important to stress an important aspect. During the trading activity the hedgerwill in general simultaneously be managing assets/amounts of cash he/she legally owns, together with assets/amounts of cash that belong to the counterparty, hence it is convenient to distinguish betweenthe following • V t ( x, ϕ, A k , C k ): this is the wealth of the hedger, representing the value of the portfolio ofassets that belong to the hedger. • V pt ( x, ϕ, A k , C k ): this is the value of the (full) portfolio of the hedger, including the as-sets/amounts of cash that belong to the counterparty. • V ct ( x, ϕ, A k , C k ) := V t ( x, ϕ, A k , C k ) − V pt ( x, ϕ, A k , C k ) i.e. the difference between thelegal wealth of the hedger and his/her portfolio, is called adjustment process and representsthe impact of collateralization.Let us recall that the wealth processes are expressed in units of the local currency e . We now pro-ceed to formally define the processes introduced above. It is rather clear that, in the absence ofcollateralization, we recover our previous formulation for the dynamics of the wealth process. Definition 4.3.
For any t ∈ [0 , T ] and { k , k } ∈ { , . . . , L } , we call the process V t ( x, ϕ, A k , C k ) : = L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t + X e,k t (cid:16) ξ d k +1 ,k t S d k +1 ,k t + ψ d k +1 ,k t B d k +1 ,k t + η k ,bt B c,k ,bt + η k ,lt B c,k ,lt + η d k +2 ,k t B d k +2 ,k t + η d k +3 ,k t B d k +3 ,k t (cid:17) (4.4) the extended wealth process under funding costs and collateralization, where ( x, ϕ, A k , C k ) denotesthe hedger’s collateralized trading strategy for the portfolio ϕ = ( ξ, ψ, φ ( k )) = (cid:16) ξ , , . . . , ξ d , , , ξ , , . . . , ξ d L ,L , ψ , , . . . , ψ d , , ψ , , . . . , ψ d L ,L , φ ( k ) (cid:17) (4.5) with φ ( k ) := ( ξ d k +1 ,k , ψ d k +1 ,k , η k ,b , η k ,l , η d k +2 ,k , η d k +3 ,k ) . We also introduce the following.
Definition 4.4.
Let ( x, ϕ, A k , C k ) be a collateralized trading strategy of the hedger and t ∈ [0 , T ] . (i) The value of the hedger’s portfolio V p ( x, ϕ, A k , C k ) at time t is defined by V pt ( x, ϕ, A k , C k ) : = L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t + X e,k t (cid:16) ξ d k +1 ,k t S d k +1 ,k t + ψ d k +1 ,k t B d k +1 ,k t + η d k +3 ,k t B d k +3 ,k t (cid:17) . (4.6)(ii) In addition, denote V c ( x, ϕ, A k , C k ) given by V ct ( x, ϕ, A k , C k ) : = V t ( x, ϕ, A k , C k ) − V pt ( x, ϕ, A k , C k )= X e,k t (cid:16) η k ,bt B c,k ,bt + η k ,lt B c,k ,lt + η d k +2 ,k t B d k +2 ,k t (cid:17) (4.7) as adjustment process of the hedger’s wealth. The adjustment process reflects the presence of a collateralization agreement between the hedgerand the counterparty. Let us recall that η d k +2 ,k t might be either η d k +2 ,k ,st or η d k +2 ,k ,ht , dependingon the particular convention agreed by the hedger and the counterparty. ULTI-CURRENCY SETTING 17
Remark . By using assumption (4.1), we receive for the adjustment process V ct ( x, ϕ, A k , C k ) = X e,k t (cid:16) − C k t + η d k +2 ,k t B d k +2 ,k t (cid:17) . (4.8)for any t ∈ [0 , T ].We introduce the following useful notation. Definition 4.6.
In the collateralized multi-currency model, the process of all interest generated by thecollateral account, denoted by F h under rehypothecation and F s under segregation, is given by F ht := F ct + Z t X e,k u ( B d k +2 ,k ,hu ) − ( C k u ) + dB d k +2 ,k ,hu , (4.9) where F c is the cumulative interest of the margin account defined by F ct := Z t X e,k u ( B c,k ,lu ) − ( C k u ) − dB c,k ,lu − Z t X e,k u ( B c,k ,bu ) − ( C k u ) + dB c,k ,bu . (4.10)A standard assumption consists in assuming that all cash accounts are absolutely continuous, sothat all cash accounts can be written as dB jt = r jt B jt dt for some G -adapted processes r j and anyarbitrary index j we consider in the present setting. When this is the case, one can simplify (4.10) as F ct := Z t X e,k u ( C k u ) − r c,k ,lu du − Z t X e,k u ( C k u ) + r c,k ,bu du. The above formulation explicitly features the borrowing and lending collateral rates: the interestreceived on the posted collateral has a positive impact net of the interest paid on the received collateral.We need the following generalization of the definition of a self-financing trading strategy.
Definition 4.7.
Assume ( x, ϕ, A k , C k ) to be a collateralized trading strategy, where k and k fulfilthe usual conditions. The strategy is called self financing, if the hedger’s portfolio value V p ( x, ϕ, A k , C k ) fulfils V pt ( x, ϕ, A k , C k ) : = x + L X k =1 d k X i =1 "Z (0 ,t ] X e,k u ξ i,k u (cid:16) dS i,k u + dD i,k u (cid:17) + Z (0 ,t ] ξ i,k u S i,k u d X e,k u + Z (0 ,t ] ξ i,k u d h S i,k , X e,k i u + d k X j =0 (cid:20)Z t X e,k u ψ j,k u dB j,k u + Z t ψ j,k u B j,k u d X e,k u (cid:21) + Z (0 ,t ] X e,k u ξ d k +1 ,k u (cid:16) dS d k +1 ,k u + dD d k +1 ,k u (cid:17) + Z (0 ,t ] ξ d k +1 ,k u S d k +1 ,k u d X e,k u + Z (0 ,t ] ξ d k +1 ,k u d h S d k +1 ,k , X e,k i u + Z (0 ,t ] X e,k u ψ d k +1 ,k u dB d k +1 ,k u + Z (0 ,t ] ψ d k +1 ,k u B d k +1 ,k u d X e,k u + Z t X e,k u η d k +2 ,k u dB d k +2 ,k u + Z t X e,k u η d k +3 ,k u dB d k +3 ,k u + F ct − Z (0 ,t ] C k u d X e,k u + A e,k t − V ct ( x, ϕ, A k , C k ) . (4.11) for any t ∈ [0 , T ] . Let us provide some information concerning the adjustment process and the rules for the determi-nation of the amount of collateral. In line with Bielecki and Rutkowski (2015) the adjustment process satisfies V ct ( x, ϕ, A k , C k ) = g ( C k t ( ϕ )) for some typically Lipschitz function g . In the cases consideredin the sequel we have either V ct ( x, ϕ, A k , C k ) = X e,k t C k t or V ct ( x, ϕ, A k , C k ) = −X e,k t ( C k t ) − .The amount of collateral C k can be determined in many different ways, as the determination ofsuch process is the result of a legal negotiation between the hedger and the counterparty. However itis rather common to link the collateral with the value (mark-to-market) of the contract. Remark . We let M be a G -adapted RCLL process that representsthe value of the contract expressed in units of the local currency e . One possible specification for M is given by the setting M t := V t ( x ) − V t ( x, ϕ, A k , C k ) . (4.12)The formulation above captures the natural assumption that the collateral amount is linked to thevalue of the contract from the perspective of the hedger. In particular, recall that the portfolio V ismeant to cover the liabilities of the hedger towards the counterparty, meaning that the market valueof the contract is − V . In terms of the process M one has the following specification for the collateralaccount under a generic currency k . X e,k t C k t = (1 + δ t ) M + t − (1 + δ t ) M − t , (4.13)where the processes δ and δ represent haircuts that reduce/increase in percentage the amount ofcollateral. Using (4.13) and (4.12) we write X e,k t C k t = (1 + δ t ) (cid:16) V t ( x ) − V t ( x, ϕ, A k , C k ) (cid:17) + − (1 + δ t ) (cid:16) V t ( x ) − V t ( x, ϕ, A k , C k ) (cid:17) − Remark . One particularly important case is the case known as full collater-alization . In this case the value of the collateral is continuously updated in time in order to perfectlymatch the value of the contract. This can be obtained by setting δ t = δ t = 0 for every t , which gives X e,k t C k t = V t ( x ) − V t ( x, ϕ, A k , C k ) . (4.14)Finally, in the case where the initial endowment is zero we have X e,k t C k t = − V t (0 , ϕ, A k , C k ) . (4.15)An important fact to note is that when the transaction is fully collateralized, the collateralizationscheme completely funds the trading portfolio of the hedger, so that the cost of the collateral (whichis proportional to the collateral rate) coincides with the funding rate for the trading activity.4.2. Cash collateral.
We first proceed to study the case where collateral is exchanged in cash asillustrated in Definition 4.1. This constitutes the most common collateralization covenant. Cashcollateral is the case that is also most commonly treated in the literature. From the present treatmentwe will be able to recover the findings of, among others, Moreni and Pallavicini (2017), Fujii et al.(2010b) in the case of full collateralization. The risky asset used for collateralization is of courseimmaterial and in fact we shall set ξ d k +1 ,k t = 0 in the subsequent results.4.2.1. Margin account under segregation.
