Acknowledgments: I appreciate the valuable comments and suggestions of two anonymous referees, Dr Ji Young Kim (Bank of Korea) and Carl Taljaard. Any remaining errors are mine. This research was funded by a research grant from Kwangwoon University in 2009.

# Credit Spreads with Jump Risks and Stationary Leverage Ratio*

Article first published online: 10 FEB 2010

DOI: 10.1111/j.2041-6156.2009.00003.x

© 2010 Korean Securities Association

Additional Information

#### How to Cite

Kim, H.-S. (2010), Credit Spreads with Jump Risks and Stationary Leverage Ratio. Asia-Pacific Journal of Financial Studies, 39: 53–69. doi: 10.1111/j.2041-6156.2009.00003.x

^{†}

#### Publication History

- Issue published online: 10 FEB 2010
- Article first published online: 10 FEB 2010
- Received 26 June 2009; Accepted 25 September 2009

- Abstract
- Article
- References
- Cited By

### Keywords:

- Credit spreads;
- Default;
- Jump risk;
- Stationary leverage ratio;
- Structural model

- G12;
- G13;
- G33

### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Related Work
- 3. The Model
- 4. Calibration Methodology
- 5. Calibration Results
- 6. Effects of Jump Risks
- 7. Conclusion
- References
- Appendix

Recent structural models have utilized new factors to enhance their exploratory power over credit spreads. Some studies have shown that jump risks allow us to obtain credit spreads that are more realistic. However, according to the empirical studies on capital structure, another factor that affects credit spreads is the stationary leverage ratio of a firm. The present paper develops a simple structural model and incorporates both jump risks and the stationary leverage ratio to explain credit spreads. In comparison to the existing jump-diffusion structural model, this model generates a larger credit spread, which is more consistent with observed credit spreads, especially for investment-grade bonds. This paper also shows that jump frequency and size may be significant factors determining credit spreads for firms.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Related Work
- 3. The Model
- 4. Calibration Methodology
- 5. Calibration Results
- 6. Effects of Jump Risks
- 7. Conclusion
- References
- Appendix

The study of credit risk was pioneered by Merton (1974). Many researchers have developed credit risk models and have explored how their models can explain observed credit spreads. Huang and Huang (2003) investigate how the structural models explain observed credit spreads.^{1} They show that many structural models generate smaller credit spreads than observed credit spreads. However, according to other research (e.g. see Jones *et al.*, 1984), most structural models generate zero credit spreads for risky debt with very short maturity, even though they possess strong explanatory command over credit spreads for long-maturity debt. As a result, recent structural models have considered new factors to enhance their exploratory power. For example, a new process for determining a firm value and/or realistic capital structure has been considered.

First, some researchers have incorporated jump risks, providing more realistic results. In other words, the jumps in a firm value can generate high credit spreads. Zhou (2001), Kijima and Suzuki (2001), and Huang and Huang (2003) explore the credit risk of firms using the jump-diffusion process. In structural models that use the pure diffusion process, jump risks can provide non-zero credit spreads for risky bonds with very short maturities.

Second, according to empirical studies of capital structure (see Leary and Roberts, 2005), many firms tend to keep their leverage ratio stable. Collin-Dufresne and Goldstein (2001) develop a structural model in which a firm’s leverage ratio is stationary.^{2} They assume that the firm’s leverage ratio follows a mean-reverting process, and they then derive an analytical solution for the price of corporate debt. The stationary leverage model provides higher credit spreads than the constant threshold model. The model can also provide upward sloping credit spread curves for speculative firms. This is consistent with the results of a recent empirical study on credit spread curves (see Helwege and Turner, 1999). Even though the stationary leverage model has better implications, Huang and Huang (2003) show that it still underestimates the actual credit spreads.

