Portfolio optimization of credit swap under funding costs
 Lijun Bo^{1, 2}Email author
https://doi.org/10.1186/s4154601700236
© The Author(s) 2017
Received: 27 January 2017
Accepted: 17 October 2017
Published: 4 December 2017
Abstract
We develop a dynamic optimization framework to assess the impact of funding costs on credit swap investments. A defaultable investor can purchase CDS upfronts, borrow at a rate depending on her credit quality, and invest in the money market account. By viewing the concave drift of the wealth process as a continuous function of admissible strategies, we characterize the optimal strategy in terms of a relation between a critical borrowing threshold and two solutions of a suitably chosen system of first order conditions. Contagion effects between risky investor and reference entity make the optimal strategy coupled with the value function of the control problem. Using the dynamic programming principle, we show that the latter can be recovered as the solution of a nonlinear HJB equation whose coeffcients admit singular growth. By means of a truncation technique relying on the locally Lipschitzcontinuity of the optimal strategy, we establish existence and uniqueness of a global solution to the HJB equation.
Introduction
The vast majority of literature studying portfolio allocation strategies across fixedincome securities has focused on a single source of defaults. Kraft and Steffensen (2005) analyze optimal investment in an asset defaulting when an economic state process falls below a given threshold. Bielecki and Jang (2006) derive optimal investment strategies for an investor allocating her wealth among a defaultable bond, riskfree account and stock, assuming constant default intensity. Bo et al. (2010) consider a logarithmic investor maximizing utility from consumption in market model consisting of a defaultable perpetual bond, defaultfree stock, and money market account. Capponi and FigueroaLópez (2014) study an economy consisting of a stock and of a defaultable bond whose price processes are modulated by an observable Markov chain. Jiao and Pham (2011) decompose a global optimal investment problem into subcontrol problems defined in a progressively enlarged filtration in an economy consisting of a riskfree bond and a stock subject to counterparty risk. Few other studies have considered portfolio frameworks consisting of multiple defaultable securities. Using a static model, Giesecke et al. (2014) study the portfolio selection problem of an investor who maximizes the marktomarket value of a fixed income portfolio of credit default swaps. Bielecki et al. (2008) study how dynamic investment in credit default swaps may be used to replicate defaultable contingent claims.
When financing investment strategies, the investor is typically charged a higher rate for borrowing than for lending. In the absence of default risk, the divergence between borrowing and lending rates has been analyzed both in the context of claim valuation by Korn (1995), and of hedging in incomplete markets by Cvitanić and Karatzas (1993). We also refer to El et al. (1997) who study the superhedging price of contingent claims under asymmetry of borrowing and lending interest rates via nonlinear backward stochastic differential equations. When financing credit investment strategies referencing highrisk names, the investor would need to borrow at her own refinancing rate rather than at the repo rate. This rate for unsecured borrowing depends on the credit quality of the investor who will be charged a spread reflecting her default likelihood as perceived by the market. Moreover, contagion effects generated from the default risk of the credit sensitive security introduce complex dependence patterns between the borrowing spreads and the marktomarket price of the CDS security. We also remark that the recent crisis has underscored the importance of including funding costs in optimal replication and investment strategies so to account for liquidity and counterparty risk, see also Crépey (2015) and Bielecki and Rutkowski (2015).
The main goal of the paper is to rigorously analyze the impact of borrowing costs incurred by a defaultable investor on her optimal allocation strategy. We construct a dynamic optimization framework where a defaultable investor can buy CDS upfronts, borrow at a premium over the riskfree rate, and invest in a money market account at the riskfree rate. The choice of upfront CDSs reflects regulatory market practices following the Big Bang Protocol (ISDA News Release 2009), and requiring distressed credits to trade with an upfront fee plus a running fixed coupon depending on the credit quality of the reference entity. The borrowing rate of the investor accounts both for her credit quality and for feedback effects generated by default of the reference entity. The latter are modeled using a contagion credit risk model with interacting default intensities, see also Jarrow and Yu (2001) for more explicit examples and Frey and Backhaus (2008) for a theoretical illustration. The investor maximizes her power utility from terminal wealth until the earlier of either her default time or the investment horizon.
