propProposition[section] theoremTheorem[section] lemma[theorem]Lemma
conj[theorem]Conjecture corollaryCorollary[prop]

# Distributionally Robust XVA via Wasserstein Distance:
Wrong Way Counterparty Credit and Funding Risk

Derek Singh  Department of Industrial and Systems Engineering, University of
Minnesota
<EMAIL_ADDRESS><EMAIL_ADDRESS>Shuzhong Zhang  Department of Industrial and
Systems Engineering, University of Minnesota
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper investigates calculations of robust XVA, in particular, credit
valuation adjustment (CVA) and funding valuation adjustment (FVA) for over-
the-counter derivatives under distributional uncertainty using Wasserstein
distance as the ambiguity measure. Wrong way counterparty credit risk and
funding risk can be characterized (and indeed quantified) via the robust XVA
formulations. The simpler dual formulations are derived using recent infinite
dimensional Lagrangian duality results. Next, some computational experiments
are conducted to measure the additional XVA charges due to distributional
uncertainty under a variety of portfolio and market configurations. Finally
some suggestions for future work are discussed.

Keywords— CCR, CVA, FVA, derivatives, distributional robust optimization,
Wasserstein distance, Lagrangian duality

##  1 Introduction and Overview

###  1.1 Financial Markets Context and Background

An X-Value adjustment (XVA) is a generic term used to refer to various
valuation adjustments, typically applied to over-the-counter (OTC) derivatives
held by financial institutions. The first of the XVAs, and still one of the
most significant, in terms of market exposure, is CVA. One of the more recent,
and perhaps equally significant, exposures is FVA. Both of these XVAs have
similar structure (unilateral, bilateral) and mathematical form for
computation. Other XVAs include capital valuation adjustment (KVA) and margin
valuation adjustment (MVA). Wrong way risk refers to adversely correlated
moves in the market exposures and the counterparty spreads (e.g. credit,
funding). It can materially affect the magnitude of the XVA adjustment.

Credit valuation adjustment (CVA) represents the impact on portfolio market
value due to counterparty default. Unilateral CVA can be represented
mathematically as an integral of discounted expected positive exposure times
(incremental) counterparty default probability. The market valuation is a
function of counterparty credit risk, the underlying (market) risk factors
that drive the portfolio valuation (and hence positive exposure), as well as
the correlations between these market risk factors and the counterparty credit
risk curves for a given portfolio. CVA is typically measured and reported at
the counterparty level.

The “other side” of unilateral CVA is unilateral debit valuation adjustment
(DVA). This is the benefit to the firm, of its reduced liability, as measured
by discounted expected negative exposure times firm default probability. As
above, the market valuation is a function of firm credit risk, underlying
market risk factors that drive portfolio valuation, and the correlations.
Unilateral DVA can be represented mathematically as an integral of discounted
negative exposure times (incremental) firm default probability. DVA is
typically measured at the firm level.

Bilateral CVA represents the dual impact on portfolio market value due to
counterparty default and firm default. Bilateral CVA can be represented
mathematically as the difference between two integrals: (i) discounted
expected positive exposure times (incremental) counterparty default
probability prior to firm default, (ii) discounted expected negative exposure
times (incremental) firm default probability prior to counterparty default.
Bilateral CVA is typically measured and reported at the counterparty level,
for a given firm.

Funding valuation adjustment (FVA) represents the impact on portfolio market
value due to funding exposures for the hedge on uncollateralized derivatives.
It represents the market value of funding exposure risk. Funding cost
adjustment (FCA) can be represented mathematically as an integral of
discounted expected positive exposure times funding cost (incremental)
conditional on joint counterparty and firm survival. FCA arises for a positive
portfolio exposure since this implies a negative hedge exposure which leads to
a funding cost for collateral posted. The market valuation is a function of
joint counterparty and firm credit risk, the underlying (market) risk factors
that drive the portfolio valuation (and hence positive exposure) as well as
funding cost, as well as the correlations between these market risk factors
and the credit risk curves for a given portfolio. FCA is typically measured
and reported at the funding netting set level.

The “other side” of FCA is funding benefit adjustment (FBA). This represents
the funding benefit to the firm, for interest income proceeds on received
collateral posted against counterparty exposure on the hedge, as measured by
discounted expected negative exposure times funding benefit conditional on
joint counterparty and firm survival. As above, the market valuation is a
function of counterparty and firm credit risk, underlying market risk factors
that drive portfolio valuation and funding benefit, and the correlations. FBA
can be represented mathematically as an integral of discounted negative
exposure times funding benefit conditional on joint counterparty and firm
survival. FBA is typically measured at the funding netting set level.

(Bilateral) FVA represents the dual impact on portfolio market value due to
both funding cost and funding benefit exhibited over the portfolio lifetime.
FVA can be represented mathematically as the difference (or sum) of two
integrals: (i) discounted expected positive exposure times funding cost
conditional on joint counterparty and firm survival; (ii) discounted expected
negative exposure times funding benefit conditional on joint counterparty and
firm survival. FVA is typically measured and reported at the netting set level
for a given firm.

U.S. regulatory authorities, the Federal Reserve and Office of the Comptroller
of the Currency (OCC), periodically assess national banks’ compliance with
Market Risk Capital Rule (MRR). Counterparty credit risk (CCR) and funding
risk (FR) metrics are key metrics used to evaluate bank risk profiles and
balance sheet exposures due to over the counter (OTC) derivatives, securities
financing transactions, and other transactions and exposures (Office of the
Comptroller of the Currency, , 2011). Basel Committee on Banking Supervision
has issued supervisory guidance, in the form of its Basel III framework (and
supplemental guidance), to quantify capital charges due to CCR. A new element
in Basel III was a capital charge due to degradation in CCR for a given
portfolio or book of business (Basel Committee on Banking Supervision, ,
2015). Potential revisions to the Basel framework may include elements to
quantify CCR capital charges due to deterioration in market risk exposure.

The Dodd-Frank Wall Street Reform and Consumer Protection Act (July 2010)
enacted regulations for the swaps market and authorized creation of
centralized exchanges for swaps (and other) derivatives trading. Derivatives
that trade on an exchange reference the exchange as the transaction
counterparty. Since exchanges clear multiple (typically offsetting)
transactions and hedge their risk through other third parties, exchange traded
derivatives have minimal CCR risk profile. However, OTC derivatives typically
have banks or other financial institutions as counterparties which do have
material credit risk profiles. According to International Swap Dealers
Association (ISDA) the OTC derivatives notional outstanding was 544 trillion
at year end 2018. Interest rate derivatives notional outstanding was 437
trillion at year end 2018. Recent (04/20/20) Bloomberg CDX investment grade
and high yield credit spreads are 93 and 643 basis points respectively.
Consequently the CCR and FR exposures (due to uncollateralized or partially
collateralized hedges) inherent in the OTC derivatives market represent
significant market risk exposures. This motivates the concepts of worst case
CVA, FVA, and wrong way risk (WWR) and the impact of uncertainty in
probability distribution on these exposures and risk metrics. It is these
considerations that motivate this line of research (Ramzi Ben-Abdallah and
Marzouk, , 2019), (El Hajjaji and Subbotin, , 2015).

In our study, distributional uncertainty is characterized via the Wasserstein
metric for a couple reasons. The Wasserstein metric is a (reasonably) well
understood metric and a natural, intuitive way to compare two probability
distributions using ideas of transport cost. It is also a flexible approach
that encompasses parametric and non-parametric distributions of either
discrete or continuous form. For example, one can explore distributions that
alter the shape of the marginals as well as the correlation structure.
Furthermore, recent duality results and structural results on the worst case
distributions could help us understand and/or quantify the market model
transitions as well as measure (in a relative sense) the degree of wrong way
risk inherent to a given market model.

An outline of this paper is as follows. Section 1 gives an overview of CVA,
FVA, and WWR as well as a literature review. Section 2 develops the main
theoretical results of the paper and provides proof sketches. Section 3
conducts a computational study of WWR for a representative set of derivative
portfolios and market environments. Section 4 discusses the conclusions and
suggestions for future research. All detailed proofs of propositions,
corollaries, and theorems are deferred to the Appendix.

###  1.2 Literature Review

######  Remark 1.

The authors are not aware of any substantial research that has been done on
the topic of worst case FVA. The discussion below pertains to literature
regarding worst case CVA.

In the past few years some research has been done to investigate and quantify
the effect of distributional uncertainty on CVA. Brigo et al.,  (2013)
explicitly incorporate correlation into the stochastic processes driving the
market risk and credit default factors. They quantify the effect of dependency
structure (and hence wrong way risk) on CVA for a variety of asset classes:
interest rate swaps, interest rate swaptions, commodities, equities, and
foreign exchange products. Glasserman and Yang,  (2015) bound the effect of
wrong way risk on CVA. Their approach considers a discrete setting for
portfolio exposures and counterparty default times and formulates worst case
CVA as the solution to a worst case linear program subject to certain
constraints (such as fixed marginals for portfolio exposures and default
times), where the dependency structure across the risk factors is allowed to
vary. As this approach leads to large values for worst case CVA, they
introduce a penalty term to modulate or temper the degree of wrong way risk
and run some sensitivity analysis to study the effect of the penalty term.
Kullback-Leibler (KL) divergence is used to measure the distance between the
reference (empirical) and the perturbed distribution. They remark that
determining a suitable value for the penalty term would be a topic for further
research.

Memartoluie, in his PhD thesis, uses an ordered scenario copula methodology to
quantify worst case CVA (Memartoluie, , 2017). A particular method of scenario
ordering correlates portfolio exposures to company default times (firm,
counterparty, or both) and the resulting dependency structure introduces wrong
way risk. He chooses to order exposure scenarios by increasing time averaged
total exposure and then simulates company default times conditional on the
exposure path using pre-specified correlation between the market risk
factor(s) and credit risk factor(s). For worst case correlations set to one,
he finds results for worst case CVA that are comparable to the method of
Glasserman and Yang,  (2015). In a recent paper, Ben-Abdallah et al. perform a
computational study on the effect of wrong way risk on CVA for a portfolio of
interest rate swaps, caps, and floors (Ramzi Ben-Abdallah and Marzouk, ,
2019). They find that the dependency structure between interest rates and
default intensity produces material wrong way risk whereas the dependency
structure between interest rate volatility and default intensity does not.

Recent results in Lagrangian duality were independently developed by Blanchet
and Murthy,  (2019) and Gao and Kleywegt,  (2016). These results hold under
mild assumptions such as upper semicontinuity in the objective function and
lower semicontinuity in the distance metric. Blanchet et al., 2016a applied
this duality theory to study a number of classical regression problems in
machine learning under distributional uncertainty. In that context, the
authors find that distributional uncertainty can be viewed as adding a
regularization term to the objective function, analogous to a penalized
regression setting. Similarly, Gao et al.,  (2017) apply the Lagrangian
duality theory to problems in statistical learning.

The main innovation in our work is to apply these recent results in Lagrangian
duality to worst case CVA and FVA using Wasserstein distance as the ambiguity
measure. Furthermore, analytical expressions are derived for the solutions to
the inner and outer convex optimization problems that comprise worst case CVA
and FVA via the Wasserstein approach. A computational study shows the material
impact of distributional uncertainty on worst case CVA and FVA, illustrates
the risk profiles, and computes the worst case distributions.

####  1.2.1 Restatement of Lagrangian Duality Result

In Section 2 we formulate the primal optimization problems for
distributionally robust CVA and FVA. As in our earlier work this year, (Singh
and Zhang, , 2020) a key step in the approach is to use recent Lagrangian
duality results to formulate the equivalent dual problems. The dual problems
are much more tractable than the primal problems since they only involve the
reference probability measure as opposed to a Wasserstein ball of probability
measures (of some finite radius). For real valued upper semicontinuous
objective function $f\in L^{1}$ and non-negative lower semicontinuous cost
function $c$ such that $\\{(u,v):c(u,v)<\infty\\}$ is Borel measurable and
non-empty, it holds that (Blanchet et al., 2016b, )

$\sup_{Q\in\mathcal{U}_{\delta}(Q_{N})}\mathbb{E}^{Q}[f(X)]=\inf_{\lambda\geq
0}\>[\lambda\delta+\frac{1}{N}\sum_{i=1}^{n}\Psi_{\lambda}(x_{i})]$

where

$\Psi_{\lambda}(x_{i}):=\sup_{u\in\text{dom}(f)}[f(u)-\lambda c(u,x_{i})].$

Further details, including proofs and concrete examples, can be found in the
papers by Blanchet and Murthy,  (2019), Gao and Kleywegt,  (2016), and
Esfahani and Kuhn,  (2018). These authors independently derived these results
around the same time although Blanchet and Murthy,  (2019) did so in a more
general setting.

####  1.2.2 Characterization of Worst Case Distributions

Simply put, the set of worst case distributions (when non-empty) can be
defined as
$WC(f,\delta):=\\{Q^{*}:\mathbb{E}^{Q^{*}}[f(X)]=\sup_{Q\in\mathcal{U}_{\delta}(Q_{N})}\mathbb{E}^{Q}[f(X)]\\}$.
Another recent set of results from the literature describes the structure of
the worst case distribution(s) when they exist [(Blanchet and Murthy, , 2019),
(Gao and Kleywegt, , 2016), (Esfahani and Kuhn, , 2018)]. The boundedness
conditions for existence are tied to the growth rate
$\kappa:=\limsup\limits_{d(X,X_{0})\rightarrow\infty}\frac{f(X)-f(X_{0})}{d(X,X_{0})}$
for fixed $X_{0}$ and the value of the dual minimizer $\lambda^{*}$. For
empirical reference distributions, supported on $N$ points, such that
$WC(f,\delta)$ is non-empty, there exists a worst case distribution that is
another empirical distribution supported on at most $N+1$ points. This worst
case distribution can be constructed via a greedy approach. For up to $N$
points, they can be identified as solving
$x_{i}^{*}\in\operatorname*{arg\,min}_{\tilde{x}\in
dom(f)}[\lambda^{*}c(\tilde{x},x_{i})-f(\tilde{x})]$. At most one point has
its probability mass split into two pieces (according to budget constraint
$\delta$) that solve
$x_{i_{0}}^{*},x_{i_{0}}^{**}\in\operatorname*{arg\,min}_{\tilde{x}\in
dom(f)}[\lambda^{*}c(\tilde{x},x_{i_{0}})-f(\tilde{x})]$. Details can be found
in Gao and Kleywegt,  (2016).

###  1.3 Notation and Definitions

####  1.3.1 Bilateral CVA

Notation and core definitions for bilateral CVA (BCVA) problem setup
incorporate those for unilateral CVA and DVA. Bilateral CVA measures expected
portfolio loss (or benefit) due to counterparty and/or firm default. Let
$V^{+}(\tau_{C})$ denote the discounted positive portfolio exposure at time
$\tau_{C}$ and let $R_{C}\in[0,1)$ denote the recovery rate the firm receives
upon counterparty default. Let $V^{-}(\tau_{F})$ denote the discounted
negative portfolio exposure at time $\tau$ and let $R_{F}\in[0,1)$ denote the
recovery rate the counterparty receives upon firm default. The problem setup
here assumes a fixed set of observation dates, $0=t_{0}<t_{1}<\cdots<t_{n}=T$.
Let $X^{+}$ denote the vector of recovery adjusted discounted positive
exposures and $Y_{C}$ denote the vector of counterparty default indicators.
Let $(x^{+}_{i},y^{c}_{i})$ denote realizations of $(X^{+},Y_{C})$ along
sample paths for $i=\\{1,2,\ldots,N\\}$. Let $X^{-}$ denote the vector of
recovery adjusted discounted firm negative exposures and $Y_{F}$ denote the
vector of firm default indicators. Let $(x^{-}_{i},y^{f}_{i})$ denote
realizations of $(X^{-},Y_{F})$ along sample paths for $i=\\{1,2,\ldots,N\\}$.

