Valuation of Collateralized Total Return Swaps...Collateralization and FVA for Total Return Swaps C. Fries, M. Lichtner nously prescribed value process. In this situation there is

Electronic copy available at: http://ssrn.com/abstract=2444452

Collateralization and FundingValuation Adjustments (FVA)

for Total Return Swaps

Christian P. [email protected]

Mark [email protected]

May 31, 2014

Version 0.9

Abstract

In this paper we consider the valuation of total return swaps (TRS). Since a totalreturn swap is a collateralized derivative referencing the value process of anuncollateralized asset it is in general not possible that both counter parties agreeon a unique value. Consequently it is not possible to have cash collateralizationof the total return swap matching each counterparts valuation. The total returnswap is a collateralized derivative with a natural funding valuation adjustment.

We develop a model for valuation and risk management of TRS where weassume that collateral is posted according to the mid average (or convex com-bination) of the valuations performed by both counterparts. This results in acoupled and recursive system of equations for the valuation of the TRS.

The main result of the paper is that we can provide explicit formulas forthe collateral and the FVA, eliminating the recursiveness which is naturallyencountered in such formulas, by assuming a natural collateralization scheme.

Although the paper focuses on total return swaps, the principles developedhere are generally applicable in situations where collateralized assets referenceuncollaterlized or partially collateralized underlings.

1

mailto:[email protected]

mailto:[email protected]

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Electronic copy available at: http://ssrn.com/abstract=2444452

Collateralization and FVA for Total Return Swaps C. Fries, M. Lichtner

Contents

1 Introduction 31.1 Total Return Swaps . . . . . . . . . . . . . . . . . . . . . . . 31.2 Lack of Perfect Collateralization . . . . . . . . . . . . . . . . 31.3 Collateralization Schemes . . . . . . . . . . . . . . . . . . . 4

2 Basic Model Setup 52.1 Defintion of the Total Return Swap . . . . . . . . . . . . . . . 52.2 Valuation of the Total Return Swap . . . . . . . . . . . . . . . 5

2.2.1 Valuation of the Plain LIBOR Flows . . . . . . . . . . 62.2.2 Valuation of the Total Return of the Underlying . . . . 62.2.3 Valuation of Collateralization . . . . . . . . . . . . . 6

3 Valuation of Derivatives with Exogenous Collateral Process 10

4 Collateralization Schemes 124.1 Fair partial collateralization . . . . . . . . . . . . . . . . . . . 12

4.1.1 Fair partial collateralization under a simple jump-diffusionmodel for the bond dynamics . . . . . . . . . . . . . . 16

4.1.2 Continuously Resetting Bond Notional . . . . . . . . 204.2 Simplified Explicit Collateral (Repo Style) . . . . . . . . . . . 21

5 Appendix 22

References 24

c©2014 Christian Fries, Mark Lichtner 2 Version 0.9 (20140531)http://www.christianfries.com/finmath/trs

http://www.christianfries.com/finmath/trs


1 Introduction

1.1 Total Return Swaps

A total return swap (TRS) exchanges the cash flows or total return of an un-collateralized underlying asset M against plain vanilla floating rate cash flows(LIBOR plus deal spread) with notional N . At maturity T , typically five to tenyears, the final underlying market valueM(T ) is exchanged against the TRSnotional N , see the top two legs in Figure 1.

Total return swap are usually subject to an ISDA master agreement with creditsupport annex (CSA) and daily cash margining, i.e. a collateral rate (typicallyEONIA) is paid on the cash collateral.

A total return swap is often “associated” with a set of other transactions,which - in the common setup - are as follows:

• Counterparty A is providing a loan with notational N and maturity T tocounterparty B. Note that such a loan is typically associated with floatinginterest rate payments which are given by coupon of LIBOR plus anadditional spread on the notional.

• Counterparty B is providing an assetM to counterparty A, whose initialvalue corresponds to that of the loan.

The assetM is provided to mitigate the counterparty risk in the load and inview of this setup total return swaps are used to mitigate market risk of the bondcollateral in this transaction.

The above setup with a loan and a bond collateral is not part of the totalreturn swap transaction and the TRS will be valued independently of it. On theother hand, though not specified in the TRS contract, in practice the TRS payerreplicates the cash flows by purchasing and funding the underlying asset at tradeinception time. At trade inception time typically the deal spread will be higherthan the TRS payer’s funding spread and lower than the TRS receiver fundingspread, so that both counterparties calculate a positive present value. The twocounter parties share a mutual funding benefit. Often the payer purchases thebond from the receiver: thus a TRS can be considered as a long dated repo andthe TRS deal spread is a long dated repo rate.

Since the total return swap is a collateralized derivative referencing a fundingtransaction, it constitutes a collateralized derivative where an FVA naturallyenters in to the valuation.

