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Generally, the trader starts the trade at a benefit. This situation is uncommon in the currency markets. Yet can happen once in a while, particularly when high volatility is existent or tenuous liquidity. Also, it has gotten increasingly uncommon as of late because of high-recurrence trading.

Where computer calculations have made pricing progressively productive and decreased the time-frame certain trading to happen. The result of that imbalance is called triangular arbitrage. These are infrequent opportunities.

Usually, traders with advanced computer equipment or programs to automate the method may take advantage of these opportunities. Arbitrage opportunities may emerge less often in the market than some other gain-making benefits, yet they do arrive on purpose. Financial analysts believe arbitrage to be a key component in keeping up the liquidity of market situations. As arbitrageurs help bring costs across the markets into balance.

Arbitrage action ought to guarantee that values of similar securities converge, in case boundless risk-free gains may emerge. Publish on AtoZ Markets. Get Free Trading Signals Your capital is at risk. Broker of the month. Test Plus Now Why Plus?

Triangular Arbitrage Procedure. The way toward finishing a triangular arbitrage system with three currencies includes a few stages:. To distinguish an arbitrage scope, traders can utilize the accompanying fundamental cross-currency esteem condition:. Hence, financial market studies are of interest from fundamental and practical point of view for a wide community of scientists, engineers and professionals, bringing together many branches of mathematics, physics, economy and computer sciences to address some challenging issues.

The main research problem addressed in this paper is the following: to what extent bivariate cross-correlations on the Forex market at various levels of fluctuations of exchange rates and timescales ranging from tens of seconds up to weeks may provide important information about a possibility to observe disparities in exchange rates, which may offer potential arbitrage opportunities.

Our goal is to demonstrate prediction power of the so-called q -detrended cross-correlation coefficient stemming from the multifractal formalism when applied to historical time series of exchange rates for a set of currencies. We will investigate in detail sensitivity of various statistical properties of small and large fluctuations using natural scalability of our cross-correlation measure and follow closely the way how information and events related to financial markets may build arbitrage opportunities among a set of currency exchange rates.

We would like to emphasize that our method based on detrended cross-correlation analysis is quite novel and only recently a plethora of applications started to emerge across many fields of nonlinear correlations studies, including meteorological data [ 22 ], electricity spot market [ 23 ], effects of weather on agricultural market [ 24 ], stock markets [ 25 ], cryptocurrency markets [ 26 ], electroencephalography EEG signals [ 27 ], electrocardiography ECG and arterial blood pressure [ 28 ] as well as air pollution [ 29 , 30 ].

Such wide interest across different fields of research in application of detrended cross-correlation analysis to nonlinear time series studies serves as an additional strong motivation for elucidating such analysis in terms of its potential and limitations. The paper is organized as follows. First, we discuss the Forex data used in the present study and our methodology for the financial times series of logarithmic returns for exchange rates.

Next, we discuss a degree of triangular arbitrage opportunity in a form of a convenient coefficient derived from suitable exchange rates. Then, we describe fundamental concepts of our statistical methods, which stem from multifractal formalism. We define q -dependent detrended coefficient capturing cross-correlations of two detrended time series.

The results section comprises global behavior of currency rates, logarithmic return rates statistics with the focus on large fluctuations, discussion of hierarchy among currency exchange rates and the role of abrupt cross-correlation changes and large fluctuations in detecting arbitrage opportunities. Finally, we summarize and draw some general conclusions. In view of the above-mentioned interdisciplinary research by other authors [ 22 , 24 , 25 , 27 , 28 , 29 , 30 ], through the conclusions from our present work, we would like also to support advantages of multifractal detrended cross-correlation method and its wide applications to study any time series with nonlinear correlations, not only in the foreign exchange market but also across other fields of pure and applied sciences.

The data used in the present study have been obtained from the Dukascopy Swiss Banking Group [ 31 ]. For the purpose of this study, we consider the arithmetic average out of the bid and ask price for each exchange rate:. Then, we consider the following time series of logarithmic returns of such exchange rates for each pair of currencies:.

Let us first consider a model situation where we can instantly carry out a sequence of transactions with exchange rates, which all of them are known for a given time instance t. Let us assume that a trader holds initially euros EUR. One possibility to use arbitrage opportunity would be to do a sequence of transformations including those 3 currencies [ 13 , 14 ]:.

That is we could end up with more currency EUR than we had initially. In practice, however, this is difficult on real markets and in fact after the first leg of such multiple transactions, remaining trades would not be possible to complete or the price will be changed by the time they will be completed. Another alternative possible minimal exchange currency path in our case again assuming all the transactions are done in the same time instance or with the frozen exchange rates would be the following.

Again its value compared to one would indicate a theoretical possibility of executing arbitrage opportunity. Let us consider multiple time series of exchange rates recorded simultaneously. Both time series are synchronized in time and have the same number N of data points.

