# A Binary Nature of Funding Impacts in Bilateral Contracts ††thanks:
Acknowledgments: We thank Stéphane Crépey for spending time to help us improve
this paper and we took into account his comments in this paper. We are also
grateful to Dylan Possamaï for interesting discussion at Workshop on Risk
Measures, XVA Analysis, Capital Allocation and Central Counterparties (2016).

Junbeom Lee Junbeom Lee is funded from the Singapore MOE AcRF Grant
R-146-000-255-114. Email<EMAIL_ADDRESS>Department of Mathematics
National University of Singapore  Chao Zhou Chao Zhou is supported by
Singapore MOE AcRF Grants R-146-000-255-114 and R-146-000-219-112. Email:
<EMAIL_ADDRESS>Department of Mathematics
National University of Singapore

We discuss a binary nature of funding impacts. Under some conditions, funding
is either cost or benefit, i.e., one of the lending/borrowing rates does not
play any role in pricing derivatives. When we price derivatives, considering
different lending/borrowing rates leads to semi-linear BSDEs and PDEs, so we
need to solve the equations numerically. However, once we can guarantee that
only one of the rates affects pricing, we can recover linear equations and
derive analytic formulae. Moreover, as a byproduct, our results explain how
debt value adjustment (DVA) and funding benefits are different. It is often
believed that DVA and funding benefits are overlapped but it will be shown
that the two components are affected by different mathematical structures of
derivative transactions. We will see later that FBA occurs where the payoff is
non-increasing, but this relationship becomes weaken as the funding choices of
underlying assets are transferred to repo markets.

Key words:  BSDEs, Malliavin calculus, bilateral contracts, incremental cash-
flows, funding benefits

AMS subject classifications:  60H07, 91G20, 91G40

##  1 Introduction

The financial crisis in 2007-2008 has forced us to examine many parts of
general practice to price derivatives. In the crisis, the defaults of big
firms heightened default risk. Moreover, banks became more reluctant to lend
money and, as a result, the gap between London inter-bank offered rate (LIBOR)
and overnight indexed swap (OIS) rate was widened. By the changed market
conditions, banks began to make several adjustments in derivative prices. It
has long been a standard to make credit value adjustment (CVA). CVA is
correcting derivative prices for the risk of counterparty’s default. On the
other hand, debt value adjustment (DVA) is a deduction for the hedger’s own
default. If one party defaults earlier than the contractual maturity, a
promised payment should be settled as close-out amount. However, because of
the default, the obligation may not fully honored. For the risk, it is general
that collateral is pledged. Funding value adjustment (FVA) is the adjustment
of derivative prices for the excessive cost and benefit in maintaining the
hedging portfolio and posting collateral.

Considering entity-specific information in derivative prices has required many
changes in classical pricing theory. First, since law of one price does not
hold, when a price is given for a contract, it may be an arbitrage opportunity
for a trader, but may not be for others. For the issue, entity-specific
definitions of arbitrage opportunity and fair values were suggested by
Bielecki Rutkowski (2015); Bielecki . (2018); Bichuch . (2018) and those
arguments were applied to various derivatives (Kim ., 20182, 20181, for
example). In addition, many other pricing methodologies have been developed. A
replication pricing under FVA with collateral was introduced by Piterbarg
(2010). Wu (2015); Li Wu (2016) also discussed replication pricing with CVA,
FVA with collateral. For risk-neutral valuation principle, readers can refer
to Brigo . (2011). Crépey (20152, 20151) discussed min-variance pricing under
default risks, funding spreads, and collateral.

Still, there have remained many puzzles among the adjustments. Especially, DVA
and FVA have their own but intertwined issues. First, there is a doubt on
reporting DVA, which is to say that an aggravation of own default risk can be
beneficial to the shareholders. As we will see later, theoretically, the
increase of own default risk can be monetized by buying back their bonds.
However, it is often impossible in practice because the bank should really
default to realize the benefit. Indeed, DVA is accepted by both IFRS and GAAP,
but excluded from common equity tier 1 capital (CET1), which is a proxy of
shareholder’s wealth. Second, it is often believed that DVA double-counts
funding benefits (see Remark 2.21). FVA has two parts: funding benefit
adjustment (FBA) and funding cost adjustment (FCA). Both DVA and FBA may
originate from banks’ own default risk. DVA is a deduction from liabilities of
a bank due to its creditworthiness. On the other hand, FBA, as a counterpart
of FCA, may countervail FCA which is also from the bank’s own default risk.
Thus, including both DVA and FBA may inflate the bank’s reported profit (see
Cameron, 2013) as well as deflate the price charged to counterparties.
Possibly due to these reasons, seller’s DVA is often excluded in derivative
transactions.

FVA is more arguable. According to Modigliani-Miller (MM) theorem, which is a
long-established financial principle, choices of funding should not be
considered in pricing. On the other hand, in practice, with the increased
interest rates offered by funding desks, it will be loss to the traders
without recouping the funding costs from counterparties. Indeed, traders feel
confident that funding costs are observable in derivative transactions (see
Andersen ., 2017)). If the traders’ belief is true, the inclusion of FVA in
derivative prices may be justified by market frictions, which is a violation
of the assumptions of MM theorem (see Modigliani Miller, 1958; Stiglitz,
1969). However, as pointed out by Hull White (2012), if FVA is really an
element to determine derivative prices, the existence of Treasury bonds is
mysterious that banks buy a bond that returns less than their funding rates.

The above issues will be discussed by the main results of this paper. Our main
contribution is showing a binary nature of FVA that FVA is either FCA or
$-\text{FBA}$ for many derivative contracts. For example, we shall see later
that, based on our model, when buying bonds, the trader will never enter a
borrowing position, i.e., $\text{FCA}=0$ and $\text{FVA}=-\text{FBA}$. If we
assume the lending rate is equal to OIS rate (for example, as in Burgard
Kjaer, 2010), we have $\text{FVA}=0$. This may explain the reason why banks do
not require FVA when buying Treasury bonds, while FVA is observable in other
derivative transactions.

The switching between funding costs and benefits depends on the structures of
derivative payoffs, close-out conventions, and choices of funding, e.g., from
a repo market or treasury department. The structure to determine the binary
nature of funding impacts is whether the payoffs of derivatives increase or
decrease in underlying assets. On the other hand, since DVA is a deduction on
derivative payables, DVA occurs where the payoffs are positive. Therefore, DVA
and FBA are affected by different mathematical structures of the payoffs.
Indeed, it will be shown later by an example that DVA and FBA are clearly
separated and this result is in line with a part of the conclusion in Andersen
. (2017) that DVA should be considered in pricing.

The difference between FBA and DVA were also pointed out by Albanese Andersen
(2014). It was shown that FBA is larger than DVA mainly because of the
effective discounting rate and absence of default indicator in the definition
even though FBA and DVA are quite similar. In their case study, FBA was 20%
larger than DVA. However, we will see later that FBA and DVA are not even
close in some cases (see Table 1, for example). The values of FBA and DVA
often resemble because of the funding choices commonly taken in the literature
that all underlying assets can be acquired from repo markets. When available,
repo markets are preferred by dealers, but it is not always possible. As we
will see later, FBA occurs where the payoff is non-increasing, but the more
assets are traded through repo markets, the weaker this relationship becomes.
Readers may want to refer to Remark 3.2-(iv, v) in advance.

Our results are also related to accounting with FVA. In FCA/FBA accounting,
which is endorsed by some leading banks, DVA is recorded in Contra-Liability
(CL) account and

$\displaystyle\text{FCA}-(\text{FBA}-\text{DVA})$

is recorded in Contra-Asset (CA) account (see Castagna, 2011). It has been
pointed out that this accounting method engenders large asset/liability
asymmetry. According to the binary nature of funding impacts, the large
asymmetry is inevitable since, for some derivative contracts, either FCA$=0$
or FBA$=0$, and DVA$\not=$FBA, i.e., FCA does not countervail FBA and FBA is
not overlapped with DVA. FVA/FDA accounting, as an alternative, was suggested
to recover asset/liability symmetry as well as protect CET1 capital, which is
a proxy of shareholder’s wealth (see Albanese Andersen, 2014, for example).
Readers can refer to Andersen . (2017) for the issue of shareholder’s wealth
protection with funding spreads.

Another benefit of our results is that we can recover linear equations to
price derivatives. Because different lending/borrowing rates make the pricing
equations semi-linear, analytic solutions are not generally allowed.
Therefore, we need to solve the equations numerically, and sometimes the
computational cost becomes expensive. For an attempt to approximate FVA of
contracts with short maturity, readers may want to refer to Gobet Pagliarani
(2015). Moreover, there have been several attempts to find closed-form
solutions under funding spreads (Piterbarg, 2010; Bichuch ., 2018; Brigo .,
2017). Still, in the arguments, the crucial assumption was the same
lending/borrowing rates. This itself can bee seen as a strong assumption.
However, once we can guarantee that one of funding rates does not play any
role in pricing derivatives, we can assume the same lending and borrowing
rates without loss of generality (because the dealer will never switch from
one to another). Put differently, our results allow their results widely
applicable.

Even though the binary nature allows us to find analytic formulae for a large
class of derivatives, depending on the close-out conventions, the analytic
formulae may or may not be represented by a closed-form. In this paper, we
deal with two close-out conventions: clean price and replacement cost. clean
price (resp. replacement cost) is the risk-neutral price without (resp. with)
with value adjustments. Under clean close-out, because pricing measures are
not matched, the analytic formulae can not be represented in a closed-form.
Indeed, to avoid the inconsistency, Bichuch . (2018) assumed a flexibility to
choose a pricing measure for calculating close-out amount, and Brigo . (2017)
considered only un-collateralized contracts with null cash-flow at defaults.
In the case of replacement close-out, the mismatch does not appear, because,
in our model, the funding rates are tacitly embedded to the replacement cost.
Therefore, we can provide closed-form solutions under replacement close-out.

In our model, we include incremental CVA, DVA, FVA, and variation margin. The
reference filtration is generated by a Brownian motion. Then we progressively
enlarge the filtration by default times of the two parties. The default times
are assumed to have intensities. We do not assume that interest rates are
deterministic, so our results can be applied to interest rate derivatives. In
the main theorems, we assume volatility and default intensities are
deterministic to avoid heavy calculations, but stochastic parameters do not
necessarily change the main results. We report one example that intensities
are not deterministic in Appendix.

This paper is organized as follows. In Section 2, we introduce our setup on
the filtration, intensities, and construct the incremental hedging portfolio.
In Section 2.4, we introduce a BSDE to price derivatives on the enlarged
filtration. Instead of dealing with the BSDE, we define XVA (x-value
adjustment) process in Section 2.3.2 and the XVA process is reduced to a BSDE
on the reference filtration as in Crépey Song (2015). Then our main results
are provided in Section 3. For proving the main theorems, iterative
transformations of the XVA process are needed and the transformations depend
on the close-out conventions. The proofs for the main theorems are reported in
Appendix. In Section 4, examples are examined and we provide a closed-form
solution for a stock call option with replacement close-out.

##  2 Modeling

###  2.1 Mathematical Setup

We consider two parties entering a bilaterally cleared contract. We call one
party a “hedger” and the other party a “counterparty”. We sometimes address
the hedger (resp. counterparty) “she” (resp. “he”). An index $H$ (resp. $C$)
will be used to represent the hedger (resp. counterparty). The argument of
this paper is conducted in view of the hedger. The hedger is a financial firm
that holds a portfolio to hedge the exchanged cash-flows of the contract. The
counterparty may or may not be a non-financial firm. Note that when we say a
“dealer”, “bank”, and “trader”, it is not necessarily addressing the hedger
since the counterparty can be also a bank.

Let $(\Omega,\mathcal{G},\mathbb{Q})$ be a probability space, where
$\mathbb{Q}$ is a risk-neutral probability measure. Let $\mathbb{E}$ denote
the expectation under $\mathbb{Q}$. We consider random times $\tau^{i}$,
$i\in\\{H,C\\}$,

$\displaystyle\tau^{i}\colon(\Omega,\mathcal{G})\to(\mathbb{R}_{+},\mathcal{B}(\mathbb{R}_{+}))),$

which represent the default times of the hedger and counterparty. For
$i\in\\{H,C\\}$, we assume that $\mathbb{Q}(\tau^{i}=0)=0$ and
$\mathbb{Q}(\tau^{i}>t)>0,~{}\forall t\in\mathbb{R}_{+}$. We also denote

$\displaystyle\tau\coloneqq\tau^{H}\wedge\tau^{C},~{}~{}~{}\bar{\tau}\coloneqq\tau\wedge
T,$

where $T$ is the maturity of the bilateral contract.

Let $W=(W^{1},\dots,W^{n})$ be a standard $n$-dimensional Brownian motion
under $\mathbb{Q}$. Let $\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$ be the usual
natural filtration of $(W_{t})_{t\geq 0}$. Then we define

$\displaystyle\mathbb{G}=(\mathcal{G}_{t})_{t\geq
0}\coloneqq\Big{(}\mathcal{F}_{t}\vee\sigma\big{(}\\{\tau^{i}\leq
u\\}~{}\colon~{}u\leq t,i\in\\{H,C\\}\big{)}\Big{)}_{t\geq 0}.$

We call $\mathbb{F}$ (resp. $\mathbb{G}$) the reference filtration (resp. full
filtration).

Then we consider a filtered probability space
$(\Omega,\mathcal{G},\mathbb{G},\mathbb{Q})$. Note that $\tau^{i}$,
$i\in\\{H,C\\}$, are $\mathbb{G}$-stopping times but may not be
$\mathbb{F}$-stopping times. As a convention, for any
$\mathbb{G}$-progressively measurable process $u$,
$(\mathbb{G},\mathbb{Q})$-semimartingale $U$, and $s\leq t$,

$\displaystyle\int_{s}^{t}u_{s}\mathop{}\\!\mathrm{d}U_{s}\coloneqq\int_{(s,t]}u_{s}\mathop{}\\!\mathrm{d}U_{s},$

where the integral is well-defined. In addition, for any $\mathbb{G}$-stopping
time $\theta$ and process $(\xi_{t})_{t\geq 0}$, we denote

$\displaystyle\xi^{\theta}_{\cdot}\coloneqq\xi_{\cdot\wedge\theta},$

and when $\xi_{\theta-}$ exists,
$\Delta\xi_{\theta}\coloneqq\xi_{\theta}-\xi_{\theta-}$. In what follows, for
$i\in\\{H,C\\}$, $t\geq 0$, we denote
$G^{i}_{t}\coloneqq\mathbb{Q}(\tau^{i}>t|\mathcal{F}_{t})$, and

$\displaystyle G_{t}\coloneqq\mathbb{Q}(\tau>t|\mathcal{F}_{t}).$

The following assumption stands throughout this paper.

######  Assumption 2.1.

  1. (i)

$(G_{t})_{t\geq 0}$ is non-increasing and absolutely continuous with respect
to Lebesgue measure.

  2. (ii)

For any $i\in\\{H,C\\}$, there exists a process $h^{i}$, defined as

$\displaystyle h^{i}_{t}\coloneqq\lim_{u\downarrow
0}\frac{1}{u}\frac{\mathbb{Q}(t<\tau^{i}\leq
t+u,\tau>t|\mathcal{F}_{t})}{\mathbb{Q}(\tau>t|\mathcal{F}_{t})},$

and the process $M^{i}$, given by

$\displaystyle M^{i}_{t}\coloneqq\mathds{1}_{\tau^{i}\leq
t\wedge\tau}-\int_{0}^{t\wedge\tau}h^{i}_{s}\mathop{}\\!\mathrm{d}s,$

is a $(\mathbb{G},\mathbb{Q})$-martingale.

For any $i\in\\{H,C\\}$, there exists a process $h^{i}$, defined as

$\displaystyle h^{i}_{t}\coloneqq\lim_{u\downarrow
0}\frac{1}{u}\frac{\mathbb{Q}(t<\tau^{i}\leq
t+u,\tau>t|\mathcal{F}_{t})}{\mathbb{Q}(\tau>t|\mathcal{F}_{t})},$

and the process $M^{i}$, given by

$\displaystyle M^{i}_{t}\coloneqq\mathds{1}_{\tau^{i}\leq
t\wedge\tau}-\int_{0}^{t\wedge\tau}h^{i}_{s}\mathop{}\\!\mathrm{d}s,$

is a $(\mathbb{G},\mathbb{Q})$-martingale.

By (i) in 2.1, there exists an $\mathbb{F}$-progressively mesaurable process
$(h^{0}_{t})_{t\geq 0}$ such that

$\displaystyle h^{0}_{t}=\lim_{u\downarrow
0}\frac{1}{u}\frac{\mathbb{P}(t<\tau\leq
t+u|\mathcal{F}_{t})}{\mathbb{P}(\tau>t|\mathcal{F}_{t})},$

and

$\displaystyle M_{t}\coloneqq\mathds{1}_{\tau\leq
t}-\int_{0}^{t\wedge\tau}h^{0}_{s}\mathop{}\\!\mathrm{d}s$

is also a $(\mathbb{G},\mathbb{P})$-martingale. Let us denote $h\coloneqq
h^{H}+h^{C}$. If $\tau^{H}$ and $\tau^{C}$ are independent on $\mathbb{F}$,
$h^{0}=h$. It is in general not the case. Moreover, by (i) in 2.1, $\tau$
avoids any $\mathbb{F}$-stopping time (see Coculescu Nikeghbali, 2012,
Corollary 3.4). In other words, for any $\mathbb{F}$-stopping time
$\tau^{\mathbb{F}}$,

$\displaystyle\mathbb{Q}(\tau=\tau^{\mathbb{F}})=0.$ (2.1)

The next lemma is borrowed from Bielecki . (2008) and Chapter 5 in Bielecki
Rutkowski (2013).

######  Lemma 2.2.

Let $i\in\\{H,C\\}$.

  1. (i)

Let $U$ be an $\mathcal{F}_{s}$-measurable, integrable random variable for
some $s\geq 0$. Then, for any $t\leq s$,

$\displaystyle\mathbb{E}(\mathds{1}_{s<\tau}U|\mathcal{G}_{t})=$
$\displaystyle\mathds{1}_{t<\tau}G_{t}^{-1}\mathbb{E}(G_{s}U|\mathcal{F}_{t}),$
$\displaystyle\mathbb{E}(\mathds{1}_{s<\tau^{i}}U|\mathcal{G}_{t})=$
$\displaystyle\mathds{1}_{t<\tau^{i}}(G^{i}_{t})^{-1}\mathbb{E}(G^{i}_{s}U|\mathcal{F}_{t}).$

  2. (ii)

Let $(U_{t})_{t\geq 0}$ be a real-valued, $\mathbb{F}$-predictable process and
$\mathbb{E}|U_{\bar{\tau}}|<\infty$. Then,

$\displaystyle\mathbb{E}(\mathds{1}_{\tau=\tau^{i}\leq
T}U_{\tau}|\mathcal{G}_{t})=\mathds{1}_{t<\tau}G_{t}^{-1}\mathbb{E}\Big{(}\int_{t}^{T}h^{i}_{s}G_{s}U_{s}\mathop{}\\!\mathrm{d}s\Big{|}\mathcal{F}_{t}\Big{)}.$

We define spaces of random variables, and stochastic processes as follows.

######  Definition 2.3.

Let $m\in\mathbb{N}$ and $p\geq 2$.

  * •

$\mathbb{L}^{p}_{T}$: the set of all $\mathcal{F}_{T}$-measurable random
variables $\xi$, such that

$\displaystyle\|\xi\|_{p}\coloneqq\mathbb{E}[|\xi|^{p}]^{\frac{1}{p}}<\infty.$

  * •

$\mathbb{S}^{p}_{T}$: the set of all real valued, $\mathbb{F}$-adapted,
càdlàg111Right continuous and left limit processes $(U_{t})_{t\geq 0}$, such
that

$\displaystyle\|U\|_{\mathbb{S}_{T}^{p}}\coloneqq\mathbb{E}\big{(}\sup\limits_{t\leq
T}|U_{t}|^{p}\big{)}^{\frac{1}{p}}<\infty.$

  * •

$\mathbb{H}^{p,m}_{T}$: the set of all $\mathbb{R}^{m}$-valued,
$\mathbb{F}$-predictable processes $(U_{t})_{t\geq 0}$, such that

$\displaystyle\|U\|_{\mathbb{H}^{p}_{T}}\coloneqq\mathbb{E}\Big{(}\int_{0}^{T}\big{|}U_{t}\big{|}^{p}\mathop{}\\!\mathrm{d}t\Big{)}^{\frac{1}{p}}<\infty.$

  * •

$\mathbb{H}^{p,m}_{T,loc}$ : the set of all $\mathbb{R}^{m}$-valued,
$\mathbb{F}$-predictable processes $(U_{t})_{t\geq 0}$, such that

$\displaystyle\int_{0}^{T}\big{|}U_{t}\big{|}^{p}\mathop{}\\!\mathrm{d}t<\infty,\quad\text{a.s.}$

Moreover, we let $D_{\theta}=(D^{1}_{\theta},\dots,D^{n}_{\theta})$ denote
Malliavin derivative at $\theta\geq 0$, and $\mathbb{D}^{1,2}$ denote the set
of Malliavin differentiable random variables. For Malliavin calculus, readers
can refer to Di Nunno . (2009) and Section 5.2 in El Karoui . (1997). In the
next section, we describe hedging portfolios under incremental CVA, DVA, FVA,
and collateral. These aggregated adjustment is often called XVA. For
simplicity in notations, when $n=1$, we denote $D_{\theta}=D^{1}_{\theta}$,
$W=W^{1}$, and $\mathbb{H}^{p}_{T}\coloneqq\mathbb{H}^{p,1}_{T}$,
$\mathbb{H}^{p}_{T,loc}\coloneqq\mathbb{H}^{p,1}_{T,loc}$.