1.2 Lack of Perfect Collateralization

The role of collateralization in a TRS is - at least in some situation - differentfrom a classical collateralized swap in that the collateral may come as an exoge-




nously prescribed value process. In this situation there is no equilibrium betweencollateralization and valuation - as in perfect cash collateralization (which leadsto so call OIS discounting). In this situation we find the TRS is a cash col-lateralized derivative for which both counterparties still calculate a differentpresent value and full collateralization for a TRS is impossible. This makescollateral management and valuation nontrivial. Partial collateralization meansthat funding valuation adjustments need to be taken into account. The fundingvaluation adjustments will be calculated differently by both counterparties.

1.3 Collateralization Schemes

In contrast to classical swaps (referencing cash flows and not value processes),the type of collateralization is a degree of freedom for the TRS. Therefore wewill discuss different collateralization schemes. One such scheme considered isthat collateral is posted according to the mid average (or convex combination)of the valuations performed by both counterparts. We will show that thiscollateralization scheme can be considered as a fair partial collateralizationscheme, where funding benefits are shared equally among both counterparts.Since the valuation is itself a function of the collateralization, we arrive at asystem of recursive equations, for which we derive simple analytic valuationformulas.

While this type of collateralization is “fair” since mutual funding benefitsare shared, it is required that both counter parties have knowledge of the othercounterparts funding costs / FVA. To circumvent this requirement and avoidcollateral management conflicts, in some TRS contracts a repo style collateralagreement is stated. According to these contracts the collateral is calculated asthe difference of the TRS notional amount (plus possibly last period accruedLIBOR plus deal spread) and the underlying asset market value. Although thissimple collateral calculation ignores the economic value of the deal, becausefuture deal spread margin and funding costs are neglected, in many marketsituations it yields quite similar results and can sometimes even be preferable tothe total return payer than a fair economic value collateral scheme.

Acknowledgment

We are grateful to Marius Rott, Jörg Zinnegger, Uwe Schreiber, Susanne Schäfer,Bruce Hunt and Klara Milz for stimulating discussions.




2 Basic Model Setup

2.1 Defintion of the Total Return Swap

A total return swap exchanges the total return of an underlying assetM withvanilla LIBOR plus spread payments Lj + spr on the TRS notional N . Atmaturity T the final value of the underlyingM(T ) is exchanged with the TRSnotional N , which is usually equal to the markt valueM(t0) of the underlyingasset at trade inception t0.

The total return of the underlying asset means that the TRS exchanges thecash flows of this asset against vanilla flows. These cash flows are depicted inFigure 1 (top two legs)1. In other words the market value of the underlying isexchanged against vanilla flows.

Additional cash flows are exchanged due to the cash collateral agreement.Thus the TRS exchanges future changes in market value of the underlying assetin cash.

2.2 Valuation of the Total Return Swap

We perform the valuation of the total return swap in two steps. We will firstvalue the TRS as an uncollateralized derivative. We will then value the collateralcash flows. All valuations are performed as (so called risk neutral) replicationcost.

Since we first consider the TRS as an uncollateralized derivative, we valueit with respects to each counterparts “funding numeraire” (i.e., using fundedreplication, [4]).

It might be tempting to directly consider an OIS discounted cash flow valua-tion, since the TRS is a OIS-collateralized instrument. However, this approachdoes not work out cleanly. Valuing the TRS as an uncollateralized derivativehas the striking advantage that we immediately obtain the value of the totalreturns of the underlying assetM in terms of its valueM(t) since that asset isuncollateralized too.2

1 Although it is not part of the TRS contract, in the bottom to legs of Figure 1 the bondpurchase and funding transactions are shown. For discussion of funding strategies andvaluation of the total package (TRS plus funding of purchased bond) see section ??.

2 For the time being, we make the assumption that any intrinsic funding capabilities (likebeing itself repo-able) are included in the valuation of the underlying assetM and that thetwo counter parties agree on a unique market priceM(t).




2.2.1 Valuation of the Plain LIBOR Flows

The funded replication cost for the plain vanilla leg is

LegrPlain(t) :=

n∑i=i(t)

Et

[e−

∫ ti+1t r(s)dsτi (Li(ti) + spr)

]M(t0)

+ Et

[e−

∫ Tt r(s)ds

]M(t0)

where r denotes the (unsecured) funding curve3, ti and ti+1 denote LIBORfixing and payment dates, spr is the deal spread,

i(t) := min{j | tj+1 ≥ t}

denotes the index of the first LIBOR period which will have payment time aftervaluation time t.

2.2.2 Valuation of the Total Return of the Underlying

The replication costs for the underlying asset in practice is simply the marketvalue or priceM(t) of the underlying asset at valuation time t.

Hence the value of a uncollateralized total return swap Vu is simply

Vu(t) = LegrPlain(t)−M(t).

We note that Vu does not depend on the future dynamics of the underlyingprice processM(s), s ≥ t. In contrast the cash collateralized value will dependon the future dynamicsM(s), s ≥ t.

2.2.3 Valuation of Collateralization

Next, we consider the cash flows generated by the cash collateral agreement, i.e.funding costs or benefit created by the collateral for both counterparties. LetC(s) denote the amount of cash collateral at time s ≥ t.

