  We price American options using kernel-based approximations of the Volterra
Heston model. We choose these approximations because they allow
simulation-based techniques for pricing. We prove the convergence of American
option prices in the approximating sequence of models towards the prices in the
Volterra Heston model. A crucial step in the proof is to exploit the affine
structure of the model in order to establish explicit formulas and convergence
results for the conditional Fourier-Laplace transform of the log price and an
adjusted version of the forward variance. We illustrate with numerical examples
our convergence result and the behavior of American option prices with respect
to certain parameters of the model.
