02-SEP-2023
The Israeli option, introduced by Kifer (2000), is a financial contract in which both the holder and the writer possess the right to terminate the contract at any time, making it a zero-sum Dynkin game. We analyse the perpetual Israeli put on a geometric Brownian motion under the risk-neutral measure, and show that its value function satisfies a double obstacle variational inequality with two free boundaries $x_1^* < x_2^*$. In the continuation region $(x_1^*, x_2^*)$ the value solves the Black–Scholes ODE, yielding the explicit form $V(x) = Ax + Bx^{-\alpha}$ with $\alpha = 2r/\sigma^2$. We derive the smooth-pasting system that determines all four unknowns $(A, B, x_1^*, x_2^*)$, recover the perpetual American put as the limit $\delta \to \infty$, and identify the two Doob–Meyer compensators of the discounted value process as the local times of the stock price at each free boundary.