The valuation of the funding benefits or costs depends on the individualfunding accounts (funding rate processes) of the counterparties. We introducethe corresponding notation: We denote the two counterparties by A and B andassume that A is the total return payer, i.e. A pays all cashflows generated by theunderlying asset (coupons, principal repayments, final market valueM(T ) orrecovery at default) to B. In return A receives from B vanilla payments, LIBORplus deal spread, and at maturity T the value N .

3 For simplicity we assume that borrowing and investing is done with respect to the samecurve r.




Li + s*

Ci M(T)

M(t0)

Li + sfunding

Ci M(t0) M(T)

M(t0) M(t0)

TRS

Cas

hflow

sBu

y Bo

ndFu

ndin

g

Figure 1: Cash flows of a total return swap (top two legs) together with theassociated replication (bottom two leg) where the buying of the underlyingasset is financed via a funding transaction. The funding spread depicted issmaller than the deal spread of the total return swap. The picture representsa more classical view on a total return swap, neglecting an important part ofthe product: The total return swap is collateralized and cash-flows generatedfrom the collateral contract do represent an important aspect of the product.

Through this paper we will write down all valuations from the perspective ofA. That is to say, a cash flow payed from A to B will be valued negative (even ifwe calculate its value for B).

Let rA and rB denote the funding rate processes for A and B, respectively.4

For A the cash collateral C(s) ≥ 0 (C(s) < 0) at time s ≥ t is invested (funded)at rate rAds and the collateral rate rcds is paid (earned).

Thus the value of the future cash collateral flows for A is:

E

[∫ T

t

e−∫ st rA(x)dx (rA(s)− rc(s))C(s)ds | Ft

].

At this point one clearly sees that, in contrast to an uncollateralized TRS, thevaluation of a cash collateralized TRS will depend on the future dynamics of

4 To be precise we assume that for counterparty A all funded cash flows are valued w.r.t. anumeraire NA, where dNA = rAN

Adt, and respectively for B.




the underlying assetM(·). As we will consider in more detail in section 4.1and 4.2 There are different ways to define the collateral process C(s). EitherC(s) may be exogeneously given by a simple formula depending directly onthe market priceM(·) of the underlying, see section 4.2, or C(s) may dependon a replicated future underlying price computed using a repo or counterpartyspecific funding rate rA or rB. Using a Risk neutral replication cost approach,this means that the rate of growth for the underlyingM(s), s ≥ t is the repo orfunding rate for the underlying (minus an underlying coupon rate plus a defaultintensity rate), see [5] and the references there. For total return swaps, whichhave typical maturities from five to ten years, in general, there will be no uniquerepo rate applicable to both counterparties A and B. Often A and B need toapply their (unsecured) funding rate rA and rB to finance the underlying. Aninteresting example is that for certain underlyings A can perform cheap centralbank funding short term and switch to unsecured funding long term as centralbank changes its liquidity policy, but B can not do that because central bankdoes not accept the underlying from B (for example the underlying might beissued by B). For that example A could compute an attractive effective repo /funding rate for the underyling by combining the central bank curve and A′sunsecured funding curve5. One notes that an obvious funding benefit existshere.

Hence in general we need to take into account two dynamic underlyingprocessesMA(·) andMB(·)6. Thus we obtain the following system of valuationequations

VA(t) = LegrAPlain(t)−MA(t)

+ E

[∫ T

t


] (1)

VB(t) = LegrBPlain(t)−MB(t)

+ E

[∫ T

t

e−∫ st rB(x)dx (rB(s)− rc(s))C(s)ds | Ft

].

(2)

5 We assume that haircuts are included in the effective rate of growth of the underlying. Forexample a 10% haircut means that 90% can be funded using cheap central bank fundingwhere the remaining 10% have to be funded unsecured

6 There might be other reasons to consider different underlying processeMA(·) andMB(·).For example the underlying might be a default free fix coupon bond which has maturityequal to the TRS maturity. Then A could replicate the cash flows of the underlying bondby buying back zero bonds (issued by A or an entity which has the same risk as A). Theprocess would be given asMA(s) := e−

∫ tis

rA(x)dxci, where ti are coupon (including finalnotional) payment dates. This process completely ignores the present and future marketpriceM(s) of the underlying. This replication can be much cheaper than purchasing theunderlying, which might be obervalued (for example German government bond recentlywere sold at negative yield).




Both counterparties discount all occurring future cash flows with their ownfunding rate and subtract the replicated bond valueMA(t) andMB(t), respec-tively. Note that if C(s) is a function of VA(s) and VB(s), then we have a systemof recursive equations for VA and VB, where the coupling of the equations is dueto the collateral C(s).7

Remark 1 (Relation to perfectly collateralized derivatives): IfM(t) woulddenote the value of an uncollateralized future cash flow, the value of thatcashflow would depend on the funding, hence collateralization itself. This is theclassical situation and in this case it is possible to find an equilibrium collateralsuch that both counter parties agree on the same value. This is not possible here,sinceM(t) is considered as an exogenously prescribed process.

