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

# Distributionally Robust XVA via Wasserstein Distance
Part 2: Wrong Way 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 funding valuation adjustment
(FVA) for over the counter (OTC) derivatives under distributional uncertainty
using Wasserstein distance as the ambiguity measure. Wrong way funding risk
can be characterized via the robust FVA formulation. The simpler dual
formulation of the robust FVA optimization is derived. Next, some
computational experiments are conducted to measure the additional FVA charge
due to distributional uncertainty under a variety of portfolio and market
configurations. Finally some suggestions for future work, such as robust
capital valuation adjustment (KVA) and margin valuation adjustment (MVA), are
discussed.

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

##  1 Introduction and Overview

###  1.1 Financial Markets Context and Background

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.

As mentioned in Part 1 (Singh and Zhang, , 2019), U.S. regulatory authorities,
the Federal Reserve and Office of the Comptroller of the Currency (OCC),
periodically assess national banks’ compliance with the Market Risk Capital
Rule (MRR). Counterparty credit risk (CCR) and funding risk (FR) metrics are
key metrics used to evaluate bank risk profiles due to OTC derivatives. Basel
Committee on Banking Supervision, through Basel III, has developed criteria to
quantify capital charges due to CCR. According to International Swap Dealers
Association (ISDA) the current OTC derivatives notional outstanding is over
500 trillion. 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 FVA and wrong way risk (WWR) and the impact of uncertainty in
probability distribution on FR and FVA. It is these considerations that
motivate this line of research. (Ramzi Ben-Abdallah and Marzouk, , 2019), (El
Hajjaji and Subbotin, , 2015)

An outline of this paper is as follows. Section 1 represents an introduction
and overview of FVA and wrong way funding risk. 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
instruments, 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 research has been done to investigate 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 measure the effect of dependency structure
(and hence wrong way risk) on CVA for a variety of asset classes. Glasserman
and Yang,  (2015) bound the effect of wrong way risk on CVA. Their approach
considers a discrete setting and formulates worst case CVA as the solution to
a worst case linear program subject to certain constraints, where the
dependency structure is allowed to vary. They introduce a penalty term,
measured via Kullback-Leibler (KL) divergence, to control the degree of wrong
way risk. Memartoluie, in his PhD thesis, uses an ordered scenario copula
methodology to quantify worst case CVA (Memartoluie, , 2017). For worst case
correlations set to one, he finds results that are comparable to the method of
Glasserman and Yang.

Lagrangian duality results were recently (and independently) developed by
Blanchet and Murthy,  (2019) and Gao and Kleywegt,  (2016). These results hold
under mild assumptions. The main innovation in our work is to apply these
recent results to worst case FVA using Wasserstein distance as the ambiguity
measure. Furthermore, analytical expressions are derived for the solutions to
the inner and outer convex optimization problems. A computational study shows
the material impact of distributional uncertainty on worst case FVA and
illustrates the risk profile.

###  1.3 Notation and Definitions

####  1.3.1 FCA

Notation and core definitions for FCA problem setup follow conventions in
Glasserman and Yang,  (2015) and Lichters et al.,  (2015). Those for the
robust FCA problem formulation follow conventions in Blanchet et al.,  (2018).
FCA measures expected funding cost over the lifetime of the portfolio. Let
$V^{+}(t)$ denote the positive 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 $Y_{C}$ and $Y_{F}$ denote the vectors of counterparty
and 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 $X^{+}$. Let $(z^{+}_{i},y^{cf}_{i})$ denote
realizations of $(Z^{+},Y_{CF})$ along sample paths for
$i=\\{1,2,\ldots,N\\}$.

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

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

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

$Z^{+}=(f_{c}(t_{0},t_{1})X^{+}(t_{1}),\ldots,f_{c}(t_{n-1},t_{n})X^{+}(t_{n}))\quad\text{and}\quad
Y_{CF}=(\mathbbm{1}_{\\{\tau_{C}>t_{1}\\}\cap\\{\tau_{F}>t_{1}\\}},\dots,\mathbbm{1}_{\\{\tau_{C}>t_{n}\\}\cap\\{\tau_{F}>t_{n}\\}}).$

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, $P_{N}$, is

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

Under the empirical measure, $P_{N}$, FCA is an expectation of an inner
product

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

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

$\mathcal{U}_{\delta_{1}}(P_{N})=\\{P:D_{c}(P,P_{N})\leq\delta_{1}\\}$

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

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

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 $c_{S_{1}}$ where

$c_{S_{1}}((u,v),(z,y))=S_{1}\langle v-y,v-y\rangle+\langle u-z,u-z\rangle.$

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

####  1.3.2 FBA

Notation and core definitions for FBA problem setup follow conventions in
Glasserman and Yang,  (2015) and Lichters et al.,  (2015). Those for the
robust FBA problem formulation follow conventions in Blanchet et al.,  (2018).
FBA measures expected funding benefit over the lifetime of the portfolio. 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
negative exposures and $Y_{C}$ and $Y_{F}$ denote the vectors of counterparty
and 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
benfits incurred on exposures $X^{-}$. Let $(z^{-}_{i},y^{cf}_{i})$ denote
realizations of $(Z^{-},Y_{CF})$ along sample paths for
$i=\\{1,2,\ldots,N\\}$.

The FBA associated with funding benefits $Z^{-}$ and joint survival indicator
$\mathbbm{1}_{\\{\tau_{C}>t\\}\cap\\{\tau_{F}>t\\}}$ is (Lichters et al., ,
2015), (Green, , 2015)

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

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

$Z^{-}=(f_{b}(t_{0},t_{1})X^{-}(t_{1}),\ldots,f_{b}(t_{n-1},t_{n})X^{-}(t_{n}))\quad\text{and}\quad
Y_{CF}=(\mathbbm{1}_{\\{\tau_{C}>t_{1}\\}\cap\\{\tau_{F}>t_{1}\\}},\dots,\mathbbm{1}_{\\{\tau_{C}>t_{n}\\}\cap\\{\tau_{F}>t_{n}\\}}).$

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, $Q_{N}$, can be written
as

