Standard CDS pricing theory

I summarize below standard CDS pricing formulas.

Credit Curves

In CDS pricing, credit default events are modelled using a Poisson process, with an intensity (or hazard rate) $\lambda(t)$. If the default time is $\tau$, then the probability of default over an infinitesimal time period $dt$, given no default to $t$ is:

\[ \begin{align*} \mathbb{P}(t < \tau < t + dt | \tau > t) = \lambda(t)dt \end{align*} \] {#eq-instantaneous probability of default}

The probability of surviving to at least time $T > t$ (assuming no default has occurred until time $t$) is given by:

\[ \begin{align*} Q(t,T) = \mathbb{P}(\tau > T | \tau > t) = \mathbb{E}[1_{\tau > T}|\mathcal{F}_t] = \exp\left(-\lambda(s)ds\right) \end{align*} \tag{1}\]

Up until this point, we have assumed that the intensity is deterministic - if it is extended to be a stochastic process, then the survival probability is given by:

\[ \begin{align*} Q(t,T) = \mathbb{E}\left[e^{-\int_{t}^T \lambda(s)ds}|\mathcal{F}_t\right] \end{align*} \tag{2}\]

It is quite clear that the survival probability $Q(t,T)$ plays the same role as the discounting factor (risk-free zero-coupon bond) $P(t,T)$, as is the intensity $\lambda(t)$ and the instantaneous short rate $r(t)$. We may extend this analogy and define the forward hazard rate $h(t,T)$ as:

\[ \begin{align*} Q(t,T) = e^{-\int_{t}^T h(t,s) ds} \implies h(t,s) = -\frac{\partial}{\partial s}(\ln Q(t,s)) = -\frac{1}{Q(t,s)} \frac{\partial Q(t,s)}{\partial s} \end{align*} \tag{3}\]

and the zero hazard rates $\Lambda(t,T)$ as:

\[ \begin{align*} Q(t,T) = e^{-(T-t)\Lambda(t,T)}, \quad \Lambda(t,T) = -\frac{1}{T-t}\ln[Q(t,T)] \end{align*} \tag{4}\]

The survival probability curve $Q(t,T)$, the forward hazard rate curve $h(t,T)$ and the zero hazard rate curve $\Lambda(t,T)$ are equivalent and we refer to them generically as credit curves.

The forward hazard rate represents the (infinitesimal) probability of default between times $T$ and $T+dt$, conditional on survival to time $T$ as seen from time $t < T$. The unconditional probability of default between times $T$ and $T+dT$ (as seen from time $t$) is given by:

\[ \mathbb{P}(T < \tau \leq T + dT | \tau > t ) = Q(t,T)h(t) \]

Pricing a CDS

The Protection Leg

THe protection leg of a CDS consists of a (random) payment of $N(1 - RR(\tau))$ at default time $\tau$ if this is before expiry of the CDS (time $T$) and nothing otherwise. The present value of this leg can be written as:

\[ \begin{align*} PV_{prot} = N \mathbb{E}[e^{-\int_0^\tau r(s) ds} (1 - RR(\tau))1_{\tau < T}] \end{align*} \tag{5}\]

Under the assumption of a flat recovery curve, this can be rewritten as:

\[ \begin{align*} PV_{prot} = N(1-RR)\mathbb{E}[e^{-\int_0^\tau r(s) ds} 1_{\tau < T}] \end{align*} \tag{6}\]

Consider first a contract that pays $N(1-RR)$, if the default takes place in the small time interval $[u,u+du]$. The value of this cash-flow at time $0$:

\[ N(1-RR)\mathbb{E}[e^{-\int_0^u r(s)ds } 1_{\tau\in[u,u+du]}] \]

We can rewrite it as:

\[ N(1-RR)\mathbb{E}[e^{-\int_0^u r(s)ds } 1_{\tau\in[u,u+du]}] = N(1-RR)\mathbb{E}[ \lambda(u) e^{-\int_0^u (r(s) + \lambda(s))ds }] \]

Integrating over $u$ from $0$ to $T$, we find that:

\[ V_{prot}(0,T) = N(1-RR)\mathbb{E}\left[\int_0^T \lambda(s) e^{-\int_0^s (r(u) + \lambda(u))du} ds \right] \]

If the short rate process and the credit default intensity processes are independent, we can write this expression as:

\[ V_{prot}(0,T) = N(1-RR) \int_0^T P(0,s) Q(0,s)\lambda(s) ds \]

The last integral can be easily approximated numerically.

