A central problem in modern mathematical
finance is that of estimating the volatility
of financial time series, whether they are
equity prices, exchange rates, interest rates
or something else, such as options. A recent trend is to try to
estimate the implied volatility of an asset from
the fluctuations in the price of derivatives
whose underlying it is. This is the volatility
surface estimation problem. I will review briefly the
background and status of this problem, including
computational issues, and I will present a variational
theory for volatility surface estimation within
stochastic volatilty models. I will show the form
this theory takes under a fast mean reverting hypothesis
and I will conclude with a calibration of the theoretical
framework using SP500 options data.