Due to the linkage, one can write $X=X^{+}+X^{-}$ and decompose sample
realizations of $X$ accordingly. Therefore, let $(x_{i},y^{c}_{i},y^{f}_{i})$
denote realizations of $(X,Y_{C},Y_{F})$ along sample paths for
$i=\\{1,2,\ldots,N\\}$. The relation $x_{i}=x^{+}_{i}+x^{-}_{i}$ can be used
to decompose $x_{i}$ into its positive and negative exposures respectively.

The bilateral CVA associated with discounted positive exposure
$V^{+}(\tau_{C})$, counterparty default indicator
$\mathbbm{1}_{\\{\tau_{C}\leq T\\}\cap\\{\tau_{C}<\tau_{F}\\}}$, discounted
negative exposure $V^{-}(\tau_{F})$, firm default indicator
$\mathbbm{1}_{\\{\tau_{F}\leq T\\}\cap\\{\tau_{F}<\tau_{C}\\}}$, is

$\text{CVA}\textsuperscript{B}=\mathbb{E}[(1-R_{C})V^{+}(\tau_{C})\mathbbm{1}_{\\{\tau_{C}\leq
T\\}\cap\\{\tau_{C}<\tau_{F}\\}}]+\mathbb{E}[(1-R_{F})V^{-}(\tau_{F})\mathbbm{1}_{\\{\tau_{F}\leq
T\\}\cap\\{\tau_{F}<\tau_{C}\\}}].$

Equivalently, one can write

$\text{CVA}\textsuperscript{B}=(1-R_{C})\int_{0}^{T}\mathbb{E}[V^{+}(t)|\tau_{C}=t,\tau_{F}>t]d\Pi^{\prime}_{C}(t)+(1-R_{F})\int_{0}^{T}\mathbb{E}[V^{-}(t)|\tau_{F}=t,\tau_{C}>t]d\Pi^{\prime}_{F}(t),$

where the joint counterparty and firm default time distributions are given by
$\Pi^{\prime}_{C}(t)=P(\tau_{C}\leq t,\tau_{F}>\tau_{C})$ and
$\Pi^{\prime}_{F}(t)=P(\tau_{F}\leq t,\tau_{C}>\tau_{F})\>\>$ [(Green, ,
2015), (Lichters et al., , 2015), (Memartoluie, , 2017)]. The pair of vectors
$(X^{+},Y_{C})\in(\mathbb{R}^{n}_{+}\times B^{1}_{n})$ is

$X^{+}=((1-R_{C})V^{+}(t_{1}),\ldots,(1-R_{C})V^{+}(t_{n}))\quad\text{and}\quad
Y_{C}=(\mathbbm{1}_{\\{\tau_{C}=t_{1}\\}\cap\\{\tau_{F}>\tau_{C}\\}},\dots,\mathbbm{1}_{\\{\tau_{C}=t_{n}\\}\cap\\{\tau_{F}>\tau_{C}\\}}),$

and the pair of vectors $(X^{-},Y_{F})\in(\mathbb{R}^{n}_{-}\times B^{1}_{n})$
is

$X^{-}=((1-R_{F})V^{-}(t_{1}),\ldots,(1-R_{F})V^{-}(t_{n}))\quad\text{and}\quad
Y_{F}=(\mathbbm{1}_{\\{\tau_{F}=t_{1}\\}\cap\\{\tau_{C}>\tau_{F}\\}},\dots,\mathbbm{1}_{\\{\tau_{F}=t_{n}\\}\cap\\{\tau_{C}>\tau_{F}\\}}).$

Here $B^{1}_{n}$ denotes the set of default time vectors: binary vectors of
ones and zeros with $n$ components, and at most one non-zero element. Note
that counterparty or firm default occurs on at most one observation date
within the fixed set of dates in the problem setup. The empirical measure,
$\Phi_{N}$, is

$\Phi_{N}(dz)=\frac{1}{N}\sum_{i=1}^{N}\mathbbm{1}_{(x_{i},y^{c}_{i},y^{f}_{i})}(dz).$

Under the empirical measure $\Phi_{N}$, bilateral CVA is a sum of expectations
of inner products

$\text{CVA}\textsuperscript{B}=\mathbb{E}^{\Phi_{N}}[\langle
X^{+},Y_{C}\rangle]+\mathbb{E}^{\Phi_{N}}[\langle X^{-},Y_{F}\rangle].$

In the context of this work, the uncertainty set for probability measures is

$\mathcal{U}_{\delta_{3}}(\Phi_{N})=\\{P:D_{c}(\Phi,\Phi_{N})\leq\delta_{3}\\}$

where $D_{c}$ is the optimal transport cost or Wasserstein discrepancy for
cost function $c$ (Blanchet et al., , 2018). For convenience the definition
for $D_{c}$ is given as

$D_{c}(\Phi,\Phi^{\prime})=\inf\\{\mathbb{E}^{\pi}[c(A,B)]:\pi\in\mathcal{P}(\mathbb{R}^{d}\times\mathbb{R}^{d}),\pi_{A}=\Phi,\pi_{B}=\Phi^{\prime}\\}$

where $\mathcal{P}$ denotes the space of Borel probability measures and
$\pi_{A}$ and $\pi_{B}$ denote the distributions of $A$ and $B$. Here $A$
denotes $(X_{A},Y^{C}_{A},Y^{F}_{A})\in(\mathbb{R}^{n}\times B^{1}_{n}\times
B^{1}_{n})$ and $B$ denotes
$(X_{B},Y^{C}_{B},Y^{F}_{B})\in(\mathbb{R}^{n}\times B^{1}_{n}\times
B^{1}_{n})$ respectively. The analysis in this work uses the cost function
$c_{S_{3}}$ where

$c_{S_{3}}((u,v_{1},v_{2}),(x,y_{1},y_{2}))=S_{3}\langle
v_{1}-y_{1},v_{1}-y_{1}\rangle+S_{3}\langle
v_{2}-y_{2},v_{2}-y_{2}\rangle+\langle u-x,u-x\rangle.$

The scale factor $S_{3}>0$ is used to compensate for different domains:
$(u,v_{1},v_{2})\in(\mathbb{R}^{n}\times B^{1}_{n}\times
B^{1}_{n}),(x,y_{1},y_{2})\in(\mathbb{R}^{n}\times B^{1}_{n}\times
B^{1}_{n})$.

####  1.3.2 Unilateral CVA, DVA

Bilateral CVA can be reduced to express unilateral CVA as

$\text{CVA}\textsuperscript{U}=\mathbb{E}[(1-R_{C})V^{+}(\tau_{C})\mathbbm{1}_{\\{\tau_{C}\leq
T\\}}]=(1-R_{C})\int_{0}^{T}\mathbb{E}[V^{+}(t)|\tau_{C}=t]d\Pi_{C}(t),$

where the counterparty default time distribution is given by
$\Pi_{C}(t)=P(\tau_{C}\leq t).\>\>$ Note the assumption here is that
$\tau_{C}<\tau_{F}$. Similarly, it can be reduced to express unilateral DVA
(note the minus sign), assuming $\tau_{F}<\tau_{C}$, as

$\text{DVA}\textsuperscript{U}=-\mathbb{E}[(1-R_{F})V^{-}(\tau_{F})\mathbbm{1}_{\\{\tau_{F}\leq
T\\}}]=-(1-R_{F})\int_{0}^{T}\mathbb{E}[V^{-}(t)|\tau_{F}=t]d\Pi_{F}(t),$

where firm default time distribution is given by $\Pi_{F}(t)=P(\tau_{F}\leq
t)\>\>$ [(Green, , 2015), (Lichters et al., , 2015), (Memartoluie, , 2017)].

####  1.3.3 FVA

Notation and core definitions for (bilateral) FVA problem setup incorporate
those for FCA and FBA. FVA measures expected funding costs and benefits over
portfolio lifetime. Let $V^{+}(t)$ denote the positive portfolio exposure at
time $t$. Let $V^{-}(t)$ denote the negative portfolio exposure at time $t$.
The problem setup here assumes a fixed set of observation dates,
$0=t_{0}<t_{1}<\cdots<t_{n}=T$. Let $X^{+}$ denote the vector of discounted
positive exposures and $Y_{C}$ denote the vector of counterparty survival
indicators. Let $X^{-}$ denote the vector of discounted negative exposures and
$Y_{F}$ denote the vector of firm survival indicators. Further, let $Y_{CF}$
denote the Hadamard product $Y_{C}\odot Y_{F}$ which represents the vector of
joint survival indicators. To incorporate funding, let $Z^{+}$ denote the
vector of funding costs incurred on exposures $X^{+}$. And similarly for
$Z^{-}$ with respect to exposures $X^{-}$. Due to the linkage between $Z^{+}$
and $Z^{-}$, one can write $Z=Z^{+}+Z^{-}$ and decompose sample realizations
of $Z$ into $Z^{+}$ and $Z^{-}$ accordingly. Therefore, let
$(z_{i},y^{cf}_{i})$ denote realizations of $(Z,Y_{CF})$ along sample paths
for $i=\\{1,2,\ldots,N\\}$. The relation $z_{i}=z^{+}_{i}+z^{-}_{i}$ can be
used to decompose $z_{i}$ into its positive and negative exposures
respectively.

The FVA associated with funding costs $Z(t)$, joint survival indicator
$\mathbbm{1}_{\\{\tau_{C}>t\\}\cap\\{\tau_{F}>t\\}}$ is [(Lichters et al., ,
2015), (Green, , 2015)]:

$\text{FVA}=\text{FCA}+\text{FBA}=\int_{0}^{T}\mathbb{E}[Z^{+}(t)\mathbbm{1}_{\\{\tau_{C}>t\\}\cap\\{\tau_{F}>t\\}}]dt+\int_{0}^{T}\mathbb{E}[Z^{-}(t)\mathbbm{1}_{\\{\tau_{C}>t\\}\cap\\{\tau_{F}>t\\}}]dt=\int_{0}^{T}\mathbb{E}[Z(t)\mathbbm{1}_{\\{\tau_{C}>t\\}\cap\\{\tau_{F}>t\\}}]dt.$

The pair of vectors $(Z,Y_{CF})\in(\mathbb{R}^{n}\times B^{1}_{n})$ is

$Z=(Z^{+}(t_{1})+Z^{-}(t_{1}),\ldots,Z^{+}(t_{n})+Z^{-}(t_{n}))\quad\text{and}\quad
Y_{CF}=(\mathbbm{1}_{\\{\tau_{C}>t_{1}\\}\cap\\{\tau_{F}>t_{1}\\}},\dots,\mathbbm{1}_{\\{\tau_{C}>t_{n}\\}\cap\\{\tau_{F}>t_{n}\\}}),$

and the pair of vectors
$(Z^{+},Z^{-})\in(\mathbb{R}^{n}_{+}\times\mathbb{R}^{n}_{-})$ is

$Z^{+}=(f_{c}(t_{0},t_{1})X^{+}(t_{1}),\ldots,f_{c}(t_{n-1},t_{n})X^{+}(t_{n}))\quad\text{and}\quad
Z^{-}=(f_{b}(t_{0},t_{1})X^{-}(t_{1}),\ldots,f_{b}(t_{n-1},t_{n})X^{-}(t_{n})).$

Here $B^{1}_{n}$ denotes the set of survival time vectors: binary vectors of
ones and zeros with $n$ components, and at most one block of ones followed by
a complementary block of zeros. The empirical measure, $\Phi_{N}$, is

$\Phi_{N}(dz)=\frac{1}{N}\sum_{i=1}^{N}\mathbbm{1}_{(z_{i},y^{cf}_{i})}(dz).$

Under the empirical measure $\Phi_{N}$, FVA is a sum of expectations of inner
products

$\text{FVA}=\mathbb{E}^{\Phi_{N}}[\langle
Z^{+},Y_{CF}\rangle]+\mathbb{E}^{\Phi_{N}}[\langle
Z^{-},Y_{CF}\rangle]=\mathbb{E}^{\Phi_{N}}[\langle Z,Y_{CF}\rangle].$

In the context of this work, the uncertainty set for probability measures is

$\mathcal{U}_{\delta_{3}}(\Phi_{N})=\\{P:D_{c}(\Phi,\Phi_{N})\leq\delta_{3}\\}$

where $D_{c}$ is the optimal transport cost or Wasserstein discrepancy for
cost function $c$ (Blanchet et al., , 2018). For convenience the definition
for $D_{c}$ is

$D_{c}(\Phi,\Phi^{\prime})=\inf\\{\mathbb{E}^{\pi}[c(A,B)]:\pi\in\mathcal{P}(\mathbb{R}^{d}\times\mathbb{R}^{d}),\pi_{A}=\Phi,\pi_{B}=\Phi^{\prime}\\}$

where $\mathcal{P}$ denotes the space of Borel probability measures and
$\pi_{A}$ and $\pi_{B}$ denote the distributions of $A$ and $B$. Here $A$
denotes $(Z_{A},Y_{A})\in(\mathbb{R}^{n}\times B^{1}_{n})$ and $B$ denotes
$(Z_{B},Y_{B})\in(\mathbb{R}^{n}\times B^{1}_{n})$ respectively. This work
uses the cost function $c_{S_{3}}$ where

$c_{S_{3}}((u,v),(z,y))=S_{3}\langle v-y,v-y\rangle+\langle u-z,u-z\rangle.$

The scale factor $S_{3}>0$ is used to compensate for different domains:
$(u,v)\in(\mathbb{R}^{n}\times B^{1}_{n}),(z,y)\in(\mathbb{R}^{n}\times
B^{1}_{n})$.

##  2 Theory: Robust XVA and Wrong Way Risk

###  2.1 Unilateral CVA, DVA

The robust unilateral CVA can be written as

$\sup_{P\in\mathcal{U}_{\delta_{1}}(P_{N})}\mathbb{E}^{P}[\langle
X^{+},Y_{C}\rangle].$ (P1)

Similarly, the robust unilateral DVA is

$-\sup_{Q\in\mathcal{U}_{\delta_{2}}(Q_{N})}\mathbb{E}^{Q}[\langle
X^{-},Y_{F}\rangle].$ (P2)

As such, the dual formulations and solutions to the above primal optimization
problems are special cases of the solutions to the bilateral CVA optimization
problems, to be described next.