In the present paper, a structural model is proposed that incorporates the two aspects mentioned above. To put it differently, a structural model is set up where the firm value is affected by jump risks and also by the stationary leverage ratio. To derive an analytical solution for the price of a risky bond, it is assumed here that the default occurs at the date of maturity, as does Merton. Although this assumption is restrictive, in the present paper, it is shown that the calculated credit spreads are closer to the observed credit spreads for the investment-grade bonds than those calculated by Huang and Huang (2003), or Collin-Dufresne and Goldstein (2001). The present study also investigates the effects of jump risks on credit spreads. It is found that jump frequency and jump size are important factors that affect credit risk as well.

This paper is organized as follows. Section 2 discusses related works. Section 3 provides a model for the price of a risky bond when both the firm’s leverage ratio is stationary and the firm value is affected by jump risks. In section 4, the calibration method is explained. Section 5 illustrates the calibration results. In section 6 the effects of jump risks on credit spreads are examined. Section 7 concludes.

### 2. Related Work

- Top of page
- Abstract
- 1. Introduction
- 2. Related Work
- 3. The Model
- 4. Calibration Methodology
- 5. Calibration Results
- 6. Effects of Jump Risks
- 7. Conclusion
- References
- Appendix

As stated in the previous section, most structural models provide very low credit spreads for short-maturity debt. There are two approaches to overcoming this fault. The first approach links structural models with reduced-form models by using incomplete-information structure. Traditional structural models are based on complete information about the firm value and the default boundary. In reality, however, the information one obtains is incomplete. Therefore, many researchers have considered a structural model based on incomplete information, upon which reduced-form models are defined. This approach is the incomplete-information model. These models were developed by Duffie and Lando (2001), Giesecke (2001), Collin-Dufresne *et al.* (2003), and Çetin *et al.* (2004). Duffie and Lando (2001) and Giesecke (2001) assume that one can indirectly calculate the value of the firm and the default barrier. Collin-Dufresne *et al.* (2003) assume that one can use only delayed information. The incomplete information on the firm’s value and default boundary leads to investors requiring a higher risk premium, and can generate higher credit spreads for short-maturity debt.

The second approach considers that jump risks can explain observed credit spreads. Zhou (2001) extends the first-passage structural model of Black and Cox (1976), who model firm value using a pure-diffusion process. Zhou (2001) provides the simulation method for pricing a risky bond. This is because the first-passage time distribution of the jump-diffusion process is not known in the case of a log-normally distributed jump size. Even though there is no closed-form solution for a risk bond, Zhou demonstrates many interesting observations about theoretical credit spreads. According to Zhou (2001), jump risks allow us to obtain non-zero credit spreads even when the maturity of a bond is very short. Kijima and Suzuki (2001) explore credit risks using a jump-diffusion process under the Merton model. They analyze bond prices in a complex capital structure. Huang and Huang (2003) investigate the amount of credit risk by studying credit spreads. In doing so, they deal with several main structural models, including the jump-diffusion structural model of Kou and Wang (2003). First, Kou and Wang (2003) derive the analytical formula for a first-passage time distribution of a jump-diffusion process, where the jump size follows double exponential distribution. They then develop the path-dependent option-pricing model using the analytical solution for a first-passage time distribution of the jump-diffusion process. Huang and Huang (2003) apply the Kou and Wang (2003) option-pricing model in a structural model and analyze the effects of jump risks on credit risk. They find that jump risks have a higher impact on credit risk relative to other structural models, even though the jump-diffusion structural model still underestimates observed credit spreads. Cremers *et al.* (2008) explore the effects of an option-implied jump risk premium on credit spreads. They modify the structural model of Huang and Huang (2003) by incorporating both systematic and firm-specific jump risks. They show that the option-implied jump premium generates higher credit spreads that are closer to observed ones.

In contrast to other structural models, Ahn *et al.* (2005) develop a hybrid credit risk model by extending the model of Madan and Unal (2000), using the jump-diffusion process.^{3} Even after controlling the unconditional variance under the jump-diffusion model to match the unconditional variance under the pure-diffusion one, they show that the effects of jump risks on credit spreads are still significant.