The premium leg

Consider now the premium leg of a CDS maturing at $T$ with the premium consisting of the periodic coupon payments only (no upfront fee).

The premium leg consists of two parts : Regular premium (or coupon) payments (e.g. every three months) up to the expiry of the CDS, which cease if a default occurs, and a single payment of the accrued premium in the event of a default.

If there are $M$ remaining payments, with payment times $t_1,t_2,\ldots,t_i,\ldots,t_M$, period end times $e_1,e_2,\ldots,e_M$ and year fractions $\Delta_1, \Delta_2,\ldots,\Delta_M$, then the present value of the premiums only is:

\[ V_{\text{premiums-only}}(0,T) = NC\mathbb{E}\left[\sum_{i=1}^M \Delta_i e^{-\int_0^{t_i} r(s) ds 1_{e_i < \tau}}\right] = NC\sum_{i=1}^M \Delta_i P(0,t_i) Q(0,e_i) \tag{7}\]

Forward Starting CDS

A forward starting CDS entered into at time $t$ will give protection against the default of an obligor for the period $T_e > t$ to $T_m$, in return for premium payments in that period. If the obligor defaults before the start of the protection $\tau < T_e$, the contract cancels worthless. This can easily be replicated by entering a long protection CDS with maturity $T_m$, and a short protection position with maturity $T_e$, leaving only the coupons between $T_e$ and $T_m$ to pay. Furthermore, if a default occurs before $T_e$, the protection payments will exactly cancel.

\[ V(t,T_e, T_m) = V(t, T_m) - V(t,T_e) \]

