1. 3.2 Black and Cox Model 學 生:施嘉紋 指導老師:戴天時

2. 3.2 Black and Cox Model • The original Merton model does not allow for a premature default, in the sense that the default may only occur at the maturity of the claim. • In most of these models, the time of default is given as the first passage time of the value process V to a deterministic or random barrier. The default may thus occur at any time before or on the bond's maturity date T.

3. First passage model Merton model FPM P T

4. 3.2 Black and Cox Model • The challenge here is to appropriately specify the lower threshold v, the recovery process Z, and to compute the corresponding functional that appears on the right-hand side of (2.3).

5. 3.2 Black and Cox Model • The practical problem of the lack of direct observations of the value process V largely limits the applicability of the first-passage-time models. • In most cases examined below,the default time is denoted by τ; the symbols ,and being reserved to some auxiliary random times.

6. 3.2.1 Corporate Zero-Coupon Bond • Black and Cox (1976) extend Merton's (1974) research in several directions. • In particular, they make account for specific features of debt contracts as: safety covenants debt subordination restrictions on the sale of assets. • Assume: the firm's stockholders (or bondholders) receive a continuous dividend payment, proportional to the current value of the firm.

7. 3.2.1 Corporate Zero-Coupon Bond • κ:constant κ≥0 ,the payout ratio • :>0 ,the constant volatility coefficient • :,the short-term interest rate is assumed to be non-random, specifically, where r is a constant. • →interest rate risk is disregarded in the • original Black and Cox (1976) model

8. 3.2.1 Corporate Zero-Coupon Bond Safety covenants • Safety covenants provide the firm’sbondholders with the right to force the firm to bankruptcy or reorganization if the firm is doing poorly according to a set standard. • The standard for a poor performance is set in Black and Cox (1976) in terms of a time-dependent deterministic barrier for some constants K, >0.

9. 3.2.1 Corporate Zero-Coupon Bond • They postulate that as soon as the value of firm's assets crosses this lower threshold, the bondholders take over the firm. Otherwise, default takes place at debt's maturity or not depending on whether VT< L or not. • Letusset

10. 3.2.1 Corporate Zero-Coupon Bond

11. 3.2.1 Corporate Zero-Coupon Bond

12. 3.2.1 Corporate Zero-Coupon Bond

13. 3.2.1 Corporate Zero-Coupon Bond • Since the interest rate r is assumed to be constant, the pricing function u = u(V, t) of a defaultable bond solves the following PDE:

14. 3.2.1 Corporate Zero-Coupon Bond

15. 3.2.1 Corporate Zero-Coupon Bond

16. 3.2.1 Corporate Zero-Coupon Bond • Proposition 3.2.1

18. 3.2.1 Corporate Zero-Coupon Bond • Before proceeding to the proof of Proposition 3.2.1, we state an elementary lemma.

24. Proof of proposition 3.2.1

27. Proof of proposition 3.2.1

28. Proof of proposition 3.2.1

38. 3.2.1 Corporate Zero-Coupon Bond

39. 3.2.1 Corporate Zero-Coupon Bond • Similarly as in the case of the Merton model, the Black and Cox model produces credit spreads close to zero for small maturities, a feature that is inconsistent with empirical studies. The reason again is that the default time is predictable with respect to the natural filtration of the value process V.

40. Strict priority rule

41. Strict priority rule

42. Strict priority rule

43. Strict priority rule

44. Special cases

45. Case γ=r

46. Case γ=r

47. Case γ=r

48. Case γ>r

50. 3.2.2 Corporate Coupon Bond • We shall assume now that r > 0 and that a default able bond of fixed maturity T and face value L pays continuously coupons at a constant rate c, so that The coupon payments stop as soon as default occurs. Formally, we consider a defaultable claim specified as follows: with the barrier v given by (3.12).