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

Under the empirical measure, $Q_{N}$, FBA is an expectation of an inner
product

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

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

$\mathcal{U}_{\delta_{2}}(Q_{N})=\\{Q:D_{c}(Q,Q_{N})\leq\delta_{2}\\}$

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

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

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 $c_{S_{2}}$ where

$c_{S_{2}}((u,v),(x,y))=S_{2}\langle v-y,v-y\rangle+\langle u-z,u-z\rangle.$

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

####  1.3.3 FVA

Notation and core definitions for (bilateral) FVA problem setup incorporate
those above 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

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

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 FVA and Wrong Way Funding Risk

###  2.1 FCA

####  2.1.1 Inner Optimization Problem

The robust FCA can be written as

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

Now 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

$\inf_{\gamma\geq
0}\>H(\gamma):=\bigg{[}\gamma\delta_{1}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\gamma}(z^{+}_{i},y^{cf}_{i})\bigg{]}$
(D1)

where

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

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

$\Psi_{\gamma}(z^{+}_{i},y^{cf}_{i})=\sup_{w_{1}\geq-z^{+}_{i},w_{2}\in{B^{2}_{n}}}[\langle
w_{1}+z^{+}_{i},w_{2}+y^{cf}_{i}\rangle-\gamma(\langle
w_{1},w_{1}\rangle+S_{1}\langle w_{2},w_{2}\rangle)]$

where $B^{2}_{n}$ denotes the set of ternary vectors of ones, zeros, and minus
ones with $n$ components, and at most one block of ones or minus ones. Note
that $\sup_{w_{1}}[\,\,;\,]$ is attained for $w_{1}^{*}\in\mathbb{R}^{n}_{+}$
(as will become evident in the proof) hence it suffices to consider this space
for $w_{1}$. It turns out that $\Psi_{\gamma}$ can be expressed as original
FCA 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 a hyperbola
plus a line of negative slope. $\Psi_{\gamma}$ quantifies the adversarial move
in FCA across both time and spatial dimensions while accounting for the cost
via the $K$ terms.

Let $\Psi_{\gamma}(z^{+}_{i},y^{cf}_{i})=\,\langle
z^{+}_{i},y^{cf}_{i}\rangle+\big{[}\frac{l^{*}}{4\gamma}+(\sum_{k=1}^{l^{*}}z^{+}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{+}_{ik})-\gamma
S_{1}K\big{]}$
where $l^{*}=\operatorname*{arg\,max}_{l\geq
0}[\frac{l}{4\gamma}+\sum_{k=1}^{l}z^{+}_{ik}-\gamma S_{1}K]$ 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, $\Psi_{\gamma}(z^{+}_{i},y^{cf}_{i})=\,\langle
z^{+}_{i},y^{cf}_{i}\rangle+\bigvee_{l=0}^{n}h_{\gamma}(l)$ for
$h_{\gamma}(l):=\big{[}\frac{l}{4\gamma}+(\sum_{k=1}^{l}z^{+}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{+}_{ik})-\gamma
S_{1}K\big{]}$. Finally, note $\bigvee_{l=0}^{n}h_{\gamma}(l)$ denotes
$\max_{l\in\\{0,\dots,n\\}}h_{\gamma}(l)$.

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

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

Applying the Cauchy-Schwarz Inequality gives

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

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

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

$\mathbf{Case\,2}$ 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

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

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

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

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

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

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

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

Therefore one can write

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

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

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

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

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

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

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

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

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

and write

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

Alternatively, one can write

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

.

####  2.1.2 Outer Optimization Problem

The goal now is to evaluate

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

where

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

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

Let $\gamma^{*}\in\,\left\\{\gamma\geq 0:0\in\partial H(\gamma)\right\\}\\\
where\,\,\partial\Psi_{\gamma}=\mathbf{Conv}\cup\\{\partial
h_{\gamma}(l)\>|\>\langle
z^{+}_{i},y^{cf}_{i}\rangle+h_{\gamma}(l)=\Psi_{\gamma};\,l\in\\{0,\dots,n\\}\\}$
and $\partial
H(\gamma)=\delta_{1}+\frac{1}{N}\sum_{i=1}^{N}\partial\Psi_{\gamma}$.  This
follows from application of standard properties of subgradients as well as
inspection of the asymptotic properties of $\Psi_{\gamma}$ and
$\partial\Psi_{\gamma}$. For $\gamma$ sufficiently small, $\Psi_{\gamma}$ has
a large positive value and $\partial\Psi_{\gamma}$ has a large negative
derivative. For $\gamma$ sufficiently large, for optimal $l^{*}$, either
$l^{*}=0\implies 0\in\partial\Psi_{\gamma}$ or
$l^{*}=\|y^{cf}_{i}\|_{1}>0\implies\partial\Psi_{\gamma}$ approaches zero
$\implies\partial H(\gamma)$ crosses zero.

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

Putting together the results of these two propositions, we arrive at our first
theorem.

The primal problem P1 has solution
$\big{[}\gamma^{*}\delta_{1}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\gamma^{*}}(z^{+}_{i},y^{cf}_{i})\big{]}$
where $\gamma^{*}\in\,\left\\{\gamma\geq 0:0\in\partial H(\gamma)\right\\}$
and $\Psi_{\gamma^{*}}(z^{+}_{i},y^{cf}_{i})=\,\langle
z^{+}_{i},y^{cf}_{i}\rangle+\bigvee_{l=0}^{n}h_{\gamma^{*}}(l)\>\>\text{for}\>\>h_{\gamma^{*}}(l):=\big{[}\frac{l}{4\gamma^{*}}+(\sum_{k=1}^{l}z^{+}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{+}_{ik})-\gamma^{*}S_{1}K\big{]}$.
Expressed in terms of original FCA, this says

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

where the additional terms represent a penalty due to uncertainty in
probability distribution.  This follows directly from the previous two
propositions.  This follows by direct substitution of $\gamma^{*}$ as
characterized in Proposition 2.2 into Proposition 2.1 and then the dual
problem D1.

