Statistical properties of the rate of return per tick in the FX market

<p>This paper explores and compares empirical distribution of the euro &ndash; US dollar exchange rate returns normalized per tick and per fixed time intervals using a mixture of normal distributions. In accordance with this purpose the following objectives were defined:</p><p style="margin-left:36.0pt;">&middot;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; to examine the efficient market hypothesis as a premise that price changes is a random process with normal Gaussian distribution;</p><p style="margin-left:36.0pt;">&middot;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; to substantiate the use of the rate of return normalized to the number of deals;</p><p style="margin-left:36.0pt;">&middot;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; to study out the concept of mixture distributions and the general solution schemes of splitting of the mixture distributions;</p><p style="margin-left:36.0pt;">&middot;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; understand basic parameter estimation procedures;</p><p style="margin-left:36.0pt;">&middot;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; develop a model selecting the best statistical hypothesis from several competing with each other, using Diebold&rsquo;s method;</p><p style="margin-left:36.0pt;">&middot;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; explore differences in statistical properties of the rate of returns, normalized by different ways.</p><p>This paper was made on the basis of publications of such authors as Greg MacKinnon, Howard Nemiroff, Leonidov, A.V, Diebold, F.X.. To prepare a theoretical part of this writing the following books were used here: &laquo;Practical statistics. Classification and decrease in dimension&raquo; (Ayvazyan, S.A., Bukhshyaber, V.M. and others), &laquo;The Statistical Mechanics of Financial Markets&raquo; (Johannes Voit).</p><p>In conclusion, this paper presents evidence that empirical distribution of returns behaves differently at different ways of normalization. Moreover, using the rate of return per tick helps convert &quot;raw&quot; values (i.e. timed) ​​to the &quot;purified&quot; ones, i.e. to approach an ideal design of the concept of an efficient market. Furthermore, the fact that in the case of normalization to the number of deals we achieved good results by creating a trading strategy based on the statistical properties of the &quot;victory&quot; local distribution makes it possible to believe that yield curves are normally distributed, and therefore, the concept of an efficient market has a chance to exist.</p>