Building on (Kraft and Steffensen 2005), we use the dynamic programming principle to characterize the HJB equations corresponding to the different default states. When the investor defaults, the solution to the HJB equation is specified as a boundary condition. Due to borrowing costs incurred by the investor when financing her purchases, the wealth process admits a concave drift, which is not differentiable with respect to the investment strategy. This prohibits the use of firstorder conditions for analyzing the optimal strategy. Moreover, because of defaults and contagion effects, we have to deal with the presence of jumps both in the price and in the wealth dynamics, and with concave drift appearing only in the wealth but not in the price dynamics. This leads us to apply a different method which considers the concave drift as a continuous function of the admissible strategy ψ, differentiable everywhere except at a critical point ℓ. Such a critical point defines two regions where the optimal admissible control may lie. More specifically, we decompose the component H(ψ) of the HJB equation yielding the optimal strategy as H(ψ)=1 _{ ψ≤ℓ } H _{1}(ψ)+1 _{ ψ>ℓ } H _{2}(ψ), where H _{1}(ψ) and H _{2}(ψ) are two continuously differentiable functions. We analyze the solutions of the system of first order conditions associated with H _{1}(ψ) and H _{2}(ψ), both seen as functions on their entire domains. The similar method has been proposed in Bo and Capponi (2017) for studying the fixedincome portfolio optimization with borrowing costs. We then characterize the optimal strategy in terms of a relation between the critical point and the previously established solutions. Default contagion makes the optimal strategy coupled with the value function, which is recovered as the solution of the corresponding nonlinear ordinary differential HJB equation. Moreover, we can prove that the coefficients of this equation admit a singular growth. By means of a delicate analysis, we establish the existence and uniqueness of a global solution to the HJB equation in two main steps. First, we prove the locally Lipschitzcontinuity of the optimal strategy when seen as a function of the value function (see Lemma 8 for details). We then show that if a solution to the HJB equation exists, then it must have a strictly positive uniform lower bound. This allows us to develop a novel truncation technique to remove the singularity in the HJB equation, and subsequently prove the existence and uniqueness of a global solution to the HJB equation.
The rest of the paper is organized as follows. “The model” section develops the default and market model. “Price dynamics and borrowing rates” section provides the dynamics of the CDS price process and analyzes borrowing rates. “Dynamic investment problem” section formulates the dynamic investment problem. “Optimal feedback strategy” section studies optimal investment strategies, while “Solvability of HJB equations” section establishes existence and uniqueness of solutions for the HJB equations. Some technical proofs are delegated to the Appendix.
The model
We describe the default model in “Default contagion model” section and the market model in “The market model” section. Throughout the paper, we refer to “1” as the reference entity of the CDS, and to “I” as the investor.
Default contagion model
The default state is described by a 2dimensional default indicator process H(t)=(H _{1}(t),H _{ I }(t)), t≥0, supported by a filtered probability space \((\Omega,{\mathcal {G}},{\mathbb {Q}})\). Here \(\mathbb {Q}\) denotes the riskneutral probability measure. Further, H _{ i }(t)=1 if i has defaulted by time t and H _{ i }(t)=0 otherwise, i∈{1,I}. This means that the state space of the default indicator process H=(H(t);t≥0) is given by \(\mathcal {S}:=\{0,1\}^{2}\). The default time of the ith name is defined by τ _{ i }= inf{t≥0; H _{ i }(t)=1} for i∈{1,I}. Then it holds that \(H_{i}(t)={\mathbf {1}}_{\{\tau _{i}\leq t\}}\phantom {\dot {i}\!}\) for t≥0.
is a martingale. At time 0, we place ourselves in the scenario where all names do not default, i.e. H(0)=0, where 0 is a two dimensional vector consisting of zero entries.
The market model

Lending account. The investor lends at a constant riskfree rate r>0. The timet price of one share of his lending account is B _{ t }=e ^{ r t } for t≥0.