###  2.2 BSDEs under Incremental XVA

####  2.2.1 Cash-flows

We consider a hedger and counterparty entering a “new” contract which
exchanges promised dividends. First, we proceed our argument with the
assumption that the two parties have not made any contract before the new
contract. Later, this assumption will be relaxed by slightly modifying our
model so that incremental effects can be considered.

Let $\textfrak{D}^{N}_{t}$ denote the accumulated amount of the promised
dividends up to $t\geq 0$. We assume that $\textfrak{D}^{N}$ is an
$\mathbb{F}$-adapted càdàg process, and the value is determined by an
$n$-dimensional underlying asset process $S=(S^{1},\dots,S^{n})$ that follows
the next stochastic differential equation (SDE):

$\displaystyle\mathop{}\\!\mathrm{d}S^{i}_{t}=$ $\displaystyle
r_{t}S^{i}_{t}\mathop{}\\!\mathrm{d}t+\sigma^{i}_{t}S^{i}_{t}\mathop{}\\!\mathrm{d}W_{t},~{}~{}i\in\\{1,\dots,n\\},$
(2.2)

for some $\mathbb{F}$-progressively measurable processes $r$ and
$(\sigma^{i})^{\top}\in\mathbb{R}^{n}$. In addition, we assume that the
$\mathbb{F}$-adapted process $\textfrak{D}^{N}$ is independent of the
information of defaults.

######  Remark 2.4.

Note that an $\mathbb{F}$-adapted process may depend on default risks, since
the default intensities are $\mathbb{F}$-adapted.

If a default occurs before the maturity of the contract $T$, two parties stop
exchanging $\textfrak{D}^{N}$, and the derivative contract is marked-to-
market. The method to calculate the close-out amount is determined before
initiation of the contract and documented in Credit Support Annex (CSA)222A
part of ISDA master agreement. Let $e^{N}_{t}$ denote the close-out amount at
$t\leq T$. In this paper, we deal with two conventions for the close-out
amount $e^{N}$: clean close-out and replacement close-out. We postpone
explaining the conventions to the next section after we define the hedger’s
hedging portfolio. As conventions,
$\mathop{}\\!\mathrm{d}\textfrak{D}^{N}_{t}\geq 0,~{}e^{N}_{t}\geq 0$ (resp.
$<0$) means that the hedger pays to (resp. is paid by) the counterparty at
$t\leq T$.

######  Example 2.5.

If the hedger buys a zero coupon bond of unit notional amount,
$\textfrak{D}^{N}=-\mathds{1}_{\llbracket T,\infty\llbracket}$.

The obligation to settle $e^{N}$ may not be fully honored because of the
default. To mitigate the risk, the two parties post or receive collateral
(often referred to as margin). The amount of the collateral posted at $t\geq
0$ (only for the new contract) is denoted by $m^{N}_{t}$. We assume that
$(m^{N}_{t})_{t\geq 0}$ is an $\mathbb{F}$-adapted process. By 2.1,
$\tau^{i}$, $i\in\\{H,C\\}$, are totally inaccessible, which means that the
defaults arrive with surprise. Margins are posted because we do not know full
information of the defaults, and this is why $m^{N}$ is calculated on the
observable information $\mathbb{F}$. The exact forms of $m^{N}$ will be given
later after the conventions for $e^{N}$ are introduced. We assume that the
close-out payment is settled at the moment of default without delay and
$m^{N}$ is posted continuously. As conventions, if $m^{N}_{t}\geq 0$ (resp.
$<0$), it means that the hedger posts (receives) the collateral at $t\leq T$.

######  Remark 2.6.

In practice, there exists a gap between the day of default and actual
settlement. The delay is required to check whether the default really
happened, collect information of the contract, find the best candidate to
replace the defaulting party (Murphy, 2013). Gap risk is the risk from the gap
between the close-out amount at the day of settlement and the last day that
variation margin is posted. For gap risk, two parties post initial margin
which is often calculated by a risk measure. Note that we ignore the gap risk
and initial margin. If we consider initial margin, we encounter anticipative
backward stochastic differential equations (ABSDEs) under replacement close-
out. For the main result of this paper, Malliavin calculus for BSDEs will be
used. However, to the best of our knowledge, Malliavin differentiability of
ABSDEs has not been studied. Moreover, It is a challenging problem to solve
ABSDEs numerically (see Agarwal ., 2018). The continuous posting of variation
margin can also be seen as a simplification. One may want to model $m$ as a
càdlàg step process. For the discussion, readers may want to refer to Brigo,
Liu . (2014).

At default, collateral is not exchanged. Thus, we set the collateral amount
posted at $\tau\leq T$, as $m^{N}_{\tau-}$. Therefore, the cash-flow at
default can be

$\displaystyle\Delta\textfrak{D}^{N}_{\tau}+e^{N}_{\tau}-m^{N}_{\tau-}.$

However, it is immaterial whether we separate $\Delta\textfrak{D}^{N}_{\tau}$
from $e^{N}$ or not in the modeling, because jumps of $\mathbb{F}$-adapted
càdlàg processes are exhausted by $\mathbb{F}$-stopping times (see He Yan,
1992, Theorem 4.21). Thus, by (2.1),

$\displaystyle\Delta\textfrak{D}^{N}_{\tau}=0,~{}~{}\text{a.s.}$

Let $\mathfrak{C}^{N}$ denote the accumulated cash-flows. Then, for any $t\leq
T$ a.s,

$\displaystyle\mathfrak{C}^{N}_{t}\coloneqq\mathds{1}_{\tau>t}\textfrak{D}^{N}_{t}+\mathds{1}_{\tau\leq
t}(\textfrak{D}^{N}_{\tau}+e^{N}_{\tau})-\mathds{1}_{\tau=\tau^{H}\leq
t}L^{H}(e^{N}_{\tau}-m^{N}_{\tau-})^{+}+\mathds{1}_{\tau=\tau^{C}\leq
t}L^{C}(e^{N}_{\tau}-m^{N}_{\tau-})^{-},$

where $0\leq L^{H}\leq 1$ (resp. $0\leq L^{C}\leq 1$ ) is the loss rate of the
hedger (resp. counterparty). Recall that $\mathfrak{C}^{N}$ is derived from
the assumption that the new contract is the first contract between the hedger
and counterparty. Now, we relax the assumption so that we can consider
incremental cash-flows.

####  2.2.2 Incremental Cash-flows

Assume that the two parties has made contracts given by some endowed càdlàg
$\mathbb{F}$-adapted processes $(\textfrak{D}^{E},e^{E},m^{E})$ before
initiation of the new contract. If the two parties did not enter the contract,
for the bank of the hedger, the cash-flows would be

$\displaystyle\mathfrak{C}^{E}_{t}\coloneqq\mathds{1}_{\tau>t}\textfrak{D}^{E}_{t}+\mathds{1}_{\tau\leq
t}(\textfrak{D}^{E}_{\tau}+e^{E}_{\tau})-\mathds{1}_{\tau=\tau^{H}\leq
t}L^{H}(e^{E}_{\tau}-m^{E}_{\tau-})^{+}+\mathds{1}_{\tau=\tau^{C}\leq
t}L^{C}(e^{E}_{\tau}-m^{E}_{\tau-})^{-}.$

On the other hand, when entering the new contract, the exposure and margin
would be $e^{E}+e^{N}$ and $m^{E}+m^{N}$, respectively. In this case, the
summed cash-flows for the bank are

$\displaystyle\mathfrak{C}^{S}_{t}\coloneqq$
$\displaystyle\mathds{1}_{\tau>t}(\textfrak{D}^{E}+\textfrak{D}^{N}_{t})+\mathds{1}_{\tau\leq
t}(\textfrak{D}^{E}_{\tau}+\textfrak{D}^{N}_{\tau}+e^{E}_{\tau}+e^{N}_{\tau})$
$\displaystyle-\mathds{1}_{\tau=\tau^{H}\leq
t}L^{H}(e^{E}_{\tau}+e^{N}_{\tau}-m^{E}_{\tau-}-m^{N}_{\tau-})^{+}+\mathds{1}_{\tau=\tau^{C}\leq
t}L^{C}(e^{E}_{\tau}+e^{N}_{\tau}-m^{E}_{\tau-}-m^{N}_{\tau-})^{-}.$

Thus, the amount that should be handled by the hedger to enter the “new”
contract can be given by

$\displaystyle\mathfrak{C}_{t}\coloneqq$
$\displaystyle\mathfrak{C}^{S}_{t}-\mathfrak{C}^{E}_{t}$ $\displaystyle=$
$\displaystyle\mathds{1}_{\tau>t}\textfrak{D}^{N}_{t}+\mathds{1}_{\tau\leq
t}(\textfrak{D}^{N}_{\tau}+e^{N}_{\tau})-\mathds{1}_{\tau=\tau^{H}\leq
t}L^{H}\big{(}(e^{E}_{\tau}+e^{N}_{\tau}-m^{E}_{\tau-}-m^{N}_{\tau-})^{+}-(e^{E}_{\tau}-m^{E}_{\tau-})^{+}\big{)}$
$\displaystyle+\mathds{1}_{\tau=\tau^{C}\leq
t}L^{C}\big{(}(e^{E}_{\tau}+e^{N}_{\tau}-m^{E}_{\tau-}-m^{N}_{\tau-})^{-}-(e^{E}_{\tau}-m^{E}_{\tau-})^{-}\big{)}.$
(2.3)

We denote the cash-flow process after $\tau$ by $\Theta$, i.e.,

$\displaystyle\Theta\coloneqq\mathfrak{C}-(\textfrak{D}^{N})^{\tau}.$ (2.4)

We assume that the hedger can access to (defaultable) zero coupon bonds of
hedger and counterparty. Let $S^{H}$ (resp. $S^{C}$) denote the defaultable
bond of the hedger (resp. counterparty), where $S^{H}$ and $S^{C}$ follow the
next SDE:

$\displaystyle\mathop{}\\!\mathrm{d}S^{i}_{t}=$ $\displaystyle
r_{t}S^{i}_{t}\mathop{}\\!\mathrm{d}t+\sigma^{i}_{t}S^{i}_{t}\mathop{}\\!\mathrm{d}W_{t}-S^{i}_{t-}\mathop{}\\!\mathrm{d}M^{i}_{t},~{}~{}i\in\\{H,C\\},$
(2.5)

where $(\sigma^{i})^{\top}\in\mathbb{R}^{n}$ are $\mathbb{F}$-progressively
measurable processes. $S^{1},\dots,S^{n}$ are used to hedge
$\textfrak{D}^{N}$, while $S^{H}$ and $S^{C}$ are used to hedge $\Theta$. We
define the $(n+2)\times n$ matrix $\sigma$ as

$\displaystyle\sigma\coloneqq\begin{bmatrix}\sigma^{1}\\\ \vdots\\\
\sigma^{n}\\\ \sigma^{H}\\\ \sigma^{C}\end{bmatrix}.$

Note that the considered financial markets are complete, since there are $n+2$
sources of randomness, $W,\tau^{H},\tau^{C}$, and $n+2$ traded assets
$S,S^{H},S^{C}$.

######  Remark 2.7.

By (2.2), underlying assets do not depend on default risk, which means that we
do not deal with credit derivatives. For modeling with emphasis on contagion
risk, readers can refer to Jiao . (2013); Bo . (2017); Brigo, Capponi
Pallavicini (2014); Bo . (2018).

####  2.2.3 Accounts and Hedging Strategy

In this section, we introduce several saving accounts and the hedger’s hedging
strategy. In what follows, we denote

$\displaystyle I\coloneqq\\{1,2,\dots,H,C\\}.$ (2.6)

For any $i\in I$, let $\eta^{S,i}$ denote the number of units of $S^{i}$ held
by the hedger, and we assume $\eta^{S,i}$ is $\mathbb{G}$-predictable. We
denote

$\displaystyle\varphi\coloneqq$
$\displaystyle(\eta^{S,1},\dots,\eta^{S,n,},\eta^{S,H},\eta^{S,C}),$
$\displaystyle\pi^{i}\coloneqq$
$\displaystyle\eta^{S,i}S^{i},~{}~{}i\in\\{1,\dots,n\\},$
$\displaystyle\pi^{i}\coloneqq$
$\displaystyle\eta^{S,i}S^{i}_{-},~{}~{}i\in\\{H,C\\},$
$\displaystyle\pi\coloneqq$
$\displaystyle(\pi^{1},\dots,\pi^{n},\pi^{H},\pi^{C}).$

We call the $(n+2)$-dimensional $\mathbb{G}$-predictable process $\varphi$ the
hedger’s hedging strategy.

######  Remark 2.8.

$\varphi$ is chosen to be $\mathbb{G}$-adapted only to describe an immediate
action taken at default. We shall see later that on $[0,\tau)$, $\varphi$ is
$\mathbb{F}$-adapted.

If the collateral is pledged, the posting party is remunerated by the
receiving party according to a certain interest rate. When $m^{N}_{t}\geq 0$
(resp. $<0$), the counterparty (resp. hedger) pays the interest rate
$R^{m,\ell}_{t}$ (resp. $R^{m,b}_{t}$) at $t\leq T$. We assume that the
collateral is posted as cash and the interest rate is accrued to a margin
account of the hedger. We denote the lending and borrowing accounts
$B^{m,\ell}$ and $B^{m,b}$ respectively, i.e., $B^{m,i}$, $i\in\\{\ell,b\\}$,
are given by

$\displaystyle\mathop{}\\!\mathrm{d}B^{m,i}_{t}=R^{m,i}_{t}B^{m,i}_{t}\mathop{}\\!\mathrm{d}t.$
(2.7)

Let $\eta^{m,\ell}$ (resp. $\eta^{m,b}$) denote the number of units of
$B^{m,\ell}$ (resp. $B^{m,b}$). Then the following equations hold:

$\displaystyle\eta^{m,\ell}\geq
0,~{}~{}\eta^{m,b}<0,~{}~{}\eta^{m,\ell}\eta^{m,b}=0,$ (2.8)
$\displaystyle\eta^{m,\ell}B^{m,\ell}+\eta^{m,b}B^{m,b}=m^{N}.$ (2.9)

We assume that the variation margin $m$ can be rehypothecated, i.e., $m^{N}$
is used by the hedger to maintain her portfolio.

######  Remark 2.9.

The margin account for $m^{E}$ may be dealt with by other dealers, so it is
not a part of the hedging portfolio.

Some underlying assets can be traded through repo markets. We denote the set
of indices for which a repo market is available by $\rho\subseteq
I\coloneqq\\{1,2,\dots,n,H,C\\}$. We assume that the borrowing and lending
repo market rates are the same, and for $i\in\rho$, let $R^{\rho,i}$ denote
the repo rate. Moreover, for any $i\in\rho$, let $B^{\rho,i}$ denote the
account that $R^{\rho,i}$ accrues, i.e., $B^{\rho,i}$ follows

$\displaystyle\mathop{}\\!\mathrm{d}B^{\rho,i}_{t}=R^{\rho,i}_{t}B^{\rho,i}_{t}\mathop{}\\!\mathrm{d}t.$
(2.10)

For $i\in\rho$, we denote the number of units of $B^{\rho,i}$ by
$\eta^{\rho,i}$. Then it follows that for any $i\in\rho$,

$\displaystyle\eta^{\rho,i}B^{\rho,i}+\eta^{S,i}S^{i}=0.$ (2.11)

If the hedger has any surplus cash, she can earn the lending rate $R^{\ell}$,
while for borrowing money, she needs to pay the borrowing rate $R^{b}$. For
$i\in\\{\ell,b\\}$, let $B^{i}$ denote the hedger’s funding account and
$\eta^{i}$ denote the number of units of $B^{i}$. Therefore, it follows that

$\displaystyle\eta^{\ell}\geq 0,~{}~{}\eta^{b}<0,$ (2.12)
$\displaystyle\mathop{}\\!\mathrm{d}B^{i}_{t}=R^{i}_{t}B^{i}_{t}\mathop{}\\!\mathrm{d}t,~{}~{}i\in\\{\ell,b\\}.$
(2.13)

###  2.3 Hedger’s Incremental Hedging Portfolio

Now, we are almost ready to define dealer’s incremental portfolio. Another
ingredient in defining hedging portfolios is incremental funding effects.
These effects will be considered by imposing some conditions on the hedging
portfolio. The detail will be followed after the next definition.

######  Definition 2.10.

If $V=V(\varphi,\mathfrak{C})$ defined on $t\in\mathbb{R}_{+}$, by

$\displaystyle
V_{t}=\eta^{\ell}_{t}B^{\ell}_{t}+\eta^{b}_{t}B^{b}_{t}+\eta^{m,\ell}_{t}B^{m,\ell}_{t}+\eta^{m,b}_{t}B^{m,b}_{t}+\displaystyle\sum_{i\in
I}\eta^{S,i}_{t}S^{i}_{t}+\displaystyle\sum_{i\in\rho}\eta^{\rho,i}_{t}B^{\rho,i}_{t},$
(2.14)

satisfies

$\displaystyle V_{t}=$ $\displaystyle
V_{0}+\displaystyle\sum_{i=\ell,b}\int_{0}^{t\wedge\bar{\tau}}\eta^{i}_{s}\mathop{}\\!\mathrm{d}B^{i}_{s}+\displaystyle\sum_{i=\ell,b}\int_{0}^{t\wedge\bar{\tau}}\eta^{m,i}_{s}\mathop{}\\!\mathrm{d}B^{m,i}_{s}+\displaystyle\sum_{i\in
I}\int_{0}^{t\wedge\bar{\tau}}\eta^{S,i}_{s}\mathop{}\\!\mathrm{d}S^{i}_{s}$
$\displaystyle+\displaystyle\sum_{i\in\rho}\int_{0}^{t\wedge\bar{\tau}}\eta^{\rho,i}_{s}\mathop{}\\!\mathrm{d}B^{\rho,i}_{s}-\mathfrak{C}_{t\wedge\bar{\tau}},$
(2.15)

for any $t\in\mathbb{R}_{+}$, then $V$ is called the hedger’s incremental
hedging portfolio.

######  Remark 2.11.

Note that by (2.3), $\mathfrak{C}_{t}=\mathfrak{C}_{t\wedge\bar{\tau}}$,
$\forall t\geq 0$, and by (2.15), $V_{t}=V_{t\wedge\bar{\tau}}$, $\forall
t\geq 0$.

Our goal is to find a proper price charged to the counterparty and hedging
strategy $\varphi$ satisfying Definition 2.10 and a certain terminal
condition. We seek to impose the terminal condition so that an incremental
funding effect should be considered. The incremental funding effect means the
difference between the funding cost/benefit of two choices: entering or not
entering the new contract. To explain the necessity of the incremental effect
briefly, consider a situation that the dealer wants to enter a new contract
that makes the hedging portfolio fall in borrowing state. If there have been
no business of the bank before the contract, the treasury department would
finance the dealer with the borrowing rate, and the excessive borrowing cost
might be charged to the counterparty. However, if the bank’s initial position
was in lending state before the contract, the treasury department should
consider deduction of lending profit rather than excessive borrowing cost. The
mentioned activity can be seen that the dealer borrows and keeps the bank’s
initial portfolio for funding the portfolio, and returns it to treasury at
maturity.

To explain the mathematical detail, let $B^{\epsilon}$ denote the endowed
bank’s portfolio without entering the new contract, for some
$\epsilon\in\mathbb{R}$ such that $B^{\epsilon}_{0}=\epsilon$. We sometimes
call $B^{\epsilon}$ legacy portfolio. In reality, $B^{\epsilon}$ is a massive
combination of numerous portfolios. We assume that the legacy portfolio is
approximately risk-neutral and grows with respect to their funding rates.
Therefore,

$\displaystyle B^{\epsilon}_{t}\coloneqq$
$\displaystyle\epsilon\exp{\Big{(}\int_{0}^{t}R^{\epsilon}_{s}\mathop{}\\!\mathrm{d}s\Big{)}},~{}\forall
t\in\mathbb{R}_{+},$ $\displaystyle R^{\epsilon}\coloneqq$
$\displaystyle\mathds{1}_{\epsilon\geq
0}R^{\ell}+\mathds{1}_{\epsilon<0}R^{b},$

and we denote $s^{\epsilon}\coloneqq R^{\epsilon}-r$.

