  We study the optimal timing of derivative purchases in incomplete markets. In
our model, an investor attempts to maximize the spread between her model price
and the offered market price through optimally timing her purchase. Both the
investor and the market value the options by risk-neutral expectations but
under different equivalent martingale measures representing different market
views. The structure of the resulting optimal stopping problem depends on the
interaction between the respective market price of risk and the option payoff.
In particular, a crucial role is played by the delayed purchase premium that is
related to the stochastic bracket between the market price and the buyer's risk
premia. Explicit characterization of the purchase timing is given for two
representative classes of Markovian models: (i) defaultable equity models with
local intensity; (ii) diffusion stochastic volatility models. Several numerical
examples are presented to illustrate the results. Our model is also applicable
to the optimal rolling of long-dated options and sequential buying and selling
of options.
