Portfolio optimization of credit swap under funding costs

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, risk-free account and stock, assuming constant default intensity. Bo et al. (2010) 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 . 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≤ H 1 (ψ) + 1 ψ> 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 fixed-income 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 Lipschitz-continuity 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 2-dimensional default indicator process H(t) = (H 1 (t), H I (t)), t ≥ 0, supported by a filtered probability space ( , G, Q). Here Q denotes the risk-neutral 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 S := {0, 1} 2 . The default time of the i-th name is defined by We model default contagion through a Markovian model with interacting intensities. The default indicator process H is assumed to follow a continuous time Markov chain on S, where H(t) transits to the neighbouring state H 1 (t) : is a positive bounded measurable function with a strictly positive lower bound. We also refer the reader to Frey and Backhaus (2008) for explicit probabilistic models under this setup. Hence, H admits the following Q-infinitesimal generator given by where g z (t) is an arbitrary measurable function and the default state vectors The market filtration is given by G t = σ (H(s); s ≤ t). We take the right continuous version of G t , i.e. G t := ∩ >0 G t+ (see also Belanger et al. (2004)). Using the Dynkin's formula by choosing g z (t) = z i , it follows that 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
Consider a financial market consisting of a lending account, a borrowing account and an upfront CDS.
-Lending account. The investor lends at a constant risk-free rate r > 0. The time-t price of one share of his lending account is B t = e rt 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 byB t = e 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.
Upfront CDSs greatly reduce the counterparty exposure of the protection seller to the protection buyer. Currently, most CDS trades follow the upfront mechanism, especially for distressed credits. For a given loss rate L > 0 paid at default of the reference entity "1", we choose the contractual running spread premium ν > 0 so that it satisfies the following upfront condition: The above condition states that the running premium ν is decided in such a way that the minimum expected loss rate paid throughout the life of the transaction L inf t≥0 h 1,00 (t) is always higher than the spread premium ν. Consequently, a positive money amount would have to be posted by the protection buyer at the inception of the trade. Let T 1 > 0 be the maturity of the CDS. Then, the dividend process is given by, for t ∈ [0, T 1 ], The ex-dividend price of the CDS at time-t is given by Here E t denotes the expectation operator conditional on G t under Q. Throughout the paper, we consider the following contagion condition.  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 self-exciting 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
From (5), the price process of the CDS can be rewritten as, for t ∈ [0, T 1 ], Here z (t), (t, z) ∈ [0, T 1 ] × S, is the price function of the CDS contract given by where Using Feymann-Kac formula,

Lemma 2
The Q-dynamics of the CDS price process is given by, for t ∈ [0, T 1 ), The investor wishes to optimize his expected utility under the measure that describes the actual distribution of risk factors governing the market value. We provide a formula which allows for identifying the historical measure P from the risk neutral measure Q under which the price processes are observed. Let λ i,z (t) be a bounded measurable function defined on (t, z) ∈ R + × S, which takes values on Assume that the process X = (X t ; t ≥ 0) satisfies the following SDE given by where the Q-default martingale process ξ i is defined by (3). Define a new probability measure P Q on G T by dP = X T dQ. Then, for i ∈ {1, I }, is a P-martingale, where the relation between the P-default intensity of the i-th default indicator H i (t) and its Q-default intensity is given by Then from Lemma 2, it follows that the P-dynamics of the CDS price process is given by for t ∈ [0, T 1 ),

Borrowing rates
The default state dependent borrowing rate function of the investor r z (t), (t, z) ∈ [0, T ] × S, is given by the yield spread of a zero coupon bond written by the investor and expiring at T. For (t, z) ∈ [0, T ] × S, we use Φ z (t, T ) to denote the time-t price function of the investor bond with maturity T > t, given by Following Duffie and Singleton (2003), Section 5.3, define the default state dependent borrowing rate of the investor when the investor does not default at time t as, for When time to maturity approaches zero, we define r z 1 0 (T ) with z 1 ∈ {0, 1} as the limiting values, i.e.
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 ].
We next develop an explicit representation of Φ z (t, T ). Recall the operator A given by (1). It follows from (14) and In the following, we analyze the borrowing rate functions r z 1 0 (t) defined by (15). Using (15), we have 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.

