CDS premium and protection legs

What a CDS is

A credit default swap (CDS) is a bilateral derivative on a named reference entity. The protection buyer pays a running premium to the protection seller. If a covered credit event occurs before maturity, the seller makes a loss-related protection payment. Unlike buying the entity’s bond, entering a CDS does not itself lend principal to the reference entity. Tuckman and Serrat describe the fee leg and the contingent leg (in this course, the premium leg and the protection leg) in Chapter 14[1]; Hull describes the same cash-flow directions for buyer and seller, and the quarterly premium convention[2].

This lesson reports both leg present values as positive magnitudes. From the protection buyer’s perspective, the net present value is the protection-leg present value minus the premium-leg present value:

$$\explain{cds-protection-buyer-net-present-value}{PV_0^{\mathrm{buyer}}} =\explain{cds-protection-leg-present-value}{PV_0^{\mathrm{prot}}} -\explain{cds-premium-leg-present-value}{PV_0^{\mathrm{prem}}}$$

(7.1.1)

The seller has the opposite signed value in this two-party model.

Start from disjoint events

Let the random default-time be $\explain{default-time}{\tau}$. Use the local cds-period-index $\explain{cds-period-index}{i}$ across the number-of-cds-periods $\explain{number-of-cds-periods}{n}$. The schedule has dates $0=\explain{payment-time}{t}_0<\explain{payment-time}{t}_1<\cdots<\explain{payment-time}{t}_{\explain{number-of-cds-periods}{n}}=\explain{maturity-time}{T}$. Define the disjoint interval events

$$\explain{cds-default-event}{G_i} =\{\explain{payment-time}{t_{\explain{cds-period-index}{i}-1}}< \explain{default-time}{\tau}\leq \explain{payment-time}{t_{\explain{cds-period-index}{i}}}\}, \qquad \explain{cds-period-index}{i}=1,\ldots,\explain{number-of-cds-periods}{n}$$

(7.1.2)

and add $\explain{cds-default-event}{G_0}=\{\explain{default-time}{\tau}>\explain{maturity-time}{T}\}$ for survival through maturity. These events are disjoint and exhaustive under the one-default assumption.

For a local cds-event-payoff $\explain{cds-event-payoff}{Y}$ that is constant at $\explain{cds-event-payoff-value}{y_{\explain{cds-period-index}{i}}}$ on each event, the risk-neutral expectation is:

$$\explain{expectation}{\mathbb{E}}^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}[\explain{cds-event-payoff}{Y}] =\sum_{\explain{cds-period-index}{i}=0}^{\explain{number-of-cds-periods}{n}} \explain{cds-event-payoff-value}{y_i}\, \explain{risk-neutral-probability-measure}{\mathbb{Q}} (\explain{cds-default-event}{G_i})$$

(7.1.3)

This is the expectation from disjoint events of the probability lesson, with risk-neutral pricing weights. If $\explain{cds-event-payoff}{Y}$ is not constant within an interval, the expectation uses the conditional means:

$$\explain{expectation}{\mathbb{E}}^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}[\explain{cds-event-payoff}{Y}] =\sum_{\explain{cds-period-index}{i}=0}^{\explain{number-of-cds-periods}{n}} \explain{risk-neutral-probability-measure}{\mathbb{Q}} (\explain{cds-default-event}{G_i}) \,\explain{expectation}{\mathbb{E}}^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}[\explain{cds-event-payoff}{Y}\mid\explain{cds-default-event}{G_i}]$$

(7.1.4)

For this reason, one payment at the end of the period is not an exact treatment of a default in the middle of the period. If every interval is divided into smaller default-time intervals, the limit of the sum is an integral. Shreve gives the finite disjoint-event expectation rule used here[3]. Tuckman and Serrat’s CDS table also weights cash flows by survival and default events[4].

