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 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

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

Fractal Dimension

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) ]

leptokurtic distribution 尖扁型分布 尖扁型分布  probability density function Gaussian distribution price difference (return)

probability density function price difference (return)

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 3-year 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)

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

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.

Power law scaling behavior We study the ”probability of return to the origin” as function of

Non-overlapping

Non-overlapping

Overlapping

Overlapping

TAIEX S&P500 0.58745 0.712 1.7022 1.4044

Non-overlapping ~

Overlapping

Lévy distribution

Lévy distribution small α S&P500 large α TAIEX

Standard deviation σ(Δt) of P(Z) TAIEX Theoretical value S&P500 MIB 0.534 1/2 0.53 0.57 This value show the presence of a weak long-range correlation. 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. ).

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

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 S&P 500 index. This nature seems to be show that Taiwan stock market is a less efficient market.