Let us recall that under segregation, if the hedger receivescollateral from the counterparty, he/she is not allowed to use it as a source of funding for the tradingactivity: this means that ( C k ) + (i.e. the received collateral) is immaterial in the hedger’s wealth,only the collateral posted by the hedger ( C k ) − will have a role in the hedger’s wealth. Concerningthe received collateral ( C k ) + , we notice that this loan, received from the counterparty, must beremunerated according to the cash account B d k +2 ,k ,st , so that this remuneration will have an impact ULTI-CURRENCY SETTING 19 via the self-financing condition. On the other hand, the posted collateral ( C k ) − is borrowed from theaccount B d k +1 ,k t and is remunerated by the counterparty with interest from the cash account B c,k ,l . Proposition 4.10.
We assume the hedger operates under segregation, hence he/she is posting orreceiving collateral in form of cash. Let ( x, ϕ, A k , C k ) be a self-financing strategy and the followingconditions hold for t ∈ [0 , T ] : ξ d k +1 ,k t = ψ d k +1 ,k t = 0 ,η d k +3 ,k t = − ( B d k +3 ,k t ) − ( C k t ) − , η d k +2 ,k ,st = ( B d k +2 ,k ,st ) − ( C k t ) + . (4.16) Then hedger’s wealth process V ( x, ϕ, A k , C k ) is given by V t ( x, ϕ, A k , C k ) = V pt ( x, ϕ, A k , C k ) + X e,k t ( C k t ) − = L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t + X e,k t η d k +3 ,k t B d k +3 ,k t + X e,k t ( C k t ) − (4.17) for every t ∈ [0 , T ] . In addition, the dynamics of the hedger’s portfolio wealth are as follows for any t ∈ [0 , T ] and V p ( ϕ ) := V p ( x, ϕ, A k , C k ) and V ct ( ϕ ) := V ct ( x, ϕ, A k , C k ) : dV pt ( ϕ ) = ˜ V pt ( ϕ ) dB et + L X k =1 d k X i =1 h ξ i,k t dK i,e,k t + X e,k t ζ i,k t ( ˜ B i,k t ) − d ˜ B i,k t + ψ i,k t B i,k t d X e,k t i + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t − X e,k t ( B d k +3 ,k t ) − ( C k t ) − dB d k +3 ,k t + X e,k t ( B d k +2 ,k t ) − ( C k t ) + dB d k +2 ,k t + dF ct − C k t d X e,k t + dA e,k t − dV ct ( ϕ ) . (4.18) Under the repo constraint (2.14) , the dynamics of the hedger’s wealth are given by dV t ( ϕ ) = ˜ V t ( ϕ ) dB et + L X k =1 d k X i =1 ξ i,k t h dK i,e,k t − S i,k t d X e,k t i + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t + dA e,k t + d ˆ F st (4.19) by using the notation V ( ϕ ) := V ( x, ϕ, A k , C k ) and d ˆ F st := dF st − C k t d X e,k t − X e,k t ( B d k +3 ,k t ) − ( C k t ) − dB d k +3 ,k t (4.20) Proof.
By combining the assumptions made in (4.16) with equality (4.4) and (4.1), we derive V t ( x, ϕ, A k , C k ) = L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t + X e,k t (cid:16) − C k t + ( C k t ) + + η d k +3 ,k t B d k +3 ,k t (cid:17) = V pt ( x, ϕ, A k , C k ) + X e,k t ( C k t ) − which proves (4.17). If we take a closer look at the hedger’s portfolio value V p ( x, ϕ, A k , C k ) andrecall that for some k = 1 , . . . , L we have k = e and hence X e,e ≡ B ,e := B e , we have that V pt ( x, ϕ, A k , C k ) = L X k =1 X e,k t d k X i =1 ζ i,k t + L X k =1 ,k = e X e,k t ψ ,k t B k t + ψ ,et B et − X e,k t ( C k t ) − for any t ∈ [0 , T ], where the quantity ζ i,k t was defined in (2.14). Hence we get ψ ,et = ˜ V pt ( x, ϕ, A k , C k ) − L X k =1 X e,k t d k X i =1 ( B et ) − ζ i,k t − L X k =1 ,k = e X e,k t ψ ,k t ( B et ) − B k t + X e,k t ( B et ) − ( C k t ) − (4.21)with ˜ V p ( x, ϕ, A k , C k ) := ( B e ) − V p ( x, ϕ, A k , C k ). By using the dynamics of the self financing condi-tion (4.7) in combination with (4.21), the notation V p ( x, ϕ, A k , C k ) := V p ( ϕ ), V c ( x, ϕ, A k , C k ) := V c ( ϕ ) and V ( x, ϕ, A k , C k ) := V ( ϕ ), we derive by using dK i,e,k t = B i,k t d ˆ S i,cld,e,k t = B i,k t d (cid:16) ˆ S i,k t X e,k t (cid:17) + ( B i,k t ) − X e,k t dD i,k t = X e,k t ( dS i,k t + dD i,k t ) − ( B i,k t ) − X e,k t S i,k t dB i,k t + S i,k t d X e,k t + d h S i,k , X e,k i t (4.22)from equation (2.11) for i = 1 , . . . , d k , k = 1 , . . . , L , that dV pt ( ϕ ) = L X k =1 d k X i =1 h X e,k t ξ i,k t ( dS i,k t + dD i,k t ) + ξ i,k t S i,k t d X e,k t + ξ i,k t d h S i,k , X e,k i t − ξ i,k t ( B i,k t ) − X e,k t S i,k t dB i,k t + ξ i,k t ( B i,k t ) − X e,k t S i,k t dB i,k t + X e,k t ψ i,k t dB i,k t | {z } =( B i,k t ) − X e,k t ζ i,k t dB i,k t + ψ i,k t B i,k t d X e,k t i + L X k =1 ,k = e h X e,k t ψ ,k t dB k t + ψ ,k t B k t d X e,k t i + ˜ V pt ( ϕ ) − L X k =1 X e,k t d k X i =1 ( B et ) − ζ i,k t − L X k =1 ,k = e X e,k t ψ ,k t ( B et ) − B k t + X e,k t ( B et ) − ( C k t ) − dB et − X e,k t ( B d k +3 ,k t ) − ( C k t ) − dB d k +3 ,k t + X e,k t ( B d k +2 ,k t ) − ( C k t ) + dB d k +2 ,k t + dF ct − C k t d X e,k t + dA e,k t − dV ct ( ϕ )= ˜ V pt ( ϕ ) dB et + L X k =1 d k X i =1 ξ i,k t dK i,e,k t + X e,k t ζ i,k t (( B i,k t ) − dB i,k t − ( B et ) − dB et ) | {z } ( ˜ B i,k t ) − d ˜ B i,k t + ψ i,k t B i,k t d X e,k t + L X k =1 ,k = e X e,k t ψ ,k t B k t (( B k t ) − dB k t − ( B et ) − dB et ) | {z } =( ˜ B k t ) − d ˜ B k t + ψ ,k t B k t d X e,k t + X e,k t ( B et ) − ( C k t ) − dB et − X e,k t ( B d k +3 ,k t ) − ( C k t ) − dB d k +3 ,k t + X e,k t ( B d k +2 ,k t ) − ( C k t ) + dB d k +2 ,k t + dF ct − C k t d X e,k t + dA e,k t − dV ct ( ϕ ) ULTI-CURRENCY SETTING 21 = ˜ V pt ( ϕ ) dB et + L X k =1 d k X i =1 h ξ i,k t dK i,e,k t + X e,k t ζ i,k t ( ˜ B i,k t ) − d ˜ B i,k t + ψ i,k t B i,k t d X e,k t i + L X k =1 ,k = e (cid:16) X e,k t ψ ,k t B k t ( ˜ B k t ) − d ˜ B k t + ψ ,k t B k t d X e,k t (cid:17) + X e,k t ( B et ) − ( C k t ) − dB et − X e,k t ( B d k +1 ,k t ) − ( C k t ) − dB d k +1 ,k t + dF st − C k t d X e,k t + dA e,k t − dV ct ( ϕ )We obtain (4.18) by noticing that we can perform the following simplification L X k =1 ,k = e (cid:16) X e,k t ψ ,k t B k t ( ˜ B k t ) − d ˜ B k t + ψ ,k t B k t d X e,k t (cid:17) = L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t Furthermore, if condition (2.14) holds, meaning that ζ i,k t = 0 for all t ∈ [0 , T ] i = 1 , . . . , d k , k =1 , . . . , L and so ψ i,k t B i,k t = − ξ i,k t S i,k t , then the dynamics of the hedger’s wealth process are given by dV t ( ϕ ) = dV pt ( ϕ ) + dV ct ( ϕ )= ˜ V t ( ϕ ) dB et + L X k =1 d k X i =1 ξ i,k t (cid:16) dK i,e,k t − S i,k t d X e,k t (cid:17) + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t + dA e,k t + dF st − X e,k t ( B d k +1 ,k t ) − ( C k t ) − dB d k +1 ,k t − C k t d X e,k t where we also used ˜ V pt ( ϕ ) = ˜ V t ( ϕ ) − X e,k t ( B et ) − ( C k t ) − (cid:3) Margin account under rehypothecation.
