3. First Passage Time Model

3. First Passage Time Model 指導教授：戴天時學生：王薇婷

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.

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.

3.1 Properties of first passage times

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.

Lemma 3.1.1

3.1.1 probability Law of the first passage Time

Lemma 3.1.2

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

Example 3.1.1

Example 3.1.2

3.1.2 Joint Probability Low of Y and τ

Lemma 3.1.3 • Corollary 3.1.2

Lemma 3.1.4

Example 3.1.3

Example 3.1.4

Remarks

3. First Passage Time Model