  Transition probability density functions (TPDFs) are fundamental to
computational finance, including option pricing and hedging. Advancing recent
work in deep learning, we develop novel neural TPDF generators through solving
backward Kolmogorov equations in parametric space for cumulative probability
functions. The generators are ultra-fast, very accurate and can be trained for
any asset model described by stochastic differential equations. These are
"single solve", so they do not require retraining when parameters of the
stochastic model are changed (e.g. recalibration of volatility). Once trained,
the neural TDPF generators can be transferred to less powerful computers where
they can be used for e.g. option pricing at speeds as fast as if the TPDF were
known in a closed form.
  We illustrate the computational efficiency of the proposed neural
approximations of TPDFs by inserting them into numerical option pricing
methods. We demonstrate a wide range of applications including the
Black-Scholes-Merton model, the standard Heston model, the SABR model, and
jump-diffusion models. These numerical experiments confirm the ultra-fast speed
and high accuracy of the developed neural TPDF generators.