7 Note again that all valuation are perform from A’s perspective, hence there is no change insign.




3 Valuation of Derivatives with ExogenousCollateral Process

The valuation of collateralized and partially collateralized derivatives is wellknown, see Lemma 12 in the Appendix and references there. Here we give ageneralization of this result, which will be used in the valuation of total returnswaps in Section 4

Lemma 2: Let X denote a Levy process (jump-diffusion process)8 withdX(t) = 0 for t > T . Let gr(t) denote the valuation of the cash flows dX(t) int over the time [t, T ], i.e.,9

gr(t) := E

[∫ ∞t

e−∫ st r(x)dxdX(t) | Ft

].

Let f denote another Levy process (jump-diffusion process) (i.e., we assumedt df = 0) and let fT := f1t<T . Then we have:

Suppose C satisfies for all t ≤ T the following equation

C(t) = f(t) + gr(t) + E

[∫ T

t

e−∫ st r(x)dx (r(s)− rc(s))C(s)ds | Ft

],

where r,rc are Ito processes and Xi are fixed cash flows in ti. Then C has theequivalent representations

C(t) = f(t) + grc(t) + E

[∫ ∞t

e−∫ st rc(x)dx (r(s)− rc(s)) fT (s)ds | Ft

](3)

= grc(t) + E

[∫ ∞t

e−∫ st rc(x)dx

(r(s)fT (s)ds− dfT (s)

)| Ft]

. (4)

Remark 3: The form (3)-(4) is just a notationally more elegant version of

8 This lemma holds under more general assumptions. In fact, we only need that the stochasticprocesses are semi-martingales such that an integration by parts formula holds, see, e.g.,[7, Theorem 21, page 278] and that s 7→ e−

∫ str(x)dx, s 7→ e−

∫ strc(x)dx are continuous

semi-martingales.9 The process X is just a more general form of defining cash flows. For example, if X

is the piecewise constant function X(t) =∑Xi1t>Ti

, then∫∞te−

∫ str(x)dxdX(t) =∑

e−∫ Tit r(x)dxXi, i.e., the integral denotes a sum of discounted cash flows.




(using f in place of fT )

C(t) = f(t) + grc(t) + E

[∫ T

t

e−∫ st rc(x)dx (r(s)− rc(s)) f(s)ds | Ft

]= grc(t) + E

[e−

∫ Tt rc(x)dxf(T )| Ft

]+ E

[∫ T

t

e−∫ st rc(x)dx (r(s)f(s)ds− df(s)) | Ft

].

Proof: Define C̃(t) := C(t)− f(t). Then we have

C̃(t) = E

[∫ T

t

e−∫ st r(x)dx (r(s)− rc(s))

(C̃(s) + f(s)

)ds | Ft

].

Applying Lemma 12 with V = C = C̃ we get

C̃(t) = E

[∫ T

t

e−∫ st rc(x)dx (r(s)− rc(s)) f(s)ds | Ft

]= E

[∫ T

t

e−∫ st r(x)dxf(s) de−

∫ st (rc(x)−r(x))dx | Ft

]and with integration by parts we find

C̃(t) = E[e−

∫ Tt rc(x)dxf(T )| Ft

]− f(t)

+ E

[∫ T

t

e−∫ st rc(x)dx (r(s)f(s)ds− df(s)) | Ft

].

2|




4 Collateralization Schemes

Given equations (1)-(2) we are now in the situation that the collateral process Cis a free parameter.

In some cases the amount of the collateral is explicitly specified as part ofthe contract, see Section 4.2 for an example. But often the collateral is notexplicitly prescribed and C is the result of an (undefined) day-to-day consensusbetween the two counterparts. In any case, the valuations will come with aresidual funding valuation adjustment.

In this section we will thus consider different collateralization schemes anddiscuss their properties.

4.1 Fair partial collateralization

Let us consider the model of future collateral given as a convex combination(for example mid average) of the two counterparty valuations

C(s) := pVA(s) + (1− p)VB(s). (5)

Our following result gives the solution to the model equations (1)-(2) withthe collateral model (5) explicitly.

Before we state the result we need to introduce a combined funding rate rAB.Let us define the stochastic discount factor for the combined rate as the weightedaverage of the stochastic discount factors for rA and rB:

e−∫ Tt rAB(s)ds = p e−

∫ Tt rA(s)ds + (1− p) e−

∫ Tt rB(s)ds,

i.e. the rate rAB is a convex combination of the rates rA and rB with timedependent weights:

rAB(t) :=p e−

∫ Tt rA(s)ds rA(t) + (1− p) e−

∫ Tt rB(s)ds rB(t)

p e−∫ Tt rA(s)ds + (1− p) e−

∫ Tt rB(s)ds

.

Moreover let us define the convex combined underlying process

MAB := pMA + (1− p)MB.

In the following we will use the notational short form A/B for A or B,respectively.