Of the two approaches mentioned above, the present paper adopts jump risks, but not the incomplete-information structure. That is, this paper uses the jump-diffusion process to represent the dynamics of a firm value. This paper also assumes that firm’s leverage ratio is stationary, as in Collin-Dufresne and Goldstein (2001). The structural models mentioned above assume that the firm’s leverage is not stationary, because the default threshold is either constant or time-dependent. In the next section, a simple structural model is developed in which the firm value is affected by jump risks under the assumption that its leverage ratio is stationary.

### 3. The Model

- Top of page
- Abstract
- 1. Introduction
- 2. Related Work
- 3. The Model
- 4. Calibration Methodology
- 5. Calibration Results
- 6. Effects of Jump Risks
- 7. Conclusion
- References
- Appendix

It is assumed that the dynamics of the logarithm of firm value are governed by

- (1)

where μ is the expected drift under the physical probability measure, *q* is the payout ratio, *σ* is the volatility of firm return, and *W*_{t} is a standard Brownian motion. Here, μ, *q*, and *σ* are constant. {*Z*_{t}, *t* ≥ 0} is a compound Poisson process with intensity *λ*, and the random variable of the jump size is ln(1 + *J*). That is, , where *N*_{t} is a Poisson process with intensity *λ*. Here, {ln(1 + *J*_{k}), *k* = 1,2,…} have independent and identical normal distributions, with mean ln(1 + *α*) – *δ*^{2}/2 and variance *δ*^{2}. For *t ≥ 0*, *W*_{t}*, N*_{t} and {ln(1 + *J*_{k}), *k* = 1,2,…} are mutually independent.

Zhou (2001), Kijima and Suzuki (2001), and Ahn *et al.* (2005) assume that the jump size is log-normally distributed. This assumption has been made in studies on option pricing (e.g. Merton, 1976; Bates, 1991, 1996; Bakshi *et al.*, 1997). In contrast to those studies, Huang and Huang (2003) and Cremers *et al.* (2008) adopt the double exponential distribution as the jump size distribution.

To consider the stationary leverage ratio, as in Collin-Dufresne and Goldstein (2001), the dynamics of the log-default threshold are represented by

- (2)

where *v* is the constant adjustment factor, as in Collin-Dufresne and Goldstein (2001). As explained by Collin-Dufresne and Goldstein (2001), the firm increases its leverage when the firm value is higher than the threshold level, and vice versa.

Let *l*_{t} be the logarithm of leverage (*l*_{t} = ln *K*_{t} – ln *V*_{t}), then the dynamics of the log-leverage is as follows:

- (3)

where

- (4)

Here, if is interpreted as the long-term mean log-leverage, *θ* represents the speed adjustment to the long-term mean log-leverage.

Because the price of risky bonds is derived under the risk-neutral probability measure, we need to transform the dynamics of firm value under the physical probability measure into those of firm value under the risk-neutral probability measure. Following Bates (1991, 1996) and Zhou (2001), if the representative agent has time-separable power utility, then under the risk-neutral probability measure, the firm value follows:

- (5)

where *r* is the risk-free rate,

- (6)

and the jump mean size is

- (7)

where *J*_{W} is the marginal utility of wealth and Δ*J*_{W} is the jump in marginal utility of wealth when a jump in firm value occurs.

As a result, under the risk-neutral probability measure, the dynamics of log-leverage ratio follow:

- (8)

where

- (9)

Then, the log-leverage is

- (10)

Although it is a more realistic assumption that the default might occur before the maturity of the debt, it is assumed that the default occurs on the maturity date, as in Merton (1974). This assumption allows us to obtain closed-form solutions for the price of corporate debt and credit spreads. The firm’s default occurs when the level of threshold *K*_{T} (i.e. the market value of debt) is greater than that of the firm on the debt maturity date. In other words, the default occurs when *l*_{T} = ln *K*_{T} – ln *V*_{T} > 0 or *K*_{T} > *V*_{T}.