###  2.2 FBA

####  2.2.1 Inner Optimization Problem

The robust FBA can be written as

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

Now 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

$\inf_{\beta\geq
0}\>G(\beta):=\bigg{[}\beta\delta_{2}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})\bigg{]}$
(D2)

where

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\sup_{u\in\mathbb{R}^{n}_{-},v\in{B^{1}_{n}}}[\langle
u,v\rangle-\beta
c_{S_{2}}((u,v),(z^{-}_{i},y^{cf}_{i}))]=\sup_{u\in\mathbb{R}^{n}_{-},v\in{B^{1}_{n}}}[\langle
u,v\rangle-\beta(\langle u-z^{-}_{i},u-z^{-}_{i}\rangle+S_{2}\langle
v-y^{cf}_{i},v-y^{cf}_{i}\rangle)].$

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

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

where sets $B^{1}_{n}$ and $B^{2}_{n}$ are defined as before. Following a
similar approach as for FCA, it turns out that $\Psi_{\beta}$ can be expressed
as original FBA 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 a (convex) piecewise line then hyperbola function plus a line of negative
slope. $\Psi_{\beta}$ quantifies the adversarial move in FBA across both time
and spatial dimensions while accounting for the cost via the $K$ terms.

We have $\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+h_{i}(\beta)$
where
$h_{i}(\beta)=\big{[}h_{i}(\beta,l^{*})+\big{(}\sum_{k=1}^{l^{*}}z^{-}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{-}_{ik}\big{)}-\beta
S_{2}K\big{]}=\big{[}\bigvee_{l=0}^{n}h_{i}(\beta,l)+\big{(}\sum_{k=1}^{l}z^{-}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{-}_{ik}\big{)}-\beta
S_{2}K\big{]}$. 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, $K:=|l^{*}-\|y^{cf}_{i}\|_{1}|=\|w_{2}^{*}\|_{1}$.
Continuing with the notational setup,

$h_{i}(\beta,l)=\sum_{k=1}^{l}g_{ik}(\beta)=\sum_{k=1}^{l}\begin{cases}-z_{ik}^{-}-\beta(z_{ik}^{-})^{2},&-z_{ik}^{-}\leq\frac{1}{2\beta}\\\
\frac{1}{4\beta},&-z_{ik}^{-}>\frac{1}{2\beta}\,.\end{cases}$

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

$l^{*}=\operatorname*{arg\,max}_{l\in\\{0,\dots,n\\}}[h_{i}(\beta,l)+\sum_{k=1}^{l}z^{-}_{ik}-\beta
S_{2}K].$

Recall $\bigvee_{l=0}^{n}h_{i}(\beta,l)$ denotes
$\max_{l\in\\{0,\dots,n\\}}h_{i}(\beta,l)$.  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 constraint $w_{1}\leq-z_{i}^{-}$ leads
to a convex but piecewise structure for $h_{i}(\beta,l)$. 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

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

First look at the unconstrained problem,

$\tilde{\Psi}_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+\sup_{w_{1}}[\langle
w_{1},y^{cf}_{i}\rangle-\beta\langle w_{1},w_{1}\rangle].$

Applying the Cauchy-Schwarz Inequality gives

$\tilde{\Psi}_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+\sup_{\|w_{1}\|}[\|w_{1}\|\|y^{cf}_{i}\|-\beta\|w_{1}\|^{2}].$

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

$\tilde{\Psi}_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+\frac{\|y^{cf}_{i}\|^{2}}{4\beta}=\langle
z^{-}_{i},y^{cf}_{i}\rangle+\frac{\|y^{cf}_{i}\|_{1}}{4\beta}.$

Now let us return to the constrained problem, $\Psi_{\beta}$.
Observing that only the first $l=\|y^{cf}_{i}\|_{1}$ components of
$w_{1}\text{ inside the sup are positive gives }\forall k\in\\{1,\ldots,l\\}$

$\sup_{w_{1}\leq-z_{i}^{-}}[\sum_{k=1}^{l}(w_{1k})-\beta\langle
w_{1},w_{1}\rangle]=\sum_{k=1}^{l}\big{[}\sup_{w_{1k}\leq-z^{-}_{ik}}w_{1k}-\beta(w_{1k})^{2}\>\big{]}.$

Deduce that

$w_{1k}^{*}=\bigg{[}-z_{ik}^{-}\wedge\frac{1}{2\beta}\bigg{]}\quad\forall
k\in\\{1,\ldots,l\\}.$

Therefore

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+\sum_{k=1}^{l}\bigg{[}-z_{ik}^{-}\wedge\frac{1}{2\beta}\bigg{]}-\beta\bigg{[}-z_{ik}^{-}\wedge\frac{1}{2\beta}\bigg{]}^{2}.$

Next, let us do some simplification for

$g_{i}(\beta)=\sum_{k=1}^{l}\bigg{[}-z_{ik}^{-}\wedge\frac{1}{2\beta}\bigg{]}-\beta\bigg{[}-z_{ik}^{-}\wedge\frac{1}{2\beta}\bigg{]}^{2}.$

Considering the two cases, it follows that:

$g_{i}(\beta)=\sum_{k=1}^{l}g_{ik}(\beta)=\sum_{k=1}^{l}\begin{cases}-z_{ik}^{-}-\beta(z_{ik}^{-})^{2},&-z_{ik}^{-}\leq\frac{1}{2\beta}\\\
\frac{1}{4\beta},&-z_{ik}^{-}>\frac{1}{2\beta}\,.\end{cases}$

Note that $g_{i}(\beta)$ is a convex function!
In the degenerate case, $-z_{ik}^{-}=0$, then
$g_{ik}(\beta)=0\,\,\forall\beta\geq 0$, where $g_{ik}$ denotes the $k^{th}$
term in the sum.
Otherwise, $g_{ik}(\beta)$ is piecewise (line of negative slope for part 1,
hyperbola for part 2) but still convex.

$g_{ik}^{\prime}(\beta)=\begin{cases}-(z_{ik}^{-})^{2},&-z_{ik}^{-}\leq\frac{1}{2\beta}\\\
-\frac{1}{4\beta^{2}},&-z_{ik}^{-}>\frac{1}{2\beta}\,.\end{cases}$

Remarkably, these slopes are equal when $-z_{ik}^{-}=\frac{1}{2\beta}$ hence
the convexity of $g_{ik}$ and thus $g_{i}$ holds. Proceed to rewrite
$\Psi_{\beta}$ as

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+g_{i}(\beta).$

$\mathbf{Case\,2}$ 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

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\sup_{w_{1}\in\mathbb{R}^{n}_{-},l\in\\{0,\ldots,n\\},l\neq\|y^{cf}_{i}\|_{1}}[\sum_{k=1}^{l}(w_{1k}+z^{-}_{ik})-\gamma(\langle
w_{1},w_{1}\rangle+S_{2}K)].$