###  2.2 Bilateral CVA

####  2.2.1 Inner Optimization Problem

The robust bilateral CVA is

$\sup_{\Phi\in\mathcal{U}_{\delta_{3}}(\Phi_{N})}\mathbb{E}^{\Phi}[\langle
X^{+},Y_{C}\rangle+\langle X^{-},Y_{F}\rangle].$ (P3)

Similar to before, use recent duality results, noting that the inner product
$\langle\>;\rangle$ satisfies the upper semicontinuous condition of the
Lagrangian duality theorem, and cost function $c_{S}$ satisfies the non-
negative lower semicontinuous condition (see Blanchet and Murthy,  (2019)
Assumptions 1 & 2, Gao and Kleywegt,  (2016)). Hence the dual problem can be
written as

$\inf_{\alpha\geq
0}\>F(\alpha):=\bigg{[}\alpha\delta_{3}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})\bigg{]}$
(D3)

where

$\displaystyle\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{u\in\mathbb{R}^{n},v_{1}\in{B^{1}_{n}},v_{2}\in{B^{1}_{n}}}$
$\displaystyle[\langle
u^{+},\mathbbm{1}_{\\{v_{1}<v_{2}\\}}v_{1}\rangle+\langle
u^{-},\mathbbm{1}_{\\{v_{2}<v_{1}\\}}v_{2}\rangle-\alpha
c_{S_{3}}((u,v_{1},v_{2}),(x_{i},y^{c}_{i},y^{f}_{i}))]$
$\displaystyle=\sup_{u\in\mathbb{R}^{n},v_{1}\in{B^{1}_{n}},v_{2}\in{B^{1}_{n}}}$
$\displaystyle[\langle
u^{+},\mathbbm{1}_{\\{v_{1}<v_{2}\\}}v_{1}\rangle+\langle
u^{-},\mathbbm{1}_{\\{v_{2}<v_{1}\\}}v_{2}\rangle-\alpha(\langle
u-x_{i},u-x_{i}\rangle+S_{3}\langle v_{1}-y^{c}_{i},v_{1}-y^{c}_{i}\rangle$
$\displaystyle+S_{3}\langle v_{2}-y^{f}_{i},v_{2}-y^{f}_{i}\rangle)].$

Note that default times $(v_{1},v_{2})$ are compared via the indicator
function $\mathbbm{1}_{\\{v_{1}\lessgtr v_{2}\\}}$ by comparing indices (into
the fixed dates array $0<t_{1}<\cdots<t_{n}=T$) of the respective default
times. So if $v_{1}$ has a one element in index $i$ and either $\|v_{2}\|=0$
or $v_{2}$ has a one element in index $j$ and $i<j$ then
$\mathbbm{1}_{\\{v_{1}<v_{2}\\}}=1$ else if $i>j$ or $\|v_{1}\|=0$ then
$\mathbbm{1}_{\\{v_{1}<v_{2}\\}}=0$. The probability that $i=j\>$ for any
$i,j\in\\{1,\dots,n\\}$ is zero in continuous time, hence this case is not
considered here. Also $\|v_{1}\|=1$ implies default time $v_{1}\leq t_{n}=T$,
the maturity date of the CVA calculation. Similar analysis applies to $v_{2}$.

Now apply change of variables $w_{1}=(u-x_{i})$, $w_{2}=(v_{1}-y^{c}_{i})$,
and $w_{3}=(v_{2}-y^{f}_{i})$ to get

$\displaystyle\begin{split}\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{2}\in{B^{2}_{n}},w_{3}\in{B^{2}_{n}}}&\big{[}\langle(w_{1}+x_{i})^{+},\mathbbm{1}_{\\{w_{2}+y^{c}_{i}<w_{3}+y^{f}_{i}\\}}w_{2}+y^{c}_{i}\rangle+\langle(w_{1}+x_{i})^{-},\mathbbm{1}_{\\{w_{3}+y^{f}_{i}<w_{2}+y^{c}_{i}\\}}w_{3}+y^{f}_{i}\rangle\\\
&-\alpha(\langle w_{1},w_{1}\rangle+S_{3}\langle
w_{2},w_{2}\rangle+S_{3}\langle w_{3},w_{3}\rangle)\big{]}.\end{split}$

It turns out that $\Psi_{\alpha}$ can be expressed as the pointwise max of
four functions of more complex forms. The four functions represent the four
logical cases for $w_{2}$ and $w_{3}$ each being zero or non-zero.
Furthermore, we need to consider the sub-cases where the counterparty defaults
before the firm, as in $\Psi_{\alpha}^{a}$ or vice-versa as in
$\Psi_{\alpha}^{b}$. Again, $\Psi_{\alpha}$ quantifies the adversarial moves
in CVA and DVA across both time and spatial dimensions while accounting for
the associated cost via the $K$ terms.

######  Remark 2.

Note that this result involves some lengthy and tedious derivations and
requires some time to go through. However, there are some patterns across the
various cases and sub-cases which does simplify the analysis to some extent.

Table 1: Lookup table of optimization sub-problems Optimization | Objective Function | Solution
---|---|---
$\sup_{w_{1}\in\mathbb{R}^{n}}$ | $\langle w_{1},y_{i}\rangle-\alpha\langle w_{1},w_{1}\rangle$ | $\frac{\|y_{i}\|^{2}}{4\alpha}$
$\sup_{w_{1}\leq x_{i\tau_{2}}}$ | $\langle w_{1},y_{i}\rangle-\alpha\langle w_{1},w_{1}\rangle$ | $[x_{i\tau_{2}}\wedge\frac{\|y_{i}\|}{2\alpha}]-\alpha[x_{i\tau_{2}}\wedge\frac{\|y_{i}\|}{2\alpha}]^{2}$
$\sup_{w_{1}\in\mathbb{R}^{n}}$ | $\langle(w_{1}+x_{i})^{+},y_{i}\rangle-\alpha\langle w_{1},w_{1}\rangle$ | $[\frac{1}{4\alpha}+\langle x_{i},y_{i}\rangle]^{+}$
$\sup_{w_{1}\in\mathbb{R}^{n}}$ | $\langle(w_{1}+x_{i})^{-},y_{i}\rangle-\alpha\langle w_{1},w_{1}\rangle$ | $\mathbbm{1}_{\\{(x_{i\tau_{2}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{2}}>0)\\}}\big{[}\frac{1}{4\alpha}+\langle x_{i},y_{i}\rangle\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq x_{i\tau_{2}}\leq 0\\}}\big{[}\alpha(\langle x_{i},y_{i}\rangle)^{2}\big{]}$
$\sup_{w_{1}\in\mathbb{R}^{n}}$ | $(w_{1}+x_{i\tau_{1}})^{-}-\alpha\langle w_{1},w_{1}\rangle$ | $\mathbbm{1}_{\\{(x_{i\tau_{1}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{1}}>0)\\}}\big{[}\frac{1}{4\alpha}+\langle x_{i},y_{i}\rangle+(x_{i\tau_{1}}-x_{i\tau_{2}})\big{]}^{-}$
|  | $-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq x_{i\tau_{1}}\leq 0\\}}\big{[}\alpha(x_{i\tau_{1}})^{2}\big{]}$

We have
$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigvee_{k=1}^{4}\Psi_{\alpha}^{k}(x_{i},y^{c}_{i},y^{f}_{i})$
where
$\Psi_{\alpha}^{1}(x_{i},y^{c}_{i},y^{f}_{i})=\mathbbm{1}_{(y_{i}^{c}<y_{i}^{f})}\Psi_{\alpha}^{1a}(x_{i},y^{c}_{i},y^{f}_{i})+\mathbbm{1}_{(y_{i}^{f}<y_{i}^{c})}\Psi_{\alpha}^{1b}(x_{i},y^{c}_{i},y^{f}_{i})$,
$\Psi_{\alpha}^{2}(x_{i},y^{c}_{i},y^{f}_{i})=\mathbbm{1}_{(w_{2}+y_{i}^{c}<y_{i}^{f})}\Psi_{\alpha}^{2a}(x_{i},y^{c}_{i},y^{f}_{i})+\mathbbm{1}_{(y_{i}^{f}<w_{2}+y_{i}^{c})}\Psi_{\alpha}^{2b}(x_{i},y^{c}_{i},y^{f}_{i})$,
$\Psi_{\alpha}^{3}(x_{i},y^{c}_{i},y^{f}_{i})=\mathbbm{1}_{(y_{i}^{c}<w_{3}+y_{i}^{f})}\Psi_{\alpha}^{3a}(x_{i},y^{c}_{i},y^{f}_{i})+\mathbbm{1}_{(w_{3}+y_{i}^{f}<y_{i}^{c})}\Psi_{\alpha}^{3b}(x_{i},y^{c}_{i},y^{f}_{i})$,
$\Psi_{\alpha}^{4}(x_{i},y^{c}_{i},y^{f}_{i})=\mathbbm{1}_{(w_{2}+y_{i}^{c}<w_{3}+y_{i}^{f})}\Psi_{\alpha}^{4a}(x_{i},y^{c}_{i},y^{f}_{i})+\mathbbm{1}_{(w_{3}+y_{i}^{f}<w_{2}+y_{i}^{c})}\Psi_{\alpha}^{4b}(x_{i},y^{c}_{i},y^{f}_{i})$,

and (suppressing arguments for brevity):
$\Psi_{\alpha}^{1a}=\bigg{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{c}\rangle\bigg{]}^{+}$,
$\Psi_{\alpha}^{1b}=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{2}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{2}}>0)\\}}\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{f}\rangle\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{2}}\leq 0\\}}\big{[}\alpha(\langle
x_{i},y_{i}^{f}\rangle)^{2}\big{]}\bigg{]}$,
$\Psi^{2a}=\bigg{[}\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{c}\rangle+(x_{i\tau_{1}^{*}}-x_{i\tau_{2}})\big{]}^{+}-\alpha
S_{3}K^{2a}\bigg{]}$, $\Psi_{\alpha}^{2b}=\bigg{[}\Psi_{\alpha}^{1b}-\alpha
S_{3}K^{2b}\bigg{]}$,
$\Psi^{3a}=\bigg{[}\Psi_{\alpha}^{1a}-\alpha S_{3}K^{3a}\bigg{]}$,
$\Psi^{3b}=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{1}^{*}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{1}^{*}}>0)\\}}\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{f}\rangle+(x_{i\tau_{1}^{*}}-x_{i\tau_{2}})\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{1}^{*}}\leq 0\\}}\big{[}\alpha(x_{i\tau_{1}^{*}})^{2}\big{]}-\alpha
S_{3}K^{3b}\bigg{]}$,
$\Psi^{4a}=\bigg{[}\Psi^{2a}-\alpha S_{3}(K^{4a}-K^{2a})\bigg{]}$,
$\Psi^{4b}=\bigg{[}\Psi^{3b}-\alpha S_{3}(K^{4b}-K^{3b})\bigg{]}$.
Note parameter $\tau_{1}^{*}$ and constant $K$ are defined within the proof by
$\mathbf{cases}$ (see Supplementary Material), and are omitted here for
brevity. Recall $\tau_{2}$ is index $\tau$ such that $y_{i\tau}^{\\{c,f\\}}=1$
else it is 0 if $\|y_{i}^{\\{c,f\\}}\|=0$. The selection in $\\{c,f\\}$ is
determined by context.  This result follows from jointly maximizing the
adversarial exposure $w_{1}$ and the default time indices $w_{2},w_{3}$. The
structure of $B^{2}_{n}$ allows us to decouple this joint maximization and
find the critical point to maximize the quadratic in $w_{1}$ and write down
the condition to select the optimal default time index $\tau_{1}^{*}$ for
either the counterparty (in sub-case a) or the firm (in sub-case b), as
determined by first to default. Finally, take the max over the four logical
cases for $w_{2}$ and $w_{3}$ to arrive at the solution. The $K$ terms
represent the cost associated with the worst case BCVA.

The particular structure of $B^{1}_{n}$ and $B^{2}_{n}$ will be exploited to
evaluate the $\sup$ above. The analysis proceeds by considering different
cases for optimal values $(w_{1}^{*},w_{2}^{*},w_{3}^{*})$.

$\mathbf{Case\,1}$ Suppose $w_{2}^{*}=0,w_{3}^{*}=0$. Then

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n}}\big{[}\langle(w_{1}+x_{i})^{+},\mathbbm{1}_{\\{y^{c}_{i}<y^{f}_{i}\\}}y^{c}_{i}\rangle+\langle(w_{1}+x_{i})^{-},\mathbbm{1}_{\\{y^{f}_{i}<y^{c}_{i}\\}}y^{f}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle)\big{]}.$

  1. a)

Suppose $\mathbbm{1}_{(y_{i}^{c}<y_{i}^{f})}=1$. Then

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n}}\big{[}\langle(w_{1}+x_{i})^{+},y^{c}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle)\big{]}.$

Therefore $\|y_{i}^{c}\|=1$. Let $\tau_{2}$ denote default time for
$y^{c}_{i}$. Simplify further to get

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1\tau_{2}}\in\mathbb{R}}\big{[}(w_{1\tau_{2}}+x_{i\tau_{2}})^{+}-\alpha(w_{1\tau_{2}})^{2}\big{]}.$

Now follow the approach in Bartl17 to write down the first order optimality
condition:

$\mathbbm{1}_{[0,\infty)}(w_{1\tau_{2}}+x_{i\tau_{2}})-2\alpha
w_{1\tau_{2}}\leq
0\leq\mathbbm{1}_{(0,\infty)}(w_{1\tau_{2}}+x_{i\tau_{2}})-2\alpha
w_{1\tau_{2}}.$

    1. i)

Suppose $(w_{1\tau_{2}}^{*}+x_{i\tau_{2}})<0$. Then $w_{1\tau_{2}}^{*}=0$. So
$x_{i\tau_{2}}<0\implies w_{i\tau_{2}}^{*}=0$.

    2. ii)

Suppose $(w_{1\tau_{2}}^{*}+x_{i\tau_{2}})>0$. Then
$w_{1\tau_{2}}^{*}=\frac{1}{2\alpha}$. So
$x_{i\tau_{2}}>-\frac{1}{2\alpha}\implies
w_{1\tau_{2}}^{*}=\frac{1}{2\alpha}$.

    3. iii)

Note $(w_{1\tau_{2}}^{*}+x_{i\tau_{2}})=0$ is not possible (does not satisfy
first order optimality condition).

Considering the intervals for $x_{i\tau_{2}}$ above, there are three cases as
below.

    1. i)

$x_{i\tau_{2}}\geq 0\implies
w_{1\tau_{2}}^{*}=\frac{1}{2\alpha}\implies\Psi_{\alpha}=[\frac{1}{4\alpha}+x_{i\tau_{2}}]$.

    2. ii)

$x_{i\tau_{2}}\leq-\frac{1}{2\alpha}\implies
w_{1\tau_{2}}^{*}=0\implies\Psi_{\alpha}=0$.

    3. iii)

$(-\frac{1}{2\alpha}<x_{i\tau_{2}}<0)\implies\Psi_{\alpha}=[\frac{1}{4\alpha}+x_{i\tau_{2}}]^{+}$.

In summary, considering all cases above, conclude that

$\Psi_{\alpha}^{1a}(x_{i},y^{c}_{i},y^{f}_{i})=\big{[}\frac{1}{4\alpha}+x_{i\tau_{2}}\big{]}^{+}.$

This can also be expressed as

$\Psi_{\alpha}^{1a}(x_{i},y^{c}_{i},y^{f}_{i})=\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{c}\rangle\big{]}^{+}.$

  2. b)

Suppose $\mathbbm{1}_{(y_{i}^{f}<y_{i}^{c})}=1$. Then

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n}}\big{[}\langle(w_{1}+x_{i})^{-},y^{f}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle)\big{]}.$

Therefore $\|y_{i}^{f}\|=1$. Let $\tau_{2}$ denote default time for
$y^{f}_{i}$. Simplify further to get

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1\tau_{2}}\in\mathbb{R}}\big{[}(w_{1\tau_{2}}+x_{i\tau_{2}})^{-}-\alpha(w_{1\tau_{2}})^{2}\big{]}.$

Now follow the approach in Bartl17 to write down the first order optimality
condition:

$\mathbbm{1}_{(-\infty,0]}(w_{1\tau_{2}}+x_{i\tau_{2}})-2\alpha
w_{1\tau_{2}}\leq
0\leq\mathbbm{1}_{(-\infty,0)}(w_{1\tau_{2}}+x_{i\tau_{2}})-2\alpha
w_{1\tau_{2}}.$

    1. i)

Suppose $(w_{1\tau_{2}}^{*}+x_{i\tau_{2}})>0$. Then $w_{1\tau_{2}}^{*}=0$. So
$x_{i\tau_{2}}>0\implies w_{1\tau_{2}}^{*}=0$.