Then the price of a risky zero-coupon bond with unit face value and the initial leverage ratio, *l*_{0}, is:

- (11)

where *w* is the loss rate and Pr^{Q}(*l*_{0},*T*) is the risk-neutral probability of default occurrence on the maturity date, *T*.

To obtain the price of risky bonds, Pr^{Q}(*l*_{0}, *T*) must be determined. To obtain an analytical default probability, the Bakshi and Madan’s option pricing methodology for pricing credit risks is applied.^{4} Let the characteristic function of *l*_{T} be *f*(0, *T*; *φ*) with parameter *φ*. Following Bakshi and Madan (2000), an analytical default probability using the characteristic function, *f*(0, *T*; *φ*) is obtained as follows^{5}:

- (12)

Therefore, to obtain a closed-form solution for the risky zero coupon bond, an analytical characteristic function, *f*(0,*T*;*φ*) should be derived. The analytical characteristic function is given by the following proposition.

**Proposition 1.** The characteristic function, *f*(0,*T*;*φ*) of *l*_{T} under the risk-neutral probability measure is

- (13)

where the function *g*(*x*) is the characteristic function of a normal random variable,

- (14)

and

- (15)

**Proof.** See Appendix.

To obtain credit spreads, the value of a coupon bond with semiannual coupons (c) and unit face value at the maturity date, *T*, are considered, as in Eom *et al.* (2004):

- (16)

where *T*_{j} (*j *= 1,2,…,2*T*) is the jth coupon date.

The yield to maturity for this risky coupon bond, *y* can be calculated using this equation:

- (17)

Therefore, the credit spread is calculated by the difference between the yield to maturity and the risk-free rate; that is, *y* – *r*.

### 4. Calibration Methodology

- Top of page
- Abstract
- 1. Introduction
- 2. Related Work
- 3. The Model
- 4. Calibration Methodology
- 5. Calibration Results
- 6. Effects of Jump Risks
- 7. Conclusion
- References
- Appendix

In this section our structural model is calibrated following the calibration method of Huang and Huang (2003). To compare credit spreads calculated through our model with those implied from the Collin-Dufresne and Goldstein (2001) model, and with the jump-diffusion model of Kou and Wang (2003), we must select the parameters used by Huang and Huang (2003) (see Table 1).

Coupon rate | 8.13% |

Riskless rate | 8% |

Recovery rate | 51.31% |

Mean-reversion coefficient | 0.2 |

Long-term mean leverage ratio | 38% |

Credit rating | Aaa | Aa | A | Baa | Ba | B |
---|---|---|---|---|---|---|

Initial leverage ratio (%) | 13.1 | 21.2 | 32.0 | 43.3 | 53.5 | 65.7 |

Diffusion volatility (%) (10-year maturity) | 31.0 | 27.2 | 24.3 | 24.5 | 31.3 | 38.7 |

Diffusion volatility (%) (4-year maturity) | 36.0 | 33.5 | 28.6 | 27.7 | 33.4 | 38.9 |

As reported by Huang and Huang (2003), the risk-free interest rate, *r*, is 8% and the coupon rate is 8.13%. The values of parameters associated with stationary leverage ratios are as follows: The mean-reversion coefficient (*θ*) is 0.2 and the long-term mean leverage ratio is 38% (from table 6 of Huang and Huang). Let *π** be the asset risk premium. Because the value of risky bonds is calculated under the risk-neutral probability measure, the long-term log-leverage ratio () under the risk-neutral probability measure is used. To back out from given parameters, we use the following equation:

- (18)

The initial leverage ratios across credit ratings are from table 8 of Huang and Huang.^{6}

Next, we focus on the set of parameters associated with jumps: *α*, *δ*, and *λ*. The intensity parameter (*λ*) represents the number of jump occurrences over 1 year. Huang and Huang (2003) adopt Kou and Wang’s (2003) jump-diffusion model, in which the jump size is assumed to follow a double exponential distribution.^{7}Huang and Huang (2003) consider a jump intensity of 3 and symmetric jumps; that is, the probability of an upward jump is the same as that of a downward jump. In addition, they assign the probability of an upward jump occurrence (*p*_{u}) at 0.5, the upward jump size parameter (*η*_{u}) at 30, and the downward jump size parameter (*η*_{d}) at 30.