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

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\max_{l\in\\{0,\ldots,n\\},l\neq\|y^{cf}_{i}\|_{1}}\sup_{w_{1}\in\mathbb{R}^{n}_{-}}[\sum_{k=1}^{l}(w_{1k}+z^{-}_{ik})-\beta(\langle
w_{1},w_{1}\rangle+S_{2}K)].$

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

$\sup_{w_{1}\leq-z_{i}^{-}}[\sum_{k=1}^{l}(w_{1k})-\beta\langle
w_{1},w_{1}\rangle]=\sum_{k=1}^{l}\big{[}\sup_{w_{1k}\leq-z^{-}_{ik}}w_{1k}-\beta(w_{1k})^{2}\>\big{]}.$

Following the approach in Case 1 above, define

$h_{i}(\beta,l)=\sum_{k=1}^{l}g_{ik}(\beta)=\sum_{k=1}^{l}\begin{cases}-z_{ik}^{-}-\beta(z_{ik}^{-})^{2},&-z_{ik}^{-}\leq\frac{1}{2\beta}\\\
\frac{1}{4\beta},&-z_{ik}^{-}>\frac{1}{2\beta}\,.\end{cases}$

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

$l^{*}=\operatorname*{arg\,max}_{l\in\\{0,\dots,n\\},l\neq\|y^{cf}_{i}\|_{1}}[h_{i}(\beta,l)+\sum_{k=1}^{l}z^{-}_{ik}-\beta
S_{2}K].$

Proceed to write $\Psi_{\beta}$ as

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=[h_{i}(\beta,l^{*})+\sum_{k=1}^{l^{*}}z^{-}_{ik}-\beta
S_{2}K].$

This can be rewritten as

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+\bigg{[}h_{i}(\beta,l^{*})+\bigg{(}\sum_{k=1}^{l^{*}}z^{-}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{-}_{ik}\bigg{)}-\beta
S_{2}K\bigg{]}.$

Introducing
$h_{i}(\beta):=\big{[}h_{i}(\beta,l^{*})+\bigg{(}\sum_{k=1}^{l^{*}}z^{-}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{-}_{ik}\bigg{)}-\beta
S_{2}K\big{]}$,

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+h_{i}(\beta).$

$\mathbf{Max}$

Taking the max values for $\Psi_{\beta}$ over cases $w_{2}^{*}=0$ and
$w_{2}^{*}\neq 0$ gives

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+g_{i}(\beta)\vee h_{i}(\beta).$

Inspection suggests that $\Psi_{\beta}$ can be simplified further. Observe
that for $l^{*}=\|y^{cf}_{i}\|_{1},\>h_{i}(\beta)$ evaluates to
$g_{i}(\beta)$. Let $l^{*}$ be determined as

$l^{*}=\operatorname*{arg\,max}_{l\in\\{0,\dots,n\\}}[h_{i}(\beta,l)+\sum_{k=1}^{l}(z^{-}_{ik})-\beta
S_{2}K]$

and write

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+h_{i}(\beta).$

Finally, note an alternate expression for $h_{i}(\beta)$ is
$h_{i}(\beta)=\big{[}\bigvee_{l=0}^{n}h_{i}(\beta,l)+\bigg{(}\sum_{k=1}^{l}z^{-}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{-}_{ik}\bigg{)}-\beta
S_{2}K\big{]}$.

####  2.2.2 Outer Optimization Problem

The goal now is to evaluate

$\inf_{\beta\geq
0}\>G(\beta):=\bigg{[}\beta\delta_{2}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})\bigg{]}$

where

$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+h_{i}(\beta)$

for $h_{i}(\beta)=\big{[}h_{i}(\beta,l^{*})+\sum_{k=1}^{l^{*}}z^{-}_{ik}-\beta
S_{2}K\big{]}=\big{[}\bigvee_{l=0}^{n}h_{i}(\beta,l)+\sum_{k=1}^{l}z^{-}_{ik}-\beta
S_{2}K\big{]}$.

The convexity of the objective function $G(\beta)$ simplifies the task of
solving this optimization problem. The first order optimality condition
suffices. As $\Psi_{\beta}$ and hence $G(\beta)$ may have non-differentiable
kinks due to the max functions, $\vee$, we characterize the optimality
condition via subgradients. In particular, we look for $\beta^{*}\geq 0$ such
that $0\in\partial G(\beta^{*})$. Inspection of the asymptotic properties of
$\Psi_{\beta}$ and its subgradients reveals that two cases are possible. Case
1 is $\partial G(\beta)$ consists of strictly positive elements hence
$\beta^{*}=0$. Case 2 is $\partial G(\beta)$will cross zero (as $\beta$ sweeps
from $0$ to $\infty$) and hence $\beta^{*}\geq 0$.

Let $\beta^{*}\in\left\\{\beta\geq 0:0\in\partial
G(\beta)\right\\}\cup\\{\beta=0:0\notin\partial G(\beta)\\}\,$ where
$\partial\Psi_{\beta}=\partial h_{i}(\beta)$
and $\partial
G(\beta)=\delta_{2}+\frac{1}{N}\sum_{i=1}^{N}\partial\Psi_{\beta}$; $\partial
h_{i}(\beta)=\mathbf{Conv}\cup\\{\partial
h_{i}(\beta,l)-S_{2}K\>|\>h_{i}(\beta)=h_{i}(\beta,l)+\big{(}\sum_{k=1}^{l}z^{-}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{-}_{ik}\big{)}-\beta
S_{2}K;\,l\in\\{0,\dots,n\\}\\}$.

This follows from application of standard properties of subgradients as well
as inspection of the asymptotic properties of $\Psi_{\beta}$ and
$\partial\Psi_{\beta}$. For Case 1, if $\partial G(\beta)$ consists of
strictly positive elements then it is clear that $\beta^{*}$ attains the
minimum. For Case 2, the asymptotic properties can be used to show that
$\partial G(\beta)$ can’t consist of strictly negative elements. For $\beta$
sufficiently large, for optimal $l^{*}$, either $l^{*}=0\implies
0\in\partial\Psi_{\beta}$ or
$l^{*}=\|y^{cf}_{i}\|_{1}>0\implies\partial\Psi_{\beta}$ approaches
$0\implies\partial G(\beta)$ crosses zero.

