Optimal shouting policies of options with strike reset right

The reset right embedded in an option contract is the privilege given to the option holder to reset certain terms in the contract according to specified rules at the moment of shouting, where the time to shout is chosen optimally by the holder. For example, a shout option with strike reset right entitles its holder to choose the time to take ownership of an at-the-money option. This paper develops the theoretical framework of analyzing the optimal shouting policies to be adopted by the holder of an option with reset right on the strike price. It is observed that the optimal shouting policy depends on the time dependent behaviors of the expected discounted value of the at-the-money option received upon shouting. During the time period when the theta of the expected discounted value of the new option received is positive, it is never optimal for the holder to shout at any level of asset value. At those times when the theta is negative, we show that there exists a threshold value for the asset price above which the holder of a reset put option should shout optimally. For the shout floor, we obtain an analytic representation of the price function. For the reset put option, we derive the integral representation of the shouting right premium and analyze the asymptotic behaviors of the optimal shouting boundaries at time close to expiry and infinite time from expiry. We also provide numerical results for the option values and shouting boundaries using the binomial scheme and recursive integration method. Accuracy and run time efficiency of these two numerical schemes are compared.