The elements of the probabilistic analysis of the Forex market

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Introduction.

FOREX – is the abbreviation from the English words FOReign EXchange market. Forex – is the largest financial market in the world, whose day turnover makes one and a half billion dollars. In contrast to other financial markets, Forex has no central stock exchange. It functions by means of an electronic network (including the Internet), whose units are banks, corporations and the private persons trading in currencies with each other. Absence of a central unit allows Forex market to work 24h/7days [http://russian.mgforex.com/].

MG Financial Group(http://www.mgforex.com/) - the leader of the Forex on-line technologies - has published the first version of its on-line trading platform (Deal Station) in April 1997. This program allows traders to view currency rates, to make transactions and to trace open positions in real time mode. The distinctive feature of DealStation is that it constantly refreshes all its information (continually). The last development - DealStation 2000 is constructed on the basis of the newest technology "Push" Java which allows "to push" new information on a trader’s computer as soon as it becomes accessible. One of competitors of the Deal Station is a program complex WinChart, which belongs to Straits Index company (http://www.straitsindex.com/). One of the advantages of this program is an opportunity to study the elements of the technical analysis.

There are two approaches to the analysis of the currency market – fundamental and technical. With the help of the fundamental analysis one can determine forces of supply and demand on the basis of financial and economic theories, which are based on the political-economical situation. The technical analysis examines trading rates and volumes on the basis of graphic representation of currency rates in time and it is directed on a tendency diagnosis in the future. The technical analysis allows predicting exchange rates movements on the basis of the last trading rates and volumes data research. This type of analysis relies on heuristic formulas for tracing the rate movement tendencies and allows to estimate opportunities for currency sale or currency purchases. Diagrams are: with 5-minute, 15-minute, 60-minute and 24 hour interval. Diagrams with a week and a month intervals are also applied. The last mentioned diagrams serve for an estimation of long-term tendencies.

I. The examples of technical analysis.

1. 1. Levels of a relative minimum and maximum.

Levels of a relative minimum and maximum are points where the diagram passes from decrease to increase and on the contrary (fig.1). The probability of exceeding these points is considered to be insignificant, therefore purchase or sale is more preferable at the moments of relative minima and maxima.


Fig.1.

2. Direct lines and channels of the tendency.

Direct lines are a simple, but powerful tool for trend revealation, i.e. the market tendencies. They connect some consecutive maxima or minima which belong to some local trend. Continuation of the line shows the most probable direction of the market movement in the future. The channel represents a corridor of quotation changes and is defined as a part of a plane between the parallel straight linees constructed on maxima and on minima. Fig.2 represents the classical example of increase trend, but fig.3 gives an example of the channel localization.


Fig.2.


Fig.3.

3. Current (dynamical) averages.

Current average allows generalize trends and show the average price for a certain period of time. Fig.4 contains three curves of average values, which depend on the period of averaging - a day, a week and a month. There are three types of dynamic average indices: usual, linearly weighed and exponentially smoothed. The exponential smoothing is considered to be more exact, from the probability of a prediction point of view, since it gives a greater weight to the rather recent data. The usual average is counted up under the formula:


where n, for example, stands for the amount of days.


Fig.4.

4. Bands.

The Current average characterizes the process of the quotation changes on average. To estimate statistics of local extremums (maxima and minima) according to the average, one calculates an average square of deviations (RMS). Fig.5. represents an example of RMS diagrams, which form the band. Thus RMS can serve as a measure of probability. The formulas for calculation are represented below.



where D* - is a standard deviation, n is the amount of days,


Fig.5.

I I. The elements of the probabilistic analysis of the Forex market.

The stated above things illustrate the fact, that all efforts of the technical analysis are aimed at an estimation of the probability of a forthcoming event. The technical analysis operates with the most important statistical characteristics - average, RMS, statistics (moments) of higher orders. However, the actual probability remains unclaimed. At the same time, the probabilistic analysis elements can be successfully used both for the calculation of probabilities, and as the graphic tool which is well known to traders.