This follows from standard application of properties of convex functions and
subgradients. First note that function $h_{i}(\beta)$ is convex in $\beta$
since it is the pointwise max of a finite set of convex functions plus a
constant plus a negative linear term. So then $\Psi_{\beta}$ is also convex.
Using properties of subgradients, $\partial\Psi_{\beta}=\partial
h_{i}(\beta)$; $\partial h_{i}(\beta)=\mathbf{Conv}\cup\\{\partial
h_{i}(\beta,l)-S_{2}K\>|\>h_{i}(\beta)=h_{i}(\beta,l)+\big{(}\sum_{k=1}^{l}z^{-}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{-}_{ik}\big{)}-\beta
S_{2}K;\,l\in\\{0,\dots,n\\}\\}$. Continuing, $G(\beta)$ is convex since it is
a linear term plus a sum of convex functions. It follows that $\partial
G(\beta)=\delta_{2}+\frac{1}{N}\sum_{i=1}^{N}\partial\Psi_{\beta}$. Finally,
we argue that $\beta^{*}\in\\{\beta\geq 0:0\in\partial
G(\beta)\\}\cup\\{\beta=0:0\notin\partial G(\beta)\\}$. Observe that for non-
empty $\\{\beta\geq 0:0\in\partial G(\beta)\\}$, this defines $\beta^{*}$ due
to convexity of $G(\beta)$. The claim is that for empty $\\{\beta\geq
0:0\in\partial G(\beta)\\}$ then $\beta^{*}=0$. For the easier case,
$0<z_{\beta}\>\forall z_{\beta}\in\partial G(\beta),\>\forall\beta\>\geq 0$,
it is clear that $\beta^{*}=0$ minimizes $\inf_{\beta\geq 0}G(\beta)$. It
remains to show that $0>z_{\beta}\>\forall\,z_{\beta}\in\partial
G(\beta),\>\forall\beta\>\geq 0$, does not occur. To elaborate, for $\beta>0$
sufficiently large, $\|y^{cf}_{i}\|_{1}>0\implies
l^{*}=\|y^{cf}_{i}\|_{1}\implies K=0\implies\exists\,z_{\beta}>-\delta_{2}$
such that $z_{\beta}\in\partial\Psi_{\beta}$. To elaborate, for $\beta>0$
sufficiently large, $\|y^{cf}_{i}\|_{1}=0\implies l^{*}=0\implies
K=0,\Psi_{\beta}=0,0=z_{\beta}>-\delta_{2}$ such that
$z_{\beta}\in\partial\Psi_{\beta}$. Hence we deduce $\exists\,z_{\beta}$ such
that $0<z_{\beta}+\delta_{2}\in\partial G(\beta)$ and by continuity and
convexity of $G(\beta)$ the claim holds.

The primal problem P2 has solution
$\big{[}\beta^{*}\delta_{2}+\frac{1}{N}\sum_{i=1}^{N}\Psi_{\beta^{*}}(z^{-}_{i},y^{cf}_{i})\big{]}$
where $\beta^{*}\in\,\left\\{\beta\geq 0:0\in\partial
G(\beta)\right\\}\cup\\{\beta=0:0\notin\partial G(\beta)\\}$ and
$\Psi_{\beta}(z^{-}_{i},y^{cf}_{i})=\,\langle
z^{-}_{i},y^{cf}_{i}\rangle+h_{i}(\beta)$ where
$h_{i}(\beta)=\big{[}\bigvee_{l=0}^{n}h_{i}(\beta,l)+\big{(}\sum_{k=1}^{l}z^{-}_{ik}-\sum_{k=1}^{\|y^{cf}_{i}\|_{1}}z^{-}_{ik}\big{)}-\beta
S_{2}K\big{]}$. Expressed in terms of original FBA, this says

$\begin{split}\sup_{Q\in\mathcal{U}_{\delta_{2}}(Q_{N})}\mathbb{E}^{Q}[\langle
Z^{-},Y_{CF}\rangle]=\mathbb{E}^{Q_{N}}[\langle
Z^{-},Y_{CF}\rangle]+\beta^{*}\delta_{2}\>+\mathbb{E}^{Q_{N}}\big{[}\bigvee_{l=0}^{n}h(\beta^{*},l)+\big{(}\sum_{k=1}^{l}Z^{-}_{k}-\sum_{k=1}^{\|Y_{CF}\|_{1}}Z^{-}_{k}\big{)}-\beta^{*}S_{2}K\big{]}\end{split}$

where the additional terms represent a penalty due to uncertainty in
probability distribution, and

$h(\beta,l)=\sum_{k=1}^{l}\begin{cases}-Z_{k}^{-}-\beta(Z_{k}^{-})^{2},&-Z_{k}^{-}\leq\frac{1}{2\beta}\\\
\frac{1}{4\beta},&-Z_{k}^{-}>\frac{1}{2\beta}\,.\end{cases}$

This result follows directly from the previous two propositions.  This follows
by direct substitution of $\beta^{*}$ as characterized in Proposition 2.4 into
Proposition 2.3 and then the dual problem D2.

###  2.3 FVA

####  2.3.1 Inner Optimization Problem

The robust FVA can be written as

$\sup_{\Phi\in\mathcal{U}_{\delta_{3}}(\Phi_{N})}\mathbb{E}^{\Phi}[\langle
Z,Y_{CF}\rangle].$ (P3)

Similar to before, 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

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

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 a hyperbola plus a line 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

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

####  2.3.2 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 $\alpha^{*}\geq
0$.

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 P3 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.  This follows by direct substitution of $\alpha^{*}$ as
characterized in Proposition 2.6 into Proposition 2.5 and then the dual
problem D3.

##  3 Computational Study: Robust FVA and Wrong Way Funding Risk

This computational study uses the Matlab Financial Instruments Toolbox and
extends WWR portfolio analysis (Brigo et al., , 2013, section 5.3) to consider
uncertainty in probability distribution. Other key concepts that will be
leveraged in this section are the measure concentration results and
association of Wasserstein radius $\delta$ with confidence level $1-\beta$ for
some $\beta\in(0,1)$. As of July 10, 2019, 5y par interest rate swaps are
1.88% (see www.interestrateswapstoday.com). The full table is shown below.
Furthermore, Bloomberg shows U.S. CDX investment grade and high yield 5y
credit default swap spreads as below.

