Optimal Timing to Purchase Options

TL;DR

This paper investigates the optimal timing for purchasing options in incomplete markets by modeling the problem as an optimal stopping problem influenced by market risk premia and payoff structure, with explicit solutions in specific Markovian models.

Contribution

It introduces a novel framework for optimal option purchase timing considering different market views and risk premia, with explicit solutions for certain Markovian models.

Findings

Explicit characterization of optimal purchase timing in Markovian models

Analysis of the delayed purchase premium and its impact

Numerical illustrations demonstrating the model's applications

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 long-dated options and sequential buying and selling of options.