華爾街的物理學家
混沌碰上華爾街
混沌碰上華爾街
•每天有數以兆計的熱錢在全球的金
融市場中快速流動。無數的投機客
與避險者投身金融市場，期望透過
股票、債券、期貨、選擇權及其他
衍生性金融商品的買賣來賺取龐大
利潤。金融遊戲的規則很簡單：買
低賣高。但怎麼做？或者說，金融
市場可以預測嗎？或者說，真有一
套系統或方法，可以從混沌的金融
市場中找出模式，預測金融市場未
來的變化嗎?
混沌碰上華爾街
• 股票市場會變動是它的本質；它會受到政
治情勢、謠傳、一時的消息所影響。但是
全球金融市場就像大海一樣，在表面浪潮
之下，還隱藏著更深的暗流，隨時會湧上
表面掀起波濤，以無法理解、難以阻擋的
規律循環不已。
華爾街的物理學家
• 一群不些邊幅的美國物理學家，視財經學
者的諄諄教誨於無物，利用混沌理論為自
己賺進可觀的創業資金，他們是怎麼辦到
的﹖這些違反市場規則、創造股市奇蹟的
天才們不是別人，就是混沌派——法默和
派卡德，他們利用這個新興的科學理論，
進軍全球最詭譎多變、變幻莫測的金融市
場，不但建立起獲利模型，還創辦了公司、
打算從中獲取龐大利潤。
混沌碰上華爾街
• 最早的股價物理模型→
布朗運動(愛因斯坦)
混沌碰上華爾街
• 股市大漲大跌的機會比預期的多!!
• 暴漲暴跌就像大地震[地震模型]
• 股市何時崩盤?
崩盤的徵兆
• 洛杉磯加州大學地質系D. Sornette認為:
• 崩盤應該會有前兆, 他發現1929, 1962 1987 的華
爾街崩盤, 1997, 1998香港恆生指數的崩盤都有相
同的徵兆。
• 他認為股市的崩盤是因為投資人間長期累積的[長
程相關力量]所造成的集體行動，而非一般經濟學
者認為的由某些[新聞事件]所造成。
VIX(恐慌)指數
•芝加哥選擇權交易所波動率指數
1987年黑色星期一
• 人類歷史上的兩次大崩盤:跳樓的絕不僅是股價
• 1987：“華爾街歷史上最壞的日子”
1987年10月19日是美股著名的「黑色星期一」，盤中一度
下挫25.3％，最後收盤時重挫22.6％，是史上單日最大的
跌幅。這一天美國股市又一次大崩盤，道瓊斯指數一天之
內便重挫508.32點，裝了特殊程式的計算機不停地在賣，
任何試圖使其穩定下來的努力都失敗了。僅僅一天時間，
美國的股票市場就大幅度縮水，其價值超過五千億美元。
這是一個“黑色星期一”，一個“華爾街歷史上最壞的日
子”。受美國股市崩盤的影響，倫敦、法蘭克福、東京、
悉尼、香港、新加坡等地股市也開始狂跌，“1929年的股
災又來了嗎？”巨大的恐慌在投機者心中蔓延，昔日的情
景再次重現，很多人由百萬富翁淪為赤貧，精神崩潰、自
殺的消息不絕於耳。
• 根據他的理論後來他更成功預測了1999年
的的日本股市大漲，乃至2000年的大跌，
以及2000年美國科技股的崩盤。
• 對他來說[股市模型]與[地震模型]並沒有什
麼不同。
各種股票價格模型
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.
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.
碎形曲線
• Mandelbrot 認為股票價格的上上下下，就像海
岸線，其實就是一種碎形曲線。
碎形曲線
• Walking" Along a Coastline
Fractal dimensions of time sequences
2009
碎形曲線
Fractal dimensions
碎形曲線
• Weierstrass function
n
W (t )   2 i ( 2 D ) cos(2 i t )
i 0
60
f(t)
Weierstrass function D=1.4 N=218
50
0.0
0.2
0.4
0.6
t
0.8
1.0
Dow Jones Industrial Average stock index
( 1900 ~ 2007 )
M = 11 (green), 12 (blue), and 13 (red).
Fractal Dimension
D = 1.321, 1.486, 1.449
• 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.
台灣加權股價指數(2001~2003)
741x 271=200811
6500
6000
Index
5500
5000
4500
4000
3500
3000
0
50000
100000
Time
150000
200000
Fractal Dimension
3.5
TAIEX
3.0
s=0.12787, D=1+s/log2=1.42477
log 
k
2.5
2.0
1.5
Weierstrass function
1.0
s=0.12041, D=1+s/log2=1.4
0.5
0.0
-2
0
2
4
6
8
10
k
12
14
16
18
20
台灣加權股價指數的歧異現象
2009
活用數學交易選擇權
The Mathematics of Options Trading
Scales in Taiwan stock
index data
F.T. Lee (2004)
St. John’s & St. Mary’s Institute of
Technology
• 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 S&P 500 index by Mantegna and Stanley.
[ Nature, 376, 46-49(1995) ]
probability density function
leptokurtic distribution
尖扁型分布
Gaussian distribution
price difference (return)
price difference (return)
probability density function
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.
• We consider a study of the statistical properties of time
evolution of Taiwan stock indexes (TAIEX) over the 3year period January 2001 to December 2003.
741x271=200811
• 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)
Zt (t )  Y (t  t )  Y (t )
t : trading time
Non-overlapping
t
t+Δt
Non-overlapping
1
3
5
7
9
Δt=2
Overlapping
t
t+Δt
Overlapping
1
2
3
4
5
6
7
8
9
Δt=2
2500
t  5 min
2000
PDF
1500
1000
500
0
-1.0
Fig.3
-0.5
0.0
Z
0.5
1.0
0
-1
1 mins
120 mins
-3
10
log P(Z)
-2
-4
-5
-6
-200
-100
0
100
200
Z/
The pdfs is
‧almost symmetric, and spread as Δt increases as in any
random process
‧highly leptokurtic, and
‧characterized by a non-Gaussian profile for small index
changes.
Semi-logarithmic plot shows the leptokurtic nature.
2
Levy
10
log P(Z)
0
t  1 min
Gaussian
-2
-4
-30
-20
-10
0
Z/
10
20
30
Power law scaling behavior
• We study the ”probability of return to the origin”
Pt (Z  0) as function of t.
•
Pt (Z  0)  (t )