The results of the estimation researches of the quotation Forex market currencies probabilities are represented below. These researches include the special software developing and the realization (on basis of the software) of the physical simulation and the calculation of the probabilistic distributions of real exchange rates.

1. The Simulation of distribution of DM quotations.


Fig.6.

Fig.6 shows the modelling distribution of the daily diagram of the DM quotations. There are three highlighted trends a, b, c on the diagram. Trends a, b are increasing, c is neutral. The value of DM =1.8376 in a range of trend b determination is marked with a red mark. Fig.7 shows a diagram of the distribution of probabilities. Right from the beginning it is necessary to note, that the trends represent a regular mistake and basically they should be removed. However, in a context of accomplishing a task, they serve as a helpful information. Fig.8 shows a smoothed curve of distribution of probabilities. From the analysis of distributions in fig.7 and fig.8 it becomes clear that there is a visible trend division (localization). It is not possible in the traditional technical analysis.


Fig.7.


Fig.8.

2. The localization of trends of real distribution of quotations DM.

Fig.9 shows the daily diagram of the DM quotations from the 30th of April 1997 until the 14th of June 1998. The total amount is 297 reports. DM =1.7646 is marked with a red mark. Fig.11 and fig.12 show the diagrams of the probability distribution. From the examination of the abovementioned diagrams it is clear that


Fig.9.


Fig.10.


Fig.11.

that the analyzable value belongs to the group of trends of the transition period, which goes from the low values of quotations to the higher ones. The probability of getting into this group of trends is insignificant. For graphic calculation of probabilities we shall transform distribution on fig.11 into the integral of probabilities, which is shown on fig.12.


fig.12.

From the examination of the above mentioned diagrams it is clear that the probability of DM quotations can be found within the given interval [1.6748; 1.7646] and it makes about 30 %.

3. The removal of "a regular mistake".

To make a calculation of probabilities with the given (high) accuracy, we will need to remove the trends (fig.13). By the way, the removal of trends is also practiced in traditional technical analysis. Fig. 14 shows required distribution of the casual process probabilities of absolute quotation changes in fig.13. The appearance of distribution is a good evidence that the analyzed, random process is normal or at least quasi- normal. To get stronger evidence it is necessary to apply the " chi-square".


fig.13.


fig.14.

If in the diagram belonging tî the fig.14 one counts up an integral of probabilities in an interval from -0.0370 up to .0006 (red mark), it will be equal to 0.1986. Thus, it is possible to assert, that the quotation change of DM in the limits from -0.0370 up to .0006 is expected with the probability about 20 %. If we shall count up an integral of probabilities for a symmetric DM interval, the general probability will be about 38 %. The latter will allow to assert that the change of DM quotations in an interval [-0.0060; + 0.0060] is necessary to expect with the confidential probability of 62 %.

4. Application Markoff’s transitional probabilities.

Processes of change of exchange rates are processes stochastic, i.e. represent set both determined (trend) and casual events (fig.13, fig.14). The conclusion about expediency of application Markoff’s transitional probabilities from here follows, which characterize process of transitions (for example, in time) random variable χ from one condition χi in other condition χj.

If to speak about process of change of quotations of currencies transitional probabilities it is conditional probabilities pi,j of that at the moment of time t the current value of an exchange rate is j provided that at the moment t-1 it was equal i. The example Markoff’s distributions of probabilities P (i, j) is submitted on fig.15.


Fig.15.


The basic property Markoff’s probabilities is memory of previous transitions. This property in a context of a considered problem can be formulated as follows: distribution of probabilities Pt(i,j) characterizes probability of that the quotation of currency will accept value j, under condition of, that after t steps (for example, t hours) the quotation, was equal i.

