To explain how to calibrate hazard rates to CDS levels we shall follow closely O'Kane and Turnbull [65], to this day one of the clearest discussions on the topic.

We have defined a credit default swap as a contract in which one side pays a premium in the form of a coupon with a certain frequency up to a default (premium leg) and the other side, in case of default, exchanges the principal for the recovery value (i.e., it pays (1 – R) and we call this the recovery leg).

The premium leg is dehned as

where Di is the discount factor at time Ti, Si is the survival probability at time Tj, and . As far as the premium leg is concerned we need to know whether, in case of default, the party buying protection (by paying the premium) is required to pay the fraction of coupon accrued from the last payment date up to the default date. This information would be specified in the agreement. Should the payment of the accrued coupon be necessary, this involves the calculation of an integral for each payment frequency. It can be shown (see O'Kane and Turnbull [65]) that this can be approximated and the premium leg can be written as

(3.14)

where 1PA is equal to 1 if we need to consider the accrued premium and 0 otherwise. The impact of considering or ignoring the accrued premium is small when valuing the present value of the CDS and, of course, is a function of the frequency of payments: the more frequent, the smaller.^{[1]}

The recovery leg needs to assess the present value of the payment of (1 – R) in case of default. Since default can happen at any time, we are going to approximate this (technically continuous) calculation with a discrete set of time steps by writing where we have used a different suffix counter n and we assume, in order to better approximate a continuous integral, that , that is, the summation frequency is higher in the recovery leg. Because we are approximating a continuous integral, we describe the possibility of defaulting at any time t by bracketing this time within a short interval In this case then,is the probability of defaulting in that interval given the survival up to Tn-1.

The only market information traded^{[2]} in a CDS is the coupon premium C. Since at the beginning of the contract there should be no initial gain for either side, we must have

(3.16)

Our goal is the calibration of a hazard rate term structure. By using market information, a strip of CDS quotes for increasing maturities, we calculate the hazard rate for the equivalent maturity. We use a bootstrapping method similar to the one we have used in Section 2.5 to build a discount factors term structure, that is, we start by calculating the shortest maturity, then we move on to the next one, and so forth. Let us imagine that we have CDS quotes for one, two, and three years, respectively C1, C2, and C3. Starting with the first maturity, by using Equation 3.13 we rewrite the above equation as

(3.17)

where b01 is the hazard rate from the beginning of the contract up to 1 year, i has a quarterly frequency (per definition of CDS contract), that is, Ti=1 = 0.25, Ti=2 = 0.5, Ti=3 = 0.75, ..., and we can decide to have m running at a monthly frequency, that is, Tm=1 = 0.08333, Tm=2 = 0.16667, Tт=з = 0.25,... Equation 3.17 can be solved using a solver that tries to guess values of b0,1 until both sides of the equation are equal.

Now that we have found b0,1 we can use the information contained in the second piece of market information, that is, the rate C2 for a CDS with maturity two years. We apply a solver algorithm to

where we have used Equation 3.13, in order to find h1,2. We repeat this process one final time by applying a solver to

and we complete our simple term structure of hazard rates by finding h2,3. (The only practical difference between the last two equations is the fact that the summation of the hazard rates is one step longer in the latter.) In the chapter About the Web Site we direct the reader to a web site where we offer a spreadsheet with an implementation of the process outlined above.

Could this process lead to negative hazard rates? In practice yes, particularly for heavily inverted CDS rate curves (in Section 4.2.3 we will see some examples of these in the case of emerging markets) but it should be taken as a numerical error. Absence of arbitrage (and in the case of credit, the absence of arbitrage that applies to short rate models is reinforced by common sense) dictates that we cannot have negative hazard rates since they would imply a probability of survival greater than one. With real, liquid, and up-to-date data, negative rates cannot occur.

If the data is stale, the bid-offer is wide, and the situation extreme (highly inverted curve and/or very large CDS rates), it can happen that hazard rates become negative. A typical example is CDS spreads for emerging market entities where one or two specific points are very liquid and the neighboring ones rarely traded. For example, Turkey's five-year point is very liquid, but the four-year and the six-year are certainly not as liquid. In the neighborhood of the five-year point, then we could encounter negative hazard rates.

Our approach should be to doubt the data for the illiquid points and assume that the true rate is a different one. In a real-life situation, if the hazard rate hk,k+1 implied by CDS levels for maturities Tk and Tk+1 is negative, two things can be done. We can overwrite the value for CDSTk+1 by shifting it a little upward until the implied hazard rate is positive. Otherwise, in our algorithm we can decide to floor the hazard rates at a small positive value (i.e., whenever the solution is negative we discard it and pick a small positive value, say, 0.00001). The result is broadly the same in both cases, but the advantage of the latter consists of being automatic; it has, however, the disadvantage of not communicating to what assumed-to-be-true new value of CDS rate the fictitious hazard rate corresponds. While tempting, hazard rates should not be floored at zero: if this were the case, that is, hk,k+i = 0, we would be saying that 5¾ = Sk+i, implying an impossibility of defaulting between Tk and Tk+1.

[1] This dependency is, however, not very important considering that the great majority of CDS contracts have quarterly premium payments.

[2] The recovery R is also an input but it is not traded in a CDS, it is assumed. When it is traded it is done through recovery swaps, a different type of instrument.

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