To calculate the credit spreads generated by this model, we need the values of the parameters associated with jumps under the risk-neutral probability measure. We only have to calculate the risk-neutral jump mean, the risk-neutral jump volatility, and the risk-neutral jump intensity because values of the jump parameters under the physical probability measure are provided by Huang and Huang (2003). We also have to calculate the values of *α*, *δ*, and *λ* under the risk-neutral probability measure to match the jump mean and volatility in this model to those in Huang and Huang (2003).

To complete the full analysis, we provide the framework of Huang and Huang’s (2003) jump part before calculating *α*, *δ*, and *λ* under the risk-neutral probability measure. Let , , , and *λ*^{Q} be the upward (downward) jump probability, the upward jump size, the downward jump size, and the intensity under the risk-neutral probability measure, respectively. The relationship between parameters under the physical probability measure and parameters under the risk-neutral probability measure is as follows (see equation (14) of Huang and Huang (2003)):

- (19)

where *γ* is a constant relative risk aversion under a Lucas-type equilibrium framework (for more detail see Kou, 2002). In addition, let *ξ* and *ξ*^{Q} be the mean percentage jump sizes under the physical probability measure and under the risk-neutral measure, respectively. From equations (9) and (11) of Huang and Huang (2003), *ξ* (*ξ*^{Q}) is a function of *p*_{u}, *η*_{u}, and *η*_{d} (, , and ):

- (20)

- (21)

The intensity under the risk-neutral probability measure is as follows^{8}:

- (22)

Now we move onto the procedure for estimating *α*, *δ*, and *λ* under the risk-neutral probability measure. First, we extract *γ* from the jump risk premium *λξ – λ*^{Q}*ξ*^{Q}, the values of which are represented in table 8 of Huang and Huang (2003). Given the values of *p*_{u}, *η*_{u}, and *η*_{d}, the jump risk premium *λξ – λ*^{Q}*ξ*^{Q} is a function of the value of *γ* if we use equations (19)–(22). We can thus extract the value of *γ*. Second, if we use the value of *γ* and equations (19) and (22), then we can calculate the upward (downward) jump probability , the upward jump size , the downward jump size, , and the intensity, *λ*^{Q}, under the risk-neutral probability measure, *Q*. Third, we calculate the jump mean , the jump volatility and the jump intensity under the risk-neutral probability measure, *Q*. Finally, we calculate *α*, *δ*, and *λ* to match the jump mean and variance in this model with their counterparts in the Huang and Huang model. As a result, *α* takes values between 0.017 and 0.022 across various credit ratings. The value of *δ* is approximately 5%, irrespective of credit ratings. Note that we observe that the value of *λ* is approximately similar to that of *λ*^{Q}, which is consistent with Bates (1991) and Zhou (2001). For 10-year maturity bonds, the value of *λ* is 3 according to Huang and Huang, and the values of *λ*^{Q} range from 3.16 to 3.26. The next section provides the calibration results.

### 5. Calibration Results

- Top of page
- Abstract
- 1. Introduction
- 2. Related Work
- 3. The Model
- 4. Calibration Methodology
- 5. Calibration Results
- 6. Effects of Jump Risks
- 7. Conclusion
- References
- Appendix

It is well known that there is a big difference between credit spreads calculated by most structural models and observed credit spreads, especially for investment-grade firms. According to the empirical study by Jones *et al.* (1984), for investment-grade bonds with very short maturities, the structural models generate zero credit spreads, and these are inconsistent with the observed spreads. Even though the maturity is not very short (e.g. 4-year maturity bonds in Collin-Dufresne and Goldstein (2001)), credit spreads for investment-grade bonds are almost zero. In contrast to these results, the model calibrated in the present paper generates high credit spreads for 4-year maturity bonds.

