  We introduce signature payoffs, a family of path-dependent derivatives that
are given in terms of the signature of the price path of the underlying asset.
We show that these derivatives are dense in the space of continuous payoffs, a
result that is exploited to quickly price arbitrary continuous payoffs. This
approach to pricing derivatives is then tested with European options, American
options, Asian options, lookback options and variance swaps. As we show,
signature payoffs can be used to price these derivatives with very high
accuracy.
