Free boundaries of credit rating migration in switching macro regions

. In this paper, under the structure framework, a valuation model for a corporate bond with credit rating migration risk and in macro regime switch is established. The model turns to a free boundary problem in a partial diﬀerential equation (PDE) system. By PDE techniques, the existence, uniqueness and regularity of the solution are obtained. Furthermore, numerical examples are also presented.

1. Introduction. Regime switch was introduced by Hamilton in 1989, [8]. In his paper, he described an autoregressive regime switching process. This topic lighted wide interests as it well described a general phenomenon that the situations might change in switching macro atmosphere. Later, the topic became very popular and was extended to various areas such as energy, economics etc.. Many researchers developed the Hamilton's model in different ways, e.x. [5,21]. Especially, in financial area, valuations of the financial products, such as stocks, options etc. were also considered in regime switching models (see [2,3,7]).
The Financial Crisis in 2008 and later the European Debt Crisis gave a lesson to the financial market that managing the credit risk was important, where the credit risks included both default risk and credit rating migration one. People realized that in different macro atmospheres, the credit risks behaved totally differently. However, the academic researches were behind legs.
There are two traditional ways to study credit risks, known as structural and reduced-form ones, where the first one treats the credit event is an exogenous variant, (e.x. see [11,14,6]), while the second one is assumed that the credit event occurs once the firm's value touches a threshold, (e.x. see [19,1,15]).

It satisfies
in the high rating, r Mt S t dt + σ Mt L S t dW t , in the low rating, where r Mt is the risk free interest rate in different macro environment regions, and σ Mt H , σ Mt L : κ → R + represent the volatilities of the firm asset, M t is the region switching process mentioned above, W t is the Brownian motion which generates the filtration {F t }. W t and M t are assumed to be independent. Also It is reasonable to assume (2.1), namely, that the volatility in the high rating region is lower than the one in the low rating region.  are the first moment when the firm's grade is downgraded and upgraded respectively as follows: is a contingent claim with respect to S t and is a positive constant representing the threshold proportion of the debt and value of the firm's rating.
Assumption 2.5 (the macro region changing time). The probability that the credit rating and the macro region transfer in the same time is zero. Denote τ ij is the macro region changing time, which the state i turns to j.

Cash flow.
Once the credit rating migrates or macro region switches before the maturity T , though there is no cash flow, a virtual substitute termination happens, i.e., the bond is virtually terminated and substituted by a new one with a new credit rating (new macro region). There is a virtual cash flow of the bond. Denoted by Φ i H (y, t) and Φ i L (y, t) the values of the bond on the M t = i ∈ κ state and in high and low grades respectively. Then, they are the conditional expectations as follows: 2.3. PDE problem. By Feynman-Kac formula, it is not difficult to drive that Φ Mt H and Φ Mt L are the functions of the firm value S and time t. They satisfy the following partial differential equations in their regions: with the terminal condition: where i ∈ κ. Also, if we construct a risk free portfolio π by longing a bond and shorting ∆ amount asset value S, i.e., π i t = Φ i t − ∆ i t S t and such that dπ t = rπ t , this portfolio is also continuous when it passes the rating migration boundary, i.e., π i H = π i L on the rating migration boundary, (2.9) or by ( 3. Free boundary problem. Using the standard change of variables x = log S and rename T − t as t, for i ∈ κ, define in the low rating, using also (2.8) and (2.11), we then derive the following equation from (2.5), (2.6): where σ i is a function of φ i and x, i.e., (3. 2) The constants γ, σ i H , σ i L are defined in (2.1), (2.2). Without losing generality, we assume F = 1. Equation (2.5) is supplemented with the initial condition (derived from (2.7)) For any i, in i− macro environment state, the domain is divided into the high rating region Ω i H where φ i < γe x and a low rating region Ω i L where φ i > γe x . We shall prove that these two domains will be separated by a free boundary x = s i (t), and In another word, s i (t) is apriorily unknown since it should be solved by the equation where the solution φ i is apriorily unknown.
Since we have assumed that equation (2.5) is valid across the free boundary x = s i (t), we can derive from (2.8), (2.11):

