In this paper we provide evidence that financial option markets for equity indices give rise to non-trivial dependency structures between its constituents. Thus, if the individual constituent distributions of an equity index are inferred from the single-stock option markets and combined via a Gaussian copula, for example, one fails to explain the steepness of the observed volatility skew of the index. Intuitively, index option prices are encoding higher correlations in cases where the option is particularly sensitive to stress scenarios of the market. As a result, more complex dependency structures emerge than the ones described by Gaussian copulas or (state-independent) linear correlation structures. In this paper we "decode" the index option market and extract this correlation information in order to extend the multi-asset version of Dupire's "local volatility" model by making correlations a dynamic variable of the market. A "local correlation" model (LCM) is introduced for the pricing of multi-asset derivatives. We show how consistency with the index volatility data can be achieved by construction. LCM achieves consistency with both the constituent- and index option markets by construction while preserving the efficiency and easy implementation of Dupire's model.
