Israeli options are another variant of the American option. These options give the option seller the ability to cancel the option early - at the expense of a payment to the holder of the option whilst maintaining the early exercise feature for the holder of the option.

The more common comparison to the Israeli option is that of a callable convertible bond. The convertible bond can be converted into N number of shares of the underlying firm at a particular Strike price whilst the issuer can recall the bonds by paying some sort of compensation premium to the holder.

Pricing:

Pricing of such contracts involve determining the optimal conversion and cancellation of the contract, which we can relate to that of callable convertible bonds, as looked at in Sirbu, Pikovsky & Shreve (2002) and McConnell & Schwartz (1986).

Kifer (2000) introduced the Israeli option and examined the pricing of such contracts using game theory (determining the value of a Dynkin game) under a slightly modified general Black Scholes non-arbitrage framework, see also Kühn & Kyprianou (2003). However, Kifer's approach was only applicable in complete markets. Kifer shows that the valuation can be reduced to evaluating a stochastic saddle point problem associated with a Dynkin game.

Kallsen & Kühn (2003) expanded the above approach for incomplete markets by use of indifference arguments, again by determining the value of a Dynkin game and adjusting the measure.

Although certain cases [such as that of perpetual put options and the case of an Israeli-Russian option - see Kühn & Kyprianou (2003)] of this option can be solved explicitly via closed form analytical methods, the free boundary problem of American options is transposed to Israeli options. Therefore, approximations and simulation methods are generally required for the pricing of these contracts.

Jump Diffusion

Kühn & Kyprianou examine the valuation of Israeli options - more notably, the example of callable convertible bonds under a Levy jump diffusion framework where the underlying asset exhibits a number of jumps.

Monte Carlo Simulation

Rogers (2002) considered a dual way method to price American options under a Monte Carlo framework by taking particular choices of Lagrangian martingales. This method was found to be surprisingly efficient even under simple choices of Lagrangian martingales. Due to the similarly between American options and Israeli options, this approach was applied towards the pricing of Israeli options by Kühn & Kyprianou (2003).

Other Known Names / Variants:

Game Options
Israeli-Russian Options
Perpetual American Options
Recall Options

Additional/Useful List of resources

Papers:

Ekstrom, E., "Properties of Game Options" (2005)
Kallsen, J., & Kühn, C., "Pricing Derivatives of American and game type in Incomplete Markets", Finance & Statistics (2003)
Kyprianou, A., & Baurdoux, C., "Further Calculations for Israeli Options" Stoch. Stoch. Rep., 76, pp. 549-569 (2004)
Kyprianou, A., & Kühn, C., "Israeli Options as composite Exotic Options" (2003)
Kyprianou, A., "Some Calculations for Israeli Options, Journal of Finance & Stochastics" (2002)
Rogers, L.C.G., "Monte Carlo Valuation of American Options ", Mathematical Finance, Vol. 12, pp. 271-286 (2002)
Sirbu, M.,Pikovsky, I. & Shreve, S., "Perpetual Convertible Bonds" (2002)