you areyou are talking credit-PVBP vs. IR-PVBPBut don't forget that there will be a small IR PVBP most of the time b/c the contract is not likely to remain at par

salah, i call this carry/accrual (or cash management), not theta. if one day goes by, my MtM goes down because of 1 day accrual. theta in my opinion is pure time decay, which applies to equity options values going down with time...MtM = (Change is Spread + Accrual) * RiskyAnnuity

i meant:- Accrualnot +

Jarod,I think that a theta still exist. let's say for example that when you buy a CDS you pay it UpFront (no running coupon). so there is non accrual to take into account.then one day later and even if the spread remains the same, your CDS doeant worth the same value as yesterday!do you agree?

there is thetat here bc u pay upfront - if u pay quarterly it is different.

I think it is erroneous to use theta to describe any carry related effect. Theta is really supposed to be the loss in the value of an option-like financial asset that you have to pay to benefit from gamma P&L. It's convenient to blend DP/dt under the theta lable, but it makes things complicated to to BE analyses on options. How would you differentiate what you call theta below from true option theta if you bought a credit swaption on an upfront premium CDS? I like to use two greeks, a pure theta, and carry measure.

theta is more for equity options. i agree we should use carry here.

OK I agree with you! thanks for these remarks!

Let's go back to an important point made below, and also in the Schonbucher copula thread. The sub-topic being spread risk and default risk of a FTD position, and related hedges. It seems to me that in order to be default neutral to all names, then you must have a short-dated CDS to each name in a notional equal to DefaultDelta(i) = X(i)-SpreadDelta(i) [Equation 1], where SpreadDelta(i) is the hedge ratio to name i that the model generates, and X(i) is the notional of the FTD (usually all X(i) are equal across names, but not always, especially managed HY CBOs). The problem I have with this aproach is that we have hedges that are not the mathematical first derivative of the pricing model, unless the pricing model does use the short-dated CDS prices as an input (but the point was that they don't really depend on spread that much). Or did I miss the point, and did you state that the FTD model uses both? In this case, are the respective hedges, partial first order derivatives to respective long and short dated CDS, satisfying Equation 1 above? It seems to me that this may not be an arbitrage-free framework because the hedges are not derived straight from the model, otherwise, they may not satisfy euqation 1.Another way to look at it is that if you buy protection on say 3 month CDS to hedge the "bullet-out-of-the-blue" default, and assuming (i) credit curves are flat, and (ii) you keep rolling the 3M hedge on all names to maturity of the FTD, then Equation 1 practically imposes that you will be paying carry over the term of the FTD on 100% delta to each name. So that pretty much implies that the pricing on an FTD should be the sum of the spread, or that all assets are uncorrelated. So why using a model at all? If you assume curves are steep, say the 3M spread is only 50% of the 5y spread (which is pretty steep), you'll still have substantial carry by rolling the short maturity CDS, and this should again imply a very high pricing, and low asset correlation. So I am at odds with the approach. it seems nice in theory, but I can't see how an arbitrage-free model can price and hedge to it? On the other hand, a spread-only model that puts you in 100% delta to a name that approaches default seems to make more sense to me.

With further thinking, I don't see that this discussion is specific to Nth-TD's. I think the same issue is present with any option type product with a credit-risky bond or CDS as the underlier, like a credit default swaption. My point is that a default is nothing less than a jump to recovery. If you try to hedge a default swaption using frictionless assumptions, and hedge to your delta in the underlying CDS, you also have exposure to instantaneous default. So in a sense, I view this whole discussion as simply proving that an option-type instrument on a non pure-diffusion process cannot be perfectly hedged with a position in the underlying instrument and dynamic rebalancing. This is really nothing new. Many of us have ran books of exotics based on adiffusion-jump Merton process. Some of the inputs, such as the frequency of jump, and the size of jump (and possibly the volatility of the size of jump) are not directly tradeable, and can only be obtained with another option on a jumpy asset. You may obtain some sensitivity to such inputs from a position in the underlier, but it will not match the option exposure. The only way to get out this situation is to diversify such risk, and grow a large book. Same goes with a book of digital options (infinite gamma only replicable with other digitals, but strike not always available, so diversify strikes). At the end, traders and risk-managers wind-up using ad-hoc techniques to try and create a more "static" hedge. Maybe the combination of long-dated and short-dated CDS is an attempt to do that for FTD's, but the implication on pricing FTD's remains to be resolved if pursuing that approach, as hedges are not the first deriv of prices, so daily P&L will be wacko.

