آنالیز روش فوکر - پلانک برای مطالعه آمار بازار ارز

In this work we discuss the problem of price definition when using high frequency foreign exchange data. If one uses the spot mid price a strong autocorrelation of returns, at one lag, is found which is only due to microstructure effect and does not capture the real behavior of price dynamics. This autocorrelation increases the intraday volatility estimated from this type of data. To solve this problem we introduce an algorithm which is able, by using the no-arbitrage principle, of eliminating every microstructure effects.

As it was mentioned earlier, τ=+∞τ=+∞ corresponds to Δt=0Δt=0, thus common sense dictates that this distribution needs to be singular and the singularity should be in the point Δx=0Δx=0, as this means that for the zero time intervals the exchange rate cannot change. We can see that obtained stationary solution meets the first condition, and in order to meet the second, D(2)(Δx,τ)D(2)(Δx,τ) parameter c2c2 (see formula (5)) should be equal to zero. This means that while approximating empiric data one needs to take this constraint into account. Examining formula (3) taking into account (8) one can see that in the vicinity of τ=+∞τ=+∞ the distribution function will have major term proportional to |Δx+c2|a1/b2|Δx+c2|a1/b2, which means that the distribution is not Gaussian, but has much fatter tails than Gauss. The fact of fat tails for larger values of ττ was numerically shown in paper [1] and is now confirmed analytically.