# Pricing homogeneous Basket Default Swap

Consider a basket with $K=10$ names. Default times of the names, $\tau_k$, are i.i.d. random variables with distribution $P(\tau_k \leq t) = 1 - e^{-\lambda t}$. Suppose that each name in the basket has a notional $N_k = N$, the basket notional, $N=KN$, the discount factors, $D(t) = exp(-rt)$. Denote by $s$ the payment rate (also called the spread). Denote by $T$ the maturity of the $m$-to-default BDS written on this basket. Assume that the number of payment dates in $M$.

Find the value of the default and premium legs for $m=1$ and $m=2$.

My attempt: For $m=1$: Let the random variable $\tau^* = \min_{i\in \{1,...,K \}}\tau_i$ denote the first order statistic. The distribution of $\tau^*$ is easily calculated: $$f_{\tau_1^*}(t) = {K \choose 1} f_{\tau}(t) F_{\tau}(t)(1-F_{\tau}(t))^{k-1}$$

Let $T_1,...T_M$ denote the term structure of payment dates.

The event {$\tau^*<T_i$} implies that the first default occurred before $T_i$.

For the default leg, I think: $$V_0^{Def} = E_0^{Q}[\sum_{i=1}^Me^{-rT_i} s *N*\mathbb{1}_{{0<\tau ^* <T_i}}]$$

Let $\tau_{(1)} = \min(\tau_1, \ldots, \tau_K)$ be the first-to-default time. Moreover, for $1< m \le K$, let \begin{align*} \tau_{(m)} = \min\left(\tau_k: k=1, \ldots, K, \tau_{k} > \tau_{(m-1)}\right). \end{align*} be the $m^{\rm th}$-to-default time. In particular, $\tau_{(K)} = \max(\tau_1, \ldots, \tau_K)$. Note that, for $t \ge 0$, \begin{align*} P\left(\tau_{(1)} > t\right) &= \prod_{i=1}^K P(\tau_i > t)\\ &=e^{-K \lambda t}. \end{align*} Then, the density function is of the form \begin{align*} \frac{d P\left(\tau_{(1)} \le t\right)}{dt} &= K \lambda\, e^{-K \lambda t}. \end{align*} Generally, for $1 \le m \le K$, the event $\left(\tau_{(m)} > t\right)$ happens as long as there are $K-m+1$ defaults happen later than time $t$, while the remaining $m-1$ defaults happens earlier than $t$. That is, \begin{align*} P\left(\tau_{(m)} > t\right) &= \sum_{j=K-m+1}^K \sum_{\pi \in \Pi_j}\prod_{i_k \in \pi}P(\tau_{i_k} > t) \prod_{i_l \not\in \pi}P(\tau_{i_l} \le t)\\ &=\sum_{j=K-m+1}^K {K \choose j} e^{-j\lambda t} \Big(1-e^{-(K-j)\lambda t} \Big), \end{align*} where $\Pi_j$ denotes the family of subsets of $(1, \ldots, K)$ consisting of $j$ elements. Here, $\prod_{i_l \not\in \pi}P(\tau_{i_l} \le t)=1$, if $j=K$. The density function is then given by \begin{align*} \sum_{j=K-m+1}^K {K \choose j} \lambda\, e^{-j\lambda t}\Big(j-Ke^{-(K-j)\lambda t} \Big). \end{align*} We consider the first-to-default case below, that is, $m=1$. The general $m^{\rm th}$-to-default case is similar based on the density function above.
Let $R$ be the recovery rate (e.g., R = 40 %). Note that the default leg is also called the protection leg.
Premium Leg. For $j=1, \ldots, M$, let $\Delta T_j = T_j - T_{j-1}$. We assume that the premium $s$, for the payment period $(T_{j-1}, \, T_j]$ $(j=1, \ldots, M)$, is paid at the end date $T_j$. Moreover, the accrued interest to default, $s(\tau_{(1)}-T_{j-1}) 1_{T_{j-1} < \tau_{(1)} \le T_j}$, is also paid at $T_j$. Then value of the premium leg is then given by \begin{align*} &\ N s E\left(\sum_{j=1}^M D(T_j) \Big[\Delta T_j 1_{\tau_{(1)} > T_j} + \big(\tau_{(1)}-T_{j-1}\big) 1_{T_{j-1} < \tau^* \le T_j} \Big] \right) \\ =& \ N s\sum_{j=1}^M e^{-r T_j} \bigg[\Delta T_j e^{-K \lambda T_j} + K \lambda \int_{T_{j-1}}^{T_j}(t-T_{j-1})e^{-K \lambda t} dt \bigg]\\ \approx& \ N s\sum_{j=1}^M e^{-r T_j} \bigg[\Delta T_j e^{-K \lambda T_j} +\frac{\Delta T_j}{2} \left(e^{-K \lambda T_{j-1}} - e^{-K \lambda T_j}\right) \bigg]. \end{align*} For the last step, we basically assume that, if the default happens, it happens in the middle of the payment period.
• Changed $\sum_{(i_1, \ldots, i_{K-m+1}) \in \pi_m}$ to $\sum_{(i_1, \ldots, i_{K-m+1}) \in \pi_{K-m+1}}$ – Gordon Jan 13 '16 at 21:13