An iterative method for solving a PDE with free boundary arising from pricing corporate bond with credit rating migration

: In this paper an iterative method is proposed to solve a partial di ff erential equation (PDE) with free boundary arising from pricing corporate bond with credit grade migration risk. A iterative algorithm is designed to construct two sequences of fixed internal boundary problems, which produce two weak solution sequences. It is proved that both weak solution sequences are convergent. In each iteration step, an implicit-upwind di ff erence scheme is used to solve the fixed internal boundary problem. It is shown that the scheme is stable and first-order convergent. Numerical experiments verify that the limit of the weak solution sequence is the solution of the free boundary problem. This method simplifies the free boundary problem solving, ensures the stability of the discrete scheme and reduces the amount of calculation.


Introduction
In recent years, with the frequent occurrence of financial risk events, more and more attention has been paid to the credit risks of financial products. Credit risks of financial products include both default risks and credit grade migration risks. The previous research pays more attention to default risk, but now credit grade migration risk has become an important role in the bond risk managements. The upgrade or downgrade of credit rating will affect the value of corporate bond. The free boundary models have been established in [2,6,8,[10][11][12]15] for pricing corporate bonds with the characteristic of credit grade migration risk, in which the free boundary is determined by the ratio of corporate debt to corporate value.
In this paper we study the following PDE with free boundary for pricing a corporate bond with the characteristic of credit grade migration risk [6,13] ∂v L ∂τ Here S is the corporate asset value, τ is the time, v L (S , τ) and v H (S , τ) are the bond values in low and high credit grades respectively, σ L and σ H (0 < σ H < σ L ) are volatilities of the corporate asset value under the low and high credit rating respectively, γ (0 < γ < 1) is the threshold ratio of corporate debt to corporate asset value, r is the risk-free rate of interest, and F is the face value of the bond. Generally, it can be assumed that F = 1. The defined domain of the free boundary problem is divided into a low rating region Ω L where 0 < S < 1 γ v L and a high rating region Ω H where S > 1 γ v H . It has been proved that two domains are separated by a free boundary s(τ), and At the credit rating migration boundary s(τ), the values of the bond in low and high credit rating satisfy The above problem requires not only solving the value of the bond, but also solving the free boundary. In financial engineering, pricing financial products with free boundary has long been recognized as a very challenging problem. A few numerical methods have been used to solve such problems. Explicit difference schemes are used in [6,11,15] to solve free boundary problems for pricing corporate bonds with credit grade migration risks. A front fixing method is derived in [7] to solve problems (1.1)-(1.5), which transforms the free boundary into a fixed boundary by including the unknown boundary into the equation, resulting in the differential equation becoming a nonlinear equation. For the transformed fixed boundary problem, the predictor-corrector algorithm and Newton-like iterative algorithms are used to solve the difference equations in [7]. The predictor-corrector algorithm is also an explicit discrete scheme that needs to satisfy the stability conditions, while the Newton-like iterative method needs a lot of computation to solve the nonlinear difference equations.
In this paper, we propose a novel method to solve the PDE with free boundary (1.1)-(1.5). An iterative algorithm is designed to generate weak solution sequences of fixed internal boundary problems. It is proved that both weak solution sequences are convergent. Since it is not easy to obtain analytical solutions of the fixed internal boundary problems, numerical methods are used to solve them. In each iteration step, an implicit-upwind difference scheme is applied to solve the fixed internal boundary problem. The stability and convergence order of the discrete scheme are given. Numerical experiments verify that the limit of the weak solution sequence is the solution of the free boundary problem and also verify that the discrete scheme is stable and first-order convergent. The advantages of this method are reflected in three aspects: first, the free boundary problem is transformed into a sequence of fixed internal boundary problems, which simplifies the problem and deepens the understanding of this free boundary problem; second, the implicit scheme is used to solve the fixed internal boundary problem in each iteration step, so as to ensure the stability of the discrete scheme without additional constraints; third, this method only involves solving the root of a single nonlinear equation without solving the system of nonlinear equations, which reduces the amount of calculation.