The hazard rate gives the probabilities in the formulas

Let $\explain{cds-general-hazard-rate}{\lambda(u)}$ be the risk-neutral default intensity at time $\explain{cds-default-integration-time}{u}$, given survival to $\explain{cds-default-integration-time}{u}$. It is a deterministic function of time, measured per model-year under the risk-neutral-probability-measure $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$. It generalizes the constant Constant hazard rate $\explain{hazard-rate}{\lambda}$ of this lesson. Its meaning is the conditional intensity of the credit lesson, but it need not be constant: $\explain{cds-general-hazard-rate}{\lambda(u)}$ is the proportional rate at which the survival probability decreases at $\explain{cds-default-integration-time}{u}$, given survival to $\explain{cds-default-integration-time}{u}$. As a differential equation, this meaning is:

$$-\,d\explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,\explain{cds-default-integration-time}{u})} =\explain{cds-general-hazard-rate}{\lambda(u)}\, \explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,\explain{cds-default-integration-time}{u})}\,d \explain{cds-default-integration-time}{u}, \qquad \explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,0)}=1$$

(7.1.5)

Dividing by the (positive) survival probability turns the left side into a derivative of its logarithm, $-d\ln\left(\explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}\left(0,\explain{cds-default-integration-time}{u}\right)}\right)= \explain{cds-general-hazard-rate}{\lambda(u)}\,d \explain{cds-default-integration-time}{u}$. Integrating from $0$ to $\explain{payment-time}{t}$ and using $\ln\left(\explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}\left(0,0\right)}\right)=\ln(1)=0$ gives the survival probability:

$$\explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,t)} =\exp\left(-\int_0^{\explain{payment-time}{t}} \explain{cds-general-hazard-rate}{\explain{hazard-rate}{\lambda}\left(\explain{cds-default-integration-time}{u}\right)}\,d \explain{cds-default-integration-time}{u}\right)$$

(7.1.6)

Integrating the same differential equation over one period, not from $0$, gives the interval default probability for a general hazard rate:

$$\explain{interval-default-probability}{\Delta q_{\explain{cds-period-index}{i}}^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}} =\int_{\explain{payment-time}{t}_{\explain{cds-period-index}{i}-1}}^{\explain{payment-time}{t}_{\explain{cds-period-index}{i}}} \explain{cds-general-hazard-rate}{\lambda(u)}\, \explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,\explain{cds-default-integration-time}{u})}\,d \explain{cds-default-integration-time}{u} =\explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,t_{\explain{cds-period-index}{i}-1})} -\explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,t_{\explain{cds-period-index}{i}})}$$

(7.1.7)

The constant Constant hazard rate $\explain{hazard-rate}{\lambda}$ of this lesson is the special case $\explain{cds-general-hazard-rate}{\lambda(u)}\equiv\explain{hazard-rate}{\lambda}$. Then $\int_0^{\explain{payment-time}{t}}\explain{cds-general-hazard-rate}{\lambda(u)}\,d\explain{cds-default-integration-time}{u}=\explain{hazard-rate}{\lambda} \explain{payment-time}{t}$, and the two results above become:

$$\explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,t)}=\exp(-\explain{hazard-rate}{\lambda} \explain{payment-time}{t}), \qquad \explain{interval-default-probability}{\Delta q_{\explain{cds-period-index}{i}}^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}} =\exp(-\explain{hazard-rate}{\lambda} \explain{payment-time}{t}_{\explain{cds-period-index}{i}-1})-\exp(-\explain{hazard-rate}{\lambda} \explain{payment-time}{t}_{\explain{cds-period-index}{i}})$$

(7.1.8)

The next two sections, on the premium leg and on the protection leg, use the general $\explain{cds-general-hazard-rate}{\lambda(u)}$. The section on closed forms after them uses the constant $\explain{hazard-rate}{\lambda}$.

These probabilities are pricing probabilities, not forecasts, for a constant and for a general hazard rate. A hazard rate implied from CDS prices can be a risk-neutral quantity rather than a prediction of realized defaults[5]. The constant-hazard survival relation is given in §14.6 and Appendix A14.1[6].

Default in the middle of a period

Suppose default occurs at $\explain{cds-default-integration-time}{u}$ in $(\explain{payment-time}{t}_{\explain{cds-period-index}{i}-1},\explain{payment-time}{t}_{\explain{cds-period-index}{i}}]$. The input recovery-rate is $\explain{recovery-rate}{R}$.

