2.1 DEFAULTABLE CLAIMS

1. 指導教授：戴天時 學生：王薇婷 2.1 DEFAULTABLE CLAIMS

2. T*>0, a finite horizon date (Ω,F,P): underlying probability space :real world probability :spot martingale measure (the risk-neutral probability) • The short-term interest rate process r • The firm’s value process V • The barrier process v • The promised contingent claim Xthe firm’s liabilities to be redeemed at T<T* • The process A (promised dividends) • The recovery claim (recovery payoff received at T, if default occurs ≦T) • The recovery process Z

3. Technical assumptions • V, Z, A and v are measurable with respect to the filtration • r.v : X and -measurable • All random objects introduced above satisfy suitable integrability conditions that are needed for evaluating the functionals defined in the sequel.

4. Default time • In structure approach, the default time will be defined in terms of the value process V and the barrier process v. • (2.1) It’s means that there exists a sequence of increasing stopping times announcing the default time, the default time can be forecasted withsome degree of certainty.

5. 資料來源http://www.barra.com/support/library/credit/trends_compensators.pdf(Kay Giesecke Lisa Goldberg)

6. Default time • In the intensity-based approach, the default time will not be a predictable stopping time with respect to the ‘enlarged’ filtration • The occurrence of the default event comes as a total surprise. • For any date t, the PV of the default intensity yields the conditional probability of the occurrence of default over an infinitesimally small time interval [t,t+dt].

7. Recovery rules • If default occurs after time T, the promised claim X is paid in full at time T. • Otherwise, depending on the adopted model, • In general, • In most practical, • The date T,

8. 2.1.1 Risk-Neutral Valuation Formula • Suppose the underlying financial market model is arbitrage-free. • the price process (no coupons or dividends, follows an F-martingale under P*) discounted by the savings accountB.

9. Definition 2.1.1 The dividend process D of a defaultable contingent claim , which settles at time T, equals ------------------------------------------------------------------ The default occurs at some date t, the promised dividend At-At- .

10. The promised payoff X could be considered as a part of the promised dividends process A. However, such a convention would be inconvenient, since in practice the recovery rules concerning the promised dividends A and the promised claim Xare generally different. • r.v: Xd(t,T) -- At any time t<T, the current value of all future CFs associated with a given defaultable claim DCT. (set Xd(T,T)= Xd(T). )

11. Definition 2.1.2 • The risk-neutral valuation formula is known to give the arbitrage price of attainable contingent claims. • Structural models typically assume that assets of the firm represent a tradable security. (In practice, the total market value of firm’s shares is usually taken as V) • The issue of existence of replicating strategies for defaultable claims can be analyzed in a similar way as in standard default-free financial models.

12. Assume is generated by the price processes of tradable asset. • Otherwise, when the default time τ is the first passage time of Vto a lower threshold, which does not represent the price of a tradable asset (so that τ is not a stopping time with respect to the filtration generated by some tradable assets), the issue of attainability of defaultable contingent claims becomes more delicate.

14. Recovery at maturity ( ) • In absence of the promised dividends (A=0), • Under a set of technical assumptions, a suitable version of the martingale representation theorem with respect to the Brownian filtration will ensure the attainability of the terminal payoff .

15. Recovery at default ( ) • In absence of the promised dividends (A=0), (2.3) defines only the pre-default value of a defaultable claim. • The value process vanishes identically on the random interval [τ,T].

16. Another possible solution • Assuming that the recovery payoff Zτis invested in default-free zero-coupon bonds of maturity T. • When we search for the pre-default value of a defaultable claim, such a convention does not affect the valuation problem for DCT2.

17. 2.1.2 Self-Financing Trading Strategies A formal justification of Definition 2.1.2 based on the no-arbitrage argument. • Price process of k primary securities Si, i=1,…,k • Si –semimartingales , i=1,…,k-1and non-dividend-paying assets • Sk : saving account • A trading strategy process:

18. Assume that we have an additional security that pays dividends during its lifespan according to a process of finite variation D, with D0 =0. Let S0denote the yet unspecified price process of this security. • Since we do not assume a priori that S0 follows a semimartingale, we are not yet in a position to consider general trading strategies involving the defaultable claim anyway.

19. Suppose that we purchase one unit of the 0th asset

20. Lemma 2.1.1

22. 2.1.3 Martingale Measures • Goal: derive the risk-neutral valuation formula for the ex-dividend price • Assume:

23. Proposition 2.1.1

24. Proposition 2.1.1

26. Corollary 2.1.1

28. Remarks • It is worth noticing that represents the discounted cum-dividend price at time t of the 0th asset. • Under the assumption of uniqueness of a spot martingale measure , any -integrable contingent claim is attainable, and the valuation formula can be justified by means of replication. Otherwise - that is, when a martingale probability measure is not unique - the right-hand side of (2.10) may depend on the choice of a particular martingale probability. • In this case, a process defined by (2.10) for an arbitrarily chosen spot martingale measure can be taken as the no-arbitrage price process of a defaultable claim.