Numerical Result

In this paper, we propose a difference between a Parisian option framework and a barrier option framework. Also, we want to present a difference between the double exponential jump diffusion model and the Merton jump-diffusion model. We follow the

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concept of Zhou’s approach (2001). We control the overall mean and volatility of the firm’s value to be constants as we change the parameter values which domain the random component of asset value. Therefore we know that the variations of bond values are truly caused by different combinations of parameter values rather than by the changes in overall mean and volatility of the firm’s value. To do this, we have to know under the risk-neutral measure P, the mean and volatility of return in these models. point is simulated 1 million times for precise value.

First, we want to present the difference of structure model between a barrier option and a Parisian option framework. In this case, we simulate the asset value under the double exponential jump diffusion. We control the parameter settings that total variance

0.09 1 months, 6 months, and 1 year to observe the effects caused by these changes. Because there are no apparent differences after 15 days caution time, figure 2 and 3 only presents the result of no caution time to 15 days caution time. Figure 2. presents the relationship between cumulative default probability and maturity under different caution time. It shows that longer caution time has less cumulative default probability. Figure 3 shows

4 Appendix C presents the moments in Merton jump-diffusion model and the double exponential jump diffusion model.

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Figure 2. The relationship between cumulative default probability and maturity in different caution time: ( ) : no caution time; (--): 5 days; ( ) : 10 days; ( ) : 15 days.

Figure 3. The relationship between credit spreads and maturity in different caution time: ( ) : no caution time; (--): 5 days; ( ) : 10 days; ( ) : 15 days.

0 1 2 3 4 5 6 7 8 9 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Maturity

Cumulative Default Probability

0 1 2 3 4 5 6 7 8 9 10

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Maturity

Credit Spread

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Figure 4. The relationship between average recovery value and maturity in different caution time: ( ) : no caution time; (--): 5 days.

that credit spread after 2 years maturity under barrier option framework is lower than Parisian option framework with 5 days caution time. The credit spreads decrease under Parisian framework with caution time increasing. To examine the result, we check the average recovery value of no caution time and 5 days caution time. We save the recovery value of no caution time and 5 days caution time in the same iteration. The result of average recovery value in figure 4. is consistent with figure 3. that no caution time has more average recovery value than 5 days caution time after 2 years maturity.

The high credit spread under 5 days caution is due to the low recovery value. This combination of parameter settings leads this result that the process of asset value usually goes down in 5 days after first hit time.

Second, we want to present the flexibility of the double exponential jump diffusion model. We compare the double exponential jump diffusion model with the Merton

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25 30 35 40 45 50 55

Maturity

Average Recovery Value

19 parameter settings to generate different skewness. In Merton jump-diffusion model, the flexibility of skewness is limited. In this case, we let skewness 0 and variance of pure diffusion  ² 0.01 generate a setting combination so that mean of jump size

0.016250257

  and variance of jump size  0.399738782 under the Merton jump-diffusion model. Also, we generate three setting combinations with different skewness under the double exponential jump diffusion model. First, we control

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Figure 6. the relationship between credit spreads and maturity in different model and skewness.

( ) : Merton jump-diffusion model with skewness 0 , 0.2 ,  ² 0.01 , skewness with different parameter settings. Figure 6 shows the relationship between credit spreads and maturity with different skewness under the Merton model and the double exponential jump diffusion model. The result of figure 6, that lower skewness

0 1 2 3 4 5 6 7 8 9 10

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Figure 7. The relationship between credit spreads and maturity in different model and skewness.

( ) : Merton jump-diffusion model with skewness 0 ,  =0.05,  0.03403074 ,

has lower credit spread, is not consist with the comment sense. The lower skewness has larger lose in a short maturity. It should have higher credit spread due to more probability to makes bonds default. In this case, We observe that the lower skewness has higher credit spread in a very short maturity. Therefore, we infer that the weight of variance between pure diffusion and jump size generate this result instead of skewness.

To check our inference, we let  =0.05 so that we can change the weight of variance between pure diffusion and jump size with the same skewness, p and q. We generate a setting combination that  0.03403074,  0.825697084 and  ² 0.48714694 under Merton jump-diffusion model. In addition, we generate the three parameter

0 1 2 3 4 5 6 7 8 9 10

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settings under double exponential jump diffusion, 1.  _u 1.83694595044413 , 1.58459906319395

 d and  ² 0.0552696595816093 with skewness  0.5, 2.

1.92390672760078

 u ,  _d 1.92390672759791 and  ² 0.0629838660328502 with skewness 0 , 3.  _u 2.07131120647897 ,  _d 3.54067693331008 and

2 0.0743575057803415

  with skewness 0.5. We increase the weight of variance in pure diffusion part in each setting. Figure 7. shows that the credit spread increases after 0.5 year maturity and decrease in a very short maturity. In the long term, the credit spreads under the same EX and Var X are very close. This result is consistent with ( ) our inference. Although the skewness is not a main reason affecting the shape of credit spread, skewness still limits the varieties of parameter combinations. However, the double exponential jump diffusion model has more flexibility of parameter setting if we control the moments of models. We can use this model generate more shapes of credit spreads.