An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients. (English) Zbl 1355.60072
Summary: We consider the approximation of one-dimensional stochastic differential equations (SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modified explicit Euler-Maruyama discretization scheme that allows us to prove strong convergence, with a rate. Under some regularity and integrability conditions, we obtain the optimal strong error rate. We apply this scheme to SDEs widely used in the mathematical finance literature, including the Cox-Ingersoll-Ross (CIR), the 3/2, and the Ait-Sahalia models, as well as to a family of mean-reverting processes with locally smooth coefficients. We numerically illustrate the strong convergence of the scheme and demonstrate its efficiency in a multilevel Monte Carlo setting.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 65C05 | Monte Carlo methods |
| 91G60 | Numerical methods (including Monte Carlo methods) |
| 91G80 | Financial applications of other theories |
Keywords:
stochastic differential equations; non-Lipschitz coefficients; explicit Euler-Maruyama scheme; projection; CIR model; Ait-Sahalia model; multilevel Monte Carlo methodReferences:
| [1] | K. Abe, {\it Pricing exotic options using MSL-MC}, Quant. Finance, 11 (2011), pp. 1379-1392. · Zbl 1277.91191 |
| [2] | Y. Ait-Sahalia, {\it Testing continuous-time models of the spot interest rate}, Rev. Financial Stud., 9 (1996), pp. 385-426. |
| [3] | A. Alfonsi, {\it On the discretization schemes for the CIR (and Bessel squared) processes}, Monte Carlo Methods Appl., 11 (2005), pp. 355-384. · Zbl 1100.65007 |
| [4] | A. Alfonsi, {\it Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process}, Statist. Probab. Lett., 83 (2013), pp. 602-607. · Zbl 1273.65007 |
| [5] | A. Berkaoui, M. Bossy, and A. Diop, {\it Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence}, ESAIM Probab. Statist., 12 (2008), pp. 1-11. · Zbl 1183.65004 |
| [6] | M. Bossy and A. Diop, {\it An Efficient Discretization Scheme for One Dimensional SDEs with a Diffusion Coefficient of the Form \(|x|^α, α∈[1/2,1]\)}, Tech. report, INRIA, Paris, 2004. |
| [7] | M. Bossy and H. O. Quinteros, {\it Strong Convergence of the Symmetrized Milstein Scheme for Some CEV-like SDEs}, preprint, 2015, . |
| [8] | D. Brigo and F. Mercurio, {\it Interest Rate Models–Theory and Practice: With Smile, Inflation and Credit}, Springer Finance, Springer-Verlag, Berlin, 2006. · Zbl 1109.91023 |
| [9] | S. Burgos and M. Giles, {\it Computing Greeks using multilevel path simulation}, in Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer Proc. Math. Statist. 23, Springer, Berlin, 2012, pp. 281-296. · Zbl 1270.91091 |
| [10] | J. Cox, J. Ingersoll, and S. Ross, {\it A theory of the term structure of interest rates}, Econometrica, 53 (1985), pp. 385-407. · Zbl 1274.91447 |
| [11] | J. Cox and S. Ross, {\it The valuation of options for alternative stochastic processes}, J. Financial Econom., 3 (1976), pp. 145-166. |
| [12] | S. De Marco, {\it Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions}, Ann. Appl. Probab., 21 (2011), pp. 1282-1321. · Zbl 1246.60082 |
| [13] | S. Dereich, A. Neuenkirch, and L. Szpruch, {\it An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process}, Proc. Roy. Soc. Math. Phys. Eng. Sci., 468 (2012), pp. 1105-1115. · Zbl 1364.65013 |
| [14] | W. Feller, {\it Diffusion processes in one dimension}, Trans. Amer. Math. Soc., 77 (1954), pp. 1-31. · Zbl 0059.11601 |
| [15] | M. Giles, {\it Improved multilevel Monte Carlo convergence using the Milstein scheme}, in Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, Berlin, 2008, pp. 343-358. · Zbl 1141.65321 |
| [16] | M. Giles, {\it Multilevel Monte Carlo path simulation}, Oper. Res., 56 (2008), pp. 607-617. · Zbl 1167.65316 |
| [17] | M. Giles, D. Higham, and X. Mao, {\it Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff}, Finance Stoch., 13 (2009), pp. 403-413. · Zbl 1199.65008 |
