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A comparison of fuzzy regression methods for the estimation of the implied volatility smile function

机译：模糊回归方法在隐含波动率微笑函数估计中的比较

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摘要

The information content of option prices on the underlying asset has a special importance in finance. In particular, with the use of option implied trees, market participants may price other derivatives, estimate and forecast volatility (see e.g. the volatility index VIX), or higher moments of the underlying asset distribution. A crucial input of option implied trees is the estimation of the smile (implied volatility as a function of the strike price), which boils down to fitting a function to a limited number of existing knots. However, standard techniques require a one-to-one mapping between volatility and strike price, which is not met in the reality of financial markets, where, to a given strike price, two different implied volatilities are usually associated (coming from different types of options: call and put). In this paper we compare the widely used methodology of discarding some implied volatilities and interpolating the remaining knots with cubic splines, to a fuzzy regression approach which does not require an a-priori choice of implied volatilities. To this end, we first extend some linear fuzzy regression methods to a polynomial form and we apply them to the financial problem. The fuzzy regression methods used range from the possibilistic regression method of Tanaka et al. to the least squares fuzzy regression method of Savic and Pedrycz and to the hybrid method of Ishibuchi and Nii.
机译：基础资产的期权价格信息内容在财务中具有特别重要的意义。尤其是，使用期权隐含树，市场参与者可以对其他衍生产品定价，估计和预测波动率（例如，参见波动率指数VIX）或标的资产分配的较高时刻。期权隐含树的关键输入是微笑的估计（隐含波动率是行使价的函数），其简化为使函数适合有限数量的现有节点。但是，标准技术需要在波动率和行使价之间建立一对一的映射，这在金融市场的现实中是无法满足的，在金融市场中，对于给定的行使价，通常会关联两个不同的隐含波动率（来自不同类型的隐含波动率）。选项：看涨期权和看跌期权）。在本文中，我们将舍弃某些隐含波动率并用三次样条插值剩余结的广泛使用的方法与不需要先验选择隐含波动率的模糊回归方法进行比较。为此，我们首先将线性模糊回归方法扩展为多项式形式，并将其应用于财务问题。使用的模糊回归方法的范围从Tanaka等人的可能性回归方法开始。 Savic和Pedrycz的最小二乘模糊回归方法以及Ishibuchi和Nii的混合方法。

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