论文标题
用不确定的波动率计算黑色学者 - 机器学习方法
Computing Black Scholes with Uncertain Volatility-A Machine Learning Approach
论文作者
论文摘要
在金融数学中，这是一种典型的方法，可以通过连续时间模型（例如Black Scholes模型）在离散时间内运行的金融市场。由于市场数据的离散性质，因此拟合此模型会引起困难。因此，我们通过黑色Scholes方程对金融导数的定价过程进行建模，其中波动率是有限数量的随机变量的函数。这反映了确定波动率时不确定因素的影响。目的是在计算衍生品价格时量化这种不确定性的效果。我们的基本方法是通用多项式混乱（GPC）方法，以通过随机盖尔金方法和有限的差异方法来数值计算溶液的不确定性。我们提出了该方法的有效数值变化，该方法基于机器学习技术，即所谓的BIFIDELITY方法。用数值示例说明了这一点。
In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.