Following the idea of a new cross-correlation coefficient defined in terms of detrended fluctuation analysis DFA and detrended cross-correlation analysis DCCA —[ 32 ], which has been put forward in [ 33 ], we use in the present study a multifractal detrended cross-correlation analysis MFCCA [ 34 ] with q -dependent cross-correlation coefficient.

The cross-correlations of stock markets have been also investigated with a time-delay variant of DCCA method [ 35 ]. Multiscale multifractal detrended cross-correlation analysis MSMF—DXA has been proposed and subsequently employed to study dynamics of interactions in the stock market [ 36 ]. Other methods, including weighted multifractal analysis of financial time series [ 37 ] and multiscale properties of time series based on the segmentation [ 38 ], allow for multifractal and multiscale nonlinear effects investigations.

The approach adopted in the present study has been introduced by [ 41 ]. In what follows, we briefly state main points of this approach. We define a new time series X k of partial sums of the original time series elements x i. In an analogous way, the second time series of interest, Y k , is obtained out of the original time series y i. Depending on the nature of the signal, we may expect in time series trends and seasonal periodicities.

For a given timescale s , we may repeat the partitioning procedure from the other end of the time series, thus obtaining in total 2 M s time intervals, each of which will contain s data points. In general, it could be a polynomial of any finite degree, depending on the nature of the signal. Now we are ready to construct an estimate of the covariance of both newly derived time series, each of which has those polynomial trends removed interval by interval:.

If both time series are the same, we obtain an estimate of the detrended variance in the k -th interval from partitioning with the scale s. The definition of the family of fluctuation functions given by Eq. In such a case, we obtain, as it should be, the q -th order fluctuation function, which follows from the multifractal detrended fluctuation analysis MFDFA for a single time series [ 42 ].

The family of fluctuation functions of order q , defined by Eq. Therefore, one can define a q -dependent detrended cross-correlation q DCCA coefficient using a family of such fluctuation functions [ 41 ]:. The definition of the cross-correlation given by Eq. The parameter q helps to identify the range of detrended fluctuation amplitudes corresponding to the most significant correlations for these two time series [ 41 ].

Hence, by choosing a range of values in q we may filter out correlation coefficient for either small or large fluctuations. Based on the methodology described, we will investigate multiscale properties for cross-correlations among time series corresponding to currency exchange rates for the whole period — as well as for some sub-periods. We also apply standard methods of MATLAB source codes validation and surrogate data checking against artefacts or robustness of nonlinear correlations within our data sets [ 41 ].

Although in our study we focus on different signatures and statistical properties of multivariate time series with respect to triangular arbitrage, one may envisage a broader picture of such analysis, whereby one would like to uncover a specific kind of cross-correlations in these time series which would help us to detect underlying interconnections useful for the system behavior prediction in future.

In this subsection, we will present a general picture for the financial time series dynamics with an emphasis on events which have an impact on the Forex market. Color online Currency indexes as defined by Eq.

For a reference, some important global political and economic events have been indicated over the timescale. The currency index given by Eq. With such a single averaged characteristics, one may have a general overview of a global temporal behavior and performance of any currency in the Forex market.

In Fig. For this plot, we take logarithmic returns arising from average bid and ask exchange rates. In the figure, we have indicated some political or economic events on the timescale with dotted vertical line which in principle could have impact on the Forex market performance during that period of time.

These labels may serve as an intuitive explanation of features observed on the curves related to the currency indexes. In general, one observes significant variations of considered currency indexes over the period of 8 years. In the following, we will explore in some more details statistical properties of the Forex market data in the vicinity of these events. We will discuss to what extent our proposed statistical analysis corroborates these features, when looking from the hindsight with the help of historical data from the Forex market.

Another interesting feature of the Forex market is related to the cumulative distribution of absolute logarithmic returns:. Each solid line of different color demonstrates the tail behavior for the corresponding currency. The inset shows these distributions when a short period of an extreme volatility in currency exchange rates has been removed.

This removed period a half an hour corresponds to the wake of the SNB intervention on January 15, We note that all currency exchange rates with the Swiss franc CHF as base currency yield higher probability of larger absolute logarithmic returns than other exchange rates.

These outliers could be attributed to two instances of the SNB interventions in and In order to demonstrate the origin of these deviations, we remove from our data sets a period of a half an hour in the morning on January 15, , when a significant volatility of currencies exchange rates has been observed in the wake of the SNB intervention [ 9 ]. The resultant tails of the probability distributions are shown in the inset of Fig. Hence without this single, short-termed event on the market, the tails of the distributions approximately follow the inverse-cubic behavior.

We already know that the outliers of the cumulative distributions document an increased level of larger fluctuations in absolute log-returns. This means also a higher chance to encounter fluctuations yielding larger returns.

The question arises to what extent these fluctuations are cross-correlated among exchange rates. Such cross-correlations at least between two exchange rate time series would offer a potential opportunity of triangular arbitrage. We verify indeed that the scaling according to Eq.