Let us recall that, under rehypothecation, when thehedger receives the collateral amount (cid:0) C k (cid:1) + he/she can use it to fund his/her trading activity.Interest is paid by the hedger to the counterparty based on (cid:0) C k (cid:1) + and the cash account B c,k ,b .Instead, in case the hedger posts the amount (cid:0) C k (cid:1) − to the counterparty, then the hedger will receivefrom the counterparty an interest amount based on (cid:0) C k (cid:1) − and the cash account B c,k ,l . As thehedger needs to rise the amount of cash (cid:0) C k (cid:1) − he/she borrows such amount from the dedicated cashaccount B d k +1 ,k that might coincide with the unsecured cash account in currency k i.e. B k .The present case is the most common one in the market for bilateral trades (i.e. trades not involvinga central counterparty) and, when the collateral is perfect (as in (4.14)) then we will obtain in thesequel useful valuation formulas based on the present case. Proposition 4.11.
Consider the market model, where the hedger delivers or posts collateral in formof cash under rehypothecation. Let ( x, ϕ, A k , C k ) be a self financing trading strategy and the followingconditions hold for any t ∈ [0 , T ] : ξ d k +1 ,k t = ψ d k +1 ,k t = 0 , η d k +3 ,k t = − ( B d k +3 ,k t ) − ( C k t ) − , η d k +2 ,k ,ht = 0 . (4.23) Consequently, the hedger’s wealth process V ( x, ϕ, A k , C k ) is given by V t ( x, ϕ, A k , C k ) = L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t − X e,k t ( C k t ) + = V pt ( x, ϕ, A k , C k ) − X e,k t C k t . (4.24) and the dynamics of the hedger’s portfolio value V pt ( ϕ ) := V pt ( x, ϕ, A k , C k ) are dV pt ( ϕ ) = ˜ V pt ( ϕ ) dB et + L X k =1 d k X i =1 h ξ i,k t dK i,e,k t + X e,k t ζ i,k t ( ˜ B i,k t ) − d ˜ B i,k t + ψ i,k t B i,k t d X e,k t i + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t − X e,k t ( B d k +3 ,k t ) − ( C k t ) − dB d k +3 ,k t + X e,k t ( B et ) − ( C k t ) + dB et + dF ct − C k t d X e,k t + dA e,k t − dV ct ( ϕ ) . (4.25) Hence the dynamics of the hedger’s wealth process under the repo constraint (2.14) can be denoted by dV t ( ϕ ) = ˜ V t ( ϕ ) dB et + L X k =1 d k X i =1 ξ i,k t h dK i,e,k t − S i,k t d X e,k t i + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t + dA e,k t + d ˆ F ht (4.26) where d ˆ F ht = dF ct − C k t d X e,k t − X e,k t ( B d k +3 ,k t ) − ( C k t ) − dB d k +3 ,k t + X e,k t ( B et ) − ( C k t ) + dB et (4.27) Proof.
By using the assumptions (4.23) combined with (4.1) and (4.4), we get for any t ∈ [0 , T ] V t ( x, ϕ, A k , C k ) = L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t + X e,k t (cid:16) − ( C k t ) − − C k t (cid:17) = L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t − X e,k t ( C k t ) + = V pt ( x, ϕ, A k , C k ) − X e,k t C k t . Hence the hedger’s portfolio wealth gives us ψ ,et = ˜ V pt ( x, ϕ, A k , C k ) − L X k =1 X e,k t d k X i =1 ( B et ) − ζ i,k t − L X k =1 ,k = e X e,k t ψ ,k t ( B et ) − B k t + X e,k t ( B et ) − ( C k t ) − . (4.28)and combining (4.28) with the self financing condition (4.7) and (4.22), we receive dV pt ( ϕ ) = ˜ V pt ( ϕ ) dB et + L X k =1 d k X i =1 h ξ i,k t dK i,e,k t + X e,k t ζ i,k t ( ˜ B i,k t ) − d ˜ B i,k t + ψ i,k t B i,k t d X e,k t i + L X k =1 ,k = e (cid:16) X e,k t ψ ,k t B k t ( ˜ B k t ) − d ˜ B k t + ψ ,k t B k t d X e,k t (cid:17) − X e,k t ( B d k +3 ,k t ) − ( C k t ) − dB d k +3 ,k t + X e,k t ( B et ) − ( C k t ) − dB et − C k t d X e,k t + dF ct + dA e,k t − dV ct ( ϕ ) , ULTI-CURRENCY SETTING 23 where V ( ϕ ) , V p ( ϕ ) and V c ( ϕ ) are defined as before. In addition, let the repo constraint (2.14) befulfilled and the dynamics of the hedger’s wealth process V ( ϕ ) are given by dV t ( ϕ ) = ˜ V pt ( ϕ ) dB et + L X k =1 d k X i =1 ξ i,k t h dK i,e,k t − S i,k t d X e,k t i + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t − X e,k t ( B d k +3 ,k t ) − ( C k t ) − dB d k +3 ,k t + X e,k t ( B et ) − ( C k t ) − dB et − C k t d X e,k t + dF ct + dA e,k t By observing that ˜ V pt ( ϕ ) = ˜ V t ( ϕ ) + X e,k t ( B et ) − ( C k t ) + − X e,k t ( B et ) − ( C k t ) − we conclude. (cid:3) Risky asset collateral.
Formally, there is no need to distinguish between the case where thehedger posts or receives collateral in form of shares of the risky asset S d k +1 ,k under segregation orrehypothecation, since the hedger’s wealth process behaves in the the same way modulo the differentreinvestment rates B d k +2 ,k ,s and respectively B d k +2 ,k ,h . In the following, the index h can bereplaced by s without loss of generality to formally make a distinction between the underlying collateralconventions.4.3.1. Risky asset collateral under segregation and rehypothecation.
Proposition 4.12.
Consider the hedger posting or receiving collateral in form of shares of the riskyasset S d k +1 ,k with no further restrictions concerning the underlying collateral conventions. Let ( x, ϕ, A k , C k ) be a self financing trading strategy and assume that the following conditions hold for t ∈ [0 , T ] : ξ d k +1 ,k t = ( S d k +1 ,k t ) − ( C k t ) − , ψ d k +1 ,k t = − ( B d k +1 ,k t ) − ( C k t ) − ,η d k +2 ,k ,ht = ( B d k +2 ,k ,ht ) − ( C k t ) + , η d k +3 ,k t = 0 . (4.29) The hedger’s wealth process is now given by V t ( x, ϕ, A k , C k ) = L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t + X e,k t ( C k t ) − = V pt ( x, ϕ, A k , C k ) + X e,k t ( C k t ) − (4.30) and V ct ( ϕ ) = X e,k t ( C k t ) − . The dynamics of the hedger’s portfolio value V pt ( ϕ ) := V pt ( x, ϕ, A k , C k ) are dV pt ( ϕ ) = ˜ V pt ( ϕ ) dB et + L X k =1 d k X i =1 h ξ i,k t dK i,e,k t + X e,k t ζ i,k t ( ˜ B i,k t ) − d ˜ B i,k t + ψ i,k t B i,k t d X e,k t i + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t + ξ d k +1 ,k t dK d k +1 ,e,k t + ψ d k +1 ,k t B d k +1 ,k t d X e,k t + X e,k t ( B d k +2 ,k ,ht ) − ( C k t ) + dB d k +2 ,k ,ht + dF ct − C k t d X e,k t + dA e,k t − dV ct ( ϕ )(4.31) It follows that the dynamics of the hedger’s wealth process are given by dV t ( ϕ ) = ˜ V t ( ϕ ) dB et + L X k =1 d k X i =1 ξ i,k t h dK i,e,k t − S i,k t d X e,k t i + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t + ξ d k +1 ,k t h dK d k +1 ,e,k t − S d k +1 ,k t d X e,k t i + dA e,k t + d ¯ F ht (4.32) under the repo constraint (2.14) with d ¯ F ht = dF ct − C k t d X e,k t + X e,k t ( B d k +2 ,k ,ht ) − ( C k t ) + dB d k +2 ,k ,ht − X e,k t ( B et ) − ( C k t ) − dB et (4.33) Proof.
