Sunday, September 6, 2015

Pricing homogeneous Basket Default Swap



Consider a basket with K=10 names. Default times of the names, τk, are i.i.d. random variables with distribution P(τkt)=1eλt. 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: fτ1(t)=(K1)fτ(t)Fτ(t)(1Fτ(t))k1


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<mK, let τ(m)=min(τk:k=1,,K,τk>τ(m1)).

be the mth-to-default time. In particular, τ(K)=max(τ1,,τK). Note that, for t0, P(τ(1)>t)=Ki=1P(τi>t)=eKλt.
Then, the density function is of the form dP(τ(1)t)dt=KλeKλt.
Generally, for 1mK, the event (τ(m)>t) happens as long as there are Km+1 defaults happen later than time t, while the remaining m1 defaults happens earlier than t. That is, P(τ(m)>t)=Kj=Km+1πΠjikπP(τik>t)ilπP(τilt)=Kj=Km+1(Kj)ejλt(1e(Kj)λt),
where Πj denotes the family of subsets of (1,,K) consisting of j elements. Here, ilπP(τilt)=1, if j=K. The density function is then given by Kj=Km+1(Kj)λejλt(jKe(Kj)λt).
We consider the first-to-default case below, that is, m=1. The general mth-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.


Default Leg. The value of the default leg, if we assume that the default payment is made at the default time, is given by (1R)NE(D(τ(1))10<τ(1)T)=(1R)NKλT0e(r+Kλ)tdt=(1R)NKλr+Kλ(1e(r+Kλ)T).

However, if we assume that the default payment is made at the next premium payment date, then the value of the default leg is given by (1R)NE(Mj=1D(Tj)1Tj1<τ(1)Tj)=(1R)NKλMj=1erTjTjTj1eKλtdt=(1R)NMj=1erTj(eKλTj1eKλTj).


Premium Leg. For j=1,,M, let ΔTj=TjTj1. We assume that the premium s, for the payment period (Tj1,Tj] (j=1,,M), is paid at the end date Tj. Moreover, the accrued interest to default, s(τ(1)Tj1)1Tj1<τ(1)Tj, is also paid at Tj. Then value of the premium leg is then given by  NsE(Mj=1D(Tj)[ΔTj1τ(1)>Tj+(τ(1)Tj1)1Tj1<τTj])= NsMj=1erTj[ΔTjeKλTj+KλTjTj1(tTj1)eKλtdt] NsMj=1erTj[ΔTjeKλTj+ΔTj2(eKλTj1eKλTj)].

For the last step, we basically assume that, if the default happens, it happens in the middle of the payment period.


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