Additionally, for the reference both insets in Fig. This average is defined as follows [ 53 ]:. It is worth noting that in this shown example, the scaling of the fluctuation functions for the case of the triangular relation among two exchange rates the top panel in Fig. Two examples of the cross-correlation between two series of returns for exchange rates are shown in Fig.

From the data shown in Fig. This seems to be expected as in the former case there is a common base currency JPY. In such a way, a pair of returns is intrinsically correlated by JPY currency performance due to the triangular constraint in the exchange rates. This is an example of cross-correlations among 3 currencies.

In this case, the cross-correlations are in the triangular relation the top panel of Fig. In the case shown in the bottom panel of Fig. On the bottom-left and bottom-right panels, the results are unsorted making an easier task to identify particularly high cross-correlations shown with labels for each of the both cases, the triangular and non-triangular relationship.

The magnitude of this cross-correlation measure is weakly dependent on the timescale and only slightly grows with time. Its growth is more pronounced for larger fluctuations cf. Let us investigate in some more detail the cross-correlations between relatively small and large fluctuations of two exchange rate return series.

Note that the cross-correlation pairs are grouped into two classes. One class of exchange rate pairs in black, left top and bottom panels which pertain to the triangular relation and the second class, where cross-correlated pairs are outside the triangular relation in red, right top and bottom panels. This gives an idea about the range of obtained values of cross-correlation coefficient distributions for the currency pairs which are in or out the triangular relation.

The black dotted horizontal line on the top-left panel shows the average cross-correlation of different pairs pertaining to the triangular relation. The value of that overall average is about 0. It indicates a possibility of observing stronger correlations in exchange rates among four currencies in comparison with what we would expect on average in the case of exchange rates linked with the triangular relations.

This somewhat unexpected result could be ascribed to mechanisms coupling economies of these two countries. Hence, from our study it follows that indeed some cross-correlations of the pairs, which are not linked by a common currency and are traded on the Forex market, may reach that overall average cross-correlation of exchange rates with a common base. This seems to be a surprising conclusion, since typically we would expect stronger correlations between explicitly correlated two series by means of a common, base currency rather than in a case where there is no such common base.

However, we have to appreciate the fact that cross-correlations between any pair of exchange rates will have some impact on the cross-correlations of other pairs through mutual connections arising from different combinations of currencies being exchanged. Nonetheless, the small fluctuations in logarithmic returns would be difficult to use in viable trading strategies, mainly due to finite spreads in bid and ask rates.

From the practical point of view, correlations of large fluctuations seem to be more promising in finding and exploiting arbitrage opportunities. The order of currency pairs in the bottom panels is kept the same as in the top panels. For the case of the larger fluctuations, the level of overall average of cross-correlations is marked again with horizontal black dotted line at a value of 0.

The cross-correlations of the large fluctuations are therefore approximately two times smaller than in the case of small correlations. In the case of cross-correlations outside the triangular relations, the strong cross-correlations arise when we take AUD as base currency on the one side and NZD on the other. The time evolution of averaged cross-correlations over currency pairs with the common base for large fluctuations will be discussed later cf. As we have already seen, studying quantitative levels of cross-correlations may uncover some less obvious connections among currencies than just the explicit link through a common base currency.

Thus, the distance for agglomerative hierarchical trees takes the following form:. It is worth noting that to the best of our knowledge, it is the first ever such use of the cross-correlation q -coefficient as a way to induce measure for creating a hierarchy tree a dendrogram.

As a result of adopting the distance given by Eq. An interesting observation follows that Australlian AUD and New Zealand NZD dollars are strongly correlated—they appear together in the same clusters of exchange rates for both the small and large fluctuations. This indicates a possibility of building strong cross-correlations between exchange rate pairs which do not have the same common base. Such findings are important when designing the trading strategies, both for optimizing portfolio and for its hedging.

We would like to stress the fact that our method is not limited only to time series from the Forex and it may well be applied to the signals in a form of time series arising in other fields of research and applications. Color online Dendrograms corresponding to Fig. We have looked already into the cross-correlations within fluctuation magnitude domain. Let us now investigate the cross-correlations in the time domain. The results are shown in Fig. We also show the overall average for the currency exchange rate pairs complying to the triangular relation the black dotted line and for currency exchange rate pairs which are not bounded by the triangular relation red dotted line.

The most striking feature when comparing the small and large fluctuation cross-correlations over different timescales is that in the former case little is happening over different timescales considered. The plots indicate nearly static cross-correlations, almost independent on the timescale for the small fluctuations. Specifically, the overall average denoted by the black dotted horizontal line grows from a value which is less than 0.