    2. ii)

Suppose $(w_{1\tau_{2}}^{*}+x_{i\tau_{2}})<0$. Then
$w_{1\tau_{2}}^{*}=\frac{1}{2\alpha}$. So
$x_{i\tau_{2}}<-\frac{1}{2\alpha}\implies
w_{1\tau_{2}}^{*}=\frac{1}{2\alpha}$.

    3. iii)

Note $(w_{1\tau_{2}}^{*}+x_{i\tau_{2}})=0$ is not possible (does not satisfy
first order optimality condition).

Considering the intervals for $x_{i\tau_{2}}$ above, there are three cases as
below.

    1. i)

$x_{i\tau_{2}}>0\implies w_{1\tau_{2}}^{*}=0\implies\Psi_{\alpha}=0$.

    2. ii)

$x_{i\tau_{2}}<-\frac{1}{2\alpha}\implies
w_{i\tau_{2}}^{*}=\frac{1}{2\alpha}\implies\Psi_{\alpha}=[\frac{1}{4\alpha}+x_{i\tau_{2}}]$.

    3. iii)

$[-\frac{1}{2\alpha}\leq x_{i\tau_{2}}\leq 0]\implies
w_{1\tau_{2}}^{*}=|x_{i\tau_{2}}|$.
Note the slope $(1-2\alpha w_{1\tau_{2}})$ is positive for $0\leq
w_{1\tau_{2}}<\frac{1}{2\alpha}$, and equals zero at
$w_{1\tau_{2}}=\frac{1}{2\alpha}$.
However, $(w_{1\tau_{2}}+x_{i\tau_{2}})^{-}$ attains its max value of zero for
$w_{i\tau_{2}}=|x_{i\tau_{2}}|$ so stop there.

In summary, considering all cases above, conclude that

$\Psi_{\alpha}^{1b}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{2}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{2}}>0)\\}}\big{[}\frac{1}{4\alpha}+x_{i\tau_{2}}\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{2}}\leq 0\\}}\big{[}\alpha(x_{i\tau_{2}})^{2}\big{]}\bigg{]}.$

This can also be expressed as

$\Psi_{\alpha}^{1b}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{2}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{2}}>0)\\}}\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{f}\rangle\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{2}}\leq 0\\}}\big{[}\alpha(\langle
x_{i},y_{i}^{f}\rangle)^{2}\big{]}\bigg{]}.$

  3. c)

Suppose $\mathbbm{1}_{(\|y_{i}^{f}\|=\|y_{i}^{c}\|=0)}=1$.
In this trivial case, $\Psi_{\alpha}=0$. Note there is no third subcase for
Cases 2-4 below since that would imply $w_{2}^{*}=0,w_{3}^{*}=0$.

Finally, to sum up Case 1, considering parts a) and b), let us write:

$\Psi_{\alpha}^{1}(x_{i},y^{c}_{i},y^{f}_{i})=\mathbbm{1}_{(y_{i}^{c}<y_{i}^{f})}\Psi_{\alpha}^{1a}(x_{i},y^{c}_{i},y^{f}_{i})+\mathbbm{1}_{(y_{i}^{f}<y_{i}^{c})}\Psi_{\alpha}^{1b}(x_{i},y^{c}_{i},y^{f}_{i}).$

$\mathbf{Case\,2}$ Suppose $w_{2}^{*}\neq 0,w_{3}^{*}=0$.
Then $w_{2}^{*}$ has +1 in position $\tau_{1}^{*}$ and -1 in position
$\tau_{2}$, where $\tau_{j}=0$ means the value $\pm 1$ does not occur.
Furthermore, $\tau_{1}^{*}\neq\tau_{2}$ otherwise $w_{2}^{*}=0$.

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{2}\in
B_{n}^{2}}\big{[}\langle(w_{1}+x_{i})^{+},\mathbbm{1}_{\\{w_{2}+y^{c}_{i}<y^{f}_{i}\\}}w_{2}+y^{c}_{i}\rangle+\langle(w_{1}+x_{i})^{-},\mathbbm{1}_{\\{y^{f}_{i}<w_{2}+y^{c}_{i}\\}}y^{f}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{2},w_{2}\rangle)\big{]}.$

  1. a)

Suppose $\mathbbm{1}_{(w_{2}+y_{i}^{c}<y_{i}^{f})}=1$. Then

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{2}\in
B_{n}^{2}}\big{[}\langle(w_{1}+x_{i})^{+},w_{2}+y^{c}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{2},w_{2}\rangle)\big{]}.$

Recall
$\langle(w_{1}+x_{i}),(w_{2}+y_{i}^{c})\rangle=(w_{1\tau_{1}}+x_{i\tau_{1}})$.
Also recall $\tau_{1}$ and $\tau_{2}$ are associated with $y^{c}_{i}$. Let
$\tau_{2,f}$ denote default time (index) for $y_{i}^{f}$. The default time
constraint implies $\tau_{1}<\tau_{2,f}$. Therefore $\tau_{1}>0$. The
structure of finite set $B^{2}_{n}$ implies

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},0<\tau_{1}<\tau_{2,f},\tau_{1}\neq\tau_{2}}\big{[}(w_{1\tau_{1}}+x_{i\tau_{1}})^{+}-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{2},w_{2}\rangle)\big{]}.$

Observe the only positive component for $w_{1}\in\mathbb{R}^{n}$ in $\sup$
above is $\tau_{1}$.

$\sup_{w_{1}\in\mathbb{R}^{n}}\big{[}(w_{1\tau_{1}}+x_{i\tau_{1}})^{+}-\alpha\langle
w_{1},w_{1}\rangle\big{]}=\sup_{w_{1\tau_{1}}\in\mathbb{R}}\big{[}(w_{1\tau_{1}}+x_{i\tau_{1}})^{+}-\alpha(w_{1\tau_{1}}^{2})\big{]}.$

Evaluating at the critical point
$w^{*}_{1\tau_{1}}=\frac{1}{2\alpha}\in\mathbb{R}$ for the above quadratic
gives

$\sup_{w_{1\tau_{1}}\in\mathbb{R}}\big{[}(w_{1\tau_{1}}+x_{i\tau_{1}})^{+}-\alpha(w_{1\tau_{1}}^{2})\big{]}=\big{[}\frac{1}{4\alpha}+x_{i\tau_{1}}\big{]}^{+}.$

Therefore one can write

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\max_{0<\tau_{1}<\tau_{2,f},\tau_{1}\neq\tau_{2}}\big{[}\frac{1}{4\alpha}+x_{i\tau_{1}}\big{]}^{+}-\alpha
S_{3}K^{2a}$

where $K^{2a}:=(1+\mathbbm{1}_{\\{\tau_{2}\neq 0\\}})$. Furthermore,
$\tau_{1}^{*}$ is determined as

$\tau_{1}^{*}=\operatorname*{arg\,max}_{0<\tau_{1}<\tau_{2,f},\tau_{1}\neq\tau_{2}}[x^{+}_{i\tau_{1}}].$

Substituting back into expression for $\Psi_{\alpha}$ gives

$\Psi^{2a}_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\big{[}\frac{1}{4\alpha}+x_{i\tau_{1}^{*}}\big{]}^{+}-\alpha
S_{3}K^{2a}\bigg{]}.$

This can also be expressed as

$\Psi^{2a}_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{c}\rangle+(x_{i\tau_{1}^{*}}-x_{i\tau_{2}})\big{]}^{+}-\alpha
S_{3}K^{2a}\bigg{]}.$

  2. b)

Suppose $\mathbbm{1}_{(y_{i}^{f}<w_{2}+y_{i}^{c})}=1$. Then

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{2}\in
B_{n}^{2}}\big{[}\langle(w_{1}+x_{i})^{-},y^{f}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{2},w_{2}\rangle)\big{]}.$

Recall $\tau_{1}$ and $\tau_{2}$ are associated with $y^{c}_{i}$. Let
$\tau_{2,f}$ denote the default time (index) for $y_{i}^{f}$. The default time
constraint implies $\tau_{2,f}<\tau_{1}$. Therefore $\tau_{2,f}>0$ and
$\|y_{i}^{f}\|=1$. Note the only non-zero component of $\|y^{f}_{i}\|$ is
$\tau_{2,f}$. Hence set $w^{*}_{1\tau}=0\>\forall\tau\neq\tau_{2,f}$.
Simplifying further

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1\tau_{2,f}}\in\mathbb{R},w_{2}\in
B_{n}^{2}}\big{[}(w_{1\tau_{2,f}}+x_{i\tau_{2,f}})^{-}-\alpha((w_{1\tau_{2,f}})^{2}+S_{3}K^{2b})\big{]}.$

where $K^{2b}:=(\mathbbm{1}_{\\{\tau_{1}\neq
0\\}}+\mathbbm{1}_{\\{\tau_{2}\neq 0\\}})$ = 1. For $K^{2b}$, if $\tau_{2}=0$,
then $\tau_{1}\neq 0$ since $w_{2}^{*}\neq 0$. Otherwise set $\tau_{1}=0$ if
$\tau_{2}\neq 0$ to maximize $\sup_{w_{2}}$ above. Following the calculations
in $\text{Case}\>\text{1b})$ above, conclude that

$\Psi_{\alpha}^{2b}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{2}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{2}}>0)\\}}[\frac{1}{4\alpha}+x_{i\tau_{2}}]^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{2}}\leq 0\\}}\big{[}\alpha(x_{i\tau_{2}})^{2}\big{]}-\alpha
S_{3}K^{2b}\bigg{]}.$

This can also be expressed as

$\Psi_{\alpha}^{2b}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{2}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{2}}>0)\\}}\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{f}\rangle\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{2}}\leq 0\\}}\big{[}\alpha(\langle
x_{i},y_{i}^{f}\rangle)^{2}\big{]}-\alpha S_{3}K^{2b}\bigg{]}.$

Finally, to sum up Case 2, considering parts a) and b), let us write:

$\Psi_{\alpha}^{2}(x_{i},y^{c}_{i},y^{f}_{i})=\mathbbm{1}_{(w_{2}+y_{i}^{c}<y_{i}^{f})}\Psi_{\alpha}^{2a}(x_{i},y^{c}_{i},y^{f}_{i})+\mathbbm{1}_{(y_{i}^{f}<w_{2}+y_{i}^{c})}\Psi_{\alpha}^{2b}(x_{i},y^{c}_{i},y^{f}_{i}).$

$\mathbf{Case\,3}$ Suppose $w_{2}^{*}=0,w_{3}^{*}\neq 0$.
Then $w_{3}^{*}$ has +1 in position $\tau_{1}^{*}$ and -1 in position
$\tau_{2}$, where $\tau_{j}=0$ means the value $\pm 1$ does not occur.
Furthermore, $\tau_{1}^{*}\neq\tau_{2}$ otherwise $w_{3}^{*}=0$.

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{3}\in
B_{n}^{2}}\big{[}\langle(w_{1}+x_{i})^{+},\mathbbm{1}_{\\{y^{c}_{i}<w_{3}+y^{f}_{i}\\}}y^{c}_{i}\rangle+\langle(w_{1}+x_{i})^{-},\mathbbm{1}_{\\{w_{3}+y^{f}_{i}<y^{c}_{i}\\}}w_{3}+y^{f}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{3},w_{3}\rangle)\big{]}.$

  1. a)

Suppose $\mathbbm{1}_{(y_{i}^{c}<w_{3}+y_{i}^{f})}=1$. Then

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{3}\in
B_{n}^{2}}\big{[}\langle(w_{1}+x_{i})^{+},y^{c}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{3},w_{3}\rangle)\big{]}.$

Recall $\langle(w_{1}+x_{i}),y_{i}^{c}\rangle=(w_{1\tau_{2}}+x_{i\tau_{2}})$.
Also recall $\tau_{1}$ and $\tau_{2}$ are associated with $y^{f}_{i}$. Let
$\tau_{2,c}$ denote the default time (index) for $y_{i}^{c}$. The default time
constraint implies $\tau_{2,c}<\tau_{1}$. Therefore $\tau_{2,c}>0$ and
$\|y_{i}^{c}\|=1$. Note the only positive component of $\|y^{c}_{i}\|$ is
$\tau_{2,c}$. Hence set $w^{*}_{1\tau}=0\>\forall\tau\neq\tau_{2,c}$. Simplify
further to get

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1\tau_{2,c}}\in\mathbb{R},w_{3}\in
B_{n}^{2}}\big{[}(w_{1\tau_{2,c}}+x_{i\tau_{2,c}})^{+}-\alpha((w_{1\tau_{2,c}})^{2}+S_{3}K^{3a})\big{]}$

where $K^{3a}:=(\mathbbm{1}_{\\{\tau_{1}\neq
0\\}}+\mathbbm{1}_{\\{\tau_{2}\neq 0\\}})$ = 1, following logic in Case 2b)
above. Evaluating at the critical point
$w^{*}_{1\tau_{2,c}}=\frac{1}{2\alpha}\in\mathbb{R}$ gives

$\sup_{w_{1\tau_{2,c}}\in\mathbb{R}}\big{[}(w_{1\tau_{2,c}}+x_{i\tau_{2,c}})^{+}-\alpha(w_{1\tau_{2,c}}^{2})\big{]}=\big{[}\frac{1}{4\alpha}+x_{i\tau_{2,c}}\big{]}^{+}.$

Therefore one can write

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\big{[}\frac{1}{4\alpha}+x_{i\tau_{2,c}}\big{]}^{+}-\alpha
S_{3}K^{3a}.$

This can also be expressed as

$\Psi^{3a}_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{c}\rangle\big{]}^{+}-\alpha S_{3}K^{3a}\bigg{]}.$

  2. b)

Suppose $\mathbbm{1}_{(w_{3}+y_{i}^{f}<y_{i}^{c})}=1$. Then

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{3}\in
B_{n}^{2}}\big{[}\langle(w_{1}+x_{i})^{-},w_{3}+y^{f}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{3},w_{3}\rangle)\big{]}.$

Recall
$\langle(w_{1}+x_{i}),(w_{3}+y_{i}^{f})\rangle\,=\,(w_{1\tau_{1}}+x_{i\tau_{1}})$.
Also recall $\tau_{1}$ and $\tau_{2}$ are associated with $y^{f}_{i}$. Let
$\tau_{2,c}$ denote default time (index) for $y_{i}^{c}$. The default time
constraint implies $\tau_{1}<\tau_{2,c}$. Therefore $\tau_{1}>0$ and

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},0<\tau_{1}<\tau_{2,c},\tau_{1}\neq\tau_{2}}\big{[}(w_{1\tau_{1}}+x_{i\tau_{1}})^{-}-\alpha((w_{1\tau_{1}})^{2}+S_{3}K^{3b})\big{]}$

where $K^{3b}:=(1+\mathbbm{1}_{\\{\tau_{2}\neq 0\\}})$. Following the
calculations in $\text{Case}\>\text{2b})$ above, conclude that

