Optimal multiple stopping models of reload options and shout options

The reload provision in an employee stock option entitles its holder to receive one new (reload) option from the employer for each share tendered as the payment of strike upon the exercise of the stock option. The number of reloads allowed can be finite or infinite. The shout feature in a call option allows its holder to reset the option's strike price to the prevailing stock price upon shouting. We explore a symmetry relationship between the price functions of the employee reload options and shout call options. When the dividend yield of the underlying stock is zero, the value of the employee reload option can be expressed in terms of the price functions of a shout call option and a forward contract. For an employee reload option with an infinite number of allowable reloads, the payoff of the employee reload option can be related to the lookback feature of the stock price process. We also examine the optimal exercise policies of the multi-reload employee stock options. The behavior of the critical stock price at which the holder should exercise optimally is shown to depend on the relative magnitude of the dividend yield, interest rate and volatility. Our analysis of the multi-reload and multi-shout options contributes to the literature on optimal stopping policies of contingent claims with multiple stopping rights.