Optimal Timing to Purchase Options

by Tim Leung of the Johns Hopkins University, and
Mike Ludkovski of the University of California, Santa Barbara

April 5, 2011

Abstract: 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 longdated options and sequential buying and selling of options.

JEL Classification: G12, G13, D52, D81.

Keywords: optimal stopping, delayed purchase premium, martingale measures, risk premia

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