Estimating Correlation Vega of a Basket
June 15, 2010
Of late, we have been having a lot of discussions in the CFE class regarding the impact of correlation on basket structures. Since there are many traders in our CFE class in all the five locations, these discussions turn out to be quite productive and illuminating. It is now universally accepted by all and sundry in the field of quantitative finance that despite all fancy modelling no one can get a complete grip on the issue of correlation and more importantly, how it can be hedged in the case of basket products.
We would like to point out one risk measure which has been sometimes used, though not very effectively, by the risk managers to measure the impact of correlation on a basket structure. The measure is known as Correlation Vega.
Strictly speaking, Correlation Vega corresponds to the change in the price of a basket (any two or multi-asset product) due to the change in the correlation between the two assets in the basket. An operator can shift the correlation and estimate the price using a numerical technique, such as Monte Carlo or closed form solution, if it exists.
However, many risk managers in the past have defined a Correlation Vega as the sensitivity of the basket volatility with respect to the correlation between the assets. Correlation vega is estimated by taking the first mathematical derivative of the basket volatility with respect to the correlation between the pairs. In that sense, Correlation Vega can be defined as the change in basket volatility with respect to the change in the correlation between the assets of the basket.
If the basket volatility is given by:
Then, the closed form estimate of Correlation Vega of the basket with respect to the correlation, between asset and asset is given by:
This approach of viewing Correlation Vega as the sensitivity of the basket volatility with respect to the correlation between the underlying assets stems from the approach that many traders in the good old days would try to reduce a complex product to a "pseudo-vanilla" option. For example, a "pseudo-vanilla" for a basket call option is a vanilla call option that has a volatility of the basket. But this approach has severe limitations and only a few sensitivities can be tested using this method.
Of course, these days no trader or risk manager would be inclined to follow a pseudo-vanilla approach to pricing or risk management. This is because the products have become so complex and the sensitivities (greeks) so diverse and varied that a pseudo-vanilla approach will simply not work.
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