The growth of the overall average of cross-correlation is even more convincing for the class of pairs of currency exchange rates which are not in a strict triangular relation. What is more, for the shortest timescale shown here, the difference between cross-correlations for pairs that are in the triangular relations and those that are not, is the biggest. The cross-correlation for currency exchange pairs outside the triangular relation in the case of large fluctuations in logarithmic return rates grows in time, which indicates propagation of correlations in time.

This explains why averaged cross-correlations for such currency pairs may be unexpectedly high cf. This gives us some idea about the information propagation time through the Forex market, which is the time needed to reflect the maximum average cross-correlation between any pair of exchange currency rates.

As we have already mentioned above, in the Forex market all currency rates are connected through mutual exchange rate mechanism. However, in some cases the inherently stronger correlations e. This time lag could be regarded as an estimate for the time duration of window of opportunity to execute an arbitrage opportunity. The result is consistent for a range of timescales s taken in our approach. A similar conclusion is valid when considering GBP or JPY taken as the base currency—corresponding curves have a maximum in This could mean that the sudden overnight increase in the rates by the Bank of Canada in did not have longer lasting effect and was only causing very short term effect.

In order to identify promising arbitrage opportunities e. In view of the above findings where we have already identified an important role of the large fluctuations, a question arises to what extent even briefly occurring in time such extreme events fluctuations in currency exchange returns may influence the detrended cross-correlations. It is interesting to see how these extreme events manifest themselves as far as cross-correlations are concerned.

These exchange rates exhibit substantial volatility during considered years. The dashed line corresponds to the cross-correlation results with rejected periods of time with large volatility and existence of triangular arbitrage opportunities. The periods of extreme variation of exchange rates are shown in the corresponding insets of Fig. The insets show that in fact the exchange rates compared red and black curves were changing so rapidly that they could not follow each other.

In such a way, the possible arbitrage opportunities have arisen. Finally, let us investigate closely these brief in time periods of arbitrage opportunities we have identified by our data analysis. Color online Deviations from the triangular relations. In , existed a big arbitrage opportunity CHF , moderate arbitrage opportunity GBP in and no such opportunity in In this case, we use ask and bid prices for exchange rates instead of averaged ones in order to show this in more details.

All events indicated by values greater than 0 in fact could potentially offer triangular arbitrage opportunities. The top panel shows an example of potentially significant arbitrage opportunity which is related to the SNB intervention in and fluctuations in the CHF exchange rates. The middle panel of Fig.

Finally, the bottom panel illustrates rather weak chance of exploiting triangular arbitrage opportunity—there is only one very brief in time instance when in theory this might be possible. The arbitrage opportunities are very closely related to large fluctuations which tend to be more pronounced in the longer timescales s.

This is the case for exchange rates related to CHF and GBP, and this is precisely what opens windows of opportunities for the triangular arbitrage. We have investigated currency exchange rates cross-correlations within the basket of 8 major currencies. The tails of the cumulative distributions of the high-frequency intra-day quotes exhibit non-Gaussian distribution of the rare events by means of the so-called fat tails large fluctuations.

This clearly documents that large fluctuations in the logarithmic rate returns occur more frequently than one may expect from the Gaussian distribution. We have found that on average the cross-correlations of exchange rates for currencies in the triangular relationship are stronger than cross-correlations between exchange rates for currencies outside the triangular relationship.

Such dendrograms may have important applications related to hedging, risk optimization, and diversification of the currency portfolio in the Forex market. Such abrupt changes of cross-correlations combined with the presence of relatively large fluctuations may signal potential triangular arbitrage opportunities. Finally, our conjecture is that during significant events e.

Such events and the resultant opportunities indeed have been identified in the historical trading data for the period —

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Bruno solnik international investments pdf free | Although in our study forex correlation triangle focus on disinvestment in public sector pptx signatures and statistical properties of multivariate time series with respect to triangular arbitrage, one may envisage a broader picture of such analysis, whereby one would like to uncover a specific kind of cross-correlations in these time series which would help us to detect underlying interconnections useful for the system behavior prediction in future. ST] S17 Fig. However, hierarchical organization of ties expressed in terms of dendrograms, with a novel application of the multiscale cross-correlation coefficient, is more pronounced at large fluctuations. S13 Fig. Third, the Arbitrager Model does not achieve its goal by directly modelling cross-currency correlations. Major companies, importers and exporters, governments, investors, and tourists, all needed a method to simultaneously transact business in euros while allowing for money and profits to repatriate back to their home currencies. |

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The order of currency pairs in the bottom panels is kept the same as in the top panels. For the case of the larger fluctuations, the level of overall average of cross-correlations is marked again with horizontal black dotted line at a value of 0. The cross-correlations of the large fluctuations are therefore approximately two times smaller than in the case of small correlations. In the case of cross-correlations outside the triangular relations, the strong cross-correlations arise when we take AUD as base currency on the one side and NZD on the other.

The time evolution of averaged cross-correlations over currency pairs with the common base for large fluctuations will be discussed later cf. As we have already seen, studying quantitative levels of cross-correlations may uncover some less obvious connections among currencies than just the explicit link through a common base currency. Thus, the distance for agglomerative hierarchical trees takes the following form:.