• The scheduled premium at $\explain{payment-time}{t}_{\explain{cds-period-index}{i}}$ is not paid, because the reference entity did not survive to that date.
• The buyer pays premium accrued from $\explain{payment-time}{t}_{\explain{cds-period-index}{i}-1}$ to $\explain{cds-default-integration-time}{u}$. In this model its year fraction is $\explain{cds-accrued-year-fraction-at-default}{a_i(u)}=\explain{cds-default-integration-time}{u}-\explain{payment-time}{t}_{\explain{cds-period-index}{i}-1}$.
• The seller pays protection of $\explain{loss-given-default}{\mathrm{LGD}}\explain{cds-notional}{N}=(1-\explain{recovery-rate}{R})\explain{cds-notional}{N}$.
• Both amounts triggered by default are discounted from the modeled default time $\explain{cds-default-integration-time}{u}$, not from $\explain{payment-time}{t}_{\explain{cds-period-index}{i}}$.

Hull notes that a final accrued premium is paid when default occurs between scheduled dates[2]. The half-period and midpoint calculations in Hull and in Tuckman and Serrat are discrete approximations[7][8]. This lesson integrates over the modeled default time exactly, for a general hazard rate and a general discount curve. The closed forms later in the lesson use a constant hazard rate and a constant interest rate.

Exact premium-leg expectation in this model

Let $\explain{discount-factor}{D(0,\explain{cds-default-integration-time}{u})}$ be the general dated discount curve of the rates lesson; this section does not assume a flat rate. The cds-scheduled-premium-annuity is:

$$\explain{cds-scheduled-premium-annuity}{A_0^{\mathrm{sched}}} =\sum_{\explain{cds-period-index}{i}=1}^{\explain{number-of-cds-periods}{n}} \explain{cds-accrual-year-fraction}{\alpha_i} \explain{discount-factor}{D(0,t_{\explain{cds-period-index}{i}})} \explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,t_{\explain{cds-period-index}{i}})}$$

(7.1.9)

The exact cds-accrued-premium-annuity is:

$$\explain{cds-accrued-premium-annuity}{A_0^{\mathrm{accr}}} =\sum_{\explain{cds-period-index}{i}=1}^{\explain{number-of-cds-periods}{n}} \int_{\explain{payment-time}{t}_{\explain{cds-period-index}{i}-1}}^{\explain{payment-time}{t}_{\explain{cds-period-index}{i}}} \explain{cds-accrued-year-fraction-at-default}{a_i(u)} \explain{discount-factor}{D(0,\explain{cds-default-integration-time}{u})} \explain{cds-general-hazard-rate}{\lambda(u)} \explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,\explain{cds-default-integration-time}{u})}\,\mathrm{d}\explain{cds-default-integration-time}{u}$$

(7.1.10)

The cds-premium-annuity is the sum of the two annuities. The premium-leg present value is the annual spread, as a decimal, multiplied by the notional and by the premium annuity:

$$\explain{cds-premium-annuity}{A_0^{\mathrm{prem}}} =\explain{cds-scheduled-premium-annuity}{A_0^{\mathrm{sched}}} +\explain{cds-accrued-premium-annuity}{A_0^{\mathrm{accr}}}, \qquad \explain{cds-premium-leg-present-value}{PV_0^{\mathrm{prem}}} =\explain{cds-contract-spread}{s} \explain{cds-notional}{N} \explain{cds-premium-annuity}{A_0^{\mathrm{prem}}}$$

(7.1.11)

A spread quoted in basis-point per year is divided by 10,000 before it is used in (7.1.11).