Table 1: Swap Rates Swap Tenor | 1y | 2y | 3y | 5y | 7y | 10y | 30y
---|---|---|---|---|---|---|---
Swap Rate | 2.13% | 1.95% | 1.89% | 1.88% | 1.94% | 2.05% | 2.27%
Table 2: CDS Spreads CDX Index | IG | HY
---|---|---
CDS Spread | 53 | 323

Referencing current (as of August 26, 2019) MarkIt funding spreads, the
funding spread curves are set as below. Unavailable quotes for high yield
spreads are displayed as “N/A”.

Table 3: Funding Spreads Funding Tenor | 1y | 2y | 3y | 5y | 7y | 10y
---|---|---|---|---|---|---
IG Spread | 0.13% | 0.21% | 0.31% | 0.59% | 0.84% | 1.07%
HY Spread | N/A | N/A | 3.02% | 3.62% | 4.01% | 4.09%

The computational studies in this section will investigate (and quantify)
worst case FCA, FBA, and FVA for different market environments and portfolios
of interest rate swaps. The current swaps curve (shown above) will be used in
conjunction with monte carlo simulation of a one factor Hull-White model for
interest rates. The funding spreads will be used in conjunction with a Libor
Market Model (LMM) simulation of forward funding spreads. For this analysis,
the same funding spreads will be used for both FCA and FBA calculations. The
counterparty credit curve selection will vary between investment grade and
high yield (as shown above). The different portfolio setups will be described
in the following sections. All calculations are done in Matlab as an extension
of the example provided in the financial instruments toolbox (Matlab, , 2019).
The default Matlab settings for volatility and correlation parameters are used
for the Hull-White and LMM term structure models. Independence between the
funding cost, interest rate, and credit default factors is assumed for the
joint simulation.

###  3.1 Wasserstein Radius and Significance Levels

As mentioned in Part 1 (Singh and Zhang, , 2019), a natural question to ask
when computing worst case FVA is how to interpret the size of the Wasserstein
radius $\delta$. Substantial research has been done and some key results are
mentioned here. The following result is due to Fournier and Guillin,  (2015)
and the constants $c_{1},c_{2}$ below can be calculated explicitly by
following the proof:

$P[D_{c}(\Phi,\Phi_{N})\geq\delta)]\leq\begin{cases}c_{1}\exp(-c_{2}N\delta^{\max\\{n,2\\}})&\text{if}\>\delta\leq
1,\\\ c_{1}\exp(-c_{2}N\delta^{a})&\text{if}\>\delta>1\end{cases}$

$\forall N\geq 1,n\neq 2,\,\text{and}\,\delta>0$ where $c_{1}>0,c_{2}>0$
depend only on $a$, $A$, and $n$.
Esfahani and Kuhn,  (2018) discuss how equating the RHS above to $\beta$ and
solving for $\delta$ gives

$\delta_{N}(\beta)=\begin{cases}(\frac{\log(c_{1}\beta^{-1}}{c_{2}N})^{1/\max\\{n,2\\}}&\text{if}\>N\geq\frac{\log(c_{1}\beta^{-1})}{c_{2}},\\\
(\frac{\log(c_{1}\beta^{-1}}{c_{2}N})^{1/a}&\text{if}\>N<\frac{\log(c_{1}\beta^{-1})}{c_{2}}\end{cases}$

however these bounds are overly conservative, and result in a radius
$\delta^{*}$ much larger than necessary.

As an alternative approach, we follow a method that provides a more explicit
mapping between $\delta$ and $\beta$ (Carlsson et al., , 2018, section 3).
Theorem 6.15 of Villani,  (2008) gives a bound on Wasserstein distance between
two pdfs $\Phi,\Phi^{\prime}$ as

$D(\Phi,\Phi^{\prime})\leq\iint\limits_{\mathcal{R}}\|x_{0}-x\|\cdot|\Phi(x)-\Phi^{\prime}(x)|dA.$

Carlsson et al.,  (2018) get the result

$P(D(\Phi,\Phi_{N})\geq\delta)\lesssim\exp\left(-N\frac{8r-2\sqrt{16r^{2}+16r\delta+24r+12\delta+9}+4\delta+6}{3+4r}\right)$

where

$r=\max_{x_{0}\in\mathcal{R},x\in\mathcal{R}}\|x-x_{0}\|$

denotes the radius of domain $\mathcal{R}$. This is the characterization which
is used in this study. Therefore, for a desired significance (confidence)
level $\beta\in(0,1)$, find $\delta_{\beta}$ such that

$1-\beta=\exp\left(-N\frac{8r-2\sqrt{16r^{2}+16r\delta_{\beta}+24r+12\delta_{\beta}+9}+4\delta_{\beta}+6}{3+4r}\right).$

In our problem setting we use $r_{N}$ (computed using the empirical domain
$\mathcal{R}_{N}$) as the discrete approximation to $r$, which is difficult to
bound.

###  3.2 FCA

####  3.2.1 Portfolio of Interest Rate Swaps, Investment Grade Counterparty
and Firm

The portfolio here consists of a dozen interest rate swaps, with a mix of
receving fixed and paying fixed swaps, at different coupons, maturities, and
notionals. The fixed coupons range between 2% and 2.5%, the maturities range
between 4y and 12y, the notionals range between 400k USD and 1mm USD. The
investment grade counterparty and firm credit spreads are set to 50 basis
points. The table of confidence levels $\beta$ and their corresponding
Wasserstein radii $\delta$ follows.

Table 4: FCA Investment Grade Wasserstein Radii Confidence Level | 0.80 | 0.85 | 0.90 | 0.95 | 0.99 | 0.999
---|---|---|---|---|---|---
W Radius delta | 1.2 | 1.3 | 1.5 | 1.7 | 2.1 | 2.6

The scale factor $S_{1}$ is set (by default) to 1 and the portfolio exposures
are scaled to be in units of thousands of dollars. Again, the intent of
scaling is to provide appropriate penalty to the adversarial change in joint
distribution of portfolio funding exposures and default times that promotes
worst case FCA and wrong way risk. Further work may conduct a sensitivity
analysis regarding the pairings of $S_{1}$ and units of portfolio exposures to
investigate suitable (unsuitable) ranges that preserve (distort) the shape of
the robust FCA profile. Matlab plots characterizing the FCA exposure profile
and trajectory of worst case FCA as a function of Wasserstein radius are
shown. Again, we think about worst case FCA (which incorporates joint survival
probability) as compared to the funding PFE (potential future exposure) which
shows tail percentiles of funding exposure (not scaled by joint survival
probability).