Borrowing account. We proxy the borrowing costs incurred by investor with her bond credit spreads. We also refer to Chen (2010) for a dynamic model addressing the credit spread puzzle along with its relation to borrowing costs. In our model, credit spreads account both for the default risk of the investor and for contagion effects induced by the reference entity underlying the CDS. More specifically, the borrowing account of the investor is given by \(\bar {B}_{t} = e^{\int _{0}^{t} r_{\mathbf {H}(s)}(s)ds}\), t≥0, where r _{ H(t)}(t) denotes the default state dependent borrowing rate. The latter will be analyzed in detail in “Borrowing rates” section.

Upfront CDS. In a traditional running CDS contract a spread is paid throughout the life of the contract. The Big Bang Protocol introduced by ISDA News Release (2009) requires the premium leg to perform one of the following actions:
 1.
Make one single payment at the initiation of the CDS contract for protection until maturity.
 2.
Make one upfront payment plus pay a running premium until the earlier of a credit event or maturity. The running premium is set much lower than it would be under the traditional method.
 1.

L inft≥0h _{1,00}(t)−ν>0.

The default intensities satisfy$$\begin{array}{@{}rcl@{}} h_{1,01}(t) > h_{1,00}(t)\ \text{for}\ t\in[0,T_{1}],\ \text{and}\ h_{I,10}(t) > h_{I,00}(t)\ \text{for}\ t\in[0,T], \end{array} $$
where T _{1}>0 is the maturity of the CDS, and T∈(0,T _{1}) denotes the terminal time of the borrowing account. Such a condition captures the selfexciting feature of default events, empirically tested in several studies, see, for instance, Azizpour et al. (2017) for the case of U.S. corporate defaults.
Price dynamics and borrowing rates
We price CDS in “CDS pricing” section and analyze the investor’s borrowing costs in “Borrowing rates” section.
CDS pricing
where the operator \({\mathcal {A}}\) is given by (1). It is not difficult to get that
Lemma 1
Also for all t∈[0,T _{1}), Γ _{01}(t)>Γ _{00}(t), and Γ _{01}(T _{1})=Γ _{00}(T _{1})=0.
Lemma 2
Borrowing rates
However, when the investor has defaulted at time t, he can not invest the borrowing account anymore, and hence the borrowing rates of the investor \(r_{z_{1}1}(t)\) with z _{1}∈{0,1} are not needed. In order to be consistent mathematically, we set \(r_{z_{1}1}(t)=0\) for all t∈[0,T].
Also for all t∈[0,T _{1}), Φ_{10}(t,T)<Φ_{00}(t,T), and Φ_{10}(T,T)= Φ_{00}(T,T)=1.
When time to maturity approaches zero, r _{10}(T) and r _{00}(T) are given by the limiting values (16). The exact expressions are given in the following lemma whose proof is reported in the Appendix.
Lemma 3
It holds that r _{10}(T)=r+h _{ I,10}(T) and r _{00}(T)=r+h _{ I,00}(T). Moreover \(r + \underline {m}_{I,10}\leq r_{10}(t) \leq r + \bar {m}_{I,10}\) and \(r+\underline {m}_{I,00}\leq r_{00}(t)< r+\bar {m}_{1,00}+\bar {m}_{I,00}\) for all t∈[0,T]. Here \(\underline {m}_{i,\mathbf {z}}:=\inf _{t\in [0,T]}h_{i,\mathbf {z}}(t)\) and \(\bar {m}_{i,\mathbf {z}}:=\sup _{t\in [0,T]}h_{i,\mathbf {z}}(t)\).
Dynamic investment problem
We derive the wealth dynamics in “Wealth process” section and formulate the optimal control problem in “The optimal control problem” section.
Wealth process
As usual, we require the portfolio process ϕ to be \(\mathbb {G}\)predictable. Moreover, for t∈[0,T], we use \(\pi ^{l}(t):=\frac {\phi ^{l}(t)B_{t}}{V_{t}^{\boldsymbol {\phi }}}\) to denote the proportion of wealth invested in the lending account. We use \(\pi ^{b}(t):=\frac {\phi ^{b}(t)\bar {B}_{t}}{V_{t}^{\boldsymbol {\phi }}}\) to denote the proportion of wealth invested in the borrowing account. Then π ^{ l }(t)≥0, π ^{ b }(t)≤0 and π ^{ l }(t)π ^{ b }(t)=0.