Proposition 4 (Valuation under Fair Partial Collateralization): The valueof a total return swap onM with with the equilibrium collateral model (5) isgiven as

VA/B(t) = C(t) + FVAA/B, (6)




where the collateral process is given by

C(t) = LegrcPlain(t)−MAB(t)

− E[∫ T

t

e−∫ st rc(x)dx (rAB(s)− rc(s))MAB(s)ds | Ft

]= Legrc

Plain(t)− E[e−

∫ Tt rc(x)dxMAB(T )| Ft

]− E

[∫ T

t

e−∫ st rc(x)dx (rAB(s)MAB(s)ds− dMAB(s)) | Ft

],

(7)

and the funding valuation adjustment is

FVAA/B(t) :=MAB(t)−MA/B(t)

+ E

[∫ T

t

e−∫ st rA/B(x)dx (rAB(s)− rA/B(s))MAB(s)ds | Ft

].

(8)

Remark 5 (Formulas at initial valuation time): The formulas above hold forany tv ≤ t ≤ T , where tv denotes inital valuation time. At initial valuation timethe underlying processesMAB,MA andMB are all equal to the underlyingmarket valueM. I.e. if t = tv is initial valuation time, then

MAB(t) =MA(t) =MB(t) =M(t)

and the FVA and collateral formulas are

FVAA/B(t) = E

[∫ T

t

e−∫ st rA/B(x)dx (rAB(s)− rA/B(s))MAB(s)ds | Ft

],

C(t) = LegrcPlain(t)−M(t)

− E[∫ T

t


].

Remark 6 (Explicity of Collateral and FVA): The remarkable feature forProposition 4 is that we give the collateral C and the FVAs FVAA and FVAB

explicitly in terms ofM. Note that the standard expression of partially collater-alized products involve implicit equations for the collateral (as in Lemma 12).

Remark 7 (Mutual Funding Benefit): Before we turn to the proof, note thatProposition 4 immediately shows that for rB > rAB > rA the two counter partiesenter into a win-win situation. Due to rAB > rA we have that FVAA > 010,hence counterpart A positively values a funding benefit.

10 We assumed thatMA/B > 0.




Due to rAB < rB we have that FVAB < 0, and since all valuation are per-formed from A’s perspective, the negative value implies that counterpart Bpositively values a funding benefit too.

In other words: the two counterparts share a mutual funding benefit byagreeing on an average funding rate rAB and this agreement is done via thecollateral contract!

The TRS provides a mutual funding benefit as long as rB > rA, see Figure 2.

rA

Tmutual funding benefits

rB

rAB

r

mutual funding costs

Figure 2: Funding curves of the two counterparts and the average fundingcurve rAB. The TRS provides a mutual funding benefit as long as rB > rA

Remark 8 (Equal sharing of funding benefit (p = 12)): By construction we

have

C(t) = pVA(t) + (1− p)VB(t)

= C(t) + pFVAA + (1− p)FVAB.

HencepFVAA = −(1− p)FVAB.

Suppose A has cheaper funding than B, i.e. rA(s) < rB(s) for t ≤ s ≤ T .Consider p = 2, i.e. both counterparties agree that collateral is posted accordingto the average of the valuations calculated by A and B. Then funding benefitsare shared equally among both counterparties, i.e. both counterparties calculatethe same funding benefit and FVA adjustment, respectively, to the fair valuecollateral C(t):

FVAA = −FVAB.




Proof: Multiplying (1) with p and (2) with 1− p and adding we get

C(s) = pLegrAPlain(t)− pMA(s)

+ pE

[∫ T

s

e−∫ us rA(x)dx (rA(u)− rc(u))C(u)du | Ft

]+ (1− p)LegrB

Plain(t)− (1− p)MB(s)

+ (1− p)E[∫ T

s

e−∫ us rB(x)dx (rB(u)− rc(u))C(u)du | Ft

].

From the definition of rAB we find

pLegrAPlain(t) + (1− p) LegrB

Plain(t) = LegrABPlain(t)

and hence

C(t) = LegrABPlain(t)−MAB(t)

+ E

[∫ T

t

e−∫ st rAB(x)dx (rAB(s)− rc(s))C(s)ds | Ft

]Applying Lemma 2 with gr = grAB = LegrAB

Plain(t) and f = −MAB(t) gives

C(t) = LegrcPlain(t)−MAB(t)

− E[∫ T

t


]= Legrc

Plain(t)− E[e−


]− E

[∫ T

t


],

(9)

i.e. (7).We now define the FVA for counterpart A as FVAA and for counterparty B as

FVAB viaFVAA/B(t) := VA/B(t)− C(t).