Now, let us think of $V$ as the bank’s summed profit/loss and consider two
cases. First, if the dealer does not enter the contract exchanging
$\mathfrak{C}$, the bank will have $B^{\epsilon}_{\bar{\tau}}$ at
$\bar{\tau}$. Second, the dealer can enter the contract with a certain initial
price, say $p\in\mathbb{R}$, for the contract from the counterparty. Then, the
bank’s initial wealth is $\epsilon+p$, namely

$\displaystyle V_{0}=\epsilon+p.$

The dealer gains from investing in risky assets and accounts. These revenues
are used to pay the cash-flows $\mathfrak{C}$. Thus, in (2.15), represents the
hedging error between the investment and cash-flows. Recall that we consider
complete markets (there are $n+2$ sources of randomness,
$(W,\tau^{H},\tau^{C})$, and $n+2$ hedging assets $\pi$). Therefore, we can
expect that we can find a hedging strategy that replicates the cash-flows with
null hedging error up to termination of the contract, either by the maturity
or default. For example, if $\epsilon=0$, we can expect to find $(V,\pi)$ such
that $V_{\bar{\tau}}=0$. However, since the opportunity cost for the bank is
the profit/loss from legacy portfolio, the portfolio value at $\bar{\tau}$
should be compared with $B^{\epsilon}_{\bar{\tau}}$, i.e., we seek to find
$(V,\pi)$ such that

$\displaystyle V_{\bar{\tau}}=B^{\epsilon}_{\bar{\tau}},$ (2.16)

and the dealer may want to charge

$\displaystyle p=V_{0}-\epsilon$ (2.17)

to the counterparty. Therefore, by (2.15) together with (2.2), (2.5), (2.7),
(2.8), (2.9), (2.10), (2.11), (2.13), we need to solve the following BSDE:

$\displaystyle\left\\{\begin{array}[]{r@{}l}\mathop{}\\!\mathrm{d}V_{t}=&\bigg{[}\big{(}V_{t}-m^{N}_{t}-\displaystyle\sum_{i\in
I\setminus\rho}\pi^{i}_{t}\big{)}^{+}R^{\ell}_{t}-\big{(}V_{t}-m^{N}_{t}-\displaystyle\sum_{i\in
I\setminus\rho}\pi^{i}_{t}\big{)}^{-}R^{b}_{t}+r_{t}\displaystyle\sum_{i\in
I}\pi^{i}_{t}-\displaystyle\sum_{i\in\rho}R^{\rho,i}_{t}\pi^{i}_{t}\bigg{]}\mathop{}\\!\mathrm{d}t\\\
&+\bigg{[}R^{m,\ell}m^{+}_{t}-R^{m,b}m^{+}_{t}\bigg{]}\mathop{}\\!\mathrm{d}t+\displaystyle\sum_{i\in
I}\pi^{i}_{t}\sigma^{i}_{t}\mathop{}\\!\mathrm{d}W_{t}-\displaystyle\sum_{i=H,C}\pi^{i}_{t}\mathop{}\\!\mathrm{d}M^{i}_{t}-\mathop{}\\!\mathrm{d}\mathfrak{C}_{t},\\\
V_{\bar{\tau}}=&B^{\epsilon}_{\bar{\tau}}.\end{array}\right.$ (2.21)

For now, we do not examine the existence and uniqueness of (2.21). The
solvability will be examined by a reduced form of (2.21). The financial
interpretation of each component in (2.21) will be provided later in the
following section. Before the detail, we first discuss the incremental funding
impacts by a simple example.

######  Example 2.12 (Incremental FVA).

Let $n=1$, $\rho=\emptyset$, $\textfrak{D}^{N}=\mathds{1}_{\llbracket
T,\infty\llbracket}\xi B_{T}$ for some $\xi\in\mathbb{L}^{2}_{T}$. We ignore
default risks and set $\pi^{H}=\pi^{C}=m^{N}=0$. In this case, $(V,\pi^{1})$
is given by

$\displaystyle V_{t}=\xi
B_{T}+B_{T}^{\epsilon}-\int_{t}^{T}\Big{[}(V_{s}-\pi^{1}_{s})^{+}R^{\ell}_{s}-(V_{s}-\pi^{1}_{s})^{-}R^{b}_{s}+r_{s}\pi^{1}_{s}\Big{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}\pi^{1}_{s}\sigma^{1}_{s}\mathop{}\\!\mathrm{d}W_{s}.$

To consider a net profit/loss to the hedger without the legacy portfolio,
consider

$\displaystyle v\coloneqq B^{-1}(V-B^{\epsilon}),$ (2.22)

and we denote that $\tilde{\pi}^{1}\coloneqq B^{-1}\pi^{1}$,
$\tilde{B}^{\epsilon}=B^{-1}B^{\epsilon}$. Then $(v,\tilde{\pi}^{1})$ is given
by

$\displaystyle
v_{t}=\xi-\int_{t}^{T}\Big{[}(v_{s}+\tilde{B}^{\epsilon}_{s}-\tilde{\pi}^{1}_{s})^{+}s^{\ell}_{s}-(v_{s}+\tilde{B}^{\epsilon}_{s}-\tilde{\pi}^{1}_{s})^{-}s^{b}_{s}-s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s}\Big{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}\tilde{\pi}^{1}_{s}\sigma^{1}_{s}\mathop{}\\!\mathrm{d}W_{s}.$
(2.23)

Assuming there exists a unique solution
$(v,\tilde{\pi}^{1})\in\mathbb{S}^{2}_{T}\times\mathbb{H}^{2}_{T}$ of (2.23),
for any $t\geq 0$, we define

$\displaystyle\text{FBA}^{\Delta}_{t}\coloneqq$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{T}\big{[}(v_{s}+\tilde{B}^{\epsilon}_{s}-\tilde{\pi}^{1}_{s})^{+}s^{\ell}_{s}-(s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s})^{+}\big{]}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{F}_{t}\bigg{]},$
$\displaystyle\text{FCA}^{\Delta}_{t}\coloneqq$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{T}\big{[}(v_{s}+\tilde{B}^{\epsilon}_{s}-\tilde{\pi}^{1}_{s})^{-}s^{b}_{s}-(s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s})^{-}\big{]}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{F}_{t}\bigg{]}.$

$\text{FBA}^{\Delta}$ (resp. $\text{FCA}^{\Delta}$) represents the incremental
funding benefits (resp. costs) for entering the new contract. Notice that as
$\epsilon$ increases, $v+\tilde{B}^{\epsilon}-\pi^{1}$ is more likely to
positive. Consider a case that $v-\pi^{1}$ is negative but
$v+\tilde{B}^{\epsilon}-\pi^{1}$ is non-negative. If we ignore the incremental
effect, i.e., $\epsilon=0$, the dealer should consider the increased funding
cost. However, in the view of incremental effects, instead, the deduction of
funding benefit should be included in the derivative price. However, the
dealer also need to consider the opportunity cost for not entering the new
contract, e.g., if $\epsilon\geq 0$, the lost (discounted) benefit at $t\leq
T$ would be

$\displaystyle\tilde{B}^{\epsilon}_{T}-\tilde{B}^{\epsilon}_{t}=\int_{t}^{T}\mathop{}\\!\mathrm{d}\tilde{B}^{\epsilon}_{s}=\int_{t}^{T}s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s}\mathop{}\\!\mathrm{d}s=\int_{t}^{T}(s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s})^{+}\mathop{}\\!\mathrm{d}s.$
(2.24)

The difference between the two impacts is the actual net benefit and cost,
$\text{FBA}^{\Delta}$ and $\Delta\text{FCA}^{\Delta}$, that should be charged
to the counterparty. Indeed, by (2.17), (2.22) and (2.23), the dealer would
want to charge

$\displaystyle
p=\mathbb{E}[\xi]-\text{FBA}^{\Delta}_{0}+\text{FCA}^{\Delta}_{0}.$ (2.25)

In addition, if $R^{\ell}=R^{b}$, (2.23) becomes

$\displaystyle
v_{t}=\xi-\int_{t}^{T}(v_{s}-\tilde{\pi}^{1}_{s})s^{\ell}_{s}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}\tilde{\pi}^{1}_{s}\sigma^{1}_{s}\mathop{}\\!\mathrm{d}W_{s}.$
(2.26)

Thus, under linear funding models, $v$ does not depend on $B^{\epsilon}$.

In what follows, for simplicity, we impose some realistic assumptions on
interest rates, and the endowed processes, $e^{E}$ and $m^{E}$. In practice,
$R^{m,i}$, $i\in\\{\ell,b\\}$, are chosen as Federal funds or EONIA rates,
approximately $r$. In addition, the difference between OIS and repo market
rates can be interpreted as the liquidity premium of the repo markets. We
assume the repo markets are liquid enough for the difference to be small.
Moreover, we assume that OIS rate, $(r_{t})_{t\geq 0}$, is the smallest among
all interest rates. In addition, $e^{E}$ and $m^{E}$ are given before the new
contract, so they are chosen exogenously, i.e., they do not depend on $V$.
These assumptions are summarized as follows.

######  Assumption 2.13.

  1. (i)

$R^{\rho,i}=R^{m,\ell}=R^{m,b}=r$, for $i\in\rho$.

  2. (ii)

$R^{\ell}\geq r$ and $R^{b}\geq r$.

  3. (iii)

$e^{E}$ and $m^{E}$ are exogenous processes.

######  Remark 2.14.

The assumption on repo market rates given in 2.13-(i) is merely for simplicity
in representing (2.21). Mathematically, it does not play any crucial role.

Recall that we have not yet specified the amount of close-out $e^{N}$ in
$\mathfrak{C}$. In the next section, two important close-out conventions are
introduced: clean price and replacement cost. Clean price is basically the
risk-neutral price of $\textfrak{D}^{N}$. By using the SDE that the clean
price follows, we can remove $\textfrak{D}^{N}$ from $\mathfrak{C}$. After the
elimination of $\textfrak{D}^{N}$, the rest of cash-flows is $\Theta$ which is
exchanged only at one point $\tau$. Thus, by subtracting the clean price from
$V$, we can recover a standard BSDE with
$(\mathbb{G},\mathbb{Q})$-martingales, $W$ and $M^{i}$, $i\in\\{H,C\\}$. Then
we further reduce the BSDE to a BSDE only with the Brownian motions by the
typical argument of filtration reduction.

####  2.3.1 Close-out Conventions

When the contract terminates earlier than the contractual maturity by one
party’s default, the defaulting party should settle the close-out amount. The
close-out amount is calculated by a determining party which will act in good
faith (ISDA, 2009). As stated in ISDA (2009, p.15), “the Determining Party may
consider any relevant information, including, without limitation, one or more
of the following types of information: quotations (either firm or indicative)
for replacement transactions supplied by one or more third parties that may
take into account the creditworthiness of the Determining Party at the time
the quotation is provided and the terms of any relevant documentation …”. The
statement leaves some rooms for interpretation. ISDA recommends to consider
creditworthiness of the surviving party, but it is not mandatory. Moreover, it
is unclear whether funding costs should be considered in the replacement
transaction. Therefore, we consider two close-out conventions: clean price and
replacement cost. Clean price is the risk-neutral price without XVA. The
close-out with clean price is often called clean close-out and risk-free
close-out. Let $p^{N}$ denote the clean price, i.e.,

$\displaystyle p^{N}_{t}\coloneqq
B_{t}\mathbb{E}\bigg{(}\int_{t}^{T}B_{s}^{-1}\mathop{}\\!\mathrm{d}\textfrak{D}^{N}_{s}\bigg{|}\mathcal{F}_{t}\bigg{)},~{}~{}\forall
t\in\mathbb{R}_{+}.$ (2.27)

Clean close-out (or risk-free close-out) has been often chosen in literature
(for example, Crépey (20152, 20151)).

######  Remark 2.15.

Bichuch . (2018) also considered expected value of discounted cash-flow in
absence of default risk, but in the calculation, they assumed flexibility to
choose the discounting rate and probability measure. Indeed, they chose the
pricing measure as an equivalent probability measure such that
$(B^{i})^{-1}S$, $i\in\\{\ell,b\\}$, are martingales. In other words, the
amount is “clean price$+$FVA”. This choice is for avoiding mismatch of pricing
measures to obtain a closed-form solution. We will explain this point later
with examples.

Replacement cost (or replacement close-out) means the price under XVA. In this
case, it is not clear which funding rate should be chosen. We assume that the
replacing party has similar credit spreads to the hedger. Recall that
$V-B^{\epsilon}$ is the value for calculating the derivative price in view of
the hedger. In other words, one may want to choose the value of
$V-B^{\epsilon}$ at the default for the replacement cost. However, at the
default, there is a jump in $V$ by $\pi^{i}$, $i\in\\{H,C\\}$. Moreover,
$\pi^{i}$ is retained in $V$ for the default risk, i.e., the close-out
payment. Therefore, if we take

$\displaystyle e^{N}_{\tau}=$ $\displaystyle
V_{\tau}-B^{\epsilon}_{\tau}=V_{\tau-}-B^{\epsilon}_{\tau}+\Delta V_{\tau}$
$\displaystyle=$ $\displaystyle
V_{\tau-}-B^{\epsilon}_{\tau}-\mathds{1}_{\tau=\tau^{H}}\pi^{H}_{\tau}-\mathds{1}_{\tau=\tau^{C}}\pi^{C},$

the defaulting party would pay the remainder after the deduction of the same
amount, basically nothing. Hence, for replacement close-out, we should set

$\displaystyle e^{N}_{\tau}=V_{\tau-}-B^{\epsilon}_{\tau}.$ (2.28)

In both conventions, we assume that the collateral is a portion of the close-
out amount. More precisely, for $0\leq L^{m}\leq 1$ (margin loss),

$\displaystyle m^{N}=(1-L^{m})e^{N}.$ (2.29)

Note that (2.29) is consistent with our financial modeling. Indeed, if there
is a $\mathbb{G}$-adapted process satisfying (2.21), in our filtration setup
$\mathbb{G}$, there is an $\mathbb{F}$-adapted process $V^{\mathbb{F}}$ such
that

$\displaystyle V=\mathds{1}_{\llbracket
0,\bar{\tau}\llbracket}V^{\mathbb{F}}.$

Therefore, under replacement close-out, the margin process

$\displaystyle m^{N}=$
$\displaystyle(1-L^{m})e^{N}=(1-L^{m})(V_{-}-B^{\epsilon})$ $\displaystyle=$
$\displaystyle(1-L^{m})(V^{\mathbb{F}}_{-}-B^{\epsilon})$

is $\mathbb{F}$-adapted before $\bar{\tau}$. In what follows, we assume that
the endowed margin also follows the same convention as (2.29), i.e.,

$\displaystyle m^{E}=(1-L^{m})e^{E}.$ (2.30)

######  Remark 2.16.

  1. (i)

The two close-out conventions have pros and cons in financial modeling. Clean
price may be disadvantageous to the defaulting party because the default risk
of surviving party is not considered. However, the surviving party’s default
risk can be negatively affected by the default, especially when defaulting
party has a systemic impact. In that case, replacement close-out may heavily
penalize the surviving party. Readers can refer to Brigo Morini (2011) for the
comparison.

  2. (ii)

A similar collateral convention was discussed by Burgard Kjaer (2010). For
BSDEs approach on general endogenous collateral, readers can refer to Nie
Rutkowski (2016).

Before further argument, we provide a lemma on properties of clean price
$p^{N}$. The following lemma will be used to present an XVA process and
simplify the representation of the amount of cash-flow at default $\Theta$.
Readers may want to recall (2.27), the definition of $p^{N}$, before the
following lemma.

######  Lemma 2.17.

  1. (i)

$p^{N}_{T}=0$.

  2. (ii)

$p^{N}_{\bar{\tau}}=\mathds{1}_{\tau\leq T}p^{N}_{\tau}$.

  3. (iii)

$\mathop{}\\!\mathrm{d}p^{N}_{t}=r_{t}p^{N}_{t}\mathop{}\\!\mathrm{d}t+B_{t}(Z^{N}_{t})^{\top}\mathop{}\\!\mathrm{d}W_{t}-\mathop{}\\!\mathrm{d}\textfrak{D}^{N}_{t}$,
$\forall t\leq T$, for some $Z^{N}\in\mathbb{H}^{2,n}_{T,loc}$.

  4. (iv)

$p^{N}_{\tau-}=p^{N}_{\tau}$ almost surely.

###### Proof.

(i) is from the definition and (ii) is a directly obtained from (i). For
(iii), notice that
$B^{-1}p^{N}+\int_{0}^{\cdot}B^{-1}_{s}\mathop{}\\!\mathrm{d}\textfrak{D}^{N}_{s}$
is an $(\mathbb{F},\mathbb{Q})$-local martingale. Thus, by (local) martingale
representation property, there exists $Z^{N}\in\mathbb{H}^{2,n}_{T,loc}$ such
that for any $t$,

$\displaystyle
B_{t}^{-1}p^{N}_{t}+\int_{0}^{t}B^{-1}_{s}\mathop{}\\!\mathrm{d}\textfrak{D}^{N}_{s}=\int_{0}^{t}(Z^{N}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$

Therefore, $p^{N}_{t}$ follows the SDE:

$\displaystyle\mathop{}\\!\mathrm{d}p^{N}_{t}=$ $\displaystyle
r_{t}p^{N}_{t}\mathop{}\\!\mathrm{d}t+B_{t}(Z^{N}_{t})^{\top}\mathop{}\\!\mathrm{d}W_{t}-\mathop{}\\!\mathrm{d}\textfrak{D}^{N}_{t}$
(2.31)

By (iii), $p^{N}$ is an $\mathbb{F}$-adapted càdlàg process, but $\tau$ avoids
any $\mathbb{F}$-stopping time. Thus, $\Delta p^{N}_{\tau}=0$ almost surely,
equivalently $p^{N}_{\tau-}=p^{N}_{\tau}$ a.s. ∎

######  Remark 2.18.

Note that $Z^{N}$ is the (discounted) delta risk of clean price. For example,
consider $n=1$ and a stock forward contract with exercise price $K$, and
assume $r,~{}\sigma^{1}$ are deterministic. Then
$B^{-1}p^{N}=B^{-1}S^{1}-B_{T}^{-1}K$. Thus, for $t<T$,

$\displaystyle
Z^{N}_{t}=D_{t}(B^{-1}_{t}p^{N}_{t})=\sigma^{1}_{t}B^{-1}_{t}S^{1}_{t}.$

Recall that $e^{E}$ and $B^{\epsilon}$ are $\mathbb{F}$-adapted, so they do
not jump at $\tau$, i.e., $e^{E}_{\tau}=e^{E}_{\tau-}$ and
$B^{\epsilon}_{\tau}=B^{\epsilon}_{\tau-}$, a.s. Then, by (2.4) together with
(iv) in Lemma 2.17, under clean close-out, a.s,

$\displaystyle\Theta_{t}=$ $\displaystyle\mathds{1}_{\tau=\tau^{H}\leq
t}\big{[}p^{N}_{\tau-}-L^{H}L^{m}\big{(}(p^{N}_{\tau-}+e^{E}_{\tau-})^{+}-(e^{E}_{\tau-})^{+}\big{)}\big{]}$
$\displaystyle+\mathds{1}_{\tau=\tau^{C}\leq
t}\big{[}p^{N}_{\tau-}+L^{C}L^{m}\big{(}(p^{N}_{\tau-}+e^{E}_{\tau-})^{-}-(e^{E}_{\tau-})^{-}\big{)}\big{]}.$
(2.32)

On the other hand, under replacement close-out,

$\displaystyle\Theta_{t}=$ $\displaystyle\mathds{1}_{\tau=\tau^{H}\leq
t}\big{[}V_{\tau-}-B^{\epsilon}_{\tau-}-L^{H}L^{m}\big{(}(V_{\tau-}-B^{\epsilon}_{\tau-}+e^{E}_{\tau-})^{+}-(e^{E}_{\tau-})^{+}\big{)}\big{]}$
$\displaystyle+\mathds{1}_{\tau=\tau^{C}\leq
t}\big{[}V_{\tau-}-B^{\epsilon}_{\tau-}+L^{C}L^{m}\big{(}(V_{\tau-}-B^{\epsilon}_{\tau-}+e^{E}_{\tau-})^{-}-(e^{E}_{\tau-})^{-}\big{)}\big{]}.$
(2.33)

####  2.3.2 Incremental XVA process

We can remove $\textfrak{D}^{N}$ in (2.21) by using (iii) in Lemma 2.17. To
this end, we introduce an incremental XVA process. In both close-out
conventions, we will deal with the XVA process instead of (2.21). The XVA
process is defined by the discounted difference between $V-B^{\epsilon}$ and
$p^{N}$. Let $X$ denote the (incremental) XVA process, i.e.,

$\displaystyle X\coloneqq B^{-1}[V-B^{\epsilon}-(p^{N})^{\bar{\tau}}].$

Moreover, let $\tilde{p}^{N}\coloneqq B^{-1}p^{N}$, $\tilde{\pi}\coloneqq
B^{-1}\pi$, $c\coloneqq B^{-1}m^{N}$, $\tilde{\Theta}\coloneqq B^{-1}\Theta$,
$\tilde{B}^{\epsilon}\coloneqq B^{-1}B^{\epsilon}$, and

$\displaystyle s^{i}\coloneqq R^{i}-r,~{}~{}i\in\\{\ell,b\\}.$

Note that $s^{\ell}$ (resp. $s^{b}$) represents the lending (resp. borrowing)
spread of the hedger. We can easily check that for $t\geq 0$,
$\mathop{}\\!\mathrm{d}\tilde{p}^{N}_{t\wedge\bar{\tau}}=\mathds{1}_{t\leq\bar{\tau}}\mathop{}\\!\mathrm{d}\tilde{p}^{N}_{t}$.
Assuming there exists $(V,\pi)$ satisfying (2.21), by applying Itô’s formula
to $X$, we attain that for $t\leq\bar{\tau}$, $(X,\tilde{\pi})$ follows

$\displaystyle\left\\{\begin{array}[]{r@{}l}\mathop{}\\!\mathrm{d}X_{t}=&\bigg{[}\big{(}X_{t}+\tilde{p}^{N}_{t}+\tilde{B}^{\epsilon}_{s}-c_{t}-\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}_{t}\big{)}^{+}s^{\ell}_{t}-\big{(}X_{t}+\tilde{p}^{N}_{t}+\tilde{B}^{\epsilon}_{s}-c_{t}-\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}_{t}\big{)}^{-}s^{b}_{t}-s^{\epsilon}_{t}\tilde{B}^{\epsilon}_{t}\bigg{]}\mathop{}\\!\mathrm{d}t\\\
\vspace{0.2cm}\hfil&+\big{[}\tilde{\pi}^{\top}\sigma_{t}-(Z^{N}_{t})^{\top}\big{]}\mathop{}\\!\mathrm{d}W_{t}-\tilde{\pi}^{H}_{t}\mathop{}\\!\mathrm{d}M^{H}_{t}-\tilde{\pi}^{M}_{t}\mathop{}\\!\mathrm{d}M^{m}_{t},\\\
X_{\bar{\tau}}=&\tilde{\Theta}_{\bar{\tau}}-\tilde{p}^{N}_{\bar{\tau}}.\end{array}\right.$
(2.37)

