t t t t r t t
T T
,
t 1 h e E he RE
e-r (16)
where hs is the conditional probability at time s under a risk-neutral probability measure of default between s and s+1, given the information available at time s in the event of no default by s. Besides, r is the risk-free rate.
From Figure 5, it can be shown how the asymmetric jumps and noisy information create impacts on the term structure of credit spreads. First of all, noisy information tends to reshape the term structure from a hump to a monotone; in other words, the short spread will ascend and the long spread will descend while noise increases. This phenomenon results from the fact that the uncertainty causes the default probability having a more moderate variation with maturity, as illustrated in Figure 6. Secondly, as the arrival rate of jump increases, the term structure is still a humped-like curve, but the spread will be ascended over the whole range of maturity. Finally, the term structure of credit spreads is monotone if the both the asymmetric jump and the noisy information are simultaneously considered and large enough. The term structure of credit spreads therefore can be more enriched by including jumps and noise.
4 Conclusion
The pure diffusion approach for structural model with noisy information is generalized in this paper by including jumps in the firm-value processes. The explicit solution of the default
probability for this structural model is given, and numerical results show how the combination of the noisy information and the asymmetric jump brings the impacts on the credit risk.
Firstly, the opposite ways of impact of noise on the default intensity depends on whether the lower or higher range of the reported asset level is considered, and the noisy information will hence make the intensity with jump less sensitive with the reported asset level. Secondly, for the term structure of credit spread, the noisy information tends to reshape the term structure from a hump-like curve to a more flattened one; however, as the arrival rate of jump increases, the term structure is still hump-like, but the spread will be ascended over the whole range of maturity. Furthermore, it is found that the term structure of credit spreads is monotone if the two factors are simultaneously considered and large enough.
In short, the term structure of credit spreads is enriched by simultaneously incorporating the asymmetric jump and the noisy information into our credit risk model, and this generalization could be more potential to interpret empirical data in real world where rare events beyond the realm of normal expectations happen.
Reference
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[2] Duffie, D., K. Singleton. Credit Risk: Pricing, Measurement, and Management.
[3] Duffie, D., and D. Lando. “Term structures of credit spreads with incomplete accounting information”, Econometrica, 69, 3, pp. 633-664, May 2001.
[4] Guo, X., R. A. Jarrow, and Y. Zeng. “Credit Risk Models with Incomplete Information”, working paper, June 2008.
[5] John C. H.. Option, Futures, and other Derivatives. Pear.
[6] Kou, S. G.. “A Jump-Diffusion Model for Option Pricing”, Management Science, 48, 8, pp.
1086-1101, August 2002.
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Probab, 35, pp. 504-531, 2003.
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Table 1a: Comparison between Monte Carlo simulation and explicit solution of default probability. (λ=0.01). The simulation is based on 1,000,000 simulation runs.
5% 10% 25% 40%
0.25 year Formula 0.0086 0.0148 0.0170 0.0173 Simulation 0.0085 0.0153 0.0174 0.0178
Error (%) 1.176 3.27 2.30 2.81
0.5 year Formula 0.0235 0.0323 0.0352 0.0356 Simulation 0.0234 0.0323 0.0357 0.0355
Error (%) 0.427 0 1.40 0.282
1 year Formula 0.0591 0.0681 0.0706 0.0709 Simulation 0.0580 0.0682 0.0706 0.0708
Error (%) 1.90 0.147 0 0.141
Parameter setting:
86.3 86.3;
0.5;
0.03;
0.02;
0.01;
0.05;
0.01;
0 t
2 1
2
Vˆ V
η p 1 η
λ 1 2 σ
.
Table 1b: Comparison between Monte Carlo simulation and explicit solution of default probability. (λ=3). The simulation is based on 1,000,000 simulation runs.
