  In the paper, we characterize the asymptotic behavior of the implied
volatility of a basket call option at large and small strikes in a variety of
settings with increasing generality. First, we obtain an asymptotic formula
with an error bound for the left wing of the implied volatility, under the
assumption that the dynamics of asset prices are described by the
multidimensional Black-Scholes model. Next, we find the leading term of
asymptotics of the implied volatility in the case where the asset prices follow
the multidimensional Black-Scholes model with time change by an independent
increasing stochastic process. Finally, we deal with a general situation in
which the dependence between the assets is described by a given copula
function. In this setting, we obtain a model-free tail-wing formula that links
the implied volatility to a special characteristic of the copula called the
weak lower tail dependence function.