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{1}^{*}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{1}^{*}}>0)\\}}\big{[}\frac{1}{4\alpha}+x_{i\tau_{1}^{*}}\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{1}^{*}}\leq 0\\}}\big{[}\alpha(x_{i\tau_{1}^{*}})^{2}\big{]}-\alpha
S_{3}K^{3b}\bigg{]}.$

Furthermore, $\tau_{1}^{*}$ is determined as

$\tau_{1}^{*}=\operatorname*{arg\,max}_{0<\tau_{1}<\tau_{2}^{c},\tau_{1}\neq\tau_{2}}[x_{i\tau_{1}}].$

Therefore one can write

$\Psi^{3b}_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{1}^{*}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{1}^{*}}>0)\\}}\big{[}\frac{1}{4\alpha}+x_{i\tau_{1}^{*}}\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{1}^{*}}\leq 0\\}}\big{[}\alpha(x_{i\tau_{1}^{*}})^{2}\big{]}-\alpha
S_{3}K^{3b}\bigg{]}.$

This can also be expressed as

$\Psi^{3b}_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{1}^{*}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{1}^{*}}>0)\\}}\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{f}\rangle+(x_{i\tau_{1}^{*}}-x_{i\tau_{2}})\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{1}^{*}}\leq 0\\}}\big{[}\alpha(x_{i\tau_{1}^{*}})^{2}\big{]}-\alpha
S_{3}K^{3b}\bigg{]}.$

Finally, to sum up Case 3, considering parts a) and b), let us write:

$\Psi_{\alpha}^{3}(x_{i},y^{c}_{i},y^{f}_{i})=\mathbbm{1}_{(y_{i}^{c}<w_{3}+y_{i}^{f})}\Psi_{\alpha}^{3a}(x_{i},y^{c}_{i},y^{f}_{i})+\mathbbm{1}_{(w_{3}+y_{i}^{f}<y_{i}^{c})}\Psi_{\alpha}^{3b}(x_{i},y^{c}_{i},y^{f}_{i}).$

$\mathbf{Case\,4}$ Suppose $w_{2}^{*}\neq 0,w_{3}^{*}\neq 0$.
Then $w_{2}^{*}$ has +1 in position $\tau_{1,c}^{*}$ and -1 in position
$\tau_{2,c}$, where $\tau_{j,c}=0$ means the value $\pm 1$ does not occur.
Furthermore, $\tau_{1,c}^{*}\neq\tau_{2,c}$ otherwise $w_{2}^{*}=0$.
And $w_{3}^{*}$ has +1 in position $\tau_{1,f}^{*}$ and -1 in position
$\tau_{2,f}$, where $\tau_{j,f}=0$ means the value $\pm 1$ does not occur.
Furthermore, $\tau_{1,f}^{*}\neq\tau_{2,f}$ otherwise $w_{3}^{*}=0$.

$\begin{split}\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{2}\in{B^{2}_{n}},w_{3}\in{B^{2}_{n}}}\bigg{[}\langle(w_{1}+x_{i})^{+},\mathbbm{1}_{\\{w_{2}+y^{c}_{i}<w_{3}+y^{f}_{i}\\}}w_{2}+y^{c}_{i}\rangle+\langle(w_{1}+x_{i})^{-},\mathbbm{1}_{\\{w_{3}+y^{f}_{i}<w_{2}+y^{c}_{i}\\}}w_{3}+y^{f}_{i}\rangle\\\
-\alpha(\langle w_{1},w_{1}\rangle+S_{3}\langle
w_{2},w_{2}\rangle+S_{3}\langle w_{3},w_{3}\rangle)\bigg{]}.\end{split}$

  1. a)

Suppose $\mathbbm{1}_{(w_{2}+y_{i}^{c}<w_{3}+y_{i}^{f})}=1$. Then

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{2}\in
B_{n}^{2},w_{3}\in
B_{n}^{2}}\big{[}\langle(w_{1}+x_{i})^{+},w_{2}+y^{c}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{2},w_{2}\rangle+S_{3}\langle
w_{3},w_{3}\rangle)\big{]}.$

Recall
$\langle(w_{1}+x_{i}),(w_{2}+y_{i}^{c})\rangle=(w_{1\tau_{1,c}}+x_{i\tau_{1,c}})$.
The default time constraint implies $\tau_{1,c}<\tau_{1,f}$. Therefore
$\tau_{1,c}>0$.
The structure of finite set $B^{2}_{n}$ implies

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},0<\tau_{1,c}<\tau_{1,f},\tau_{1,c}\neq\tau_{2,c}}\big{[}(w_{1\tau_{1,c}}+x_{i\tau_{1,c}})^{+}-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{2},w_{2}\rangle+S_{3}\langle
w_{3},w_{3}\rangle)\big{]}.$

Observe the only positive component for $w_{1}\in\mathbb{R}^{n}$ in $\sup$
above is $\tau_{1,c}$.

$\sup_{w_{1}\in\mathbb{R}^{n}}\big{[}(w_{1\tau_{1,c}}+x_{i\tau_{1,c}})^{+}-\alpha\langle
w_{1},w_{1}\rangle\big{]}=\sup_{w_{1\tau_{1,c}}\in\mathbb{R}}\big{[}(w_{1\tau_{1,c}}+x_{i\tau_{1,c}})^{+}-\alpha(w_{1\tau_{1,c}}^{2})\big{]}.$

Evaluating at the critical point
$w^{*}_{1\tau_{1,c}}=\frac{1}{2\alpha}\in\mathbb{R}$ for the above quadratic
gives

$\sup_{w_{1\tau_{1,c}}\in\mathbb{R}}\big{[}(w_{1\tau_{1,c}}+x_{i\tau_{1,c}})^{+}-\alpha(w_{1\tau_{1,c}}^{2})\big{]}=\big{[}\frac{1}{4\alpha}+x_{i\tau_{1,c}}\big{]}^{+}.$

Therefore one can write

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\max_{0<\tau_{1,c}<\tau_{1,f},\tau_{1,c}\neq\tau_{2,c}}\big{[}\frac{1}{4\alpha}+x_{i\tau_{1,c}}\big{]}^{+}-\alpha
S_{3}K^{4a}.$

where $K^{4a}:=(\mathbbm{1}_{\\{\tau_{1,c}\neq
0\\}}+\mathbbm{1}_{\\{\tau_{2,c}\neq 0\\}}+\mathbbm{1}_{\\{\tau_{1,f}\neq
0\\}}+\mathbbm{1}_{\\{\tau_{2,f}\neq 0\\}})=(2+\mathbbm{1}_{\\{\tau_{2,c}\neq
0\\}})$ following logic as in Case 3a) above.
Furthermore, $\tau_{1}^{*}$ is determined as

$\tau_{1}^{*}=\operatorname*{arg\,max}_{0<\tau_{1,c}<\tau_{1,f},\tau_{1,c}\neq\tau_{2,c}}[x^{+}_{i\tau_{1,c}}].$

Substituting back into expression for $\Psi_{\alpha}$ gives

$\Psi^{4a}_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\big{[}\frac{1}{4\alpha}+x_{i\tau_{1}^{*}}\big{]}^{+}-\alpha
S_{3}K^{4a}\bigg{]}.$

Let $\tau_{2}=\tau_{2,c}$. Then this can also be expressed as

$\Psi^{4a}_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{c}\rangle+(x_{i\tau_{1}^{*}}-x_{i\tau_{2}})\big{]}^{+}-\alpha
S_{3}K^{4a}\bigg{]}.$

  2. b)

Suppose $\mathbbm{1}_{(w_{3}+y_{i}^{f}<w_{2}+y_{i}^{c})}=1$. Then

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{2}\in
B_{n}^{2},w_{3}\in
B_{n}^{2}}\big{[}\langle(w_{1}+x_{i})^{-},w_{3}+y^{f}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{2},w_{2}\rangle+S_{3}\langle
w_{3},w_{3}\rangle)\big{]}.$

Recall
$\langle(w_{1}+x_{i}),(w_{3}+y_{i}^{f})\rangle=(w_{1\tau_{1,f}}+x_{i\tau_{1,f}})$.
The default time constraint implies $\tau_{1,f}<\tau_{1,c}$. Therefore
$\tau_{1,f}>0$.
The structure of finite set $B^{2}_{n}$ implies

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},0<\tau_{1,f}<\tau_{1,c},\tau_{1,f}\neq\tau_{2,f}}\big{[}(w_{1\tau_{1,f}}+x_{i\tau_{1,f}})^{-}-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{2},w_{2}\rangle+S_{3}\langle
w_{3},w_{3}\rangle)\big{]}.$

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},0<\tau_{1,f}<\tau_{1,c},\tau_{1,f}\neq\tau_{2,f}}\big{[}(w_{1\tau_{1,f}}+x_{i\tau_{1,f}})^{-}-\alpha((w_{1\tau_{1,f}})^{2}+S_{3}K)\big{]}.$

where $K:=(\mathbbm{1}_{\\{\tau_{1,c}\neq 0\\}}+\mathbbm{1}_{\\{\tau_{2,c}\neq
0\\}}+\mathbbm{1}_{\\{\tau_{1,f}\neq 0\\}}+\mathbbm{1}_{\\{\tau_{2,f}\neq
0\\}})=(2+\mathbbm{1}_{\\{\tau_{2,f}\neq 0\\}})$ following logic as in Case
4a) above.
Following the calculations in $\text{Case}\>\text{3b})$ above, conclude that

$\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{1}^{*}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{1}^{*}}>0)\\}}\big{[}\frac{1}{4\alpha}+x_{i\tau_{1}^{*}}\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{1}^{*}}\leq 0\\}}\big{[}\alpha(x_{i\tau_{1}^{*}})^{2}\big{]}-\alpha
S_{3}K\bigg{]}.$

Furthermore, $\tau_{1}^{*}$ is determined as

$\tau_{1}^{*}=\operatorname*{arg\,max}_{0<\tau_{1,f}<\tau_{1,c},\tau_{1,f}\neq\tau_{2,f}}[x_{i\tau_{1,f}}].$

Therefore one can write

$\Psi^{4b}_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{1}^{*}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{1}^{*}}>0)\\}}\big{[}\frac{1}{4\alpha}+x_{i\tau_{1}^{*}}\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{1}^{*}}\leq 0\\}}\big{[}\alpha(x_{i\tau_{1}^{*}})^{2}\big{]}-\alpha
S_{3}K^{4b}\bigg{]}.$

Let $\tau_{2}=\tau_{2,f}$. Then this can also be expressed as

$\Psi^{4b}_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})=\bigg{[}\mathbbm{1}_{\\{(x_{i\tau_{1}^{*}}<-\frac{1}{2\alpha})\vee(x_{i\tau_{1}^{*}}>0)\\}}\big{[}\frac{1}{4\alpha}+\langle
x_{i},y_{i}^{f}\rangle+(x_{i\tau_{1}^{*}}-x_{i\tau_{2}})\big{]}^{-}-\mathbbm{1}_{\\{-\frac{1}{2\alpha}\leq
x_{i\tau_{1}^{*}}\leq 0\\}}\big{[}\alpha(x_{i\tau_{1}^{*}})^{2}\big{]}-\alpha
S_{3}K^{4b}\bigg{]}.$

Finally, to sum up Case 4, considering parts a) and b), let us write:

$\Psi_{\alpha}^{4}(x_{i},y^{c}_{i},y^{f}_{i})=\mathbbm{1}_{(w_{2}+y_{i}^{c}<w_{3}+y_{i}^{f})}\Psi_{\alpha}^{4a}(x_{i},y^{c}_{i},y^{f}_{i})+\mathbbm{1}_{(w_{3}+y_{i}^{f}<w_{2}+y_{i}^{c})}\Psi_{\alpha}^{4b}(x_{i},y^{c}_{i},y^{f}_{i}).$

####  2.2.2 Outer Optimization Problem

The goal now is to evaluate

$\inf_{\alpha\geq
0}\>F(\alpha):=\bigg{[}\alpha\delta_{3}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})\bigg{]}$

where the $\Psi_{\alpha}$ functions are given as the solutions to Proposition
2.1. Although the Lagrangian duality implies the convexity of $F(\alpha)$, due
to its complexity, computational methods and solvers are used to evaluate this
expression. Nonetheless, the solution can be expressed as below. Note that for
$\delta_{3}=0$ one recovers the expression for original CVAB given in Section
1.3.1.

The primal problem P3 has solution
$\big{[}\alpha^{*}\delta_{3}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\alpha^{*}}(x_{i},y^{c}_{i},y^{f}_{i})\big{]}$
where $\alpha^{*}=\,\operatorname*{arg\,min}_{\alpha\geq
0}\big{[}\alpha\delta_{3}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\alpha}(x_{i},y^{c}_{i},y^{f}_{i})\big{]}$
and
$\Psi_{\alpha^{*}}(x_{i},y^{c}_{i},y^{f}_{i})=\bigvee_{k=1}^{4}\Psi_{\alpha^{*}}^{k}(x_{i},y^{c}_{i},y^{f}_{i})$.
Expressed in terms of original BCVA, this says

$\sup_{\Phi\in\mathcal{U}_{\delta_{3}}(\Phi_{N})}\mathbb{E}^{\Phi}[\langle
X^{+},Y^{C}\rangle+\langle X^{-},Y^{F}\rangle]=\mathbb{E}^{P_{N}}[\langle
X^{+},Y^{C}\rangle+\langle
X^{-},Y^{F}\rangle]+\alpha^{*}\delta_{3}+\mathbb{E}^{P_{N}}\big{[}\Psi_{\alpha^{*}}(X,Y^{C},Y^{F})-[\langle
X^{+},Y^{C}\rangle+\langle X^{-},Y^{F}\rangle]\big{]}^{+}$

where the additional terms represent a penalty due to uncertainty in
probability distribution.  This follows directly from the previous
proposition. $\delta_{3}=0$ reduces to original BCVA.  This follows by direct
substitution of $\alpha^{*}$ as characterized above into the dual problem D3.

####  2.2.3 Recovering the Worst Case Distribution

The process of recovering the worst case CVA distribution involves evaluating
the $\operatorname*{arg\,min}$ expressions given in Section 1.2.2. The
procedure is a bit tedious but one can go through the various cases and
subcases discussed in Proposition 2.1, and compute the value of the dual
minimizer $\alpha^{*}$ as given in Theorem 2.1, to recover the worst case
distribution $\\{(x^{*}_{i},y^{c*}_{i},y^{f*}_{i}):i\in\\{1,...,N+1\\}\\}$ for
a given $\delta$. This procedure is done for a few concrete examples in
Section 3.

####  2.2.4 Discussion

One limitation in the current approach is the omission of a risk neutral
measure constraint on the underlying interest rate and credit default
distributions that generate the portfolio exposure distributions described by
the Wasserstein ball $\mathcal{U}_{\delta_{3}}(\Phi_{N})$. It is not clear how
to (either explicitly or implicitly) incorporate such a constraint in a
solvable way. We highlight this as an opportunity for improvement and a
direction for further research. Empirical results for our worst case CVA
studies are provided in Section 3. From the authors’ perspective the
computational study was illuminating to understand the magnitude and shape of
worst case CVA profiles as a function of uncertainty. Some recent work was
done to map Wasserstein radii into lower and upper bounds on the distance
between the true and empirical distributions. See the discussion on this topic
in Section 3.2.

