v 0
0
0 b xz,x t, dx
dx t, x , z x t, b
x , z x
g . (11)
Finally, the H –conditional default probability is given by t
s t
v
st,xx
g
xz,x0 t,
dxp (12)
where
st,xx
Ρ
tminhsXh xXt x
ΡtmaxhsXh 0X0 0
, V0 v0 (13)
where Xs Xs, X0 0.
3. An Analysis of the Credit Risk Model
In order to analyze how the combination of the noisy information and the asymmetric jump brings the impacts on the credit risk model, we provide several cases for the default probability, the default intensity, and the credit spread with different arrival rates of jump (λ) and standard deviations of noise (a). In these case, the initial asset level given one year ago
V =86.3 and the default boundary 0 v78 . We also suppose in all case that U has t expectation
2 a2
, so that E
eUt 1 implies an unbiased asset report. For the jump-diffusion process, r=0.05, σ0.05, 0.02η1
1 , and 0.03
η12
1 . Our basic case is
% 10
a . All parameters refer to Duffie and Lando(2001) and Kou and Wang(2003).
3.1 Default Probability
This section presents the numerical illustration for the default probability with varying
standard deviation of noise and different jumps to analyze the cooperative impacts of the asymmetric jump and noisy information on the default probability.
Table 1a and 1b compare the explicit solution computed by (14) with the Monte Carlo simulation in the two cases of λ0.01 and λ3. The simulation is based on 1,000,000 simulation runs.
Figure 1 shows the probability of first passage of a double exponential jump diffusion from an initial condition Vt v0 given v to a level below 0 v before time s. In order to
emphasize the result affected by the jump frequency and the probability of upward jump, the case of λ=3 with symmetric jump is compared with the case of λ=0.01 with symmetric jump, which is near to without jump and the case of λ=3 with asymmetric jump (p=0.7). The previous year’s asset level is set to be 86.3 arbitrarily. As can be seen from Figure 1, given a jump-condition, the probability of first passage time decreases in the actual asset level v at time t, which is in line with our intuition. Furthermore, given the same actual asset level v at time t, the probability of first passage time increases in jump frequency but decreases in the probability of upward jump, which is also in line with our intuition.
Figure 2a, Figure 2b, and Figure 2c show the conditional density of the current asset level V , given t H , that would be realized in the event that the bond has not yet default and t the current asset level reported Vˆt has a outcome equal to the previous year’s report. There are three cases of different jump-conditions considered here. One is λ=0.01, p=0.5 (shown
in Figure 2a); another is λ=3, p=0.5 (shown in Figure 2b); still another is λ=3, p=0.7 (shown in Figure 2c). And the standard deviation of noise is set as 5% and 40% for all cases.
The previous year’s asset level is set to be 86.3 arbitrarily. It can be shown that in all cases, the density becomes flattened as the standard deviation of noise (a) increases, and the peak density is shifted to the right with jump, p=0.7, which is asymmetric and favoring upward. It is worthy to note that the amount of this shift due to asymmetric jump is magnified by the dubious level of information. It seems that the asymmetric jump cooperates with the information noise and creates a composite effect on the conditional density.
Figure 3 illustrates outcomes of the conditional default probability for cases of different jumps and various levels a of noise. One is for λ0.01and p=0.5; another λ3 and p=0.5;
and the other λ3 and p=0.7, given the same previous year asset level of 86.3. For the case of λ0.01, representing a jump of once per hundred year, which is considered rare to happen, the curve we obtained is quite close to that obtained by Duffie and Lando (2001). It means that our generalized model can be reduced to Duffie and Lando’s model. The default probability will increase first and then converge to a certain level as the standard deviation of noise increases. This is because the noise affection will eventually saturate if the standard deviation of noise is large enough. For λ3and p=0.5, the whole curve of the default probability is raised by the jump, and the point probability of the curve increases first as in the case of λ=0.01, reaches the peak value, and then decreases to converge to a saturation level
as the information become more and more dubious. For the case of λ3and p=0.7, the whole curve of the default probability is lowered with respect to the case of λ3and p=0.5, because the asymmetric jump favors upward by assumption; there is also a decrease-to-saturation phenomenon in this case of asymmetric jump. The decrease-to-saturation phenomenon is the result of cooperation of the jump and noisy information. As shown in the equation (12), the default probability is equal to the inner product of the probability of first passage time
st,xx
and the conditional probability
xz,x t,
dxg 0 . From Figure 2a-2c, it can be seen that noisy information makes the conditional
density more flattened. At the lower and higher actual asset level, the conditional density increases in the standard deviation of noise, which generates an affection that increases the default probability, whereas the conditional density decreases in the standard deviation of noise at the middle actual asset level, which generates a contrary affection that decreases the default probability. The default probability is determined by these two contrary powers.
Because the probability of first passage time decreases and converges to zero more slowly, the affection that decreases the default probability can be displayed more obviously in the case of
3
λ than in the case of λ=0.01, which result in the decrease-to-saturation phenomenon.
3.2 Default Intensity
We turn now to the implications of asymmetric jumps and noisy information for the default
intensity of the modeled firm.
By the definition, the default intensity is a local default rate, in that
h h t, t t lim p
0 t h
Ht
. (14) Figure 4a illustrates the different between the default intensity of the modeled firm, the asset value of which has an asymmetric jump, with and without perfect information; besides, the special case of one-sided jump is also given. From this figure, we have three findings about the default intensity. First, the intensity with noise (a=10%) is less sensitive with reported asset level than the intensity without noise (a=0) as long as the jump is considered in the asset process. However, in the case that the value of asset follows a pure diffusion ( =0), the intensity with noise is always greater than the intensity without noise, which is zero at any reported asset level (Duffie and Singleton 2001). Second, if only the upward jump is included, the intensity is close to zero regardless of information quality. Third, at the same probability of upward jump, the intensity increases over the whole level of the reported asset as the arrival rate of jump increases, as illustrated in Figure 4b, whereas the intensity decreases for the low reported asset level but increase for the high reported asset level as information become more dubious, as illustrated in Figure 4a. It is suggested that there is a neutral point where the intensity is not affected.
From Figure 4c, it may be concluded that the arrival rate of jump creates greater impact on the intensity as the standard deviation of noise increases (a) from 0 to 25% at the reported
asset level (86) higher than the neutral point. In contrast, from Figure 4d, at the reported asset level (80) lower than the neutral point, the impact of jump initially increases as a increases from 0 to 5% and then decrease as a increases from 5% to 25%. In addition, the impact of noise on intensities will be saturated as the standard deviation of noise is large enough for both higher and lower reported asset level.
3.3 Credit Spreads
According to Kou and Wang(2004), under the risk-neutral probability measure , the return process given by
Nt
1 i
i t
2
t t W Y
2 r 1
X , 0X0 . (15)
Where Wt is a standard Brownian motion under ,
Nt ;t0
is a Poisson process with intensity . The log jump size
Y1,Y2
still from a sequence of i.i.d. random variables with a new double exponential density fY(y) p1e1yIy0q2e2yIy0 The constants1
0, 0;
1
, 2 p ,q p q
1
, and :
1 1
1 q 1 e p
E
2 2 1
Y 1
. All sources of
randomness, Nt, Wt, and Ys, are still independent under .
For a given time to maturity T, the yield spread on a given zero-coupon bond sold at a price 0 is the real number such that t,T e-rTt. We assume the bond holder can receive some fraction R
0,1 of its market value Vt at default. According to Duffieat time t is given by
r t t
t t,T
t T , 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.