خطر ابتلا به داوری ناشی از هزینه های معامله

We discuss the time evolution of quotation of stocks and commodities and show that they form an Ising chain. We show that transaction costs induce arbitrage risk that is usually neglected. The full analysis of the portfolio theory is computationally complex but the latest development in quantum computation theory suggests that such a task can be performed on quantum computers.

One can simply define arbitrage as an opportunity of making profit without any risk [1]. But this definition has one flaw: it neglects transaction costs. And any market activity involves costs (e.g. brokerage, taxes and others depending on the established rules). Therefore, there is always some uncertainty and an arbitrageur cannot avoid risk. Below we will describe an extremely profitable manipulation of a one asset market that certainly fits this definition and show how brokerage can induce risk. The method allows to make maximal profits in a fixed interval [0,k] (short-selling allows to make profits with arbitrary price changes). We will analyze the associated risk by introducing canonical arbitrage portfolios that admit Ising model like description. Investigation of such models is difficult from the computational point of view (the complication grows exponentially in k) but the latest development in quantum computation seems to pave the way for finding effective methods of solving the involved computational problems [2].

Simulations are usually perceived as modelling of real processes by Turing machines. But complexity of various phenomena shows the limits of effective polynomial algorithms. It seems that the future will reverse the roles: we will compute by simulations perceived as measurements of appropriate (physical?) phenomena. In fact such methods have been used for centuries.4 The model discussed above can be easily associated with quantum computation (and games). Calculations for portfolios should take into consideration all available pure strategies whose number grows exponentially in k (2k). Therefore the classical Turing machines are of little use. One of the future possibilities might be exploration of nano-structures having properties of Ising chains. Changes of local magnetic fields hm and controlling temperature may allow for effective determination of profits and strategies for players of various abilities (measured by their temperature [3] and [4]). The values of the parameters nm would be found by measurements of magnetic moments. Another effective method might consist in using quantum parallelism for simultaneous determination of all 2k components of the statistical sum. Quantum computation would use superpositions of k qubit quantum states View the MathML source with arbitrary phases ϕn1…nk. Measurements of the states |ψ〉 would allow to identify for a given portfolio all important leading terms in the statistical sum. The paper [11] presents analysis of the problem of simulation of an Ising chain on a quantum computer. One can easily identify the unitary transformations used there with transition matrices for probability amplitudes. Details of such computations and their interpretation in term of quantum market games will be presented in a separate paper (cf Ref. [12]).