The Dependence Structure of Running Maxima and Minima:Results and Option Pricing Applications

We provide general results for the dependence structure of running maxima (minima) of sets of variables in a model based on i) Markov dynamics; ii) no Granger causality; iii) cross-section dependence. At the time series level, we derive recursive formulas for running minima and maxima. These formulas enable to use a ``bootstrapping'' technique to recursively recover the pricing kernels of barrier options from those of the corresponding European options. We also show that the dependence formulas for running maxima (minima) are completely defined from the copula function representing dependence among levels at the terminal date. The result is applied to multivariate discrete barrier digital products. Barrier Altiplanos are simply priced by i) bootstrapping the price of univariate barrier products; ii) evaluating a European Altiplano with these values.