It is worth noting that to the best of our knowledge, it is the first ever such use of the cross-correlation q -coefficient as a way to induce measure for creating a hierarchy tree a dendrogram. As a result of adopting the distance given by Eq. An interesting observation follows that Australlian AUD and New Zealand NZD dollars are strongly correlated—they appear together in the same clusters of exchange rates for both the small and large fluctuations.

This indicates a possibility of building strong cross-correlations between exchange rate pairs which do not have the same common base. Such findings are important when designing the trading strategies, both for optimizing portfolio and for its hedging. We would like to stress the fact that our method is not limited only to time series from the Forex and it may well be applied to the signals in a form of time series arising in other fields of research and applications.

Color online Dendrograms corresponding to Fig. We have looked already into the cross-correlations within fluctuation magnitude domain. Let us now investigate the cross-correlations in the time domain. The results are shown in Fig. We also show the overall average for the currency exchange rate pairs complying to the triangular relation the black dotted line and for currency exchange rate pairs which are not bounded by the triangular relation red dotted line.

The most striking feature when comparing the small and large fluctuation cross-correlations over different timescales is that in the former case little is happening over different timescales considered. The plots indicate nearly static cross-correlations, almost independent on the timescale for the small fluctuations. Specifically, the overall average denoted by the black dotted horizontal line grows from a value which is less than 0.

The growth of the overall average of cross-correlation is even more convincing for the class of pairs of currency exchange rates which are not in a strict triangular relation. What is more, for the shortest timescale shown here, the difference between cross-correlations for pairs that are in the triangular relations and those that are not, is the biggest.

The cross-correlation for currency exchange pairs outside the triangular relation in the case of large fluctuations in logarithmic return rates grows in time, which indicates propagation of correlations in time. This explains why averaged cross-correlations for such currency pairs may be unexpectedly high cf. This gives us some idea about the information propagation time through the Forex market, which is the time needed to reflect the maximum average cross-correlation between any pair of exchange currency rates.

As we have already mentioned above, in the Forex market all currency rates are connected through mutual exchange rate mechanism. However, in some cases the inherently stronger correlations e. This time lag could be regarded as an estimate for the time duration of window of opportunity to execute an arbitrage opportunity. The result is consistent for a range of timescales s taken in our approach. A similar conclusion is valid when considering GBP or JPY taken as the base currency—corresponding curves have a maximum in This could mean that the sudden overnight increase in the rates by the Bank of Canada in did not have longer lasting effect and was only causing very short term effect.

In order to identify promising arbitrage opportunities e. In view of the above findings where we have already identified an important role of the large fluctuations, a question arises to what extent even briefly occurring in time such extreme events fluctuations in currency exchange returns may influence the detrended cross-correlations.

It is interesting to see how these extreme events manifest themselves as far as cross-correlations are concerned. These exchange rates exhibit substantial volatility during considered years. The dashed line corresponds to the cross-correlation results with rejected periods of time with large volatility and existence of triangular arbitrage opportunities. The periods of extreme variation of exchange rates are shown in the corresponding insets of Fig.

The insets show that in fact the exchange rates compared red and black curves were changing so rapidly that they could not follow each other. In such a way, the possible arbitrage opportunities have arisen. Finally, let us investigate closely these brief in time periods of arbitrage opportunities we have identified by our data analysis. Color online Deviations from the triangular relations. In , existed a big arbitrage opportunity CHF , moderate arbitrage opportunity GBP in and no such opportunity in In this case, we use ask and bid prices for exchange rates instead of averaged ones in order to show this in more details.

All events indicated by values greater than 0 in fact could potentially offer triangular arbitrage opportunities. The top panel shows an example of potentially significant arbitrage opportunity which is related to the SNB intervention in and fluctuations in the CHF exchange rates. The middle panel of Fig. Finally, the bottom panel illustrates rather weak chance of exploiting triangular arbitrage opportunity—there is only one very brief in time instance when in theory this might be possible.

The arbitrage opportunities are very closely related to large fluctuations which tend to be more pronounced in the longer timescales s. This is the case for exchange rates related to CHF and GBP, and this is precisely what opens windows of opportunities for the triangular arbitrage. We have investigated currency exchange rates cross-correlations within the basket of 8 major currencies.

The tails of the cumulative distributions of the high-frequency intra-day quotes exhibit non-Gaussian distribution of the rare events by means of the so-called fat tails large fluctuations. This clearly documents that large fluctuations in the logarithmic rate returns occur more frequently than one may expect from the Gaussian distribution. We have found that on average the cross-correlations of exchange rates for currencies in the triangular relationship are stronger than cross-correlations between exchange rates for currencies outside the triangular relationship.