Exact protection-leg expectation in this model

The protection-leg present value, as a positive magnitude, is:

$$\explain{cds-protection-leg-present-value}{PV_0^{\mathrm{prot}}} =\explain{cds-notional}{N} \explain{loss-given-default}{\mathrm{LGD}} \sum_{\explain{cds-period-index}{i}=1}^{\explain{number-of-cds-periods}{n}}\int_{\explain{payment-time}{t}_{\explain{cds-period-index}{i}-1}}^{\explain{payment-time}{t}_{\explain{cds-period-index}{i}}} \explain{discount-factor}{D(0,\explain{cds-default-integration-time}{u})} \explain{cds-general-hazard-rate}{\lambda(u)} \explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,\explain{cds-default-integration-time}{u})}\,\mathrm{d}\explain{cds-default-integration-time}{u}$$

(7.1.12)

Equation (7.1.12) is an expectation. Each possible payoff at a default time is discounted and multiplied by the risk-neutral probability density of that default time. The integral then adds over all default times.

Closed forms and the zero-upfront par spread

This section uses a constant Constant hazard rate, $\explain{cds-general-hazard-rate}{\lambda(u)}\equiv\explain{hazard-rate}{\lambda}$, which is a special case of the cds-general-hazard-rate. It also uses a constant continuously-compounded-risk-free-rate $\explain{continuously-compounded-risk-free-rate}{r}$. So the general discount curve of the last two sections is $\explain{discount-factor}{D(0,\explain{cds-default-integration-time}{u})}=\exp(-\explain{continuously-compounded-risk-free-rate}{r}\explain{cds-default-integration-time}{u})$.

For the regular model-year schedule, $\explain{cds-accrual-year-fraction}{\alpha_{\explain{cds-period-index}{i}}}=\explain{payment-time}{t}_{\explain{cds-period-index}{i}}-\explain{payment-time}{t}_{\explain{cds-period-index}{i}-1}$ and the maturity-time is $\explain{maturity-time}{T}=\explain{payment-time}{t}_{\explain{number-of-cds-periods}{n}}$. Define the abbreviation $\explain{cds-combined-decay-rate}{\kappa}=\explain{continuously-compounded-risk-free-rate}{r}+\explain{hazard-rate}{\lambda}$. Then the scheduled-premium annuity and the accrued-premium annuity are:

$$\explain{cds-scheduled-premium-annuity}{A_0^{\mathrm{sched}}} =\sum_{\explain{cds-period-index}{i}=1}^{\explain{number-of-cds-periods}{n}}\explain{cds-accrual-year-fraction}{\alpha_{\explain{cds-period-index}{i}}}\exp(-\explain{cds-combined-decay-rate}{\kappa} \explain{payment-time}{t}_{\explain{cds-period-index}{i}})$$

(7.1.13)

$$\explain{cds-accrued-premium-annuity}{A_0^{\mathrm{accr}}} =\sum_{\explain{cds-period-index}{i}=1}^{\explain{number-of-cds-periods}{n}} \explain{hazard-rate}{\lambda}\exp(-\explain{cds-combined-decay-rate}{\kappa} \explain{payment-time}{t}_{\explain{cds-period-index}{i}-1}) \frac{1-\exp(-\explain{cds-combined-decay-rate}{\kappa}\explain{cds-accrual-year-fraction}{\alpha_{\explain{cds-period-index}{i}}})(1+\explain{cds-combined-decay-rate}{\kappa}\explain{cds-accrual-year-fraction}{\alpha_{\explain{cds-period-index}{i}}})}{\explain{cds-combined-decay-rate}{\kappa}^2}$$

(7.1.14)

The cds-protection-factor, per unit of notional, is:

$$\explain{cds-protection-factor}{B_0^{\mathrm{prot}}} =\explain{loss-given-default}{\mathrm{LGD}}\,\explain{hazard-rate}{\lambda} \frac{1-\exp(-\explain{cds-combined-decay-rate}{\kappa} \explain{maturity-time}{T})}{\explain{cds-combined-decay-rate}{\kappa}}$$

(7.1.15)

At $\explain{cds-combined-decay-rate}{\kappa}=0$, the continuous limits replace the two fractions by $\explain{cds-accrual-year-fraction}{\alpha}_{\explain{cds-period-index}{i}}^2/2$ and $\explain{maturity-time}{T}$, respectively. The implementation uses numerically stable forms of those limits.

