A Computational Spectral Approach to Interest Rate Models

L. Di Persio, Gregorio Pellegrini, M. Bonollo
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引用次数: 3

Abstract

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 {\xi}, 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.
利率模型的计算谱方法
多项式混沌展开(PCE)技术利用给定随机量{\xi}的函数正交多项式的适当线性组合来恢复有限二阶随机变量,从而作为一种随机基。PCE方法是一种数学上严格的不确定性量化(UQ)方法,旨在为定义某些工程模型及其相关仿真动态的某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方法来分析一些考虑一维情况的股票和利率模型,而不失去一般性。特别地,我们将考虑那些基于几何布朗运动(gBm)的模型,例如Vasicek模型,CIR模型等。我们还提供了几个数值应用和结果,讨论了一组波动值。后者允许我们在相当大的一组不同的场景中测试PCE技术,从而提供了对PCE近似的特征和属性的相当完整和详细的研究,例如统计,分布和分位数的收敛性。此外,通过将我们的输出与采用蒙特卡罗方法的标准形式及其增强版本的输出进行比较,我们给出了关于我们方法的效率和准确性研究的结果。
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