Such dendrograms may have important applications related to hedging, risk optimization, and diversification of the currency portfolio in the Forex market. Such abrupt changes of cross-correlations combined with the presence of relatively large fluctuations may signal potential triangular arbitrage opportunities.

Finally, our conjecture is that during significant events e. Such events and the resultant opportunities indeed have been identified in the historical trading data for the period — The evidence we have shown clearly indicates that the multifractal cross-correlation methodology should contribute significantly to predictive modeling of temporal and multiscale patterns in time series analysis. We believe that our present study, where we consider currencies interaction through their mutual exchange rates and the dynamics of the rates adjustment to a new conditions due to a sudden event, may encourage future research in studying the information propagation through complex networks of interacting entities.

This in turn may have some consequences for design of new smart learning methods for neural networks and a general computational intelligence in predicting a future behavior of complex systems. For example, since we have demonstrated feasibility of financial time series analysis against favorable patterns, we may expect future advancement in computer algorithms for financial engineering when trading tick-by-tick data are available in real time.

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Mantegna, R. B 11 , — E 95 , Download references. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. The data used in the present study has been obtained from the Dukascopy Swiss Banking Group [ 31 ]. Thus, we have in the data set all 28 exchange rates bid and ask prices among the set of 8 currencies. Hence, the foreign exchange rates are the following:.

The indicative and executable prices differ typically by a few basis points [ 13 , 14 ]. The ask price is greater than or equal to the bid price. The spread is dependent on the liquidity a number and volume of transactions as well as on some other factors. We filter out such raw data time series by removing periods when for any given pair there was no quote available or no trading e.

Typically, we thus have approximately 2. Validation methods included a simple model parameter variation as well as surrogate data methods. We can apply our procedure to randomly shuffled original data. We can also create Fourier surrogate time series. In the latter validation method, the Fourier transform of the original time series is computed and then the inverse Fourier transform is applied to the retained amplitudes, but randomly mixed phases [ 1 , 41 ].

Reprints and Permissions. At any time t we expect the following equality to hold 1 that is, the costs of a direct and indirect purchase of the same amount of a given currency must be the same. Clearly, Eq 1 can be generalized to any currency triplet. However, several datasets [ 50 , 51 , 53 , 55 ] reveal narrow time windows in which Eq 1 does not hold.

In this scenario, traders might try to exploit one of the following misprices 2a 2b by implementing a triangular arbitrage strategy. Assuming that the arbitrager completes each transaction at the best quotes i. Following [ 37 , 50 , 51 ], the presence of triangular arbitrage opportunities is detected whenever one of the following processes 4a 4b exceeds the unit.

EBS is an important inter-dealer electronic platform for FX spot trading [ 46 ]. The EBS dataset provides a coverage of the trading activity from The shortest time window between consecutive records is millisecond ms. Events occurring within ms are aggregated and recorded at the nearest available timestamp.

The tick size has changed two times within the considered four years window, see [ 56 ] and S1 Table in S1 File for further details. The EBS dataset, in virtue of its features, is a reliable source of granular market data. First, EBS directly collects data from its own trading platform. This prevents the common issues associated to the presence of third parties during the recording process, such as interpolations of missing data and input errors e. Second, the EBS dataset offers a continuous record of LOB events across a wide spectrum of currencies, thus becoming a natural choice for cross-sectional studies e.

Third, in spite of the increasing competition, the EBS platform has remained a key channel for accessing FX markets for more than two decades by connecting traders across more than 50 countries [ 57 , 58 ]. The enduring relevance of this platform has been guaranteed by the fairness and the competitiveness of the quoted prices. To meet the goals of this study, a model Arbitrager Model henceforth of three co-existing inter-dealer FX markets is introduced. The scope of this framework is to mimic the interactions between different trading strategies across multiple FX markets and capture the mechanisms through which these interactions shape the documented cross-correlation among FX rate fluctuations [ 47 , 48 ].

In the Arbitrager Model, each market hosts a fixed number of agents who interact by exchanging a given FX rate. Trading is organized in simplified LOBs where prices move in a continuous grid. Agents provide liquidity to the market by adjusting limit orders through which they quote a bid and an ask price, thus acting as market makers.

To set these prices, market makers adopt simple trend-based strategies. Furthermore, market makers cannot interact across markets, that is, they can only trade in the market they have been assigned to. Finally, echoing [ 37 ], the ecology hosts a special agent i. The ecology comprises three independent FX markets represented by the red, yellow and green areas. Trading is organized in continuous price grid LOBs as in [ 42 ], see S3.

Market makers black agents maintain bid and ask quotes by adopting trend-based strategies. Transactions occur when the best bid matches or exceeds the best ask. Market makers engaging in a trade close the deal at the mid point between the two matching prices i. Finally, an arbitrager blue agent exclusively submits market orders across the three markets black arrows to exploit triangular arbitrage opportunities emerging now and then, see S5 Fig.

