Delay Ait-Sahalia-type interest rate model with jumps and its strong approximation
-
Emmanuel Coffie
Abstract
In this paper, we study the analytical properties of the true solution to the generalised delay Ait-Sahalia-type interest rate model with Poisson-driven jumps. Since this model does not have a closed-form solution, we employ several new truncated Euler-Maruyama (EM) techniques to investigate the finite-time strong convergence theory of the numerical solution under the local Lipschitz condition plus the Khasminskii-type condition. We justify the strong convergence result for Monte Carlo calibration and valuation of some debt and derivative instruments.
Appendix
The proof for Lemma 3.2 is provided below.
Proof of Lemma 3.2.
If we assume that
and
It then follows that
Noting from Assumption 2.3 that
then for any
as required, where
Here is the proof for Lemma 3.3.
Proof of Lemma 3.3.
By Assumptions 2.1 and 2.3, we get
where
Acknowledgements
The author would like to thank the University of Strathclyde for the scholarship and his first PhD supervisor, Professor Xuerong Mao.
References
[1] Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Rev. Financ. Stud. 9 (1996), no. 2, 385–426. 10.1093/rfs/9.2.385Suche in Google Scholar
[2] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics 116, Cambridge University, Cambridge, 2009. 10.1017/CBO9780511809781Suche in Google Scholar
[3] M. Arriojas, Y. Hu, S.-E. Mohammed and G. Pap, A delayed Black and Scholes formula, Stoch. Anal. Appl. 25 (2007), no. 2, 471–492. 10.1080/07362990601139669Suche in Google Scholar
[4] J. Bao, B. Böttcher, X. Mao and C. Yuan, Convergence rate of numerical solutions to SFDEs with jumps, J. Comput. Appl. Math. 236 (2011), no. 2, 119–131. 10.1016/j.cam.2011.05.043Suche in Google Scholar
[5] F. E. Benth, Option Theory with Stochastic Analysis: An Introduction to Mathematical Finance, Universitext, Springer, Berlin, 2003. 10.1007/978-3-642-18786-5Suche in Google Scholar
[6] S.-R. Cheng, Highly nonlinear model in finance and convergence of Monte Carlo simulations, J. Math. Anal. Appl. 353 (2009), no. 2, 531–543. 10.1016/j.jmaa.2008.12.028Suche in Google Scholar
[7] S. Deng, C. Fei, W. Fei and X. Mao, Generalized Ait-Sahalia-type interest rate model with Poisson jumps and convergence of the numerical approximation, Phys. A 533 (2019), Article ID 122057. 10.1016/j.physa.2019.122057Suche in Google Scholar
[8] S. Deng, W. Fei, W. Liu and X. Mao, The truncated EM method for stochastic differential equations with Poisson jumps, J. Comput. Appl. Math. 355 (2019), 232–257. 10.1016/j.cam.2019.01.020Suche in Google Scholar
[9] N. T. Dung, Tail probabilities of solutions to a generalized Ait-Sahalia interest rate model, Statist. Probab. Lett. 112 (2016), 98–104. 10.1016/j.spl.2016.01.026Suche in Google Scholar
[10] C. Emmanuel and X. Mao, Truncated EM numerical method for generalised Ait-Sahalia-type interest rate model with delay, J. Comput. Appl. Math. 383 (2021), Paper No. 113137. 10.1016/j.cam.2020.113137Suche in Google Scholar
[11] D. J. Higham and P. E. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math. 101 (2005), no. 1, 101–119. 10.1007/s00211-005-0611-8Suche in Google Scholar
[12] D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, J. Comput. Financ. 8 (2005), no. 3, 35–61. 10.21314/JCF.2005.136Suche in Google Scholar
[13] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2130, 1563–1576. 10.1098/rspa.2010.0348Suche in Google Scholar
[14] S. G. Kou, A jump-diffusion model for option pricing, Manag. Sci. 48 (2002), no. 8, 1086–1101. 10.1287/mnsc.48.8.1086.166Suche in Google Scholar
[15] M.-K. Lee and J.-H. Kim, A delayed stochastic volatility correction to the constant elasticity of variance model, Acta Math. Appl. Sin. Engl. Ser. 32 (2016), no. 3, 611–622. 10.1007/s10255-016-0588-3Suche in Google Scholar
[16] X. Mao, The truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), 370–384. 10.1016/j.cam.2015.06.002Suche in Google Scholar
[17] X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl. 23 (2005), no. 5, 1045–1069. 10.1080/07362990500118637Suche in Google Scholar
[18] X. Mao and S. Sabanis, Delay geometric Brownian motion in financial option valuation, Stochastics 85 (2013), no. 2, 295–320. 10.1080/17442508.2011.652965Suche in Google Scholar
[19] R. C. Merton, Option pricing when underlying stock returns are´discontinuous, J. Financ. Econ. 3 (1976), no. 1–2, 125–144. 10.1016/0304-405X(76)90022-2Suche in Google Scholar
[20] L. Szpruch, X. Mao, D. J. Higham and J. Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model, BIT 51 (2011), no. 2, 405–425. 10.1007/s10543-010-0288-ySuche in Google Scholar
[21] F. Wu, X. Mao and K. Chen, Strong convergence of Monte Carlo simulations of the mean-reverting square root process with jump, Appl. Math. Comput. 206 (2008), no. 1, 494–505. 10.1016/j.amc.2008.09.040Suche in Google Scholar
[22] F. Wu, X. Mao and K. Chen, The Cox–Ingersoll–Ross model with delay and strong convergence of its Euler–Maruyama approximate solutions, Appl. Numer. Math. 59 (2009), no. 10, 2641–2658. 10.1016/j.apnum.2009.03.004Suche in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