Note that under replacement close-out,

$\displaystyle\tilde{\Theta}_{t}=$ $\displaystyle\mathds{1}_{\tau\leq
t}\tilde{p}^{N}_{\tau-}+\mathds{1}_{\tau=\tau^{H}\leq
t}\big{[}X_{\tau-}-L^{H}L^{m}\big{(}(X_{\tau-}+\tilde{p}^{N}_{\tau-}+\tilde{e}^{E}_{\tau-})^{+}-(\tilde{e}^{E}_{\tau-})^{+}\big{)}\big{]}$
$\displaystyle+\mathds{1}_{\tau=\tau^{C}\leq
t}\big{[}X_{\tau-}+L^{C}L^{m}\big{(}(X_{\tau-}+\tilde{p}^{N}_{\tau-}+\tilde{e}^{E}_{\tau-})^{-}-(\tilde{e}^{E}_{\tau-})^{-}\big{)}\big{]},$
$\displaystyle\tilde{e}^{E}\coloneqq$ $\displaystyle B^{-1}e^{E}$

while under clean close-out, $\tilde{\Theta}$ is independent of $X$. In both
cases, we denote

$\displaystyle\Theta(X_{-})\coloneqq\Theta.$

Moreover, we define $\Theta^{H}(X_{-})$ and $\Theta^{C}(X_{-})$ such that

$\displaystyle\tilde{\Theta}_{\bar{\tau}}-\tilde{p}^{N}_{\bar{\tau}}=-\mathds{1}_{\tau=\tau^{H}\leq
T}(\tilde{\Theta}^{H}_{\tau}(X_{\tau-})-L^{H}L^{m}(\tilde{e}^{E}_{\tau-})^{+})+\mathds{1}_{\tau=\tau^{C}\leq
T}(\tilde{\Theta}^{C}_{\tau}(X_{\tau-}-L^{C}L^{m}(\tilde{e}^{E}_{\tau-})^{-}).$

where $\tilde{\Theta}^{i}\coloneqq B^{-1}\Theta^{i}$, $i\in\\{H,C\\}$. For
example, under replacement close-out,

$\displaystyle\tilde{\Theta}^{H}_{t}(X_{t-})=$ $\displaystyle-
X_{t-}+L^{H}L^{m}(X_{t-}+\tilde{p}^{N}_{t-}+\tilde{e}^{E}_{t-})^{+},$ (2.38)
$\displaystyle\tilde{\Theta}^{C}_{t}(X_{t-})=$ $\displaystyle
X_{t-}+L^{C}L^{m}(X_{t-}+\tilde{p}^{N}_{t-}+\tilde{e}^{E}_{t-})^{-}.$ (2.39)

while under clean close-out,

$\displaystyle\tilde{\Theta}^{H}_{t}(X_{t-})=$ $\displaystyle
L^{H}L^{m}(\tilde{p}^{N}_{t-}+\tilde{e}^{E}_{t-})^{+},$
$\displaystyle\tilde{\Theta}^{C}_{t}(X_{t-})=$ $\displaystyle
L^{C}L^{m}(\tilde{p}^{N}_{t-}+\tilde{e}^{E}_{t-})^{-}.$

In the case of clean close-out, $\tilde{\Theta}^{i}$, $i\in\\{H,C\\}$,
represent the amount of breach of the contract. Recall $B$ and $p^{N}$ are
independent of $V$. Therefore, showing the existence and uniqueness of the
hedger’s hedging portfolio and hedging strategy, $(V,\pi)$, reduces to
investigating the BSDE of the XVA process (2.37). Before examining the
solvability, assuming the existence and integrability, we define each
component in the incremental XVA and give some remarks on them.

######  Definition 2.19.

Assume that there exists $(X,\tilde{\pi}$) satisfying (2.37). Then for
$t<\bar{\tau}$, we define adjustment processes: FCA, FBA, CVA, DVA, and
incremental adjustment processes: $\emph{FBA}^{\Delta}$,
$\emph{FCA}^{\Delta}$, $\emph{DVA}^{\Delta}$, $\emph{CVA}^{\Delta}$ processes
as follows:

$\displaystyle\emph{FCA}_{t}\coloneqq$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{\bar{\tau}}\big{(}X_{s}+\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-c_{s}-\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}_{s}\big{)}^{-}s^{b}_{s}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{G}_{t}\bigg{]},$
(2.40) $\displaystyle\emph{FBA}_{t}\coloneqq$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{\bar{\tau}}\big{(}X_{s}+\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-c_{s}-\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}_{s}\big{)}^{+}s^{\ell}_{s}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{G}_{t}\bigg{]},$
(2.41) $\displaystyle\emph{DVA}_{t}\coloneqq$
$\displaystyle\mathbb{E}\bigg{[}\mathds{1}_{\bar{\tau}=\tau=\tau^{H}}\tilde{\Theta}^{H}_{\tau}(X_{\tau-})\bigg{|}\mathcal{G}_{t}\bigg{]},$
(2.42) $\displaystyle\emph{CVA}_{t}\coloneqq$
$\displaystyle\mathbb{E}\bigg{[}\mathds{1}_{\bar{\tau}=\tau=\tau^{C}}\tilde{\Theta}^{C}_{\tau}(X_{\tau-})\bigg{|}\mathcal{G}_{t}\bigg{]},$
(2.43)

and denoting
$\mathcal{O}_{t}\coloneqq\big{[}\int_{t}^{\bar{\tau}}s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s}\mathop{}\\!\mathrm{d}s\big{|}\mathcal{G}_{t}\big{]}$,

$\displaystyle\emph{FCA}^{\Delta}_{t}\coloneqq$
$\displaystyle\emph{FCA}_{t}-\mathcal{O}_{t}^{-},$ (2.44)
$\displaystyle\emph{FBA}^{\Delta}_{t}\coloneqq$
$\displaystyle\emph{FBA}-\mathcal{O}_{t}^{+},$ (2.45)
$\displaystyle\emph{DVA}^{\Delta}_{t}\coloneqq$
$\displaystyle\emph{DVA}_{t}-\mathbb{E}\Big{[}\int_{t}^{\bar{\tau}}L^{H}L^{m}(\tilde{e}^{E}_{s})^{+}\mathop{}\\!\mathrm{d}s\Big{|}\mathcal{G}_{t}\Big{]},$
(2.46) $\displaystyle\emph{CVA}^{\Delta}_{t}\coloneqq$
$\displaystyle\emph{CVA}_{t}-\mathbb{E}\Big{[}\int_{t}^{\bar{\tau}}L^{C}L^{m}(\tilde{e}^{E}_{s})^{-}\mathop{}\\!\mathrm{d}s\Big{|}\mathcal{G}_{t}\Big{]},$
(2.47)

where (2.40)-(2.47) are well-defined. In this case, we also define

$\displaystyle\emph{FVA}\coloneqq$ $\displaystyle\emph{FCA}-\emph{FBA},$
$\displaystyle\emph{FVA}^{\Delta}\coloneqq$
$\displaystyle\emph{FCA}^{\Delta}-\emph{FBA}^{\Delta}.$

######  Remark 2.20.

  1. (i)

Assume that the local-martingales in (2.37) are true martingales. Then

$\displaystyle X=$
$\displaystyle\text{FVA}^{\Delta}-\text{DVA}^{\Delta}+\text{CVA}^{\Delta}$
$\displaystyle=$
$\displaystyle\text{FCA}^{\Delta}-\text{FBA}^{\Delta}-\text{DVA}^{\Delta}+\text{CVA}^{\Delta}.$
(2.48)

  2. (ii)

$\mathcal{O}$ is the opportunity cost of not entering the new contract. In
addition, FCA and FBA is the aggregated funding cost and benefit together with
the legacy portfolio. Recalling the definitions

$\displaystyle\text{FCA}^{\Delta}=$ $\displaystyle\text{FCA}-\mathcal{O}^{-},$
$\displaystyle\text{FBA}^{\Delta}=$ $\displaystyle\text{FBA}-\mathcal{O}^{+},$

the incremental funding impacts are the differences between aggregated funding
adjustments and the opportunity cost.

  3. (iii)

Under replacement close-out,

$\displaystyle\text{FCA}^{\Delta}_{t}=$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{\bar{\tau}}\Big{[}\big{(}L^{m}X_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}_{s}\big{)}^{-}s^{b}_{s}-(s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s})^{-}\Big{]}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{G}_{t}\bigg{]},$
(2.49) $\displaystyle\text{FBA}^{\Delta}_{t}=$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{\bar{\tau}}\Big{[}\big{(}L^{m}X_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}_{s}\big{)}^{+}s^{\ell}_{s}-(s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s})^{+}\Big{]}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{G}_{t}\bigg{]},$
(2.50)

while under clean close-out,

$\displaystyle\text{FCA}^{\Delta}_{t}=$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{\bar{\tau}}\Big{[}\big{(}X_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}_{s}\big{)}^{-}s^{b}_{s}-(s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s})^{-}\Big{]}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{G}_{t}\bigg{]},$
(2.51) $\displaystyle\text{FBA}^{\Delta}_{t}=$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{\bar{\tau}}\Big{[}\big{(}X_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}_{s}\big{)}^{+}s^{\ell}_{s}-(s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s})^{+}\Big{]}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{G}_{t}\bigg{]}.$
(2.52)

It is often stated that there is no FVA when contracts are fully
collateralized. Indeed, European Banking Authority (EBA) requires banks to
assess FCA and FBA for derivatives that “are not strongly collateralized” (see
Cameron, 2013). To examine this, assume full collateralization, i.e.,
$L^{m}=0$, replacement close-out, and $\rho=I$. The condition of full repo
markets is commonly chosen in literature. Moreover, when $R^{\ell}\geq r$ and
$R^{b}\geq r$, we can see that $\text{FCA}^{\Delta}=\text{FBA}^{\Delta}=0$
from (2.49) and (2.50). Therefore, based on our model, $\text{FVA}^{\Delta}$
is not necessary when the close-out amount is the replacement cost and all
repo markets are fully liquid. However, even in full collateralization,
$\text{FVA}^{\Delta}$ still exists under clean close-out. This is one of the
reasons why replacement close-out should be discussed.

  4. (iv)

Because of FVA, the BSDE of XVA (2.37) becomes semi-linear. When $s^{\ell}$
and $s^{b}$ are bounded, the generator is uniformly Lipsitch, so the existence
and uniqueness are not hard to obtain. However, we need to solve the BSDE
numerically, and this can be costly when a large netting set and long maturity
are considered. There have been several attempts to obtain a closed-form
solution. However, for the closed-form solution, it was necessary to assume
$s^{\ell}=s^{b}$ so that one can recover a linear equation as in Piterbarg
(2010); Brigo . (2017); Bichuch . (2018). At the stage, it was an assumption,
but we will show that we do not need such assumption by proving that FVA is
either FCA or $-\text{FBA}$, by obtaining either

$\displaystyle
X_{t}+\tilde{p}^{N}_{t}+\tilde{B}^{\epsilon}_{t}-c_{t}-\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}_{t}\geq$ $\displaystyle
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t~{}\text{a.s,
~{} or}$ (2.53) $\displaystyle
X_{t}+\tilde{p}^{N}_{t}+\tilde{B}^{\epsilon}_{t}-c_{t}-\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}_{t}\leq$ $\displaystyle
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t~{}\text{a.s,}$
(2.54)

for many derivative contracts. Then the BSDE becomes linear regardless of the
assumption, $s^{\ell}=s^{b}$, because one of the spreads does not play any
role in solving the BSDE.

  5. (v)

Considering different lending/borrowing not only makes the BSDEs for
replication pricing semi-linear but also makes the associated Hamiltonians
non-smooth in optimal investment problems (see Bo, 2017; Bo Capponi, 2016;
Yang ., 2017, for example).

  6. (vi)

For modeling incremental XVA with capital value adjustment and initial margin,
readers may want to refer to Albanese . (2018).

######  Remark 2.21.

It is worth mentioning how FVA and DVA related. To see this, we briefly review
the results of Burgard Kjaer (2010, 2011). Let $n=1$, $\rho=\\{1,C\\}$,
$L^{m}=1$, $e^{N}=p^{N}$. As XVA was not incremental in Burgard Kjaer (2010,
2011), in this example, we also set $e^{N}=0$ and $\epsilon=0$.

First, we guess the amount of $\tilde{\pi}^{H}$. At $\tau=\tau^{H}\leq T$,
$X_{\tau^{H}}=-L^{H}(\tilde{p}^{N}_{\tau^{H}-})^{+}$. Therefore,

$\displaystyle\Delta
X_{\tau^{H}}=-X_{\tau^{H}-}+\tilde{\pi}^{H}_{\tau^{H}}-L^{H}(\tilde{p}^{N}_{\tau^{H}-})^{+}.$

The hedger may want to hedge the jump risk at $\tau^{H}$ using
$\tilde{\pi}^{H}$ so that $\Delta X_{\tau^{H}}=0$, i.e., the hedger may choose

$\displaystyle\tilde{\pi}^{H}=X_{-}-L^{H}(\tilde{p}^{N})^{+}.$ (2.55)

Indeed, it will be shown later that (2.55) is the right choice. For now, we
accept (2.55).

Assuming the local martingales in (2.37) are true martingales, by Definition
2.19, for $t<\bar{\tau}$,

$\displaystyle\text{DVA}_{t}=$
$\displaystyle\mathbb{E}\bigg{[}\mathds{1}_{\bar{\tau}=\tau=\tau^{H}}L^{H}(\tilde{p}^{N}_{\tau})^{+}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{G}_{t}\bigg{]},$
$\displaystyle\text{FCA}_{t}=$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{\bar{\tau}}s^{b}_{s}\big{[}\tilde{p}^{N}_{s}-L^{H}(\tilde{p}^{N}_{s})^{+}\big{]}^{-}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{G}_{t}\bigg{]},$
$\displaystyle\text{FBA}_{t}=$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{\bar{\tau}}s^{\ell}_{s}\big{[}\tilde{p}^{N}_{s}-L^{H}(\tilde{p}^{N}_{s})^{+}\big{]}^{+}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{G}_{t}\bigg{]}.$

However, since $L^{H}\leq 1$,

$\displaystyle[\tilde{p}^{N}-L^{H}(\tilde{p}^{N})^{+}]^{-}=$
$\displaystyle(\tilde{p}^{N})^{-},$
$\displaystyle[\tilde{p}^{N}-L^{H}(\tilde{p}^{N})^{+}]^{+}=$
$\displaystyle(\tilde{p}^{N})^{+}.$

Then by Lemma 2.2,

$\displaystyle\text{DVA}_{t}=$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{T}G_{s}h^{H}_{s}L^{H}(\tilde{p}^{N}_{s})^{+}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{F}_{t}\bigg{]},$
$\displaystyle\text{FCA}_{t}=$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{T}G_{s}s^{b}_{s}(\tilde{p}^{N}_{s})^{-}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{F}_{t}\bigg{]},$
$\displaystyle\text{FBA}_{t}=$
$\displaystyle\mathbb{E}\bigg{[}\int_{t}^{T}G_{s}s^{\ell}_{s}(\tilde{p}^{N}_{s})^{+}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{F}_{t}\bigg{]}.$

If we assume that the hedger’s borrowing rate is higher than $r$ only because
of the own default risk, i.e., there is no liquidity premium, then we can
approximate the hedger’s borrowing spread as

$\displaystyle s^{b}=h^{H}L^{H}.$

With this assumption, FCA becomes a counterpart of DVA, i.e.,

$\displaystyle\text{DVA}_{t}=$
$\displaystyle\int_{t}^{T}G_{s}h^{H}_{s}L^{H}\mathbb{E}[(\tilde{p}^{N}_{s})^{+}|\mathcal{F}_{t}]\mathop{}\\!\mathrm{d}s,$
(2.56) $\displaystyle\text{FCA}_{t}=$
$\displaystyle\int_{t}^{T}G_{s}h^{H}_{s}L^{H}\mathbb{E}[(\tilde{p}^{N}_{s})^{-}|\mathcal{F}_{t}]\mathop{}\\!\mathrm{d}s.$
(2.57)

The above two equations show the financial relationship between DVA and FCA.
DVA is a benefit to the shareholders because the hedger may default on
derivative payables. On the other hand, the bondholders will receive a partial
amount of the derivative receivables, namely $(1-L^{H})(\tilde{p}^{N})^{-}$.
Therefore, the hedger should compensate the funding provider for the expected
loss.

When the hedger has redundant money, she can use it to buy back loans that
were already issued. In this case, we may inspect $s^{\ell}=h^{H}L^{H}$. This
inspection leads to

$\displaystyle\text{DVA}=\text{FBA}.$ (2.58)

It is important to avoid double-counting for both pricing and accounting.
However, in this case, recalling that (2.48), it seems that we have two
adjustments with the approximately same value. If (2.58) is valid for many
contracts, i.e., DVA is overlapped with FBA, then one of them should be
ignored.

Indeed, it often believed that recording both FBA and DVA in bank’s account
engenders a double-counting paradox. On the other hand, IFRS and GAAP accept
DVA. To remedy this, in FCA/FBA accounting, which is endorsed by some banks,
DVA is recorded in Contra-Liability (CL) account and

$\displaystyle\text{FCA}-(\text{FBA}-\text{DVA})$

is recorded in Contra-Asset (CA) account (see Castagna, 2011; Albanese
Andersen, 2014). However, it has been pointed out that FCA/FBA accounting
produces large asset/liability asymmetry.

The large asymmetry is partly attributed to the binary nature of FVA. Based on
the marginal effect of entering a contract, the FCA term in CA account does
not countervail the FBA term in CL account. It will be shown later that the
binary nature of FVA is related to whether the payoff functions are increasing
or decreasing with respect to underlying assets. On the other hand, because
DVA occurs from derivative payables, e.g., $p^{N}\geq 0$ in (2.56), DVA arises
where the payoffs are positive. Therefore, FBA and DVA are affected by
different mathematical structures of derivative contracts, i.e., FBA is not
overlapped with DVA in CA account. Thus, FCA/FBA accounting inevitably leads
to large asset/liability asymmetry. To avoid the asymmetry, in FVA/FDA
accounting, FVA is recorded in CA account and funding debt value adjustment
(FDA) is recorded in CL account. FDA is a benefit that the bank can default on
its liability. It was named DVA2 in Hull White (2012). If liquidity is not
considered, the value of FVA can be approximated by FDA. For the detail,
readers may want to refer to Albanese Andersen (2014).

In the next section, we represent (2.37) as a standard form, and reduce it to
a BSDE on the reference filtration $\mathbb{F}$.

###  2.4 BSDE formulation

For a BSDE representation, we begin this section with considering a family of
maps $(\phi_{t})_{t\geq 0}$ such that

$\displaystyle\phi_{t}~{}\colon~{}\displaystyle\sum_{i\in
I}\tilde{\pi}\sigma^{i}_{t}-Z^{N}_{t}~{}\to~{}\displaystyle\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}.$

The form of $(\phi_{t})_{t\geq 0}$ varies depending on parameters and
accessibility of repo markets. Before giving the general form
$(\phi_{t})_{t\geq 0}$, we examine some examples.

######  Example 2.22.

(i) Consider $\rho=I$, which is commonly assumed in literature. In this case,

$\displaystyle\phi_{t}\colon z\to 0.$ (2.59)

(ii) Consider $n=1$, and constant parameters. Then $S^{1},~{}S^{H},~{}S^{C}$
follow

$\displaystyle\mathop{}\\!\mathrm{d}S^{1}_{t}=$ $\displaystyle
rS^{1}_{t}\mathop{}\\!\mathrm{d}t+\sigma^{1}S^{1}_{t}\mathop{}\\!\mathrm{d}W_{t},$
$\displaystyle\mathop{}\\!\mathrm{d}S^{i}_{t}=$ $\displaystyle
rS^{i}_{t}\mathop{}\\!\mathrm{d}t-S_{t-}\mathop{}\\!\mathrm{d}M^{i}_{t},~{}~{}i\in\\{H,C\\}.$

It follows that $\sum_{i\in
I}\tilde{\pi}^{1}\sigma^{i}=\tilde{\pi}^{1}\sigma^{1}$. When $\rho=\\{H,C\\}$,

$\displaystyle\phi_{t}\colon
z\to(\sigma^{1})^{-1}(z+Z^{N}_{t})=(\sigma^{1})^{-1}z+(\sigma^{1})^{-1}Z^{N}_{t}.$
(2.60)

On the other hand, when $\rho=\\{1,C\\}$, $\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}=\tilde{\pi}^{H}$. Thus,

$\displaystyle\phi_{t}\colon z\to\tilde{\pi}^{H}_{t}.$ (2.61)

This case was discussed by Burgard Kjaer (2010).