From the definition of VA(t) we have

FVAA(t) = LegrAPlain(t)−MA(t)

+ E

[∫ T

t


]− C(t)

(10)

To eliminate the integral term which includes C(s), observe that we can apply




Lemma 2 to (9) again in reverse, but now with

gr = grc = LegrcPlain(t)− E[e−


]− E

[∫ T

t


],

and f = 0 to get

C(t) = LegrAPlain(t)− E

[e−

∫ Tt rA(x)dxMAB(T )| Ft

]− E

[∫ T

t

e−∫ st rA(x)dx (rAB(s)MAB(s)ds− dMAB(s)) | Ft

]+ E

[∫ T

t


]= LegrA

Plain(t)−MAB(t)

− E[∫ T

t

e−∫ st rA(x)dx (rAB(s)− rA(s))MAB(s)ds | Ft

]+ E

[∫ T

t


]

(11)

Plugging (11) into (10)

FVAA(t) :=MAB(t)−MA(t)

+ E

[∫ T

t

e−∫ st rA(x)dx (rAB(s)− rA(s))MAB(s)ds | Ft

].

The corresponding expression for FVAB(t) follows likewise. 2|

4.1.1 Fair partial collateralization under a simple jump-diffusion modelfor the bond dynamics

We will now derive a corollary to proposition 4 making an explicit modelassumption on the underlying dynamics M. For the underlying bond priceprocessM we assume that

M(s) = M(s)1τM>s,




where τM is the first jump of a Cox process with stochastic intensity λM11 andM(s) is the pre-default bond value process with dynamics

dM(s) = rM(s)M(s)ds+ σM(s,M(s))dWM ,

i.e. the full dynamics is

dM(s) =rM(s)M(s)1τM>sds+ σM(s,M(s))1τM>sdWM +M(s)d1τM>s

in risk neutral measure.The drift rM is used to model the pull-to-par of an underlying bond: rM(s),

s ≥ t is negative (positive) if the bond value M(s) is over (under) par at timet. Clearly, the drift will be an important model parameter driving the collateraldynamics of the TRS. For a zero coupon bond rM is the zero bond yield, whichis obtained from the associated zero coupon bond curve. For a coupon payingbond the continuously compounding bond coupon rate has to be subtracted fromthe zero bond yield, i.e., coupons are modeled via a continuously compoundingdividend rate.

Suppose A and B associate the specific funding (repo) rates rM,A, rM,B forfunding the underlying. For example rM,A (rM,B) is A’s (B’s) unsecured fundingrate. Another example is that rM,A is equal to a central bank rate, assuming Acan obtain cheap central bank funding for the underlying12.

Let us consider a funded replication risk neutral approch: then in the economyof counterparty A (B) the bond needs to have the effective growth rate (drift)rM,A (rM,B) [5].

Hence we define the associated counterparty bond dynamics by applying thecorresponding drift adjustment:13

MA(s) :=M(s)e∫ st rM,A(x)

e∫ st r0(x)dx

, MB(s) :=M(s)e∫ st rM,B(x)

e∫ st r0(x)dx

. (12)

The drift adjustments in (12) can be interpreted as fx rates using a cross currencyanalogy [4, 2]. The fx rates are

fxA(s) :=e∫ st rM,A(x)dx

e∫ st r0(x)dx

, fxB(s) :=e∫ st rM,B(x)dx

e∫ st r0(x)dx

.

Let us define the convex combined fx rate

fxAB(s) := p fxA(s) + (1− p) fxB(s).

11 If the reader prefers he can for simplicity assume that λM is deterministic, so that τM is thefirst jump of an inhomogeneous Poisson process.

12 If A can obtain central bank funding, but B can not due to regulatory reasons, this is anexample of regulatory arbitrage. The arbitrage or funding benefit can be shared by adjustingthe deal spread in the TRS.

13 here we assume that t is initial valuation time, i.e. t = tv




Corollary 9: The value of a total return swap on M with the equilibriumcollateral model (5) and whereM follows the above model is given as

VA/B(t) = C(t) + FVAA/B, (13)

where

C(t) = LegrcPlain(t)− E

[e−

∫ Tt (rc(x)+λM(x))dxfxAB(T )M(T ) | Ft

]1τM>t

− E

[∫ T

t

e−∫ st (rc(x)+λM(x))dx(rAB(s)− (rM(s)− r0(s)− λM

+ p rM,A + (1− p) rM,B))fxAB(s)M(s)ds | Ft

]1τM>t

= LegrcPlain(t)−M(t)− E

[∫ T

t

e−∫ st (rc(x)+λM(x))dx

(rAB(s)− rc(s)) fxAB(s)M(s)ds | Ft

]1τM>t,

(14)

and the funding valuation adjustment is

FVAA/B :=E

[∫ T

t

e−∫ st (rA/B(x)+λM(x))dx (rAB(s)− rA/B(s))

fxAB(s)M(s)ds | Ft

]1τM>t.

(15)

Proof: The result follows from Proposition 4 and the definition ofM and therelation (see [8, page 122])

E [d1τM>s | Fs] = −1τM>sλM(s)ds.

