  We provide an efficient method to approximate the covariance between decision
variables and uncertain parameters in solutions to a general class of
stochastic nonlinear complementarity problems. We also develop a sensitivity
metric to quantify the uncertainty propagation in the problem by determining
the change in the variance of the output variables due to a change in the
variance of an input parameter. The covariance matrix of the solution variables
quantifies the uncertainty in the output and pairs correlated variables and
parameters. The sensitivity metric helps in identifying the parameters that
cause maximum fluctuations in the output. The method developed in this paper
optimizes the use of gradients and matrix multiplications which makes it
particularly useful for large-scale problems. Having developed this method, we
extend the deterministic version of the North American Natural Gas Model
(NANGAM) to incorporate effects due to uncertainty in the parameters of the
demand function, supply function, tariffs, and investment costs. We then use
the sensitivity metrics to identify the parameters that impact the equilibrium
the most.