The formulation of a real situation can look as follows. At the order of the trader there are data on 15-minute change of rate EURO/USD for the last 3 - 4 months. (for example, Bid Price). The sequence of values EURO/USD forms Markoff’s circuit P(i,j). 15-minute change of rate EURO/USD we shall interpret as transition of a random variable χ of a condition χi in a condition χj. Set of all transitions forms a matrix of transitions (or relative frequency Ni,j), for example:

  [ 4  5  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
  [ 5 25  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
  [ 0  1 29  7  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
  [ 0  0  7 27  8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
  [ 0  0  0  8 49 20  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
  [ 0  0  0  0 21 67  6  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
  [ 0  0  0  0  0  6 26  2  0  0  0  0  0  0  0  0  0  0  0  0  0]
  [ 0  0  0  0  0  0  1  5  2  0  0  0  0  0  0  0  0  0  0  0  0]
  [ 0  0  0  0  0  0  0  1  5  1  0  0  0  0  0  0  0  0  0  0  0]
  [ 0  0  0  0  0  0  0  0  0  8  2  0  0  0  0  0  0  0  0  0  0]
  [ 0  0  0  0  0  0  0  0  0  1  9  1  0  0  0  0  0  0  0  0  0]
  [ 0  0  0  0  0  0  0  0  0  0  0  2  1  0  0  0  0  0  0  0  0]
  [ 0  0  0  0  0  0  0  0  0  0  0  0  5  6  0  0  0  0  0  0  0]
  [ 0  0  0  0  0  0  0  0  0  0  0  0  5 48  8  0  0  0  0  0  0]
  [ 0  0  0  0  0  0  0  0  0  0  0  0  0  7 17  3  1  0  0  0  0]
  [ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  3 16  2  0  0  0  0]
  [ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2  5  2  0  0  0]
  [ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  4  3  0  0]
  [ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2 17  4  0]
  [ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  3 54  0]
  [ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0].
Fig.16.

Problem. It is required to reply: if current value EURO/USD is in a condition j=5 in what condition i=k this value will be after 15 minutes? Or after 30, 45, 60. minutes?

Solution. Let's take advantage of a matrix of transitions Ni,j and we shall calculate transitive probabilities pi,j with a condition:
M-1   
   Σpi,j = 1,
j =0   

where M - the greatest value of transitions in system EURO/USD. The appropriate matrix of transitive probabilities will look like:

  [ .4444  .5556      0      0      0      0      0      0      0      0      0 . . .
  [ .1613  .8065  .0323      0      0      0      0      0      0      0      0 . . .
  [     0   .027  .7838  .1892      0      0      0      0      0      0      0 . . .
  [     0      0  .1667  .6429  .1905      0      0      0      0      0      0 . . .
  [     0      0      0  .1026  .6282  .2564  .0128      0      0      0      0 . . .
  [     0      0      0      0  .2234  .7128  .0638      0      0      0      0 . . .
  [     0      0      0      0      0  .1765  .7647  .0588      0      0      0 . . .
  [     0      0      0      0      0      0   .125   .625    .25      0      0 . . .
  [     0      0      0      0      0      0      0  .1429  .7143  .1429      0 . . .
  . . .
  . . .
  . . .
Fig.17.

The condition of system EURO/USD at the present moment of time is equal to a vector of probabilities:

s1 = [ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0].


The condition of system EURO/USD in 15 minutes is described by expression and a vector:

s2 = P2(i,j) = s1 x P(i,j) =

= [0 0 0 .02292 .2996 .5766 .0971 .0038 0 0 0 0 0 0 0 0 0 0 0 0 0 0].


Decision Making. If as algorithm for the Decision Making we shall use search of a maximum of probability (.5766) the condition of quotation EURO/USD will be former, since i=5.

Below the table of 15-minute predication (Predict) EURO/USD is submitted during 8 hours June, 11 2003. The probability Pmax an expected condition i, is maximal for all transitions    j=>i,     j=0..M-1. In last column of the table values of mistakes of prediction (Err [%]) in percentage.