| [18] | M. Giles and L. Szpruch, {\it Multilevel Monte Carlo methods for applications in finance}, in Recent Developments in Computational Finance, World Scientific, Singapore, 2013, pp. 3-47. · Zbl 1277.91193 |
| [19] | P. Glasserman, {\it Monte Carlo Methods in Financial Engineering}, Stochast. Model. Appl. Probab. 53, Springer-Verlag, New York, 2003. · Zbl 1038.91045 |
| [20] | I. Gyöngy, {\it A note on Euler’s approximations}, Potential Anal., 8 (1998), pp. 205-216. · Zbl 0946.60059 |
| [21] | I. Gyöngy and M. Rásonyi, {\it A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients}, Stochast. Process. Appl., 121 (2011), pp. 2189-2200. · Zbl 1226.60095 |
| [22] | S. Heston, {\it A closed-form solution for options with stochastic volatility with applications to bond and currency options}, Rev. Financial Stud., 6 (1993), pp. 327-343. · Zbl 1384.35131 |
| [23] | S. Heston, {\it A Simple New Formula for Options with Stochastic Volatility}, course notes, Washington University, St. Louis, MO, 1997. |
| [24] | D. J. Higham, X. Mao, and A. M. Stuart, {\it Strong convergence of Euler-type methods for nonlinear stochastic differential equations}, SIAM J. Numer. Anal., 40 (2002), pp. 1041-1063, . · Zbl 1026.65003 |
| [25] | M. Hutzenthaler and A. Jentzen, {\it Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients}, Mem. Amer. Math. Soc., 236 (2015), 1112. · Zbl 1330.60084 |
| [26] | M. Hutzenthaler, A. Jentzen, and P. Kloeden, {\it Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients}, Proc. Roy. Soc. A Math. Phys. Eng. Sci., 467 (2011), pp. 1563-1576. · Zbl 1228.65014 |
| [27] | M. Hutzenthaler, A. Jentzen, and P. Kloeden, {\it Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients}, Ann. Appl. Probab., 22 (2012), pp. 1611-1641. · Zbl 1256.65003 |
| [28] | M. Hutzenthaler, A. Jentzen, and M. Noll, {\it Strong Convergence Rates and Temporal Regularity for Cox-Ingersoll-Ross Processes and Bessel Processes with Accessible Boundaries}, preprint, 2014, . |
| [29] | A. Jentzen, P. Kloeden, and A. Neuenkirch, {\it Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients}, Numer. Math., 112 (2009), pp. 41-64. · Zbl 1163.65003 |
| [30] | I. Karatzas and S. Shreve, {\it Brownian Motion and Stochastic Calculus}, Grad. Texts Math. 113, Springer, New York, 1998. · Zbl 0638.60065 |
| [31] | P. Kloeden and A. Neuenkirch, {\it Convergence of Numerical Methods for Stochastic Differential Equations in Mathematical Finance}, preprint, 2012, . · Zbl 1277.91194 |
| [32] | P. Kloeden and E. Platen, {\it Numerical Solution of Stochastic Differential Equations}, Stochast. Model. Appl. Probab. 23, Springer-Verlag, Berlin, 1992. · Zbl 0752.60043 |
| [33] | N. V. Krylov, {\it A simple proof of the existence of a solution of Itô’s equation with monotone coefficients}, Theory Probab. Appl., 35 (1991), pp. 583-587, . · Zbl 0735.60061 |
| [34] | A. Neuenkirch and L. Szpruch, {\it First order strong approximations of scalar SDEs defined in a domain}, Numer. Math., 128 (2014), pp. 103-136. · Zbl 1306.60075 |
| [35] | H.-L. Ngo and D. Taguchi, {\it Strong Rate of Convergence for the Euler-Maruyama Approximation of Stochastic Differential Equations with Irregular Coefficients}, preprint, 2013, . |
| [36] | S. Sabanis, {\it Euler Approximations with Varying Coefficients: The Case of Superlinearly Growing Diffusion Coefficients}, preprint, 2013, . · Zbl 1352.60101 |
| [37] | S. Sabanis, {\it A note on tamed Euler approximations}, Electron. Comm. Probab., 18 (2013), pp. 1-10. · Zbl 1329.60237 |
| [38] | L. Szpruch, X. Mao, D. Higham, and J. Pan, {\it Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model}, BIT, 51 (2011), pp. 405-425. · Zbl 1230.65011 |
| [39] | L. Yan, {\it The Euler scheme with irregular coefficients}, Ann. Probab., 30 (2002), pp. 1172-1194. · Zbl 1020.60054 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
