A pairwise local correlation model

Frank Koster and Daniel Oeltz

Abstract

We develop a local correlation model that uses a given correlation matrix and a generic function g.t ; mi ; mj / to compute the local correlation between any asset– asset pair .i; j / of a basket of underlyings. The arguments mi , mj are spot moneynesses. The generic function is calibrated to fit the implied volatilities of an equity index such as the DAX or EURO STOXX 50. The advantage of this approach is that we do not need to simulate the complete index basket when pricing options on a (usually small) subset of this index. The approach does not suffer from the so-called chewing gum effect of correlation models, where the local correlation depends on just the index value. We first show how to calibrate the generic function for each time step with constraints on the positive definiteness of the resulting correlation matrixes and the smoothness of g to allow for stable evaluation. We then present numerical experiments that show the impact on prices and deltas for synthetic autocallable instruments.