论文标题
在多因素黑色旋风模型下使用正交样条小波的选项定价
Option Pricing under Multifactor Black-Scholes Model Using Orthogonal Spline Wavelets
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
The paper focuses on pricing European-style options on several underlying assets under the Black-Scholes model represented by a nonstationary partial differential equation. The proposed method combines the Galerkin method with $L^2$-orthogonal sparse grid spline wavelets and the Crank-Nicolson scheme with Rannacher time-stepping. To this end, we construct an orthogonal cubic spline wavelet basis on the interval satisfying homogeneous Dirichlet boundary conditions and design a wavelet basis on the unit cube using the sparse tensor product. The method brings the following advantages. First, the number of basis functions is significantly smaller than for the full grid, which makes it possible to overcome the so-called curse of dimensionality. Second, some matrices involved in the computation are identity matrices, which significantly simplifies and streamlines the algorithm, especially in higher dimensions. Further, we prove that discretization matrices have uniformly bounded condition numbers, even without preconditioning, and that the condition numbers do not depend on the dimension of the problem. Due to the use of cubic spline wavelets, the method is higher-order convergent. Numerical experiments are presented for options on the geometric average.