5% 10% 25% 40%
0.25 year Formula 0.0491 0.0584 0.0601 0.0602 Simulation 0.0483 0.0574 0.0598 0.0592
Error (%) 1.66 1.74 0.50 1.69
0.5 year Formula 0.1053 0.1131 0.1127 0.1124
Simulation 0.1027 0.1109 0.1091 0.1083
Error (%) 2.53 1.98 3.30 3.79
1 year Formula 0.2064 0.2061 0.2005 0.1994 Simulation 0.2033 0.2028 0.1958 0.1963
Error (%) 1.52 1.63 2.40 1.58
Parameter setting:
86.3 86.3;
0.5;
0.03;
0.02;
3;
0.05;
0.01;
0 t
2 1
2
Vˆ V
η p 1 η
λ 1 2 σ
.
Figure 1: Probability of first passage time, for different arrival rate of jump and the probability of upward jump.
Parameter setting: 0.01; 0.05; 0.02; 0.03; 0 86.3; t 86.3
2 1
2
Vˆ η V
1 η
σ 1 2
.
Figure 2a: Conditional density of Vt, for different standard deviation of noise. ( 0.01,
p=0.5)
Parameter setting:
86.3 86.3;
0.03;
0.02;
0.5;
0.01;
0.05;
0.01;
0 t
2 1
2
Vˆ η V
1 η
p 1
2 σ
Figure 2b: Conditional density of Vt, for different standard deviation of noise. ( =3, p=0.5)
Parameter setting:
86.3 86.3;
0.03;
0.02;
0.5;
3;
0.05;
;
0
0 t
2 1
2
Vˆ η V
1 η
p 1 σ
01
2 .
Figure 2c: Conditional density of Vt, for different standard deviation of noise. ( =3, p=0.7)
Parameter setting:
86.3 86.3;
0.03;
0.02;
0.7;
3;
0.05;
0.01;
0 t
2 1
2
Vˆ η V
1 η
p 1
2 σ
.
Figure 3: Default probability for time horizon 1 year, varying standard deviation of noise.
Parameter setting: 0.01; 0.05; 0.02; 0.03; 0 86.3; t 86.3
2 1
2
Vˆ η V
1 η
σ 1 2
.
Figure 4a: Default intensity, for different probabilities of upward jump, with noiseless and noisy information.
Parameter setting: 0.01; 0.05; 3; 0.02; 0.03; 0 86.3; t 86.3
2 1
2
Vˆ η V
1 η
σ 1
2
Figure 4b: Default intensity, for different arrival rates of jump, with noisy information.
Parameter setting: 0.01; 0.05; 0.02; 0.03; 0 86.3; t 86.3
2 1
2
Vˆ η V
1 η
σ 1 2
Figure 4c: Default intensity, for different arrival rates of jump and varying standard deviation of noise. (Vˆt 86)
Parameter setting:
86 86.3;
0.03;
0.02;
0.7;
3;
0.05;
0.01;
0 t
2 1
2
Vˆ η V
1 η
p 1
2 σ
.
Figure 4d: Default intensity, for different arrival rates of jump and varying standard deviation of noise. (Vˆt 80)
Parameter setting:
80 86.3;
0.03;
0.02;
0.7;
3;
0.05;
0.01;
0 t
2 1
2
Vˆ η V
1 η
p 1
2 σ
.
Figure 5: Credit Spread, for different arrival rates of jump, with noiseless and noisy information.
Parameter setting: 0.05; 0.05; 0.7; 0.02; 0.03; 0 86.3; t 86.3
2 1
Vˆ η V
1 η
p 1 σ
r
Figure 6: The default probability, with noiseless and noisy information.
Parameter setting: 0.05; 0.05; 3; 0.7; 0.02; 0.03; 0 86.3; t 86.3
2 1
Vˆ η V
1 η
p 1 σ
r .
Appendix
Appendix A. Notations Notation 1: (for equation (4))
Note that to simplify the notation, we shall drop the superscript ′ in parameters, i.e., using
, , p , q , , 1 rather than 2
,, p , q , 1, 2.Note that to compute the Hh function, the three-term recursion of Hh which is 1
Notation 2: (for equation (5))
Note that to compute the H function, the three-term recursion of H which is
H
a,b,c;n
(1) The density functions are given by
(2) The tail probabilities are given by
Similarly,
Similarly,
Therefore,