No.DateRealPredict.PmaxErr [%]
1Wed Jun 11 04:39:43 20031.16770 0.00100.00
2Wed Jun 11 04:54:47 20031.16721.1677 0.19 -0.04
3Wed Jun 11 05:09:52 20031.16711.1672 0.45 -0.01
4Wed Jun 11 05:24:57 20031.16711.1671 0.48 0.00
5Wed Jun 11 05:40:02 20031.16701.1671 0.52 -0.01
6Wed Jun 11 05:55:06 20031.16591.1671 0.18 -0.10
7Wed Jun 11 06:10:11 20031.16681.1670 0.33 -0.02
8Wed Jun 11 06:25:16 20031.16731.1671 0.31 0.02
9Wed Jun 11 06:40:20 20031.16801.1677 0.15 0.03
10Wed Jun 11 06:55:25 20031.16731.1673 0.19 0.00
11Wed Jun 11 07:10:30 20031.16711.1677 0.16 -0.05
12Wed Jun 11 07:25:35 20031.16751.1671 0.45 0.03
13Wed Jun 11 07:40:39 20031.16771.1675 0.15 0.02
14Wed Jun 11 07:55:44 20031.16751.1671 0.22 0.03
15Wed Jun 11 08:10:49 20031.16761.1675 0.17 0.01
16Wed Jun 11 08:25:54 20031.17121.1671 0.20 0.35
17Wed Jun 11 08:40:59 20031.17111.1713 0.19 -0.02
18Wed Jun 11 08:56:03 20031.17151.1713 0.19 0.02
19Wed Jun 11 09:11:08 20031.17081.1715 0.20 -0.06
20Wed Jun 11 09:26:13 20031.17051.1715 0.16 -0.09
21Wed Jun 11 09:41:18 20031.17071.1701 0.42 0.05
22Wed Jun 11 09:56:24 20031.17151.1715 0.13 0.00
23Wed Jun 11 10:11:31 20031.17091.1715 0.21 -0.05
24Wed Jun 11 10:24:41 20031.17271.1709 0.22 0.15
25Wed Jun 11 10:39:02 20031.17201.1725 0.12 -0.04
26Wed Jun 11 10:54:07 20031.17231.1715 0.21 0.07
27Wed Jun 11 11:09:18 20031.17091.1715 0.11 -0.05
28Wed Jun 11 11:24:23 20031.17081.1709 0.19 -0.01
29Wed Jun 11 11:39:32 20031.16901.1715 0.14 -0.21
30Wed Jun 11 11:54:42 20031.16981.1691 0.15 0.06
31Wed Jun 11 12:09:48 20031.17041.1701 0.37 0.03
32Wed Jun 11 12:24:54 20031.17071.1709 0.14 -0.02
33Wed Jun 11 12:32:47 20031.17071.1715 0.12 -0.07


Values of real quotations and values of predictions are illustrated with the help of the diagram (fig. 18), where t - 15-minutes period.


Fig.18.


The diagram of mistakes is shown on fig. 19, where t - 15-minutes period.


Fig.19.


I I I. Preliminary conclusion.

The represented results of the probability distribution of currency quotations research for the market Forex show the following.

a. They allow to interprete the state of Forex market both as a graphic form of representation of probability distributions, and as the numerical characteristics.

b. The represented numerical characteristics of the probability analysis (integrals of probabilities and confidential probabilities) have generalizing character and can be used for a long-term forecasts. The software, in a context of long-term forecasts, can be static i.e. to use some database of currency quotations. The software, in a context of long-term forecasts, can be static i.e. use some database of currency quotations. The database can be refilled by a user in a manual style. The static version development of such a program will take 2- 3 months. The dynamic version development will take extra man-hours. The difference between the dynamic software and the static one is that for getting the currency quotations in a real time the dynamic software has to contain an additional functional block. Example:

c. The use of Markoff’s series device is expedient for the short-term forecasts, which is actualfor traders. The development of such a program research version and its realization by means of probabilistic distribution research will take from 1,5 up to 2 months. The demo version of software CPS (Currency Prediction Software ) can be looked Here.