References

Pricing Single-name and Multi-name credit derivatives, Dominic O’ Kane

--- title: "Standard CDS Pricing Theory" author: "Quasar" date: "2024-05-08" categories: [Credit Derivatives] image: image.jpg toc: true toc-depth: 3 --- # Standard CDS pricing theory I summarize below standard CDS pricing formulas. ## References {.appendix} *[Pricing Single-name and Multi-name credit derivatives](https://www.amazon.co.uk/Modelling-Single-Multi-Wiley-Finance/dp/0470519282), Dominic O' Kane* ## Credit Curves In CDS pricing, credit default events are modelled using a Poisson process, with an intensity (or hazard rate) $\lambda(t)$. If the default time is $\tau$, then the probability of default over an infinitesimal time period $dt$, given no default to $t$ is: $$ \begin{align*} \mathbb{P}(t < \tau < t + dt | \tau > t) = \lambda(t)dt \end{align*} $$ {#eq-instantaneous probability of default} The probability of surviving to at least time $T > t$ (assuming no default has occurred until time $t$) is given by: $$ \begin{align*} Q(t,T) = \mathbb{P}(\tau > T | \tau > t) = \mathbb{E}[1_{\tau > T}|\mathcal{F}_t] = \exp\left(-\lambda(s)ds\right) \end{align*} $$ {#eq-survival-probability-deterministic-intensity} Up until this point, we have assumed that the intensity is deterministic - if it is extended to be a stochastic process, then the survival probability is given by: $$ \begin{align*} Q(t,T) = \mathbb{E}\left[e^{-\int_{t}^T \lambda(s)ds}|\mathcal{F}_t\right] \end{align*} $${#eq-survival-probability-stochastic-intensity} It is quite clear that the survival probability $Q(t,T)$ plays the same role as the discounting factor (risk-free zero-coupon bond) $P(t,T)$, as is the intensity $\lambda(t)$ and the instantaneous short rate $r(t)$. We may extend this analogy and define the forward hazard rate $h(t,T)$ as: $$ \begin{align*} Q(t,T) = e^{-\int_{t}^T h(t,s) ds} \implies h(t,s) = -\frac{\partial}{\partial s}(\ln Q(t,s)) = -\frac{1}{Q(t,s)} \frac{\partial Q(t,s)}{\partial s} \end{align*} $${#eq-forward-hazard-rate} and the zero hazard rates $\Lambda(t,T)$ as: $$ \begin{align*} Q(t,T) = e^{-(T-t)\Lambda(t,T)}, \quad \Lambda(t,T) = -\frac{1}{T-t}\ln[Q(t,T)] \end{align*} $${#eq-zero-hazard-rate} The survival probability curve $Q(t,T)$, the forward hazard rate curve $h(t,T)$ and the zero hazard rate curve $\Lambda(t,T)$ are equivalent and we refer to them generically as credit curves. The forward hazard rate represents the (infinitesimal) probability of default between times $T$ and $T+dt$, conditional on survival to time $T$ as seen from time $t < T$. The unconditional probability of default between times $T$ and $T+dT$ (as seen from time $t$) is given by: $$ \mathbb{P}(T < \tau \leq T + dT | \tau > t ) = Q(t,T)h(t) $$ ## Pricing a CDS ### The Protection Leg THe protection leg of a CDS consists of a (random) payment of $N(1 - RR(\tau))$ at default time $\tau$ if this is before expiry of the CDS (time $T$) and nothing otherwise. The present value of this leg can be written as: $$ \begin{align*} PV_{prot} = N \mathbb{E}[e^{-\int_0^\tau r(s) ds} (1 - RR(\tau))1_{\tau < T}] \end{align*} $${#eq-pv-protection-leg-1} Under the assumption of a flat recovery curve, this can be rewritten as: $$ \begin{align*} PV_{prot} = N(1-RR)\mathbb{E}[e^{-\int_0^\tau r(s) ds} 1_{\tau < T}] \end{align*} $${#eq-pv-protection-leg-2} Consider first a contract that pays $N(1-RR)$, if the default takes place in the small time interval $[u,u+du]$. The value of this cash-flow at time $0$: $$ N(1-RR)\mathbb{E}[e^{-\int_0^u r(s)ds } 1_{\tau\in[u,u+du]}] $$ We can rewrite it as: $$ N(1-RR)\mathbb{E}[e^{-\int_0^u r(s)ds } 1_{\tau\in[u,u+du]}] = N(1-RR)\mathbb{E}[ \lambda(u) e^{-\int_0^u (r(s) + \lambda(s))ds }] $$ Integrating over $u$ from $0$ to $T$, we find that: $$ V_{prot}(0,T) = N(1-RR)\mathbb{E}\left[\int_0^T \lambda(s) e^{-\int_0^s (r(u) + \lambda(u))du} ds \right] $$ If the short rate process and the credit default intensity processes are independent, we can write this expression as: $$ V_{prot}(0,T) = N(1-RR) \int_0^T P(0,s) Q(0,s)\lambda(s) ds $$ The last integral can be easily approximated numerically. ### The premium leg Consider now the premium leg of a CDS maturing at $T$ with the premium consisting of the periodic coupon payments only (no upfront fee). The premium leg consists of two parts : Regular premium (or coupon) payments (e.g. every three months) up to the expiry of the CDS, which cease if a default occurs, and a single payment of the accrued premium in the event of a default. If there are $M$ remaining payments, with payment times $t_1,t_2,\ldots,t_i,\ldots,t_M$, period end times $e_1,e_2,\ldots,e_M$ and year fractions $\Delta_1, \Delta_2,\ldots,\Delta_M$, then the present value of the premiums only is: $$ V_{\text{premiums-only}}(0,T) = NC\mathbb{E}\left[\sum_{i=1}^M \Delta_i e^{-\int_0^{t_i} r(s) ds 1_{e_i < \tau}}\right] = NC\sum_{i=1}^M \Delta_i P(0,t_i) Q(0,e_i) $${#eq-pv-premiums-only} ## Forward Starting CDS A forward starting CDS entered into at time $t$ will give protection against the default of an obligor for the period $T_e > t$ to $T_m$, in return for premium payments in that period. If the obligor defaults before the start of the protection $\tau < T_e$, the contract cancels worthless. This can easily be replicated by entering a long protection CDS with maturity $T_m$, and a short protection position with maturity $T_e$, leaving only the coupons between $T_e$ and $T_m$ to pay. Furthermore, if a default occurs before $T_e$, the protection payments will exactly cancel. $$ V(t,T_e, T_m) = V(t, T_m) - V(t,T_e) $$