  The implied volatility surface (IVS) is a fundamental building block in
computational finance. We provide a survey of methodologies for constructing
such surfaces. We also discuss various topics which can influence the
successful construction of IVS in practice: arbitrage-free conditions in both
strike and time, how to perform extrapolation outside the core region, choice
of calibrating functional and selection of numerical optimization algorithms,
volatility surface dynamics and asymptotics.