###  2.3 FCA, FBA

The robust FCA can be written as

$\sup_{P\in\mathcal{U}_{\delta_{1}}(P_{N})}\mathbb{E}^{P}[\langle
Z^{+},Y_{CF}\rangle].$ (P4)

Similarly, the robust FBA can be written as

$\sup_{Q\in\mathcal{U}_{\delta_{2}}(Q_{N})}\mathbb{E}^{Q}[\langle
Z^{-},Y_{CF}\rangle].$ (P5)

As such, the dual formulations and solutions to the above primal optimization
problems are special cases of the solutions to the FVA optimization problems,
to be described next.

###  2.4 FVA

####  2.4.1 Inner Optimization Problem

The robust FVA is

$\sup_{\Phi\in\mathcal{U}_{\delta_{3}}(\Phi_{N})}\mathbb{E}^{\Phi}[\langle
Z,Y_{CF}\rangle].$ (P6)

Similar to before, we use recent duality results, noting the inner product
$\langle\>;\rangle$ satisfies the upper semicontinuous condition of the
Lagrangian duality theorem, and cost function $c_{S}$ satisfies the non-
negative lower semicontinuous condition (see Blanchet and Murthy,  (2019)
Assumptions 1 & 2, Gao and Kleywegt,  (2016)). Hence the dual problem (to sup
above) can be written as

$\inf_{\alpha\geq
0}\>F(\alpha):=\bigg{[}\alpha\delta_{3}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\alpha}(z_{i},y^{cf}_{i})\bigg{]}$
(D6)

where

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\sup_{u\in\mathbb{R}^{n},v\in{B^{1}_{n}}}[\langle
u,v\rangle-\alpha
c_{S_{3}}((u,v),(z_{i},y^{cf}_{i}))]=\sup_{u\in\mathbb{R}^{n},v\in{B^{1}_{n}}}[\langle
u,v\rangle-\alpha(\langle u-z_{i},u-z_{i}\rangle+S_{3}\langle
v-y^{cf}_{i},v-y^{cf}_{i}\rangle)].$

Now apply change of variables $w_{1}=(u-z_{i})$ and $w_{2}=(v-y^{cf}_{i})$ to
get

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},w_{2}\in{B^{2}_{n}}}[\langle
w_{1}+z_{i},w_{2}+y^{cf}_{i}\rangle-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}\langle w_{2},w_{2}\rangle)]$

where the sets $B^{1}_{n}$ and $B^{2}_{n}$ are defined as before. It turns out
that $\Psi_{\alpha}$ can be expressed as original FVA plus the pointwise max
of $(n+1)$ convex functions. The degenerate case $l=0$ is just a line of
negative slope. The other $n$ cases are hyperbolas plus lines of negative
slope. $\Psi_{\alpha}$ quantifies the adversarial move in FVA across both time
and spatial dimensions while accounting for the cost via the $K$ terms.

We have $\Psi_{\alpha}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\big{[}\frac{l^{*}}{4\alpha}+\big{(}\sum_{k=1}^{l^{*}}z_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z_{ik}\big{)}-\alpha
S_{3}K\big{]}$
where $l^{*}=\operatorname*{arg\,max}_{l\geq
0}[\frac{l}{4\alpha}+\sum_{k=1}^{l}z_{ik}-\alpha S_{3}K]$ and
$l=\|w_{2}+y^{cf}_{i}\|_{1}\geq 0,\>l\in\mathbb{Z^{+}}$. Also
$\|y^{cf}_{i}\|_{1}\in\mathbb{Z^{+}}$, and
$K=|l-\|y^{cf}_{i}\|_{1}|=\|w_{2}\|_{1}\geq 0,K\in\mathbb{Z^{+}}$. Once
$l^{*}$ is selected, $K:=|l^{*}-\|y^{cf}_{i}\|_{1}|=\|w_{2}^{*}\|_{1}$.
Alternatively, $\Psi_{\alpha}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\bigvee_{l=0}^{n}h_{\alpha}(l)$ for
$h_{\alpha}(l):=\big{[}\frac{l}{4\alpha}+\big{(}\sum_{k=1}^{l}z_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z_{ik}\big{)}-\alpha
S_{3}K\big{]}$.  This result follows from jointly maximizing the adversarial
funding exposure $w_{1}$ and the survival time index $w_{2}$. The structure of
$B^{2}_{n}$ allows us to decouple this joint maximization and find the
critical point to maximize the quadratic in $w_{1}$ and write down the
condition to select the optimal survival time index $l^{*}$. Finally, consider
the two cases $w_{2}=0$ and $w_{2}\neq 0$ and take the max to arrive at the
solution. The $K$ terms represent the cost associated with the worst case.

The particular structure of $B^{1}_{n}$ and $B^{2}_{n}$ will be exploited to
evaluate the $\sup$ above. The analysis proceeds by considering different
cases for optimal values $(w_{1}^{*},w_{2}^{*})$.

$\mathbf{Case\,1}$ Suppose $w_{2}^{*}=0\implies l=\|y^{cf}_{i}\|_{1}$. Then

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\sup_{w_{1}\in\mathbb{R}^{n}}[\langle
w_{1},y^{cf}_{i}\rangle-\alpha\langle w_{1},w_{1}\rangle].$

Applying the Cauchy-Schwarz Inequality gives

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\sup_{\|w_{1}\|}[\|w_{1}\|\|y^{cf}_{i}\|-\alpha\|w_{1}\|^{2}].$

Evaluating the critical point
$\|w_{1}^{*}\|=\frac{\|y^{cf}_{i}\|}{2\alpha}\in\mathbb{R}_{+}$ for the
quadratic gives

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\frac{\|y^{cf}_{i}\|^{2}}{4\alpha}=\langle
z_{i},y^{cf}_{i}\rangle+\frac{\|y^{cf}_{i}\|_{1}}{4\alpha}.$

$\mathbf{Case\,2}$ Now consider $w_{2}^{*}\neq 0\implies
l\neq\|y^{cf}_{i}\|_{1}$.
Observe for $l=\|w_{2}+y^{cf}_{i}\|_{1}\geq 0$,

$\langle w_{1}+z_{i},w_{2}+y_{i}^{cf}\rangle=\sum_{k=1}^{l}(w_{1k}+z_{ik}).$

The structure of finite set $B^{2}_{n}$ implies

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\sup_{w_{1}\in\mathbb{R}^{n},l\in\\{0,\ldots,n\\},l\neq\|y^{cf}_{i}\|_{1}}[\sum_{k=1}^{l}(w_{1k}+z_{ik})-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}K)].$

Again, using that $B^{2}_{n}$ is a finite set, one can write

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\max_{l\in\\{0,\ldots,n\\},l\neq\|y^{cf}_{i}\|_{1}}\sup_{w_{1}\in\mathbb{R}^{n}}[\sum_{k=1}^{l}(w_{1k}+z_{ik})-\alpha(\langle
w_{1},w_{1}\rangle+S_{3}K)].$

Observing that only the first $l$ components of $w_{1}\text{ inside the sup
are positive gives }\forall k\in\\{1,\ldots,l\\}$

$\sup_{w_{1}\in\mathbb{R}^{n}}[\sum_{k=1}^{l}(w_{1k})-\alpha\langle
w_{1},w_{1}\rangle]=l\times\sup_{w_{1k}\in\mathbb{R}}[w_{1k}-\alpha(w_{1k})^{2}].$

Evaluating at the critical point
$w^{*}_{1k}=\frac{1}{2\alpha}\in\mathbb{R}_{+}$ for the above quadratic gives

$\sup_{w_{1k}\in\mathbb{R}}[w_{1k}-\alpha(w_{1k}^{2})]=\frac{1}{4\alpha}.$

Therefore one can write

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\max_{l\in\\{0,\ldots,n\\},l\neq\|y^{cf}_{i}\|_{1}}[\frac{l}{4\alpha}+\sum_{k=1}^{l}(z_{ik})-\alpha
S_{3}K].$

Furthermore, $l^{*}$ is determined as

$l^{*}=\operatorname*{arg\,max}_{l\in\\{0,\dots,n\\},l\neq\|y^{cf}_{i}\|_{1}}[\frac{l}{4\alpha}+\sum_{k=1}^{l}(z_{ik})-\alpha
S_{3}K].$

Substituting back into expression for $\Psi_{\alpha}$ gives

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\bigg{[}\frac{l^{*}}{4\alpha}+\bigg{(}\sum_{k=1}^{l^{*}}z_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z_{ik}\bigg{)}-\alpha
S_{3}K\bigg{]}.$

Finally, taking the max values for $\Psi_{\alpha}$ over cases $w_{2}^{*}=0$
and $w_{2}^{*}\neq 0$ gives

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\bigg{[}\frac{\|y^{cf}_{i}\|_{1}}{4\alpha}\bigg{]}\vee\bigg{[}\frac{l^{*}}{4\alpha}+\bigg{(}\sum_{k=1}^{l^{*}}z_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z_{ik}\bigg{)}-\alpha
S_{3}K\bigg{]}.$

Observe that for $l^{*}=\|y^{cf}_{i}\|_{1}$, the last term in brackets $[\,;]$
above evaluates to $\big{[}\frac{\|y^{cf}_{i}\|_{1}}{4\alpha}\big{]}$. Let
$l^{*}$ be determined as

$l^{*}=\operatorname*{arg\,max}_{l\in\\{0,\dots,n\\}}[\frac{l}{4\alpha}+\sum_{k=1}^{l}(z_{ik})-\alpha
S_{3}K]$

and write

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\bigg{[}\frac{l^{*}}{4\alpha}+\bigg{(}\sum_{k=1}^{l^{*}}z_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z_{ik}\bigg{)}-\alpha
S_{3}K\bigg{]}.$

Alternatively, one can write

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\bigvee_{l=0}^{n}\bigg{[}\frac{l}{4\alpha}+\bigg{(}\sum_{k=1}^{l}z_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z_{ik}\bigg{)}-\alpha
S_{3}K\bigg{]}.$

####  2.4.2 Outer Optimization Problem

The goal now is to evaluate

$\inf_{\alpha\geq
0}\>F(\alpha):=\bigg{[}\alpha\delta_{3}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\alpha}(z_{i},y^{cf}_{i})\bigg{]}$

where

$\Psi_{\alpha}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\bigvee_{l=0}^{n}h_{\alpha}(l)\>\>\text{for}\>\>h_{\alpha}(l):=\big{[}\frac{l}{4\alpha}+\big{(}\sum_{k=1}^{l}z_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z_{ik}\big{)}-\alpha
S_{3}K\big{]}.$

The convexity of the objective function $F(\alpha)$ simplifies the task of
solving this optimization problem. The first order optimality condition
suffices. As $\Psi_{\alpha}$ and hence $F(\alpha)$ may have non-differentiable
kinks due to the max functions, $\vee$, we characterize the optimality
condition via subgradients. In particular, we look for $\alpha^{*}\geq 0$ such
that $0\in\partial F(\alpha^{*})$. Inspection of the asymptotic properties of
$\Psi_{\alpha}$ and its subgradients reveals that $\partial F(\alpha)$ will
cross zero (as $\alpha$ sweeps from $0$ to $\infty$) and hence $\alpha^{*}\geq
0$. Note that for $\delta_{3}=0$ one recovers the expression for original FVA
given in Section 1.3.3.

Let $\alpha^{*}\in\,\left\\{\alpha\geq 0:0\in\partial F(\alpha)\right\\}\\\
where\,\,\partial\Psi_{\alpha}=\mathbf{Conv}\cup\left\\{\partial
h_{\alpha}(l)\>|\>\langle
z_{i},y^{cf}_{i}\rangle+h_{\alpha}(l)=\Psi_{\alpha};\,l\in\\{0,\dots,n\\}\right\\}$
and $\partial
F(\alpha)=\delta_{3}+\frac{1}{N}\sum_{i=1}^{N}\partial\Psi_{\alpha}$ .

This follows from application of standard properties of subgradients as well
as inspection of the asymptotic properties of $\Psi_{\alpha}$ and
$\partial\Psi_{\alpha}$. For $\alpha$ sufficiently small, $\Psi_{\alpha}$ has
a large positive value and $\partial\Psi_{\alpha}$ has a large negative
derivative. For $\alpha$ sufficiently large, for optimal $l^{*}$, either
$l^{*}=0\implies 0\in\partial\Psi_{\alpha}$ or
$l^{*}=\|y^{cf}_{i}\|_{1}>0\implies\partial\Psi_{\alpha}$ approaches zero
$\implies\partial F(\alpha)$ crosses zero.

This follows from standard application of properties of convex functions and
subgradients. First note that function $h_{\alpha}$ is convex in $\alpha$
since (for fixed $l$) it is the sum of a hyperbola plus a constant plus a
negative linear term. So then $\Psi_{\alpha}$ is convex since it is the
pointwise max of a finite set of convex functions plus a constant. Using
properties of subgradients, one can write
$\partial\Psi_{\alpha}=\mathbf{Conv}\cup\\{\partial h_{\alpha}(l)\>|\>\langle
z_{i},y^{cf}_{i}\rangle+h_{\alpha}(l)=\Psi_{\alpha};\,l\in\\{0,\dots,n\\}\\}$.
Furthermore $F(\alpha)$ is convex in $\alpha$ since it is a linear term plus a
sum of convex functions, so one can write
$\alpha^{*}\in\,\left\\{\alpha:0\in\partial F(\alpha)\right\\}$ and it follows
that $\partial
F(\alpha)=\delta_{3}+\frac{1}{N}\sum_{i=1}^{N}\partial\Psi_{\alpha}$. Finally,
we argue that $\alpha^{*}\geq 0$. For $\alpha>0$ sufficiently small,
$\exists\,z<-\delta_{3}$ such that $z\in\partial\Psi_{\alpha}$ and for
$\alpha>0$ sufficiently large, $\exists\,z>-\delta_{3}$ such that
$z\in\partial\Psi_{\alpha}$. To elaborate, for $\alpha>0$ sufficiently large,
$\|y^{cf}_{i}\|_{1}>0\implies l^{*}=\|y^{cf}\|_{1}\implies
K=0\implies\exists\,z>-\delta_{3}$ such that $z\in\partial\Psi_{\alpha}$. To
elaborate, for $\alpha>0$ sufficiently large, $\|y^{cf}_{i}\|_{1}=0\implies
l^{*}=0\implies K=0,\Psi_{\alpha}=0,0=z>-\delta_{3}$ such that
$z\in\partial\Psi_{\alpha}$. Hence we deduce $\partial F(\alpha)$ crosses the
origin ( as $\alpha$ sweeps from $0$ to $\infty$ ).

The primal problem P6 has solution
$\big{[}\alpha^{*}\delta_{3}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\alpha^{*}}(z_{i},y^{cf}_{i})\big{]}$
where $\alpha^{*}\in\,\left\\{\alpha\geq 0:0\in\partial F(\alpha)\right\\}$
and $\Psi_{\alpha^{*}}(z_{i},y^{cf}_{i})=\,\langle
z_{i},y^{cf}_{i}\rangle+\bigvee_{l=0}^{n}h_{\alpha^{*}}(l)\>\>for\>\>h_{\alpha^{*}}(l):=\big{[}\frac{l}{4\alpha^{*}}+\big{(}\sum_{k=1}^{l}z_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z_{ik}\big{)}-\alpha^{*}S_{3}K\big{]}$.
Expressed in terms of original FVA, this says

$\sup_{\Phi\in\mathcal{U}_{\delta_{3}}(\Phi_{N})}\mathbb{E}^{\Phi}[\langle
Z,Y_{CF}\rangle]=\mathbb{E}^{\Phi_{N}}[\langle
Z,Y_{CF}\rangle]+\alpha^{*}\delta_{3}+\mathbb{E}^{\Phi_{N}}\big{[}\bigvee_{l=0}^{n}\frac{l}{4\alpha^{*}}+\big{(}\sum_{k=1}^{l}Z_{k}-\sum_{k=1}^{\|Y_{CF}\|_{1}}Z_{k}\big{)}-\alpha^{*}S_{3}K\big{]}$

where the additional terms represent a penalty due to uncertainty in
probability distribution.  This follows directly from the previous two
propositions. $\delta_{3}=0$ reduces to original FVA.  This follows by direct
substitution of $\alpha^{*}$ as characterized in Proposition 2.3 into
Proposition 2.2 and then the dual problem D6.

