Journal of Probability and Statistics

Volume 2015 (2015), Article ID 369053, 24 pages

http://dx.doi.org/10.1155/2015/369053

## Polynomial Chaos Expansion Approach to Interest Rate Models

^{1}Department of Computer Science, University of Verona, Strada le Grazie 15, 37134 Verona, Italy^{2}Iason Ltd., Milan, Italy^{3}IMT Lucca, Piazza San Francesco 19, 55100 Lucca, Italy

Received 30 June 2015; Accepted 19 October 2015

Academic Editor: Z. D. Bai

Copyright © 2015 Luca Di Persio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Polynomial Chaos Expansion (PCE) technique allows us to recover a finite second-order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantity , 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 use the PCE approach in order to analyze some equity and interest rate models. In particular, we take into consideration those models which are based on, for example, the Geometric Brownian Motion, the Vasicek model, and the CIR model. We present theoretical as well as related concrete numerical approximation results considering, without loss of generality, the one-dimensional case. We also provide both an efficiency study and an accuracy study of our approach by comparing its outputs with the ones obtained adopting the Monte Carlo approach, both in its standard and its enhanced version.

#### 1. Introduction

In this paper, we use the Polynomial Chaos Expansion (PCE) technique to study the strong solution of the following Itô stochastic differential equation (SDE):where is the initial value of the unknown stochastic process , and the terms and represent the* drift* and the* volatility* of the process, respectively, while is a standard Brownian motion and it is considered up to a certain finite time to which we refer, taking into account the financial setting, as the maturity time, for example, of some underlying. According to [1, Section 4.5], if the coefficients and are smooth enough, then we have the existence and uniqueness of the solution to (1).

The PCE approach to the aforementioned problem belongs to the so-called* Uncertain Quantification* (UQ) methods, which constitute a fundamental step towards the validation of numerical techniques to be used for critical decisions; see, for example, [2, 3] and the references therein. The term* polynomial chaos* was originally introduced by Wiener in his 1938 paper [4] in which he applies his generalized harmonic analysis (see, e.g., [5]). Then, Ghanem and Spanos in [6, Section 2.4, Subsection 3.3.6] combined the PCE with finite element method to compute the solution of PDEs in the presence of uncertain parameters. Then, such an approach has been applied in several frameworks. For example, for the simulation of probabilistic chemical reactions, see [7], for the stochastic optimal trajectories generation, see [8], for the sensitivity analysis, see [9]. In order to refine specific numerical methods, see [10], and also for a rather large set of problems arising in engineering and computational fluid dynamics (CFD), see, for example, [11, Chapter 6], [12, 13], and references therein.

Strictly speaking, PCE recovers a random variable in terms of a linear combination of functionals whose entries are known random variables called* germs* or* basic variables*. Several methods are available to compute these coefficients. In particular, we choose the Nonintrusive Spectral Projection (NISP) method which employs a set of deterministic realizations; see, for example, [11, Chapter 3] for further details.

In order to show the advantages of the PCE approach, the results of our PCE implementation have been compared* versus* the ones obtained using both the standard Monte Carlo (MC) and the* quasi*-Monte Carlo (QMC) techniques. The former is based on pseudorandom sampling and its convergence properties basically rely on the* law of large numbers* and the* central limit theorem*. The latter uses low-discrepancy sequences to simulate the process and it is theoretically based on the Koksma-Hlawka inequality; see [14], which provides error bounds for computations involving QMC method; see, for example, [15, Chapter 5].

Our analysis has been focused on three well known one-dimensional SDEs whose solutions are related to the Geometric Brownian Motion equity model; see, for example, [16, Chapter 11, Section 3], the Vasicek model, [17] and the Cox-Ingersoll-Ross (CIR) interest rate model, and [18]. We also refer to [19, Chapter 3, Section 3.2.3], for more details and references about such models.

We underline that the first two cases are meaningful because they have an analytical solution to the related SDE, so that we provide a rigorous study on convergence property of the PCE method applied to them. In the CIR case, as we do not have an analytical solution, the application of the PCE technique allows us also to derive some considerations on the convergence properties of the related approximating solutions.

The paper is organized as follows: in Section 2, the PCE machinery is described in a general setting, while Section 3 deals with the NISP approach. Section 4 points out how to apply PCE in order to decompose the solution of the considered SDE, while Section 5 is focused on the relation between the usual numerical method for solving (1) and the PCE technique. In Sections 6, 7, and 8, we consider numerical applications of the PCE approach to approximate the GBM equity and the Vasicek and the CIR model. In Section 9, we give an overview of the PCE results in terms of accuracy and computational time cost for everyone of the aforementioned considered models. Eventually, in the Appendix, we provide a table with the evaluation of the first six Hermite polynomials on a set of 40 equally spaced points in the interval .

#### 2. Polynomial Chaos Expansion

Let be a probability space, where is the set of elementary events, is a -algebra of subsets of , and is a probability measure on .

We consider the Hilbert space of scalar real-valued random variables , whose generic element is a random variable defined on , such that

Notice that, as for Lebesgue spaces, the elements are equivalent classes of random variables. Note that is a Hilbert space endowed with the following scalar product: withbeing the usual norm. To shorten the notation, , and the related convergence has been always referred to as* mean square convergence* or* strong convergence*.

Among elements in , there is the class of* basic random variables*, which is used to decompose, as entries of functionals, the quantity of interest such as a random variable of interest, the solution of the SDE at time . We notice that not all the functions can be used to such a decomposition because they have to satisfy, for example, [20, Section 3], at least the following two properties:

(i)has finite raw moments of all orders;(ii)the distribution function of the basic random variable is continuous, with being its probability density function (pdf).

Because the increments of the Brownian motion are independent and normally distributed, from now on, we consider as basic random variable. The latter choice is motivated in Remark 1 in Section 3.1.

The elements in can be gathered in two groups: in the first one, we have the* basic random variables*; let us indicate them by , ruling the decomposition; for simplicity, let us call it , while the second is composed of* generic elements*; let us say , which we want to decompose using the elements of the first set, which is referred to as .

Let us denote by the -algebra generated by the basic random variable ; hence, . If we want to polynomially decompose the random variable in terms of , then has at least to be measurable with respect to the -algebra . Exploiting the Doob-Dynkin lemma, [21, Lemma 1.13], we have that is measurable, by detecting a Borel measurable function , such that . In what follows, without loss of generality, we restrict ourselves to consider the decomposition in . Moreover, the basic random variable determines the class of orthogonal polynomials indicated as the* generalized polynomial chaos* (gPC) basis. We underline that their orthogonality property is detected by means of the measure induced by in the image space , where denotes the Borel -algebra of . In particular, for each , we have

Because , the related set is represented by the family of Hermite polynomials defined on the whole real line; namely, , andFigure 1 provides the graph of the first six orthonormal polynomials, achieved by scaling each in (6) by its norm in ; namely, , and is divided by . See also the Appendix where these polynomials are evaluated on equally spaced points in .