Transactions occur when the i -th market maker is willing to buy at a price that matches or exceeds the ask price of the j -th market maker i. The arbitrager is a liquidity taker i. As soon as one of these processes exceeds the unit, the arbitrager submits market orders to exploit the current opportunity predatory market orders henceforth. Contrary to limit orders, market orders trigger an immediate transaction between the arbitrager and the market maker providing the best quote on the opposite side of the LOB.

This implies that transactions involving the arbitrager are always settled at the bid or ask quote offered by the matched market maker, which are by the definition the current best bid or ask quote of the LOB. The Arbitrager Model builds on various existing studies. The structure of each market mimics, with few exceptions, the one introduced in the Dealer Model [ 42 ], where a number of autonomous market makers interact in a continuous price-grid LOB by managing limit orders.

In the Arbitrager Model, the strategic behavior of market makers is driven by a simple process, see Eq 5 , that is reminiscent of those proposed in the Dealer Model [ 42 ] and, more recently, in the HFT Model [ 59 ]. Finally, the idea of an arbitrager acting as a connection between otherwise independent markets was introduced in the Aiba and Hatano Model [ 37 ].

In particular, the authors advanced the intriguing comparison between an ecology comprising multiple markets, such as the Arbitrager Model, and a spring-mass system in which the dynamics of three random walkers i. First, it is composed by several actors i. Second, the decision making processes, the available trading strategies and the rules governing the interactions among agents retain a remarkable simplicity. This reduces the computational effort required to build and simulate the dynamics of the model and facilitates the understanding and interpretation of its outcomes.

Third, the Arbitrager Model does not achieve its goal by directly modelling cross-currency correlations. Instead, this statistical regularity of FX markets is conceived as a macroscopic phenomenon which emerges from the iteration of simple, antagonistic interactions occurring on a more microscopic level. However, the trading data-based cross-correlation functions presented in this study stabilize on much shorter time-scales. This variability might be related to the different tick sizes adopted by EBS during the four years covered in this empirical analysis, see [ 56 ] and S1 Table in S1 File.

Details on the initialization of the model and the conversion between simulation time i. To support this hypothesis, an extended version of the Arbitrager Model which includes additional features of real FX markets is presented and examined in S3. Addressing this open question is one of the main objectives of the present study. When the arbitrager is not included in the system, two markets have the same probability of being in the same and opposite state, see first column of Fig 6.

This occurs because price trends are driven by transactions triggered by endogenous decisions, that is, events occurring in different markets remain completely unrelated. As a consequence, market states flip independently and at the same rate. In these settings, the dynamics of the mid price of FX rate pairs do not present any significant correlation, see third column of Fig 6.

The red solid line in the histograms marks the value of 0. Simulations are performed under the same settings of the experiment presented in Fig 5 b , bottom panel. Furthermore, the active presence of this special agent intertwines the dynamics of different FX rates, creating cross-correlations functions that resemble those emerging in real trading data.

Statistics are collected from simulations of the Arbitrager Model with active violet and inactive grey arbitrager. The presence of an active arbitrager increases the average lifetimes a and appearance probabilities b of certain configurations and reduces the same statistics for others. Statistics in a are expressed in real time i.

The inclusion of the arbitrager has a major impact on the overall behavior of the model. Imbalances in the probability of observing two markets in the same or opposite state emerge in each FX rate pair. The sign and stabilization levels of these functions are consistent with the sign and size of the probabilities imbalances, suggesting that these two results are two faces of the same coin.

The statistical properties of the eight ecology configurations shall be examined in order to understand how the findings presented in Fig 6 unfold. The presence of the arbitrager introduces a degree of heterogeneity in both the expected lifetimes and appearance probabilities of ecology configurations, see Fig 7. This reveals three interesting facts. First, the average lifetime of every ecology configuration is smaller than its counterpart in an arbitrager-free system.

To explain this feature, recall that predatory market orders trigger three simultaneous transactions i. When this occurs, the arbitrager weakens the trend-following behaviors of market makers in at least one of the three markets, thus increasing the likelihood of a transition to another ecology configuration. As triangular arbitrage opportunities of both types appear, with different incidences, during any ecology configuration, see S15 Fig , the expected lifetimes of these configurations are, to different extents, shorter than in an arbitrager-free system.

Second, certain ecology configurations are expected to last more than others i. As reactions to triangular arbitrage opportunities increase the likelihood of flipping a market state, the average lifetime of a given configuration relate to the time required for the first triangular arbitrage opportunity to emerge. This difference can be intuitively explained by looking at the combination of market states. In this case, both the FX rate and the implied FX cross rate move in the same direction, extending the time required by these prices to create a gap that can be exploited by the arbitrager.

The third and final interesting fact emerged in Fig 7 is that certain configurations are more likely to appear than others. To understand this aspect, consider the significant differences between the probabilities of transitioning from a configuration to another, see S5 Table in S1 File.

