Optimal behavior of an investor in option market

The paper deals with the problem of optimal behavior of an investor in the option market with his 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. A modified continuous version of VaR-criterion is introduced that reflects market players' preferences more precisely. An increasing function of critical incomes, possibly with some parameters, given for all critical probabilities in the segment [0,1] is considered. A VaR-criterion for every point in this segment as far as possible starting from the point zero is required to be satisfied. 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.