Property and numerical simulation of the Ait-Sahalia-Rho model with nonlinear growth conditions

Feng Jiang; Hua Yang; Tianhai Tian

doi:10.3934/dcdsb.2017005

Article Contents

2017, Volume 22, Issue 1: 101-113. Doi: 10.3934/dcdsb.2017005

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Property and numerical simulation of the Ait-Sahalia-Rho model with nonlinear growth conditions

1.
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
2.
School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China
3.
School of Mathematical Sciences, Monash University, Melbourne VIC 3800, Australia

* Corresponding author
* Corresponding author

Received: July 31, 2015

Revised: March 31, 2016

Published: November 2017

The work is supported by the National Natural Science Foundation of China (61304067 and 11571368), the Natural Science Foundation of Hubei Province of China (2013CFB443) and the Australian Research Council Future Fellowship (FT100100748)

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Abstract

The Ait-Sahalia-Rho model is an important tool to study a number of financial problems, including the term structure of interest rate. However, since the functions of this model do not satisfy the linear growth condition, we cannot study the properties for the solution of this model by using the traditional techniques. In this paper we overcome the mathematical difficulties due to the nonlinear growth condition by using numerical simulation. Thus we first discuss analytical properties of the model and the convergence property of numerical solutions in probability for the Ait-Sahalia-Rho model. Finally, an example for option pricing is given to illustrate that the numerical solution is an effective method to estimate the expected payoffs.

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Mathematics Subject Classification: Primary:60H10, 65C35;Secondary:65C05.

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Figure 1. Simulated discrete Euler-Maruyama approximation. (A) Parameter value $\sigma=1.329\times 10^{-2}.$ (B) Parameter value $\sigma=0.1.$ (Solid-line: $\gamma =1.025, \rho=1.01$; dash-line: $\gamma =2, \rho=2$; dash-dot-line: $\gamma =3, \rho=3.$

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