The cds-par-spread $\explain{cds-par-spread}{s^\star}$ is the running spread that makes the legs equal with no upfront amount:

$$\explain{cds-par-spread}{s^\star} =\frac{\explain{cds-protection-factor}{B_0^{\mathrm{prot}}}} {\explain{cds-premium-annuity}{A_0^{\mathrm{prem}}}}, \qquad \explain{cds-premium-leg-present-value}{PV_0^{\mathrm{prem}}}(\explain{cds-par-spread}{s^\star})=\explain{cds-protection-leg-present-value}{PV_0^{\mathrm{prot}}}$$

(7.1.16)

Tuckman and Serrat state the condition that the fair spread makes the two legs equal, and the relationship between spread and upfront[9]. The next lesson distinguishes this model par spread from a standard running coupon and a signed upfront amount.

Example 7.1.1 Exact default-time CDS legs

Open the set, then follow the event partition into the exact leg calculation.

A finite event partition

Suppose a two-period protection payoff, simplified to one value per event, is USD 6,000 on $\explain{cds-default-event}{G_1}$, USD 5,800 on $\explain{cds-default-event}{G_2}$, and zero on the survival event $\explain{cds-default-event}{G_0}$. The risk-neutral probabilities of $\explain{cds-default-event}{G}_1$, $\explain{cds-default-event}{G}_2$, and $\explain{cds-default-event}{G}_0$ are 0.02, 0.03, and 0.95. The expected payoff before discounting is:

$$0(0.95)+6{,}000(0.02)+5{,}800(0.03)=\text{USD }294$$

This finite sum is exact only because the payoff is constant on each event. In an exact CDS valuation, the discounted payoff varies with the default time inside an event. So the exact valuation uses an integral over all default times.

One-year exact legs

Take notional $\explain{cds-notional}{N}=\text{USD }1{,}000{,}000$, running spread $\explain{cds-contract-spread}{s}=0.01$ per year ($100\ \text{bp/year}$), recovery $\explain{recovery-rate}{R}=0.40$, constant risk-neutral hazard $\explain{hazard-rate}{\lambda}=0.02$ per model-year, continuously compounded risk-free rate $\explain{continuously-compounded-risk-free-rate}{r}=0.04$ per model-year, two equal periods, and maturity $\explain{maturity-time}{T}=1$ model-year.

The tested domain code gives:

Factor	Value
Scheduled premium annuity	0.956105034 model-years
Accrued-premium annuity	0.004828691 model-years
Premium annuity	0.960933725 model-years
Protection factor	0.011647093 per unit notional

The premium-leg present value is:

$$\explain{cds-premium-leg-present-value}{PV_0^{\mathrm{prem}}} =0.01(1{,}000{,}000)(0.960933725) \approx\text{USD }9{,}609.34$$

The protection-leg present value is:

$$\explain{cds-protection-leg-present-value}{PV_0^{\mathrm{prot}}} =1{,}000{,}000(0.011647093) \approx\text{USD }11{,}647.09$$

The present value of the accrued premium is USD 48.29. It is small in this example. The model calculates it at every possible default time; it does not approximate it by one half of a period.

Solve the exact par spread

With the same inputs, the par spread is:

$$\explain{cds-par-spread}{s^\star} =\frac{0.011647093}{0.960933725} \approx0.012120600 =121.206\text{ bp/year}$$

At the unrounded par spread, the present value of each leg is approximately USD 11,647.09. The notional cancels in the par spread, because both legs are proportional to the same notional.

Generalizing the premium leg: an extinguishing swap

This lesson uses the name extinguishing swap for the construction below. The name belongs to this course; it is not a standard market term or textbook term. An extinguishing swap is any schedule of cash flows that stops permanently at the default of the reference entity. The premium leg above has this property.

