Can anybody explain this puzzle to me?Suppose we are hedging a call option. The world is a Black-Scholes world (constant vol and IR, no transaction costs, bid=offer, lognormal distribution etc etc); here we can be sure that the delta-hedging strategy on the stock, if done perfectly, will perfectly replicate the call option.So is this also true of a credit derivative product like an iTraxx?As an example, if we have an iTraxx index swap on 125 names with notional N and a spread of S. Can we perfectly replicate this by hedging with individual CDS’s? Suppose we try, assuming a simple theoretical world where spreads are constant, recovery is 0%. We set up 125 CDS’s to hedge with a notional of N/125. Then a name defaults. The index contingent payment is N/125 and likewise is the CDS hedge. Fine; we replicated this event on the contingent payment. But what happens next? The index spread remains unchanged but on a smaller notional (N*124/125). But the CDS portfolio? The spread payments reduce by what ever the spread was on that particular single name, be it large or small. It seems that we are not replicating our derivative in the same manner as our vanilla call above. Am I to suppose it is impossible to perfectly replicate simple index CDS with single CDS’s?BM

This 'disperson bias' should improve by using risky durations as index weightings.KDan

Thx for the post.If we choose the CDS notionals to better replicate the spread payments then we will not be hedged on default.I am supposing that even under simple assumptions (flat hazard rates, zero interest, zero recovery), it is impossible perfectly replicate the index CDS with single name CDS’s (as we do in a Black-Scholes world when delta-hedging the underlying); and consequently it is impossible perfectly replicate synthetic tranches either… Could this possibly be true?

Well mohamed, I think that in BS world, when delta hedging the underlying, we are hedging against market risk but not against default risk. In this sense the basket of CDS would correctly replicate the index, apart from the basis intrinsic in the index (consider it as a liquidity premium, which sets a 1-2 bp difference between the index spread and the weighted average of the CDS basket).Default event is a jump process. Introducing it, is (somewhat) like releasing the continuum hedge opportunity in the BS world, which would also imply the failure of your "perfect" replica.A very stupid example showing this difference is the following:consider a 3M CDO ITRAXX equity tranche, delta-hedged. you sell protection on the tranche and buy protection on the index. (delta ratio approx 25 -- this means that in order to offset market risk I have to do 25 times the notional of the tranche (75M) by the index). Then a default occurs. I have to pay (RR = 40 %) 480k euros and I receive 75/125*(100% - 40%) = 360K euros. therefore my JtD (Jump to Default) is still negative.Delta hedging in credit means hedging against market risk, but not agains default risk, I think.

Zub, you bring up an interesting point - JtD in BS world. Default is a binary event, yet the pricing of default risk is a continuous process. In theory we can assume that the world in continuous and default must be preceded by the widening of the spread to the level of default. If we look at your example under this assumption for name to default it's spread would have to widen to the level that would make the PV of the CDS equal to it's LGD. While this is happening we can trade and in case of the equity tranche the delta to this name is increasing.Of course even under this assumption delta replication of an equity tranche isn't likely to work, since tranche pricing models don't reflect risk neutral return distribution.

Thx StructCred & Zub.In my simple example, I have excluded "market risk" by asserting that spreads are constant. The only stochastic process is the default event as defined by a homogoneous Poisson process (i.e. constant hazard rate). Hence JtD is the only process at work. Yet I am unable to achieve perfect replication on paper...

Some other things to considerIndex restructuring type may differ from underlying (I believe this only affects US)Recovery rates for underlying may differ from Index (XOver is priced at 40% recovery byt contains a few subordinated)Indices roll every six months whereas single name roll every three months - so current 5y S7 index is now 4.75 yearsRisky duration differs from average spread by around 4.5 bps on 5 yr XOver S7

Should it not work if we increase the notionals of the non-defaulting entities by a proportional amount, it would be the same as saying that the deltas of the individual names have increased by that much and now we use the new (scaled up) delats?

But what if a non-defaulting names subsequently defaults? It no longer pays N/125 and so the index and CDS are no longer offsetting for defaults.

NoelWatson:agree with you. One possibility is considering all the issues you mention as "basis components" and therefore consider them as a whole in the basis between the index and the underlying CDS pool

After some thought, I believe the problem lies in the nature of the underlying. The underlying is a default;. but what event happens upon default? For a CDS there isn't just one event - there are two. Firstly upon default, an annuity ceases. Secondly upon default, a payment is made.To explain better, imagine that a bank making markets in two "primative" credit products: firstly a "default call"; this pays $1 if a credit survives until maturity of the option. Secondly, a "default put"; this pays $1 if a default happens before maturity.We could then construct a CDS as a series of default calls (e.g. for 5Y annual we have 5 calls with maturity = 1, 2... 5) each with a notional of Si*Ni plus we sell a default put with notional N * (1-R). The portfolio replicates a CDS on notional Ni and spread Si (paying annually).We could also construct an index by (a) buying a series of calls with notional S*N/125 on each credit (b) selling puts on each credit with notional N * (1-R)/ 125 where S is the index spread and N the index notional (assuming 125 names in the index).A CDS hedge alone does not replicate an index because there are two events associated with default (viz annuity cessation and default payment) and we are locked into hedging both with a fixed ratio.A default put is the same as buying protection with all up-front. And a binary call is short put + long risk-free zero-coupon bond. This it seems that if CDS's were liquid upfront and at all maturities then the problem is fixed in my paper world, notwithstanding the points raised on other posts (viz recovery assumptions, restructuring types, index basis, market risk)Does this make any sense to anyone?

There was a nice introduction to default swap done in late nineties by Duffie dealing with its perfect replication in a simple setting.It is equivalent to going long a risk free floating rate note maturing on the same date than the CDS, and shorting the bond (in FRN format with a spread above the risk free rate) underlying the CDS.Same pay-off upon default, plus your risky bond bears the spread risk.Don't know whether this helps ...----------------slym