Table 2 shows the comparison between credit spreads calculated in the present paper and in Huang and Huang (2003). Because the performance of the Huang and Huang model is better than that of the Collin-Dufresne and Goldstein model (2001), we focus on the comparison with the Huang and Huang (2003) model. To evaluate how the models compare against the observed credit spreads, we measure the percentage errors between the calculated credit spreads from the models and the observed credit spreads.

Credit rating | Average yield spread (bp) | Credit spread by this model (bp) | Percentage error (%) | Credit spread by HH (bp) | Percentage error (%) | Credit spread by CDG (bp) | Percentage error (%) |
---|---|---|---|---|---|---|---|

Panel A: Maturity = 10 years | |||||||

Aaa | 63 | 96.3 | 52.86 | 11.0 | 82.54 | 11.4 | 81.90 |

Aa | 91 | 108.3 | 19.01 | 15.8 | 82.64 | 14.9 | 83.63 |

A | 123 | 122.1 | 0.73 | 26.1 | 78.78 | 22.5 | 81.71 |

Baa | 194 | 139.3 | 28.20 | 61.4 | 68.35 | 52.3 | 73.04 |

Ba | 320 | 160.9 | 49.72 | 198.9 | 37.84 | 182.7 | 42.91 |

B | 470 | 187.5 | 60.11 | 394.9 | 15.98 | 371.6 | 20.94 |

Panel B: Maturity = 4 years | |||||||

Aaa | 55 | 67.8 | 23.27 | 1.7 | 96.91 | 0.0 | 100 |

Aa | 65 | 106.9 | 64.46 | 6.8 | 89.54 | 6.3 | 90.31 |

A | 96 | 147.8 | 53.96 | 11.4 | 88.13 | 9.9 | 89.69 |

Baa | 158 | 204.0 | 29.11 | 35.9 | 77.28 | 31.1 | 80.32 |

Ba | 320 | 277.1 | 13.41 | 180.7 | 43.53 | 168.0 | 47.50 |

B | 470 | 356.2 | 24.21 | 463.0 | 1.49 | 435.3 | 7.38 |

Interestingly, for investment-grade bonds, credit spreads calculated in this model are closer to the observed credit spreads than those calculated by Huang and Huang (2003). For 10-year maturity bonds, the relative error of this model is smaller than that of the Huang and Huang model in the case of investment-grade bonds. Although the relative errors of this model range from approximately 0.73% to approximately 53%, those of the Huang and Huang model are between 68% and 83%.

Next, we compare the 4-year credit spreads calculated using this model with the corresponding spreads calculated by Huang and Huang. For investment-grade bonds, the model in the present paper still overestimates credit spreads. However, the Huang and Huang model generates the opposite result. For AAA-rated firms, a 4-year observed credit spread is 55 basis points. The Huang and Huang model generates a 4-year AAA credit spread of 1.7 basis points, while the model calibrated in the present paper generates a spread of around 68 basis points. The relative errors generated by this model are smaller than those of the Huang and Huang model for all ratings except B-rated bonds. The relative errors of the Huang and Huang model are more than three times as great as those of this model. For only B-rated bonds, the credit spreads of the Huang and Huang model are very similar to actual data.

In addition, we can observe that the Collin-Dufresne and Goldstein model provides a similar result to the Huang and Huang one. That is, for speculative-grade bonds, the relative error of our model is greater than that of the Collin-Dufresne and Goldstein model. We infer that this is because the default can occur before the maturity of bonds in their model, even though no jump risks are considered.