For each configuration, the absolute value of this statistics is sampled at the emergence of any triangular arbitrage opportunity. This happens when the arbitrager responds to a type 2 triangular arbitrage opportunity i.

The conditional transition probability matrix displayed in S5 Table in S1 File reveals the presence of another configuration triplet i. S16 Fig shows this mechanism in action by displaying the sequence of ecology configurations during a segment of the model simulation. It is easy to observe how the system tends to move across configurations belonging to the same looping triplet for long, uninterrupted time windows.

To sum up, the Arbitrager Model elucidates how the interplay between different trading strategies entangles the dynamics of different FX rates, leading to the characteristic shape of the cross-correlation functions observed in real trading data. The Arbitrager Model restricts its focus to the interactions between two types of strategies, namely triangular arbitrage and trend-following.

Despite the simplicity of this framework, the interplay between these two strategies alone satisfactorily reproduces the cross-correlation functions observed in real trading data. In particular, trend-following strategies preserve the current combination of market states for some time while reactions to triangular arbitrage opportunities influence the behavior of trend-following market makers by altering the price trend signals used in their dealing strategies.

The interactions between these two strategies constantly push the system towards certain configurations and away from others through multiple mechanisms. This can be easily seen in Fig 7 as two distinct statistics, the average expected lifetimes and the appearance probability, put the eight configurations in the same order. This force shapes the features of the statistical relationships between currency pairs.

FX rates traded in markets that share the same state in configurations with higher lower appearance probabilities and longer shorter expected lifetimes are more likely to fluctuate in the same opposite direction. These two markets have the same states in the four configurations with higher probabilities i. In these settings, the mid price dynamics of two FX rates become permanently entangled, leading to the cross-correlation functions displayed in Figs 5 b and 6.

The purpose of this study was to obtain further insights into the microscopic origins of the widely documented cross-correlations among currencies. To take up this challenge, a new ABM, the Arbitrager Model, has been proposed as a simple tool to describe the interplay between trend-following and triangular arbitrage strategies across three FX markets. In these settings, the model reproduced the characteristic shape of the cross-correlation function between fluctuations of FX rate pairs under the assumption that triangular arbitrage is the only mechanism through which the different FX rates become synchronized.

This suggests that triangular arbitrage plays a primary role in the entanglement of the dynamics of currency pairs in real FX markets. In particular, triangular arbitrage influences the trend-following behaviors of liquidity providers, driving the system towards certain combinations of price trend signs and away from others. This affects the probabilities of observing two FX rates drifting in the same or opposite direction, making one of the two scenarios more likely than the other.

Ultimately, this entangles the dynamics of these prices, creating the significant cross-currency correlations that are reproduced in our model and observed in real trading data. The present study, finding a common ground between previous microscopic ABMs of the FX market and triangular arbitrage [ 37 , 42 , 59 ], sets a new benchmark for further investigations on the relationships between agent interactions and market interdependencies.

In particular, it is the first ABM to provide a complete picture on the microscopic origins of cross-currency correlations. The outcomes of this work open different research paths and raise new challenges that shall be considered in future studies:. The model introduced in the present study could be subject of meaningful extensions and enhancements aimed to turn this framework into a valuable tool that could be used by exchanges, regulators and market designers.

In particular, its simple settings would allow these entities to make predictions on how regulations or design changes could affect the relationships between FX rates and the properties e. Furthermore, its applicability might attract the attention of other actors operating in the FX market, such as central banks. The ultimate objective of this work and its potential future extensions shall remain the provision of useful means to enhance the understanding of financial market dynamics, assisting the aforementioned entities in conceiving safer and more efficient trading environments.

Browse Subject Areas? Click through the PLOS taxonomy to find articles in your field. Abstract Foreign exchange rates movements exhibit significant cross-correlations even on very short time-scales. Download: PPT. Fig 3. Profitable misprices and associated triangular arbitrage strategies.

Fig 5. Trading data vs. Fig 6. Statistical relationships between different FX markets. Fig 7. Expected lifetime and appearance probability of the eight ecology configurations. The outcomes of this work open different research paths and raise new challenges that shall be considered in future studies: The Arbitrager Model could be further generalized by including a larger number of currencies, allowing traders to monitor different currency triangles.

A potential extension of this model should consider the active presence of special agents operating in FX markets. Another interesting path leads to market design problems. Calling for further investigations, an extended version of the present model should examine how different tick sizes affect the correlations between FX rates.

Future works shall also consider established e. Such investigations might reveal additional statistical relationships whose mechanistic origins can be studied in an augmented version of the Arbitrager Model. Supporting information. S1 Fig. S2 Fig. S3 Fig. S4 Fig. S5 Fig. S6 Fig. S7 Fig. S8 Fig. S9 Fig. S10 Fig. S11 Fig. S12 Fig. S13 Fig. S14 Fig. S15 Fig. S16 Fig. S17 Fig. S1 File.

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