Iterative method
By using the variable transformations x = ln S and t = T − τ, and defining we can derive the following equation from (1.1)-(1.5) and the assumption F = 1 where x * (t) is the free boundary transformed from s(τ). Here x * (t) is an apriorily unknown function since it should be solved by the following equation where the solution u is also an apriorily unknown function.
Let u H (x, t) and u L (x, t) be the solutions of problems respectively. Then, using the method for solving the classical Black-Scholes equation [9], we can get the solutions of problems (2.3) and (2.4) as follows Similar results have been given in the literature [12]. Furthermore, the following result can be obtained, which also has been proved in [ Next, according to the theory of the linear parabolic equation [4], we construct two weak solution sequences ū (k) and for ρ > 0 and Q ρ = (−ρ, ρ) × 0, ρ 2 , which are generated as follows When the solutionū (k−1) of the k − 1 iteration is known, the existence and uniqueness ofx (k) can be derived from the results in Lemmas 2.2 and 2.3. Whenx (k) is known, the iteration equation of the k-th order is a fixed internal boundary problem. For the fixed internal boundary problems in (2.5), they are parabolic equations with discontinuous coefficients as discussed in [3,5,14]. By using a construction method as used in [5] we can prove that the parabolic equation with discontinuous coefficients exists a solution. Letū (k) 1 (x, t) andū (k) 2 (x, t) be particular solutions of the following differential equations respectively Consider the following function where ϕ 1 (x, t) and ϕ 2 (x, t) are the solutions of the following parabolic problems respectively By imposing the conditions we can get A(t) and B(t). From this we conclude that the parabolic equation with discontinuous coefficients in (2.5) exists a solution. Similar results can be obtained for (2.6). Next, we give some properties of iterative solutions. Lemma 2.2 The weak solutionsū (k) and u (k) of problems (2.5) and (2.6) satisfy Proof. Hu et al. [6] regard the free boundary problem (2.1) as a parabolic equation with discontinuous coefficients, and apply the maximum principle to prove in Lemmas 4.2 and 4.5 and Theorem 5.1 of [6] that the properties (2.7)-(2.10) hold true for the solution of the free boundary problem (2.1).
For the fixed internal boundary problems in (2.5) and (2.6), they are also parabolic equations with discontinuous coefficients. The only difference between the two equations is that the coefficient σ is different.
it can be proved by the same method that the results (2.7)-(2.10) for the fixed internal boundary problems in (2.5) and (2.6) also hold true for 0 < t ≤ T and the inequalities (2.7)-(2.10) become equations for t = 0, where H(ξ) is the Heaviside function and u is the solution of the free boundary problem. □ In order to simplify the expression, we introduce the following problems for j = 1, 2, which are any two iterative equations in the iterative problems (2.5) and (2.6). Furthermore, let s j (t) with j = 1, 2 be the solutions of the following problems v j s j (t), t = γe s j (t) , 0 ≤ t ≤ T, j = 1, 2, (2.12) respectively. By making the variable transformation v j = e x w j , problems (2.11) and (2.12) can be reduced to and w j s j (t), t = γ, 0 ≤ t ≤ T (2.14) for j = 1, 2, respectively. Applying Lemma 2.2 we can get and for j = 1, 2. Combining the variable transformation v j (x, t) = e x w j (x, t) and inequalities (2.15) and (2.16) we obtain and Proof. For each j, it is assumed that there exist two solutions s 1 j (t) and s 2 j (t) to the problem (2.12). Suppose there exists t 0 ∈ [0, T ] such that Since s 1 j (0) = s 2 j (0) = 0, we have t 0 0. Moreover, from (2.14) and (2.17) we have which is a contradiction. Hence, for each j we have s 1 j (t) ≤ s 2 j (t). Similarly, for each j we also can get s 2 j (t) ≤ s 1 j (t). Therefore, for each j we have s 1 j (t) = s 2 j (t), which implies that the problem (2.12) have a unique solution s j (t) for each j.
Furthermore, it is easy to prove that each iterative equation in problems (2.5) and (2.6) has a unique solution by using the maximum principle as given in [3,5,14]. □ Lemma 2.4 For the solutions v j (x, t) of problems (2.11), the following results hold true: Proof. Set z(x, t) = v 2 (x, t) − v 1 (x, t) and

(2.19)
Then, if y 1 (t) ≥ y 2 (t) for t ∈ [0, T ], from (2.11) and (2.19) we have where we have used Lemma 2.2. Obviously, z(x, 0) = 0. Hence, it follows by the maximum principle in the sense of weak solution that (2.20) By making the variable transformations v j = e x w j , problems (2.11) can be reduced to problem (2.13). Set ψ(x, t) = w 2 (x, t) − w 1 (x, t) and Then, if y 1 (t) ≥ y 2 (t) for t ∈ [0, T ], from (2.13) and (2.21) we have where we have used (2.18). It is also obvious that ψ(x, 0) = 0. Hence, it follows by the maximum principle in the sense of weak solution that which implies Furthermore, we can prove the following inequality Then, from (2.14), (2.17) and (2.26) we have which is a contradiction.
Combining inequalities (2.20) and (2.27), we conclude that the results in (i) hold true.
Proof. From Lemma 2.1 we have Then, from (2.4) and (2.5) we havē where we have used Lemma 2.2. It is also obvious thatv (1) (x, 0) = 0. Hence, it follows by the maximum principle in the sense of weak solution thatv (1) ≥ 0, i.e., Then, by using Lemma 2.4 and (2.34) we can get (2) and Then, from (2.3) and (2.5) we havē where we also have used Lemma 2.2. It is also obvious thatv (2) (x, 0) = 0. Hence, it follows by the maximum principle in the sense of weak solution thatv (2) ≥ 0, i.e., Then, by using Lemma 2.4 and (2.37) we can get Next we assume that the inequalities (2.28) and (2.29) hold true when the number of iterations is not greater than 2k. Then we havē Thus, from (2.41) and Lemma 2.4 we havē Hence, it can be seen from the induction that the inequalities (2.28) and (2.29) hold true for all k. Thus, by using Arzelà-Ascoli Theorem, we can prove that the monotone bounded sequences x (2k−1) and ū (2k−1) are convergent respectively. Similarly, we can also prove that the monotone bounded sequences x (2k) and ū (2k) are convergent respectively. □ Remark. Although Theorem 2.5 does not prove that the limit of the solution sequence is the solution of Eq (2.1) in the classical sense, it can be considered to prove that the limit satisfies Eq (2.1) in the sense of distribution [1]. Considering that this paper focuses on numerical calculation, we use numerical experiments to verify that the limit is the solution of Eq (2.1). Using the same method for proving Theorem 2.5 we also can obtain the following results. Theorem 2.6 The solution sequence u (k) , x (k) of problem (2.6) satisfies which imply that the weak solution sequence u (k) and the internal boundary sequence x (k) are convergent respectively.

