In his project for the autumn, 2000, fractal geometry course, Enoch Wu used driven IFS to compare the Taiwan stock market (TAIEX) and the Shanghai Securities Exchange (SSE). Wu proposed a novel application of IFS:
| If both markets rise, apply T1(x,y) = (x/2,y/2) |
| If TAIEX falls and SSE rises, apply T2(x,y) = (x/2,y/2) + (1/2,0) |
| If TAIEX rises and SSE falls, apply T3(x,y) = (x/2,y/2) + (0,1/2) |
| If both markets fall, apply T4(x,y) = (x/2,y/2) + (1/2,1/2). |
That is, in relation to the driven IFS
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If the markets are tied together, we should see many applications of T1 and T4, thus many points lying along the diagonal from lower left to upper right. To test this, here is a graph for the Dow Jones and NASDAQ, from July 3, 1997 to December 1, 2000, a total of 861 data points.
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Indeed, we see a strong trend along this diagonal. For the TAIEX and the SSE, we see this.
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While we see no such clear trend as with the DJA and NASDAQ, some more subtle patterns are revealed. First, Wu divided the data at May 22, 2000, the inauguration of Taiwan's new president.
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In the longer record, Wu observed the subsquare with address 323 is empty. This is plausible, because this address signals a rise, fall, and rise of the TAIEX, and corresponding fall, rise, and fall of the SSE.
After the inauguration of the new president, Wu observes most points land in squares with address 2 and 4, signaling declines in the TAIEX. The small population of points in addresses 33 and 44 indicate few consecutive falls in the SSE, while the larger number of points in square 22 shows there were many consecutive falls in the TAIEX, simultaneous with rises in the SSE. Although we must be careful with deductions made on such small data sets, the general observation matches known trends: China's market continued to grow, while aiwan's lost over half its value in this time period.
Wu analyzed the individual markets, applying the transformations in this fashion.
| If the market drops more than 2.5% of its value, apply T1 |
| If the market drops less than 2.5% of its value, apply T2 |
| If the market rises less than 2.5% of its value, apply T3 |
| If the market rises more than 2.5% of its value, apply T4 |
That is, in relation to the driven IFS
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With this binning of the data, here are the driven IFS for the TAIEX and the SSE. We see immediately that the SSE fluctuations are more often in the 2.5% or under range. The TAIEX has more off-diagonal fluctuations, and Wu notes in particular the number of points in the 14 subsquare. Because his data were entered from the present to the past, this means a large fall often was followed by a large rise. The near emptiness of the 41 subsquare means large gains were only rarely followed by large losses.
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Finally, Wu noted a weakness of his data binning scheme: it is good only for detecting simultaneous movement of the markets. After Taiwan's former president gave his "Theory of Two States" speech, the ensuing polotical crisis caused both markets to fall over 20%. Yet the driven IFS shows little correlation, presumably because each market reacted also to internal forces not synchronized with those of the other market. This could be explored by comparing movement of one stock today with that of the other one day in the past, then two days in the past, and so on. If a strong pattern appeared, that would be evidence that an important time lag had been found in the interactions between the markets.
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