I suppose Paul's ITO33 approach, described in this thread "Option pricing on assets which follow generalised motions", can provide some answer to my last sentence. HERO would represent the daily P&L volatility that may occur, but the framework provides a risk-neutral pricing as well. Time to apply this to credit portfolios and CDOs!

QuoteOriginally posted by: j20056With further thinking, I don't see that this discussion is specific to Nth-TD's. I think the same issue is present with any option type product with a credit-risky bond or CDS as the underlier, like a credit default swaption. My point is that a default is nothing less than a jump to recovery. If you try to hedge a default swaption using frictionless assumptions, and hedge to your delta in the underlying CDS, you also have exposure to instantaneous default. So in a sense, I view this whole discussion as simply proving that an option-type instrument on a non pure-diffusion process cannot be perfectly hedged with a position in the underlying instrument and dynamic rebalancing. This is really nothing new. Many of us have ran books of exotics based on adiffusion-jump Merton process. Some of the inputs, such as the frequency of jump, and the size of jump (and possibly the volatility of the size of jump) are not directly tradeable, and can only be obtained with another option on a jumpy asset. You may obtain some sensitivity to such inputs from a position in the underlier, but it will not match the option exposure. The only way to get out this situation is to diversify such risk, and grow a large book. Same goes with a book of digital options (infinite gamma only replicable with other digitals, but strike not always available, so diversify strikes). At the end, traders and risk-managers wind-up using ad-hoc techniques to try and create a more "static" hedge. Maybe the combination of long-dated and short-dated CDS is an attempt to do that for FTD's, but the implication on pricing FTD's remains to be resolved if pursuing that approach, as hedges are not the first deriv of prices, so daily P&L will be wacko.ITO 33 cannot agree more!- We alrealy apply optimal dynamic hedging (not just with the underlying but any derivative instrument), and estimation of HERO (or residual unhedgeable risk), to derivatives pricing under general jump-diffusion / default risk / stochastic volatility processes. Our I CARE program has already applications in FX, equity options, and Convertible Bond pricing.- And yes! default is nothing but a jump to recovery.

J20056 says"So going back to the main argument, I am not sure if one should hedge to default correlations, or total correlation, or model both correlations separately?"How do you calcul this total correlation according to the correlation of spreads and of correlation of default times ?Yassin

Actually, in my thinking, spread correlation is the total correlation. Given that spreads incorporate default, liquidity and tax premiae, I think that one approach is to call all of the latter the "non-default" component. As a result, the 2 correlations to model are default correlation and non-default correlation. The trick is that allocation of non-default premium on the assets virtually cannot be modeled properly when applied to the liabilities, because it is well known that BB/BBB tranches have a negative non-default premium (i.e. you get paid less than the risk-neutral spread based model tells you if you sell protection), whereas the AA and above tranches have a positive non-default premium. Short of using ad-hoc methods to calibrate such non-default premium for liabilities (or Nth-to-default), I'm not sure if there is a consistent framework that can produce such a non-default spread basis curve across the capital structure of a CDO, or i-th-to-default where i belongs to {1...N}

More generally, I have 2 questions :- when you have determined spread correlation of each name "two by two", how do you deduce basket's correlation ?- and how do you determine hedge ratio for each name in the basket ?Thanks