  The aim of this paper is to provide a mathematical contribution on the
semi-static hedge of timing risk associated to positions in American-style
options under a multi-dimensional market model. Barrier options are considered
in the paper and semi-static hedges are studied and discussed for a fairly
large class of underlying price dynamics. Timing risk is identified with the
uncertainty associated to the time at which the payoff payment of the barrier
option is due. Starting from the work by Carr and Picron (1999), where the
authors show that the timing risk can be hedged via static positions in plain
vanilla options, the present paper extends the static hedge formula proposed in
Carr and Picron (1999) by giving sufficient conditions to decompose a
generalized timing risk into an integral of knock-in options in a
multi-dimensional market model. A dedicated study of the semi-static hedge is
then conducted by defining the corresponding strategy based on positions in
barrier options. The proposed methodology allows to construct not only first
order hedges but also higher order semi-static hedges, that can be interpreted
as asymptotic expansions of the hedging error. The convergence of these higher
order semi-static hedges to an exact hedge is shown. An illustration of the
main theoretical results is provided for i) a symmetric case, ii) a one
dimensional case, where the first order and second order hedging errors are
derived in analytic closed form. The materiality of the hedging benefit gain of
going from order one to order two by re-iterating the timing risk hedging
strategy is discussed through numerical evidences by showing that order two can
bring to more than 90% reduction of the hedging 'cost' w.r.t. order one
(depending on the specific barrier option characteristics).
