Utility Indifference Valuation for Defaultable Corporate Bond with Credit Rating Migration

Credit risk modeling by debt pricing has been a popular theme in both academia and practice since the subprime crisis. In this paper, we devote our study to the indifferent price of a corporate bond with credit risk involving both default risk and credit rating migration risk in an incomplete market. The firm’s stock and a financial index on the market as tradable assets are introduced to hedge the credit risk, and the bond price is determined by the indifference of investors’ utilities with and without holding the bond. The models are established under the structural framework and result in Hamilton–Jacobi–Bellman (HJB) systems regarding utilities subject to default boundary and multiple migration boundaries. According to dynamic programming theory, closed-form solutions for pricing formulas are derived by implementing an inverted iteration program to overcome the joint effect of default and multiple credit rating migration. Therefore, with the derived explicit pricing formulas for the corporate bond, the models can be easily applied in practice, and investors can generate their strategies of hedging the credit risk by easily analyzing the impacts of the parameters on the bond price.


Introduction
Corporate bonds play important roles in the financing channels of firms. They are very popular in financial markets. Corporate bonds contain credit risk, which has been empirically verified as seriously influential on financial stability and the business cycle [1][2][3]. Especially after the financial crisis of 2008 and the European debt crisis of 2010, credit information has been more attractive to market participants. Investors demand trading strategies to hedge credit risk, and regulators are anxious to control credit risk contagion [4][5][6]. Credit risk refers not only to default risk [7][8][9], but also to credit rating migration risk [10][11][12], where the latter has received more and more attention. Credit ratings influence the prices of corporate bonds [13][14][15]. How do we price the corporate bond in different credit ratings, and how does the bond change its price when its rating migrates? These problems deserve answers. Accordingly, in this paper, we focus on the valuation of defaultable corporate bonds with credit rating migration, where the bond is supposed to be a zero-coupon.
We assume that the market is incomplete. Thus, if the corporate bond is considered as the claim written on the firm's asset value, theoretically, there is no longer a unique claim price. In this situation, the utility indifference valuation methodology, initiated by Hodges and Neuberger [16], is considered as a useful tool to ensure the unique claim price in the incomplete market, with the advantage including economic justification and the incorporation of risk aversion. Thus, in this paper, the prices of the corporate bonds are determined according to the utility indifference principle. Meanwhile, the credit between the firm's debt and the asset value, which results in a free boundary problem. Subsequently, Liang et al. [22] incorporated a risk discount factor, which measures the sensibility of credit rating migration to the proportion of the debt and asset value, into the model and showed a profile of an asymptotic traveling wave exists in the free boundary problem. Liang et al. [42] and Yin et al. [43] considered the situations of the time-varying risk-free interest rate [44,45]. Wu and Liang [46] provided some numerical results for multiple credit rating migration problems. More papers on credit rating migration modeling by the structural approach are Wu and Liang [47], Wu et al. [48], Wang et al. [49], Huang et al. [50,51], Liang and Zhou [52], and so forth.
We model the credit risk, involving default risk and credit rating migration risk, by using the Black and Cox's structural model (known as the first-passage-time model) and obtain the explicit formula of the corporate bond price with both default and credit rating migration probabilities in an incomplete market according to the utility indifference principle. Our work contributes to the literature on the indifference pricing methodology, the structural approach, and credit rating migration modeling.

Modeling
In this section, we develop the structural model to embed the utility indifference valuation framework for the defaultable corporate bond with credit rating migration. We follow the Black-Cox assumption on the default risk (Black and Cox [18] and Liang and Jiang [19]). The credit rating migration risk is captured according to Hu et al. [21] and Liang et al. [22], where the firm's credit rating is downgraded or upgraded as the proportion of the firm's debt and the firm's value passes some pre-set threshold.

The Market
Let (Ω, F , P) be a complete probability space. The market is built on four assets: a risk-free asset with constant interest rate r, a financial index, a stock, and a corporate bond, the latter two of which are issued by the firm. Let S t be the stock's value at time t, which satisfies: where the constants µ S and σ S represent the appreciation rate and the volatility of the stock, respectively, and (B S t ) t≥0 is the Brownian motion with its natural filtration {F S t } t≥0 . The financial index, whose price satisfies: dP t = µ P P t dt + σ P P t dB P t , is traded on the market and is non-defaultable, where the constants µ P and σ P represent the appreciation rate and the volatility of the stock, respectively, and (B P t ) t≥0 is the Brownian motion with its natural filtration {F P t } t≥0 .

