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stampeding
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Joined: July 14th, 2002, 3:00 am

Local Correlation in Rainbow Option context?

October 2nd, 2019, 3:36 pm

Hi

As I understand it, there are three existing methods for calculation of Local Correlation:

 1. Local in index volatility of the index (Lagnau 2010, Kovrizhkin 2012)
 2. Local in index correlation matrix (Guyon & Henry-Labordère 2011)
 3. "Building all admissible local correlation models" (Guyon 2013)

Number 1 & 2 are special cases of 3.
Number 1 is used for "FX Triangle" (as I understand) and the other two are used for calibrating Local Correlations to Index Smile.

Now, my question is this:

Suppose we have some kind of Rainbow Option with only a few of the Stocks in an Index as Underlying, and we want to use a Local Volatility + Local Correlation model. I see two problems, one not so serious (?) and one more serious:

Not so serious (?): To calibrate the Local Correlations of the few Underlying Stocks, we need to use the smile of the entire Index, and the result is all Local Correlations between all the Stocks in the Index.

More serious: It seems to me -- if I'm correct? -- that in the simulation itself, one needs to simulate all the Stocks in the Index to get the correct Index Value, which is used for Local Correlation retrieval/calculation, since every individual Local Correlation pair depends on (potentially among other things) the Index Level.

A possible solution to both these problems would perhaps be to treat the "remaining Stocks" (i.e. the Stocks in Index not affecting the Rainbow Option) as one single Stock or a "Partial Index", with would have a very heavy weight in the Index. One would then need to calculate the Smiles etc for this "Partial Index", and (I assume) re-calibrate all the Local Correlations. The simulation would simulate N+1 stocks, with N Underlyings, and 1 "Partial Index" which would only affect the Local Correlations. But this seems to me like an extremely cumbersome method, which I assume also likely would have a lot of potential error and inaccuracies.


So my question is if there has been any work done by anyone on this particular problem? I.e. Local Correlation used on Rainbow or Basket Options, where the Underlyings are just a few Stocks in an Index which have many more constituents?

And if the answer is no, which kind of Correlation is then commonly used for Rainbow or Basket options in conjunction with Local Volatility? Deterministic Correlation? (Like LVCC, Local Volatility Constant Correlation?) And if that is the case, how is then the Decorrelation problem handled? Not at all? (I.e. Decorrelation is considered part of the model?) By rule-of-thumb? Other?

Regards,
/Samuel, Stockholm, Sweden