We used the notation x ^{−}:= min{x,0} and x ^{+}:= max{x,0} for . We next define the class of admissible strategies for the investor.
Definition 1
Let . The admissible control set \(\tilde {\mathcal {U}}_{t}=\tilde {\mathcal {U}}_{t}(v,\mathbf {z})\), is a class of \(\mathbb {G}\)predictable locally bounded feedback trading strategies given by \({\psi (u)=\psi _{\mathbf {H}(u)}(u,V_{u}^{v,\psi })}\) for u∈[t,T]. Here \(V_{u}^{v,\psi }>0\) for u∈[t,T] denotes the positive wealth process associated with the strategy ψ when \(V_{t}^{v,\psi }=v\) and H(t)=z. In particular, in the state z=(z _{1},1) with z _{1}∈{0,1}, we set the corresponding optimal feedback strategy to \(\psi _{z_{1}1}^{*}(u,v)=\pi _{z_{1}1}^{l,*}(u,v)=\pi _{z_{1}1}^{b,*}(u,v)=0\) for . Here \(\pi _{\mathbf {z}}^{l,*}(\cdot)\) and \(\pi _{\mathbf {z}}^{b,*}(\cdot)\) denote the optimal feedback fractions of wealth invested in the borrowing and the lending account respectively. We use \({\mathcal {U}}_{t}\) to denote the set of all locally bounded feedback functions ψ _{ z }(u,v) for .
Using (13), it follows that
Lemma 4
The optimal control problem
Here the stopping time \(\tau _{I}^{t}:=\inf \{s\geq t;\ H_{I}(s)=1\}\).
Consider the default state z=(z _{1},1) for z _{1}∈{0,1}, i.e., corresponding to a defaulted investor. This implies that she will not invest in any security. Hence, her terminal wealth will be the same as her current wealth. As a result, the value function in this state is given by, for z∈{0,1},
Optimal feedback strategy
The aim of this section is to find the optimal feedback strategy to our portfolio optimization problem (25)–(25).
Equation (25) in the state z=(0,0) can then be rewritten as
where the last equality above follows from Lemma 1.
It is easy to see that g _{ i } is the firstorder derivative of γ ^{−1} H _{ i } w.r.t. ψ for i=1,2. Then we have the following lemma whose proof is reported in the Appendix.
Lemma 5
Using Lemma 5, we characterize the optimal feedback strategy in terms of a relation between the critical point ℓ(t) and the solutions \(\psi _{i}^{foc}\) of the system of first order condition equations.
Proposition 1

The optimum \(\psi ^{*}(t,\varphi)=\psi _{2}^{foc}(t,\varphi)\) if and only if the critical point \(\ell (t)\leq \psi _{2}^{foc}(t,\varphi)\). Correspondingly, the optimal feedback borrowing strategy is given by \(\pi ^{b,*}(t,\varphi)=1\psi _{2}^{foc}(t,\varphi)\mathit {\Gamma }_{00}(t)\) and the optimal feedback lending strategy is π ^{ l,∗}(t,φ)=0;

The optimum ψ ^{∗}(t,φ)=ℓ(t) if and only if the critical point \(\psi _{2}^{foc}(t,\varphi)\leq \ell (t)\leq \psi _{1}^{foc}(t,\varphi)\). Correspondingly, the optimal feedback borrowing and lending strategies are given by π ^{ b,∗}(t,φ)=π ^{ l,∗}(t,φ)=0;

The optimum \(\psi ^{*}(t,\varphi)=\psi _{1}^{foc}(t,\varphi)\) if and only if the critical point \(\ell (t)\geq \psi _{1}^{foc}(t,\varphi)\). Correspondingly, the optimal feedback borrowing strategy is given by π ^{ b,∗}(t,φ)=0 and the optimal feedback lending strategy is \(\pi ^{l,*}(t,\varphi)=1\psi _{1}^{foc}(t,\varphi)\mathit {\Gamma }_{00}(t)\).
