Constant-Hazard CDS Valuation and Mark-to-Market Assumptions
Summary
The document derives a simplified fair spread for a credit default swap using a constant hazard rate, constant recovery, and constant discount rate. It approximates scheduled premium payments with a continuous premium leg and assumes protection is paid at default. Equating the legs yields a spread equal to the loss given default times the hazard rate under these assumptions.
It then proposes valuing an existing contract from the protection buyer’s perspective by comparing protection and premium legs, and asks whether the hazard rate should be inferred from the original contract spread or the current market spread. The proposed mark-to-market formula appears to use a remaining-maturity annuity, but the setup describes a new contract with maturity T at time t, so the time convention and remaining tenor need careful treatment. The analysis also omits accrued premium, scheduled payment details, and market term structures; it is a simplified framework rather than a complete CDS valuation model.
Key ideas
- With constant hazard and recovery assumptions, the simplified fair CDS spread is proportional to hazard and loss given default.
- The premium leg and protection leg can be valued using survival probabilities and discounted default probabilities.
- A CDS mark-to-market compares the value of protection with the value of contractual premium payments.
- The market hazard input should reflect current market CDS quotes and consistent contract conventions.
- Accrued premium, discrete payment dates, and term structures can change the valuation.
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Full text
# CDS Mark-to-Market
# CDS Mark-to-Market
I am trying to calculate the Mark-to-Market of a CDS, and I want to know if what I did is correct.
Let a CDS of maturity $T$, we suppose that the recovery $RR$, the discount rate $r$, and the hazard rate $\lambda > 0$ are constant. The default time $\tau$ is defined such that $\mathbb{P}(\tau > t)$ $=$ $e^{-\lambda t}$. Let $0 = T_0 < ... < T_N = T$ the contractual payment dates. We ignore the payment accrued.
I'm going to calculate the fair spread $s$*, this is the spread such that $PVPremiumLeg = PVProtectionLeg$. We can write the present value of the premium leg of an existing CDS contract as :
$PVPremiumLeg = s\sum_{i=0}^{N-1} (T_{i+1} - T_i) \mathbb{E}[e^{-rT_i}1_{\tau > T_{i+1}}] = s\sum_{i=0}^{N-1} (T_{i+1} - T_i)e^{-(r+\lambda)T_{i+1}}$. We also suppose that the premium payments are continuous. So :
$PVPremiumLeg \simeq s\int_{0}^{T} e^{-(r+\lambda)u} du = s \frac{1-e^{-(r+\lambda)T}}{r+\lambda}$
If we suppose that the protection leg is payed at the defaut time $\tau$, we have, $PVProtectionLeg = (1-RR)\mathbb{E}[e^{-r\tau}1_{\tau \le T}] = (1-RR)\int_{0}^{T} \lambda e^{-(r+\lambda)u} du = (1-RR) \frac{\lambda}{\lambda +r}(1-e^{-(\lambda + r)T})$
We deduce that the fair spread $s$* is $s$* $= (1-RR)\lambda$
Now we want to compute the Mark-to-Market (protection buyer point of view) of this CDS, at a time $0 < t \le T$.
We have $MtM(t) = PVProtection(t) - PVPremiumLeg(t)$. But at this time $t$, we know that the market spread (the spread quoted in the market) $s_t$ is the spread which makes the Mark-to-Market of a new CDS (of maturity $T$) starting at time $t$ equal to $0$, i.e $PVProtectionLeg(t) - s_t\frac{1-e^{-(r+\lambda)T}}{r+\lambda} = 0$.
So $PVProtectionLeg(t) = s_t\frac{1-e^{-(r+\lambda)T}}{r+\lambda}$. And then we conclude that :
$MtM(t) = (s_t - s_0)\frac{1-e^{-(r+\lambda)T}}{r+\lambda}$ with $s_0$ the contractual spread.
I want to implement this MtM function in $R$. My question is how do you calibrate the $\lambda$ ? Do we still have $\lambda = \frac{s_0}{1-RR}$, or do we calibrate it at each time $t$, i.e $\lambda_t = \frac{s_t}{1-RR}$ ? Thank your for you answer and sorry for my poor english.
Adam.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.