# American Institute of Mathematical Sciences

October  2013, 18(8): 2083-2100. doi: 10.3934/dcdsb.2013.18.2083

## Convergence, non-negativity and stability of a new Milstein scheme with applications to finance

 1 Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, United Kingdom 2 Department of Statistics and Modelling Science, University of Strathclyde, Glasgow, G1 1XH, Scotland 3 School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3JZ, United Kingdom

Received  April 2012 Revised  February 2013 Published  July 2013

We propose and analyse a new Milstein type scheme for simulating stochastic differential equations (SDEs) with highly nonlinear coefficients. Our work is motivated by the need to justify multi-level Monte Carlo simulations for mean-reverting financial models with polynomial growth in the diffusion term. We introduce a double implicit Milstein scheme and show that it possesses desirable properties. It converges strongly and preserves non-negativity for a rich family of financial models and can reproduce linear and nonlinear stability behaviour of the underlying SDE without severe restriction on the time step. Although the scheme is implicit, we point out examples of financial models where an explicit formula for the solution to the scheme can be found.
Citation: Desmond J. Higham, Xuerong Mao, Lukasz Szpruch. Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2083-2100. doi: 10.3934/dcdsb.2013.18.2083
##### References:
 [1] Assyr Abdulle and Adrian Blumenthal, Stabilized multilevel Monte Carlo method for stiff stochastic differential equations,, EPFL-ARTICLE-183502, (2013). doi: 10.1016/j.jcp.2013.05.039. [2] D. H. Ahn and B. Gao, A parametric nonlinear model of term structure dynamics,, Review of Financial Studies, 12 (1999). doi: 10.1093/rfs/12.4.721. [3] Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate,, Review of Financial Studies, 9 (1996), 385. [4] J. A. D. Appleby, M. Guzowska, C. Kelly and A. Rodkina, Preserving positivity in solutions of discretised stochastic differential equations,, Applied Mathematics and Computation, 217 (2010), 763. doi: 10.1016/j.amc.2010.06.015. [5] E. Buckwar and T. Sickenberger, A comparative linear mean-square stability analysis of Maruyama-and Milstein-type methods,, Mathematics and Computers in Simulation, 81 (2011), 1110. doi: 10.1016/j.matcom.2010.09.015. [6] T. C. Gard, "Introduction to Stochastic Differential Equations,", Marcel Dekker, (1988). [7] M. Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme,, Monte Carlo and Quasi-Monte Carlo Methods, 2006 (2008), 343. doi: 10.1007/978-3-540-74496-2_20. [8] M. B. Giles, Multilevel Monte Carlo path simulation,, Operations Research-Baltimore, 56 (2008), 607. doi: 10.1287/opre.1070.0496. [9] J. Goard and M. Mazur, "Stochastic Volatility Models and the Pricing of Vix Options,", Mathematical Finance, (2011). doi: 10.1111/j.1467-9965.2011.00506.x. [10] I. Gyöngy, A note on Euler's approximations,, Potential Analysis, 8 (1998), 205. doi: 10.1023/A:1008605221617. [11] S. L. Heston, "A Simple New Formula for Options with Stochastic Volatility,", Course Notes of Washington University in St. Louis, (1997). [12] D. J. Higham, A-stability and stochastic mean-square stability,, BIT Numerical Mathematics, 40 (2000), 404. doi: 10.1023/A:1022355410570. [13] D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method,, SIAM Journal on Numerical Analysis, 38 (2000), 753. doi: 10.1137/S003614299834736X. [14] 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,, Proceedings of the Royal Society A, 467 (2011), 1563. [15] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients,, The Annals of Applied Probability, 22 (2012), 1611. doi: 10.1214/11-AAP803. [16] A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients,, Numerische Mathematik, 112 (2009), 41. doi: 10.1007/s00211-008-0200-8. [17] C. Kahl, M. Gunther and T. Rosberg, Structure preserving stochastic integration schemes in interest rate derivative modeling,, Applied Numerical Mathematics, 58 (2008), 284. doi: 10.1016/j.apnum.2006.11.013. [18] C. Kahl and H. Schurz, Balanced Milstein methods for ordinary SDEs,, Monte Carlo Methods and Applications, 12 (2006), 143. doi: 10.1515/156939606777488842. [19] I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus,", Springer, (1991). doi: 10.1007/978-1-4612-0949-2. [20] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,", Springer, (1992). [21] A. L. Lewis, "Option Valuation Under Stochastic Volatility,", Finance Press, (2000). [22] R. S. Liptser and A. N. Shiryayev, "Theory of Martingales,", Kluwer Academic Publishers, (1989). doi: 10.1007/978-94-009-2438-3. [23] R. Lord, R. Koekkoek and D. J. C. Van Dijk, A comparison of biased simulation schemes for stochastic volatility models,, Quantitative Finance, 10 (2010), 177. doi: 10.1080/14697680802392496. [24] X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales,", Longman Scientific & Technical, (1991). [25] X. Mao, "Stochastic Differential Equations and Their Applications,", Horwood Pub Ltd, (1997). [26] X. Mao and L. Szpruch, Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients,, Stochastics, 85 (2012), 144. doi: 10.1080/17442508.2011.651213. [27] X. Mao and L. Szpruch, Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients,, J. Comput. Appl. Math., 238 (2013), 14. doi: 10.1016/j.cam.2012.08.015. [28] G. N. Milstein, E. Platen and H. Schurz, Balanced implicit methods for stiff stochastic systems,, SIAM Journal on Numerical Analysis, 35 (1998), 1010. doi: 10.1137/S0036142994273525. [29] G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics. Scientific Computation,", Springer-Verlag, (2004). [30] Tetsuya Misawa, A lie algebraic approach to numerical integration of stochastic differential equations,, SIAM Journal on Scientific Computing, 23 (2001), 866. doi: 10.1137/S106482750037024X. [31] A. Neuenkirch and L. Szpruch, First order strong approximations of scalar sdes with values in a domain,, Preprint. , (). [32] H. Schurz, Convergence and stability of balanced implicit methods for systems of SDEs,, Int. J. Numer. Anal. Model, 2 (2005), 197. [33] Y. Shen, Q. Luo and X. Mao, The improved LaSalle-type theorems for stochastic functional differential equations,, Journal of Mathematical Analysis and Applications, 318 (2006), 134. doi: 10.1016/j.jmaa.2005.05.026. [34] A. N. Shiryaev, "Probability,", Springer-Verlag, (1996). [35] L. Szpruch, X. Mao, D. J. Higham and J. Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia type interest rate model,, BIT Numerical Mathematics, 51 (2011), 405. doi: 10.1007/s10543-010-0288-y. [36] X. Wang, S. Gan and D. Wang, A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise,, BIT, 52 (2012), 741. doi: 10.1007/s10543-012-0370-8. [37] E. Zeidler, "Nonlinear Functional Analysis and Its Applications,", Springer Verlag, (1990).

