Quick theoretical question ...I understand that a copula is used to model nonlinear dependency between variables. But if that is the case, why in credit derivatives pricing is it common practice to specify a flat default correlation, which presumably is then run through the copula to get a price?In other words why do we parameterize a copula which is supposed to embody nonlinear dependence with a number that embodes linear dependence?Rgds

the copula function has another nice feature. you can specify the margins first independently, i.e. calibrate each credit curve, and then put it all together into the copula to get the joint distribution. otherwise you would have to specifiy the joined distribution at once, and there is limited choice of "nice" multivariate distributions.this is the main reason copulas are used in practice and not the point you made, which is of course perfectly correct. in practice a full specification of the complete correlation matrix would be too time-consuming. furthermore it does not really change a lot for the price (I have tried it). Another option would be to use other copulas than the gaussian, which have less parameters. this is sometimes done, also in practice. e.g. you can chose the clayton copula in the bloomberg basket pricer if you want... i am sure that investment banks also use other copulas to evaluate their model risk. but the gaussian is somehow the standard (we also know that equity returns does not fit a gaussian distribution, still the black and scholes formula remains the choice...)

Now, andersen and sidenius don't suppose that correlation is flat it's the random factor loading

Hi folksThanks very much for the replies. This is good info.I do want to take it a bit further ... if you don't mind. Aside from the practical aspects ... what I am really interested in is the theoretical underpinnings of the copula model. There is something I am missing with respect to the relationship between correlations and copulas.Even if for example I assume a heterogeneous correlation matrix ... what am I accomplishing by inputting a matrix of correlations to create a copula? In other words ...1. Why do we use a linear measure of correlation to parameterize something that is supposed to be nonlinear?2. What have I accomplished by inputting a certain matrix of correlations in rather than another (aside from its impact on price)?3. What is the relationship between the correlation parameters and the resulting copula?Thanks for the help ...Rgds

hi stylz,ragarding your questions:1) this is just out of convenience. the same reason as to use the normal distribution for everything, even if you know it is not the best choice. the normal distribution is well known, easlily calculated and easily simulated. Same applies to the normal copula, you can handle it easily even in ms excel. there are some papers recommending to use the t-copula (mostly by mashal et al) but same is true in the univariate case for the t-distribution, still everybody uses the normal.2) i do not really understand your question. when you price some product, say a cdo tranche, you could put in the true correlation matrix of the underlying names. what people do is, just to allow for one equal correlation parameter. this has the advantage that you can invert the pricing formula and calculate an "implied" correlation, similar to the concept of implied volatility in option pricing. if the market becomes liquid, you could use this to price new products.3)this i also do not understand. do you mean with correlation parameter pearsons correlation coefficient or the parameter of the copula. for the guassian the coincite, but this is not the case in general.

Thanks for your replies. I am obviously doing an exceedingly poor job of phrasing my question, since I can't seem to get it answered. Many apologies for that.Let me try again in the most basic terms I know, maybe someone can pound the idea into my head. Practically, this isn't a problem. I can draw normal variates, correlate them with Cholesky, convert them back to uniforms, and get default times, this isn't a problem. I just don't understand why I am doing that.So, to rephrase, in the case of a Gaussian copula, why do I input default correlations (which I thought were linear measures of correlation) in order to produce a copula (which I thought was a nonlinear measure of co-movement)?Thanks againRgds

The reason you are doing that is that you want to convert N one-factor distributions into one N-factor distribution. That is what using a copula wins you. But in using a Gaussian copula, which is very easy to implement, you are making certain implicit assumptions about joint correlations of large (tail) events.Using other copulas you will be making other assumptions about the joint tail events. There is a big literature on this.No matter which copula you use, you will still use correlation of "time to default" as an input. By the way IMO "correlation of time to default" is a highly flaky parameter!But if you view this model as simply a way to express tranche prices, fine.

QuoteOriginally posted by: StylzThanks for your replies. I am obviously doing an exceedingly poor job of phrasing my question, since I can't seem to get it answered. Many apologies for that.Let me try again in the most basic terms I know, maybe someone can pound the idea into my head. Practically, this isn't a problem. I can draw normal variates, correlate them with Cholesky, convert them back to uniforms, and get default times, this isn't a problem. I just don't understand why I am doing that.So, to rephrase, in the case of a Gaussian copula, why do I input default correlations (which I thought were linear measures of correlation) in order to produce a copula (which I thought was a nonlinear measure of co-movement)?Thanks againRgds- you gained the fact that your default probabilities can be anything you like: they are not normally distributed.- the linear co-movement is between the normal variates you draw. But then, you convert to uniforms and...... so your default times are not linearly correlated. You don't have tail dependence but that is a different story.

I think you're just mistaken in what copulas are for. They provide correlation between marginals: both linear and nonlinear correlation. They're not just about nonlinear correlation. The point of using copulas isn't always to get nonlinear correlation effects into a model: it's to get *any* kind of correlation. They're used in the cases where you know marginal distributions of a bunch of variables but don't know the joint distribution; they provide that joint distribution.In your example you're using a Gaussian copula, which ends up giving a pretty linear correlation structure (unless the marginals are particularly distorted in some way). That's not necessarily bad; it depends on what the pricing problem is. If the problem isn't too sensitive to correlation skew then it might not matter much.

Thanks again for the replies. This is very helpful.Just to confirm my understanding, can I say each of the following:1. Pearson correlation is a linear measure. It refers to the tendency of two variables to move together in first order only. So it assumes that the movement of X given Y will exhibit a certain correlation regardless of the value of Y.2. Copulas are a way to join certain marginal distributions into a joint density. The key attribute is that the marginal distributions and the copula choice can be independent. For example we can assume uniform margins and a normal copula, or an archimedean copula for that matter. That is one reason for their popularity in pricing credit derivatives; the individual survival probabilities are exponential but we assume a normal copula for the basket survival.3. Copulas provide a way to model the fact that co-movement varies depending on the state of the world. So the distribution of X given Y can be very different (depending on the choice of copula, of course) according to where exactly Y is. One reason this is so important is that it is unclear how to parameterize the tail of the joint default of two companies. Because the companies have never defaulted we have a scarcity of data and cannot possibly parameterize a joint distribution. Therefore we utilize a copula because we think it best captures a reasonable way that things might begin to go sour "in the tail."4. By inputting a correlation of time to default between reference credits, we are stating what linear correlation between the marginals might be. But the copula itself will overlay its own co-movement structure; so that for a "large move" that linear correlation will no longer hold.5. One reason Gaussian copulas are popular is ease of implementation. But most feel they are not a good representation of reality because of symmetry and thin tails. For example they might have the "degree of co-movement" being the same for both high and low credit spread environments, when in reality we feel that in a weak credit cycle co-movement is likely to increase. Other copulas might introduce asymmetry or simply fatter tails, which we think might be better representations of reality.How does that sound? Anything I am missing?Thx and Rgds