股價衝擊隨機過程：應用阻尼簡諧振盪

Abstract

效率市場假說是經濟學很重要的假說。在效率市場的架構下，關於股票的所有資訊都已經反映在股價上，股價只會因為未來的、隨機的資訊而改變。此假說為股價的隨機漫步行為定下了基礎，使得財務學家開始需要利用隨機過程去描述股價的變化。 其中最簡單又經典的模型稱作幾何布朗運動，他能捕捉到股價的隨機性，也能保證過程不會讓股價低於0元，且其常態假設讓此模型很便於計算。然而此模型也並非完美，許多實證研究證實股價的走勢並不如幾何布朗運動所描述。 舉例而言，股價偶爾會有飆漲或崩跌等情況，而幾何布朗運動是連續的，出現如此大幅度的波動的機率微乎其微。根據歷史資料，飆漲或崩跌的次數早已遠遠超出幾何布朗運動所預測的。 另一個幾何布朗運動無法描述的特色是均值回歸現象。當股價出現太高或太低的價位時，市場通常會將價格修正回均值附近。然而幾何布朗運動的增幅是對稱的常態，在出現過高或過低價位時，股價上漲或下跌的機率還是相同，與實際市場狀況不符。 財務學家發明很多其他的模型去修正幾何布朗運動這些缺點，例如Ornstein-Uhlenbeck 模型使得隨機過程擁有均值回歸的現象，jump-diffusion模型使得股價能有不連續的跳點，affine jump-diffusion使股價既能夠均值回歸，又能夠有不連續的跳點。 近年來行為財務學成為顯學，越來越多人相信市場的無效率性，投資人往往會過度反應、過度交易，使得股價不會均值回歸，而有振盪的行為。本論文將基於阻尼簡諧振盪的架構，建立一個新的隨機過程，承襲以往模型之精神，同時描述此類振盪行為。

In financial economics, the efficient-market hypothesis is well known for stating market behavior. Under this hypothesis, asset prices fully reflect all historical information, which implies that only new relevant information affects market prices. Investors’ reactions to the information is random and in a normally distributed pattern so that the change on the market price is also normally distributed. This is a strong argument for the use of geometric Brownian motion (GBM) on modeling stock prices. However, GBM is not a completely realistic model, in particular it fails to describe some properties of stock prices. One is that GBM is a continuous path through time, but in real life, stock price often show jumps. The other is the mean-reverting property. When stock price is far from its equilibrium due to some shocks, it will have a high chance to be adjusted to its equilibrium nearby, but GBM will still follow the trend even in an unreasonable price level. There have been several models conducted to modify GBM, some examples like Ornstein-Uhlenbeck model for mean-reverting property, jump-diffusion model for discontinuity, and affine jump-diffusion model for both. Recently, more and more economists believes the inefficiency of the market. Investors predictably overreact to new information, creating a large effect on the stock price, making the price oscillate. This kind of oscillation has not been described by those classical models. My thesis is to discuss the dynamic of the oscillation, and introducing a process in the framework of damped harmonic oscillation.