  We propose the deep parametric PDE method to solve high-dimensional
parametric partial differential equations. A single neural network approximates
the solution of a whole family of PDEs after being trained without the need of
sample solutions. As a practical application, we compute option prices in the
multivariate Black-Scholes model. After a single training phase, the prices for
different time, state and model parameters are available in milliseconds. We
evaluate the accuracy in the price and a generalisation of the implied
volatility with examples of up to 25 dimensions. A comparison with alternative
machine learning approaches, confirms the effectiveness of the approach.