The Firm
The firm's asset value, denoted by (V t ) t≥0 , is observed from the balance sheet, but cannot be traded on the market. We suppose that its value satisfies: where the constant µ V is the surplus of the expected return, that is the expected return subtracted by dividends, and σ V is the volatility of the firm's value. (B V t ) t≥0 is the Brownian motion, which generates the filtration {F V t } t≥0 . The firm's value V is correlated with its stock price S and the financial index price P. Their correlations are modeled by the correlations of the associated Brownian motions: It is natural to assume that the firm's value V and the stock price S are positively correlated. However, the financial index P may include financial institutions that are the counterparty of such a firm. Hence, the correlation between V and P might be negative. We

Default Risk
We suppose that the corporate bond has face value F and maturity T. Its price at time t is denoted by Θ t , which is the key quantity we would like to value in this paper. We follow the Black-Cox assumption on the contract. Default will occur at the first time when the firm's value V falls below the default barrier, denoted by D t , which evolves as: where D and α are positive constants. The default time is then the following stopping time: and the payoff of the contract is: where the constant ω ∈ [0, 1] is the recovery rate that the firm will pay for the holders of the corporate bond when default occurs.

Credit Rating Migration Risk
We follow the assumption of Hu et al. [21] on the modeling of credit rating migration. The credit rating of the firm migrates at the first time the proportion of the firm's debt and the firm's value, denoted by: passes some pre-set threshold. Suppose that there are n credit ratings considered in this paper and then n − 1 thresholds dividing the credit region. Denote these thresholds by: and then, the credit rating migration times are defined as follows: for i = 2, 3, · · · , n − 1, and: The volatility of the firm's value with a high credit rating is usually weaker than that with a low credit rating (Hu et al. [21]). Thus, denote by σ Vi the corresponding volatility of the firm's value in the i'th credit rating, i = 1, 2, · · · , n. These volatilities satisfy the order: Meanwhile, the volatility of the firm's value may spill over to its stock. This results in that the volatility of the stock is also heterogeneous in different credit ratings of the firm. Denote by σ Si the volatility of the stock in the i'th credit rating, i = 1, 2, · · · , n. They also satisfy the order:

The Investor
The holder of the corporate bond has a CARAutility function, which depends on his/her terminal wealth W: where γ ≥ 0 denotes the risk aversion parameter. More general and practice utility functions, such as the mean-variance utility function [53], used in [54,55], can be approximated by this CARA utility function. The investor with initial wealth W invests in either the stock S or the financial index P by following the admissible trading strategy π ∈ A ad [0, T]: If the investor invests in the stock S, then his/her wealth W S follows: where the wealth is divided into two parts: the first part W S t − π t is risk free, and the second part π t satisfies (1) by self-financing. Likewise, if the investor invests in the financial index P, then the wealth W P follows: dW P t = rW P t dt + (µ P − r)π t dt + σ P π t dB P t .

