Scales in Taiwan Stock market Prices

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VIX(??)??

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Random Walk Hypothesis

• The random walk hypothesis is a financial theory
  stating that stock market prices evolve according
  to a random walk and thus the prices of the stock
  market cannot be predicted.

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Non-Random Walk Hypothesis

• There are other economists, professors, and
  investors who believe that the market is
  predictable to some degree. These people believe
  that prices may move in trends and that the study
  of past prices can be used to forecast future
  price direction. There have been some economic
  studies that support this view, and a book has
  been written by two professors of economics that
  tries to prove the random walk hypothesis wrong.

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• Mandelbrot ???????????,?????,???????????

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• Walking" Along a Coastline

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Fractal dimensions of time sequences
2009
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Fractal dimensions
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• Weierstrass function

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Dow Jones Industrial Average stock index ( 1900
2007 )
M 11 (green), 12 (blue), and 13 (red).
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Fractal Dimension
D 1.321, 1.486, 1.449
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• In conclusion, we have presented that the DJIA
  index is not a random walk for most of the time
  (recall that a random walk has a fractal
  dimension 1.5).
• That is, by calculating the fractal dimension of
  a stock index, we have shown clearly that the
  assumption of efficient market is false in
  general.

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741x 271200811
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Fractal Dimension
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2009
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?????????The Mathematics of Options Trading
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Scales in Taiwan stock index data

• F.T. Lee (2004)
• St. Johns St. Marys Institute of Technology

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• In this talk, we will analyze the time evolution
  of the Taiwan stock index over the 3-year period
  (2001-2003).
• We observe an interesting power-law scaling
  behavior.
• We show that the empirical distribution function
  (pdf) of index changes have weak leptokurtic
  wings.
• Our results are different from the results of the
  analysis of the SP 500 index by Mantegna and
  Stanley. Nature, 376, 46-49(1995)

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leptokurtic distribution ????? 
probability density function
Gaussian distribution
price difference (return)
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probability density function
price difference (return)
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• In summary
• We have seen a change in the distribution of
  price returns that evolves according to the
  relative timescales. 
•  
• There is a gradual transition from a leptokurtic
  to a Gaussian distribution.  
• What statistics of price fluctuations does one
  assume over various timescales?
• No model exists for the stochastic process
  describing the time evolution of price change
  that is accepted by all researchers.
• The random walk is by far the most easiest
  stochastic modeling of stock prices.

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• We consider a study of the statistical properties
  of time evolution of Taiwan stock indexes (TAIEX)
  over the 3-year period January 2001 to December
  2003.
• 741x271200811
• We label the times series of the index as Y(t)
  for every minute.
• We calculate the probability density function
  (pdf) P(Z) of index changes (return)

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Non-overlapping
t t?t
Non-overlapping
1 3 5 7
9
?t2
Overlapping
t t?t
Overlapping
1 2 3 4 5 6 7 8
9
?t2
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The pdfs is ?almost symmetric, and
spread as ?t increases as in any
random process ?highly leptokurtic, and
?characterized by a non-Gaussian profile
for small index changes.
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Semi-logarithmic plot shows the leptokurtic
nature.
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Power law scaling behavior

• We study the probability of return to the
  origin
• as function of

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Non-overlapping
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Non-overlapping
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Overlapping
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Overlapping
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TAIEX SP500
0.58745 0.712
1.7022 1.4044
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Non-overlapping


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Overlapping


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Lévy distribution
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Lévy distribution

• small a
  SP500
• large a
  TAIEX

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Standard deviation s(?t) of P(Z)
TAIEX Theoretical value SP500 MIB
0.534 1/2 0.53 0.57

1. This value show the presence of a weak long-range
   correlation.
2. The strength of the long-range correlation is
   market-dependent and seems to be larger for less
   efficient markets (The market information is
   passed on to all investors instantaneously, so no
   one has an advantage over others when it comes to
   decisions on buying/selling. ).

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By extrapolating the and , we
estimate the breakdown of non-Gaussian scaling
occurs at mins. ( SP 500 index occurs at
mins.)
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summary

• We find a non-Gaussian pdf in the probability of
  price change from TAIEX. ( mins)
• We observe a scaling regime spanning a time
  interval of three orders of magnitude.
• The empirical pdf of TAIEX have weaker
  leptokurtic wings than SP 500 index.
• This nature seems to be show that Taiwan stock
  market is a less efficient market.