Preliminaries.
4.1. Approximation. Let H(ξ) be the Heaviside function, i.e., H(ξ) = 0 for ξ < 0 and H(ξ) = 1 for ξ > 0. Then we can rewrite (3.2) as Consider the approximated system for i ∈ κ, 3) As σ i ε has uniform upper and lower positive bounds, it is not difficult to uncouple the system. Then by a suitable Fixed Point Theorem ( [9]), we can prove that the system (4.1)-(4.2) admits a unique classical solution {φ i ε , i ∈ κ} for any ε > 0. We now proceed to derive estimates for φ i ε . 4.2. Estimates for the approximating system. For i ∈ κ, for the sake of the simplicity of (4.1), denote the operator We prove Lemma 4.1-Lemma 4.3 which are applied on a more general operator G first. The lemmas are treated as maximum principles and can be extended as comparison lemmas for a system.
Proof. Let u j = v j e αt , where α > 0 is big enough. We want to prove v j < 0. In fact, from conditions of the Lemma, If the conclusion is not true, then when 0 < t < t, v(x, t) 0 and there exists which is a contradiction.
respectively, then the conclusion turns to be u(x, t) 0( 0).
We have proved the maximum principle for our problem in bounded domain. Next, we will prove that the conclusion still holds in unbounded domain by using above lemmas.
By calculation, we could obtain that 0. Then the conclusion holds by letting L → +∞.
Proof. It is not difficult to verify the conclusion for φ i ε satisfying (4.1) by Lemma 4.3 we have proved.
Proof. Differentiating equation (4.1) with respect to x, we obtain − γe x to be a given function,which is bounded and −λ ij < 0, i = j, we are able to use Lemma 4.3 with checking Using (4.1), we find that It is also clear that initially w i = 0 for x < 0 and w i = −1 for x > 0. It follows by Lemma 4.6.
Proof. Differentiating equation (4.1) with respect to t, we obtain Combining the equations for φ i ε , ∂φ i ε ∂x and ∂φ i ε ∂t , we obtain