Consider the same dates $0=\explain{payment-time}{t}_0<\explain{payment-time}{t}_1<\cdots<\explain{payment-time}{t}_{\explain{number-of-cds-periods}{n}}=\explain{maturity-time}{T}$ and the same reference entity. Replace the scheduled premium $\explain{cds-contract-spread}{s}\explain{cds-notional}{N} \explain{cds-accrual-year-fraction}{\alpha_i}$ at each date with an arbitrary deterministic, signed net cash flow $\explain{cds-extinguishing-swap-cash-flow}{c_i}$ for one named perspective. Under the signed cash-flow convention of this course, a positive $\explain{cds-extinguishing-swap-cash-flow}{c_{\explain{cds-period-index}{i}}}$ is received and a negative $\explain{cds-extinguishing-swap-cash-flow}{c_{\explain{cds-period-index}{i}}}$ is paid. As with the premium leg, a scheduled amount is paid only on survival to its date. Every later scheduled amount is extinguished at the default time. In this lesson, $\explain{cds-extinguishing-swap-cash-flow}{c_{\explain{cds-period-index}{i}}}$ is an input and deterministic; it does not depend on a floating rate. If a cash flow accrues continuously between dates, an accrued-at-default term can be added with the same construction as the cds-accrued-premium-annuity. This lesson does not repeat that term.

The expectation from disjoint events, used for both CDS legs above, gives the cds-extinguishing-swap-present-value of the scheduled cash flows. It is the same survival-weighted discounting that gave $\explain{cds-scheduled-premium-annuity}{A_0^{\mathrm{sched}}}$:

$$\explain{cds-extinguishing-swap-present-value}{PV_0^{\mathrm{ext}}} =\sum_{\explain{cds-period-index}{i}=1}^{\explain{number-of-cds-periods}{n}} \explain{cds-extinguishing-swap-cash-flow}{c_i}\, \explain{discount-factor}{D(0,t_{\explain{cds-period-index}{i}})}\, \explain{survival-probability}{S^{\explain{risk-neutral-probability-measure}{\mathbb{Q}}}(0,t_{\explain{cds-period-index}{i}})}$$

(7.1.17)

With $\explain{cds-extinguishing-swap-cash-flow}{c_i}= \explain{cds-contract-spread}{s}\explain{cds-notional}{N} \explain{cds-accrual-year-fraction}{\alpha_i}$ for every $\explain{cds-period-index}{i}$, the present value is $\explain{cds-contract-spread}{s}\explain{cds-notional}{N}\explain{cds-scheduled-premium-annuity}{A_0^{\mathrm{sched}}}$, the scheduled-payment term of the premium leg. So the premium leg is the special case of an extinguishing swap whose cash flows are one running spread paid on a fixed notional.

Example 7.1.2 A two-date extinguishing swap

Open the set, then reuse the discounted-survival factors of the one-year example.

A two-date extinguishing swap

Use the model of the “One-year exact legs” example: $\explain{cds-combined-decay-rate}{\kappa}=\explain{continuously-compounded-risk-free-rate}{r}+\explain{hazard-rate}{\lambda}=0.06$ per model-year, with semiannual dates $\explain{payment-time}{t}_1=0.5$ and $\explain{payment-time}{t}_2=1$. The discounted-survival factors are:

$$\explain{discount-factor}{D}(0,\explain{payment-time}{t}_1)\explain{survival-probability}{S}(0,\explain{payment-time}{t}_1)=\exp(-0.06\times0.5)\approx0.970445534, \qquad \explain{discount-factor}{D}(0,\explain{payment-time}{t}_2)\explain{survival-probability}{S}(0,\explain{payment-time}{t}_2)=\exp(-0.06\times1)\approx0.941764534$$

Suppose this extinguishing swap’s schedule is a USD 30,000 receipt at $\explain{payment-time}{t}_1$ and a USD 12,000 payment at $\explain{payment-time}{t}_2$, so $\explain{cds-extinguishing-swap-cash-flow}{c}_1=\text{USD }30{,}000$ and $\explain{cds-extinguishing-swap-cash-flow}{c}_2=-\text{USD }12{,}000$. The present value of the extinguishing swap is:

$$\explain{cds-extinguishing-swap-present-value}{PV_0^{\mathrm{ext}}} =30{,}000(0.970445534)+(-12{,}000)(0.941764534) \approx\text{USD }17{,}812.19$$

Both terms use the same $\explain{discount-factor}{D}(0,\explain{payment-time}{t}_{\explain{cds-period-index}{i}})\explain{survival-probability}{S}(0,\explain{payment-time}{t}_{\explain{cds-period-index}{i}})$ factors as the premium leg. Only the signed cash flows $\explain{cds-extinguishing-swap-cash-flow}{c_{\explain{cds-period-index}{i}}}$ are different from the premium leg’s $\explain{cds-contract-spread}{s}\explain{cds-notional}{N}\explain{cds-accrual-year-fraction}{\alpha_{\explain{cds-period-index}{i}}}$.

