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

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 X the 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

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.

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 with some degree of certainty.

資料來源 (Kay Giesecke Lisa Goldberg)

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].

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,

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 account B.

Definition 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 A t -A t-.

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 X are generally different. r.v: X d (t,T) -- At any time t<T, the current value of all future CFs associated with a given defaultable claim DCT. (set X d (T,T)= X d (T). )

Definition 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.

Assume is generated by the price processes of tradable asset. Otherwise, when the default time τ is the first passage time of V to 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.

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.

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]. Recovery at default ( )

Another possible solution Assuming that the recovery payoff Z τ is 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 DCT 2.

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

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

Suppose that we purchase one unit of the 0 th asset

Lemma 2.1.1

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

Proposition 2.1.1

Corollary 2.1.1

Remarks It is worth noticing that represents the discounted cum- dividend price at time t of the 0 th 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.