  This paper is devoted to the application of an $l_1$ -minimisation technique
to construct an arbitrage-free call-option surface. We propose a
nononparametric approach to obtaining model-free call option surfaces that are
perfectly consistent with market quotes and free of static arbitrage. The
approach is inspired from the compressed-sensing framework that is used in
signal processing to deal with under-sampled signals. We address the problem of
fitting the call-option surface to sparse option data. To illustrate the
methodology, we proceed to the construction of the whole call-price surface of
the S\&P500 options, taking into account the arbitrage possibilities in the
time direction. The resulting object is a surface free of both butterfly and
calendar-spread arbitrage that matches the original market points. We then move
on to an FX application, namely the HKD/USD call-option surface.
