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:
[1]

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

[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.  Google Scholar

[3]

M. BardiA. 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

[4]

K. C. ChanG. A. KarolyiF. 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

[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.  Google Scholar

[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.  Google Scholar

[7] P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, Berlin, 2004.   Google Scholar
[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.  Google Scholar

[9]

F. JiangY. 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

[10]

X. Mao, Stochatic Differential Equations and Applications, Horwood, 1997. Google Scholar

[11]

X. MaoA. 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

[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.  Google Scholar

[13]

L. SzpruchX. MaoD. 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

[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.   Google Scholar

[15]

F. WuX. 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

[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.  Google Scholar

[17]

X. ZongF. 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

[18]

X. ZongF. 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:
[1]

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

[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.  Google Scholar

[3]

M. BardiA. 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

[4]

K. C. ChanG. A. KarolyiF. 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

[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.  Google Scholar

[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.  Google Scholar

[7] P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, Berlin, 2004.   Google Scholar
[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.  Google Scholar

[9]

F. JiangY. 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

[10]

X. Mao, Stochatic Differential Equations and Applications, Horwood, 1997. Google Scholar

[11]

X. MaoA. 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

[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.  Google Scholar

[13]

L. SzpruchX. MaoD. 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

[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.   Google Scholar

[15]

F. WuX. 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

[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.  Google Scholar

[17]

X. ZongF. 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

[18]

X. ZongF. 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

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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