Why there is no volatility input here

Risk-neutral valuation does not remove volatility. A change from the real-world measure to $\explain{risk-neutral-probability-measure}{\mathbb{Q}}$ changes the pricing probabilities, and in common diffusion models it changes the drift. It does not change the size of the random fluctuations. Shreve shows that the Brownian volatility term remains after the risk-neutral change of measure[10].

This CDS model needs no separate volatility parameter for two reasons. First, its risk-free rate, hazard rate, and recovery rate are deterministic. Second, both legs are expectations that are linear in the cash flows. In other cases, the distributions, the dependence, and the volatility of the inputs could matter: if rates, the hazard rate, or recovery were stochastic, or if the product were a CDS option, had wrong-way risk, or had nonlinear exposure. The risk-neutral measure gives the pricing weights. The restrictive model assumptions, not the measure, remove the need for a separate volatility input.

Try it

Change the running spread, the constant risk-neutral hazard rate, the recovery rate, the risk-free rate, and the term. The explorer integrates the accrued premium and the protection payment over the exact modeled default time. It shows the contribution of every period.

Explore the CDS premium and protection legs

The notional is fixed at $10,000,000 and the synthetic schedule has four equal model periods per year. Default-triggered cash flows are integrated at their exact modeled default times. Rates entered as percentages or basis points are converted to decimals at this UI boundary.

Interactive controls and the chart require JavaScript. The default result, assumptions, and complete period table remain available below.

Contractual spread 120.0 bp/yearExact contractual spread in basis points per year

Constant risk-neutral hazard 2.5%/yearExact constant risk-neutral hazard rate in percent per model-year

Recovery rate 40%Exact recovery rate in percent of notional

Continuously compounded risk-free rate 4%Exact continuously compounded risk-free rate in percent per model-year

Maturity 5 yearsExact maturity in whole model-years

Scheduled premium PV

$508,106

Accrued-on-default premium PV

$1,596

Premium-leg magnitude

$509,703

Protection-leg magnitude

$640,321

Protection-buyer net PV before upfront

$130,619

Par spread

150.8 bp/year

At this contractual spread, protection exceeds premium, so the model value is positive to the protection buyer.

Positive leg magnitudes at the selected contractual spread. The signed buyer value is protection minus premium, as reported above.

Model assumptions beside this output

• Exact equal quarterly model periods; no dates, day count, stubs, or business-day adjustments.
• Supplied constant risk-neutral hazard and a deterministic continuously compounded risk-free rate; no curve calibration.
• Protection and accrued premium are integrated over every possible default time and discounted from that modeled time.
• Accrued premium uses elapsed model time since the preceding premium date; it is not a half-period approximation.
• Deterministic recovery, one modeled default, running spread only, and no quote calibration, counterparty risk, funding, or actual settlement mechanics.

Period-by-period factor table

The annuity columns are per unit notional and per unit annual spread. The protection factor already includes loss given default. Summing each contribution column reproduces the corresponding tested domain total.

