The fundamental theorem behind financial markets is that stock prices are intrinsically complex and stochastic. One of the complexities is the volatility associated with stock prices. Volatility is a tendency for prices to change unexpectedly [1]. Price volatility is often detrimental to the return economics, and thus, investors should factor it in whenever making investment decisions, choices, and temporal or permanent moves. It is, therefore, crucial to make necessary and regular short and long-term stock price volatility forecasts for the safety and economics of investors returns. These forecasts should be accurate and not misleading. Different models and methods, such as ARCH GARCH models, have been intuitively implemented to make such forecasts. However, such traditional means fail to capture the short-term volatility forecasts effectively. This paper, therefore, investigates and implements a combination of numeric and probabilistic models for short-term volatility and return forecasting for high-frequency trades. The essence is that one-day-ahead volatility forecasts were made with Gaussian Processes (GPs) applied to the outputs of a Numerical market prediction (NMP) model. Firstly, the stock price data from NMP was corrected by a GP. Since it is not easy to set price limits in a market due to its free nature and randomness, a Censored GP was used to model the relationship between the corrected stock prices and returns. Forecasting errors were evaluated using the implied and estimated data.

翻译：金融市场的基本定理在于股票价格本质上是复杂且随机的。其复杂性之一在于与股票价格相关的波动率。波动率是价格意外变化的趋势[1]。价格波动通常对收益经济性不利，因此投资者在做出投资决策、选择以及临时或永久性操作时，应将其纳入考量。因此，为保障投资者收益的安全性和经济性，定期对股票价格进行必要的短期和长期波动率预测至关重要。这些预测应准确且无误导性。尽管不同模型和方法（如ARCH GARCH模型）已被直观地用于此类预测，但传统手段难以有效捕捉短期波动率预测。为此，本文研究并实施了一种结合数值模型与概率模型的方法，用于高频交易的短期波动率与收益预测。其核心在于：将高斯过程（GPs）应用于数值市场预测（NMP）模型的输出，以实现一日前瞻的波动率预测。首先，通过GP对NMP输出的股票价格数据进行校正。鉴于市场自由性与随机性导致定价限制困难，采用截断高斯过程（Censored GP）建模校正后的股票价格与收益之间的关系。预测误差通过隐含数据与估计数据进行评估。