Maximal Expected Utility Problem
If the investor holds a corporate bond, he/she invests in either the stock S or the financial index P to hedge the credit risk (involving default risk and credit rating migration risk) from holding such a corporate bond. The indifference price of the corporate bond is determined by the comparison of two utility maximization problems with and without holding the corporate bond.
The first case is that the investor invests in the stock S. Suppose that the investor holds the corporate bond, i.e., he/she hedges the credit risk of the corporate bond by trading the stock S. The investor buys the corporate bond at time t with the price Θ t and invests in the stock with the remaining wealth W S t − Θ t . The investor faces both the default risk and credit rating migration risk during the time period [t, T]. If the firm defaults before migrating to another credit rating in the time period [t, T], i.e., on the set {τ D ≤ min{τ i , T}} for some 1 ≤ i ≤ n, meaning that the firm is in the i'th credit rating at time t, the investor gets W S τ D + H τ D . Since the firm collapses, the investor has no stock to trade after the default and then will put the remaining wealth W S τ D + H τ D into the bank account to earn the risk-free interest rate r. If the credit rating migrates before defaulting in the time period [t, T], a virtual substitute termination happens, i.e., on the set {τ i ≤ min{τ D , T}} for some 1 ≤ i ≤ n, the contract is virtually terminated and substituted by a new one with a new credit rating. If both the default and credit rating migration do not happen before maturity, i.e., on the set {min{τ D , τ i } ≥ T} for some 1 ≤ i ≤ n, the investor gets W S T + H T at maturity. Overall, the investor will maximize his/her utility: where: and: where: for i = 2, 3, · · · , n − 1, and: where: Suppose that the investor does not hold the corporate bond and only invests in the stock. As the stock is issued by the firm, the wealth of the investor is also exposed to both default risk and credit rating migration risk, despite he/she not investing in the corporate bond. The investor will maximize his/her utility: where:û and: for i = 2, 3, · · · , n − 1, and: If the bond holder hedges the credit risk of the corporate bond by trading the stock, the indifference price of the corporate bond Θ t is such that the value functions coincide, i.e., The second case is that the investor invests in the financial index P. Suppose that the investor holds the corporate bond, i.e., he/she hedges the credit risk of the corporate bond by trading the financial index P. The difference from investing in the stock is that after default, the investor can continue to trade the financial index, but with a shifted initial wealth depending on his/her loss from the default. The investor buys the corporate bond with the price Θ t and invests in the financial index with the remaining wealth W P t − Θ t at time t. If the firm defaults before credit rating migration during the time period [t, T], i.e., on the set {τ D ≤ min{τ i , T}} for some 1 ≤ i ≤ n, meaning that firm is in the i'th credit rating at time t, the investor gets W P τ D + H τ D at the default time τ D . After the default, he/she continues to trade the financial index on the market, but with the shifted initial wealth W P τ D + H τ D , and maximizes the utility: where W P still follows the wealth process (2), but with the shifted initial wealth W P τ D = W P τ D + H τ D . Then, (4) is the standard exponential utility maximization problem, whose solution is known as: which is the value function after the default on the set {τ D ≤ min{τ i , T}} for some 1 ≤ i ≤ n. If the credit rating migrates before default during the time period [t, T], a virtual substitute termination happens, i.e., the contract is virtually terminated and substituted by a new one with a new credit rating on the set {τ i ≤ min{τ D , T}} for some 1 ≤ i ≤ n. If both the default and credit rating migration do not happen before maturity, i.e., on the set {min{τ D , τ i } ≥ T} for some 1 ≤ i ≤ n, the investor gets W P T + H T at maturity. Overall, the investor will maximize his/her utility: where: and: where: for i = 2, 3, · · · , n − 1, and: where: Suppose that the investor does not hold the corporate bond. In this situation, both the default risk and credit rating migration risk will not impact the trading strategy of the investor. The problem reduces to a standard exponential utility problem: which can be solved explicitly as: If the bond holder hedges the credit risk of the corporate bond by trading the financial index, the indifference price of the corporate bond Θ t is such that the value function coincides, i.e.,

Pricing Characteristics by the HJB Equation System
The HJB equations are derived to characterize the indifference price of the corporate bond Θ t . The dynamic programming principle is employed. Before starting the lemma, let us define the following operators: and: Lemma 1. The value functions (U S i , i = 1, 2, · · · , n) are the viscosity solutions of the following HJB equation system: and: for i = 2, 3, · · · , n − 1, and: with default boundary: and terminal condition: In addition, on the credit rating migration boundaries, it holds that: The value functions ( U S i , i = 1, 2, · · · , n) are the viscosity solutions of the following HJB equation system: and: for i = 2, 3, · · · , n − 1, and: with default boundary: and terminal condition: In addition, on the credit rating migration boundaries, it holds that: The value functions (U P i , i = 1, 2, · · · , n) are the viscosity solutions of the following HJB equation system: and: for i = 2, 3, · · · , n − 1, and: with default boundary: and terminal condition: In addition, on the credit rating migration boundaries, it holds that: The proof of Lemma 1 can be found in Appendix A.

Indifference Pricing for the Corporate Bond
In this section, we solve the HJB equation systems (6)-(19) by a unified approach and derive the indifference price of the corporate bond under both situations, i.e., hedging credit risk by investing the stock issued by the firm itself or investing the financial index in the market.
Lemma 2. The HJB equation system (6)- (11) can be transformed into the following semi-linear parabolic equation system: and: for i = 2, 3, · · · , n − 1, and: with default boundary: and terminal condition: In addition, on the credit rating migration boundaries, it holds that: The HJB equation system (12)- (17) can be transformed into the following semi-linear parabolic equation system: and: for i = 2, 3, · · · , n − 1, and: with default boundary: and terminal condition: In addition, on the credit rating migration boundaries, it holds that: The HJB equation system (18)-(23) can be transformed into the following semi-linear parabolic equation system: and: for i = 2, 3, · · · , n − 1, and: with default boundary: and terminal condition: In addition, on the credit rating migration boundaries, it holds that: Now, we would like to solve system (27) for αt − log i + log Θ ≤ z ≤ αt − log i−1 + log Θ, i = 2, 3, · · · , n − 1, and: for log D ≤ z ≤ αt − log n−1 + log Θ, with default boundary: and initial condition:
Now, we are in the position of presenting the main theorem of the paper, which gives the explicit pricing formulas for the indifference price of the corporate bond. Theorem 1. Denote by (ψ i (t, Θ), i = 1, 2, · · · , n − 1) the credit rating boundaries: Denote by Θ i the price of the bond in the i'th credit rating of the firm, where i = 1, 2, · · · , n. Suppose that the bond holder hedges the credit risk of the corporate bond by trading the stock. Then, the prices of the corporate bond (Θ i , i = 1, 2, · · · , n) satisfy: where (H S i , i = 1, 2, · · · , n) are solutions of System (27)- (32) and ( H S i , i = 1, 2, · · · , n) are solutions of System (33)- (38), whose explicit formulas are given in Lemma 3. Suppose that the bond holder hedges the credit risk of the corporate bond by trading the financial index. Then, the prices of the corporate bond (Θ i , i = 1, 2, · · · , n) satisfy: for V ≥ −1 1 Θ 1 (t, V), where (H P i , i = 1, 2, · · · , n) are solutions of System (39)- (44), whose explicit formulas are given in Lemma 3.