Current exact-time quarterly schedule, survival values, discount factors, and leg-factor contributions

Period	Start–end, years	Start survival	End survival	Interval default	Payment discount factor	Scheduled annuity	Accrued annuity	Protection factor
1	0.00–0.25	1.000000	0.993769	0.006231	0.990050	0.245970	0.000773	0.003720
2	0.25–0.50	0.993769	0.987578	0.006192	0.980199	0.242006	0.000760	0.003660
3	0.50–0.75	0.987578	0.981425	0.006153	0.970446	0.238105	0.000748	0.003601
4	0.75–1.00	0.981425	0.975310	0.006115	0.960789	0.234267	0.000736	0.003543
5	1.00–1.25	0.975310	0.969233	0.006077	0.951229	0.230491	0.000724	0.003486
6	1.25–1.50	0.969233	0.963194	0.006039	0.941765	0.226776	0.000713	0.003429
7	1.50–1.75	0.963194	0.957193	0.006001	0.932394	0.223120	0.000701	0.003374
8	1.75–2.00	0.957193	0.951229	0.005964	0.923116	0.219524	0.000690	0.003320
9	2.00–2.25	0.951229	0.945303	0.005927	0.913931	0.215985	0.000679	0.003266
10	2.25–2.50	0.945303	0.939413	0.005890	0.904837	0.212504	0.000668	0.003214
11	2.50–2.75	0.939413	0.933560	0.005853	0.895834	0.209079	0.000657	0.003162
12	2.75–3.00	0.933560	0.927743	0.005817	0.886920	0.205709	0.000646	0.003111
13	3.00–3.25	0.927743	0.921963	0.005780	0.878095	0.202393	0.000636	0.003061
14	3.25–3.50	0.921963	0.916219	0.005744	0.869358	0.199131	0.000626	0.003011
15	3.50–3.75	0.916219	0.910510	0.005709	0.860708	0.195921	0.000616	0.002963
16	3.75–4.00	0.910510	0.904837	0.005673	0.852144	0.192763	0.000606	0.002915
17	4.00–4.25	0.904837	0.899200	0.005638	0.843665	0.189656	0.000596	0.002868
18	4.25–4.50	0.899200	0.893597	0.005602	0.835270	0.186599	0.000586	0.002822
19	4.50–4.75	0.893597	0.888030	0.005568	0.826959	0.183591	0.000577	0.002776
20	4.75–5.00	0.888030	0.882497	0.005533	0.818731	0.180632	0.000568	0.002732
Totals						4.234220	0.013304	0.064032

Check your understanding

The assessment provides direct and transfer evidence for leg perspective, premium-leg magnitude, protection-leg magnitude, and par spread.

Knowledge check 7.1.1 CDS legs and par spread

1. DirectQuestion 1Not answered

2. TransferQuestion 2Not answered

3. DirectQuestion 3Not answered

4. TransferQuestion 4Not answered

5. DirectQuestion 5Not answered

6. TransferQuestion 6Not answered

7. DirectQuestion 7Not answered

8. TransferQuestion 8Not answered

Model boundary and review note

The valuation is exact only inside this simplified model. The premium leg and the protection leg are derived for a general, possibly time-varying risk-neutral hazard rate and a general dated discount curve. The closed forms, the numeric examples, the explorer, and the assessment use a special case: one constant risk-neutral hazard rate, one deterministic continuously compounded risk-free rate, a deterministic recovery rate, and settlement at the default time. The extinguishing swap is a construction of this lesson that generalizes the premium leg; it is not a named market instrument. The model omits calendar schedules, day counts, business-day rules, payment delays, term-structure calibration, auctions, deliverables, credit-event legal definitions, counterparty risk, collateral, funding, restructuring, and transaction costs. All materially changed content remains `draft` pending independent human review.

References

1. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §14.5, printed pp. 361–362. draft ↩
2. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). §24.1, printed pp. 548–550. draft ↩
3. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 1 §1.3, printed pp. 13–18. draft ↩
4. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §14.6, printed pp. 368–370, Table 14.10 and eqs. 14.6–14.7. draft ↩
5. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §14.7, printed p. 371. draft ↩
6. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). printed pp. 367–368 and 505, eqs. 14.4–14.5 and A14.1–A14.2. draft ↩
7. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). §24.2, printed pp. 551–554, Tables 24.1–24.4. draft ↩
8. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §14.6, printed pp. 368–370, Table 14.10. draft ↩
9. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §14.6, printed pp. 370–371, and Appendix A14.2, printed pp. 506–507. draft ↩
10. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 5 §5.2.2, printed pp. 214–217. draft ↩