Credit Default Swaps (CDS) and Credit Derivatives › CDS Pricing Framework

The fair CDS spread is determined by the no-arbitrage condition: at inception, the present value of the premium leg must equal the present value of the protection leg. Let's develop this framework step by step, building from the basic principle that a fairly priced derivative should have zero net value at inception.

The logic is straightforward: if the premium leg were worth more than the protection leg, no rational investor would sell protection, since they would be receiving less than the value of the risk they assume. Conversely, if the protection leg were worth more, no investor would buy protection at that price. Market equilibrium occurs only when both legs have equal present value, making either side of the transaction fair.

Consider a CDS with maturity $T$, premium payments at times $t_1, t_2, \ldots, t_n$ (typically quarterly), and a CDS spread of $s$. Let:

• $Q(t_i)$ = probability of survival to time $t_i$
• $D(t_i)$ = discount factor for time $t_i$ (from the risk-free curve)
• $R$ = recovery rate
• $N$ = notional principal
• $\Delta t_i = t_i - t_{i-1}$ = time between payment dates

These building blocks allow us to express the value of each leg as a sum of expected, discounted cash flows. The survival probabilities capture the likelihood that each payment occurs, while the discount factors convert future values to present values using the time value of money.

The protection buyer pays premiums only while the reference entity survives. The expected present value of the premium leg is:

$$PV_{\text{premium}} = s \times N \times \sum_{i=1}^{n} \Delta t_i \times Q(t_i) \times D(t_i)$$

where:

• $s$: CDS spread
• $N$: notional principal
• $\Delta t_i$: time accrual factor for period $i$
• $Q(t_i)$: survival probability to payment date $t_i$
• $D(t_i)$: discount factor for payment date $t_i$
• $n$: total number of payment periods

This formula multiplies each premium payment by the probability that it will be made (survival probability) and discounts it to present value. The product $\Delta t_i \times Q(t_i) \times D(t_i)$ represents the expected, discounted value of receiving one unit of spread payment at time $t_i$. By summing over all payment dates and multiplying by the spread and notional, we obtain the total present value of the premium stream.

We also need to account for accrued premium if default occurs between payment dates. Adding this gives:

$$PV_{\text{premium}} = s \times N \times \left[\sum_{i=1}^{n} \Delta t_i \times Q(t_i) \times D(t_i) + \text{Accrual Adjustment}\right]$$

where:

• $s$: CDS spread
• $N$: notional principal
• $\Delta t_i$: time accrual factor
• $Q(t_i)$: survival probability
• $D(t_i)$: discount factor
• $\text{Accrual Adjustment}$: expected value of premium accrued between the last payment and default
• $n$: total number of payment periods

The accrual adjustment accounts for the fact that if default occurs, say, two months into a quarterly period, the protection buyer typically pays the premium for those two months. This adjustment is often small but contributes to accurate pricing, particularly for shorter-dated contracts where accrual represents a larger fraction of total payments.

For simplicity, we often use the risky annuity factor (sometimes referred to as the RPV01 on a unit notional basis):

$$\text{RPV01} = \sum_{i=1}^{n} \Delta t_i \times Q(t_i) \times D(t_i)$$

where:

• $\text{RPV01}$: risky annuity factor (sum of discounted survival probabilities)
• $\Delta t_i$: accrual period
• $Q(t_i)$: survival probability
• $D(t_i)$: discount factor
• $n$: total number of payment periods

The risky annuity factor is a powerful summary statistic. It tells us the present value of receiving 1 unit of spread per year for the life of the contract, accounting for both discounting and survival probability. The word "risky" in RPV01 distinguishes this from a risk-free annuity, which would assume certain survival. The risky annuity is always less than or equal to the corresponding risk-free annuity because there is some probability that the payments will cease due to default.

The protection seller pays $(1-R) \times N$ upon default. Since default can occur at any time, we integrate over all possible default times:

$$PV_{\text{protection}} = (1-R) \times N \times \int_0^T D(t) \times q(t) \, dt$$

where:

• $R$: recovery rate
• $N$: notional principal
• $D(t)$: discount factor at time $t$
• $q(t)$: default probability density function
• $T$: time to maturity

This integral captures the expected protection payment by weighting the discounted loss given default by the probability density of default at each instant. The protection payment $(1-R) \times N$ is fixed given our recovery assumption, so the key unknown is when default might occur. The integral sums up the contributions from all possible default times, with each contribution weighted by how likely default is at that moment.

The default density $q(t) = -\frac{dQ(t)}{dt}$ represents the likelihood of default occurring exactly at time $t$. This derivative of the survival function converts cumulative probabilities into instantaneous rates. For a constant hazard rate:

$$q(t) = \lambda e^{-\lambda t}$$

where:

• $q(t)$: default probability density
• $\lambda$: constant hazard rate
• $t$: time

Notice that this density function declines over time. This occurs not because the entity becomes safer, but because for default to happen at time $t$, the entity must first survive until then. As time passes, the pool of possible default times shrinks because some probability mass has already been "used up" by earlier potential defaults.

In practice, we often discretize this integral by assuming defaults occur at payment dates:

$$PV_{\text{protection}} \approx (1-R) \times N \times \sum_{i=1}^{n} D(t_i) \times [Q(t_{i-1}) - Q(t_i)]$$

where:

• $R$: recovery rate
• $N$: notional principal
• $D(t_i)$: discount factor at payment date $i$
• $Q(t_{i-1}) - Q(t_i)$: probability of default occurring in period $i$
• $n$: total number of payment periods

The term $Q(t_{i-1}) - Q(t_i)$ represents the probability of defaulting between times $t_{i-1}$ and $t_i$. This discretization simplifies computation considerably while remaining accurate for practical purposes, especially when payment periods are short (quarterly or more frequent). The approximation essentially assumes that all defaults within a period occur at the period end, which slightly undervalues the protection leg by deferring the expected payment date.

To find the fair spread, we equate the present value of the premium leg to the present value of the protection leg. Since the notional principal $N$ appears on both sides, it cancels out:

$$\begin{aligned} PV_{\text{premium}} &= PV_{\text{protection}} && \text{(no-arbitrage condition)} \\ s \times N \times \text{RPV01} &= (1-R) \times N \times \sum_{i=1}^{n} D(t_i) [Q(t_{i-1}) - Q(t_i)] && \text{(substitute leg values)} \\ s &= \frac{(1-R) \times \sum_{i=1}^{n} D(t_i) [Q(t_{i-1}) - Q(t_i)]}{\text{RPV01}} && \text{(solve for $s$)} \\ s &= \frac{(1-R) \times \sum_{i=1}^{n} D(t_i) [Q(t_{i-1}) - Q(t_i)]}{\sum_{i=1}^{n} \Delta t_i \times Q(t_i) \times D(t_i)} && \text{(expand RPV01)} \end{aligned}$$

where:

• $s$: fair CDS spread
• $N$: notional principal (cancels out)
• $R$: recovery rate
• $D(t_i)$: discount factor at time $t_i$
• $Q(t)$: survival probability
• $\Delta t_i$: accrual period
• $n$: total number of payment periods

This is the fundamental CDS pricing formula. The numerator represents expected losses from default, while the denominator is the risky annuity factor. The ratio tells us what annual spread payment, made contingent on survival, would have the same present value as the expected protection payment contingent on default. This formula generalizes our earlier approximation $s \approx \lambda(1-R)$ by properly accounting for discounting and the precise timing of cash flows.