Credit Derivatives
The main geometric condition of completeness (uniqueness of risk-neutral probability) requiring d+1 possible vector jumps in dimension d was not very natural for stock prices (resulting in incompleteness of many real-life models). For credit derivatives t
- PDF / 115,174 Bytes
- 5 Pages / 439.36 x 666.15 pts Page_size
- 80 Downloads / 190 Views
Credit Derivatives
16.1 Basic Model with No Simultaneous Jumps Consider a market of N securities that can default in discrete time δ , 2δ , . . .. Let Nt denote the number of securities defaulted up to time t, and set dt = N − Nt for the number of not-yet-defaulted ones. We will work mostly with the instantaneous digital credit default swaps (CDSs). Entering an instantaneous digital CDS agreement on jth (not yet defaulted) security at time t means agreeing to pay a (protection) premium α j (t)δ and to receive back the compensation of one unit of money if j defaults during the period (t,t + δ ] (and nothing otherwise). The premium α j (t)δ is chosen in such a way that there is no charge at inception at time t to enter this contract [alternatively, of course, α j (t)δ can be considered as a charge for receiving one in the case of the jth default]. Working with instantaneous digital CDSs rather than actually traded CDSs means choosing a convenient basis and is a well-accepted approach in the finance literature (e.g., Cousin et al. [55]). The infinitesimal premium α j (t) is usually assumed to depend on the whole history of defaults of our basic securities until and including time t. We start in this section with the standard (in the literature) simplifying assumption that only one default can occur in any given short period of time (t,t + δ ]. Thus we assume that, for any time t, the possible (dt + 1) outcomes at time t + δ are either no default or a default of only one of dt live (not yet defaulted) securities. These outcomes can be described symbolically by dt + 1 vectors in Rdt : zero vector e0 and dt basis vectors ei (with the ith coordinate 1 and other coordinates vanishing), so that any short-term contingent claim starting at t for the period (t,t + δ ] can be described by a function f on {e0 , e1 , . . . , edt }. Suppose that, to replicate any such claim, an investor, with an amount of capital X at time t, is allowed to enter an arbitrary amount γ i of an instantaneous digital CDS agreement on the ith security, i = 1, . . . , dt (recall that entering such an agreement is costless). Then his capital at time t + δ in case of event ei becomes
P. Bernhard et al., The Interval Market Model in Mathematical Finance, Static & Dynamic Game Theory: Foundations & Applications, DOI 10.1007/978-0-8176-8388-7 16, © Springer Science+Business Media New York 2013
285
286
16 Credit Derivatives
−δ
dt
∑ γ j α j (t) + fi + γ i + X,
j=1
where fi = f (ei ) and it assumed that γ 0 = 0 (to make this formula valid for e0 ). In other words, his capital equals fi − (γ , ηi ) + X, where η0 = δ (α1 (t), . . . , αdt (t)) and ηi = η0 − ei for i = 1, . . . , dt . By hedging price we mean, as usual, the minimal value of X needed to be able to fulfill the obligation in any case, which is then Ch = min
max [ fi − (γ , ηi )].
γ ∈Rdt i=0,1,...,dt
(16.1)
The vectors {ηi } form a strictly positively complete family, so that Proposition 12.6 holds. Hence the corresponding market is complete and Ch = E{ f. } =
dt
∑ p j (t) f j ,
j=0
Data Loading...
