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**Game options.**
*(English)*
Zbl 1066.91042

A standard \((B,S)\)-securities market consists of a nonrandom (riskless) component \(B_{t},\) which is described as a savings account (or price of a bond) at time \(t\) with an interest \(r,\) and of a random (risky) component \(S_{t},\) which can be described as the price of a stock at time \(t.\) The problem of fair pricing, for example, of American options in the \((B,S)\)-securities market leads to the optimal stopping of certain stochastic processes. The author introduces game options in which the seller of an option can cansel the contract at any time \(t.\) In this case the buyer’s gain is the sum \((K-S_{t})^{+}+\delta_{t}\) in the put and \((S_{t}-K)^{+}+\delta_{t}\) in the call option case, where \(\delta_{t}\geq 0\) is certain penalty paid by the seller and \(K\) is some specific price to sell (put option) or to buy (call option) the stock at any time \(t.\) The pricing of these options leads to a game version of the optimal stopping problem introduced in the discrete time case by E. B. Dynkin [Sov. Math., Dokl., 10, 270–274 (1969); translation from Dokl. Akad. Nauk SSSR 185, 16–19 (1969; Zbl 0186.25304)]. The author considers only basic problems concerning extension of the option pricing theory to game options (or Israeli options to put them in line with American, European, Asian, Russian etc. ones) and many problems still remain to deal with. The analysis is based on the theory of optimal stopping games (Dynkin’s games). Game options can be sold cheaper than usual American options and their introduction could diversify financial markets.

Reviewer: A. V. Swishchuk (Kyïv)

### MSC:

91G20 | Derivative securities (option pricing, hedging, etc.) |

60G40 | Stopping times; optimal stopping problems; gambling theory |

91A15 | Stochastic games, stochastic differential games |