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3.2.1 Hazard Rates and a Spread-Based Modeling of Credit

Let us introduce the main elements of the spread-based framework for credit risk modeling. Although in this chapter's introduction we said that a default is not always a clear, linear, and transparent process, we assume that there is a precise moment in time r when this takes place. From this we can define a survival indicator function A(t)

(3.4)

telling us whether the entity is alive, that is, it has not defaulted, at time t. In the previous section we described bonds as coupon-bearing instruments, let us now discuss the simpler situation of a zero coupon bond, that is, the promise of a single payment at some point in the future. Let us define as

the value at time t of a riskless, that is, default-free bond[1] paying a unit of principal at T. Let us also define asthe price of a risky, that is, defaultable, bond at time t paying a unit of principal at time T.

No arbitrage (and common sense) dictates that we must have

(3.5)

that is, the defaultable bond cannot be worth more, before maturity, than the riskless bond. At maturity we have, of course, , since,

by reaching maturity, the defaultable bond has by definition not defaulted.

Let us also assume that there is no recovery upon default, that is, the price of the defaultable bond is given by

(3.6)

and that there is independence between riskless interest rates and default time, that is, no correlation between the riskless bond value Bt(T) and τ.

We know from the assumptions behind all short rate models that the price of every contingent claim of unit principal is equal to the expected value of its discounted value, that is,

(3.7)

where rt is the short risk-free interest rate. Since, in the case of a defaultable bond, the unit payment at maturity T happens only if τ > T (i.e., the default does not occur during the life of the bond), the risky bond value should be written as

(3.8)

Since we have assumed independence between risk-free interest rates and default time, we can write the above as

or

(3.10)

where St,T is the survival probability of the borrower at time T seen at time t. Since the survival probability is the ratio between the defaultable bond and the riskless bond

(3.11)

we must have St < 1 for all t < T. We now introduce the fundamental concept of hazard rate h{t, T) (which, when there is no confusion, we will simply define as ht) as

(3.12)

provided τ > t. In a way more useful to us, we can express the relation between hazard rate and survival probability as

(3.13)

so that immediately we recognize the similarities between hazard rate modeling and risk-free interest rate modeling. In this view any (see Brigo and Mercurio [19]) positive short rate model used for interest rates r can be used to model hazard rate h. Similarly to short rate models, where inputs are not taken directly from the market but need to be calibrated to the only available quantity (the bond price), hazard rates need calibration. The most important and common instruments used for calibration are CDSs and we shall illustrate in the next section how this takes place in practice.

Throughout the remainder of the book, unless otherwise stated, we shall simplify the notation and, taking the same approach we have taken for discount factors, assume that we are observing survival probabilities as of now (t = 0), meaning that we shall only write St instead of S0,t.

  • [1] A riskless bond, of course, does not really exist. We can choose to read this as either a bond of negligible risk or a bond for which we have chosen to ignore the credit risk.
 
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