Proof
We next check sufficiency. For case (i), given that the optimum \(\psi ^{*}(t,\varphi)=\psi _{2}^{foc}(t,\varphi)\), assume by contradiction that the critical point \(\ell (t)>\psi _{2}^{foc}(t,\varphi)\). Consider first the case where \(\psi _{2}^{foc}(t,\varphi)<\ell (t)<\psi _{1}^{foc}(t,\varphi)\). Then, using the necessary result proved above in (ii), it follows that the optimum ψ ^{∗}(t,φ)=ℓ(t), which contradicts the given assumption \(\psi ^{*}(t,\varphi)=\psi _{2}^{foc}(t,\varphi)\), since \(\ell (t)>\psi _{2}^{foc}(t,\varphi)\) in this subcase. Next consider the remaining case where \(\ell (t)\geq \psi _{1}^{foc}(t,\varphi)\). Using the necessary result proved above in (iii), we conclude that the optimum \(\psi ^{*}(t,\varphi)=\psi _{1}^{foc}(t,\varphi)\), which also contradicts the given assumption \(\psi ^{*}(t,\varphi)=\psi _{2}^{foc}(t,\varphi)\) due to the inequality (34) in Lemma 5. Thus we have proven sufficiency in case (i). For cases (ii) and (iii), the corresponding proofs are similar. Here we omit them. This completes the proof of the proposition. □
Lemma 5 provides a finite lower bound for \(\psi _{i}^{foc}\). The next lemma gives a finite upper bound for \(\psi _{i}^{foc}\). These boundedness estimates will play a crucial role in proving existence and uniqueness of solutions of the HJB equation.
Lemma 6
The proof is delegated into the Appendix.
Solvability of HJB equations
We prove the existence and uniqueness of a global solution to HJB equations at each default state. We then show that it corresponds with the value function of the control problem via a verification theorem.
We next analyze the nontrivial case, i.e. when no credit event has occurred. To this purpose, recall \(\psi _{i}^{foc}=\psi _{i}^{foc}(t,\varphi)\) as obtained in Lemma 5. We have
Lemma 7
It implies that for fixed t∈[0,T], φ→ψ ^{∗}(t,φ) is continuous and decreasing. It is not difficult to claim that
Lemma 8
It holds that φ→ψ ^{∗}(t,φ) given by (39) is locally Lipschitzcontinuous uniformly in t∈[0,T].
Above, we have used the explicit representation for φ _{01}(t)=γ ^{−1} given in (28), and that φ _{10}(t) is given by (37).
with v(T)=γ ^{−1}, where \(\bar {A}(t,v):=A(t,\psi ^{*}(t,v))\) and \(\bar {C}(t,v):=C(t,\psi ^{*}(t,v))\) on . The coefficients \(\bar {A}(t,v)\) and \(\bar {C}(t,v)\) admit the following estimates.
Lemma 9
Let . Then there exist \(K,\bar {K}>0\) such that \(\bar {A}(t,v) \leq K + \bar {K} v^{\frac {1}{\gamma 1}}\) and \(0<\bar {C}(t,v) \leq K + \bar {K} v^{\frac {\gamma }{\gamma 1}}\).
The following lemma, proven in the 2, show that the solution to the HJB equation is bounded from below.
Lemma 10
If there exists a solution (v(t); t∈[0,T]) to Eq. (44), then there exists a constant η>0 such that v(t)≥η for all t∈[0,T].
Then we have
Theorem 1
There exists a unique solution (v(t); t∈[0,T]) to Eq. (45). Moreover, \(\eta \leq v(t)\leq \bar {\kappa }(t)\) for all t∈[0,T].
Proof
By the uniqueness of the solution to Eq. (50), we have v(t)=v _{ κ }(t) for all t∈[0,T], and moreover \(\eta \leq v(t)\leq \bar {\kappa }(t)\) for all t∈[0,T] in light of (51) and Lemma 10. This completes the proof of the theorem. □
We finally mention that the verification theorem also holds on our control problem, i.e., the solution of the HJB equation is the value function of our control problem. The proof is standard and it heavily depends on the bounded solutions to the HJB equations discussed above. Hence, we omit the statement of the verification theorem and the proof.
