# Quanto CDS modeling

What is the market standard for pricing quanto CDS (i.e. CDS which pays the contingent leg in different currency than the pricing leg)?

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It would be cool if the answer could specifically address this in the context of sovereign CDS and contagion. – Brian B Feb 22 '11 at 18:36
Yes, especially now :) – quant_dev Feb 25 '11 at 21:45

there is no standard approach to model quanto CDS. In practice, people look at the dynamic hedging costs over time as well as the expected loss from an fx gap in the event of a default of the ref entity. the former is modelled by some correlated brownian (for FX) and mean-reverting processes (for credit - could be Ornstein Uhlenbeck for example). In addition, you need some event correlation of the FX gapping when the reference entity defaults. You see, all a little messy. Don't get me started on the calibration.....

CDS on sovereigns in the country's own currency trade at roughly 50%-60% of their liquid spot contract while Eur-countries trade between 5% (perepherial) and 30% (core)

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Why such big discount for Eurozone countries? because of high correlation between them? – quant_dev Mar 22 '11 at 20:09

The consensus seems to be is using jump diffusion process (affine), and then using copula's and/or correlated brownian motions to handle the correlation structure.

Here's a link to a recent paper that discusses these models in great detail, and includes application of these models for modeling quanto cds:

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1153400

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I know this paper. Is this the "standard" approach? – quant_dev Mar 22 '11 at 20:08

Here is a link to a very interesting paper about the subject. The model assumes

• lognormal intensities (I think to ensure non-arbitrage as default probability must always be between 0 and 1, which is not the case if we assume Gaussian process for the intensity)
• deterministic FX volatility
• correlation between FX and the intensity
• a jump in the FX spot in the event of default

The author gives some examples of how each of these parameters affect the ratio quantoCDS/CDS (which seems to be the quotation convention for some emerging market quanto CDS).

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you do not happen to be the author of the linked paper? If yes, I would recommend you mention that, obviously you have a special interest in calling your own paper "a very interesting paper", regardless of whether its interesting or not. Its kind of market practice to do so (and I would assume that you, as Director in the Analytics team, would know about how to reference own and others' work), all making the assumption that you are the author.Nonetheless, I will want to read through the paper, interesting topic. – Matt Wolf Dec 9 '12 at 14:17
Hi, I m the author of the paper. – Rachid Dec 9 '12 at 18:24
thanks for mentioning this. – Matt Wolf Dec 9 '12 at 18:31
It went quickly before I finish my comment:) I wrote another paper for pricing quanto first-to-default as an extension of the quanto cds pricing. Any comments are welcome. – Rachid Dec 9 '12 at 18:32
Funny to have the author saying his paper is "very interesting"... – SRKX Dec 12 '12 at 18:49

This take into account three components:

• Dynamic model for hazard rates (and thus cds). The dynamic is chosen to be lognormal (so it is always positive and arbitrage free) with constant volatility and mean reversion.
• The FX spot follows a BS dynamic with Jump at time of default. More complex dynamic for the model are not essential given that quanto CDS is a quasi linear payoff as function of the FX.
• Correlation between the hazard rate and FX spot.

All these parameters are important for the pricing of quanto cds:

• The jump size affect the overall level of the ratio quanto CDS/CDS. Example: jump size of 40% mean that the quanto CDS will be traded at -40% of CDS premium level (in absence of correlation between hazard rate and FX spot).
• The correlation between hazard rates and FX spot affect more the long dated quanto CDS as the short dated quanto CDS are not affected too much by this correlation.

Calibration of the model parameters. The model is calibrated to two components:

• The FX spot volatility is calibrated to ATM FX volatilities. The paper describe a calibration algorithm using forward PDE. However the paper about pricing quanto FTD which is a generalization of the pricing of quanto cds describe another calibration algorithm which is more simple to implement.
• Calibration of lognormal hazard rate process to a CDS curve. The calibration is done using forward PDE (could be done using a trinomial tree).

I think that those are the main risks embedded in quanto CDS trades. We could add to that stochastic interest rates if the trades are long dated.

When I said that this paper is very interesting, it is because it answers exactly the initial question posted here. I hope this will help answering the original question.

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