Analytical valuation of options on joint minima and maxima

In this paper, we show how to obtain explicit formulae for a variety of popular path-dependent contracts with complex payoffs involving joint distributions of several extrema. More specifically, we give formulae for standard step-up and step-down barrier options, as well as partial and outside step-up and step-down barrier options, involving multiple integrals of dimensions ranging between three and five. Our method can be extended to other exotic path-dependent payoffs as well as to higher dimensions. Numerical results show that the quasi random integration of these formulae involving multivariate distributions of correlated Gaussian random variables provides option values more quickly and more accurately than Monte Carlo simulation.