  In this study, we generate a large number of implied volatilities for the
Stochastic Alpha Beta Rho (SABR) model using a graphics processing unit (GPU)
based simulation and enable an extensive neural network to learn them. This
model does not have any exact pricing formulas for vanilla options, and neural
networks have an outstanding ability to approximate various functions.
Surprisingly, the network reduces the simulation noises by itself, thereby
achieving as much accuracy as the Monte-Carlo simulation. Extremely high
accuracy cannot be attained via existing approximate formulas. Moreover, the
network is as efficient as the approaches based on the formulas. When
evaluating based on high accuracy and efficiency, extensive networks can
eliminate the necessity of the pricing formulas for the SABR model. Another
significant contribution is that a novel method is proposed to examine the
errors based on nonlinear regression. This approach is easily extendable to
other pricing models for which it is hard to induce analytic formulas.
