  We present a functional generative approach to extract risk-neutral densities
from market prices of options. Specifically, we model the log-returns on the
time-to-maturity continuum as a stochastic curve driven by standard normal. We
then use neural nets to represent the term structures of the location, the
scale, and the higher-order moments, and impose stringent conditions on the
learning process to ensure the neural net-based curve representation is free of
static arbitrage. This specification is structurally clear in that it separates
the modeling of randomness from the modeling of the term structures of the
parameters. It is data adaptive in that we use neural nets to represent the
shape of the stochastic curve. It is also generative in that the functional
form of the stochastic curve, although parameterized by neural nets, is an
explicit and deterministic function of the standard normal. This explicitness
allows for the efficient generation of samples to price options across strikes
and maturities, without compromising data adaptability. We have validated the
effectiveness of this approach by benchmarking it against a comprehensive set
of baseline models. Experiments show that the extracted risk-neutral densities
accommodate a diverse range of shapes. Its accuracy significantly outperforms
the extensive set of baseline models--including three parametric models and
nine stochastic process models--in terms of accuracy and stability. The success
of this approach is attributed to its capacity to offer flexible term
structures for risk-neutral skewness and kurtosis.