Combining assumption (4.29) with (4.1) and (4.4), we receive V t ( x, ϕ, A k , C k ) = L X k =1 X e,k t d k X i =1 ξ i,k t S i,k t + d k X j =0 ψ j,k t B j,k t + X e,k t ( C k t ) − − ( C k t ) − − C k t + ( C k t ) + | {z } =( C k t ) − = V pt ( x, ϕ, A k , C k ) + X e,k t ( C k t ) − and thus (4.30) for any t ∈ [0 , T ]. By using ψ ,et = ˜ V pt ( x, ϕ, A k , C k ) − L X k =1 X e,k t d k X i =1 ( B et ) − ζ i,k t − L X k =1 ,k = e X e,k t ψ ,k t ( B et ) − B k t , (4.34)the self financing condition (4.7) and the dynamics of K d k +1 ,e,k given by (4.22), the dynamics of thehedger’s portfolio wealth V p ( x, ϕ, A k , C k ) := V p ( ϕ ) are given by dV pt ( ϕ ) = ˜ V pt ( ϕ ) dB et + L X k =1 d k X i =1 h ξ i,k t dK i,e,k t + X e,k t ζ i,k t ( ˜ B i,k t ) − d ˜ B i,k t + ψ i,k t B i,k t d X e,k t i + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t + ξ d k +1 ,k t (cid:16) X e,k t ( dS d k +1 ,k t + dD d k +1 ,k t ) + d h S d k +1 ,k , X e,k i t + S d k +1 ,k t d X e,k t − ( B d k +1 ,k t ) − X e,k t S d k +1 ,k t dB d k +1 ,k t (cid:17) + ψ d k +1 ,k t B d k +1 ,k t d X e,k t + ξ d k +1 ,k t ( B d k +1 ,k t ) − X e,k t S d k +1 ,k t dB d k +1 ,k t + X e,k t ψ d k +1 ,k t dB d k +1 ,k t | {z } = X e,k t (cid:18) B dk ,k t (cid:19) − (cid:16) ξ d k +1 ,k t S d k +1 ,k t + ψ d k +1 ,k t B d k +1 ,k t (cid:17)| {z } (4.29)= 0 dB dk ,k t + X e,k t ( B d k +2 ,k ,ht ) − ( C k t ) + dB d k +2 ,k ,ht + dF ct − C k t d X e,k t + dA e,k t − dV ct ( ϕ )= ˜ V pt ( ϕ ) dB et + L X k =1 d k X i =1 h ξ i,k t dK i,e,k t + X e,k t ζ i,k t ( ˜ B i,k t ) − d ˜ B i,k t + ψ i,k t B i,k t d X e,k t i ULTI-CURRENCY SETTING 25 + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t + ξ d k +1 ,k t dK d k +1 ,e,k t + ψ d k +1 ,k t B d k +1 ,k t d X e,k t + X e,k t ( B d k +2 ,k ,ht ) − ( C k t ) + dB d k +2 ,k ,ht + dF ct − C k t d X e,k t + dA e,k t − dV ct ( ϕ )Let the repo constraint (2.14) be fulfilled. Hence the dynamics of the wealth process V ( ϕ ) := V ( x, ϕ, A k , C k ) are given by dV t ( ϕ ) = ˜ V pt ( ϕ ) dB et + L X k =1 d k X i =1 h ξ i,k t dK i,e,k t − S i,k t d X e,k t i + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t + ξ d k +1 ,k t h dK d k +1 ,e,k t − S d k +1 ,k t d X e,k t i + X e,k t ( B d k +2 ,k ,ht ) − ( C k t ) + dB d k +2 ,k ,ht + dF ct − C k t d X e,k t + dA e,k t Since we have ˜ V pt ( ϕ ) = ˜ V t ( ϕ ) − X e,k t ( B et ) − ( C k t ) − we conclude. (cid:3) Pricing under funding costs and collateralization
Pricing in the absence of collateralization was discussed in Section 3.2, where we defined the hedger’sfair price ¯ p e . In this section we want to show that pricing in a multi-currency setting can be processedsimilarly to Proposition 5.1 of Bielecki and Rutkowski (2015). Let us assume that unsecured fundingborrowing and lending rates coincide for all currencies, i.e. B k ,b = B k ,l = B k for k = 1 , . . . , L andas well for repo funding accounts B i,k ,b = B i,k ,l = B i,k for any i = 1 , . . . , d k and k = 1 , . . . , L .Pricing will be analysed from the perspective of the hedger: given the contractually agreed cumu-lative stream of cashflows A k − A k , the objective of the hedger is to find p e = A k by means ofreplication, i.e. by investing according to an admissible trading strategy. Definition 5.1 ((Bielecki and Rutkowski, 2015) Definition 5.1) . Let t ∈ [0 , T ] and p et be a G t -measurable random variable. A self financing trading strategy ( V t ( x ) + p et , ϕ, A k − A k t , C k )(5.1) replicates the collateralized contract ( A k , C k ) on the interval [ t, T ] whenever V T ( V t ( x ) + p et , ϕ, A k − A k t , C k ) = V T ( x ) . (5.2) Definition 5.2 (Bielecki and Rutkowski (2015) Definition 5.2) . Let t ∈ [0 , T ] . Any G t -measurablerandom variable p et for which there exists a replicating strategy for ( A k , C k ) over [ t, T ] is calledex-dividend price at time t of the contract A e,k associated with ϕ , also denoted by S t ( x, ϕ, A k , C k ) . The following points from Bielecki and Rutkowski (2015) are important: first, any ex-dividend price p e of A e,k is also a hedger’s fair price ¯ p e for A e,k at time 0. Secondly, the ex-dividend price in generalmight depend on the initial endowment x and the choice of ϕ . However, for the sake of the presenttreatment, the ex-dividend price will be independent of the choice of x and ϕ and equivalent to thevaluation ex dividend price defined below. Recall from section (3.1) that the future value of thehedger’s initial endowment is given as V t ( x ) = xB et for any t ∈ [0 , T ]. Definition 5.3 ((Bielecki and Rutkowski, 2015) Definition 5.3) . Assume that an admissible self-financing trading strategy ( x, ϕ, A k , C k ) replicates ( A k , C k ) on [0 , T ] . Then the process ˆ p et := V t ( x, ϕ, A k , C k ) − V t ( x ) is called the valuation ex-dividend price of A k , denoted by ˆ S t ( x, ϕ, A k , C k ) The following assumptions are crucial for the next steps:(i) The assumptions of Proposition 3.4 are met. This means in particular that the exists aprobability measure Q e on (Ω , G T ), such that the processes (3.1) and (3.2), i.e. Z (0 ,t ] X e,k u d (cid:18) S i,k B i,k (cid:19) u + X e,k u B i,k u dD i,k u + d (cid:20) S i,k B i,k , X e,k (cid:21) u !! ≤ t ≤ T , X e,k t B k t B et ! ≤ t ≤ T i = 1 , . . . , d k , k = 1 . . . , L are local martingales under the assumption that the repo con-straint (2.14) is fulfilled.(ii) The collateral process C k is independent of the hedger’s portfolio ϕ . Notation 5.4.
In the sequel we will make use of the notation ˆ A c or ¯ A c to stress out the impact of thecollateral in form of cash or respectively risky asset S d k +1 ,k independent of its collateralconvention and the contractual cash flows. In the cash collateral case we have that ˆ A c ∈ { ˆ F s + A e,k , ˆ F h + A e,k } and if the hedger posts or receives risky asset collateral, ¯ A c is determinedby ¯ A c ∈ { ¯ F s + A e,k , ¯ F h + A e,k } . To ensure that the integrals over the FX-processes X e,k are well-defined, we assume those to be finite. The cash account B e remains an increasingprocess.Let Q e be a martingale measure for the discounted cumulative dividend price processes ˆ S i,cld,e,k with i ∈ { , . . . d k } k =1 ,...,L ∪ { d k + 1 } k ∈{ ,...,L } and we denote E Q e t ( · ) := E Q e ( · | G t ) as theconditional expectation of some integrable random variable for any t ∈ [0 , T ] .For any t ∈ [0 , T ] , denote the ex-dividend price by S t ( x, ϕ, A k , C k ) := S t ( A k , C k ) . Proposition 5.5.