 (t )
1/ 
Non-overlapping
0.0
-0.5
-1.0
-1.5
10
log P(Z=0)
slope -0.62138+0.02779
-2.0
t  10 min
-2.5
-0.5
0.0
0.5
1.0
1.5
log t
10
2.0
2.5
3.0
3.5
Non-overlapping
-0.5
10
log P(Z=0)
-1.0
-1.5
-2.0
-2.5
-3.0
0.0
0.5
1.0
1.5
log t
10
2.0
2.5
3.0
Overlapping
0.0
slope -0.587450.00885
-1.0
-1.5
10
log P(Z=0)
-0.5
-2.0
-2.5
-0.5
0.0
0.5
1.0
1.5
log t
10
2.0
2.5
3.0
3.5
Overlapping
-0.5
-1.5
10
log P(Z=0)
-1.0
-2.0
-2.5
0.0
0.5
1.0
1.5
log t
10
2.0
2.5
3.0
TAIEX
S&P500
1/ 
0.58745
0.712

1.7022
1.4044
~
Z
Z
(t )1/ 
~ ~
P (Z ) 
Z
1 / 
(t )
0
-1
1 mins
120 mins
-3
10
log P(Z)
-2
-4
-5
Non-overlapping
-6
0
100
200
Z/
0
-1
-2
~ ~
log10 P (Z )
-3
10
-100
Log P()
-200
-4
-5
-6
-60
-40
-20
0
~

20
40
60
Overlapping
0
-1
Log P(Z)
-2
10
~ ~
log10 P (Z )
-3
-4
-5
-6
-60
-40
-20
0
~
Z
Z
20
40
60
2
Levy
t  1 min
10
log P(Z)
0
Gaussian
-2
-4
-30
-20
-10
0
10
20
30
Z/
Lévy
distribution
PL ( x ) 
(1   ) sin( / 2)
x
1
x
(1 )
Lévy distribution
PL ( x )  x
 (1 )
small α S&P500
large α TAIEX
Standard deviation σ(Δt) of P(Z)
2.5
slope 0.53472+0.00263
10
log (t)
2.0
t  30 min
1.5
TAIEX
Theoretical
value
S&P500
MIB
0.534
1/2
0.53
0.57
1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
log t
10
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. ).
0.0
x
1
PG ( x) 
e
2 
2 2
-1.0
-1.5
P (0)
G
10
log P(Z=0)
-0.5
2
P(0)
PG (0)
-2.0
-2.5
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
log t
10
By extrapolating the P(0) and PG (0) , we estimate
the breakdown of non-Gaussian scaling occurs
7
at 10 mins. ( S&P 500 index occurs at 104 mins.)
summary
•
We find a non-Gaussian pdf in the probability of price
change from TAIEX. ( ≈ 107 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 S&P 500 index.
•
This nature seems to be show that Taiwan stock market
is a less efficient market.