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# Error Estimates for Binomial Approximations of Game Put Options

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Abstract:

A game or Israeli option is an American style option where both the writer and the holder have the right to terminate the contract before the expiration time. Kifer (2000) shows that the fair price for this option can be expressed as the value of a Dynkin game. In general, there are no explicit formulas for fair prices of American and game options and approximations are used for their computations. The paper by Lamberton (1998) provides error estimates for binomial approximation of American put options and here we extend the approach of Lamberton (1998) in order to obtain error estimates for binomial approximations of game put options which is more complicated as it requires us to deal with two free boundaries corresponding to the writer and to the holder of the game option. 1. Introduction A put option on a stock can be interpreted as a contract between a holder and a writer which allows the former to claim from the latter at an exercise time the amount , where is a fixed amount called the option’s strike, is the stock price at time , and . In the American options case its holder has the right to choose any exercise time before the contract matures, while in the game options case the contract writer also has the right to terminate it at any time before its maturity, but then he is required to pay a cancellation fee in addition to the payoff above. The fair price of American options and of game options is defined as the minimal amount the writer needs to construct a self-financing portfolio which covers his obligation to pay according to the option’s contract. It is well known that in the American options case the fair price can be obtained as a value of an appropriate optimal stopping problem, while for game options we have to deal with an optimal stopping (Dynkin) game (see [1]). For more information about results on Dynkin games and game options we refer the reader to the survey [2]. In general, both for American options and, even more so, for game options with finite maturity explicit formulas for their price are not available and approximation methods come into the picture, while estimates of their errors become important. One of the most easily implemented methods is the binomial approximation of stock prices modelled by the geometric Brownian motion and [3] provided corresponding error estimates for American put options. In the present paper we extend this approach in order to provide error estimates of binomial approximations for game put options. We observe that for perpetual game options some explicit formulas can be obtained (see [4]), but

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