####  2.4.3 Recovering the Worst Case Distribution

The process of recovering the worst case FVA distribution is similar to that
for CVA. In fact, for the FVA case, the procedure is a bit simpler since there
are less cases and subcases to consider to recover
$\\{x^{*}_{i},y^{cf*}_{i}:i\in\\{1,...,N+1\\}\\}$. The steps to recover the
dual minimizer $\alpha^{*}$ are the same. This procedure is done for a few
concrete examples in Section 3.

####  2.4.4 Discussion

The comments regarding incorporation of risk neutral measure constraint for
the robust CVA problem formulations apply for the robust FVA problem
formulations as well. Empirical results for the worst case FVA studies are
provided in Section 3. Similar to CVA, from the authors’ perspective the
computational study was illuminating to understand the magnitude and shape of
worst case FVA profiles as a function of uncertainty.

##  3 Computational Study: Robust XVA and Wrong Way Risk

This computational study uses the Matlab Financial Instruments Toolbox and
extends WWR portfolio analysis (brigo2013counterparty, section 5.3) to
consider uncertainty in probability distribution. Other key concepts that will
be discussed in this section include suitable choice for Wasserstein radius
$\delta$, calibration of scale factor $S_{3}$, and choice of units for
exposures. The studies in this section will investigate (and quantify) worst
case bilateral CVA and FVA for different market environments and portfolios of
interest rate swaps. For CVA, the current swaps market data (see below) will
be used in conjunction with Monte Carlo simulation of a market calibrated one
factor Hull-White model for interest rates. The counterparty credit curve
selection will vary between investment grade and high yield. For FVA, the
funding spreads and volatility data is taken from Markit. The swaps portfolios
are shown as well. All calculations are done in Matlab using the financial
instruments toolbox (Matlab19).

###  3.1 Market Data

As of April 20, 2020, the 5y par interest rate swap rate is 0.47% (on
Bloomberg). The full interest rate swaps curve is shown in Table 2. All market
data displayed below is for this date.

Table 2: Swap Rates Swap Tenor | 1y | 2y | 3y | 5y | 7y | 10y | 30y
---|---|---|---|---|---|---|---
Swap Rate | 0.515% | 0.409% | 0.401% | 0.470% | 0.569% | 0.691% | 0.855%

Bloomberg shows the interest rate swaption volatility matrix (with option
expirations as rows and swap tenors as columns).

Table 3: Swaption Normal Volatilities Exp / Tenor | 2y | 3y | 5y | 7y | 10y
---|---|---|---|---|---
2y | 0.520% | 0.542% | 0.601% | 0.631% | 0.680%
3y | 0.577% | 0.592% | 0.622% | 0.640% | 0.671%
5y | 0.637% | 0.637% | 0.637% | 0.643% | 0.652%
7y | 0.640% | 0.639% | 0.636% | 0.636% | 0.636%
10y | 0.639% | 0.633% | 0.624% | 0.618% | 0.612%

Furthermore, Markit shows U.S. CDX investment grade and high yield 5y credit
default swap spreads as in Table 4. The firm and counterparty investment grade
credit spreads are set to 100 and 150 basis points respectively. The high
yield credit spreads are shown in Table 5. Referencing Markit funding spreads,
the funding spread curves are shown in Table 6. Unavailable quotes for high
yield spreads are displayed as “N/A”. This term structure of funding spreads
is used for the FVA analysis. Funding spread lognormal volatility is set to
exponential decay. For investment grade it decays from 85% down to about 31%
in 10 years. For high yield it decays from 35% down to about 13% in 10 years.

Table 4: 5y CDS Spreads CDX Index | IG | HY
---|---|---
CDS Spread | 0.933% | 6.432%
Table 5: High Yield Counterparty Credit Spreads CDS Tenor | 1y | 2y | 3y | 4y | 5y | 6y | 7y | 8y | 9y | 10y
---|---|---|---|---|---|---|---|---|---|---
HY Spread | 6.00% | 5.75% | 5.50% | 5.25% | 5.00% | 4.75% | 4.50% | 4.25% | 4.00% | 3.75%
Table 6: Funding Spreads Funding Tenor | 1y | 2y | 3y | 5y | 7y | 10y
---|---|---|---|---|---|---
IG Spread | 0.54% | 0.81% | 0.81% | 0.88% | 1.01% | 1.14%
HY Spread | N/A | N/A | 8.02% | 6.72% | 7.08% | 6.36%

The swaps portfolios for the CVA and FVA studies are shown in Tables 7 and 8.
All 10 swaps are used for the 30y Monte Carlo simulation for CVA. The last 4
are capped at 10y maturity for FVA, as we have (only) 10y of funding market
data.

Table 7: CVA Swaps Portfolio Issued | Notional | Maturity | Rec / Pay Fixed | Coupon | Freq
---|---|---|---|---|---
4/20/20 | 10 | 4/20/21 | Rec | 0.51% | quarterly
4/20/20 | 10 | 4/20/22 | Pay | 0.41% | quarterly
4/20/20 | 10 | 4/20/23 | Pay | 0.40% | quarterly
4/20/20 | 10 | 4/20/25 | Rec | 0.47% | quarterly
4/20/20 | 10 | 4/20/27 | Pay | 0.57% | quarterly
4/20/20 | 10 | 4/20/30 | Rec | 0.69% | quarterly
4/20/20 | 10 | 4/20/35 | Rec | 0.74% | quarterly
4/20/20 | 10 | 4/20/40 | Rec | 0.83% | quarterly
4/20/20 | 10 | 4/20/45 | Pay | 0.83% | quarterly
4/20/20 | 10 | 4/20/50 | Pay | 0.85% | quarterly
Table 8: FVA Swaps Portfolio Issued | Notional | Maturity | Rec / Pay Fixed | Coupon | Freq
---|---|---|---|---|---
4/20/20 | 100 | 4/20/21 | Pay | 0.51% | quarterly
4/20/20 | 100 | 4/20/22 | Rec | 0.41% | quarterly
4/20/20 | 100 | 4/20/23 | Rec | 0.40% | quarterly
4/20/20 | 100 | 4/20/25 | Pay | 0.47% | quarterly
4/20/20 | 100 | 4/20/27 | Rec | 0.57% | quarterly
4/20/20 | 100 | 4/20/30 | Pay | 0.69% | quarterly
4/20/20 | 100 | 4/20/30 | Pay | 0.74% | quarterly
4/20/20 | 100 | 4/20/30 | Pay | 0.83% | quarterly
4/20/20 | 100 | 4/20/30 | Rec | 0.83% | quarterly
4/20/20 | 100 | 4/20/30 | Rec | 0.85% | quarterly

###  3.2 Suitable Choice for Wasserstein Radius

A natural question to ask when computing worst case XVA is how to interpret
the size of the Wasserstein radius $\delta$. A discussion of some key results
is given in (Carlsson2018, Section 3). For this study, we adopt a fairly
straightforward approach to compute upper and lower bounds for the expected
Wasserstein distance between the empirical and true distributions. A rough
procedure for selecting $\delta$ involves sampling two independent data sets
$D_{1}\text{\,and\,}D_{2}$, and setting $\delta=\alpha c^{*}$ where
$\alpha\in[1/2,1]$ and $c^{*}$ denotes the cost of the minimum bipartite
matching between $D_{1}\text{\,and\,}D_{2}$ (Carlsson2018), (radius12). This
approach relies on the following theorem referenced in Carlsson2018 and
established in radius12.

######  Theorem.

Let $\hat{f}_{1}$ and $\hat{f}_{2}$ denote empirical distributions associated
with two sets of independent samples of $n$ points from a distribution $f$.
Then

$\frac{1}{2}\mathbb{E}\;[D_{c}(\hat{f}_{1},\hat{f}_{2})]\leq\mathbb{E}\;[D_{c}(f,\hat{f}_{1})]\leq\mathbb{E}\;[D_{c}(\hat{f}_{1},\hat{f}_{2})].$

As such, our approach is to sample two indepedent data sets $D_{1}$ and
$D_{2}$ of portfolio exposures and default times and compute lower and upper
bounds $\delta^{l}:=\frac{1}{2}\mathbb{E}\;[D_{c}(\hat{f}_{1},\hat{f}_{2})]$
and $\delta^{u}:=\mathbb{E}\;[D_{c}(\hat{f}_{1},\hat{f}_{2})]$ for the
expected Wasserstein distance between the empirical and true distributions.
Given these bounds, one can compute the corresponding lower and upper bounds
on the worst case XVA risk metrics and exposure and default time
distributions.

Constructing the bounds $\delta^{l}$ and $\delta^{u}$ in this way builds in a
dependency on the units of portfolio exposures (e.g. millions of dollars) and
units in the time dimension (e.g. years), through the computation of
$D_{c}(\hat{f}_{1},\hat{f}_{2})$ and the calibration of the scale factor
$S_{3}$ (see Section 3.3 below for this). Such a dependency is desirable to
assign “units” to $\delta$ as well as to conduct relative value analysis
across portfolios. See Section 3.4 below for more commentary on choice of
units for exposures.

###  3.3 Calibration of Scale Factors

####  3.3.1 Calibration of $S_{3}$ for CVA

The scale factor $S_{3}$ represents a scaling for changes to default times. A
suitable choice for $S_{3}$ is one that charges an appropriate cost for this.
Let us think about what a change in default time means in the context of CVA.
For a fixed path with index $i$, and exposure vector $x^{\pm}_{i}$, changing
the default time from $\tau_{2}$ to $\tau_{1}$ changes the value of the
realized exposure from $x^{\pm}_{i\tau_{2}}$ to $x^{\pm}_{i\tau_{1}}$ upon
default. A reasonable value for $S_{3}$, call it $s_{3}$, for this particular
path, might be $\|{x}^{\pm}_{i\tau_{1}}-{x}^{\pm}_{i\tau_{2}}\|_{\infty}$
where $\tau_{1},\tau_{2}\in\\{1,...,n\\}$. Now let us generalize this to
average over all paths $i\in\\{1,...,N\\}$ in our empirical distribution
$\Phi_{N}$. Let $\overline{{x}^{\pm}_{\tau}}$ denote
$\frac{1}{N}\sum_{i=1}^{N}x^{\pm}_{i\tau}$, the average exposure at default
time $\tau$. Substituting average exposures into our previous expression gives
the relation
$S_{3}:=\|\overline{{x}^{\pm}_{\tau_{1}}}-\overline{{x}^{\pm}_{\tau_{2}}}\|_{\infty}$.
Let us use this as our working definition for $S_{3}$ for unilateral CVA, DVA.
Calibration is straightforward given $\Phi_{N}$, the set of sample paths
$\\{(x_{i},y^{c}_{i},y^{f}_{i}):i\in\\{1,...,N\\}\\}$. For bilateral CVA, take
the average over the unilateral CVA and DVA scale factors, namely
$S_{3}:=\frac{1}{2}(\|\overline{{x}^{+}_{\tau_{1}}}-\overline{{x}^{+}_{\tau_{2}}}\|_{\infty}+\|\overline{{x}^{-}_{\tau_{1}}}-\overline{{x}^{-}_{\tau_{2}}}\|_{\infty})$.

####  3.3.2 Calibration of $S_{3}$ for FVA

Let us the follow the approach above for FVA. For a fixed path with index $i$,
the funding exposure vector is $z^{\pm}_{i}$ and the incremental change is
$\Delta z^{\pm}_{i\tau_{2}}$. A reasonable value for $S_{3}$, call it $s_{3}$,
for this particular path, might be
$\|{z}^{\pm}_{i\tau_{1}}-{z}^{\pm}_{i\tau_{2}}\|_{\infty}$. Substituting
average exposures into this expression gives the relation
$S_{3}:=\|\overline{{z}^{\pm}_{\tau_{1}}}-\overline{{z}^{\pm}_{\tau_{2}}}\|_{\infty}$.
Let us use this as our working definition for $S_{3}$ for FCA, FBA.
Calibration is straightforward given $\Phi_{N}$, the set of sample paths
$\\{(z_{i},y^{cf}_{i}):i\in\\{1,...,N\\}\\}$. For FVA, take the average over
the FCA and FBA scale factors, namely
$S_{3}:=\frac{1}{2}(\|\overline{{z}^{+}_{\tau_{1}}}-\overline{{z}^{+}_{\tau_{2}}}\|_{\infty}+\|\overline{{z}^{-}_{\tau_{1}}}-\overline{{z}^{-}_{\tau_{2}}}\|_{\infty})$.

###  3.4 Choice of Units for Exposures

Standardizing the units across portfolios is useful for relative value
analysis. The choice of units for exposures (e.g. millions of dollars) and
default times (e.g. decimal years) is up to the user, although we recommend
these conventions, and use them in our analysis in this section. Note that
different choices of units will lead to calibrated different values for
$S_{3}$ for BCVA and FVA. There is no one choice for units (as in regression
analysis, for example) although consistency is recommended as a good practice.
The same comments apply for the choices of time frequency and time horizon for
the robust XVA analysis.

###  3.5 Definitions for Exposure Calculations

The definitions for the various exposure calculations plotted in Section 3.6
for CVA and DVA are given in Table 9 below. For FVA calculations (FCA and
FBA), plotted in Section 3.7, replace portfolio exposures $V^{+}$ and $V^{-}$
with funding exposures $Z^{+}$ and $Z^{-}$ respectively.

Table 9: CVA Exposure Calculations Term |  CVAU |  DVAU
---|---|---
$\text{EE}(t)$ | $\mathbb{E}[V^{+}(t)]$ | $\mathbb{E}[V^{-}(t)]$
$\text{PFE}_{\alpha}(t)$ | $\inf\;\\{x\in\mathbb{R}:\alpha\leq F_{V^{+}(t)}(x)\\}$ | $\inf\;\\{x\in\mathbb{R}:\alpha\leq F_{V^{-}(t)}(x)\\}$
$\text{EPE}(t)$ | $\frac{1}{T}\int_{0}^{T}\text{EE}(t)dt$ | $\frac{1}{T}\int_{0}^{T}\text{EE}(t)dt$
$\text{EffEE}(t)$ | $\max\;\\{\text{EE}(\tau):\tau\in[0,t]\\}$ | $\max\;\\{\text{EE}(\tau):\tau\in[0,t]\\}$
$\text{EffEPE}(t)$ | $\frac{1}{T}\int_{0}^{T}\text{EffEE}(t)dt$ | $\frac{1}{T}\int_{0}^{T}\text{EffEE}(t)dt$

###  3.6 Bilateral CVA

####  3.6.1 Investment Grade Counterparty and Firm

The swaps portfolio shown in Table 7 is used for this analysis. The portfolio
consists of ten par coupon interest rate swaps, with a mix of receving fixed
and paying fixed swaps at different maturities. The investment grade firm and
counterparty credit spreads are set to 100 and 150 basis points respectively.
The calibrated value of $S_{3}$ is 1.4584 which results in $\delta^{l}=14.414$
and $\delta^{u}=28.828$ using a second set of Bloomberg market data (for
03/20/20) along with the first set for 04/20/20. The full range of Wasserstein
radii $\delta$ is given in Table 10.

