Modern Stochastics: Theory and Applications

The major characteristic of the cancellable American options is the existing writer’s right to cancel the contract prematurely paying some penalty amount. The main purpose of this paper is to introduce and examine a new subclass of such options for which the penalty which the writer owes for this right consists of three parts – a fixed amount, shares of the underlying asset, and a proportion of the usual option payment. We examine the asymptotic case in which the maturity is set to be infinity. We determine the optimal exercise regions for the option’s holder and writer and derive the fair option price.

In this paper we study the existence of an optimal hedging strategy for the shortfall risk measure in the game options setup. We consider the continuous time Black–Scholes (BS) model. Our first result says that in the case where the game contingent claim (GCC) can be exercised only on a finite set of times, there exists an optimal strategy. Our second and main result is an example which demonstrates that for the case where the GCC can be stopped on the whole time interval, optimal portfolio strategies need not always exist.