FREE BOUNDARIES IN SWITCHING MACRO REGIONS 265
Notice that then we havê (4.6) At t = 0, w i produces a dirac measure of intensity −1 at x = 0 and by w i (x, 0) = 0 for both x < 0 and x > 0. By further approximating the initial data with smooth functions if necessary, we derive w i < 0. Hence the lemma holds.
Lemma 4.7. There exist constants c 1 , C 2 and C 3 , independent of ε, such that Proof. From (4.5), It is also clear that initially Next, since φ i ε (0, 0) = 1 > γ, and by Hölder continuity of the solution, there exists a ρ > 0, independent of ε, such that It follows from the standard parabolic estimates that In particular, this implies that We now consider ρ 2 t T , and definê Then, for any C > 0, As an immediate corollary, we have Next, we derive the estimates of free boundaries. Denote s i ε (t) is the approximated free boundary, which is the implied solution of the equation Lemma 4.9. The approximated free boundary s i ε (t) is uniquely defined by (4.9).
Then, we have the following estimates for the free boundary.
Lemma 4.10. The approximated free boundary defined in (4.9) satisfies Proof. It is also clear that, by Maximum Principle,φ i ε < e −rt , so that, This means that the region {x > log 1 γ − rt} is in the high rating region and hence (4.13) holds.
We next derive lower bound for s i ε (t).
Lemma 4.11. The approximated free boundary defined in (4.9) satisfies, for any A > 1.
It follows that L i ε [e x − e Ax+Bt ] 0. It is obvious that φ ε (0, t) > 0 1 − e Bt . Thus we can apply Maximum Principle to obtain which means that {x < 1 A−1 log(1 − γ) − B A−1 t} is in the low rate region and Using the definition of B we conclude.
Lemma 4.12. For any T > 0, there exists C T > 0, independent of ε, such that the derivative of the approximated free boundary s i ε (t) is bounded by (4.14) Proof. Clearly, .
The estimates ∂φ i ε ∂t < 0 and ∂φ i ε ∂x −φ i ε < 0 imply that the approximated free boundary is strictly decreasing: By Lemma 4.7, there is a constant ρ > 0 (independent of ε) such that s i ε (t) ρ for 0 t ρ 2 . It follows from Lemma 4.7 that for some constant C * > 0 independent of ε. To finish the proof, it suffices to , t) c * for some positive c * independent of ε. Let L i ε be the operator defined in Lemma 4.5. As shown in Lemma 4.5, By Lemmas 4.7, 4.10 and 4.11, there exists R T > 0, independent of ε, such that and s i ε (t) ρ for 0 t ρ 2 . Consider the region The parabolic boundary of this region Ω 1 consists of 5 line segments. On the initial line segment {(x, 0), ρ/2 x R T }, w i (x, 0) = 1. The remaining 4 parabolic boundaries t T } are completely and uniformly within the high or low rating region (independent of ε). Thus by compactness and strong maximum principle, on these 4 boundaries, w i c > 0 for somec independent of ε. It follows that w i min(1,c) min(1,c) ≡ c * on Ω 1 and this establishes in (4.14).
Proof. Without loss of generality, we suppose ∂ 2 ψ i ∂x 2 − ∂ψ i ∂x 0. Besides, in this paper, two macro environment regions are considered, then κ = {1, 2}. Let Since F i ≡ 0 for |x| > M , the solution v decays exponentially fast to 0 as x → ±∞. It follows that Therefore if the conclusion is not true, one of components of w, which is denoted by w 1 must attain a negative minimum at a point (x * 1 , t * 1 ) with x * 1 finite and 0 < t *
Lemmas 4. 10-4.12 show that there is a uniform estimate in space C 1 ([0, T ]) for the approximated free boundary s i ε (t),i ∈ κ. Therefore, the limit of s i ε (t) as ε → 0 exists, which is denoted by s i (t), i ∈ κ. This s i (t), i ∈ κ is the free boundary of our problem (3.1)-(3.6).
, and s ∈ W 1,∞ [0, T ]. Furthermore, the solution satisfies By the classical parabolic theory, it is also clear that the solution is in Applying Lemma 4.13, we obtain the uniqueness of the solution directly.
is unique.
6. Numerical results. Here some numerical results are presented by explicit finite difference approach. In this section, we just consider two regimes, where bull market denotes good macro-economic situation and bear market represents bad situation.
6.1. Calibration. To apply the finite difference scheme, we need calibration first. The parameters to be calibrated are: γ i , r i , σ i H , σ i L , λ ij for i, j = 1, 2, i = j. 1. The method of estimating γ i , r i , σ i H , σ i L , i = 1, 2 could be seen in [22]. 2. The intensities λ 12 and λ 21 can be calibrated through the market quotes of the stock index. The details are shown as follows: (a) Estimating the "Bull-Bear Boundary". Bull-Bear Boundary which distinguishes bull market and bear market could be estimated by 250-day moving average of stock index. If the index is trading below its 250-day moving average, the market is said to be bear market. Otherwise, it is bull market.
(b) Estiamting the transition intensities. The transition matrix (P ij ) denotes the probabilities of moving regimes, where i, j = 1, 2 and P ij + P ii = 1 for i = j. From [8], the relationship between P ij and T ii which denotes the average length of a single run in regime i is where T ii can be estimated from the above step. On the other hand, we have P ij (t) = e − t 0 λij ds .
Then transition intensities could be estimated from the above equation. The differences of value function and free boundary between Case 1 and Case 2 are shown in Figure 3.   Figure 1 and Figure 2, the value functions in deferent regime regions are divided into two regions respectively. The value changes quite significantly across the free boundary. The free boundaries are decreasing as expected.
From Figure 3, it could be seen that for fixed x, the value in Case 2 is greater the the one in Case 1. And the high rating region of Case 2 is larger than the one in Case 1. That is because in Case 2 the transition intensity λ 21 < λ 12 , which is opposite of the situation in Case 1. That means, in Case 2, bear market is more likely to move into bull market, which is different with Case 1. In other words, "good" macro-economic situation prevails during the period in Case 2, which leads to lower volatilities and higher valuation. 7. Conclusion. By establishing a free boundary system model, we have valued a corporate bond with credit rating migration risks in regime switch probability. This is a new model for measuring credit rating migration as well as regime switch. Some theoretical results, such as existence, uniqueness and regularities are obtained. Numerical results are also graphed and discussed.