3. First Passage Time Model

1. 3. First Passage Time Model 指導教授：戴天時 學 生：王薇婷

2. Introduction • The First-passage-time approach extends the original Merton model by accounting for the observed feature. • The default not only at the debt’s maturity, but also prior to this date. • The default event with the first passage time of the firm’s value process to pre-specified barrier. • Barrier process: given endogenously, or exogenously with respect to the model.

3. Introduction • The first-passage-time model allow a greater flexibility in modeling credit events in comparison with the Merton model of corporate debt. • The time of the bankruptcy of the firm to occur before the maturity of the debt • The recovery payoff can be specified in a large variety of ways, in order to reflect more closely the real-life bond covenants and other important factors, such as bankruptcy cost or taxes.

4. 3.1 Properties of first passage times

5. Many results in the existing literature rely on the probabilistic approach. • Bond value formulae in Longstaff and Schwartz (1995) and Saa-Requejo and Santa-Clara (1999) correspond to the following generie expression • The default time is defined as the first passage time of the value process to a barrier. • The knowledge of conditional distribution of the default time τ with respect to the σ–field Ft.

7. Lemma 3.1.1

8. 3.1.1 probability Law of the first passage Time

9. Lemma 3.1.2

10. Lemma 3.1.2

11. Corollary 3.1.1 • We apply the foregoing results to specific examples of default times.

12. Example 3.1.1

13. Example 3.1.1

14. Example 3.1.2

15. Example 3.1.2

16. 3.1.2 Joint Probability Low of Y and τ

17. Lemma 3.1.3 • Corollary 3.1.2

18. Lemma 3.1.4

19. Example 3.1.3

20. Example 3.1.4

21. Example 3.1.4

22. Remarks