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.
