GARCH, Diffusion, Stochastic Volatility and Wavelets

金融随机模型有两个独立的分支：以现代金融文献为中心的连续时间模型和以经验金融文献为中心的离散时间模型。连续时间模型以扩散模型为主，它很好地适应了套利、期权定价等金融理论，但很难进行统计推断。大多数离散时间模型是自回归条件异方差(ARCH)和随机波动率(SV)模型，它们往往对观察到的离散时间数据提供简约的表示，并且相对容易进行统计推断。人们很自然地会问，离散时间模型能否与连续时间模型兼容。直到最近几年，研究人员才开始弥合这两种建模方法之间的差距，并建立了离散时间ARCH模型对连续时间扩散的弱收敛。由于弱收敛联系，金融经济学和金融数学界普遍认为，ARCH模型及其扩散极限在所有方面都是“等价的”。由于这两类模型都包含未知参数，它们的实际实现需要从数据中估计和检验参数。由于信念和ARCH更容易进行统计推断，因此将ARCH模型下导出的统计过程应用于相应的扩散是一种常见的做法。然而，所声称的统计等价性和所采用的做法在很大程度上是基于盲目相信和缺乏充分的统计理由。特别是，它们不能用弱收敛联动来严格证明。在这项提议中，PI将提出一个新的研究课题：研究这些离散时间模型和连续时间模型之间的统计关系。将调查三个相互关联的问题。由ARCH模型的观测形成的实验是否在Le Cam的亏空距离方面与由来自扩散极限的观测组成的实验渐近等价？研究模型在不同频率(如日、周、月)下的等价性或不等价性；针对广泛可用的高频数据，提出了一个小波随机波动率模型。所提出的研究具有重要的计算和实用意义。例如，如果两个模型在某些低频下渐近等价，那么基于ARCH模型的统计推断可以应用于从相应频率的扩散数据中采样的子样本；由于ARCH和SV模型描述的是平稳过程，而没有考虑高频数据中存在的局部尖峰和长记忆，因此所提出的小波模型有望更好地拟合高频数据，并容易提取尖峰、局部冲击和非平稳等高频特征以及长记忆和长期趋势等低频现象。在文献中，股市建模有两种类型的方法。一种是连续时间模型，它假设股票价格随时间连续变化，服从连续时间随机过程。从历史上看，金融经济学中发展了基于随机微分方程的连续时间模型。由于套利和期权定价等金融理论的完美融合，现代金融理论在很大程度上是建立在连续时间模型的基础上的。然而，在现实中，所有数据只以离散的间隔进行记录。连续时间模型中的未知参数需要从观测到的离散时间数据中估计和检验。由于基于离散数据的连续时间模型难以进行统计推断，使得连续时间模型的有效性难以直接检验。另一种方法是对可用的离散数据进行离散时间建模。成功的离散时间模型是自回归条件异方差(ARCH)模型和随机波动率(SV)模型。这些离散时间模型往往为观测到的离散时间数据提供简约的表示，并且它们的统计推断相对容易。但离散时间模型本质上是统计模型，不容易适应金融理论。这项建议将研究两种模型的统计兼容性，并研究高频数据的小波建模。本研究具有重要的理论和实践意义。例如，研究可以得出连续时间模型和离散时间模型在统计上等价的情况；如果等价，则易于获得的离散时间模型的统计推断过程可以应用于连续时间模型；基于小波的模型有望更好地拟合高频数据，并且容易提取尖峰、局部冲击和非平稳性等高频特征，以及长记忆和长期趋势等低频现象。

There are two independent strands of financial stochastic modeling: continuous-time models centered in the modern finance literature and discrete-time models in the empirical finance literature. The continuous-time models are dominated by the diffusion which elegantly accommodates finance theory such as arbitrage and option pricing but is very hard for statistical inference. Most of the discrete-time models are the autoregressive conditionally heteroscedastic (ARCH) and stochastic volatility (SV) models which often provide parsimonious representations for the observed discrete-time data and are relatively easier for statistical inference. It is natural to ask whether the discrete-time model can be compatible with the continuous-time model. Not until recent years did researchers begin to bridge the gap between the two modeling approaches and establish the weak convergence of the discrete-time ARCH model to continuous-time diffusion. Because of the weak convergence linkage, there is a general belief in financial economics and financial mathematics that the ARCH model and its diffusion limit are ``equivalent'' at all respects. Since both types of models involves unknown parameters, their practical implementation requires to estimate and test the parameters from the data. Because of the belief and ARCH's easier statistical inference, it is a common practice toapply statistical procedures derived under the ARCH model to the corresponding diffusion. However, the claimed statistical equivalence and the employed practice are much based on blind faith and lack of adequate statistical justification. In particular, they can not be rigorously justified by the weak convergence linkage. In this proposal PI will initiate a new research topic: study the statistical relationship between these discrete-time and continuous-time models. Three interrelated problems will be investigated. Whether the experiment formed by observations from the ARCH model is asymptotically equivalent in terms of Le Cam's deficiency distance to an experiment comprised by observationsfrom the diffusion limit ? Study model equivalence or nonequivalence at different frequencies (e.g. daily, weekly and monthly); Propose a wavelet stochastic volatility model for widely available high-frequency data. The proposed research bears important computational and practical consequence. For example, if the two models are asymptotically equivalent at certain lower frequencies, the easily obtained statistical inference based on the ARCH model can be applied to the subsample that are sampled from the diffusion data at the corresponding frequencies; because ARCH and SV models describe stationary processes and fail to account for local sharp peaks and long-memory founded in high-frequency data, the proposed wavelet model is expected to fit high-frequency data better and easily pick up high frequency features like sharp peaks, local shock, and non-stationarity as well as low frequence phenomenon such as long-memory and long term trend. Stock market modeling has two types of approaches in the literature. One is continuous-time modeling that assumes a stock price to change with time continuously and obey a continuous-time stochastic process. Historically, continuous-time models based on stochastic differential equations have been developed in financial economics. Because of elegant accommodation of finance theory such as arbitrage and option pricing, modern finance theory is much based on the continuous-time modeling. However, in reality all data are recorded only at discrete intervals. Unknown parameters in the continuous-time models need to be estimated and tested from the observed discrete-time data. Due to the difficulty in statistical inference for the continuous time model based on the discrete data, the validity of the continuous-time modeling is not straightforward to check. Another approach is discrete-time modeling of available discrete data. Successful discrete-time models are the autoregressive conditionally heteroscedastic (ARCH) and stochastic volatility (SV) models. These discrete-time models often provide parsimonious representations for the observed discrete-time data, and their statistical inference is relatively easier. But the discrete-time models are statistical models in nature and are not easy to accommodate finance theory. This proposal will study the statistical compatibility of the two types of models and investigate wavelet modeling for high-frequency data. The research bears important theoretical and practical consequences. For example, the research can yield a picture on when continuous-time and discrete-time models are statistically equivalent; if equivalent, the easily obtained statistical inference procedures for thediscrete-time models can be applied to the continuous-time models; the wavelet based model is expected to fit high-frequency data better and easily pick up high frequency features like sharp peaks, local shock, and non-stationarity as well as low frequence phenomenon such as long-memory and long term trend.