Consider a basket with K=10 names. Default times of the names, τk, are i.i.d. random variables with distribution P(τk≤t)=1−e−λt. Suppose that each name in the basket has a notional Nk=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 τ∗=mini∈{1,...,K}τi denote the first order statistic. The distribution of τ∗ is easily calculated: fτ∗1(t)=(K1)fτ(t)Fτ(t)(1−Fτ(t))k−1
Let T1,...TM denote the term structure of payment dates.
The event {$\tau^*
For the default leg, I think: $$V_0^{Def} = E_0^{Q}[\sum_{i=1}^Me^{-rT_i} s *N*\mathbb{1}_{{0<\tau ^*
But I unsure about this logic. I have little background with these financial terms.
Answer
Let τ(1)=min(τ1,…,τK) be the first-to-default time. Moreover, for 1<m≤K, let τ(m)=min(τk:k=1,…,K,τk>τ(m−1)).
Let R be the recovery rate (e.g., R = 40 %). Note that the default leg is also called the protection leg.
Default Leg. The value of the default leg, if we assume that the default payment is made at the default time, is given by (1−R)NE(D(τ(1))10<τ(1)≤T)=(1−R)NKλ∫T0e−(r+Kλ)tdt=(1−R)NKλr+Kλ(1−e−(r+Kλ)T).
Premium Leg. For j=1,…,M, let ΔTj=Tj−Tj−1. We assume that the premium s, for the payment period (Tj−1,Tj] (j=1,…,M), is paid at the end date Tj. Moreover, the accrued interest to default, s(τ(1)−Tj−1)1Tj−1<τ(1)≤Tj, is also paid at Tj. Then value of the premium leg is then given by NsE(M∑j=1D(Tj)[ΔTj1τ(1)>Tj+(τ(1)−Tj−1)1Tj−1<τ∗≤Tj])= NsM∑j=1e−rTj[ΔTje−KλTj+Kλ∫TjTj−1(t−Tj−1)e−Kλtdt]≈ NsM∑j=1e−rTj[ΔTje−KλTj+ΔTj2(e−KλTj−1−e−KλTj)].
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