# American Institute of Mathematical Sciences

January  2017, 22(1): 101-113. doi: 10.3934/dcdsb.2017005

## 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

Received  August 2015 Revised  April 2016 Published  December 2016

Fund Project: 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).

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.

Citation: Feng Jiang, Hua Yang, Tianhai Tian. Property and numerical simulation of the Ait-Sahalia-Rho model with nonlinear growth conditions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 101-113. doi: 10.3934/dcdsb.2017005
##### References:
 [1] Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Rev. Finan. Stud., 9 (1996), 385-426.  doi: 10.3386/w5346. [2] C. H. Baduraliya and X. Mao, The Euler-Maruyama approximation for the asset price in the mean-reverting-theta stochastic volatility model, Computers and Mathematics with Applications, 64 (2012), 2209-2223.  doi: 10.1016/j.camwa.2012.01.037. [3] M. Bardi, A. Cesaroni and D. Ghilli, Large deviations for some fast stochastic volatility models by viscosity methods, Discrete and Continuous Dynamical Systems, 35 (2015), 3965-3988.  doi: 10.3934/dcds.2015.35.3965. [4] K. C. Chan, G. A. Karolyi, F. A. Longstaff and A. B. Sanders, An empirical comparison of alternative models of the short-term interest rate, The Journal of Finance XLVII, 47 (1992), 1209-1227.  doi: 10.1111/j.1540-6261.1992.tb04011.x. [5] L. Chen and F. Wu, Almost sure exponential stability of the backward Euler-Maruyama scheme for stochastic delay differential equations with monotone-type condition, J. Computational Applied Mathematics, 285 (2015), 44-53.  doi: 10.1016/j.cam.2014.12.036. [6] S. Cheng, Highly nonlinear model in finance and convergence of Monte Carlo simulations, J. Math. Anal. Appl., 353 (2009), 531-543.  doi: 10.1016/j.jmaa.2008.12.028. [7] P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, Berlin, 2004. [8] D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, Journal of Computational Finance, 8 (2005), 35-61.  doi: 10.21314/JCF.2005.136. [9] F. Jiang, Y. Shen and F. Wu, Jump systems with the mean-reverting $γ$-process and convergence of the numerical approximation, Stochastics and Dynamics, 12 (2012), 1150018, 15pp.  doi: 10.1142/S0219493712003663. [10] X. Mao, Stochatic Differential Equations and Applications, Horwood, 1997. [11] X. Mao, A. Truman and C. Yuan, Euler-Maruyama approximations in mean-reverting stochastic volatility model under regime-switching, Journal of Applied Mathematics and Stochastic Analysiss, 2006 (2006), Art. ID 80967, 20 pp..  doi: 10.1155/JAMSA/2006/80967. [12] K. B. Nowman, Gaussian Estimation of single-factor continuous time models of the term structure of interest rate, The Journal of Finance, LII, 52 (1997), 1695-1706.  doi: 10.1111/j.1540-6261.1997.tb01127.x. [13] L. Szpruch, X. Mao, D. J. Higham and J. Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model, BIT Numer. Math., 51 (2011), 405-425.  doi: 10.1007/s10543-010-0288-y. [14] F. Wu and S. Hu, The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay, Discrete and Continuous Dynamical Systems, 32 (2012), 1065-1094. [15] F. Wu, X. Mao and K. Chen, A highly sensitive mean-reverting process in finance and the Euler-Maruyama approximations, J. Math. Anal. Appl., 348 (2008), 540-554.  doi: 10.1016/j.jmaa.2008.07.069. [16] X. Zong and F. Wu, Choice of $θ$ and mean-square exponential stability in the stochastic theta method of stochastic differential equations, J. Computational Applied Mathematics, 255 (2014), 837-847.  doi: 10.1016/j.cam.2013.07.007. [17] X. Zong, F. Wu and C. Huang, Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients, Applied Mathematics and Computation, 228 (2014), 240-250.  doi: 10.1016/j.amc.2013.11.100. [18] X. Zong, F. Wu and C. Huang, Theta schemes for SDDEs with non-globally Lipschitz continuous coefficients, J. Computational Applied Mathematics, 278 (2015), 258-277.  doi: 10.1016/j.cam.2014.10.014.

