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BerndSchmitz
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XVA of an uncollateralized swaption

October 25th, 2016, 10:37 am

Hi,

I was studying the papers of Burgard and Kjaer (I use "Partial differential equation representations of derivatives with bilateral counterparty risk and funing costs" as a formula reference) in the context of an uncollateralized swaption (on an uncollateralized swap).

For me the underlying [$]S[$] in formula (3.1) is a collateralized swapRate, as I can only calibrate a collateralized swapRate process to the market (to clarify: The collateralized and uncollateralized swapRates are imO assets that are very correlated but they are not the same). Furthermore, I completely agree with formula (3.17) which is the PDE of a collateralized swaption (on a collateralized swap) - unsurprisingly the value process of the collateralized swaption depends on the collateralized vol [$]\sigma[$]. However, I'm a bit surprised to find the collateralized vol [$]\sigma[$] also appears in formula (3.7), which is the PDE of the uncollateralized swaption (isn't it?). Of course I can assume that the vol of the uncollateraslized swapRate is equal to the one of the collateralized swapRate (probably that's the only goable solution as I simply have no market to calibrate the uncollateralized vol to). However, to me that is quiet an assumption that should have been made more transparent.

What are your opinions on that? 

Thanks,
Bernd
 
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cenperro

Re: XVA of an uncollateralized swaption

October 25th, 2016, 1:42 pm

First: you are right in that "The collateralized and uncollateralized swapRates are imO assets that are very correlated but they are not the same". However, under the standard hypothesis that make the calculation feasible they can be considered as being led by the same stochastic process and having the same vol. (in other words, only a discount effect, based in some basis/spreads makes the difference and, for the most common applications, those basis/spreads are considered as 'deterministic') 
Second:  With the first point in mind, the distinction you try to make between the vol of the collateralized and uncollateralized swapt rate vol , while accurate, is very hard to put into practice. You are right in that, the vol that should be present in equation 3.7 is the vol of the underlying , but, does the underlying need to be the uncollateralized swap rate? that's a matter of definition. We can enter into an uncollateralized swaption that gives you the right to enter into a collateralized swap at expiry. Or we might be talking of cash delivery in which case it would be more standard to look into the value of the collateralized swap rate at expiry. Anyway, even if we want to look at S as the uncollateralized rate, in terms of vol I do not think any practioner make that distinction as it would imply modeling some stuff very difficult to model.

By the way, if you are starting to look into this I would recommend you to take the classic approach of considering S to be an Stock an think of all the calculations under deterministic rates. The extension to stoch rates is straightforward but it is easier to grab all the ideas without that extra ingredient. 




 
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BerndSchmitz
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Re: XVA of an uncollateralized swaption

October 26th, 2016, 10:25 am

You are totally right that there is nothing to talk about for an uncollateralized physical swaption on a collateralzed swap or for an uncollateralized cash-settled swaption. I was explicitly thinking about an uncollateralized swaption physically settling on an uncollateralzed swap. Thanks for clarifying that indeed there is an implicit assumption made in this case. I'm not questioning this assumption but I wanted to be sure that one was made.

I'm now only wondering how this implicit assumption that the collaterlized swapRate and the uncollateralized one have the same volatility compares to the very common assumption that the spread between ois and funding is deterministic, leading to the result that the implied normal volatility of the swaption on the uncollateralized swap is the same as the implied normal volatility of the swaption on the collateralized swap if taken at the same moneyness.
 
mathfam
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Re: XVA of an uncollateralized swaption

November 22nd, 2016, 12:03 am

Gents, few observations:

This paper (and it's a firm favourite for reasons I'll elaborate), is very much a bird's eye view. The actual choices of dynamics are much less impactful than (2.3). To mention some of the elements omitted in the  paper (and which are 100% priced daily in my desk experience) are for example stochastic intensities (of your choice, Black - Karasinski was productionised where I last traded for example so we have mean reversion and interesting structure to the forwards as spreads vary), credit-rate diffusion correlation (highly significant if we are talking about say large cyclical European counterparties and Euro swaps), and critically bank / counterparty credit correlation driving the first to default intensities - governing who the model thinks will default first. These really are all the elephants in the room. More of this shortly.