(iii) On the other hand, let us assume that OIS rate is an
$\mathbb{F}$-adapted process. In addition, we assume that for any
$i\in\\{H,C\\}$, $(G^{i}_{t})_{t\geq 0}$ is given by

$\displaystyle\mathop{}\\!\mathrm{d}G^{i}_{t}=-h^{i}_{t}G^{i}_{t}\mathop{}\\!\mathrm{d}t,$

where $(h^{i}_{t})_{t\geq 0}$ are deterministic processes. We consider non-
defaultable and defaultable zero coupon bonds with the same maturity as $T$,
i.e., $S^{1},~{}S^{H},~{}S^{C}$ are defined as

$\displaystyle S^{1}_{t}\coloneqq$ $\displaystyle
B_{t}\mathbb{E}\big{[}B_{T}^{-1}\big{|}\mathcal{F}_{t}\big{]},$ $\displaystyle
S^{i}_{t}\coloneqq$ $\displaystyle
B_{t}\mathbb{E}\big{[}\mathds{1}_{\tau^{i}>T}B_{T}^{-1}\big{|}\mathcal{G}_{t}\big{]},~{}~{}i\in\\{H,C\\}.$

By Lemma 2.2,
$S^{i}_{t}=\mathds{1}_{t<\tau}B_{t}(G^{i}_{t})^{-1}\mathbb{E}[G^{i}_{T}B_{T}^{-1}|\mathcal{F}_{t}]$.
Since $(G^{i}_{t})_{t\geq 0}$, $i\in\\{H,C\\}$, are deterministic,

$\displaystyle
S^{i}_{t}=\mathds{1}_{t<\tau}(G^{i}_{t})^{-1}G^{i}_{T}S^{1}_{t}.$

It follows that for $t<\tau$, $\sigma^{1}=\sigma^{H}=\sigma^{C}$. Recall
$\sigma=[\sigma^{1}~{}\dots~{}\sigma^{n}~{}\sigma^{H}~{}\sigma^{C}]^{\top}$.
We define $n\times n$ matrix

$\displaystyle\Sigma\coloneqq\begin{bmatrix}\sigma^{1}\\\ \vdots\\\
\sigma^{n}\end{bmatrix},$

and assume that $\Sigma$ is invertible for any $t<T$. Then,

$\displaystyle(\Sigma^{\top})^{-1}\sigma^{\top}\tilde{\pi}=\begin{bmatrix}\tilde{\pi}^{1}+\tilde{\pi}^{H}+\tilde{\pi}^{C}\\\
\tilde{\pi}^{2}\\\ \vdots\\\ \tilde{\pi}^{n}\end{bmatrix}.$

Therefore,
$\mathds{1}^{\top}(\Sigma^{\top})^{-1}\sigma^{\top}\tilde{\pi}=\sum_{i\in
I}\tilde{\pi}$, where
$\mathds{1}\coloneqq(1,\dots,1)^{\top}\in\mathbb{R}^{n}$. Thus, if
$\rho=\emptyset$,

$\displaystyle\phi_{t}\colon
z\to\mathds{1}^{\top}(\Sigma^{\top}_{t})^{-1}(z+Z^{N}_{t}),$ (2.62)

On the other hand, consider $\rho\not=\emptyset$ and define
$\mathds{1}^{\rho}\in\mathbb{R}^{n}$ by

$\displaystyle(\mathds{1}^{\rho})_{i}\coloneqq\left\\{\begin{array}[]{r@{}l}0{}{}&i\in\rho,\\\
1{}{}&i\notin\rho,\end{array}\right.$ (2.65)

where $(\mathds{1}^{\rho})_{i}$ denote $i$-th component of
$\mathds{1}^{\rho}$. When $\rho\cap\\{1,H,C\\}=\emptyset$, $\phi_{t}\colon
z\to(\mathds{1}^{\rho})^{\top}(\Sigma^{\top}_{t})^{-1}(z+Z^{N}_{t})$. However,
if $\rho=\\{1\\}$,

$\displaystyle\phi_{t}\colon
z\to(\mathds{1}^{\rho})^{\top}(\Sigma^{\top}_{t})^{-1}(z+Z^{N}_{t})+\tilde{\pi}^{H}+\tilde{\pi}^{C}.$
(2.66)

Therefore, the form of BSDE depends on the choice of model, accessibility of
repo markets, etc. However, from (2.59)-(2.66), we can observe that $\phi$ is
linear in $z,Z^{N},\tilde{\pi}^{H},\tilde{\pi}^{C}$. As we will see later that
$\tilde{\pi}^{H}$ and $\tilde{\pi}^{C}$ can be dependent of a solution of the
BSDE under replacement close-out. For simplicity, we assume that $\phi$ is an
independent form of $\tilde{\pi}^{H}$ and $\tilde{\pi}^{C}$ such as (2.59),
(2.60), and (2.62).

######  Assumption 2.23.

For some $n$-dimensional $\mathbb{F}$-progressively measurable process
$\alpha$,

$\displaystyle\phi_{t}(z)=\alpha^{\top}_{t}(z+Z^{N}_{t}).$

######  Remark 2.24.

Note that we do not exclude the convention commonly used in the literature
that assets are traded from repo markets, i.e., $I=\rho$. In this case, we can
set $\alpha=0$ in 2.23.

Now, we denote the generator of (2.37) by $g^{\mathbb{G}}$, i.e.,

$\displaystyle
g^{\mathbb{G}}_{t}(y,z)\coloneqq-\big{(}y+\tilde{p}^{N}_{t}+\tilde{B}^{\epsilon}_{t}-c_{t}-\phi_{t}(z)\big{)}^{+}s^{\ell}_{t}+\big{(}y+\tilde{p}^{N}_{t}+\tilde{B}^{\epsilon}_{t}-c_{t}-\phi_{t}(z)\big{)}^{-}s^{b}_{t}+s^{\epsilon}_{t}\tilde{B}^{\epsilon}_{t}.$

In addition, we denote the incremental exposures by $\Theta^{\Delta,i}$,
$i\in\\{H,C\\}$, more precisely,

$\displaystyle\Theta^{\Delta,H}(y)\coloneqq$
$\displaystyle\Theta^{H}(y)-L^{H}L^{m}(e^{E}_{-})^{+},$
$\displaystyle\Theta^{\Delta,C}(y)\coloneqq$
$\displaystyle\Theta^{C}(y)-L^{C}L^{m}(e^{E}_{-})^{-}.$

Let $(Y^{\mathbb{G}},Z^{\mathbb{G}},\tilde{\pi}^{H},\tilde{\pi}^{C})$ denote
the solution, in a certain space, of the following BSDE:

$\displaystyle Y^{\mathbb{G}}_{t}=$
$\displaystyle-\mathds{1}_{\bar{\tau}=\tau=\tau^{H}}\tilde{\Theta}^{\Delta,H}_{\tau}(Y^{\mathbb{G}}_{\tau-})+\mathds{1}_{\bar{\tau}=\tau=\tau^{C}}\tilde{\Theta}^{\Delta,C}_{\tau}(Y^{\mathbb{G}}_{\tau-})$
$\displaystyle+\int_{t}^{\bar{\tau}}g^{\mathbb{G}}_{s}(Y^{\mathbb{G}}_{s},Z^{\mathbb{G}}_{s})\mathop{}\\!\mathrm{d}s-\int_{t}^{\bar{\tau}}(Z^{\mathbb{G}}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s}+\displaystyle\sum_{i=H,C}\int_{t}^{\bar{\tau}}\tilde{\pi}^{i}_{s}\mathop{}\\!\mathrm{d}M^{i}_{s}.$
(2.67)

Then $(Y^{\mathbb{G}},Z^{\mathbb{G}},\tilde{\pi}^{H},\tilde{\pi}^{C})$
provides $(X,\tilde{\pi})$ as well as $(V,\pi)$. However, instead of directly
dealing with (2.67), we will investigate a reduced BSDE on the reference
filtration $\mathbb{F}$. The idea is as follows.

It is a known fact that in the progressively enlarged filtration $\mathbb{G}$,
for any $\mathbb{G}$-optional (resp. predictable) process has an
$\mathbb{F}$-optional (resp. predictable) reduction. Therefore, if there
exists a solution of (2.67) such that $Y^{\mathbb{G}}$ is
$\mathbb{G}$-optional and $Z^{\mathbb{G}}$ is $\mathbb{G}$-predictable, there
exists an $\mathbb{F}$-adapted pair $(Y^{\mathbb{F}},Z^{\mathbb{F}})$
satisfying

$\displaystyle Y^{\mathbb{G}}=$
$\displaystyle\mathds{1}_{t<\bar{\tau}}Y^{\mathbb{F}},$ (2.68) $\displaystyle
Z^{\mathbb{G}}=$ $\displaystyle\mathds{1}_{t\leq\bar{\tau}}Z^{\mathbb{F}}.$
(2.69)

Moreover, we guess that $(Y^{\mathbb{F}},Z^{\mathbb{F}})$ is a solution of a
BSDE on the reference filtration $\mathbb{F}$, i.e., “for some”
$g^{\mathbb{F}}\colon\Omega\times[0,T]\times\mathbb{R}^{n+1}\to\mathbb{R}$,

$\displaystyle
Y^{\mathbb{F}}_{t}=\int_{t}^{T}g^{\mathbb{F}}_{s}(Y^{\mathbb{F}}_{s},Z^{\mathbb{F}}_{s})\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\mathbb{F}}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s}.$
(2.70)

Then, by the terminal condition
$Y^{\mathbb{G}}_{\bar{\tau}}=-\mathds{1}_{\bar{\tau}=\tau=\tau^{H}}\tilde{\Theta}^{\Delta,H}_{\tau}(Y^{\mathbb{G}}_{\tau-})+\mathds{1}_{\bar{\tau}=\tau=\tau^{C}}\tilde{\Theta}^{\Delta,C}_{\tau}(Y^{\mathbb{G}}_{\tau-})$,
together with (2.68), we can expect that

$\displaystyle
Y^{\mathbb{G}}_{t}=\mathds{1}_{t<\bar{\tau}}Y^{\mathbb{F}}_{t}+\mathds{1}_{t\geq\bar{\tau}}\big{[}-\mathds{1}_{\bar{\tau}=\tau=\tau^{H}}\tilde{\Theta}^{\Delta,H}_{\tau}(Y^{\mathbb{F}}_{\tau-})+\mathds{1}_{\bar{\tau}=\tau=\tau^{C}}\tilde{\Theta}^{\Delta,C}_{\tau}(Y^{\mathbb{F}}_{\tau-})\big{]}.$
(2.71)

By applying Itô’s formula to (2.71), we can find what the proper
$g^{\mathbb{F}}$ should be. At the end, finding $(V,\pi)$ reduces to
investigating the reduced BSDE (2.70). The next proposition explains the
detail.

######  Proposition 2.25.

Assume there exits a unique solution
$(Y^{\mathbb{F}},Z^{\mathbb{F}})\in\mathbb{S}^{2}_{T}\times\mathbb{H}^{2,n}_{T}$
of the following BSDE:

$\displaystyle
Y^{\mathbb{F}}_{t}=\int_{t}^{T}g^{\mathbb{F}}_{s}(Y^{\mathbb{F}}_{s},Z^{\mathbb{F}}_{s})\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\mathbb{F}}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$
(2.72) $\displaystyle g^{\mathbb{F}}_{t}(y,z)\coloneqq
g^{\mathbb{G}}_{t}(y,z)-h^{H}_{t}\tilde{\Theta}^{\Delta,H}_{t}(y)+h^{C}_{t}\tilde{\Theta}^{\Delta,C}_{t}(y)-h_{t}y.$
(2.73)

Then $(Y^{\mathbb{G}},Z^{\mathbb{G}})$ can solve (2.67) by the following
relationships:

$\displaystyle Y^{\mathbb{G}}_{t}=$
$\displaystyle\mathds{1}_{t<\bar{\tau}}Y^{\mathbb{F}}_{t}+\mathds{1}_{t\geq\bar{\tau}}\big{[}-\mathds{1}_{\bar{\tau}=\tau=\tau^{H}}\tilde{\Theta}^{\Delta,H}_{\tau}(Y^{\mathbb{F}}_{\tau-})+\mathds{1}_{\bar{\tau}=\tau=\tau^{C}}\tilde{\Theta}^{\Delta,C}_{\tau}(Y^{\mathbb{F}}_{\tau-})\big{]},$
(2.74) $\displaystyle Z^{\mathbb{G}}_{t}=$
$\displaystyle\mathds{1}_{t<\bar{\tau}}Z^{\mathbb{F}}_{t},$ (2.75)
$\displaystyle\tilde{\pi}^{H}_{t}=$ $\displaystyle
Y^{\mathbb{F}}_{t}+\tilde{\Theta}^{H}_{t}(Y^{\mathbb{F}}_{t-}),$ (2.76)
$\displaystyle\tilde{\pi}^{C}_{t}=$ $\displaystyle
Y^{\mathbb{F}}_{t}-\tilde{\Theta}^{C}_{t}(Y^{\mathbb{F}}_{t-}).$ (2.77)

Moreover, if (H)-hypothesis holds between $\mathbb{F}$ and $\mathbb{G}$, i.e.,
any $(\mathbb{F},\mathbb{Q})$-martingale is a
$(\mathbb{G},\mathbb{Q})$-martingale, then (2.74)-(2.77) is the only solution
of (2.67).

###### Proof.

By Itô’s formula,

$\displaystyle\mathop{}\\!\mathrm{d}(\mathds{1}_{t<\bar{\tau}}Y^{\mathbb{F}}_{t})=\mathds{1}_{t\leq\bar{\tau}}\mathop{}\\!\mathrm{d}Y^{\mathbb{F}}_{t}-\bm{\delta}_{\bar{\tau}}(\mathop{}\\!\mathrm{d}t)Y^{\mathbb{F}}_{\bar{\tau}}=$
$\displaystyle\mathds{1}_{t\leq\bar{\tau}}\mathop{}\\!\mathrm{d}Y^{\mathbb{F}}_{t}-\mathds{1}_{t\leq\bar{\tau}}Y^{\mathbb{F}}_{t}\mathop{}\\!\mathrm{d}M_{t}-\mathds{1}_{t\leq\bar{\tau}}h^{0}_{t}Y^{\mathbb{F}}\mathop{}\\!\mathrm{d}t,$
$\displaystyle=$
$\displaystyle\mathds{1}_{t\leq\bar{\tau}}\mathop{}\\!\mathrm{d}Y^{\mathbb{F}}_{t}-\mathds{1}_{t\leq\bar{\tau}}Y^{\mathbb{F}}_{t}\mathop{}\\!\mathrm{d}M^{H}_{t}-\mathds{1}_{t\leq\bar{\tau}}Y^{\mathbb{F}}_{t}\mathop{}\\!\mathrm{d}M^{C}_{t}-\mathds{1}_{t\leq\bar{\tau}}h_{t}Y^{\mathbb{F}}\mathop{}\\!\mathrm{d}t,$

and

$\displaystyle\mathop{}\\!\mathrm{d}\Big{(}\mathds{1}_{t\geq\bar{\tau}}\big{[}-\mathds{1}_{\bar{\tau}=\tau=\tau^{H}}\tilde{\Theta}^{\Delta,H}_{\tau}(Y^{\mathbb{F}}_{\tau-})+$
$\displaystyle\mathds{1}_{\bar{\tau}=\tau=\tau^{C}}\tilde{\Theta}^{\Delta,C}_{\tau}(Y^{\mathbb{F}}_{\tau-})\big{]}\Big{)}$
$\displaystyle=$ $\displaystyle-\mathds{1}_{\tau=\tau^{H}\leq
T}\bm{\delta}_{\tau}(\mathop{}\\!\mathrm{d}t)\tilde{\Theta}^{\Delta,H}_{t}(Y^{\mathbb{F}}_{t-})+\mathds{1}_{\tau=\tau^{C}\leq
T}\bm{\delta}_{\tau}(\mathop{}\\!\mathrm{d}t)\tilde{\Theta}^{\Delta,C}_{t}(Y^{\mathbb{F}}_{t-})$
$\displaystyle=$
$\displaystyle-\mathds{1}_{t\leq\bar{\tau}}\tilde{\Theta}^{\Delta,H}_{t}(Y^{\mathbb{F}}_{t-})\mathop{}\\!\mathrm{d}M^{H}_{t}-\mathds{1}_{t\leq\bar{\tau}}h_{t}^{H}\tilde{\Theta}^{\Delta,H}_{t}(Y^{\mathbb{F}}_{t-})\mathop{}\\!\mathrm{d}t$
$\displaystyle+\mathds{1}_{t\leq\bar{\tau}}\tilde{\Theta}^{\Delta,C}_{t}(Y^{\mathbb{F}}_{t-})\mathop{}\\!\mathrm{d}M^{C}_{t}+\mathds{1}_{t\leq\bar{\tau}}h_{t}^{C}\tilde{\Theta}^{\Delta,C}_{t}(Y^{\mathbb{F}}_{t-})\mathop{}\\!\mathrm{d}t.$

If we take $Y^{\mathbb{G}}$ as in (2.74),

$\displaystyle\mathop{}\\!\mathrm{d}Y_{t}^{\mathbb{G}}=$
$\displaystyle\mathds{1}_{t\leq\bar{\tau}}\mathop{}\\!\mathrm{d}Y^{\mathbb{F}}_{t}-\mathds{1}_{t\leq\bar{\tau}}Y^{\mathbb{F}}_{t}\mathop{}\\!\mathrm{d}M^{H}_{t}-\mathds{1}_{t\leq\bar{\tau}}Y^{\mathbb{F}}_{t}\mathop{}\\!\mathrm{d}M^{C}_{t}-\mathds{1}_{t\leq\bar{\tau}}h_{t}Y^{\mathbb{F}}\mathop{}\\!\mathrm{d}t$
$\displaystyle-\mathds{1}_{t\leq\bar{\tau}}\tilde{\Theta}^{\Delta,H}_{t}(Y^{\mathbb{F}}_{t-})\mathop{}\\!\mathrm{d}M^{H}_{t}-\mathds{1}_{t\leq\bar{\tau}}h_{t}^{H}\tilde{\Theta}^{\Delta,H}_{t}(Y^{\mathbb{F}}_{t-})\mathop{}\\!\mathrm{d}t$
$\displaystyle+\mathds{1}_{t\leq\bar{\tau}}\tilde{\Theta}^{\Delta,C}_{t}(Y^{\mathbb{F}}_{t-})\mathop{}\\!\mathrm{d}M^{C}_{t}+\mathds{1}_{t\leq\bar{\tau}}h_{t}^{C}\tilde{\Theta}^{\Delta,C}_{t}(Y^{\mathbb{F}}_{t-})\mathop{}\\!\mathrm{d}t$
$\displaystyle=$
$\displaystyle\mathds{1}_{t\leq\bar{\tau}}\Big{[}-g^{\mathbb{F}}_{t}(Y^{\mathbb{F}}_{t},Z^{\mathbb{F}}_{t})-h_{t}Y^{\mathbb{F}}_{t}-h^{H}_{t}\tilde{\Theta}^{\Delta,H}_{t}(Y^{\mathbb{F}}_{t})+h^{C}_{t}\tilde{\Theta}^{\Delta,C}_{t}(Y^{\mathbb{F}}_{t})\Big{]}\mathop{}\\!\mathrm{d}t+\mathds{1}_{t\leq\bar{\tau}}(Z^{\mathbb{F}}_{t})^{\top}\mathop{}\\!\mathrm{d}W_{t}$
$\displaystyle-\mathds{1}_{t\leq\bar{\tau}}[Y^{\mathbb{F}}_{t}+\tilde{\Theta}^{\Delta,H}_{t}(Y^{\mathbb{F}}_{t})]\mathop{}\\!\mathrm{d}M^{H}_{t}-\mathds{1}_{t\leq\bar{\tau}}[Y^{\mathbb{F}}_{t}-\tilde{\Theta}^{\Delta,C}_{t}(Y^{\mathbb{F}}_{t})]\mathop{}\\!\mathrm{d}M^{H}_{t}.$

Therefore, by (2.73), (2.74)-(2.77) give a solution for (2.67). Moreover, if
(H)-hypothesis holds, (unique) martingale representation property holds by $W$
and $M^{i}$, $i\in\\{H,C\\}$. Therefore, by Theorem 4.1 in Crépey Song (2015),
if $(Y^{\mathbb{G}},Z^{\mathbb{G}})$ solves (2.67),
$((Y^{\mathbb{G}})^{\tau-},Z^{\mathbb{G}}\mathds{1}_{\llbracket
0,\tau\llbracket})$ solves (2.70) as well. ∎

######  Remark 2.26.

This reduction argument was also used by Bichuch . (2018); Crépey (20152,
20151); Brigo . (2016).

######  Remark 2.27.

(2.76) explains how the bank’s own default can be beneficial to the
shareholders. The own default can be monetized by buying back their own bond
that becomes cheaper. However, this means that banks can realize the profit
only when they actually default. Indeed, DVA is often excluded from Common
Equity Tier 1 capital (CET1), which is a proxy of the shareholder’s wealth.