Assume that (i)-(ii) hold and the collateralized contract ( A k , C k ) can be replicatedby an admissible trading strategy ( x, ϕ, A k , C k ) on the interval [0 , T ] and the integral over ˆ A c or ¯ A c isfinite. If the stochastic integrals with respect to (3.1) , for the indices i = 1 , . . . , d k with k = 1 , . . . , L and for the risky asset collateral case k ∈ { , . . . , L } are Q e -martingales, then its correspondingex-dividend price process S ( x, ϕ, A k , C k ) is independent of ( x, ϕ ) and equals S t ( A, C ) = − B et E Q e t Z ( t,T ] ( B eu ) − d ˆ A cu ! (5.3) for the cash collateral case and respectively S t ( A, C ) = − B et E Q e t Z ( t,T ] ( B eu ) − d ¯ A cu ! (5.4) for the risky asset collateral case.Proof. We will start with the case, where the hedger posts or receives collateral in form of sharesof risky asset S d k +1 ,k since the cash collateral case will follow immediately. Assume that thereexists an admissible trading strategy ( x, ϕ, A k , C k ) replicating the collateralized contract ( A k , C k )on the interval ( t, T ]. With (2.11), i.e. dK i,e,k t = B e,k t d ˆ S i,cld,e,k t for i, k , t fulfilling the usualconditions including the risky asset collateral S d k +1 ,k t , the dynamics of the discounted wealth process ULTI-CURRENCY SETTING 27 ˜ V ( x, ϕ, A k , C k ) := ˜ V ( ϕ ) are given by d ˜ V t ( ϕ ) = d (( B et ) − V t ( ϕ )) = V t ( ϕ ) d ( B et ) − + ( B et ) − d ¯ A ct = L X k =1 d k X i =1 ξ i,k t (cid:16) ˜ B i,k t d ˆ S i,cld,e,k t − S i,k t d X e,k t (cid:17) + L X k =1 ,k = e ψ ,k t B et d (cid:18) B k X e,k B e (cid:19) t + ξ d k +1 ,k t (cid:16) ˜ B d k +1 ,k t d ˆ S d k +1 ,k ,cld,e,k t − S d k +1 ,k t d X e,k t (cid:17) + ( B et ) − d ¯ A ct (5.5)by using Ito’s formula and the dynamics of the hedger’s wealth process (4.32). By definition, thisprocess is self financing and fulfils the repo constraint (2.14). We fix some t ∈ [0 , T ). By assumption,there exists a replicating trading strategy( V t ( x ) + p et , ϕ, A k − A k t , C k )in the sense of Definition 5.1 fulfilling V T ( V t ( x ) + p et , ϕ, A k − A k t , C k ) = V T ( x ) and hence V T ( x, ϕ, A k , C k ) = V T ( x + ( B eT ) − p eT ) = ( x + ( B eT ) − p eT ) B eT leading to − ( B et ) − p et = ( B eT ) − V eT ( x, ϕ, A k , C k ) − ( B et ) − V et ( x, ϕ, A k , C k )= L X k =1 d k X i =1 Z ( t,T ] ξ i,k u (cid:16) ˜ B i,k u d ˆ S i,cld,e,k u − S i,k u d X e,k u (cid:17) + L X k =1 ,k = e Z ( t,T ] ψ ,k u B eu d (cid:18) B k X e,k B e (cid:19) u + Z ( t,T ] ξ d k +1 ,k u (cid:16) ˜ B d k +1 ,k u d ˆ S i,cld,e,k u − S d k +1 ,k u d X e,k u (cid:17) + Z ( t,T ] ( B eu ) − d ¯ A cu . by using that˜ B i,k t d ˆ S i,cld,e,k t − S i,k t d X e,k t = X e,k t d (cid:18) S i,k B i,k (cid:19) t + X e,k t B i,k t dD i,k t + d (cid:20) S i,k B i,k , X e,k (cid:21) t (5.6)for all indices mentioned above as a direct consequence out of equation (3.3) and (4.22). Since theintegrals with respect to (3.1) and (3.2) are true Q e -martingales for i = 1 , . . . , d k , k = 1 , . . . , L and k ∈ { , . . . , L } , the ex-dividend price of the collateralized contract ( A k , C k ) can be derived by S t ( A k , C k ) = − B et E Q e L X k =1 d k X i =1 Z (0 ,T ] ξ i,k u X e,k u d (cid:18) S i,k B i,k (cid:19) u + X e,k u B i,k u dD i,k u + d (cid:20) S i,k B i,k , X e,k (cid:21) u ! − L X k =1 d k X i =1 Z (0 ,t ] ξ i,k u X e,k u d (cid:18) S i,k B i,k (cid:19) u + X e,k u B i,k u dD i,k u + d (cid:20) S i,k B i,k , X e,k (cid:21) u ! | G t − B et E Q e Z (0 ,T ] ξ d k +1 ,k u X e,k u d S d k +1 ,k B d k +1 ,k ! u + X e,k u B d k +1 ,k u dD d k +1 ,k u + d " S d k +1 ,k B d k +1 ,k , X e,k u ! − Z (0 ,t ] ξ d k +1 ,k u X e,k u d S d k +1 ,k B d k +1 ,k ! u + X e,k u B d k +1 ,k u dD d k +1 ,k u + d " S d k +1 ,k B d k +1 ,k , X e,k u ! | G t ! − B et E Q e Z (0 ,T ] ( B eu ) − d ¯ A cu − Z (0 ,t ] ( B eu ) − d ¯ A cu | G t ! = − B et E Q e Z ( t,T ] ( B eu ) − d ¯ A cu | G t ! = − B et E Q e t Z ( t,T ] ( B eu ) − d ¯ A cu ! for any t ∈ [0 , T ] by using the martingale and measurability properties and is independent of ( x, ϕ ).By following the same steps for the cash collateral case, (5.3) follows immediately. (cid:3) Remark . Note that in case of absolute continuity of all repo accounts, the ex-dividend price process S t ( A k , C k ) for any t ∈ [0 , T ] is given as follows:1) Cash collateral under segregation: By using equation (4.20), we derive S t ( A k , C k ) = − B et E Q e t "Z ( t,T ] ( B eu ) − dA e,k u − B et E Q e t "Z ( t,T ] ( B eu ) − d ˆ F su = − B et E Q e t "Z ( t,T ] ( B eu ) − dA e,k u − B et E Q e t "Z ( t,T ] h(cid:16) r d k +2 ,k ,su ( C k u ) + − r d k +3 ,k u ( C k u ) − + r c,k ,lu ( C k u ) − − r c,k ,bu ( C k u ) + i X e,k u du − C k u d X e,k u (cid:17) B eu (cid:21) = − B et E Q e t "Z ( t,T ] ( B eu ) − dA e,k u − B et E Q e t "Z ( t,T ] B eu h(cid:16) r d k +2 ,k ,su − r c,k ,bu (cid:17) ( C k u ) + − (cid:16) r d k +3 ,k u − r c,k ,lu (cid:17) ( C k u ) − i X e,k u du − Z ( t,T ] C k u B eu d X e,k u (5.7) 2) Cash collateral under rehypothecation: With similar calculations by using equation (4.27), weget S t ( A k , C k ) = − B et E Q e t "Z ( t,T ] ( B eu ) − dA e,k u − B et E Q e t "Z ( t,T ] B eu h(cid:16) r eu − r c,k ,bu (cid:17) ( C k u ) + − (cid:16) r d k +3 ,k u − r c,k ,lu (cid:17) ( C k u ) − i X e,k u du − Z ( t,T ] C k u B eu d X e,k u (5.8) 3) Risky asset collateral under segregation: By using equation (4.33), we derive S t ( A k , C k ) = − B et E Q e t "Z ( t,T ] ( B eu ) − dA e,k u − B et E Q e t "Z ( t,T ] B eu h(cid:16) r d k +2 ,k ,su − r c,k ,bu (cid:17) ( C k u ) + − (cid:16) r d k +1 ,k u − r c,k ,lu (cid:17) ( C k u ) − i X e,k u du − Z ( t,T ] C k u B eu d X e,k u (5.9) 4) Risky asset collateral under rehypothecation: Analogously, replacing index s by h , we receive S t ( A k , C k ) = − B et E Q e t "Z ( t,T ] ( B eu ) − dA e,k u − B et E Q e t "Z ( t,T ] B eu h(cid:16) r d k +2 ,k ,hu − r c,k ,bu (cid:17) ( C k u ) + − (cid:16) r d k +1 ,k u − r c,k ,lu (cid:17) ( C k u ) − i X e,k u du − Z ( t,T ] C k u B eu d X e,k u (5.10) ULTI-CURRENCY SETTING 29 Diffusion models
The aim of the present section is to provide concrete examples concerning the valuation of crosscurrency products. The diffusion model we present can be thought of as a footprint to constructcross currency simulation models for the computation of various valuation adjustments known in theliterature under the acronym of xVA.We will assume that all cash accounts are absolutely continuous with respect to the Lebesguemeasure, so that they can be written in the form dB · t = r · t B · t dt for some G -adapted RCLL processes r · . We also assume, in line with the previous section, that unsecured funding borrowing and lendingrates coincide for all currencies, i.e. B k ,b = B k ,l = B k for k = 1 , . . . , L and as well for repo fundingaccounts B i,k ,b = B i,k ,l = B i,k for any i = 1 , . . . , d k and k = 1 , . . . , L . In each currency area k = 1 , . . . , L we postulate the existence of d k traded risky assets S i,k . For the collateral currency k we also postulate the existence of the traded assets S d k +1 ,k and S d k +2 ,k . Finally, we also assumethat the repo constraint (2.14) is satisfied.6.1. Model Dynamics and martingale measure.
We construct the model and the domestic mar-tingale measure Q e . In line with the single currency model of Bielecki and Rutkowski (2015) wepostulate the following dynamics for each asset S i,k under the physical measure P . Each risky assetevolves according to dS i,k t = S i,k t (cid:16) µ S i,k t dt + σ S i,k t dW S i,k , P t (cid:17) (6.1)for k = 1 , . . . , L , k ∈ { , . . . , L } , i = 1 , . . . , d k for the hedging assets and d k + 1 , d k + 2 forthe collateral assets. The drift functions µ S i,k are bounded while the volatility functions σ S i,k arestrictly positive and bounded. The Brownian motions W S i,k , P are correlated by means of correlationfunctions ρ S i,k ,S i ′ ,k ′ such that − ≤ ρ S i,k ,S i ′ ,k ′ ≤
1. The dividend processes of the risky assets aregiven by D i,k t = R t κ i,k u S i,k u du , where the bounded processes κ i,k represent dividend yields. We alsoassume that exchange rates evolve according to d X e,k t = X e,k t (cid:16) µ X e,k t dt + σ X e,k t dW X e,k , P t (cid:17) (6.2)with analogous assumptions on drifts and volatilities. We allow for correlations among exchange rates,denoted by ρ X e,k , X e,k ′ and correlations among exchange rates and risky assets, denoted by ρ X e,k ,S i,k ′ ,for k , k ′ = 1 , . . . , L and i = 1 , . . . , d k . The correlation coefficient functions are such that the resultingcorrelation matrix is positive semi-definite.The following generalizes Lemma 5.2 in Bielecki and Rutkowski (2015). Lemma 6.1.
