    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 & Continuous Dynamical Systems - B, 2017, 22 (1) : 101-113. doi: 10.3934/dcdsb.2017005
##### References:
  Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Rev. Finan. Stud., 9 (1996), 385-426.  doi: 10.3386/w5346. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, Berlin, 2004. Google Scholar  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. Google Scholar  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.  Google Scholar  X. Mao, Stochatic Differential Equations and Applications, Horwood, 1997. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar

show all references

##### References:
  Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Rev. Finan. Stud., 9 (1996), 385-426.  doi: 10.3386/w5346. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, Berlin, 2004. Google Scholar  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. Google Scholar  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.  Google Scholar  X. Mao, Stochatic Differential Equations and Applications, Horwood, 1997. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar 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: $\gamma =3, \rho=3.$
  Antonio Di Crescenzo, Maria Longobardi, Barbara Martinucci. On a spike train probability model with interacting neural units. Mathematical Biosciences & Engineering, 2014, 11 (2) : 217-231. doi: 10.3934/mbe.2014.11.217  Yinghui Dong, Guojing Wang. Ruin probability for renewal risk model with negative risk sums. Journal of Industrial & Management Optimization, 2006, 2 (2) : 229-236. doi: 10.3934/jimo.2006.2.229  Emilija Bernackaitė, Jonas Šiaulys. The finite-time ruin probability for an inhomogeneous renewal risk model. Journal of Industrial & Management Optimization, 2017, 13 (1) : 207-222. doi: 10.3934/jimo.2016012  Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069  Eric Cancès, Claude Le Bris. Convergence to equilibrium of a multiscale model for suspensions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 449-470. doi: 10.3934/dcdsb.2006.6.449  Tianyuan Xu, Shanming Ji, Chunhua Jin, Ming Mei, Jingxue Yin. Early and late stage profiles for a chemotaxis model with density-dependent jump probability. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1345-1385. doi: 10.3934/mbe.2018062  Baoyin Xun, Kam C. Yuen, Kaiyong Wang. The finite-time ruin probability of a risk model with a general counting process and stochastic return. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021032  Qingwu Gao, Zhongquan Huang, Houcai Shen, Juan Zheng. Asymptotics for random-time ruin probability in a time-dependent renewal risk model with subexponential claims. Journal of Industrial & Management Optimization, 2016, 12 (1) : 31-43. doi: 10.3934/jimo.2016.12.31  Jan Lorenz, Stefano Battiston. Systemic risk in a network fragility model analyzed with probability density evolution of persistent random walks. Networks & Heterogeneous Media, 2008, 3 (2) : 185-200. doi: 10.3934/nhm.2008.3.185  Yuebao Wang, Qingwu Gao, Kaiyong Wang, Xijun Liu. Random time ruin probability for the renewal risk model with heavy-tailed claims. Journal of Industrial & Management Optimization, 2009, 5 (4) : 719-736. doi: 10.3934/jimo.2009.5.719  Pavel Krejčí, Songmu Zheng. Pointwise asymptotic convergence of solutions for a phase separation model. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 1-18. doi: 10.3934/dcds.2006.16.1  Hans-Otto Walther. Convergence to square waves for a price model with delay. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1325-1342. doi: 10.3934/dcds.2005.13.1325  Shijin Deng, Weike Wang, Shih-Hsien Yu. Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundary. Networks & Heterogeneous Media, 2007, 2 (3) : 383-395. doi: 10.3934/nhm.2007.2.383  Youshan Tao, Michael Winkler. Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1901-1914. doi: 10.3934/dcds.2012.32.1901  Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097  Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324  Hai-Yang Jin, Tian Xiang. Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3071-3085. doi: 10.3934/dcdsb.2017197  Wenting Cong, Jian-Guo Liu. Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 307-338. doi: 10.3934/dcdsb.2017015  Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026  Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034

2020 Impact Factor: 1.327