Note that $hY^{\mathbb{F}}$ in $g^{\mathbb{F}}$ is an adjustment for an early
termination. To see this, let $\hat{Y}\coloneqq GY^{\mathbb{F}}$,
$\hat{Z}\coloneqq GZ^{\mathbb{F}}$. Then (2.72) becomes

$\displaystyle\hat{Y}^{\mathbb{F}}_{t}=\int_{t}^{T}G_{s}\big{[}g^{\mathbb{G}}_{s}(Y^{\mathbb{F}}_{s},Z^{\mathbb{F}}_{s})-h^{H}_{s}\tilde{\Theta}^{\Delta,H}_{s}(Y^{\mathbb{F}}_{s})+h^{C}_{s}\tilde{\Theta}^{\Delta,C}_{s}(Y^{\mathbb{F}}_{s})\big{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(\hat{Z}^{\mathbb{F}}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s}.$

Moreover, by Definition 2.19 and Lemma 2.2, if
$Z^{\mathbb{F}}\in\mathbb{H}^{2,n}_{T}$, for $t<\tau$,

$\displaystyle\text{FCA}^{\Delta}_{t}=$ $\displaystyle
G^{-1}_{t}\mathbb{E}\bigg{[}\int_{t}^{T}G_{s}\Big{[}\big{(}Y^{\mathbb{F}}_{s}+\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-c_{s}-\phi_{s}(Z^{\mathbb{F}}_{s})\big{)}^{-}s^{b}_{s}-(s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s})^{+}\Big{]}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{F}_{t}\bigg{]},$
(2.78) $\displaystyle\text{FBA}^{\Delta}_{t}=$ $\displaystyle
G^{-1}_{t}\mathbb{E}\bigg{[}\int_{t}^{T}G_{s}\Big{[}\big{(}Y^{\mathbb{F}}_{s}+\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-c_{s}-\phi_{s}(Z^{\mathbb{F}}_{s})\big{)}^{+}s^{\ell}_{s}-(s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s})^{-}\Big{]}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{F}_{t}\bigg{]},$
(2.79) $\displaystyle\text{DVA}^{\Delta}_{t}=$ $\displaystyle
G^{-1}_{t}\mathbb{E}\bigg{[}\int_{t}^{T}G_{s}\tilde{\Theta}^{\Delta,H}_{s}(Y^{\mathbb{F}}_{s})\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{F}_{t}\bigg{]},$
(2.80) $\displaystyle\text{CVA}^{\Delta}_{t}=$ $\displaystyle
G^{-1}_{t}\mathbb{E}\bigg{[}\int_{t}^{T}G_{s}\tilde{\Theta}^{\Delta,C}_{s}(Y^{\mathbb{F}}_{s})\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{F}_{t}\bigg{]},$
(2.81)

and $\text{XVA}=G^{-1}\hat{Y}$. Still, because of the semi-linearity in (2.78)
and (2.79), we need to solve (2.70) numerically. More importantly, it will be
interesting to investigate how much cancellation between FCA and FBA, and
between FBA and DVA, can be expected. We answer the questions in the next
section. Based on our model, the answers for the questions are both negative.

##  3 Main Results

In this section, we show a binary nature of FVA; FVA is either FCA or -FBA. In
other words, either $\text{FCA}=0$ or $\text{FBA}=0$. This switching property
of FVA is determined by some properties of payoff structures of the derivative
contract. Before the main theorem, we explain the idea by an example first.

######  Example 3.1 (Stock forward contract with clean close-out).

Consider $n=1$, $e^{N}=p^{N}$. For simplicity, we assume all parameters are
constant and let the traded assets $(S^{1},S^{H},S^{C})$ given by

$\displaystyle\mathop{}\\!\mathrm{d}S^{1}_{t}=$ $\displaystyle
r_{t}S^{1}_{t}\mathop{}\\!\mathrm{d}t+\sigma^{1}S^{1}_{t}\mathop{}\\!\mathrm{d}W_{t},$
$\displaystyle\mathop{}\\!\mathrm{d}S^{i}_{t}=$ $\displaystyle
r_{t}S^{i}_{t}\mathop{}\\!\mathrm{d}t-S^{i}_{t-}\mathop{}\\!\mathrm{d}M^{i}_{t},~{}~{}i\in\\{H,C\\}.$

Moreover, we assume that the defaultable bonds can be traded through repo
markets, i.e., $\rho=\\{H,C\\}$. Then $\sum_{i\in
I}\tilde{\pi}^{i}\sigma^{i}=\tilde{\pi}^{1}\sigma^{1}$ and $\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}=\tilde{\pi}^{1}$, therefore

$\displaystyle\phi(z)=\alpha(z+Z^{N})=(\sigma^{1})^{-1}(z+Z^{N}).$

Let $\textfrak{D}^{N}=\mathds{1}_{\llbracket
T,\infty\llbracket}(S^{1}_{T}-K)$, for some $K\geq 0$. We denote
$\tilde{S}^{1}\coloneqq B^{-1}S^{1}$. Recall the definition
$\tilde{p}^{N}=B^{-1}p^{N}$ and

$\displaystyle\tilde{\Theta}^{\Delta,H}=$ $\displaystyle
L^{H}L^{m}\big{(}(\tilde{p}^{N}+\tilde{e}^{E})^{+}-(\tilde{e}^{E})^{+}\big{)},$
$\displaystyle\tilde{\Theta}^{\Delta,C}=$ $\displaystyle
L^{C}L^{m}\big{(}(\tilde{p}^{N}+\tilde{e}^{E})^{-}-(\tilde{e}^{E})^{-}\big{)}.$

Thus, the generator $g^{\mathbb{F}}$ becomes

$\displaystyle g^{\mathbb{F}}_{t}(y,z)=$
$\displaystyle-\big{[}y+L^{m}\tilde{p}^{N}_{t}+\tilde{B}^{\epsilon}_{t}-\alpha(z+Z^{N}_{t})\big{]}^{+}s^{\ell}+\big{[}y+L^{m}\tilde{p}^{N}_{t}+\tilde{B}^{\epsilon}_{t}-\alpha(z+Z^{N}_{t})\big{]}^{-}s^{b}+s^{\epsilon}\tilde{B}^{\epsilon}_{t}$
$\displaystyle-h^{H}L^{H}L^{m}\big{(}(\tilde{p}^{N}_{t}+\tilde{e}^{E}_{t})^{+}-(\tilde{e}^{E}_{t})^{+}\big{)}+h^{C}L^{C}L^{m}\big{(}(\tilde{p}^{N}_{t}+\tilde{e}^{E}_{t})^{-}-(\tilde{e}^{E}_{t})^{-}\big{)}-hy.$

To explain the idea, the hedger should pay $S^{1}_{T}-K$ at the maturity or
$p^{N}_{t}=S_{t}^{1}-B_{t}B_{T}^{-1}K$ at an early termination $t<T$. For the
payment, she needs to retain $S^{1}$. To buy $S^{1}$, the hedger may need to
borrow money, so it is expected that $s^{\ell}$ does not play an important
role in maintaining the hedging portfolio. Therefore, we guess

$\displaystyle
Y^{\mathbb{F}}+L^{m}\tilde{p}^{N}-\alpha(Z^{\mathbb{F}}-Z^{N})\leq
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t-\text{a.s}.$
(3.1)

Unless the tendency of (3.1) is dominated by the legacy portfolio, we can
recover a linear BSDE. For simplicity, we assume that the dealer had big
enough exposure to the counterparty before the new contract and the initial
exposure dominates the new exposure, i.e., we assume that

$\displaystyle\tilde{p}^{N}+\tilde{e}^{E}\geq 0,~{}~{}\tilde{e}^{E}\geq
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t-\text{a.s}.$
(3.2)

Then we consider $(Y^{\\#},Z^{\\#})$ satisfying

$\displaystyle Y^{\\#}_{t}=$
$\displaystyle-\int_{t}^{T}\bigg{[}\big{(}Y^{\\#}_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\alpha(Z^{\\#}_{s}+Z^{N}_{s})\big{)}s^{b}-s^{\epsilon}\tilde{B}^{\epsilon}_{s}+hY^{\\#}_{s}\bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle-\int_{t}^{T}h^{H}L^{H}L^{m}\tilde{p}^{N}_{s}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\\#}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$

Then we will show that

$\displaystyle
Y^{\\#}+L^{m}\tilde{p}^{N}+\tilde{B}^{\epsilon}-\alpha(Z^{\\#}+Z^{N})\leq
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t-\text{a.s}.$
(3.3)

To show this, we take another transformation, $V^{\mathbb{F}}\coloneqq
Y^{\\#}+L^{m}\tilde{p}^{N}+\tilde{B}^{\epsilon}$ and
$\Pi^{\mathbb{F}}\coloneqq Z^{\\#}+L^{m}Z^{N}$. Then
$(V^{\mathbb{F}},\Pi^{\mathbb{F}})$ is the solution of

$\displaystyle V^{\mathbb{F}}_{t}=$ $\displaystyle
L^{m}\xi+\tilde{B}^{\epsilon}_{T}+\int_{t}^{T}F_{t}(V^{\mathbb{F}}_{s},\Pi^{\mathbb{F}}_{s})\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(\Pi^{\mathbb{F}}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$
(3.4)

where $\xi\coloneqq\tilde{S}^{1}_{T}-B^{-1}_{T}K$ and

$\displaystyle F_{t}(y,z)\coloneqq-(y-\alpha z)s^{b}-hy+\xi^{b}_{t},$ (3.5)
$\displaystyle\xi^{b}\coloneqq(h-h^{H}L^{H}L^{m})\tilde{p}^{N}+h\tilde{B}^{\epsilon}+\alpha(1-L^{m})s^{b}Z^{N}.$

Under mild conditions, we can obtain
$(V^{\mathbb{F}},\Pi^{\mathbb{F}})\in\mathbb{L}^{2}([0,T]\colon\mathbb{D}^{1,2}\times\mathbb{D}^{1,2})$,
i.e., $\forall t\leq T$,
$(V^{\mathbb{F}}_{t},\Pi^{\mathbb{F}}_{t})\in\mathbb{D}^{1,2}\times\mathbb{D}^{1,2}$
and

$\displaystyle\int_{0}^{T}\Big{(}\|V^{\mathbb{F}}_{t}\|^{2}_{1,2}+\|\Pi^{\mathbb{F}}_{t}\|^{2}_{1,2}\Big{)}\mathop{}\\!\mathrm{d}t<\infty.$
(3.6)

Moreover, $(D_{t}V^{\mathbb{F}}_{t})_{0\leq t\leq T}$ is a version of
$(\Pi^{\mathbb{F}}_{t})_{0\leq t\leq T}$. In addition, note that (3.3) is
equivalent to

$\displaystyle V^{\mathbb{F}}-(1-L^{m})s^{b}\alpha
Z^{N}-\alpha\Pi^{\mathbb{F}}\leq V^{\mathbb{F}}-(1-L^{m})s^{b}\alpha
Z^{N}-\alpha DV^{\mathbb{F}}\leq
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t-\text{a.s}.$
(3.7)

To show (3.7), for $\theta\leq t$, let
$(V^{\mathbb{F}}_{t,\theta},\Pi^{\mathbb{F}}_{t,\theta})\coloneqq(\alpha
D_{\theta}V^{\mathbb{F}}_{t},\alpha D_{\theta}\Pi^{\mathbb{F}}_{t})$.
Therefore, $(V^{\mathbb{F}}_{\cdot,\theta},\Pi^{\mathbb{F}}_{\cdot,\theta})$
is given by

$\displaystyle V^{\mathbb{F}}_{t,\theta}=\alpha
L^{m}D_{\theta}\xi+\int_{t}^{T}F_{s,\theta}(V^{\mathbb{F}}_{t,\theta},\Pi^{\mathbb{F}}_{t,\theta})\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(\Pi^{\mathbb{F}}_{s,\theta})^{\top}\mathop{}\\!\mathrm{d}W_{s},$
(3.8) $\displaystyle F_{t,\theta}(y,z)\coloneqq-(y-\alpha
z)s_{t}^{b}-h_{t}y+\alpha D_{\theta}\xi^{b}_{t}.$ (3.9)

Note that $F_{t,\theta}(y,z)=F_{t}(y,z)+\alpha
D_{\theta}\xi^{b}_{t}-\xi^{b}_{t}$. Namely, (3.8) can be written as

$\displaystyle V^{\mathbb{F}}_{t,\theta}=$ $\displaystyle\alpha
L^{m}D_{\theta}\xi+\int_{t}^{T}\Big{[}F_{s}(V^{\mathbb{F}}_{t,\theta},\Pi^{\mathbb{F}}_{t,\theta})+\alpha
D_{\theta}\xi^{b}_{t}-\xi^{b}_{t}\big{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(\Pi^{\mathbb{F}}_{s,\theta})^{\top}\mathop{}\\!\mathrm{d}W_{s},$
(3.10)

We will show $V^{\mathbb{F}}_{\cdot}\leq V^{\mathbb{F}}_{\cdot,\theta}$ by
comparing (3.4) and (3.8), and it suffices to show that for $\theta\leq t$,

$\displaystyle L^{m}\xi+\tilde{B}^{\epsilon}_{T}-\alpha
L^{m}D_{\theta}\xi\leq$ $\displaystyle 0,~{}~{}\text{a.s, }$ (3.11)
$\displaystyle\xi^{b}-\alpha D_{\theta}\xi^{b}\leq$ $\displaystyle
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t-\text{a.s}.$
(3.12) $\displaystyle Z^{N}\geq$ $\displaystyle
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t-\text{a.s}.$
(3.13)

We will show that the above inequalities hold if $\epsilon$ is not too big.
More precisely, we assume

$\displaystyle\epsilon\leq\epsilon_{*}\coloneqq
K(B^{\ell}_{T})^{-1}\min\Big{\\{}L^{m},~{}1-\frac{h^{H}L^{H}L^{m}}{h}\Big{\\}}.$
(3.14)

It is easy to check

$\displaystyle\tilde{B}^{\epsilon}_{T}+L^{m}\xi-\alpha L^{m}D_{\theta}\xi=$
$\displaystyle\tilde{B}^{\epsilon}_{T}+L^{m}(\tilde{S}^{1}_{T}-B^{-1}_{T}K)-(\sigma^{1})^{-1}L^{m}D_{\theta}(\tilde{S}^{1}_{T}-B_{T}^{-1}K)$
$\displaystyle=$
$\displaystyle\tilde{B}^{\epsilon}_{T}+L^{m}(\tilde{S}^{1}_{T}-B_{T}^{-1}K)-L^{m}\tilde{S}^{1}_{T}=\tilde{B}^{\epsilon}_{T}-L^{m}B_{T}^{-1}K\leq
0.$

Moreover, by Proposition 3.12 in Di Nunno . (2009),

$\displaystyle\tilde{p}^{N}_{t}-\alpha D_{\theta}\tilde{p}^{N}_{t}=$
$\displaystyle\mathbb{E}[\xi|\mathcal{F}_{t}]-\alpha
D_{\theta}\mathbb{E}[\xi|\mathcal{F}_{t}]$ $\displaystyle=$
$\displaystyle\mathbb{E}[\xi|\mathcal{F}_{t}]-\alpha\mathbb{E}[D_{\theta}\xi|\mathcal{F}_{t}]$
$\displaystyle=$ $\displaystyle\mathbb{E}[\xi-\alpha
D_{\theta}\xi|\mathcal{F}_{t}].$

Moreover, $Z^{N}=D\tilde{p}^{N}=(\sigma^{1})^{-1}\tilde{S}^{1}$. Then, it
follows that

$\displaystyle\xi^{b}_{t}-\alpha D_{\theta}\xi^{b}_{t}=$
$\displaystyle(h-h^{H}L^{H}L^{m})\mathbb{E}[\xi-\alpha
D_{\theta}\xi|\mathcal{F}_{t}]+h\tilde{B}^{\epsilon}_{t}$ $\displaystyle=$
$\displaystyle
h(\tilde{B}^{\epsilon}_{t}-B_{T}^{-1}K)+h^{H}L^{H}L^{m}B_{T}^{-1}K$ (3.15)

Therefore, by comparison principle, for $\theta\leq t$,
$V^{\mathbb{F}}_{t}\leq V^{\mathbb{F}}_{t,\theta}$. In particular,
$V^{\mathbb{F}}_{t}\leq V^{\mathbb{F}}_{t,t}=\alpha
D_{t}V^{\mathbb{F}}_{t}=\alpha\Pi^{\mathbb{F}}_{t}$. Thus, (3.3) is guaranteed
and $(Y^{\\#},Z^{\\#})=(Y^{\mathbb{F}},Z^{\mathbb{F}})$. Moreover,
$(Y^{\mathbb{F}},Z^{\mathbb{F}})$ follows the linear BSDE (3.4), so we can
find an analytic form of $(Y^{\mathbb{F}},Z^{\mathbb{F}})$, namely,
$(V,\tilde{\pi})$ as well. In addition, $\text{FBA}=0$. ∎

######  Remark 3.2.

  1. (i)

If $\epsilon\geq\epsilon^{*}\coloneqq K(B^{b}_{T})^{-1}$, one may want to
consider $(Y^{\\#},Z^{\\#})$ given by

$\displaystyle Y^{\\#}_{t}=$
$\displaystyle-\int_{t}^{T}\bigg{[}\big{(}Y^{\\#}_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\alpha(Z^{\\#}_{s}+Z^{N}_{s})\big{)}s^{\ell}-s^{\epsilon}\tilde{B}^{\epsilon}_{s}+hY^{\\#}_{s}\bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle-\int_{t}^{T}h^{H}L^{H}L^{m}\tilde{p}^{N}_{s}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\\#}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$

and obtain opposite inequalities of (3.11) and (3.12). In this case,
$\text{FCA}=0$. Thus, the binary funding impacts depend on the value of
initial portfolio, $\epsilon$.

  2. (ii)

If inequalities such as (3.2) are not satisfied, instead of (3.1) in the above
example, one should consider

$\displaystyle\xi^{b}\coloneqq$ $\displaystyle
h\tilde{B}^{\epsilon}+\alpha(1-L^{m})s^{b}Z^{N}+h\tilde{p}^{N}-\mathds{1}_{\tilde{p}^{N}+\tilde{e}^{E}\geq
0,\tilde{e}^{E}\geq 0}h^{H}L^{H}L^{m}\tilde{p}^{N}$
$\displaystyle-\mathds{1}_{\tilde{p}^{N}+\tilde{e}^{E}\leq 0,\tilde{e}^{E}\leq
0}h^{C}L^{C}L^{m}\tilde{p}^{N}-\mathds{1}_{\tilde{p}^{N}+\tilde{e}^{E}\geq
0,\tilde{e}^{E}\leq
0}L^{m}(h^{H}L^{H}\tilde{p}^{N}+(h^{H}L^{H}-h^{C}L^{C})\tilde{e}^{E})$
$\displaystyle-\mathds{1}_{\tilde{p}^{N}+\tilde{e}^{E}\leq 0,\tilde{e}^{E}\geq
0}L^{m}(h^{C}L^{C}\tilde{p}^{N}+(h^{C}L^{C}-h^{H}L^{H})\tilde{e}^{E}).$

If the hedger and counterparty are both major banks having similar credit
risks so that we can assume $h^{H}L^{H}=h^{C}L^{C}$, we can obtain the same
result as in Example 3.1.

  3. (iii)

Note that in the above example, $\text{DVA}\not=0$. Consider the same market
conditions but

$\displaystyle\textfrak{D}^{N}=\mathds{1}_{\llbracket
T,\infty\llbracket}(K-S^{1}_{T}).$

Namely, the hedger is in long position of the stock forward contract. We can
calculate the same way for the opposite inequality of (3.3). In this case,
$\text{FCA}=0$, and still $\text{DVA}\not=0$. Note that FVA is either $-$FBA
or FCA, so neutralization of a substantial portion between FCA and FBA is
hardly expected. This binary nature of FVA is a source of asset/liability
asymmetry of FCA/FBA accounting.

  4. (iv)

Example 3.1 also tells us how and when FBA and DVA are different. FBA, as a
counterpart of FCA, reduces FCA which originates from the bank’s default risk.
DVA is also a benefit from the bank’s default risk, but it is hard to monetize
DVA before the bank actually defaults. By these reasons, it is often believed
that FBA and DVA are overlapped, and DVA is not considered in derivative
transactions. However, Example 3.1 shows the different mathematical structure
between FBA and DVA. DVA is a benefit from the possibility that the bank may
default on its derivative payables. Thus, DVA occurs where $\xi\geq 0$. On the
other hand, FBA occurs where the opposite inequality of (3.11). To understand
the meaning of (3.11), we set $\epsilon=0$ and consider
$\xi=\psi(\tilde{S}^{1}_{T})$ for some smooth function
$\psi\colon\mathbb{R}\to\mathbb{R}_{+}$. Then, FBA occurs where

$\displaystyle\xi-\alpha D_{\theta}\xi\geq 0,$ (3.16)

where $\alpha=(\sigma^{1})^{-1}$ in Example 3.1. (3.16) can be rewritten as

$\displaystyle\psi(\tilde{S}^{1}_{T})-\psi^{\prime}(\tilde{S}^{1}_{T})\tilde{S}^{1}_{T}\geq
0,$ (3.17)

and (3.17) again can be rewritten as

$\displaystyle\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}x}\ln{\psi(x)}\leq\frac{1}{x},~{}~{}\forall
x\geq 0.$ (3.18)

One sufficient condition for (3.18) is that $\psi$ is non-increasing.
Similarly, consider $\xi=\Psi(\tilde{S}^{1}_{T})$, where $\Psi(x)\coloneqq
a(x-k)$, for some $a\in\mathbb{R}$ and $k>0$. In this case again, (3.16) is
equivalent to $a\leq 0$. In summary, DVA occurs where the payoff is positive
while FBA occurs where the payoff is non-increasing.

  5. (v)

Another reason of belief that FBA is overlapped with DVA may be the convention
in literature that all assets can be traded in repo markets, i.e., $\rho=I$.
Recall that $\alpha=0$ when $\rho=I$ (recall Remark 2.24). When $\alpha=0$, by
(3.16), FBA also occurs where $\xi\geq 0$. However, we can not guarantee that
repo markets are always available. Indeed, there are some difficulties in
using equities as repo collateral. The amounts of traded equities are smaller
than fixed-income securities and there is no generally accepted method to
valuate equities. By these reasons, equity repo markets are often limited to
equity indices and baskets including many securities.

######  Remark 3.3.

To the best of our knowledge, Malliavin differentiability of ABSDE has not
been studied. In cases of clean close-out, the absence of initial margin is
merely for simplicity. However, when we consider with initial margin under
replacement close-out, we can not use the same method as in Example 3.1.