2|

Remark 10 (Zero bond): If the underlying is a zero bond we have rM =r0 + λM. Thus we get (assuming that the spread rM,A/B − rc and the remaining




processes are independent, i.e. neglecting a convexity adjustment)

C(t) = LegrcPlain(t)

−M(t)E[e−

∫ Tt rc(x)dx

(p e

∫ Tt rM,A(x)dx + (1− p) e

∫ Tt rM,B(x)dx

)| Ft]

−M(t)E

[∫ T

t

e−∫ st rc(x)dx (rAB(s)− (p rM,A + (1− p) rM,B))

(p e

∫ st rM,A(x)dx + (1− p) e

∫ st rM,B(x)dx

)ds | Ft

]= Legrc

Plain(t)−M(t)

−M(t)E

[∫ T

t

e−∫ st rc(x)dx (rAB(s)− rc(s))

(p e


∫ st rM,B(x)dx

)ds | Ft

],

and

FVAA/B :=M(t)E

[∫ T

t

e−∫ st rA/B(x)dx (rAB(s)− rA/B(s))

(p e


∫ st rM,B(x)dx

)ds | Ft

].

In particular for p = 1 (i.e. the deal is fully collateralized for A) and rM,A = rA

we get the simple formula

VA(t) = C(t) = LegrcPlain(t)−M(t)E

[e−

∫ Tt rc(x)dxe

∫ Tt rA(x)dx | Ft

].

Remark 11 (Wrong Way Funding Risk in the Collateral): The more ex-plicit form with the bond dynamics allows for a qualitative discussion now: Theexpected local growth ofM is

E [dM(s) | Fs] = (rM(s)− λM(s))M(s)1τM>sds.

Suppose λM(s) is negatively correlated to both rA and rB and that rA << rB.Then on paths where λM(s) becomes small with high probability rA, rB andrAB become large and M(s) increases. In that case the collateral C(t) becomessignificantly negative while the magnitude of both FVAA and FVAB becomelarge as well. This means that A has to post large amounts of collateral to Bwhile at the same time funding becomes expensive. This is an example of wrongway collateral and funding valuation adjustments for counterparty A, while the




situation is in favor for counterparty B.Likewise, if λM(s) is positively correlated to both rA and rB and rA << rB,

the situation is an example of wrong way collateral and funding valuationadjustments for counterparty B, while the situation is in favor for counterpartyA. Note that such a situation can naturally arise if the underlying asset and theTRS counterparty belong to the same economic sectors (e.g., financials).

One possibility to avoid such wrong way funding risk is a resetting of thetotal return swap (like it is market practice for cross-currency swaps), wherethe notional ofM is adjusted according to its change in market value. We willdiscuss this case in Section 4.1.2.

4.1.2 Continuously Resetting Bond Notional

As became clear in Remark 11, a resetting of the notional of the underlyingasset such that it’s market value matches the notional or market value M(t0)at inception time t0 can be a desirable feature: As soon as the market valuechanges, the volume of the asset is reduced or increased, respectively, to matchthe notional amount. In other words we haveM(t) ≡M(t0) and hence

VA/B(t) = LegrA/BPlain(t)−M(t0)

+ E

[∫ T

t

e−∫ st rA(x)dx (rA/B(s)− rc(s))C(s)ds | Ft

]This is a special case (rM = σM = λM = 0, rM,A = rM,B = r0) of the more

general dynamic bond model setup of Section 4.1. For this simple special casewe obtain as a Corollary of Corollary 9:

C(t) =M(t0)n∑

i=i(t)

E[e−

∫ ti+1t rc(s)dsτi (Li(ti) + spr) | Ft

]−M(t0)E

[∫ T

t

e−∫ st rc(x)dxrAB(s)ds | Ft

].

This formula means simply that LIBOR plus spread is earned and funding ispaid, and that all flows are discounted with the collateral rate. The valuationformulas are:

VA/B(t) =C(t) + FVAA/B,

where the funding valuation adjustment is

FVAA/B :=M(t0)E

[∫ T

t

e−∫ st rA/B(x)dx (rAB(s)− rA/B(s)) ds | Ft

].




4.2 Simplified Explicit Collateral (Repo Style)

To avoid complications in collateral management due to asymmetric valuationscollateral can be explicitly specified. A common example is to use (see [1])

C(s) := M(t0)−M(s). (16)

That is, the collateral at each time s ≥ t0 is defined to be the difference of theTRS notionalM(t0) (market value at inception time) and the time s marketvalue M(s) 14. Thus market changes of the underlying asset are triviallycollateralized, neglecting TRS margin and funding levels. Note thatM(s) isan exogenous given market rate process, not a funded and replicated processMA/B(s). This means that the underlying processM(s) has to be consideredlike a quanto index for each counterparty A/B.

To value this contract one just solves (1)-(2):

VA/B(t) = LegrA/BPlain(t)−M(t)

+ E

[∫ T

t

e−∫ st rA/B(x)dx (rA/B(s)− rc(s))C(s)ds | Ft

].

This can be easily done after bond dynamics has been calibrated. A verysimple approach is to use a simple deterministic forward bond dynamics. Ingeneral one should use correlated stochastic spread models for the bond [4].

