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Estimation and Prediction of a Non-Constant Volatility

机译：非恒定波动率的估计和预测

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摘要

In this paper we study volatility functions. Our main assumption is that the volatility is a function of time and is either deterministic, or stochastic but driven by a Brownian motion independent of the stock. Our approach is based on estimation of an unknown function when it is observed in the presence of additive noise. The set up is that the prices are observed over a time interval [0, t], with no observations over (t, T), however there is a value for volatility at T. This value is may be inferred from options, or provided by an expert opinion. We propose a forecasting/interpolating method for such a situation. One of the main technical assumptions is that the volatility is a continuous function, with derivative satisfying some smoothness conditions. Depending on the degree of smoothness there are two estimates, called filters, the first one tracks the unknown volatility function and the second one tracks the volatility function and its derivative. Further, in the proposed model the price of option is given by the Black–Scholes formula with the averaged future volatility. This enables us to compare the implied volatility with the averaged estimated historical volatility. This comparison is done for three companies and has shown that the two estimates of volatility have a weak statistical relation.
机译：在本文中，我们研究了波动率函数。我们的主要假设是，波动率是时间的函数，是确定性的或随机的，但由独立于股票的布朗运动驱动。当存在附加噪声时，我们的方法基于对未知函数的估计。设置是在一个时间间隔[0，t]上观察价格，而没有观察到（t，T），但是在T处有一个波动率值。该值可以从期权推论得出或提供根据专家的意见。我们针对这种情况提出了一种预测/插值方法。主要的技术假设之一是挥发性是一个连续函数，其导数满足某些平滑条件。根据平滑度，有两个估计值，称为滤波器，第一个估计值跟踪未知的波动率函数，第二个估计值跟踪波动率函数及其导数。此外，在建议的模型中，期权的价格由布莱克-斯科尔斯公式与未来的平均波动率给出。这使我们能够将隐含波动率与平均估计历史波动率进行比较。对三家公司进行了此比较，结果表明这两种波动率估计值之间的统计关系很弱。

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