###  3.1 Main Theorems

In what follows, for the incremental effects, we focus on funding impacts, so
in what follows, we assume that

$\displaystyle e^{E}=0,$

i.e., $\text{CVA}^{\Delta}=\text{CVA}$ and $\text{DVA}^{\Delta}=\text{DVA}$.
However, $\epsilon$ may not be zero. Under this restriction, we give the main
theorems to show that either FBA$=0$ or FCA$=0$. We consider deterministic
default intensities, volatility, and funding spreads, but $(r_{t})_{t\geq 0}$
does not need to be deterministic so that we can apply the result to cases
that $(r_{t})_{t\geq 0}$ is a general $\mathbb{F}$-adapted process. These
assumptions are mainly for simplicity. One case of stochastic default
intensities is reported in Appendix. Moreover, we consider a derivatives of
European style. For derivatives that has cash-flows at multiple times, we can
divide the interval $[0,T]$ according to the time of cash-flows. For example,
if $\textfrak{D}^{N}_{t}=\sum_{i=1}^{N}\mathds{1}_{T_{i}\leq t}\xi^{i}$, for
$t\in(T_{i-1},T_{i})$, we consider the following BSDE:

$\displaystyle
Y^{\mathbb{F}}_{t}=Y^{\mathbb{F}}_{T_{i}}+\int_{t}^{T_{i}}g^{\mathbb{F}}_{s}(Y^{\mathbb{F}}_{s},Z^{\mathbb{F}}_{s})\mathop{}\\!\mathrm{d}s+\int_{t}^{T_{i}}(Z^{\mathbb{F}}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s}.$
(3.19)

Then we can apply the next theorems for each BSDE (3.19). The idea of the
proofs is similar to Example 3.1, and the proofs are reported in Appendix. We
start from clean close-out.

######  Theorem 3.4 (Clean close-out).

We assume $h^{H}$, $h^{C}$, $s^{\ell}$, $s^{b}$ are deterministic and bounded.
Moreover, $\alpha$ is also deterministic and $\alpha s^{\ell}$, $\alpha s^{b}$
are bounded. Consider clean close-out, i.e., $e^{N}=p^{N}$, and
$B^{-1}\textfrak{D}^{N}=\mathds{1}_{T\leq t}\xi$. In addition, we assume

$\displaystyle\xi\in\mathbb{L}_{T}^{2}\cap\mathbb{D}^{1,2},~{}~{}D_{\theta}\xi\in(\mathbb{D}^{1,2})^{n}~{}~{}\forall\theta\leq
T,$ (3.20)
$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\big{|}D_{\theta}\xi\big{|}^{2}\mathop{}\\!\mathrm{d}\theta\bigg{]}<\infty,$
(3.21)
$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\int_{0}^{T}\big{|}D_{t}(D_{\theta}\xi)\big{|}^{2}\mathop{}\\!\mathrm{d}\theta\mathop{}\\!\mathrm{d}t\bigg{]}<\infty.$
(3.22)

We assume that $\mathds{1}_{\tilde{p}^{N}=0}=0$,
$\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t$ a.s. Let

$\displaystyle\xi^{b}\coloneqq$ $\displaystyle
s^{b}(1-L^{m})\alpha^{\top}Z^{N}+h\tilde{B}^{\epsilon}+(h-h^{H}L^{H}L^{m})(\tilde{p}^{N})^{+}-(h-h^{C}L^{C}L^{m})(\tilde{p}^{N})^{-},$
$\displaystyle\xi^{\ell}\coloneqq$ $\displaystyle
s^{\ell}(1-L^{m})\alpha^{\top}Z^{N}+h\tilde{B}^{\epsilon}+(h-h^{H}L^{H}L^{m})(\tilde{p}^{N})^{+}-(h-h^{C}L^{C}L^{m})(\tilde{p}^{N})^{-}.$

(i) Assume that for any $\theta\leq T$,

$\displaystyle
L^{m}(\xi-\alpha_{\theta}^{\top}D_{\theta}\xi)+\tilde{B}^{\epsilon}_{T}\leq$
$\displaystyle 0,~{}~{}\text{a.s},$ (3.23)
$\displaystyle\alpha^{\top}Z^{N}\geq$ $\displaystyle
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t-\text{a.s},$
(3.24) $\displaystyle\xi^{b}-\alpha_{\theta}^{\top}D_{\theta}\xi^{b}\leq$
$\displaystyle
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t-\text{a.s},$
(3.25)

then there exists
$(Y^{\\#},Z^{\\#})\in\mathbb{S}^{2}_{T}\times\mathbb{H}^{2,n}_{T}$ that
satisfies

$\displaystyle Y^{\\#}_{t}=$
$\displaystyle\int_{t}^{T}-\bigg{[}\Big{(}Y^{\\#}_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\phi_{s}(Z^{\\#}_{s})\Big{)}s^{b}_{s}-s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s}+h_{s}Y^{\\#}_{s}\bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle+\int_{t}^{T}-\big{[}\mathds{1}_{\tilde{p}^{N}_{s}\geq
0}h^{H}_{s}L^{H}L^{m}+\mathds{1}_{\tilde{p}^{N}_{s}<0}h^{C}_{s}L^{C}L^{m}\big{]}\tilde{p}^{N}_{s}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\\#}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$

and $(Y^{\\#},Z^{\\#})=(Y^{\mathbb{F}},Z^{\mathbb{F}})$. In particular, _FBA_
=$0$.

(ii) On the other hand, assume that for any $\theta\leq T$,

$\displaystyle
L^{m}(\xi-\alpha_{\theta}^{\top}D_{\theta}\xi)+\tilde{B}^{\epsilon}_{T}\geq$
$\displaystyle 0,~{}~{}\text{a.s,}$ (3.26)
$\displaystyle\alpha^{\top}Z^{N}\leq$ $\displaystyle
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t-\text{a.s},$
(3.27)
$\displaystyle\xi^{\ell}-\alpha_{\theta}^{\top}D_{\theta}\xi^{\ell}\geq$
$\displaystyle
0,~{}~{}\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t-\text{a.s},$
(3.28)

then there exists
$(Y^{\\#},Z^{\\#})\in\mathbb{S}^{2}_{T}\times\mathbb{H}^{2,n}_{T}$ that
satisfies

$\displaystyle Y^{\\#}_{t}=$
$\displaystyle\int_{t}^{T}-\bigg{[}\Big{(}Y^{\\#}_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\phi_{s}(Z^{\\#}_{s})\Big{)}s^{\ell}_{s}-s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s}+h_{s}Y^{\\#}_{s}\bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle+\int_{t}^{T}-\big{[}\mathds{1}_{\tilde{p}^{N}_{s}\geq
0}h^{H}_{s}L^{H}L^{m}+\mathds{1}_{\tilde{p}^{N}_{s}<0}h^{C}_{s}L^{C}L^{m}\big{]}\tilde{p}^{N}_{s}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\\#}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$

and $(Y^{\\#},Z^{\\#})=(Y^{\mathbb{F}},Z^{\mathbb{F}})$. In particular,
$\emph{FCA}=0$.

(iii) If the contract is un-collateralized, i.e., $L^{m}=1$, (3.24) and (3.27)
are not required in (i) and (ii).

Recall that when we consider replacement close-out,
$\tilde{\Theta}^{\Delta,i}$, $i\in\\{H,C\\}$, depend on $Y^{\mathbb{F}}$.
However, $\tilde{\Theta}^{\Delta,i}(y)$ is not differentiable in $y$. We can
avoid the irregularity by considering contracts such that either
$\tilde{p}^{N}\geq 0$ or $\tilde{p}^{N}\leq 0$,
$\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t$ a.s, i.e.,
options.

######  Theorem 3.5 (Replacement close-out).

We assume $h^{H}$, $h^{C}$, $s^{\ell}$, $s^{b}$ are deterministic and bounded.
Moreover $\alpha$ is also deterministic and $\alpha s^{\ell}$, $\alpha s^{b}$
are bounded. Consider replacement close-out, i.e.,
$e^{N}=V_{-}-B^{\epsilon}=BY^{\mathbb{F}}+p^{N}$, and
$B^{-1}\textfrak{D}^{N}=\mathds{1}_{T\leq t}\xi$. In addition, we assume

$\displaystyle\xi\in\mathbb{L}_{T}^{2}\cap\mathbb{D}^{1,2},$ (3.29)
$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\big{|}D_{\theta}\xi\big{|}^{2}\mathop{}\\!\mathrm{d}\theta\bigg{]}<\infty.$
(3.30)

We assume that either $\xi\geq 0$ or $\xi\leq 0$ a.s, i.e., we consider
options.

(i) Assume that $\epsilon\leq 0$ and for any $\theta\leq T$,

$\displaystyle L^{m}\xi-\alpha_{\theta}^{\top}D_{\theta}\xi\leq
0,~{}~{}\text{a.s.}$ (3.31)

If $\xi\geq 0$ a.s, then there exists a solution
$(Y^{\\#},Z^{\\#})\in\mathbb{S}^{2}_{T}\times\mathbb{H}^{2,n}_{T}$ that
satisfies

$\displaystyle Y^{\\#}_{t}=$
$\displaystyle\int_{t}^{T}-\bigg{[}\Big{(}L^{m}Y^{\\#}_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\phi_{s}(Z^{\\#}_{s})\Big{)}s^{b}_{s}-s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s}\bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle+\int_{t}^{T}\big{[}-h^{H}_{s}L^{H}L^{m}(Y^{\\#}_{s}+\tilde{p}^{N}_{s})\big{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\\#}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$

and $(Y^{\\#},Z^{\\#})=(Y^{\mathbb{F}},Z^{\mathbb{F}})$. On the other hand, if
$\xi\leq 0$ a.s, then there exists a solution
$(Y^{\\#},Z^{\\#})\in\mathbb{S}^{2}_{T}\times\mathbb{H}^{2,n}_{T}$ that
satisfies

$\displaystyle Y^{\\#}_{t}=$
$\displaystyle\int_{t}^{T}-\bigg{[}\Big{(}L^{m}Y^{\\#}_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\phi_{s}(Z^{\\#}_{s})\Big{)}s^{b}_{s}-s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s}\bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle+\int_{t}^{T}\big{[}-h^{C}_{s}L^{C}L^{m}(Y^{\\#}_{s}+\tilde{p}^{N}_{s})\big{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\\#}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$

and $(Y^{\\#},Z^{\\#})=(Y^{\mathbb{F}},Z^{\mathbb{F}})$. In particular, for
both cases, $\emph{FBA}=0$.

(ii) Assume that $\epsilon\geq 0$ and for any $\theta\leq T$,

$\displaystyle L^{m}\xi-\alpha_{\theta}^{\top}D_{\theta}\xi\geq
0,~{}~{}\text{a.s.}$ (3.32)

If $\xi\geq 0$ a.s, then there exists a solution
$(Y^{\\#},Z^{\\#})\in\mathbb{S}^{2}_{T}\times\mathbb{H}^{2,n}_{T}$ that
satisfies

$\displaystyle Y^{\\#}_{t}=$
$\displaystyle\int_{t}^{T}-\bigg{[}\Big{(}L^{m}Y^{\\#}_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\phi_{s}(Z^{\\#}_{s})\Big{)}s^{\ell}_{s}-s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s}\bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle+\int_{t}^{T}\big{[}-h^{H}_{s}L^{H}L^{m}(Y^{\\#}_{s}+\tilde{p}^{N}_{s})\big{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\\#}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$

and $(Y^{\\#},Z^{\\#})=(Y^{\mathbb{F}},Z^{\mathbb{F}})$. On the other hand, if
$\xi\leq 0$ a.s, then there exists a solution
$(Y^{\\#},Z^{\\#})\in\mathbb{S}^{2}_{T}\times\mathbb{H}^{2,n}_{T}$ that
satisfies

$\displaystyle Y^{\\#}_{t}=$
$\displaystyle\int_{t}^{T}-\bigg{[}\Big{(}L^{m}Y^{\\#}_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\phi_{s}(Z^{\\#}_{s})\Big{)}s^{\ell}_{s}-s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s}\bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle+\int_{t}^{T}\big{[}-h^{C}_{s}L^{C}L^{m}(Y^{\\#}_{s}+\tilde{p}^{N}_{s})\big{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\\#}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$

and $(Y^{\\#},Z^{\\#})=(Y^{\mathbb{F}},Z^{\mathbb{F}})$. In particular, for
both cases, $\emph{FCA}=0$.

##  4 Examples and a Closed-form Solution

Many standard derivatives satisfy the conditions in Theorem 3.4 and Theorem
3.5. We will apply the main theorems to several derivatives and provide a
closed-form solution for a call option. In what follows, for $i\in I$, we
denote

$\displaystyle\tilde{S}^{i}\coloneqq B^{-1}S^{i}.$

Moreover, recall that in the main theorems, we defined $\xi$ by

$\displaystyle\xi\coloneqq B_{T}^{-1}\Delta\textfrak{D}^{N}_{T}.$

###  4.1 Clean close-out

Banks buy Treasury bonds that return less than their funding rate. It was
insisted in Hull White (2012) that this shows that FVA should not be
considered in derivative prices. We will show that when buying bonds, FCA$=0$
for the hedger. Therefore, if we assume

$\displaystyle R^{\ell}=r,$

as in Burgard Kjaer (2010), the fair price for the hedger is approximately the
same as the bond price derived from discounting with the Treasury rate. Recall
that, in the main theorem, we only assume that $s^{\ell}$ and $s^{b}$ are
deterministic. As long as the spreads are deterministic, we can apply the
theorems to interest rate derivatives.

######  Example 4.1 (Buying a Treasury bond).

Let us consider a hedger buying a Treasury bond with unit notional amount,
i.e., $\textfrak{D}^{N}=-\mathds{1}_{\llbracket T,\infty\llbracket}$. We
assume that for $i\in\\{H,C\\}$,
$\mathop{}\\!\mathrm{d}G^{i}_{t}=-h^{i}_{t}G^{i}_{t}\mathop{}\\!\mathrm{d}t$,
where $(h^{i}_{t})_{t\geq 0}$ are deterministic processes. Moreover, we assume
that OIS rate $r$ is given by

$\displaystyle\mathop{}\\!\mathrm{d}r_{t}=\kappa(\theta-
r_{t})\mathop{}\\!\mathrm{d}t+\Sigma\mathop{}\\!\mathrm{d}W_{t},$

for some $\kappa,\theta,\Sigma>0$. Thus,

$\displaystyle\sigma^{1}=$ $\displaystyle\sigma^{H}=\sigma^{C},$
$\displaystyle\sigma^{1}_{t}=$
$\displaystyle-\frac{\Sigma[1-e^{-\kappa(T-t)}]}{\kappa}.$

Moreover, we assume $\rho=\emptyset$. Then $\sum_{i\in
I}\tilde{\pi}^{i}\sigma^{i}=\sigma^{1}(\sum_{i\in I}\tilde{\pi}^{i})$.
Therefore,

$\displaystyle\phi(z)=\alpha(z+Z^{N})=(\sigma^{1})^{-1}(z+Z^{N}).$

Since

$\displaystyle r_{t}=r_{0}e^{-\kappa t}+\theta(1-e^{-\kappa
t})+\Sigma\int_{0}^{t}e^{-\kappa(t-u)}\mathop{}\\!\mathrm{d}W_{u},$

by Corollary 3.19 in Di Nunno . (2009), for any $\theta\leq t$,
$D_{\theta}r_{t}=\Sigma e^{-\kappa(t-\theta)}$, and it follows that

$\displaystyle D_{\theta}B^{-1}_{t}=$ $\displaystyle-
B_{t}^{-1}\int_{\theta}^{t}D_{\theta}r_{s}\mathop{}\\!\mathrm{d}s$
$\displaystyle=$ $\displaystyle-
B_{t}^{-1}\Sigma\int_{\theta}^{t}e^{-\kappa(s-\theta)}\mathop{}\\!\mathrm{d}s$
$\displaystyle=$ $\displaystyle-
B_{t}^{-1}\frac{\Sigma[1-e^{-\kappa(t-\theta)}]}{\kappa}.$

Recalling that $\xi=-B^{-1}_{T}$,

$\displaystyle\xi-\alpha_{\theta}D_{\theta}\xi=$
$\displaystyle\xi-(\sigma^{1}_{\theta})^{-1}D_{\theta}\xi$ $\displaystyle=$
$\displaystyle-
B_{T}^{-1}+(\sigma^{1}_{\theta})^{-1}B_{T}^{-1}\frac{\Sigma[1-e^{-\kappa(T-\theta)}]}{\kappa}=0.$

It follows that for any $\theta\leq t$,
$\tilde{p}^{N}_{t}-\alpha_{\theta}D_{\theta}\tilde{p}^{N}_{t}=0$. Moreover,

$\displaystyle\alpha_{t}^{\top}Z^{N}_{t}=(\sigma^{1}_{t})^{-1}D_{t}\tilde{p}^{N}_{t}=-S^{1}_{t}<0.$

Therefore, by (ii) in Theorem 3.4, FCA$=0$ where $\epsilon\geq 0$. Therefore,
if the initial value of legacy portfolio is non-negative, the trader does not
enter a borrowing position and there is no FVA that should be recouped. ∎

The next example is a general form of Example 3.1.

######  Example 4.2 (Combination of forward contracts).

Let $n=1$, $\rho=\\{H,C\\}$, and the traded assets are given by

$\displaystyle\mathop{}\\!\mathrm{d}S^{1}_{t}=$ $\displaystyle
rS^{1}_{t}\mathop{}\\!\mathrm{d}t+\sigma^{1}_{t}S^{1}_{t}\mathop{}\\!\mathrm{d}W_{t},$
$\displaystyle\mathop{}\\!\mathrm{d}S^{i}_{t}=$ $\displaystyle
rS^{i}_{t}\mathop{}\\!\mathrm{d}t+S^{i}_{t-}\mathop{}\\!\mathrm{d}M^{i}_{t},~{}i\in\\{H,C\\}.$

Since $\rho=\\{H,C\\}$, $\sum_{i\in
I}\tilde{\pi}^{i}\sigma^{i}=\sigma^{1}\tilde{\pi}^{1}$ and $\sum_{i\in
I\setminus\rho}\tilde{\pi}^{i}=\tilde{\pi}^{1}$. Thus,

$\displaystyle\phi_{t}(z)=$ $\displaystyle\alpha_{t}(z+Z^{N}_{t}),$
$\displaystyle\alpha_{t}=$ $\displaystyle(\sigma^{1}_{t})^{-1}.$

We consider a combination of forward contracts:
$\textfrak{D}^{N}_{t}=\mathds{1}_{T\leq
t}\sum_{i=1}^{N}\omega_{i}(S^{1}_{T}-K_{i})$, where
$\omega_{i},K_{i}\in\mathbb{R}$. We assume that parameters are deterministic
and

$\displaystyle s^{b}\geq 0,~{}~{}h^{H}\geq 0,~{}~{}h^{C}\geq 0,$
$\displaystyle\sum^{N}_{i=1}\omega_{i}\leq
0,~{}~{}\sum^{N}_{i=1}\omega_{i}K_{i}\leq 0.$

By the definition of clean price,

$\displaystyle\tilde{p}^{N}_{t}=$
$\displaystyle\sum_{i=1}^{N}\mathbb{E}\big{[}\omega_{i}B_{T}^{-1}(S^{1}_{T}-K_{i})\big{|}\mathcal{F}_{t}\big{]}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\omega_{i}(\tilde{S}_{t}-B^{-1}_{T}K_{i}).$

Since
$\mathop{}\\!\mathrm{d}\tilde{S}^{1}_{t}=\sigma^{1}_{t}\tilde{S}_{t}\mathop{}\\!\mathrm{d}W_{t}$,
by Corollary 3.19 in Di Nunno . (2009), for any $\theta\leq t$,

$\displaystyle
D_{\theta}\tilde{S}^{1}_{t}=\int_{\theta}^{t}\sigma_{\theta}^{1}D_{\theta}\tilde{S}^{1}_{s}\mathop{}\\!\mathrm{d}W_{s}+\sigma^{1}_{\theta}\tilde{S}^{1}_{\theta}.$
(4.1)

Since $\sigma_{\theta}^{1}\tilde{S}^{1}_{t}$ satisfies (4.1), by the
uniqueness,

$\displaystyle
D_{\theta}\tilde{S}^{1}_{t}=\sigma_{\theta}^{1}\tilde{S}^{1}_{t}.$

Moreover,
$\alpha_{t}Z^{N}_{t}=(\sigma_{t}^{1})^{-1}Z^{N}_{t}=\tilde{S}^{1}_{t}\sum_{i}\omega_{i}\leq
0$. By straightforward calculation, for any $\theta\leq t$,

$\displaystyle Z^{N}_{t}-\alpha_{\theta}D_{\theta}Z^{N}_{t}=$
$\displaystyle\tilde{S}^{1}_{t}\sum_{i}\omega_{i}-\alpha_{\theta}D_{\theta}(\tilde{S}^{1}_{t})\sum_{i}\omega_{i}$
$\displaystyle=$
$\displaystyle\tilde{S}^{1}_{t}\sum_{i}\omega_{i}-\tilde{S}^{1}_{t}\sum_{i}\omega_{i}=0,$
(4.2)
$\displaystyle\tilde{p}^{N}_{t}-\alpha_{\theta}D_{\theta}\tilde{p}^{N}_{t}=$
$\displaystyle\sum_{i=1}^{N}\omega_{i}(\tilde{S}_{t}-B^{-1}_{T}K_{i})-\alpha_{\theta}D_{\theta}\big{[}\sum_{i=1}^{N}\omega_{i}(\tilde{S}_{t}-B^{-1}_{T}K_{i})\big{]}$
$\displaystyle=$
$\displaystyle-B^{-1}_{T}\sum\limits_{i=1}^{N}\omega_{i}K_{i}\leq 0.$ (4.3)

