The Implication of Missing the Optimal-Exercise Time of an American Option

The optimal-exercise policy of an American option dictates when the option should be exercised. In this paper, we consider the implications of missing the optimal exercise time of an American option. For the put option, this means holding the option until it is deeper in-the-money when the optimal decision would have been to exercise instead. We derive an upper bound on the maximum possible loss incurred by such an option holder. This upper bound requires no knowledge of the optimal-exercise policy or true price function. This upper bound is a function of only the option-holders exercise strategy and the intrinsic value of the option. We show that this result holds true for both put and call options under a variety of market models ranging from the simple Black-Scholes model to complex stochastic-volatility jump-diffusion models. Numerical illustrations of this result are provided. We then use this result to study numerically how the cost of delaying exercise varies across market models and call and put options. We also use this result as a tool to numerically investigate the relation between an option-holders risk-preference levels and the maximum possible loss he may incur when adopting a target-payoff policy that is a function of his risk-preference level.