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##### References:
 [1] Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Rev. Finan. Stud., 9 (1996), 385-426.  doi: 10.3386/w5346. [2] C. H. Baduraliya and X. Mao, The Euler-Maruyama approximation for the asset price in the mean-reverting-theta stochastic volatility model, Computers and Mathematics with Applications, 64 (2012), 2209-2223.  doi: 10.1016/j.camwa.2012.01.037. [3] M. Bardi, A. Cesaroni and D. Ghilli, Large deviations for some fast stochastic volatility models by viscosity methods, Discrete and Continuous Dynamical Systems, 35 (2015), 3965-3988.  doi: 10.3934/dcds.2015.35.3965. [4] K. C. Chan, G. A. Karolyi, F. A. Longstaff and A. B. Sanders, An empirical comparison of alternative models of the short-term interest rate, The Journal of Finance XLVII, 47 (1992), 1209-1227.  doi: 10.1111/j.1540-6261.1992.tb04011.x. [5] L. Chen and F. Wu, Almost sure exponential stability of the backward Euler-Maruyama scheme for stochastic delay differential equations with monotone-type condition, J. Computational Applied Mathematics, 285 (2015), 44-53.  doi: 10.1016/j.cam.2014.12.036. [6] S. Cheng, Highly nonlinear model in finance and convergence of Monte Carlo simulations, J. Math. Anal. Appl., 353 (2009), 531-543.  doi: 10.1016/j.jmaa.2008.12.028. [7] P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, Berlin, 2004. [8] D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, Journal of Computational Finance, 8 (2005), 35-61.  doi: 10.21314/JCF.2005.136. [9] F. Jiang, Y. Shen and F. Wu, Jump systems with the mean-reverting $γ$-process and convergence of the numerical approximation, Stochastics and Dynamics, 12 (2012), 1150018, 15pp.  doi: 10.1142/S0219493712003663. [10] X. Mao, Stochatic Differential Equations and Applications, Horwood, 1997. [11] X. Mao, A. Truman and C. Yuan, Euler-Maruyama approximations in mean-reverting stochastic volatility model under regime-switching, Journal of Applied Mathematics and Stochastic Analysiss, 2006 (2006), Art. ID 80967, 20 pp..  doi: 10.1155/JAMSA/2006/80967. [12] K. B. Nowman, Gaussian Estimation of single-factor continuous time models of the term structure of interest rate, The Journal of Finance, LII, 52 (1997), 1695-1706.  doi: 10.1111/j.1540-6261.1997.tb01127.x. [13] L. Szpruch, X. Mao, D. J. Higham and J. Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model, BIT Numer. Math., 51 (2011), 405-425.  doi: 10.1007/s10543-010-0288-y. [14] F. Wu and S. Hu, The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay, Discrete and Continuous Dynamical Systems, 32 (2012), 1065-1094. [15] F. Wu, X. Mao and K. Chen, A highly sensitive mean-reverting process in finance and the Euler-Maruyama approximations, J. Math. Anal. Appl., 348 (2008), 540-554.  doi: 10.1016/j.jmaa.2008.07.069. [16] X. Zong and F. Wu, Choice of $θ$ and mean-square exponential stability in the stochastic theta method of stochastic differential equations, J. Computational Applied Mathematics, 255 (2014), 837-847.  doi: 10.1016/j.cam.2013.07.007. [17] X. Zong, F. Wu and C. Huang, Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients, Applied Mathematics and Computation, 228 (2014), 240-250.  doi: 10.1016/j.amc.2013.11.100. [18] X. Zong, F. Wu and C. Huang, Theta schemes for SDDEs with non-globally Lipschitz continuous coefficients, J. Computational Applied Mathematics, 278 (2015), 258-277.  doi: 10.1016/j.cam.2014.10.014.
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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