Appendix
Technical proofs
Proof of Lemma 3
This concludes the proof of the lemma. □
Proof of Lemma 5
Since 0<Γ _{00}(t)<Γ _{01}(t)<L, and r _{00}(t)−r>0 for all t∈[0,T] using Lemma 3, we have \({\lim }_{\psi \uparrow +\infty } g_{2}(\psi ;t,\varphi)<0\). Then applying the intermediate value theorem, there is a unique finite solution \(\psi _{2}^{foc}>\frac {1}{L\mathit {\Gamma }_{00}(t)}\) satisfying \(g_{2}(\psi _{2}^{foc};t,\varphi)=0\). Further, in light of Kumagai (1980)’s implicit function theorem, \(\psi _{2}^{foc}\), viewed as a function of (t,φ) is C ^{1} in (t,φ), since the derivative \(\frac {\partial g_{2}(\psi ;t,\varphi)}{\partial \psi }<0\) in its domain.
Since Γ _{00}(t)(r _{00}(t)−r)φ>0 for all , by (30) it follows immediately that for all , g _{2}(ψ;t,φ)<g _{1}(ψ;t,φ). This yields the validity of (34). □
Proof of Lemma 6
Proof of Lemma 7
This ends the proof. □
Proof of Lemma 9
Combining Proposition 1, Lemma 6 and the estimate 0<Γ _{00}(t)<Γ _{01}(t)<δ L for some δ∈(0,1), it follows that there exist constants \(K_{1},\bar {K}_{1}>0\) so that
where we use the inequality (x+y)^{ γ }≤x ^{ γ }+y ^{ γ } for all and γ∈(0,1). This completes the proof of the lemma. □
Proof of Lemma 10
Recall A(t,ψ) and C(t,ψ) defined in (41). It is immediately verified that ψ→A(t,ψ) is Lipschitzcontinuous and ψ→C(t,ψ) is locally Lipschitzcontinuous uniformly in time t∈[0,T]. Using Lemma 8, \(v\to \bar {A}(t,v)\) and \(v\to \bar {C}(t,v)\) are locally Lipschitzcontinuous uniformly in time t∈[0,T]. Since v(t) is a solution to Eq. (44) by assumption, it is C ^{1} on [0,T], and hence it is bounded on [0,T]. Then using the comparison theorem for firstorder ODEs, we deduce that v(t)≥u(t) for all t∈[0,T] using the inequality (42) and noting that \(u(T)=a_{0}\leq \frac {1}{\gamma }=v(T)\).
Then, it holds that u(t)≥η for all t∈[0,T]. This implies that v(t)≥u(t)≥η, for all t∈[0,T]. □
Declarations
Acknowledgments
The author gratefully acknowledges the constructive and insightful comments and suggestions provided by Prof. Agostino Capponi, Prof. Stéphane Crépey two anonymous reviewers, which contributed to improve the quality of the manuscript greatly. The research was partially supported by NSF of China (No. 11471254), The Key Research Program of Frontier Sciences, CAS (No. QYZDBSSWSYS009) and Fundamental Research Funds for the Central Universities (No. WK3470000008).
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Azizpour, S, Giesecke, K, Schwenkler, G: Exploring the sources of default clustering. J. Finan. Econom. Forthcoming (2017).Google Scholar
 Belanger, A, Shreve, S, Wong, D: A general framework for pricing credit risk. Math. Finan. 14, 317–350 (2004).View ArticleMATHMathSciNetGoogle Scholar
 Bielecki, T, Jang, I: Portfolio optimization with a defaultable security. AsiaPacific Finan. Markets. 13, 113–127 (2006).View ArticleMATHGoogle Scholar
 Bielecki, T, Jeanblanc, M, Rutkowski, M: Pricing and trading credit default swaps in a hazard process model. Ann. Appl. Probab. 18, 2495–2529 (2008).View ArticleMATHMathSciNetGoogle Scholar
 Bielecki, T, Rutkowski, M: Valuation and hedging of contracts with funding costs and collateralization. SIAM J. Finan. Math. 6, 594–655 (2015).View ArticleMATHMathSciNetGoogle Scholar
 Bloomberg News, November 8, 2013: Report available at http://www.bloomberg.com/news/20131108/pimcosaidtowager10billionindefaultswapscreditmarkets.html.