Figure 1: Swaps Portfolio Positive Exposure Profiles

Figure 2: Swaps Portfolio IG FCA Exposure Profiles

Figure 3: Swaps Portfolio Worst Case IG FCA Profile

The baseline FCA for this portfolio is small (about 2.2k USD) and represents
the dot product of the discounted portfolio funding exposure profile times
joint survival probability. The worst case FCA curve is shown below. Note the
worst case FCA is approximately 70% the size of integrated (lifetime) FCA PFE
for Wasserstein radius $\delta$ about 1.7 which maps to a significance level
around $95\%$. So the takeaway here is worst case FCA is still a significant
percentage of funding PFE for swap portfolios with low counterparty default
curves (investment grade).

####  3.2.2 Portfolio of Interest Rate Swaps, High Yield Counterparty and Firm

The reference portfolio here consists of a dozen interest rate swaps, with a
mix of receving fixed and paying fixed swaps, at different coupons,
maturities, and notionals. The fixed coupons range between 2% and 2.5%, the
maturities range between 4y and 12y, the notionals range between 400k USD and
1mm USD. The high yield counterparty and firm credit spreads are set to 320
basis points. The table of confidence levels $\beta$ and their corresponding
Wasserstein radii $\delta$ is shown below. For the same reference portolio,
and same set of monte carlo interest rate paths, the max interest rate
exposures are the same. However, the high yield credit spreads and funding
costs expand the Wasserstein radii.

Table 5: FCA High Yield Wasserstein Radii Confidence Level | 0.80 | 0.85 | 0.90 | 0.95 | 0.99 | 0.999
---|---|---|---|---|---|---
W Radius delta | 3.1 | 3.4 | 3.7 | 4.2 | 5.3 | 6.5
Figure 4: Swaps Portfolio Positive Exposure Profiles

Figure 5: Swaps Portfolio HY FCA Exposure Profiles

Figure 6: Swaps Portfolio Worst Case HY FCA Profile

The baseline FCA for this portfolio is higher, around 65k USD and represents
the dot product of the discounted portfolio funding exposure profile times
joint survival probability. The worst case FCA curve is shown. Note the worst
case FCA is still approximately 33% the size of integrated (lifetime) FCA PFE
(Potential Future Exposure) for Wasserstein radius $\delta$ about 4.2 which
maps to a significance level around $95\%$. So the takeaway here is worst case
FCA is a significant percentage of funding PFE for swap portfolios with
moderately high counterparty and firm default curves.

###  3.3 FBA

####  3.3.1 Portfolio of Interest Rate Swaps, Investment Grade Counterparty
and Firm

The portfolio here consists of a dozen interest rate swaps, with a mix of
receving fixed and paying fixed swaps, at different coupons, maturities, and
notionals. The fixed coupons range between 2% and 2.5%, the maturities range
between 4y and 12y, the notionals range between 4mm USD and 10mm USD. The
investment grade counterparty and firm credit spreads are set to 50 basis
points. The table of confidence levels $\beta$ and their corresponding
Wasserstein radii $\delta$ follows.

Table 6: FBA Investment Grade Wasserstein Radii Confidence Level | 0.80 | 0.85 | 0.90 | 0.95 | 0.99 | 0.999
---|---|---|---|---|---|---
W Radius delta | 1.0 | 1.1 | 1.2 | 1.3 | 1.7 | 2.1

The scale factor $S_{2}$ is set (by default) to 1 and the portfolio exposures
are scaled to be in units of thousands of dollars. Same comments as above,
regarding scaling, apply. Matlab plots characterizing the FBA exposure profile
and trajectory of worst case FBA as a function of Wasserstein radius are
shown.

Figure 7: Swaps Portfolio Negative Exposure Profiles

Figure 8: Swaps Portfolio IG FBA Exposure Profiles

Figure 9: Swaps Portfolio Worst Case IG FBA Profile

The baseline FBA for this portfolio is -2k USD and represents the dot product
of the discounted negative funding portfolio exposure profile times joint
survival probability. The worst case FBA plot is shown. The plot illustrates
that worst case FBA quickly attains its lower bound (in magnitude) of zero (no
funding benefit to the firm for FBA).

####  3.3.2 Portfolio of Interest Rate Swaps, High Yield Counterparty and Firm

The reference portfolio here is the same one used in the previous subsection,
albeit with notionals from 4mm to 10mm USD. The high yield counterparty and
firm credit spreads are set to 320 basis points. The table of confidence
levels $\beta$ and their corresponding Wasserstein radii $\delta$ is shown.
For the same reference portolio, and same set of monte carlo interest rate
paths, the max interest rate exposures are the same. However, the high yield
credit spreads and funding costs expand the Wasserstein radii.

Table 7: FBA High Yield Wasserstein Radii Confidence Level | 0.80 | 0.85 | 0.90 | 0.95 | 0.99 | 0.999
---|---|---|---|---|---|---
W Radius delta | 2.0 | 2.2 | 2.4 | 2.8 | 3.4 | 4.3

A series of matlab plots characterizing the FBA exposure profile and
trajectory of worst case FBA as a function of Wasserstein radius is shown.

Figure 10: Swaps Portfolio Negative Exposure Profiles

Figure 11: Swaps Portfolio HY FBA Exposure Profiles

Figure 12: Swaps Portfolio Worst Case HY FBA Profile

The baseline FBA for this portfolio is -6k USD and represents the dot product
of the discounted negative portfolio funding exposure profile times joint
survival probability. The worst case FBA plot is shown. Once again, the plot
illustrates that worst case FBA moves towards its lower bound (in magnitude)
of zero (no funding benefit to the firm for FBA). Note the worst case FBA is
just 2.1% the size of integrated (lifetime) FBA PFE (Potential Future
Exposure) for Wasserstein radius $\delta$ about 2.8 which maps to a
significance level around $95\%$. So it accelerates towards the lower bound
(in magnitude) of zero.

###  3.4 FVA

####  3.4.1 Portfolio of Interest Rate Swaps, Investment Grade Counterparty
and Firm

The corresponding FCA portfolio is used for comparison. The portfolio consists
of a dozen interest rate swaps, with a mix of receving fixed and paying fixed
swaps, at different coupons, maturities, and notionals. The fixed coupons
range between 2% and 2.5%, the maturities range between 4y and 12y, the
notionals range between 400k USD and 1mm USD. The investment grade
counterparty and firm credit spreads are set to 50 basis points. The table of
confidence levels $\beta$ and their corresponding Wasserstein radii $\delta$
follows.