Table 10: BCVA Wasserstein Radii Percentage of $\delta^{u}$ | 50% | 60% | 70% | 80% | 90% | 100%
---|---|---|---|---|---|---
W Radius delta | 14.41 | 17.30 | 20.18 | 23.06 | 25.95 | 28.83

Matlab plots characterizing the BCVA positive and negative exposure profiles
and trajectory of worst case BCVA as a function of Wasserstein radius are
shown in Figures 1,2,3.

Figure 1: Swaps Portfolio Positive Exposure Profiles

Figure 2: Swaps Portfolio Negative Exposure Profiles

Figure 3: Swaps Portfolio Worst Case BCVA Profile

The baseline BCVA for this portfolio is approximately 160k USD and represents
the dot product of the discounted positive portfolio exposure profile times
counterparty default probability plus dot product of the discounted negative
portfolio exposure times firm default probability. The worst case BCVA curve
is shown in Figure 3. The worst case CVA curve ranges from 69% to 93% the size
of Max PFE (Potential Future Exposure) which is equal to 6.43mm USD (see
Figure 1), for Wasserstein radii $\delta$ given in Table 10. So the takeaway
here is worst case BCVA can be a significant percentage of PFE for swap
portfolios with low risk counterparty default curves (investment grade).

The worst case distribution for $\delta^{u}$ is shown in Figures 4 and 5. The
first plot shows the exposures $\\{x^{*}_{i}\\}$ and the second plot shows the
joint distribution of counterparty and firm default times
$\\{y^{c*}_{i},y^{f*}_{i}\\}$. Default times beyond the portfolio maturity
date denote no default prior to portfolio maturity for those simulation paths.
This results in higher contours in the back row.

Figure 4: Worst Case Exposures

Figure 5: Worst Case Default Times

####  3.6.2 High Yield Counterparty and Investment Grade Firm

The swaps portfolio shown in Table 7 is used for this analysis. The portfolio
consists of ten par coupon interest rate swaps, with a mix of receving fixed
and paying fixed swaps at different maturities. The high yield counterparty
credit spreads are set as in Table 5. The investment grade firm credit spreads
are set to a constant 100 basis points. The calibrated value of $S_{3}$ is
1.4584 which results in $\delta^{l}=14.45$ and $\delta^{u}=28.90$ using a
second set of Bloomberg market data (for 03/20/20) along with the first set
for 04/20/20. The full range of Wasserstein radii $\delta$ is given in Table
11.

Table 11: BCVA Wasserstein Radii Percentage of $\delta^{u}$ | 50% | 60% | 70% | 80% | 90% | 100%
---|---|---|---|---|---|---
W Radius delta | 14.45 | 17.34 | 20.23 | 23.12 | 26.01 | 28.90

Matlab plots characterizing the BCVA positive and negative exposure profiles
and trajectory of worst case BCVA as a function of Wasserstein radius are
shown in Figures 6,7,8.

Figure 6: Swaps Portfolio Positive Exposure Profiles

Figure 7: Swaps Portfolio Negative Exposure Profiles

Figure 8: Swaps Portfolio Worst Case BCVA Profile

The baseline BCVA for this portfolio is approximately 106k USD and represents
the dot product of the discounted positive portfolio exposure profile times
counterparty default probability plus dot product of the discounted negative
portfolio exposure times firm default probability. The worst case BCVA curve
is shown in Figure 8. Note that for this problem instance, the worst case BCVA
results for high yield counterparty credit are similar to the previous
subsection, for investment grade counterparty credit. Note the worst case BCVA
ranges from 70% to 95% the size of Max PFE (Potential Future Exposure), which
is equal to 6.43mm USD (see Figure 6), for Wasserstein radii $\delta$ given in
Table 11. So the takeaway here is worst case BCVA can be a significant
percentage of PFE for swap portfolios with high yield counterparty default
curves as well.

Figure 9: Worst Case Exposures

Figure 10: Worst Case Default Times

The worst case distribution for delta value $\delta^{u}$ is shown in Figures 9
and 10. The first plot shows the exposures $\\{x^{*}_{i}\\}$ and the second
plot shows the joint distribution of counterparty and firm default times
$\\{y^{c*}_{i},y^{f*}_{i}\\}$. Default times beyond the portfolio maturity
date denote no default prior to portfolio maturity for those simulation paths.
This results in higher contours in the joint density plot in the back row.
Higher counterparty credit spreads lead to earlier counterparty default times.

###  3.7 FVA

####  3.7.1 Investment Grade Counterparty and Firm

The swaps portfolio shown in Table 8 is used for this analysis. The portfolio
consists of ten interest rate swaps, with a mix of receving fixed and paying
fixed swaps at different maturities. Capping maturities at 10y introduces some
positive NPV to this portfolio. The investment grade firm and counterparty
funding spreads are set as shown in Table 6. The calibrated value of $S_{3}$
is 0.082 which results in $\delta^{l}=0.387$ and $\delta^{u}=0.774$ using a
second set of Bloomberg market data (for 03/20/20) along with the first set
for 04/20/20. The full range of Wasserstein radii $\delta$ is given in Table
12.

Table 12: FVA Wasserstein Radii Percentage of $\delta^{u}$ | 0.50 | 0.60 | 0.70 | 0.80 | 0.90 | 1.0
---|---|---|---|---|---|---
W Radius delta | 0.387 | 0.464 | 0.542 | 0.560 | 0.697 | 0.774

Matlab plots characterizing the FVA positive and negative exposure profiles
and trajectory of worst case FVA as a function of Wasserstein radius are shown
in Figures 13,14,15.

Figure 11: Swaps Portfolio Positive Exposure Profiles

Figure 12: Swaps Portfolio Negative Exposure Profiles

Figure 13: Swaps Portfolio IG FCA Exposure Profiles

Figure 14: Swaps Portfolio IG FBA Exposure Profiles

Figure 15: Swaps Portfolio Worst Case IG FVA Profile

The baseline FVA for this portfolio is 240k USD and represents the dot product
of the discounted portfolio FCA exposure profile times joint survival
probability plus dot product of the discounted portfolio FBA exposure times
joint survival probability. The worst case FVA curve is shown in Figure 15.
For Wasserstein radius $\delta^{l}=0.387$, the worst case FVA is approximately
4.55, or 3.0 times the size of integrated FCA PFE which is about 1.54. For
Wasserstein radius $\delta^{u}=0.774$, the worst case FVA is approximately
5.72, or 3.7 times the size of integrated FCA PFE. In this problem instance,
worst case FVA is a multiple of integrated FCA PFE exposure, so quite
significant.

Figure 16: Worst Case Exposures

Figure 17: Worst Case Survival Times

The worst case distribution for delta value $\delta^{u}$ is shown in Figures
16 and 17. The first plot shows the exposures $\\{z^{*}_{i}\\}$ and the second
plot shows the joint distribution of counterparty and firm survival times
$\\{y^{cf*}_{i}\\}$. Survival times beyond the portfolio maturity date denote
no defaults prior to portfolio maturity for those simulation paths.

####  3.7.2 High Yield Counterparty and Investment Grade Firm

The swaps portfolio shown in Table 8 is used for this analysis. The portfolio
consists of ten par coupon interest rate swaps, with a mix of receving fixed
and paying fixed swaps at different maturities. The high yield firm and
counterparty funding spreads are set as shown in Table 6. The high yield
counterparty credit spreads are set as shown in Table 5. The investment grade
firm credit spreads are set to a constant 100 basis points. The calibrated
value of $S_{3}$ is 0.2898 which results in $\delta^{l}=1.935$ and
$\delta^{u}=3.87$ using a second set of Bloomberg market data (for 03/20/20)
along with the first set for 04/20/20. The full range of Wasserstein radii
$\delta$ is given in Table 13. Matlab plots characterizing the FVA positive
and negative exposure profiles and trajectory of worst case FVA as a function
of Wasserstein radius are shown in Figures 20,21,22.

Table 13: FVA Wasserstein Radii Percentage of $\delta^{u}$ | 0.50 | 0.60 | 0.70 | 0.80 | 0.90 | 1.0
---|---|---|---|---|---|---
W Radius delta | 1.935 | 2.322 | 2.709 | 3.096 | 3.483 | 3.87
Figure 18: Swaps Portfolio Positive Exposure Profiles

Figure 19: Swaps Portfolio Negative Exposure Profiles

Figure 20: Swaps Portfolio HY FCA Exposure Profiles

Figure 21: Swaps Portfolio HY FBA Exposure Profiles

Figure 22: Swaps Portfolio Worst Case HY FVA Profile

The baseline FVA for this portfolio is 1.18mm USD and represents the dot
product of the discounted portfolio FCA exposure profile times joint survival
probability plus dot product of the discounted portfolio FBA exposure times
joint survival probability. The worst case FVA curve is shown in Figure 22.
For Wasserstein radius $\delta^{l}=1.935$, the worst case FVA is approximately
9.35, or 1.31 times the size of integrated FCA PFE which is about 7.127. For
Wasserstein radius $\delta^{u}=3.87$, the worst case FVA is approximately
12.44, or 1.75 times the size of integrated FCA PFE. In this problem instance,
similar to the investment grade example, worst case FVA is a multiple of
integrated FCA PFE exposure, so quite significant.

Figure 23: Worst Case Exposures

Figure 24: Worst Case Survival Times

The worst case distribution for delta value $\delta^{u}$ is shown in Figures
23 and 24. The first plot shows the exposures $\\{z^{*}_{i}\\}$ and the second
plot shows the joint distribution of counterparty and firm survival times
$\\{y^{cf*}_{i}\\}$. Survival times beyond the portfolio maturity date denote
no defaults prior to portfolio maturity for those simulation paths.

##  4 Conclusions and Further Work

This work has developed theoretical results and investigated calculations of
robust CVA, FVA, and wrong way risk for OTC derivatives under distributional
uncerainty using Wasserstein distance as an ambiguity measure. The financial
market overview, foundational notation, and robust XVA primal problems were
introduced in Section 1. Using recent infinite dimensional Lagrangian duality
results (blanchetFirst), the simpler dual formulations and their analytic
solutions for BCVA and FVA were derived in Section 2. After that, in Section
3, some computational experiments were conducted to measure the additional XVA
charge due to distributional uncertainty for a variety of portfolio and market
configurations. Worst case BCVA and FVA were found to be significant relative
to their respective PFE profiles in all problem instances. Finally, we
conclude with some commentary on directions for further research.

One direction for future research, as has been previously discussed, is to
extend the problem formulations to include a risk neutral measure constraint
in a solvable way. Explicitly adding the constraint would complicate the
problem formulations no doubt, so perhaps there is a more tractable indirect
approach. Another direction for future research would be to develop (and
apply) similar theoretical machinery as used for robust CVA and FVA towards
robust KVA (Capital Valuation Adjustment) and MVA (Margin Valuation
Adjustment) and wrong way risk in that context. Intuitively, wrong way risk
arises in that context when the market cost of capital and/or funding the
margin position increases at the same time as the portfolio exposure
increases.

## Data Availability Statement

The raw and/or processed data required to reproduce the findings from this
research can be obtained from the corresponding author, [D.S.], upon
reasonable request.

## Conflict of Interest Statement

The authors declare they have no conflict of interest.

## Funding Statement

The authors received no specific funding for this work.

## References

  * Bartl et al., (2017) Bartl, D., Drapeau, S., and Tangpi, L. (2017).  Computational aspects of robust optimized certainty equivalents and option pricing.
  * Basel Committee on Banking Supervision, (2015) Basel Committee on Banking Supervision (2015).  Review of the credit valuation adjustment risk framework.
  * Blanchet et al., (2018) Blanchet, J., Chen, L., and Zhou, X. Y. (2018).  Distributionally robust mean-variance portfolio selection with wasserstein distances.
  * (4) Blanchet, J., Kang, Y., and Murthy, K. (2016a).  Robust wasserstein profile inference and applications to machine learning.  arXiv preprint arXiv:1610.05627.
  * (5) Blanchet, J., Kang, Y., and Murthy, K. (2016b).  Robust wasserstein profile inference and applications to machine learning.  arXiv preprint arXiv:1610.05627.
  * Blanchet and Murthy, (2019) Blanchet, J. and Murthy, K. (2019).  Quantifying distributional model risk via optimal transport.  Mathematics of Operations Research, 44(2):565–600.
  * Brigo et al., (2013) Brigo, D., Morini, M., and Pallavicini, A. (2013).  Counterparty credit risk, collateral and funding: with pricing cases for all asset classes, volume 478.  John Wiley & Sons.
  * Canas and Rosasco, (2012) Canas, G. and Rosasco, L. (2012).  Learning probability measures with respect to optimal transport metrics.  In Advances in Neural Information Processing Systems, pages 2492–2500.
  * Carlsson et al., (2018) Carlsson, J. G., Behroozi, M., and Mihic, K. (2018).  Wasserstein distance and the distributionally robust tsp.  Operations Research, 66(6):1603–1624.
  * El Hajjaji and Subbotin, (2015) El Hajjaji, O. and Subbotin, A. (2015).  Cva with wrong way risk: Sensitivities, volatility and hedging.  International Journal of Theoretical and Applied Finance, 18(03):1550017.
  * Esfahani and Kuhn, (2018) Esfahani, P. M. and Kuhn, D. (2018).  Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations.  Mathematical Programming, 171(1-2):115–166.
  * Gao et al., (2017) Gao, R., Chen, X., and Kleywegt, A. J. (2017).  Wasserstein distributional robustness and regularization in statistical learning.  arXiv preprint arXiv:1712.06050.
  * Gao and Kleywegt, (2016) Gao, R. and Kleywegt, A. J. (2016).  Distributionally robust stochastic optimization with wasserstein distance.  arXiv preprint arXiv:1604.02199.
  * Glasserman and Yang, (2015) Glasserman, P. and Yang, L. (2015).  Bounding wrong-way risk in measuring counterparty risk.  Office of Financial Research Working Paper, (15-16):15–76.
  * Green, (2015) Green, A. (2015).  XVA: Credit, Funding and Capital Valuation Adjustments.  John Wiley & Sons.
  * Lichters et al., (2015) Lichters, R., Stamm, R., and Gallagher, D. (2015).  Modern derivatives pricing and credit exposure analysis: theory and practice of CSA and XVA pricing, exposure simulation and backtesting.  Springer.
  * Matlab, (2019) Matlab (2019).  Matlab, counterparty credit risk and cva.  https://www.mathworks.com/help/fininst/counterparty-credit-risk-and-cva.htm.  Accessed: 2019-07-30.
  * Memartoluie, (2017) Memartoluie, A. (2017).  Computational methods in finance related to distributions with known marginals.
  * Office of the Comptroller of the Currency, (2011) Office of the Comptroller of the Currency (2011).  Interagency supervisory guidance on counterparty credit risk management.  Bulletin 2011-30a.
  * Ramzi Ben-Abdallah and Marzouk, (2019) Ramzi Ben-Abdallah, M. B. and Marzouk, O. (2019).  Wrong-way risk of interest rate instruments.  Journal of Credit Risk, pages 21–44.
  * Singh and Zhang, (2020) Singh, D. and Zhang, S. (2020).  Robust arbitrage conditions for financial markets.  arXiv preprint arXiv:2004.09432.