The game option, which is also called as the Israel option, is an American option with callable features. The option holder can exercise the option at any time up to maturity. This article studies the pricing behaviors of the look back game option where the payoff of the option depends on the maximum or minimum over the asset price movement path (i.e., the game option with the look back feature). We provide the integral expression of pricing formula under the finite horizon case. In addition, we derive optimal exercise strategies and continuation regions of options in both floating and fixed strike cases.
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