Under the measure Q e the following holds. (i) The dynamics of domestic assets S i,e are of the form dS i,et = S i,et (cid:16) ( r i,k t − κ i,k t ) dt + σ S i,e t dW S i,e , Q e t (cid:17) (6.3) Equivalently d ˆ S i,cld,et = ˆ S i,cld,et σ S i,e t dW S i,e , Q e t and dK i,e,et = dS i,et − r i,et S i,et dt + κ i,et S i,et dt = S i,et σ S i,e t dW S i,e , Q e t are local martingales under Q e . (ii) The dynamics of foreign assets S i,k are of the form dS i,k t = S i,k t (cid:16) ( r i,k t − κ i,k t − ρ S i,k , X e,k t σ S i,k t σ X e,k t ) dt + σ S i,k t dW S i,k , Q e t (cid:17) (6.4) and the processes dK i,e,k t − S i,k t d X e,k t = X e,k t S i,k t σ S i,k t dW S i,k , Q e t are local martingales under Q e . (iii) The dynamics of all exchange rates are of the form d X e,k t = X e,k t (cid:16) ( r et − r k t ) dt + σ X e,k t dW X e,k , Q e t (cid:17) (6.5) and the processes d X e,k t B k t B et ! = X e,k t B k t B et σ X e,k t dW X e,k , Q e t are local martingales under Q e .Proof. The statement on the domestic assets corresponds to that of Lemma 5.2 in Bielecki and Rutkowski(2015) and thus the proof is omitted. Let us concentrate on the foreign assets. Under Q e , the process(3.1) is a local martingale. The quadratic covariation between the repo-discounted asset price S i,k B i,k and the exchange rate is (cid:20) S i,k B i,k , X e,k (cid:21) t = (cid:28) S i,k B i,k , X e,k (cid:29) t = Z (0 ,t ] ρ S i,k , X e,k u σ S i,k u σ X e,k u S i,k u X e,k u B i,k u du We can write in explicit form dK i,e,k t − S i,k t d X e,k t = B i,k t X e,k t d (cid:18) S i,k B i,k (cid:19) t + X e,k t B i,k u dD i,k t + d (cid:20) S i,k B i,k , X e,k (cid:21) t ! = B i,k t X e,k t B i,k t S i,k t (cid:16) µ S i,k t dt + σ S i,k t dW S i,k , P t (cid:17) − r i,k t S i,k t X e,k t B i,k t dt + κ i,k t S i,k t X e,k t B i,k t dt + ρ S i,k , X e,k t σ S i,k t σ X e,k t S i,k t X e,k t B i,k t dt ! = X e,k t S i,k t (cid:16) ( µ S i,k t − r i,k t + κ i,k t + ρ S i,k , X e,k t σ S i,k t σ X e,k t ) dt + σ S i,k t dW S i,k , P t (cid:17) . If the process dW S i,k , Q e t := dW S i,k , P t + 1 σ S i,k t (cid:16) µ S i,k t − r i,k t + κ i,k t + ρ S i,k , X e,k t σ S i,k t σ X e,k t (cid:17) dt is a Brownian motion under Q e then dK i,e,k t − S i,k t d X e,k t = X e,k t S i,k t σ S i,k t dW S i,k , Q e t is a local martingale under Q e . Finally, for the dynamics of the asset we obtain dS i,k t = S i,k t (cid:16) µ S i,k t dt + σ S i,k t dW S i,k , P t (cid:17) = S i,k t (cid:16) ( r i,k t − κ i,k t − ρ S i,k , X e,k t σ S i,k t σ X e,k t ) dt + σ S i,k t dW S i,k , Q e t (cid:17) which completes the proof of the second statement. ULTI-CURRENCY SETTING 31
For the exchange rates we proceed analogously. Under Q e we require that the process (3.2) is alocal martingale. The computation is straightforward. We have d X e,k t B k t B et ! = X e,k t B k t B et (cid:16) ( µ X e,k t + r k t − r et ) dt + σ X e,k t dW X e,k , P t (cid:17) If the process dW X e,k , Q e t = dW X e,k , P t + 1 σ X e,k t (cid:16) µ X e,k t + r k t − r et (cid:17) dt is a Brownian motion under Q e then we obtain a local martingale and the resulting dynamics of theexchange rates are given by d X e,k t = X e,k t (cid:16) ( r et − r k t ) dt + σ X e,k t dW X e,k , Q e t (cid:17) which completes the proof. (cid:3) The dynamics we obtained above are of independent interest: they provide a sound framework forthe construction of hybrid models for a multitude of risky asset in a multi currency setting. Suchmodels can be used for the Monte Carlo simulation of risk factors that affect a portfolio of contingentclaims. Such high-dimensional hybrid models for a multitude of risk factors constitute the marketstandard for the computation of valuation adjustments (xVA) for a whole portfolio of claims betweenthe hedger and the counterparty. A by product of our valuation framework is then a sound derivationof multi-currency hybrid models for the generation of exposure profiles for counterparty credit risk.Hybrid models for xVA are presented in Sokol (2014), Green (2015), Lichters et al. (2015).The basic model above can be extended in multiple directions: our choice for the driving processesis rather simplicistic and mainly meant to provide an illustration of how one can construct a crosscurrency hybrid model in a multi curve framework. One natural stream of generalization is to considermore general driving processes. One possibility is to extend the market by introducing instrumentswhich are by definition fully collateralized, i.e. natively collateralized assets such as OIS bonds and(textbook) FRAs as in Cuchiero et al. (2016) and Cuchiero et al. (2019). The resulting model wouldallow for the joint evolution of interbank spreads, overnight rates, foreign exchange and risky assets.We leave such extensions to future research.6.2.
Wealth dynamics with collateral.
We can now provide explicit expressions for the wealthdynamics under any collateralization schemes thanks to Lemma 6.1. We assume again, as in Remark5.6 that all cash accounts are absolutely continuous. In line with Section 5, we assume for the momentthat the collateral C k is exogenously given.6.2.1. Cash Collateral under segregation.
In Proposition 4.10 we have that (4.19) takes now the form dV t ( ϕ ) = V t ( ϕ ) r et dt + L X k =1 d k X i =1 ξ i,k t X e,k t S i,k t σ S i,k t dW S i,k , Q e t + L X k =1 ,k = e ψ ,k t B k t X e,k t σ X e,k t dW X e,k , Q e t + dA e,k t + d ˆ F st (6.6)with ˆ F st = Z t (cid:16) r d k +2 ,k ,su − r c,k ,bu (cid:17) ( C k u ) + X e,k u du − Z t (cid:16) r d k +3 ,k u − r c,k ,lu (cid:17) ( C k u ) − X e,k u du − Z (0 ,t ] C k u d X e,k u Cash Collateral under rehypothecation.
In Proposition 4.11 we have that (4.26) takes now theform dV t ( ϕ ) = V t ( ϕ ) r et dt + L X k =1 d k X i =1 ξ i,k t X e,k t S i,k t σ S i,k t dW S i,k , Q e t + L X k =1 ,k = e ψ ,k t B k t X e,k t σ X e,k t dW X e,k , Q e t + dA e,k t + d ˆ F ht (6.7)with ˆ F ht = Z t (cid:16) r eu − r c,k ,bu (cid:17) ( C k u ) + X e,k u du − Z t (cid:16) r d k +3 ,k u − r c,k ,lu (cid:17) ( C k u ) − X e,k u du − Z (0 ,t ] C k u d X e,k u We observe that, for the case of cash collateral, the difference between segregation and rehypothecationis reflected only by the presence of r d k +2 ,k ,s and r e respectively.6.2.3. Risky asset collateral.
Risky asset collateral was treated in Proposition 4.12 both under segrega-tion and rehypothecation. In the diffusive setting of the present section (4.32) under rehypothecationtakes now the form dV t ( ϕ ) = V t ( ϕ ) r et dt + L X k =1 d k X i =1 ξ i,k t X e,k t S i,k t σ S i,k t dW S i,k , Q e t + L X k =1 ,k = e ψ ,k t B k t X e,k t σ X e,k t dW X e,k , Q e t + ( S d k +1 ,k t ) − ( C k t ) − X e,k t S d k +1 ,k t σ S dk ,k t dW S dk ,k , Q e t + dA e,k t + d ¯ F ht (6.8)with ¯ F ht = Z t (cid:16) r d k +2 ,k ,hu − r c,k ,bu (cid:17) ( C k u ) + X e,k u du − Z t (cid:16) r d k +1 ,k u − r c,k ,lu (cid:17) ( C k u ) − X e,k u du − Z (0 ,t ] C k u d X e,k u . The case of segregation is obtained by simply replacing r d k +2 ,k ,h with r d k +2 ,k ,s .6.3. Pricing with exogenous collateral.
