利率模型的计算谱方法
A computational spectral approach to interest rate models
摘要：
多项式混沌展开（PCE）技术利用正交多项式的适当线性组合来恢复有限个二阶随机变量，正交多项式是给定随机量λ的函数，因此充当一种随机基。PCE方法已经发展为数学上严格的不确定性量化（UQ）方法，其目的是为定义某些工程模型及其相关模拟的动态的一些不确定物理量提供可靠的数值估计。在本文中，我们利用PCE的方法来分析一些股票和利率模型考虑，不失一般性，一维的情况下。特别是，我们将考虑到这些模型的基础上的几何布朗运动（gBM），例如Vasicek模型，CIR模型等，我们还提供了几个数值应用和结果，讨论了一组波动率值。后者使我们能够在一个相当大的不同的情况下测试PCE技术，从而提供了一个相当完整和详细的调查PCE近似的功能和属性，如统计，分布和分位数的收敛。此外，我们给出的结果有关的效率和准确性的研究，我们的方法通过比较我们的输出所获得的采用蒙特卡罗方法在其标准形式，以及在其增强版。
英文摘要：
The Polynomial Chaos Expansion (PCE) technique recovers a finite second order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochas- tic quantity ξ, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ) method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper we exploit the PCE approach to analyze some equity and interest rate models considering, without loss of generality, the one dimensional case. In particular we will take into account those models which are based on the Geometric Brownian Motion (gBm), e.g. the Vasicek model, the CIR model, etc. We also provide several numerical applications and results which are discussed for a set of volatility values. The latter allows us to test the PCE technique on a quite large set of different scenarios, hence providing a rather complete and detailed investigation on PCE-approximation's features and properties, such as the convergence of statistics, distribution and quantiles. Moreover we give results concerning both an efficiency and an accuracy study of our approach by comparing our outputs with the ones obtained adopting the Monte Carlo approach in its standard form as well as in its enhanced version.
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