Discretization
Since it is difficult to get the analytical solution for problems (2.5) and (2.6), we use an implicitupwind difference scheme to solve them.
First, the spatial domain (−∞, ∞) is truncated into a finite domain [x min , x max ]. The boundary conditions are chosen to be u(x min , t) = u L (x min , t) and u(x max , t) = u H (x max , t). Generally, the error caused by the truncation of the domain is negligible for the value of the bond. A uniform mesh and Ω K ≡ {t j t j = j△t, △t = T/K }.
For the differential operator an upwind difference scheme on Ω N is utilized to approximate the spatial derivatives and an implicit Euler method on Ω K is utilized to approximate the time derivative: For the differential operator an upwind difference scheme also is utilized to approximate the left and right derivatives: It is easy to know that the discrete scheme satisfies the maximum principle, which can be derived from the fact that the matrix related to the discrete operator L N,K , l N,K is an M-matrix. Then, we can conclude that the discrete scheme is unconditionally stable and is first-order convergent by the maximum principle. Furthermore, the nonlinear equation

Numerical experiments
In this section we present some numerical results to indicate experimentally the efficiency and accuracy of our method. We consider the same example as given in [6,7].
To numerically calculate Eqs (2.5) and (2.6), the stopping criterion of the iterative algorithm is chosen as whereŪ and U are the numerical solutions of Eqs (2.5) and (2.6) respectively. The comparison between our numerical results and those of the explicit difference method given in [6] shows that they are very consistent, which are presented in Table 1.  Figure 1 displays the iterative solutions at t = 0 for the iterative equation (2.5), which shows that the even number of iterative solutions are above the numerical solution U and the odd number of iterative solutions are below the numerical solution U, and the iterative solutions are closer to the solution U with the increase of the number of iterations. Figure 2 displays the iterative solutions at t = 0 for the iterative equation (2.6), which shows that the odd number of iterative solutions are above the numerical solution U and the even number of iterative solutions are below the numerical solution U, and the iterative solutions are also closer to the solution U with the increase of the number of iterations. Figure 3 gives the numerical solution of the corporate bond with credit rating migration and Figure 4 gives the numerical solution of the free boundary caused by credit rating migration. It's easy to see from Figures 3 and 4 that the function of the bond value has been decomposed into two regions by a free boundary and the free boundary is decreasing as expected.     Figure 4. Free boundary.
Since the example has no analytical solution, the double-layer mesh principle is utilized to calculate the error and the corresponding convergence rate. The error in the discrete maximum norm is denoted by The maximum errors, convergence rates and number of iterations in the calculation of (2.5) and (2.6) for Example are presented in Table 2, which show that the discrete scheme is stable and first-order convergent. From the perspective of convergence order, our numerical method and the methods given in [6,7] are first-order convergent. Compared with the explicit difference methods given in [6,7], we use the implicit scheme to solve the fixed boundary problem in each iteration step, which ensures the stability of our discrete scheme without additional constraints. Compared with the front fixing method with Newton-like iterative algorithms given in [7], our method only involves solving the root of a single nonlinear equation without solving the system of nonlinear equations, which reduces the amount of calculation.

Conclusions and discussion
An iterative method for a PDE with free boundary arising from pricing corporate bond with credit grade migration risk has been proposed. The key to the success of this method is that the constructed iterative algorithm produces two weak solution sequences of fixed internal boundary problems which are proved to be convergent. Since it is not easy to obtain analytical solutions of the fixed internal boundary problems, numerical methods are used to solve them. In each iteration step, an implicitupwind difference scheme is used to solve the fixed internal boundary problem, which ensures the stability of the discrete scheme without additional constraints. Moreover, this method only involves solving the root of a single nonlinear equation without solving the system of nonlinear equations, which reduces the amount of calculation. It is shown that the scheme is stable and first-order convergent. Numerical experiments verify that the limit of the weak solution sequence is the solution of the free boundary problem, and numerical experiments also verify the stability and convergence order of the discrete scheme. The study of this paper broadens the method of solving free boundary problems. In future we extend this method to solve two dimensional models [11].