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 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. It could hardly be investor's initial desire. 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 VaRcriterion. 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. The topic of the paper is partly reflected in [1]. I. THEORETICAL ONE-PERIOD OPTION MARKET At first, the theoretical one-period option market that covers only two moments of time is considered. Transaction costs are ignored. The investor invests money at initial time and receives income at final time. In the market, standard European call options C(E) and put options P(E) of any strike E∈R (set of all real numbers) trade. U denotes the risk-free asset of unit size, and S denotes the underlying stock. Using these instruments as building blocks allows replicating various portfolios including continuous combination. So, almost any instrument G with the payoff g(x) of rather arbitrary type may be replicated. The essence of the problem is that to every linear combination of payoffs including continuous one, there corresponds the same combination of instruments. For example, if g(x) at x → ±∞ is restricted, the presentation ( ) ( ) ∫ ∫ ∞