- 學位論文
時距模型於金融市場非規律事件之分析
Irregular Event Duration Models on Financial Data of Regular Frequency
摘要
如何對財務市場的非規律 (irregular) 事件進行分析和預測 (例如極端下跌事件之預測)，在風險管理領域蔚為重要之議題。一般而言，以固定時距所取得之樣本觀察值 (像是日資料或週資料)，及基於固定時距樣本資料所發展之計量方法，不易直接適用於分析發生時間不規律之事件。透過估計時距模型(duration model)，將可能影響事件發生時間之因素，視為影響未來同類型事件發生時間間距 (簡稱為時距，duration) 之解釋變數，便是對財務市場非規律事件分析的一種可行方式。例如Engle and Russell (1998) 即將事件發生時間視為一點過程 (point process)，藉由自我迴歸條件時距模型(autoregressive conditional duration model, 簡稱 ACD model) 對高頻率交易資料中之非規律事件進行分析有別於過去財務文獻多將時距模型應用於高頻資料之估計，本論文將著重於一般常用股價日資料中的極端下跌事件 (簡稱為大跌事件，例如股價跌幅超過股票報酬分配5%分位數的下跌事件)。由於大跌事件之發生頻率，通常不具有規律性，因此本論文所包含的三篇文章 (分別安排在Chapter 2 至Chapter 4)，便透過時距模型，探討影響大跌事件時距之因素，並對大跌事件發生時間進行預測。 條件頻率強度函數 (conditional intensity function) 是時距分析時之核心 (Hautsch,2004)，在涉及分析多個樣本對象時，強度函數又常被稱為涉險函數 (hazard function) 或是條件機率 (conditional probability) (Engle and Russell, 1998; Keifer, 1998)。透過條件頻率強度函數可推算出事件時距之期望值，最基本的Poission process 乃假設事件時距之期望值為一常數。但實際上條件頻率強度函數可能和很多因素有關，因此可依據自我迴歸模型(autoregressive model, AR model) 之精神，假設所有資訊皆包含在事件時距本身之落後期變數中，將條件頻率強度假設為過去事件時距之函數，此設定方式即相當於將事件發生時間視為一種自我激發 過程 (self-exciting process)。另外，亦可根據經濟或財務理論，將其它可能影響事件時距之因素加入條件頻率強度函數之中。本論文其中一篇文章，便假設股價重大下跌事件發生前，部分公開或非公開的次要訊息可能提前反應在較小的跌幅事件上，因此提出一小跌事件密集度變數，並將之視為影響條件頻率強度函數之因素。 在設定股價日資料中大跌事件的條件頻率強度函數時，本論文分別從二個面向進行建構。由於股價大跌事件大多呈現叢聚(clustering) 之現象，本論文的第一篇及第二篇文章，首先假設大跌事件時距受到過去同類型事件時距之影響，探討大跌事件過去時距對時距期望值之影響。本論文的第三篇文章則考量第二個面向，在自我迴歸模型中加入小跌事件密集度作為解釋變數，估計小跌事件密集度對大跌事件時距之影響。 本論文的第一篇文章 ”Are Extreme Drops in Stock Prices Self-Exciting?-Evidence from the Irregular ACD Models on Regular Return Data” 即應用Engle and Russell (1998) 提出之ACD 模型，對一般股價日資料中的大跌事件時距進行估計。實證結果顯示，美股大盤指數和個股股價之大跌事件時距，皆普遍具有自我相關之特性，且ACD 模型估計所得之標準化時距無自我相關現象，表示ACD 可適用於描述大跌事件時距的自我相關特性。此實證發現指出ACD模型應可用於預測大跌事件之發生時間，對風險管理將有相當的實用價值。 因此論文的第二篇文章 ” Calibrating Value-at-Risk: An Application of the ACD Model to Risk Management”，即將ACD 模型應用在風險管理，利用ACD 模型作為校準VaR 的輔助工具。此文之研究動機源於過去許多VaR 模型的回溯測試 (backtesting) 文獻指出，VaR 超限事件 (violation) 經常不符合獨立同態分佈 (iid) 之假設，反而有叢聚之現象，於是本文藉由ACD 模型估計及預測VaR 超限事件之時距 (文中稱之為violation duration)，接著將預測出的信賴區間，定義為可能發生超限事件的高風險期間 (high-risk period)，在ACD 模型所預測出的高風險期間調高VaR。經由回溯測試的模擬發現，常用的VaR 模型 (例如歷史模擬法或平均變異數法)，在資產價格波動較大的時期，估計所得的VaR，其超限率 (violation rate) 通常高於目標水準。此時，根據ACD 模型預測出之高風險期間，調高VaR 估計值便可降低超限率至目標水準。相較於文獻中提到，部分大型商業銀行採用較複雜的VaR 模型，導致其揭示的VaR 偏高，本文提出以ACD 模型作為VaR 的校準工具，可改善大型商業銀行所揭示之VaR過於保守的問題。 論文的第三篇文章 ”Intensity of Minor Price Declines as a Precursor to Price-Drop Events”，則假設在重大消息出現之前，可觀察到一些相關的跡象，例如公司的年報或季報公佈之前，公開的月營收數據便透露了部分之訊息，另外對市場觀察較敏銳的投資人，亦可能察覺一些非公開的訊息，像是存貨的變動或是生產成本的改變等。因此股價大跌事件也許會像大地震那樣，可先觀察到一些較小的前震先發生。因此在單變數的ACD 模型中，加入小跌事件 (minor price declines) 之密集度作為解釋變數，將ACD 模型作一簡單但具經濟意涵的延伸。實證結果顯示，大跌事件前的小跌事件愈頻繁，將使將來大跌事件的時距愈短，即讓大跌事件愈密集地發生，因此小跌事件密集度可視為有助於大跌事件預測之前兆因素。另外，透過估計實證所採用的美股大盤指數樣本也發現，當小跌事件之門檻由股價報酬分配10%分位數放寬為40%或45%分位數時，小跌事件密集度對大跌事件時距之影響將趨於不顯著。 本論文指出ACD 模型在一般常用的財務資料 (例如日資料) 的分析上，仍有很大的應用空間。包含ACD 模型適合用於描述股價的大跌事件時距，且藉由ACD 模型預測VaR 超限事件可能發生的時間，可有效降低超限事件的發生次數，因此ACD 模型在風險管理上具有重要的實用價值。此外，大跌事件之前的次要下跌事件密集度，亦有助於預期大跌事件發生時間。本論文三篇文章之實證發現，對於金融市場投資人或財務風險管理者，皆提供了重要的政策意涵。
並列摘要
Analyzing and modeling processes of price-drop events is one of the most important issues in modern financial risk management practices. Irregularity is an obvious feature of price-drop events. Thus, traditional fixed interval econometrics may not be appropriate for the analysis of irregular data. Following Engle and Russell (1998), the arrival time of an irregular event is treated as a random variable following a point process and the time interval between two events is referred to as the duration. The ACD model introduced by Engle and Russell (1998) is widely applied to highfrequency data. However, there are only a limited number of studies applying ACD models to financial duration measures related to the inter-day trading process. Therefore, in this dissertation, we focus on the irregular price-drop events extracted from regular daily stock returns. Three studies included in this dissertation are concerned with the applications of the ACD model to irregular price-drop events, and the relationship between the length of the observed duration and covariates of theoretical