  This paper gives an arbitrage-free prediction for future prices of an
arbitrary co-terminal set of options with a given maturity, based on the
observed time series of these option prices. The statistical analysis of such a
multi-dimensional time series of option prices corresponding to $n$ strikes
(with $n$ large, e.g. $n\geq 40$) and the same maturity, is a difficult task
due to the fact that option prices at any moment in time satisfy non-linear and
non-explicit no-arbitrage restrictions. Hence any $n$-dimensional time series
model also has to satisfy these implicit restrictions at each time step, a
condition that is impossible to meet since the model innovations can take
arbitrary values. We solve this problem for any $n\in\NN$ in the context of
Foreign Exchange (FX) by first encoding the option prices at each time step in
terms of the parameters of the corresponding risk-neutral measure and then
performing the time series analysis in the parameter space. The option price
predictions are obtained from the predicted risk-neutral measure by effectively
integrating it against the corresponding option payoffs. The non-linear
transformation between option prices and the risk-neutral parameters applied
here is \textit{not} arbitrary: it is the standard mapping used by market
makers in the FX option markets (the SABR parameterisation) and is given
explicitly in closed form. Our method is not restricted to the FX asset class
nor does it depend on the type of parameterisation used. Statistical analysis
of FX market data illustrates that our arbitrage-free predictions outperform
the naive random walk forecasts, suggesting a potential for building management
strategies for portfolios of derivative products, akin to the ones widely used
in the underlying equity and futures markets.
