Credit default swap valuation

Definition A credit default swap on a reference issuer with default time $\tau$ has a notional $N$, a maturity $T$, an annual spread $s$ and premium dates $0 = t_0 < t_1 < \dots < t_n = T$ with accrual fractions $\alpha_i = t_i - t_{i-1}$. In the simplified contract of this lesson the payments are the following.

1. The protection buyer pays the premium $sN\alpha_i$ at $t_i$ if $\tau > t_i$.
2. If $t_{i-1} < \tau \le t_i$, the buyer pays the accrued premium $sN(\tau - t_{i-1})$ at $\tau$, and no later premium is due.
3. The protection seller pays $N(1 - R)$ at $\tau$ if $\tau \le T$, where $R \in [0, 1]$ is a fixed recovery rate.

The contract is valued under a pricing measure $\mathbb{Q}$ with a deterministic discount factor $D(t) = D(0, t)$ and a survival function $S$ with a nonnegative piecewise continuous hazard rate $\lambda$, so that $\tau$ has the density $\lambda S$ on $[0, T]$. Neither the buyer nor the seller can default. Accrual fractions equal elapsed time here, and real contracts use their own day counts, calendars and settlement rules, which enter the same cash-flow calculation. The value of the swap to the protection buyer is the value of the protection leg minus the value of the premium leg, and its value to the seller is the negative of that.

Definition Under the assumptions of the previous card, the premium leg is worth $V^{\text{prem}} = sNA$, where the risky annuity $A = A^{\text{sch}} + A^{\text{acc}}$ has a scheduled part and an accrual part,

The scheduled premium at $t_i$ is paid only on the event $\{\tau > t_i\}$, which has probability $S(t_i)$, and the accrual terms $a_i$ of the next card value the premium owed at default. The annuity is measured in years, discounted and weighted by survival, so $sNA$ is in currency units. The premium leg is the payment stream of the protection buyer.

Definition Accrual on default is the premium for the part of the current period that has elapsed at default, the amount $sN(\tau - t_{i-1})$ paid at $\tau$ when $t_{i-1} < \tau \le t_i$. Its contribution to the annuity from period $i$ is

For a constant rate $r$, a constant intensity $\lambda$ and periods of equal length $\delta$, write $k = r + \lambda$. The substitution $t = t_{i-1} + u$ gives $a_i = \lambda e^{-k t_{i-1}} \int_0^{\delta} u\, e^{-ku}\,du$, and integration by parts, differentiating $u$ and integrating $e^{-ku}$, gives for $k \ne 0$

with the limit $\delta^2/2$ as $k \to 0$. For small $k\delta$ the integral is $\tfrac12\delta^2\bigl(1 - \tfrac23 k\delta + O(k^2\delta^2)\bigr)$, so each $a_i$ is close to $\tfrac12\delta \cdot \lambda\delta\, e^{-k t_{i-1}}$. That is half a period of premium times the approximate default probability $\lambda\delta\, S(t_{i-1})$ of the period, discounted by $D(t_{i-1})$, which is where the common half-period approximation of the accrual comes from.

Definition Under the assumptions of the first card the protection leg is worth

the loss on the notional discounted from each possible default date and weighted by the default density. For constant $r$ and $\lambda$ with $k = r + \lambda \ne 0$ it equals

The recovery rate enters only through the loss given default $1 - R$, so with the survival curve held fixed the protection value is proportional to $1 - R$. Payment at the default date is an idealization of the settlement process of real contracts.

Definition The par spread of a credit default swap is

defined when $A > 0$. For a contract with spread $s$, the value to the protection buyer is then $V^{\text{prot}} - sNA = NA\,(s_{\text{par}} - s)$. A new contract struck at the par spread has zero initial value, and the notional cancels from $s_{\text{par}}$ because both legs are proportional to it. For an existing contract with $s \ne s_{\text{par}}$ the two legs no longer cancel, and the buyer’s value is positive exactly when $s < s_{\text{par}}$. For a contract that trades with a fixed coupon $c$ and an upfront payment $U$ from buyer to seller, the same balance in this model gives $U = V^{\text{prot}} - cNA = NA\,(s_{\text{par}} - c)$, and real upfront quotes need further contractual inputs. When a survival curve is fitted to a set of quotes, piecewise constant nonnegative intensities keep $S$ positive and nonincreasing through the exponential formula. A fit that needs a negative intensity signals inconsistent quotes, discounting, recovery assumption or fitting constraints, and it cannot be read as a conditional default rate.

Definition If the premium were paid continuously at the rate $sN$ per year while the issuer survives, up to $T$, the annuity would be $\int_0^T D(t) S(t)\,dt$ and the par spread would be

the loss given default times a weighted average of the intensity. For a constant intensity it equals $(1 - R)\lambda$ exactly, for every deterministic discount curve. With scheduled payments and exact accrual, the buyer pays in each period $sN$ times exactly the time the issuer was alive in that period, the same amount as with continuous payment, but at the period end or at default instead of along the way. So if $D \equiv 1$ the annuity equals the continuous one and $s_{\text{par}} = s_{\text{cont}}$, and if $D$ is strictly decreasing, as with positive rates, the annuity is smaller and $s_{\text{par}} > s_{\text{cont}}$. As the periods shrink with $T$ fixed, the scheduled sum converges to the continuous annuity and the accrual part to zero, so $s_{\text{par}} \to s_{\text{cont}}$. For scheduled premiums the relation $s_{\text{par}} \approx (1 - R)\lambda$ is therefore an approximation.