In sum, the model in the present paper generates credit spreads that are close to observed credit spreads for investment-grade bonds. In addition, this model provides better explanatory command over observed credit spreads over the jump-diffusion model of Huang and Huang (2003), when jump risks and the stationary leverage ratio are factored in.

### 6. Effects of Jump Risks

- Top of page
- Abstract
- 1. Introduction
- 2. Related Work
- 3. The Model
- 4. Calibration Methodology
- 5. Calibration Results
- 6. Effects of Jump Risks
- 7. Conclusion
- References
- Appendix

In this section the effects of jump risks on credit spreads are investigated. The focus is especially on *λ*^{Q} and *α*^{Q}. The parameter values of A-rated firms are chosen as the base case.

To show the effects of jump risks on credit spreads, the total volatility of the firm value needs to be fixed^{9}; that is, we keep the total volatility of the firm value at 0.3014 in this section. The total volatility of the firm from equation (5) is:

- (23)

First, we explore the relation between credit spreads and jump intensity. Given that the value of *σ* is 0.2860, that is, the base case value, we choose a pair of values of *λ*^{Q} and *δ* to calculate the total volatility of the firm value at 0.3014. We choose the parameter values as follows: (*λ*^{Q}*, δ*) = (1.5, 0.0776) and (*λ*^{Q}*, δ*) = (6, 0.035).

The value of *δ* decreases by half while *λ*^{Q} increases by 4. Figure 1 shows credit spread curves for various values of *λ*^{Q}. A high jump intensity and a relatively small jump volatility generates a more apparent hump-shaped credit spread curve, compared to the shape of the credit spread curve in the base case. According to Merton (1974), the speculative-grade firm generally generates a hump-shaped credit spread curve. From this figure, we infer that more frequent jump risks may generate hump-shaped credit spread curves (especially for short maturity) in spite of the investment-grade firms.

When *λ*^{Q} = 6, credit spreads are more than twice as high as the corresponding credit spreads in the base case, even though the jump intensity value is twice as high as that of the base case. In addition, we observe that as the jump frequency declines, the hump-shape of the credit spread curve becomes weaker.

Figure 2 presents credit spreads with respect to various values of (*α*^{Q}, *δ*). Holding the total firm volatility constant, we assign the values of *α*^{Q} and *δ* as follows: (*α*^{Q}, *δ*) = (0, 0.0534) and (*α*^{Q}, *δ*) = (0.0348, 0.0420). When the jump intensity is fixed, the value of *α*^{Q} is a more important factor in determining credit spreads than that of *δ*. When *δ* is 0.0534, which is similar to the value of the base case, *α*^{Q} increases sharply and the credit spreads relative to the base case. The 5-year credit spread is approximately 149 basis points in the base case; the corresponding credit spread is approximately 218 basis points in the case where *α*^{Q} is zero. This result indicates that the premium of the jump size may be required in the market.

### 7. Conclusion

- Top of page
- Abstract
- 1. Introduction
- 2. Related Work
- 3. The Model
- 4. Calibration Methodology
- 5. Calibration Results
- 6. Effects of Jump Risks
- 7. Conclusion
- References
- Appendix

The previous theoretical studies on credit risk suggest that jump risks are a significant factor that has an impact on the credit spreads of firms. In addition, several empirical studies support the fact that the firm’s leverage ratio is mean-reverted. This paper considers a structural model where the firm value is affected by jump risks under the assumption that the firm’s leverage ratio is stationary. This assumption generates a larger credit spread, especially for investment-grade bonds. Compared to the results of Huang and Huang (2003), the difference in credit spreads generated by the model in this paper and observed spreads is smaller. The model in the present paper is a better performer for Ba-rated 4-year bonds as well as for investment-grade bonds. Examining the effects of jump risks, we infer that more frequent jump risks make the credit spreads much higher, even those of investment-grade bonds. We also explore the effects of jump risks on credit spreads and show that the jump size may significantly affect credit spread levels.