14 In some contracts notional plus LIBOR and dealspread accrued minus underlying marketvalue are defined as collateral. For simplicity we have neglected accrued.




5 Appendix

The following lemma is a well known result on the valuation (i.e., replicationcosts) of a collateralized transaction.

Lemma 12: Let X denote a cash-flow at maturity time T and let V ,r,rc be Itostochastic processes with V (T ) = X 15 so that the following integral equationis satisfied for t ≤ T

V (t) = E

[e−

∫ Tt r(s)dsX +

∫ T

t

e−∫ st r(x)dx (r(s)− rc(s))C(s)ds | Ft

].

(17)Then for t ≤ T

V (t) = E

[e−

∫ Tt rc(s)dsX −

∫ T

t

e−∫ st rc (r(s)− rc(s)) (V (s)− C(s))ds | Ft

].

(18)Conversly, if t 7→ V (t) is given by (18), then t 7→ V (t) satisfies the integralequation (17).

A proof of Lemma 12 for the fully collateralized case V = C can be foundin [3, Appendix A]. Piterbarg [5, 6] gave a different proof, based on a selffinancing replication approach. We now give a simple alternative proof, whichuses a foreign market cross currency viewpoint [4, 2]:

Proof: Consider a market which is governed by the rate dynamics rc, and a(foreign) market which is governed by the rate dynamics r. Define the bankaccount numeraire Nc(s) := e

∫ st rc(x)dx and N(s) := e

∫ st r(x)dx, and the fx rate

FX(s) := e∫ st (r(x)−rc(x))dx. Assets in the rc market are transfered to the r

market by multiplying with the fx rate. Thus assets seen in r have a drift whichis increased by (r(t)− rc(t))dt. Because N(s) = FX(s)Nc(s) the risk neutralmeasures associated with N and Nc are equal. Write

V (t) = V1(t) + V2(t),

whereV1(t) := E

[e−

∫ Tt rc(s)dsX | Ft

]and

V2(t) := −E[∫ T

t

e−∫ st rc(x)dx (r(s)− rc(s)) (V (s)− C(s))ds | Ft

]According to the universal pricing theorem the Nc relative value V1(s)

Nc(s)is a

15 We assume standard regularity conditions, e.g. r, rc have bounded L2 norm on [0, T ], whichguarantee unique existence of solutions to the integral equation (17)




martingale. On the other hand

E

(V (T )

N(T )− V (t) | Ft

)= E

(∫ T

t

d

(V

N

)| Ft)

= E

(∫ T

t

d

(FX(s)FX−1(s)

V (s)

N(s)

)| Ft)

= E

(∫ T

t

d

(FX−1(s)

V (s)

Nc(s)

)| Ft)

= E

(∫ T

t

V (s)

Nc(s)d(FX−1(s)

)| Ft)

+ E

(∫ T

t

d

(V2(s)

Nc(s)

)FX−1(s) | Ft

)where the last equation holds because from the martingale property

E

(∫ T

t

FX−1(s)d

(V1(s)

Nc(s)

)| Ft)

= 0.

and d (FX−1(s)) d(V (s)Nc(s)

)= 0 due to the absence of any diffusion in FX .16

Because

d

(V (s)

Nc(s)

)=− rc(s)

V (s)

Nc(s)ds+ rc(s)

V (s)

Nc(s)ds

− (r(s)− rc(s)) (V (s)− C(s)) 1

Nc(s)ds

we have (17). 2|

Equation (17) has an intuitive meaning in terms of the universal pricingtheorem: the collateral can be viewed as an additional contract which generatesthe instantanous cash flows C(s)(r(s)−rc(s)). Equation (17) is a reformulationin terms of discounting with the collateral rate. Here the gap between value andcollateral V −C generates the instantaneous cashflows−(r(s)− rc(s))(V (s)−C(s))ds.

If the cash-flow X is fully collateralized, i.e. C(s) = V (s), then the valueV (t) is obtained by discounting the cash-flow X with the collateral rate rcaccording to (18).

16 Note that the numeraires N and Nc are locally risk free, hence is FX .




References

[1] Martin Baxter. Bond repo pricing with funding and credit. Fixed IncomeConference Vienna 2012, 2012.

[2] Marco Bianchetti. Two curves, one price. Risk August, 2010.

[3] Anders B. Trolle Damir Fillipovic. The term structure of interbank risk.2012.

[4] Christian Fries. Funded replication. valuing with stochastic funding. SSRN,2011.

[5] Vladimir Piterbarg. Funding beyond discounting: collateral agreements andderivatives pricing. Risk Magazine, (Feb 2010), 2010.

[6] Vladimir Piterbarg. Cooking with collateral. Risk Magazine August 2012,2012.

[7] Philip E. Protter. Stochastic Integration and Differential Equations.Springer, Berlin, 2003.

[8] Philipp Schoenbucher. Credit Derivatives Pricing Models. John Wiley andSons Ltd, 2003.



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