论文标题
差分神经网络学习随机微分方程和用于定价多资产选项的黑色 - choles方程
A differential neural network learns stochastic differential equations and the Black-Scholes equation for pricing multi-asset options
论文作者
论文摘要
具有足够平滑激活功能的神经网络可以近似任何平滑函数的值和衍生物，并且它们本身是可区分的。我们通过利用神经网络的不同性来提高神经网络的近似能力；神经网络的梯度和黑森州用于训练神经网络，以满足感兴趣问题的微分方程。还比较了几种激活功能在有效分化神经网络的期限中。我们将差异神经网络应用于财务选择的定价，在这种选择中，随机微分方程和黑色 - choles部分微分方程代表了期权和基本资产价格的关系，以及最初和第二个衍生产品（Greeks，Greeks）期权在金融工程中起重要作用。拟议的神经网络学习 - （a）随机微分方程产生的期权价格的样本路径以及（b）每个时间和资产价格的黑色 - choles方程。对多资产选项（例如Exchange和Basket选项）进行了期权定价实验。实验结果表明，所提出的方法给出了准确的期权值和希腊人。足够平滑的激活函数和黑色choles方程的约束对准确的期权定价产生了显着贡献。
Neural networks with sufficiently smooth activation functions can approximate values and derivatives of any smooth function, and they are differentiable themselves. We improve the approximation capability of neural networks by utilizing the differentiability of neural networks; the gradient and Hessian of neural networks are used to train the neural networks to satisfy the differential equations of the problems of interest. Several activation functions are also compared in term of effective differentiation of neural networks. We apply the differential neural networks to the pricing of financial options, where stochastic differential equations and the Black-Scholes partial differential equation represent the relation of price of option and underlying assets, and the first and second derivatives, Greeks, of option play important roles in financial engineering. The proposed neural network learns -- (a) the sample paths of option prices generated by stochastic differential equations and (b) the Black-Scholes equation at each time and asset price. Option pricing experiments were performed on multi-asset options such as exchange and basket options. Experimental results show that the proposed method gives accurate option values and Greeks; sufficiently smooth activation functions and the constraint of Black-Scholes equation contribute significantly for accurate option pricing.