As stated previously, this paper assumes that the default occurs only at the maturity date, which is a limitation of this paper. To the best of our knowledge, there are no analytical solutions for the probability of default if we assume that the default can occur before the maturity date. Therefore, to obtain the theoretical credit spreads, one must resort to the simulation method. The investigation of default correlation between two firms in our framework is left for future study.

- 1
There are two main approaches to valuing credit risk: The structural model (e.g. Merton, 1974; Black and Cox, 1976; Leland, 1994; Longstaff and Schwartz, 1995; Leland and Toft, 1996; Collin-Dufresne and Goldstein, 2001) and the reduced-form model (e.g. Jarrow and Turnbull, 1995; Jarrow

*et al.*, 1997; Duffie and Singleton, 1999). The two approaches have different starting points. In the structural model, it is an important issue to set up the dynamics of the firm value process and the boundary of the firm’s default. In the reduced-form approach the firm’s default is modeled using a counting process (e.g. a Poisson process) and not based on the assumption of the firm value process. This paper is related to the structural model. - 2
Demchuk and Gibson (2006) extend the stationary leverage ratio model of Collin-Dufresne and Goldstein by allowing firms to adjust their leverage ratios according to the state of the market.

- 3
In addition to structural and reduced-form models, Madan and Unal (2000) propose a credit risk model that reconciles the differences between structural models and reduced-form models. They assume that the intensity of reduced-form models is a function of the firm value and the interest rate.

- 4
Following Bakshi and Madan (2000), Heston (1993) and Bakshi

*et al.*(1997), we obtain the desired probability by inverting the characteristic function. - 5
As stated in Heston (1993), the characteristic function decays rapidly.

- 6
We note that the two titles of columns (5) and (6) in table 8 of Huang and Huang (2003) should be switched. This can be seen from the two titles of columns (5) and (6) in table 9 as well as the caption of table 8.

- 7
To put it differently, ln (1 +

*J*) of equation (1) (*Y*≡ ln(*Z*) as a notation of Huang and Huang (2003)) has an asymmetric double exponential distribution with the density: (*η*_{u},*η*_{d}> 0 and*p*_{u}+*p*_{d}= 1). - 8
According to equation (13) of Huang and Huang (2003),

*λ*^{Q}=*λE*[*Z*^{−γ}] (in their notation), where ln(*Z*) has an asymmetric double exponential density. - 9
Zhou (2001) and Ahn

*et al.*(2005) also control the total volatility of the firm in this way.

### References

- Top of page
- Abstract
- 1. Introduction
- 2. Related Work
- 3. The Model
- 4. Calibration Methodology
- 5. Calibration Results
- 6. Effects of Jump Risks
- 7. Conclusion
- References
- Appendix

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### Appendix

- Top of page
- Abstract
- 1. Introduction
- 2. Related Work
- 3. The Model
- 4. Calibration Methodology
- 5. Calibration Results
- 6. Effects of Jump Risks
- 7. Conclusion
- References
- Appendix

##### Proof of Proposition 1

The log-leverage is

Because and are independent, the characteristic function of *l*_{T} is

- ((A1))

because has a normal distribution with mean 0 and variance (1 –e^{−2θT})/(2*θ*), . Here, we define *B*_{x}(*T*) as (1−e^{−xT})/*T*.

Now we focus on the jump component. Let *X* denote . Then we need to calculate *E*^{Q}[exp(*iφX*)]. Conditional on , the jump part is

- ( (A2))

where *τ*_{j} is the time of the jth jump occurrence. The conditional expectation of equation (A2) is

- ( (A3))

where are the order statistics of *τ*_{1,}…*,τ*_{n} and follow independent and identical uniform distribution on [0,*T*].

Therefore,

Let denote *ϕ*(*φ*). Therefore, equation (A2) becomes

- ( (A4))

Also, let us calculate *ϕ*(*φ*) as follows:

- ( (A5))

where *g*(*x*) is the characteristic function of a normal random variable. This completes the proof of Proposition 1. Q.E.D.