Recall that as defined in Theorem 3.4,

$\displaystyle\xi=$
$\displaystyle\sum_{i=1}^{N}\omega_{i}(\tilde{S}^{1}_{T}-B_{T}^{-1}K_{i}),$
$\displaystyle\xi^{\ell}=$ $\displaystyle s^{\ell}(1-L^{m})\alpha
Z^{N}+h\tilde{B}^{\epsilon}+(h-h^{H}L^{H}L^{m})(\tilde{p}^{N})^{+}-(h-h^{C}L^{C}L^{m})(\tilde{p}^{N})^{-}.$

Note that $\mathds{1}_{\tilde{p}^{N}=0}=0$,
$\mathop{}\\!\mathrm{d}\mathbb{Q}\otimes\mathop{}\\!\mathrm{d}t$ a.s. Then by
the (4.2) and (4.3), we attain that

$\displaystyle
L^{m}(\xi-\alpha_{\theta}D_{\theta}\xi)+\tilde{B}^{\epsilon}_{T}=B_{T}^{-1}\big{(}B^{\epsilon}_{T}-L^{m}\sum_{i=1}^{N}\omega_{i}K_{i}\big{)}\geq
0,$ $\displaystyle\xi^{\ell}_{t}-\alpha
D_{\theta}\xi^{\ell}_{t}=B_{T}^{-1}\bigg{(}hB^{\epsilon}_{t}-\sum_{i}\omega_{i}K_{i}\Big{[}s^{\ell}(1-L^{m})+(h-h^{H}L^{H}L^{m})\mathds{1}_{\tilde{p}^{N}>0}+(h-h^{C}L^{C}L^{m})\mathds{1}_{\tilde{p}^{N}<0}\Big{]}\bigg{)}\geq
0.$

Assume that

$\displaystyle\epsilon\geq$
$\displaystyle(B^{b}_{T})^{-1}q\sum_{i}\omega_{i}K_{i},$ $\displaystyle
q\coloneqq$
$\displaystyle\min\Big{\\{}L^{m},~{}\frac{s^{\ell}(1-L^{m})+(h-h^{H}L^{H}L^{m})}{h}\Big{\\}}.$
(4.4)

Then, by (ii) in Theorem 3.4, $(Y^{\mathbb{F}},Z^{\mathbb{F}})$ follows

$\displaystyle Y^{\mathbb{F}}_{t}=$
$\displaystyle\int_{t}^{T}-\bigg{[}\Big{(}Y^{\mathbb{F}}_{s}+L^{m}\tilde{p}^{N}_{s}+\tilde{B}^{\epsilon}_{s}-\phi_{s}(Z^{\mathbb{F}}_{s})\Big{)}s^{\ell}_{s}-s^{\epsilon}_{s}\tilde{B}^{\epsilon}_{s}+h_{s}Y^{\mathbb{F}}_{s}\bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle+\int_{t}^{T}-\big{[}\mathds{1}_{\tilde{p}^{N}_{s}\geq
0}h^{H}_{s}L^{H}L^{m}+\mathds{1}_{\tilde{p}^{N}_{s}<0}h^{C}_{s}L^{C}L^{m}\big{]}\tilde{p}^{N}_{s}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}(Z^{\mathbb{F}}_{s})^{\top}\mathop{}\\!\mathrm{d}W_{s},$
(4.5)

and FCA$=0$. ∎

To find an analytic form of (4.5), let $V^{\mathbb{F}}\coloneqq
Y^{\mathbb{F}}+\tilde{p}^{N}$, $\Pi^{\mathbb{F}}=Z^{\mathbb{F}}+Z^{N}$, and
let $\mathbb{Q}^{\ell}$ denote an equivalent measure such that
$(B^{\ell})^{-1}S^{i}$ is $(\mathbb{Q}^{\ell},\mathbb{G})$-local martingales.
In particular,

$\displaystyle W^{\ell}_{t}\coloneqq
W_{t}-\int_{0}^{t}\alpha_{s}s^{\ell}_{s}\mathop{}\\!\mathrm{d}s,$

is an $(\mathbb{F},\mathbb{Q}^{\ell})$-Brownian motion. Then (4.5) becomes

$\displaystyle V^{\mathbb{F}}_{t}=$
$\displaystyle\xi+\int_{t}^{T}\Big{[}-(s^{\ell}_{s}+h_{s})V^{\mathbb{F}}_{s}+(s^{\epsilon}_{s}-s^{\ell}_{s})\tilde{B}^{\epsilon}_{s}+\beta_{s}\tilde{p}^{N}_{s}\Big{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}\Pi^{\mathbb{F}}_{s}\mathop{}\\!\mathrm{d}W^{\ell}_{s},$
$\displaystyle\beta_{t}\coloneqq$
$\displaystyle(1-L^{m})s^{\ell}_{t}+h_{t}-\mathds{1}_{\tilde{p}^{N}_{s}\geq
0}h^{H}_{t}L^{H}L^{m}-\mathds{1}_{\tilde{p}^{N}_{s}<0}h^{C}_{t}L^{C}L^{m}.$

Let
$A_{t}\coloneqq\exp{\big{[}-\int_{0}^{t}(s^{\ell}_{s}+h_{s})\mathop{}\\!\mathrm{d}s\big{]}}$.
Then

$\displaystyle
V^{\mathbb{F}}_{t}=A^{-1}_{t}\mathbb{E}^{\ell}\bigg{[}A_{T}\xi+\int_{t}^{T}A_{s}\big{[}(s^{\epsilon}_{s}-s^{\ell}_{s})\tilde{B}^{\epsilon}_{s}+\beta_{s}\tilde{p}^{N}_{s}\big{]}\mathop{}\\!\mathrm{d}s\bigg{|}\mathcal{F}_{t}\bigg{]},$
(4.6)

where $\mathbb{E}^{\ell}$ is the expectation under $\mathbb{Q}^{\ell}$. (4.6)
reduces the computational cost because it changes from backward simulation to
forward simulation. In other words, we can reduce the computational cost for
calculating conditional expectations at each time step in solving BSDEs
numerically. However, the advantage stops there with clean close-out
convention and we can not find a closed-form solution of $V^{\mathbb{F}}$.
This is because of the mismatch of the pricing measures in

$\displaystyle\mathbb{E}^{\ell}[\beta_{s}\tilde{p}^{N}_{s}\big{|}\mathcal{F}_{t}]=\mathbb{E}^{\ell}[\mathbb{E}[\beta_{s}\xi|\mathcal{F}_{s}]\big{|}\mathcal{F}_{t}].$

To avoid this difficulty, Brigo . (2017) considered un-collateralized
contracts with null cash-flow at defaults. On the other hand, Bichuch . (2018)
assumed that the close-out amount and collateral are calculated by the risk-
neutral price under $\mathbb{Q}^{\ell}$ (or $\mathbb{Q}^{b}$), namely

$\displaystyle e^{N}_{t}=$
$\displaystyle(B^{\ell}_{t})^{-1}\mathbb{E}^{\ell}[(B^{\ell}_{T})^{-1}\zeta|\mathcal{F}_{t}],$
(4.7) $\displaystyle\zeta\coloneqq$ $\displaystyle\Delta\textfrak{D}^{N}_{T},$
(4.8) $\displaystyle m^{N}_{t}=$ $\displaystyle(1-L^{m})e^{N}_{t}.$ (4.9)

In these cases, the pricing measures are aligned and a closed-form solution is
allowed. However, note that (4.7) is

$\displaystyle\textit{clean price}+\text{"the hedger's FVA"}.$

As we will see later, when replacement close-out is assumed, the inconsistency
of pricing measures does not appear and closed-form solutions are given.
However, recall that the hedger’s funding information is already considered in
$V_{-}$. Thus, the consistency of pricing measures is inherent in replacement
close-out.

###  4.2 Replacement Close-out

In the next example, we deal with a non-Markovian case. This is one benefit of
BSDEs and Malliavin calculus.

######  Example 4.3 (Floating strike Asian call option with replacement close-
out).

We consider the same market condition as in Example 4.2 except

$\displaystyle e^{N}=V_{-}-B^{\epsilon}.$ (4.10)

Moreover, we consider a floating strike Asian call option:

$\displaystyle\textfrak{D}^{N}=$ $\displaystyle\mathds{1}_{\llbracket
T,\infty\llbracket}B_{T}\xi,$ $\displaystyle\xi\coloneqq$
$\displaystyle(\tilde{S}^{1}_{T}-B_{T}^{-1}K\mathbb{I}_{T})^{+},$
$\displaystyle\mathbb{I}_{T}\coloneqq$
$\displaystyle\exp{\Big{(}\frac{1}{T}\int_{0}^{T}\ln{(S^{1}_{u})}\mathop{}\\!\mathrm{d}u\Big{)}}.$

We will show (3.32). By Theorem 3.5 in Di Nunno . (2009), for $\theta\leq T$,

$\displaystyle\alpha_{\theta}D_{\theta}\mathbb{I}_{T}=$
$\displaystyle\frac{\alpha_{\theta}}{T}\mathbb{I}_{T}\int_{\theta}^{T}D_{\theta}\ln{(S^{1}_{u})}\mathop{}\\!\mathrm{d}u$
$\displaystyle=$
$\displaystyle\frac{\alpha_{\theta}}{T}\mathbb{I}_{T}\int_{\theta}^{T}(S^{1}_{u})^{-1}D_{\theta}(S^{1}_{u})\mathop{}\\!\mathrm{d}u$
$\displaystyle=$
$\displaystyle\frac{\alpha_{\theta}}{T}\mathbb{I}_{T}\int_{\theta}^{T}(S^{1}_{u})^{-1}\sigma^{1}_{\theta}S^{1}_{u}\mathop{}\\!\mathrm{d}u$
$\displaystyle=$ $\displaystyle\frac{T-\theta}{T}\mathbb{I}_{T}.$

Since $\alpha_{\theta}D_{\theta}\tilde{S}^{1}_{T}=\tilde{S}^{1}_{T}$, we
attain that

$\displaystyle L^{m}\xi-\alpha_{\theta}D_{\theta}\xi=$
$\displaystyle\mathds{1}_{\tilde{S}^{1}_{T}\geq
B_{T}^{-1}K\mathbb{I}_{T}}\big{[}L^{m}(\tilde{S}^{1}_{T}-B_{T}^{-1}K\mathbb{I}_{T})-\alpha_{\theta}D_{\theta}(\tilde{S}^{1}_{T}-B_{T}^{-1}K\mathbb{I}_{T})\big{]}$
$\displaystyle=$ $\displaystyle\mathds{1}_{S^{1}_{T}\geq
K\mathbb{I}_{T}}\Big{[}(L^{m}-1)\tilde{S}^{1}_{T}-B^{-1}_{T}K\mathbb{I}_{T}\Big{(}L^{m}-\frac{T-\theta}{T}\Big{)}\Big{]}$
$\displaystyle\leq$ $\displaystyle\mathds{1}_{S^{1}_{T}\geq
K\mathbb{I}_{T}}\big{[}(L^{m}-1)\tilde{S}^{1}_{T}-B^{-1}_{T}K\mathbb{I}_{T}(L^{m}-1)\big{]}$
$\displaystyle=$
$\displaystyle(L^{m}-1)(\tilde{S}^{1}_{T}-B^{-1}_{T}K\mathbb{I}_{T})^{+}\leq
0.$

Therefore, FBA$=0$, where $\epsilon\leq 0$.

As the last example, we deal with a bond option.

######  Example 4.4 (Bond option with replacement close-out).

We assume that the same market conditions as in Example 4.1. Let
$\textfrak{D}^{N}_{t}=\mathds{1}_{T\leq t}(S^{1}_{T,U}-K)^{+}$, where $K>0$
and $S^{1}_{\cdot,U}$ is a zero coupon bond with $U>T$ as its maturity. We
consider two defaultable bonds with the same maturity, i.e.,

$\displaystyle S^{1}_{t,U}\coloneqq$ $\displaystyle
B_{t}\mathbb{E}\big{[}B_{U}^{-1}|\mathcal{F}_{t}\big{]},$ $\displaystyle
S^{i}_{t,U}\coloneqq$ $\displaystyle
B_{t}\mathbb{E}\big{[}\mathds{1}_{U<\tau^{i}}B_{U}^{-1}|\mathcal{G}_{t}\big{]},~{}i\in\\{H,C\\}.$

Recall $\phi(z)=\alpha(z+Z^{N})=(\sigma^{1})^{-1}(z+Z^{N})$, and

$\displaystyle\sigma^{1}=$ $\displaystyle\sigma^{H}=\sigma^{C},$
$\displaystyle\sigma^{1}_{t}=$
$\displaystyle-\frac{\Sigma[1-e^{-\kappa(U-t)}]}{\kappa}.$

Moreover, recall the definition of $\xi$:

$\displaystyle\xi=\mathds{1}_{S^{1}_{T,U}\geq
K}(\tilde{S}^{1}_{T,U}-KB^{-1}_{T}).$

We can see that

$\displaystyle\alpha_{\theta}D_{\theta}(KB^{-1}_{T})=$
$\displaystyle(\sigma_{\theta}^{1})^{-1}KD_{\theta}B_{T}^{-1}$
$\displaystyle=$ $\displaystyle-
KB_{T}^{-1}(\sigma_{\theta}^{1})^{-1}\frac{\Sigma[1-e^{-\kappa(T-\theta)}]}{\kappa}$
$\displaystyle=$ $\displaystyle
KB_{T}^{-1}\frac{1-e^{-\kappa(T-\theta)}}{1-e^{-\kappa(U-\theta)}}.$

In addition,

$\displaystyle\alpha_{\theta}D_{\theta}\tilde{S}^{1}_{T,U}=(\sigma_{\theta}^{1})^{-1}\sigma_{\theta}^{1}\tilde{S}^{1}_{T}=\tilde{S}^{1}_{T,U}.$

It follows that on $\\{S^{1}_{T,U}\geq K\\}$,

$\displaystyle L^{m}\xi-\alpha_{\theta}D_{\theta}\xi=$ $\displaystyle
L^{m}(\tilde{S}^{1}_{T,U}-KB^{-1}_{T})-\alpha_{\theta}D_{\theta}(\tilde{S}^{1}_{T,U}-KB^{-1}_{T})$
$\displaystyle=$
$\displaystyle(L^{m}-1)\tilde{S}^{1}_{T,U}-KB^{-1}_{T}\bigg{(}L^{m}-\frac{1-e^{-\kappa(T-\theta)}}{1-e^{-\kappa(U-\theta)}}\bigg{)}$
$\displaystyle\leq$
$\displaystyle(L^{m}-1)\tilde{S}^{1}_{T,U}-KB^{-1}_{T}(L^{m}-1)$
$\displaystyle=$ $\displaystyle(L^{m}-1)(\tilde{S}^{1}_{T,U}-KB^{-1}_{T})\leq
0.$ (4.11)

Therefore, by (i) in Theorem 3.5, $(Y^{\mathbb{F}},Z^{\mathbb{F}})$ is given
by

$\displaystyle
Y^{\mathbb{F}}_{t}=\int_{t}^{T}-\bigg{[}\Big{(}L^{m}Y^{\mathbb{F}}_{s}+L^{m}\tilde{p}^{N}_{s}-\phi_{s}(Z^{\mathbb{F}}_{s})\Big{)}s^{b}_{s}+h^{H}_{s}L^{H}L^{m}(Y^{\mathbb{F}}_{s}+\tilde{p}^{N}_{s})\bigg{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}Z^{\mathbb{F}}_{s}\mathop{}\\!\mathrm{d}W_{s},$
(4.12)

and FBA$=0$. Note that $\text{DVA}\not=0$ because $\xi\geq 0$, but
$\text{FBA}=0$, where $\epsilon\leq 0$ because $\xi$ increases in
$S^{1}_{t,U}$. ∎

Table 1 shows the effects of FBA and DVA with respect to option contracts.
Only when the hedger sells a put option, FBA and DVA are both positive, but
still $\text{FBA}\not=\text{DVA}$.

| FBA | DVA
---|---|---
buy/call | positive | nil
sell/call | nil | positive
buy/put | nil | nil
sell/put | positive | positive
Table 1: DVA and FBA with respect to option contracts

###  4.3 A Closed-form Solution of a Call Option under Replacement Close-out

Under replacement close-out, we can find a closed-form solution. As an
example, we discuss the solution of a stock call option. Let $n=1$,
$e^{N}=V_{-}-B^{\epsilon}$, $\rho=\\{H,C\\}$, $\epsilon\leq 0$,
$\textfrak{D}^{N}=\mathds{1}_{\llbracket
T,\infty\llbracket}(S^{1}_{T}-K)^{+}$, where

$\displaystyle\mathop{}\\!\mathrm{d}S^{1}_{t}=$ $\displaystyle
rS^{1}_{t}\mathop{}\\!\mathrm{d}t+\sigma^{1}S^{1}_{t}\mathop{}\\!\mathrm{d}W_{t},$
$\displaystyle\mathop{}\\!\mathrm{d}S^{i}_{t}=$ $\displaystyle
rS^{i}_{t}\mathop{}\\!\mathrm{d}t+S^{i}_{t-}\mathop{}\\!\mathrm{d}M^{i}_{t},~{}i\in\\{H,C\\},$

for some constants $r$ and $\sigma^{1}$. We also assume that $s^{b}$, $h^{H}$
are constant. Recall

$\displaystyle\xi=$ $\displaystyle B_{T}^{-1}(S^{1}_{T}-K)^{+},$
$\displaystyle\phi_{t}(z)=$ $\displaystyle\alpha(z+Z^{N}_{t})$
$\displaystyle=$ $\displaystyle(\sigma^{1})^{-1}(z+Z^{N}_{t}).$

It is easy to check that

$\displaystyle L^{m}\xi-\alpha_{\theta}D_{\theta}\xi=$ $\displaystyle
L^{m}\xi-(\sigma^{1})^{-1}D_{\theta}\xi$ $\displaystyle=$
$\displaystyle\mathds{1}_{S^{1}_{T}\geq
K}\big{[}(L^{m}-1)\tilde{S}^{1}_{T}-KL^{m}B_{T}^{-1}\big{]}\leq 0.$

Therefore, by (i) in Theorem 3.5, FBA$=0$ and
$(Y^{\mathbb{F}},Z^{\mathbb{F}})$ follows

$\displaystyle Y^{\mathbb{F}}_{t}=$
$\displaystyle\int_{t}^{T}-\bigg{[}\Big{(}L^{m}Y^{\mathbb{F}}_{s}+L^{m}\tilde{p}^{N}_{s}-\phi_{s}(Z^{\mathbb{F}}_{s})\Big{)}s^{b}_{s}\bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle+\int_{t}^{T}\big{[}-h^{H}_{s}L^{H}L^{m}(Y^{\mathbb{F}}_{s}+\tilde{p}^{N}_{s})\big{]}\mathop{}\\!\mathrm{d}s-\int_{t}^{T}Z^{\mathbb{F}}_{s}\mathop{}\\!\mathrm{d}W_{s}.$
(4.13)

Let $V^{\mathbb{F}}\coloneqq Y^{\mathbb{F}}+\tilde{p}^{N}$,
$\Pi^{\mathbb{F}}\coloneqq Z^{\mathbb{F}}+Z^{N}$. Moreover, let
$\mathbb{Q}^{b}$ denote an equivalent measure that

$\displaystyle W^{b}_{t}\coloneqq$ $\displaystyle
W_{t}-\int_{0}^{t}s^{b}\alpha_{s}\mathop{}\\!\mathrm{d}s$ (4.14)

is an $(\mathbb{F},\mathbb{Q}^{b})$-Brownian motion and $\mathbb{E}^{b}$
denote the expectation under $\mathbb{Q}^{b}$. Then (4.13) becomes

$\displaystyle
V^{\mathbb{F}}_{t}=\xi-\int_{t}^{T}(s^{b}+h^{H}L^{H})L^{m}V^{\mathbb{F}}_{s}-\int_{t}^{T}\Pi^{\mathbb{F}}_{s}\mathop{}\\!\mathrm{d}W^{b}_{s}.$

Thus, it follows that

$\displaystyle V^{\mathbb{F}}_{t}=$ $\displaystyle
A^{-1}_{t}\mathbb{E}^{b}[A_{T}\xi|\mathcal{F}_{t}].$ (4.15) $\displaystyle
A_{t}\coloneqq$ $\displaystyle e^{-\beta t},$ (4.16)
$\displaystyle\beta\coloneqq$ $\displaystyle(s^{b}+h^{H}L^{H})L^{m}.$ (4.17)

Note that there is no inconsistency of pricing measures. This is because we
considered the hedger’s funding cost and benefit in the close-out amount,
$e^{N}=V_{-}$. To represent (4.15) in an explicit form, write

$\displaystyle\xi=$ $\displaystyle(B^{b}_{T})^{-1}B^{b}_{T}B^{-1}_{T}B_{T}\xi$
$\displaystyle=$ $\displaystyle e^{s^{b}T}(B^{b}_{T})^{-1}B_{T}\xi.$

Then (4.15) becomes, for $t<\bar{\tau}$,

$\displaystyle
V^{\mathbb{F}}_{t}=A_{t}^{-1}A_{T}e^{s^{b}T}(B^{b}_{t})^{-1}B_{t}^{b}\mathbb{E}^{b}[(B^{b}_{T})^{-1}B_{T}\xi|\mathcal{F}_{t}].$

More explicitly,