Some calculations for Israeli options

Recently Kifer (2000) introduced the concept of an Israeli (or Game) option. That is a general American-type option with the added possibility that the writer may terminate the contract early inducing a payment exceeding the holder's claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating an optimal stopping problem asso-caited with Dynkin games. In this short text we give two examples of perpetual Israeli options where the solutions are explicit. 1 Israeli options Consider the Black-Scholes market. That is, a market with a risky asset S and a riskless bond, B. The bond evolves according to the dynamic dB t = rB t dt where r; t 0: The value of the risky asset is written as the process S = fS t : t 0g where S t = s expfW t + tg where s > 0 is the initial value of S and W = fW t : t 0g is a Brownian motion deened on the ltered probability space ((; F; F = fF t g t0 ; P) satisfying the usual conditions. Let 0 < T 1: Suppose that X = fX t : t 2 0; T]g and Y = fY t : t 2 0; T]g be two continuous stochastic processes deened on ((; F; F; P) such that with probability one Y t X t for all t 2 0; T]: The Israeli option, introduced by Kifer (2000), is a contract between a writer and holder at time t = 0 such that both have the right to exercise at any F-stopping time before the expiry date T. If 1