interest. The conditional intensity function is a central concept in the theory on point process. The Poission process is the simplest point process with constant intensity. In the spirit of the autoregressive (AR) model, the conditional intensity can be specified as a function of information related to its own past. Based on economic or financial theory, the conditional intensity can be specified as a function of the covariate of theoretical interest. Instead of specifying the conditional intensity function, a point process can generally be described directly by the process of duration between events. We construct the conditional intensity function in two dimensions. To account for the serial dependence and clustering in the arrival time of price-drop events, the expected price drop duration is assumed as the function of past durations. Furthermore, assuming that minor declines in stock price contains information about observed and unobserved minor bad news. The intensity of minor price declines is included as an explanatory variable for the expected price-drop duration. In the first essay, Engle and Russell’s (1998) ACD model is applied to estimate the serially dependent stochastic process in the duration between price-drop events. According to the empirical results of the first essay, we find that the duration between price-drop events is significantly influenced by past duration. A shorter duration may imply that bad news be released more frequently. Thus, the positive influence of past duration on the expected price drop duration has simple economic intuition. The ACD model could be a useful tool in forecasting price-drop events. Using the predicted duration of price-drop events from the ACD model should be an important input in risk management. This issue is addressed in the second essay. Numerous VaR backtesting results show that the VaR violations tend to be clustered when commonly used VaR methods are applied, such as the historical simulation method or the mean variance method. For this reason, the ACD model is employed to estimate the dependency of the duration of VaR violations. The predicted arrival time of VaR violations can be easily calculated from the conditional expected duration. Further, the prediction interval of the VaR violation arrival time is defined as the high-risk period. The VaRs estimated from commonly used VaR methods are called the unadjusted VaRs. By calibrating the unadjusted VaRs during high-risk periods in which VaR violations tend to occur, the VaR violation rates of several commonly used VaR methods could be lowered to the target level. Employing the ACD model to calibrate VaRs estimated from easily implemented VaR methods inherits the practicability advantage of simple VaR methods. Assuming that the minor price declines contain clues (either public or non public information) to the forthcoming price-drop events, the third essay investigates whether the intensity of minor stock price declines can be a precursor to extreme declines. A simple but meaningful extension of the standard ACD model is proposed by adding the intensity of minor declines. Empirical results of the third essay indicate that the intensity of minor declines in stock price has a significant negative influence on the duration between price-drop events. The essays of this dissertation provide evidence that the ACD model should be a method, not just for intra-day data analysis, but also for inter-day data analysis. Besides investigating the autocorrelation in price-drop duration, the ACD model can be used to empirically examine theoretical issues by defining the duration of interest. For example, the economic implication behind trade durations is the liquidity of the market (Engle and Russell, 1998; Hautsch, 2004). The duration between price-drop events may reflect the frequency of observable and unobservable bad news, or, as an alternative measure of market confidence. The evidence of this study offers many policy implications for both practitionersand regulators in the field of finance.