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
We consider an investor who wants to maximize her power utility from terminal wealth at time T by dynamically allocating her wealth across a CDS, financing her purchases using the borrowing account, and lending using the risk-free bank account. The investor has neither intermediate consumption nor capital income to support her purchase of financial assets. Denote by φ(t) the number of shares of the CDS that the investor buys (φ(t) > 0) or sells (φ(t) < 0) at time t. Similarly, φ l 0 (t) represents the number of shares invested in the lending account, and φ b (t) the number of shares invested in her borrowing account. By definition φ l 0 (t) ≥ 0, while φ b (t) ≤ 0. We assume that simultaneous borrowing and lending is not efficient, i.e., φ l (t)φ b (t) = 0. We remark that similar assumptions have been made by Bielecki and Rutkowski (2015) and, by Mercurio (2015). The wealth process of a portfolio φ = (φ, φ l , φ b ) equals As usual, we require the portfolio process φ to be G-predictable. Moreover, for t ∈ [0, T ], we use π l (t) to denote the proportion of wealth invested in the lending account. We use π b (t) 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.
Recall (4). Using the self-financing condition, we can describe the wealth process as We introduce for the CDS security, obtained dividing the number of CDS shares by the current wealth level. Using (20), ψ(t)C t + π b (t) + π l (t) = 1 on τ I ∧ T > t. Using the conditions π b 0 (t) ≤ 0, π l 0 (t) ≥ 0 and π b 0 (t)π l 0 (t) = 0, it follows that We used the notation x − := min{x, 0} and x + := max{x, 0} for x ∈ R. We next define the class of admissible strategies for the investor. , z), is a class of G-predictable locally bounded feedback trading strategies given by T ] denotes the positive wealth process associated with the strategy ψ when V v,ψ t = 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 ψ * z 1 1 (u, v) = π l, * z 1 1 (u, v) = π b, * z 1 1 (u, v) = 0 for (u, v) ∈ [t, T ]×R + . Here π l, * z (·) and π b, * z (·) denote the optimal feedback fractions of wealth invested in the borrowing and the lending account respectively. We use U t to denote the set of all locally bounded feedback functions ψ z (u, v) for (u, v, z) Using (13), it follows that The optimal control problem Consider the power utility U(v) = γ −1 v γ for the initial wealth v > 0 and the riskaversion parameter γ ∈ (0, 1). The value function of our optimal control problem is given by Here the stopping time τ t I := inf{s ≥ 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}, For the default state z = (z 1 , 0) for z 1 ∈ {0, 1}, i.e. corresponding to the alive investor, using the above dynamic programming principle we obtain the following HJB equation in these states

Optimal feedback strategy
The aim of this section is to find the optimal feedback strategy to our portfolio optimization problem (25)-(26).
The nontrivial case is when no credit event has occurred, i.e. z = (0, 0). We next characterize the optimal feedback strategy in this state. First, we postulate (and later verify) that the value function admitting w z (t, v) = v γ ϕ z (t) where ϕ z (t) is a positive function which depends on state z ∈ S. It then follows directly from (25) that Equation (26) in the state z = (0, 0) can then be rewritten as The aim is to analyze ψ → H (ψ; t, ϕ) for characterizing the optimal feedback strategy ψ * in the state z = (0, 0). Noting that for fixed (t, ϕ) ∈ [0, T ] × R + , H (ψ; t, ϕ) is not differentiable at ψ = (t) := 00 (t) −1 > 0. Hence the first-order condition for optimality cannot be applied. To this purpose, we introduce g 1 (ψ; t, ϕ) := ( 00 (t) − L)h 1,00 (t) − ( 01 (t) − 00 (t)) h I,00 (t) ϕ It can be seen from (32) that both g 1 and g 2 are strictly decreasing and continuous in ψ. On the other hand, to guarantee that the wealth process is always positive after a default in terms of Eq. (23), the feedback strategy ψ should satisfy the following admissibility condition: where the last equality above follows from Lemma 1. We next characterize the optimal feedback strategy satisfying (33). The key idea behind our method is to decompose the function as H (ψ; t, ϕ) = 1 ψ≤ (t) H 1 (ψ; t, ϕ) + 1 ψ> (t) H 2 (ψ; t, ϕ).
Here the decomposing functions It is easy to see that g i is the first-order 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.
We next check sufficiency. For case (i), given that the optimum ψ * (t, ϕ) = 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. Recall (29) and consider z 1 = 1 in Eq. (26) with the optimum given by (28). We obtain the HJB equation at default state z = (1, 0) which given by Recall that (t) = 00 (t) −1 > 0 for all t ∈ [0, T ]. Then, using Lemma 7 above, it follows that for each fixed time t ∈ [0, T ], there exists a unique b i (t) > 0 such that (t) = ψ f oc i (t, b i (t)) for i = 1, 2. Moreover, t → b i (t) is continuous and b 2 (t) < b 1 (t) for all t ∈ [0, T ] using Lemma 6. From Proposition 1, it follows that the optimum ψ * = ψ * (t, ϕ) is a continuous function of (t, ϕ) ∈ [0, T ] × R + , and admits the following representation (see also Fig. 4 for an illustration). For each t ∈ [0, T ] fixed, on ϕ ∈ R + , 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 (41) is locally Lipschitz-continuous uniformly in t ∈ [0, T ].
The following lemma, proven in the Appendix, show that the solution to the HJB equation is bounded from below.
By the uniqueness of the solution to Eq. (52), we have v(t) = v κ (t) for all t ∈ [0, T ], and moreover η ≤ v(t) ≤κ(t) for all t ∈ [0, T ] in light of (53) 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. and is bounded on [0, T ]. Then using the comparison theorem for first-order ODEs, we deduce that v(t) ≥ u(t) for all t ∈ [0, T ] using the inequality (44) and noting that u(T ) = a 0 ≤ 1 γ = v(T ). Next, we prove that there exists a constant η > 0 so that u(t) ≥ η for all t ∈ [0, T ]. To this purpose, we first estimate the coefficient A(t, (t)) and C(t, (t)). Firstly, observe that C(t, (t)) > 0 for all t ∈ [0, T ]. On the other hand, note that 0 < (L − 00 (t))h 1,00 (t) + ( 01 (t) − 00 (t))h I,00 (t) < L(m 1,00 +m I,00 ), and noting that 00 (t) on t ∈ [0, T ] is bounded, it follows that for all t ∈ [0, T ], (t) ≤ r +m 1,00 +m I,00 Lm 1,00 − ν × 1 1 − e −(r+m 1,00 +m I,00 )(T 1 −T ) =: δ T .