  A volatility surface is an important tool for pricing and hedging
derivatives. The surface shows the volatility that is implied by the market
price of an option on an asset as a function of the option's strike price and
maturity. Often, market data is incomplete and it is necessary to estimate
missing points on partially observed surfaces. In this paper, we show how
variational autoencoders can be used for this task. The first step is to derive
latent variables that can be used to construct synthetic volatility surfaces
that are indistinguishable from those observed historically. The second step is
to determine the synthetic surface generated by our latent variables that fits
available data as closely as possible. As a dividend of our first step, the
synthetic surfaces produced can also be used in stress testing, in market
simulators for developing quantitative investment strategies, and for the
valuation of exotic options. We illustrate our procedure and demonstrate its
power using foreign exchange market data.