Proof.
The results are easily derived by using the utility indifference pricing principles (3) and (5). In addition, as (H S i , i = 1, 2, · · · , n), ( H S i , i = 1, 2, · · · , n) and (H P i , i = 1, 2, · · · , n) are continuous across the credit rating migration boundaries, the price of the bond (Θ i , i = 1, 2, · · · , n) is also continuous across the credit rating migration boundaries, i.e., on the credit rating migration boundaries. Proposition 1. The credit rating regions in Theorem 1 given as: can be rewritten as: where {V S i , V P i , i = 1, 2, · · · , n − 1} are the credit rating migration boundaries. V S i stands for those in the situation that investors hedge credit risk by trading the firm's stock, while V P i stands for those in the situation that investors trade the financial index.
The existence of credit rating migration boundaries {V S i , V P i , i = 1, 2, · · · , n − 1} is achieved by using the implicit function theorem (see Appendix B). Hence, their explicit formulas cannot be derived.

Conclusions
In this paper, we devote our study to the corporate bond valuation with default probability possibly happening at any moment before maturity and the credit rating migration probability. We work under the structural framework, where the default follows Black and Cox's assumption and the credit rating migration follows the assumption that the credit rating migrates over the dynamic proportion of the firm's debt and asset value. As the market is supposed to be incomplete, then the utility indifference pricing methodology is used. Two representative assets are selected to hedge the credit risk and determine the utility indifferent price of the corporate bond, the stock issued by the firm itself, and a financial index from the market. The financial index is supposed to be independent of the firm's liquidation and can be traded continuously after the firm collapses, which is different from investing in the firm's stock. Thus, these two hedging strategies lead to different prices of the corporate bond. We present the explicit formulas for the indifferent prices of the corporate bond with default and credit rating migration in these two hedging strategies. Investors can analyze the effects of default and credit rating migration on the bond prices and credit spreads by our pricing formulas and easily test the sensibility of price with regard to the changes of parameters. In addition, the investors can compare the difference in prices in the two hedging strategies and generate their investment decisions by the credit information and price information involved in our explicit pricing formulas of the corporate bonds.
The multiplicity of credit rating migration really causes some difficulties for us to derive the closed-form solutions of the bond pricing formulas. However, this is solved by following Black and Cox's assumption and introducing a default boundary and an inverted iteration program. Finally, we prove the existence of the credit rating migration boundaries by the implicit function theorem. However, we cannot obtain their explicit formulas as functions of time. We have to acknowledge that this is a limitation of our results. By the dynamic programming theory, which means that the expected utility under the optimal strategy is no less than any other strategies, it holds that: On the other hand, the property of the supremum gives: Thus, (F i (t, V)) V | t=t 0 ,V=V 0 = 0 is equivalent to: H S i (t, Ve α(T−t) ) H S i (t, Ve α(T−t) ) V t=t 0 ,V=V 0 = γ(ρ 2 VS − 1) i e r(T−t 0 ) exp(γ(ρ 2 VS − 1) i e r(T−t 0 ) V 0 ).
However, the formulas of H S i (t, y) and H S i (t, y) imply that the left-hand side of (A3) cannot be the exponent of V 0 , i.e., (F i (t, V)) V | t=t 0 ,V=V 0 = 0. Thus, by the implicit function theorem, there exists a function V S i such that: Similarly, if the bond holder hedges the credit risk of the corporate bond by trading the financial index, we have: and: Thus, (F i (t, V)) V | t=t 0 ,V=V 0 = 0 is equivalent to: (H P i (t, Ve α(T−t) )) V | t=t 0 ,V=V 0 However, the formulas of H P i (t, y) imply that the left-hand side of (A4) also cannot be the exponent of V 0 , i.e., (F i (t, V)) V | t=t 0 ,V=V 0 = 0. Thus, by the implicit function theorem, there exists a function V P i such that: V = V P i (t) for 0 ≤ t ≤ T.