**G.A. Agasandian**

**Optimal Behavior of an Investor in Option Market
**

The work deals with the problem of optimal behavior of an investor in the option market with own opinion on market properties. We tell the difference between investor's and market probability distribution functions of future prices of underlier. In this case, the investor might gain in the average income. If the investor, however, is guided by the presently popular Value-at-Risk criterion in the traditional form, the results may on the full market prove absurd. The point is that any investor commonly chooses a critical level as parameter of VaR-criterion that is relatively low, and so the investor receives low income with the probability equal to 1.

A modified continuous version of VaR-criterion is introduced that reflects market players' preferences more precisely and is free of this shortcoming. If the matter is how to use this method in practice, it is necessary to use a multistage version of VaR-criterion. An increasing function of critical incomes, possibly with some parameters, given for all critical probabilities in the segment [0,1] is considered. VaR-criterion for every point in this segment as far as possible starting from the point zero is required to be satisfied.

To solve this problem, the Neuman-Pearson statistic criterion with the likelihood ratio formed by market and investor's probability densities is used. A clear procedure that determines whether the problem may be solved completely or partly is given. An example of two-sided exponential probability distribution with different parameters for the investor and the market and with the power function of critical incomes demonstrates peculiarities of constructions proposed.

An approximation technique is considered to adapt the method to discrete-in-strikes option market. The exposition is concluded with a generalization of the method to multi-period option markets.

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