We specialize the findings of Proposition 5.5 to the dif-fusive setting of the present section. In line with Bielecki and Rutkowski (2015) we assume that theprocess A e,k is adapted to the filtration F S, X , generated by all risky assets and all exchange rates. A c,k is a shorthand for the processes employed in Proposition 5.5. In the following we assume thatall conditional expectations considered in the sequel are well defined for all t ∈ [0 , T ]. Proposition 6.2.
In the diffusion model, a collateralized contract ( A k , C k ) with predetermined col-lateral process C k can be replicated by an admissible trading strategy. The ex-dividend price S ( A, C ) ULTI-CURRENCY SETTING 33 satisfies, for very t ∈ [0 , T ] S t ( A k , C k ) = − B et E Q e t Z ( t,T ] ( B eu ) − d ˆ A cu ! Proof.
The present result corresponds to Bielecki and Rutkowski (2015) Proposition 5.3, where thepredictable representation property of the Brownian filtration is employed. (cid:3)
At this point, we would like to show that the formulas we developed allow us to link the generalframework of the present paper with the findings of Moreni and Pallavicini (2017), Fujii et al. (2011),Fujii et al. (2010a), Fujii et al. (2010b), Fujii et al. (2012). Let us recall that process A e,k satisfies dA e,k = X e,k t dA k t . Corollary 6.3.
In the diffusion model, we have the following pricing formulas for a collateralizedcontract ( A k , C k ) with predetermined collateral process C k . Cash collateral under segregation: S t ( A k , C k ) = − B et E Q e t "Z ( t,T ] X e,k u B eu dA k u − B et E Q e t "Z ( t,T ] h(cid:16) r d k +2 ,k ,su − r c,k ,bu (cid:17) ( C k u ) + − (cid:16) r d k +3 ,k u − r c,k ,lu (cid:17) ( C k u ) − i X e,k u B eu du − Z ( t,T ] C k u X e,k u B eu ( r eu − r k u ) du (6.9) 2) Cash collateral under rehypothecation: S t ( A k , C k ) = − B et E Q e t "Z ( t,T ] X e,k u B eu dA k u − B et E Q e t "Z ( t,T ] h(cid:16) r eu − r c,k ,bu (cid:17) ( C k u ) + − (cid:16) r d k +3 ,k u − r c,k ,lu (cid:17) ( C k u ) − i X e,k u B eu du − Z ( t,T ] C k u X e,k u B eu ( r eu − r k u ) du (6.10)3) Risky asset collateral under segregation: S t ( A k , C k ) = − B et E Q e t "Z ( t,T ] X e,k u B eu dA k u − B et E Q e t "Z ( t,T ] h(cid:16) r d k +2 ,k ,su − r c,k ,bu (cid:17) ( C k u ) + − (cid:16) r d k +1 ,k u − r c,k ,lu (cid:17) ( C k u ) − i X e,k u B eu du − Z ( t,T ] C k u X e,k u B eu ( r eu − r k u ) du (6.11)4) Risky asset collateral under rehypothecation: S t ( A k , C k ) = − B et E Q e t "Z ( t,T ] X e,k u B eu dA k u − B et E Q e t "Z ( t,T ] h(cid:16) r d k +2 ,k ,hu − r c,k ,bu (cid:17) ( C k u ) + − (cid:16) r d k +1 ,k u − r c,k ,lu (cid:17) ( C k u ) − i X e,k u B eu du − Z ( t,T ] C k u X e,k u B eu ( r eu − r k u ) du (6.12) Proof.
The proof directly follows from Remark 5.6 and by observing that E Q e t "Z ( t,T ] C k u B eu d X e,k u = E Q e t "Z ( t,T ] C k u X e,k u B eu ( r eu − r k u ) du (cid:3) The existing literature focuses on the case of cash collateral with rehypothecation, for exampleMoreni and Pallavicini (2017) in their Proposition 1 obtain the analogue of (6.10). Also different borrowing and lending rates are not considered. Our Corollary 6.3 generalises most results availablein the literature since we allow for different combinations of collateralization covenants. The distinctivefeature of pricing formulas, when collateral can be posted in different currencies, lies in the further”correction” term which is proportional to the drift of the FX rate and the we could compute explicitlyin the present diffusive setting. As a final illustration, let us stress that each of the four pricing formulasabove nests the three following valuation formulas.
Remark . In the case of cash collateral with rehypothecation we have the following special cases of(6.10).2.a) Domestic cashflows collateralized in domestic currency. This corresponds to the case k = k = e and we obtain S t ( A e , C e ) = − B et E Q e t "Z ( t,T ] B eu dA eu − B et E Q e t "Z ( t,T ] h(cid:16) r eu − r c,e,bu (cid:17) ( C eu ) + − (cid:16) r d e +3 ,eu − r c,e,lu (cid:17) ( C eu ) − i B eu du (cid:21) (6.13) This is the case already treated both in Piterbarg (2010) Bielecki and Rutkowski (2015) amongothers.2.b) Domestic cashflows collateralized in foreign currency. This corresponds to the case k = e and k = e and we obtain S t ( A e , C k ) = − B et E Q e t "Z ( t,T ] B eu dA eu − B et E Q e t "Z ( t,T ] h(cid:16) r eu − r c,k ,bu (cid:17) ( C k u ) + − (cid:16) r d k +3 ,k u − r c,k ,lu (cid:17) ( C k u ) − i X e,k u B eu du − Z ( t,T ] C k u X e,k u B eu ( r eu − r k u ) du (6.14)2.c) Foreign cashflows collateralized in domestic currency. This corresponds to the case k = e and k = e and we obtain S t ( A k , C e ) = − B et E Q e t "Z ( t,T ] X e,k u B eu dA k u − B et E Q e t "Z ( t,T ] h(cid:16) r eu − r c,e,bu (cid:17) ( C k u ) + − (cid:16) r d e +3 ,eu − r c,e,lu (cid:17) ( C eu ) − i B eu du (cid:21) (6.15)6.4. Pricing with endogenous collateral.
We treat the case where the collateral depends on themarked-to-market value of the contract. We assume for simplicity that the initial endowment is zero,i.e. x = 0. We assume again that all interest rates are bounded and we let the filtration F be of theform F = F S, X , i.e. the filtration is generated by all risky assets and exchange rates. In line with theprevious section, the contract A e,k is adapted to the filtration F S, X . The collateral account is nowgiven by C k t = (1 + δ t ) ( − V t ( ϕ )) + X e,k t − (1 + δ t ) ( − V t ( ϕ )) − X e,k t , (6.16)where the bounded, RCLL F S, X -adapted processes δ , δ represent haircuts. The fact that now C k depends on V implies that the pricing equation has a recursive nature and hence is to be treated asa BSDE.We consider the case of cash collateral with rehypothecation and we further introduce the simplifi-cation r d k +3 ,k = r e , r c,k ,bu = r c,k ,lu = r c,k u . Concerning the drift of the exchange rate X e,k we can ULTI-CURRENCY SETTING 35 define the cross currency basis q e,k via r et − r k t = r c,et − r c,k t + q e,k t , where r c,e and r c,k are the collateral rates under the domestic and the k currency. Obviously wehave q e,e ≡ q e,k = − q k ,e . Expressing the dynamics of the exchange rate in terms for the crosscurrency basis in the present diffuse setting means that we write d X e,k t = X e,k t (cid:16) ( r c,et − r c,k t + q e,k t ) dt + σ X e,k t dW X e,k , Q e t (cid:17) . (6.17)Under the preceding assumptions (6.7) takes the form dV t ( ϕ ) = V t ( ϕ ) r et dt + L X k =1 d k X i =1 ξ i,k t X e,k t S i,k t σ S i,k t dW S i,k , Q e t + L X k =1 ,k = e ψ ,k t B k t X e,k t σ X e,k t dW X e,k , Q e t + dA e,k t + (cid:16) r et − r c,k t (cid:17) (cid:0) (1 + δ t ) ( − V t ( ϕ )) + − (1 + δ t ) ( − V t ( ϕ )) − (cid:1) dt − (cid:0) (1 + δ t ) ( − V t ( ϕ )) + − (1 + δ t ) ( − V t ( ϕ )) − (cid:1) (cid:16) r c,et − r c,k t + q e,k t (cid:17) dt − (cid:0) (1 + δ t ) ( − V t ( ϕ )) + − (1 + δ t ) ( − V t ( ϕ )) − (cid:1) σ X e,k t dW X e,k , Q e t , (6.18)where we substituted also the dynamics the exchange rate expressed via the cross currency basis.We view the expression above as a BSDE where the controls are given by the processes Z i,k = ξ i,k , Z ,k = ψ ,k t and with the additional control Z ,k = − (cid:0) (1 + δ ) ( − V ( ϕ )) + − (1 + δ ) ( − V ( ϕ )) − (cid:1) andzero terminal condition. We introduce: • The subspace of all R d -valued, F S, X -adapted processes X such that E Q e (cid:20)Z T k X t k dt (cid:21) < ∞ , (6.19) denoted by H ,d ( Q e ) . We set H ( Q e ) := H , ( Q e ) . • The subspace of all R d -valued, continuous F S, X -adapted processes X such that E Q e " sup t ∈ [0 ,T ] k X t k < ∞ , (6.20) denoted by S ,d ( Q e ) . We set S ( Q e ) := S , ( Q e ) . We have the following pricing result.