Table 8: FVA Investment Grade Wasserstein Radii Confidence Level | 0.80 | 0.85 | 0.90 | 0.95 | 0.99 | 0.999
---|---|---|---|---|---|---
W Radius delta | 1.4 | 1.5 | 1.7 | 1.9 | 2.4 | 2.9

The scale factor $S_{3}$ is set (by default) to 1 and the portfolio exposures
are scaled to be in units of thousands of dollars. Same comments as above, for
FCA and FBA, regarding scaling, apply. Matlab plots characterizing the FVA
positive and negative exposure profiles and trajectory of worst case FVA as a
function of Wasserstein radius are shown.

Figure 13: Swaps Portfolio Positive Exposure Profiles

Figure 14: Swaps Portfolio Negative Exposure Profiles

Figure 15: Swaps Portfolio IG FCA Exposure Profiles

Figure 16: Swaps Portfolio IG FBA Exposure Profiles

Figure 17: Swaps Portfolio Worst Case IG FVA Profile

The baseline FVA for this portfolio is small (less than 1k 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 below.
Note the worst case FVA is approximately 57% the size of FCA PFE for
Wasserstein radius $\delta$ about 1.9 which maps to a significance level
around $95\%$. So the takeaway here is worst case FVA is still a significant
percentage of funding PFE for swap portfolios with low counterparty default
curves (investment grade). It is also interesting to compare worst case FVA
vs. worst case FCA for a given delta. For example, for $\delta$ of 1.7, which
represents the $95\%$ significance level for FCA, we see FVA of around 9 which
is below FCA of 11.8. This agrees with intuition that FVA should be less than
FCA due to funding benefit from FBA.

####  3.4.2 Portfolio of Interest Rate Swaps, High Yield Counterparty and Firm

The corresponding FCA portfolio is used for comparison. The portfolio consists
of a dozen interest rate swaps, with a mix of receving fixed and paying fixed
swaps, at different coupons, maturities, and notionals. The fixed coupons
range between 2% and 2.5%, the maturities range between 4y and 12y, the
notionals range between 400k USD and 1mm USD. The high yield counterparty and
firm credit spreads are set to 320 basis points. The table of confidence
levels $\beta$ and their corresponding Wasserstein radii $\delta$ follows.

Table 9: FVA High Yield Wasserstein Radii Confidence Level | 0.80 | 0.85 | 0.90 | 0.95 | 0.99 | 0.999
---|---|---|---|---|---|---
W Radius delta | 3.8 | 4.2 | 4.6 | 5.3 | 6.6 | 8.1

The scale factor $S_{3}$ is set (by default) to 1 and the portfolio exposures
are scaled to be in units of thousands of dollars. Same comments as above, for
FCA and FBA, regarding scaling, apply. Matlab plots characterizing the FVA
positive and negative exposure profiles and trajectory of worst case FVA as a
function of Wasserstein radius are shown.

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 small (less than 1k 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 below.
Note the worst case FVA is approximately 23% the size of FCA PFE for
Wasserstein radius $\delta$ about 5.3 which maps to a significance level
around $95\%$. So the takeaway here is worst case FVA is still a significant
percentage of funding PFE for swap portfolios with high counterparty and firm
default curves. It is also interesting to compare worst case FVA vs. worst
case FCA for a given delta. For example, for $\delta$ of 4.2, which represents
the $95\%$ significance level for FCA, we see FVA of around 12.5 which is
below FCA of 19. This agrees with intuition that FVA should be less than FCA
due to funding benefit from FBA.

##  4 Conclusions and Further Work

This work has developed theoretical results and investigated calculations of
robust FVA and wrong way risk for OTC derivatives under distributional
uncerainty using Wasserstein distance as an ambiguity measure. The financial
market overview and foundational notation and wrong way risk (robust FVA)
primal problem definitions were introduced in Section 1. Using recent duality
results (Blanchet and Murthy, , 2019), the simpler dual formulation and its
analytic solutions for FCA, FBA, and FVA were derived in Section 2. After
that, in Section 3, some computational experiments were conducted to measure
the additional FCA charge (and/or FBA impairment) due to distributional
uncertainty for a variety of portfolio and market configurations for FCA, FBA,
and FVA. Using some probability results on bounding Wasserstein distance
between distributions (Carlsson et al., , 2018), a mapping between Wasserstein
radii $\delta$ and significance levels $\beta$ was devised to study the
trajectories of wrong way risk as a function of radius $\delta$. FCA increased
to a significant percentage of PFE. FBA quickly reached its lower bound of
zero funding benefit. FVA was below FCA (as expected) but still showed an
upward (apparently concave) trajectory as radius $\delta$ increased. Finally,
we conclude with some commentary on directions for further research.

One direction for future research, as has been previously discussed, is a
thorough study (including sensitivity analysis) regarding the pairings of
scale factors $(S_{1},S_{2},S_{3})$ and units of portfolio exposures to
investigate suitable (unsuitable) ranges that preserve (distort) the shape of
the robust FCA, FBA, FVA profiles (as a function of Wasserstein radii, and
hence distributional uncertainty) respectively. As a reminder, the intent of
scaling is to provide appropriate penalty to the adversarial change in joint
distribution of portfolio exposures and default times that promotes worst case
FCA, FBA, FVA and wrong way risk. Another direction for future research would
be to develop (and apply) similar theoretical machinery as used for robust FVA
and wrong way risk in this work 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.

## References

  * Blanchet et al., (2018) Blanchet, J., Chen, L., and Zhou, X. Y. (2018).  Distributionally robust mean-variance portfolio selection with wasserstein distances.
  * 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.
  * 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.
  * Fournier and Guillin, (2015) Fournier, N. and Guillin, A. (2015).  On the rate of convergence in wasserstein distance of the empirical measure.  Probability Theory and Related Fields, 162(3-4):707–738.
  * 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.
  * 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, (2019) Singh, D. and Zhang, S. (2019).  Distributionally robust xva via wasserstein distance part 1.  arXiv preprint arXiv:1910.01781.
  * Villani, (2008) Villani, C. (2008).  Optimal transport: old and new, volume 338.  Springer Science & Business Media.