On the specific point about swap vols, it's worth mentioning, if we take the ISDA 2002 incarnation of derivative documentation, as the paper rightly points out, there is no mention of uni/bilateral credit risk in the 6E closeout / termination language. It is entirely fair to interpret this as saying (gap and replacement cost risk precipitated by defaults of certain systemic counterparties notwithstanding) then that the CVA defect to collateralised valuation (again in (2.3)) refers to a claim (and hence will apply recovery to a claim) against the collateralised value, and therefore if you do not use the collateralised vol you are replicating to a payoff on the wrong asset . Counterparty defaults, dealer is owed money, they ask up to 15 banks (more likely 3) to quote to be replacement counterparties on a collateralised basis. The legal documents really are all important here - the language in the 2002 version is ok, (in the preceding 1999 version it is diabolical, check it out - totally inconsistent and frankly mathematically illogical).

Now back to the credit / risk niceties :) To give an example say you have a bank highly correlated with a counterparty and trading at the same spread and the bank is long a highly ITM derivative (of any kind such that it is model-unlikely that it will be OTM to the bank at any point over its life). Now say you have some correlation model for the FTD intensities, at high correlation, if the bank spread widens relative to the counterparty then the bank will model wise default first in many models, so there is no incurred CVA loss, but perhaps a smaller funding cost increase. It's likely in this case the bank is short risk its own name since the CVA loss could dominate. If the bank tightens relative to the counterparty, then it's in trouble - the expectation is that counterparty defaults first and potentially here there is a large payoff hit. Hence here again the bank is short risk its own name. If correlation drops between bank and counterparty, then the funding side could start to dominate and so the bank could become long risk its own name owing to funding costs becoming more important than the 'who defaults first' piece. All of this is on the credit delta side but shows I hope some of the interesting credit aspects to the treatment.  

Finally for me historically the paper was helpful for 2 reasons: a) the explicit FTD treatment (once you realise that a bank funding spread conditional on joint survival is really the same as the bank FTD) allows for a price to be agreed by both counterparties and b) it removed the DVA double count nonsense since the replication arguments are convincing that any +ve cash balances generated by an unfunded derivative liability do not accrue a funding benefit in addition to the DVA benefit on such a liability without incurring further credit risk.

In short then, the generalisations of dynamics are manifold, but the collateralised vs 'uncollateralised' argument for swaption vols, i.e. think at least for any deriv under 2002 ISDA, are not a central concern, and in fact I think the identity assumption is ok. Be interested in your thought tho, cheers! 
 
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Cuchulainn
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Re: XVA of an uncollateralized swaption

December 13th, 2016, 2:15 pm

Bernd,
i have a stupid Q on the Burgard/Kjaer (July 2012? yes?) paper where they discuss [$]V^{+}[$] and [$]V^{-}[$] 

[$]V^{+} = max(V,0)[$] 
[$]V^{-} = max(-V,0)[$] 

Is this the correct definition? BTW your equation number 3.1 etc. differ from the above reference. Do I have an older version?

We (ZH/DD) are working on this PDE model and comparing with some other approaches.
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An Application of Stochastic Mesh Method in Computing the Credit Value Adjustment of Derivatives with Multi-payments.pdf
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agnoatto
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Re: XVA of an uncollateralized swaption

December 14th, 2016, 5:19 pm

Hi Daniel, most papers on xVA use a different convention concerning positive and negative part, which is not in line with the one you usually find on books on real Analysis. The exception is given by Bichuch Capponi and Sturm (Arbitrage free pricing of xVA). To put it short $V^-$ is negative in Burgard Kjaer.
Prof Alessandro Gnoatto, PhD
Department of Economics
University of Verona
Via Cantarane 24 - 37129 Verona - Italy
37129, Verona, Italy
 
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Cuchulainn
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Re: XVA of an uncollateralized swaption

December 17th, 2016, 2:18 pm

Hi Daniel, most papers on xVA use a different convention concerning positive and negative part, which is not in line with the one you usually find on books on real Analysis. The exception is given by Bichuch Capponi and Sturm (Arbitrage free pricing of xVA). To put it short $V^-$ is negative in Burgard Kjaer.
Thanks, agnoatto

So [$]V^{-} = - max(-V, 0) = min(V,0)[$] 

This sounds more financiaily and mathematically intuitive since the B&K CVA pde (1) can now contain a combination of negative and positive terms.
A question  on the B&C&S XVA article? how is the pde linearised before applying Crank Nicolson? Is much tweaking needed to get it working?
In contrast to the SDE and PDE equations, the numerical algo.scheme gets short shrift, unfortunately. Ideally, a reader should be able to independently test the results.

An interesting remark is the the nonlinearity is in the drift term for XVA while it is in the reaction term for the CVA pde. For a UVM pde the nonlinearity is in the diffusion term. Is there a financial motivation for this?

// Nice to see nonlinear PDEs here. :)