 Bo, L, Wang, Y, Yang, X: An optimal portfolio problem in a defaultable market. Adv. Appl. Probab. 42, 689–705 (2010).View ArticleMATHMathSciNetGoogle Scholar
 Bo, L, Capponi, A: Optimal investment in credit derivatives portfolio under contagion risk. Math. Finan. 26, 785–834 (2016).View ArticleMATHMathSciNetGoogle Scholar
 Bo, L, Capponi, A: Optimal credit investment with borrowing costs. Math Oper. Res. 42, 546–575 (2017).View ArticleMATHMathSciNetGoogle Scholar
 Capponi, A, FigueroaLópez, JE: Dynamics portfolio optimization with a defaultable security and regimeswitching. Math Finan. 24, 207–249 (2014).View ArticleMATHMathSciNetGoogle Scholar
 Chen, H: Macroeconomic conditions and the puzzles of credit spreads and capital structure. J. Finan. 65, 2171–2212 (2010).View ArticleGoogle Scholar
 Crépey, S: Bilateral counterparty risk under funding constraints. Part I: pricing. Math Finan. 25, 23–50 (2015).View ArticleMATHMathSciNetGoogle Scholar
 Cvitanić, J, Karatzas, I: Hedging contingent claims with constrained portfolios. Ann. Appl. Probab. 3, 652–681 (1993).View ArticleMATHMathSciNetGoogle Scholar
 Draouil, O, Oksendal, B: A donsker delta functional approach to optimal insider control and applications to finance. Commun. Math Stats. 3, 365–421 (2015).View ArticleMATHMathSciNetGoogle Scholar
 Duffie, D, Singleton, K: Credit Risk. Princeton University Press, Princeton (2003).Google Scholar
 El Karoui, N, Peng, S, Quenez, MC: Backward stochastic differential equations in finance. Math Finan. 7, 1–71 (1997).View ArticleMATHMathSciNetGoogle Scholar
 Frey, R, Backhaus, J: Pricing and hedging of portfolio credit derivatives with interacting default intensities. Int. J. Theor. Appl. Finan. 11, 611–634 (2008).View ArticleMATHMathSciNetGoogle Scholar
 Giesecke, K, Kim, B, Kim, J, Tsoukalas, G: Optimal credit swap portfolios. Manage Sci. 60, 2291–2307 (2014).View ArticleGoogle Scholar
 ISDA News Release: ISDA Announces Successful Implementation of Big Bang’ CDS Protocol; Determinations Committees and Auction Settlement Changes Take Effect (2009). Available at http://www.isda.org/press/press040809.html.
 Jarrow, R, Yu, F: Counterparty risk and the pricing of defaultable securities. J. Finan. 56, 1765–1799 (2001).View ArticleGoogle Scholar
 Jiao, Y, Pham, H: Optimal investment with counterparty risk: a default density approach. Finan. Stoch. 15, 725–753 (2011).View ArticleMATHMathSciNetGoogle Scholar
 Korn, R: Contingent claim valuation in a market with different interest rates. Math. Meth. Oper. Res. 42, 255–274 (1995).View ArticleMATHMathSciNetGoogle Scholar
 Korn, R, Kraft, H: Optimal portfolios with defaultable securitiesa firm value approach. Int. J. Theor. Appl. Finan. 6, 793–819 (2003).View ArticleMATHMathSciNetGoogle Scholar
 Kraft, H, Steffensen, M: Portfolio problems stopping at first hitting time with application to default risk. Math. Meth. Oper. Res. 63, 123–150 (2005).View ArticleMATHMathSciNetGoogle Scholar
 Kumagai, S: An implicit function theorem: comment. J. Optim. Theor. Appl. 31, 285–288 (1980).View ArticleMATHGoogle Scholar
 Mercurio, F: Bergman, Piterbarg, and Beyond: pricing derivatives under collateralization and differential rates. In: Londoño, J, Garrido, J, HernándezHernández, D (eds.)Actuarial Sciences and Quantitative Finance. Springer Proceedings in Mathematics & Statistics, vol 135. Springer, Cham (2015).Google Scholar