Risk Profile

When w = b = 0, the Atlas option is simply a call on the average performance of a basket. The higher the volatility, the higher the price.

The analysis starts to get more interesting when we start removing good and bad performers at maturity.

For simplicity, we look at a homogeneous ptf with identical pairwise correlations given by a single number in a simple lognormal model.

For very high correlations, the basket behaves as a single asset and this removing assets has a small effect on the option payoff. Since it behaves as a single asset, the option's payoff generally increase with volatility.

For low correlations, the basket has a high dispersion at maturity, and the higher the volatility the higher the dispersion. On average, a few stocks will have a high price and the rest low. Since the expectation of the sum of asset prices at maturity is independent of both volatility and correlation, having a few high asset prices implies that many others are very low in most paths. But it is precisely these high-contribution assets that are removed from the basket, leaving the basket with low-priced assets, thus reducing the price of an option.

So for low correlation, increasing volatility does not necessarily increase the price.

Note that this simple analysis applies to a homogeneous basket. Individual volatilities, dividends, and a non-constant correlation can affect the payoff in ways not always easily explained.

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