We study the 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. We show that 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 in the related contexts of hedging long-dated options
and quasi-static hedging.