Proposition 6.5.
Assume that A e,k ∈ S ( Q e ) . Then the BSDE (6.18) with zero terminal conditionadmits a unique solution with V ( ϕ ) ∈ S ( Q e ) and all controls in the space H ( Q e ) . Also, the collat-eralized contract A e,k with collateral specification (6.16) can be replicated on [ t, T ] by an admissibletrading strategy ϕ and the price admits the representation S t ( A k , C k ) = − B et E Q e t "Z ( t,T ] X e,k u B eu dA k u − B et E Q e t "Z ( t,T ] (cid:16) r eu − r c,eu − q e,k u (cid:17) (cid:0) (1 + δ u ) ( − V u ( ϕ )) + − (1 + δ u ) ( − V u ( ϕ )) − (cid:1) B eu du . Proof.
Existence and uniqueness to (6.18) follow from Nie and Rutkowski (2016) Theorem 4.1 moduloour assumption that A e,k ∈ S ( Q e ). The rest of the claim is clear from our previous results. (cid:3) It is interesting to study the case of perfect collateralization. This can be immediately obtainedfrom our formulas by setting δ t = δ t = 0 d Q e ⊗ dt -a.s.. We observe that the BSDE (6.18) takes nowthe much simpler form dV t ( ϕ ) = V t ( ϕ )( r e,ct + q e,k t ) dt + L X k =1 d k X i =1 ξ i,k t X e,k t S i,k t σ S i,k t dW S i,k , Q e t + L X k =1 ,k = e ψ ,k t B k t X e,k t σ X e,k t dW X e,k , Q e t + dA e,k t + V t ( ϕ ) σ X e,k t dW X e,k , Q e t , (6.21)from which, with the help of Proposition 6.5, we immediately obtain the following valuation formulafor perfectly collateralized claims, namely S t ( A k , C k ) = − E Q e t "Z ( t,T ] e − R ut r c,es + q e,k s ds X e,k u dA k u , (6.22)from which we can obtain also the following special cases.2.a) Domestic cashflows collateralized in domestic currency. This corresponds to the case k = k = e and we obtain S t ( A e , C e ) = − E Q e t "Z ( t,T ] e − R ut r c,es ds dA eu , (6.23) so we discount using the domestic collateral rate.2.b) Domestic cashflows collateralized in foreign currency. This corresponds to the case k = e and k = e and we obtain S t ( A e , C k ) = − E Q e t "Z ( t,T ] e − R ut r c,es + q e,k s ds dA eu , (6.24) so that the foreign collateralization results in the appearance of the cross currency basis inthe discount factor.2.c) Foreign cashflows collateralized in domestic currency. This corresponds to the case k = e and k = e and we obtain S t ( A k , C e ) = − E Q e t "Z ( t,T ] e − R ut r c,es ds X e,k u dA k u . (6.25) References
Antonov, A., Bianchetti, M., and Mihai, I. (2015). Funding value adjustment for general financialinstruments: Theory and practice.
Risk .Biagini, F., Gnoatto, A., and Oliva, I. (2019). Pricing of counterparty risk and funding with CSAdiscounting, portfolio effects and initial margin. arXiv e-prints , page arXiv:1905.11328.Bianchetti, M. (2010). Two curves, one price.
Risk magazine , pages 74–80.Bielecki, T. and Rutkowski, M. (2015). Valuation and hedging of contracts with funding costs andcollateralization.
SIAM J. Finan. Math. , 6(1):594655.Brigo, D., Buescu, C., Pallavicini, A., and Liu, Q. (2015). A note on the self-financing conditionfor funding, collateral and discounting.
International Journal of Theoretical and Applied Finance ,18(2):1550011.
ULTI-CURRENCY SETTING 37
Brigo, D. and Pallavicini, A. (2014). Nonlinear consistent valuation of ccp cleared or csa bilateral tradeswith initial margins under credit, funding and wrong-way risks.
Journal of Financial Engineering ,01(01):1450001.Castagna, A. (2011). Pricing of derivatives contracts under collateral agreements: Liquidity andfunding value adjustments.
SSRN Electronic Journal .Cesari, G., Aquilina, J., Charpillon, N., Filipovic, Z., Lee, G., and Manda, I. (2009).
Modelling,Pricing, and Hedging Counterparty Credit Exposure: A Technical Guide . Springer Finance. Springer.Cr´epey, S. (2015a). Bilateral counterparty risk under funding constraintspart i: Pricing.
MathematicalFinance , 25(1):1–22.Cr´epey, S. (2015b). Bilateral counterparty risk under funding constraintspart ii: Cva.
MathematicalFinance , 25(1):23–50.Cr´epey, S., Grbac, Z., Nguyen, H., and Skovmand, D. (2015). A L´evy HJM multiple-curve model withapplication to CVA computation.
Quantitative Finance , 15(3):401–419.Cuchiero, C., Fontana, C., and Gnoatto, A. (2016). A general HJM framework for multiple yield curvemodelling.
Finance and Stochastics , 20(2):267–320.Cuchiero, C., Fontana, C., and Gnoatto, A. (2019). Affine multiple yield curve models.
MathematicalFinance , 29(2):568–611.Fujii, M., Shimada, A., and Takahashi, A. (2010a). Note on construction of multiple swap curves withand without collateral. Preprint (available at http://ssrn.com/abstract=1440633 ).Fujii, M., Shimada, A., and Takahashi, A. (2010b). On the term structure of inter-est rates with basis spreads, collateral and multiple currencies. Preprint (available at http://ssrn.com/abstract=1556487 ).Fujii, M., Shimada, A., and Takahashi, A. (2011). A market model of interest rates with dynamicbasis spreads in the presence of collateral and multiple currencies.
Wilmott , 54:61–73.Fujii, M., Shimada, A., and Takahashi, A. (2012). A survey on modeling and analysis of basis spreads.In Takahashi, A., Muromachi, Y., and Nakaoka, H., editors,
Recent Advances In Financial Engineer-ing 2011 . World Scientific, Singapore. Preprint (available at http://ssrn.com/abstract=1520619 ).Grbac, Z., Papapantoleon, A., Schoenmakers, J., and Skovmand, D. (2015). Affine LIBOR modelswith multiple curves: Theory, examples and calibration.
Siam Journal on Financial Mathematics ,6(1):984–1025.Green, A. (2015).
XVA: Credit, Funding and Capital Valuation Adjustments . Wiley Finance. JohnWiley and Sons Inc, Chichester.Henrard, M. (2007). The irony in the derivatives discounting.
Wilmott , pages 92–98.Henrard, M. (2014).
Interest Rate Modelling in the Multi-curve Framework . Palgrave Macmillan.Lichters, R., Stamm, R., and Gallagher, D. (2015).
Modern Derivatives Pricing and Credit ExposureAnalysis: Theory and Practice of CSA and XVA Pricing, Exposure Simulation and Backtesting .Applied Quantitative Finance. Palgrave Macmillan, first edition.Mercurio, F. (2010). Modern Libor market models: using different curves for projecting rates anddiscounting.
International Journal of Theoretical and Applied Finance , 13(1):113–137.Moreni, N. and Pallavicini, A. (2014). Parsimonious HJM modelling for multiple yield-curve dynamics.
Quantitative Finance , 14(2):199–210.Moreni, N. and Pallavicini, A. (2017). Derivative pricing with collateralization and FX market dislo-cations.
International Journal of Theoretical and Applied Finance , 20(06):1750040.Nie, T. and Rutkowski, M. (2016). Bsdes driven by multidimensional martingales and their applicationsto markets with funding costs.
Theory Probab. Appl. , 60(4):604–630.
Pallavicini, A., Perini, D., and Brigo, D. (2011). Funding Valuation Adjustment: a consistent frame-work including CVA, DVA, collateral,netting rules and re-hypothecation.
ArXiv e-prints , pagearXiv:1112.1521.Pallavicini, A., Perini, D., and Brigo, D. (2012). Funding, Collateral and Hedging: uncovering themechanics and the subtleties of funding valuation adjustments. arXiv e-prints , page arXiv:1210.3811.Piterbarg, V. (2010). Funding beyond discounting: collateral agreements and derivatives pricing.
RiskMagazine , 2:97–102.Piterbarg, V. (2012). Cooking with collateral.
Risk Magazine , 2:58–63.Sokol, A. (2014).
Long-Term Portfolio Simulation: For XVA, Limits, Liquidity and Regulatory Capital .Risk Books, London. (Alessandro Gnoatto)
University of Verona, Department of Economics,via Cantarane 24, 37129 Verona, Italy
E-mail address , Alessandro Gnoatto: [email protected] (Nicole Seiffert)
Mathematisches Institut der LMU M¨unchen,Theresienstr. 39, 80333 M¨unchen, Germany
E-mail address , Nicole Seiffert:, Nicole Seiffert: