1. An Reexamination of Jump Effect on Credit Spreads with Noisy Information Lung-fu Chang, Department of Finance, National Taipei College of Business

2. Introduction A sudden financial turmoil often causes market values of firms and bond prices to drop in a surprising manner, such as the subprime mortgage crisis. Default risk analysis plays a critical role in managing the credit risk of bank loan portfolios.

3. Two types of default risk have been widely used for modeling credit risk - jump risk and noisy information Introduction

4. Zhou (2001) and Scherer (2005) incorporate jump risk into the default process and suggest that a firm, the assets of which are perfectly observable, can default instantaneously because of a sudden drop in its value, and the short credit spread of this firm is also bounded away from zero. The advantage of the jump-diffusion model can generate a variety of the term structure of credit spreads, which is consistent with the empirical studies of these shapes of credit spread curves provided by Sarig and Warga (1989) and Fons (1994). Introduction

5. Because of imprecisely observed firm value and incomplete accounting information, Duffie and Lando (2001) assume the reported assets are modeled as the true firm value plus a normal noise term and show the effect of incomplete accounting information on credit spreads. A jump in market information, such as the corporate accounting scandal of Enron, will result in a sudden drop in firm values. The jump caused by incomplete information is regarded as a case of many kinds of jumps. Introduction

6. According to Kou (2002), the double exponential jump-diffusion can capture the empirical phenomenon of the asymmetric leptokurtic features - the return distribution is skewed to the left and has a higher peak and two heavier tails than those of pure Brownian motion. The double exponential jump-diffusion is considered asymmetric and therefore appropriate to capture more realistic jumps. Introduction

7. In this paper, the double jump-diffusion process is chosen as the basic model to represent the asset value of a firm and the noisy information assumption proposed by Duffie and Lando (2001) is incorporated for the incomplete information market. The effects of asymmetric jumps and noisy information on the credit risk are illustrated with numerical results in reference to the default probability, default intensity, and credit spread. Our analysis shows that the term structure of credit spreads is enriched by simultaneously incorporating the asymmetric jump and the noisy information into the credit risk model. Introduction

8. The Basic Model We suppose that the value of a company follow a stochastic process, and,where follows a normal distribution with mean and variance and is independent of. The return process is supposed to follow a double exponential jump-diffusion.

9. Under the physical probability measure, the process is given by:, =0 (1) where is the standard Brownian motion, is a Poisson process with rate, and is a sequence of independent identically distributed (i.i.d.) nonnegative random variables that has an asymmetric double exponential distribution with the density. The Basic Model

10. The parameters include,,, and. All randomness,,,and are assumed to be independent; the drift and the volatility are assumed to be constant, and the Brownian motion and jumps are assumed to be one dimensional. The Basic Model

11. The moment-generating function of is, where the function is defined as (2) The Basic Model

12. Where, the first time that the asset falls below. The Basic Model

13. is defined as a probability of the event that, conditional on and ; that is (3) where. The Basic Model

14. The cumulative distribution function of where,, and which have the distribution of where (4) The Basic Model

15. And the joint cumulative distribution of and (5) The Basic Model

16. The Basic Model

17. The Basic Model (6)

18. (7) The Basic Model

19. The Basic Model

20. Since the cumulative distributions of the first passage times are given in terms of Laplace transforms, numerical inversion of the Laplace transforms becomes necessary. For this purpose, the Gaver- Stehfest algorithm, which does the inversion on the real line, is used in this paper. The Basic Model

21. For any bounded and continuous function defined on,, where and is the Laplace transform of, i.e.. (8) The Basic Model

22. The density of, given and “killed” at where is set to be, conditional on the observation, can be calculated. The Basic Model

23. where denotes the density of and (9) The Basic Model

24. where (10) The Basic Model

25. The Basic Model

26. According to the result of Duffie and Lando (2001), the density of, conditional on the noisy observation and on, is given by (11) The Basic Model

27. –conditional default probability is given by (12) where where. (13) The Basic Model

28. The initial asset level given one year ago is =86.3 and the default boundary is. We also suppose in all cases that has the expectation of, so that implies an unbiased asset report. For the jump-diffusion process, r=0.05,,,and. Our basic case is. All parameters refer to Duffie and Lando (2001) and Kou and Wang (2003). Table 1a and 1b show that the default probability computed by our proposed explicit solution based on Equation (14) with that evaluated by the Monte Carlo simulation method in the two cases of and An Analysis of the Credit Risk Model

29. The simulation is based on 1,000,000 simulation paths. The error is defined as the difference between a Monte Carlo estimate and a estimate computed by our proposed model divided by a Monte Carlo estimate. The difference between a Monte Carlo simulation estimate and ours is generally small, which demonstrate the accuracy of our proposed model. An Analysis of the Credit Risk Model

30. Table 1a The default probability calculated by the Monte Carlo simulation method and our explicit solution. 5%10%25%40% 0.25 yearFormula0.00860.01480.01700.0173 Simulation0.00850.01530.01740.0178 Error (%)1.1763.272.302.81 0.5 yearFormula0.02350.03230.03520.0356 Simulation0.02340.03230.03570.0355 Error (%)0.42701.400.282 1 yearFormula0.05910.06810.07060.0709 Simulation0.05800.06820.07060.0708 Error (%)1.900.14700.141

31. Table 1b -The default probability computed by the Monte Carlo simulation method and our proposed model. 5%10%25%40% 0.25 yearFormula0.04910.05840.06010.0602 Simulation0.04830.05740.05980.0592 Error (%)1.661.740.501.69 0.5 yearFormula0.10530.11310.11270.1124 Simulation0.10270.11090.10910.1083 Error (%)2.531.983.303.79 1 yearFormula0.20640.20610.20050.1994 Simulation0.20330.20280.19580.1963 Error (%)1.521.632.401.58

32. Figure 1 shows the probability of the first passage of a double exponential jump-diffusion from an initial condition given below before time s. The previous year’s asset level is arbitrarily set at 86.3 Given the same actual asset level v at time t, the probability of the first passage time increases in jump frequency but decreases in the probability of upward jump An Analysis of the Credit Risk Model

33. Figure 1 -Probability of the first passage time for different arrival rates of jump

34. Figure 2a, 2b, and 2c show the conditional density of the current asset level One is =0.01, p=0.5 (shown in Figure 2a); another is =3, p=0.5 (shown in Figure 2b); and the other is =3, p=0.7 (shown in Figure 2c). The standard deviation of noise is set as 5% and 40% for all cases. The previous year’s asset level is arbitrarily set at 86.3. An Analysis of the Credit Risk Model

35. Figure 2a, 2b, 2c -Conditional density of for different standard deviations of noise. (2a) (2b) (2c)

36. 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 noteworthy 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. An Analysis of the Credit Risk Model

37. Figure 3 illustrates outcomes of the conditional default probability for cases of different jumps and various levels of noise a. One is for and p=0.5; another is and p=0.5; and the other is and p=0.7, given the same previous year asset level of 86.3. An Analysis of the Credit Risk Model

38. Figure 3 -Default probability for the time horizon of 1 year with a varying standard deviation of noise.

39. For the case of, representing a jump of once per hundred years, which is considered rare, the curve we obtained is quite close to that of Duffie and Lando (2001), which 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. An Analysis of the Credit Risk Model

40. For and 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 =0.01, reaches the peak value, and then decreases to converge to a saturation level as the information becomes increasingly dubious. An Analysis of the Credit Risk Model

41. For the case of and p=0.7, the whole curve of the default probability is lowered with respect to the case of and 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. An Analysis of the Credit Risk Model

42. This phenomenon is the result of combination of the jump and noisy information. Noisy information causes the conditional density to become 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. 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. An Analysis of the Credit Risk Model

43. The default probability is determined by these two contrary powers. Because the probability of the first passage time decreases and converges to zero more slowly than in the case of =0.01, the affection that decreases the default probability can be displayed more obviously in the case of than in the case of =0.01, which creates the decrease-to-saturation phenomenon. An Analysis of the Credit Risk Model

44. By definition, the default intensity is a local default rate, in that (14) An Analysis of the Credit Risk Model

45. Figure 4a -Default intensity for different probabilities of jump with noiseless and noisy information.

46. The intensity with noise (a=10%) is less sensitive with the reported asset level than the intensity without noise (a=0) as long as the jump is considered in the asset process. In the case where the value of the 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 2003). If only the upward jump is included, the intensity is close to zero regardless of the information quality. The intensity decreases for the low reported asset level, but increases for the high reported asset level, as information becomes more dubious (Figure 4a). An Analysis of the Credit Risk Model

47. 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. An Analysis of the Credit Risk Model

48. Figure 4b -Default intensity for different arrival rates of jump with noisy information.

49. Figure 4c -Default intensity for different arrival rates of jump and varying standard deviations of noise

50. The arrival rate of jump creates a greater effect on the intensity as the standard deviation of noise increases (a) from 0 to 25% at the reported asset level (86), which is higher than the neutral point. An Analysis of the Credit Risk Model

51. Figure 4d -Default intensity for different arrival rates of jump and varying standard deviations of noise.

52. At the reported asset level (80), which is 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%. The impact of noise on intensities will be saturated as the standard deviation of noise becomes large enough for both the higher and lower reported asset level. An Analysis of the Credit Risk Model

53. According to Kou and Wang (2004), under the risk-neutral probability measure, the return process is given by,. (15) An Analysis of the Credit Risk Model

54. Where is the standard Brownian motion under, and is a Poisson process with intensity. The log jump size is still from a sequence of i.i.d. random variables with a new double exponential density The constants are and. All sources of randomness,,, and, are still independent under. An Analysis of the Credit Risk Model

55. For a given time to maturity T, the yield spread on a given zero-coupon bond sold at a price is the real number such that. We assume the bond holder can receive some fraction of its market value at default. An Analysis of the Credit Risk Model

56. According to Duffie and Singleton (1999), the secondary market price of the bond in the event of no default at time t is given by Where 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. In addition, r is the risk-free rate. (16) An Analysis of the Credit Risk Model

57. Figure 5 -Credit Spread for different arrival rates of jump with noiseless and noisy information.

58. 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. An Analysis of the Credit Risk Model

59. Figure 6 -Default probability with noiseless and noisy information.

60. This phenomenon results from the fact that the uncertainty causes the default probability to have a more moderate variation with maturity An Analysis of the Credit Risk Model

61. As the arrival rate of jump increases, the term structure remains a humped-like curve, but the spread ascends over the whole range of maturity. 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 can be further enriched by including jumps and noise information. An Analysis of the Credit Risk Model

62. Conclusion The pure diffusion approach for the 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.

63. The 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 cause the intensity with jump to be less sensitive with the reported asset level. 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 ascends over the whole range of maturity. The term structure of credit spreads is monotone if the two factors are simultaneously considered and large enough. Conclusion

64. 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 has potential for use in interpreting empirical data in the real world, such as those involving Yu(2005). Conclusion