show all references

##### References:
 [1] Assyr Abdulle and Adrian Blumenthal, Stabilized multilevel Monte Carlo method for stiff stochastic differential equations,, EPFL-ARTICLE-183502, (2013). doi: 10.1016/j.jcp.2013.05.039. [2] D. H. Ahn and B. Gao, A parametric nonlinear model of term structure dynamics,, Review of Financial Studies, 12 (1999). doi: 10.1093/rfs/12.4.721. [3] Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate,, Review of Financial Studies, 9 (1996), 385. [4] J. A. D. Appleby, M. Guzowska, C. Kelly and A. Rodkina, Preserving positivity in solutions of discretised stochastic differential equations,, Applied Mathematics and Computation, 217 (2010), 763. doi: 10.1016/j.amc.2010.06.015. [5] E. Buckwar and T. Sickenberger, A comparative linear mean-square stability analysis of Maruyama-and Milstein-type methods,, Mathematics and Computers in Simulation, 81 (2011), 1110. doi: 10.1016/j.matcom.2010.09.015. [6] T. C. Gard, "Introduction to Stochastic Differential Equations,", Marcel Dekker, (1988). [7] M. Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme,, Monte Carlo and Quasi-Monte Carlo Methods, 2006 (2008), 343. doi: 10.1007/978-3-540-74496-2_20. [8] M. B. Giles, Multilevel Monte Carlo path simulation,, Operations Research-Baltimore, 56 (2008), 607. doi: 10.1287/opre.1070.0496. [9] J. Goard and M. Mazur, "Stochastic Volatility Models and the Pricing of Vix Options,", Mathematical Finance, (2011). doi: 10.1111/j.1467-9965.2011.00506.x. [10] I. Gyöngy, A note on Euler's approximations,, Potential Analysis, 8 (1998), 205. doi: 10.1023/A:1008605221617. [11] S. L. Heston, "A Simple New Formula for Options with Stochastic Volatility,", Course Notes of Washington University in St. Louis, (1997). [12] D. J. Higham, A-stability and stochastic mean-square stability,, BIT Numerical Mathematics, 40 (2000), 404. doi: 10.1023/A:1022355410570. [13] D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method,, SIAM Journal on Numerical Analysis, 38 (2000), 753. doi: 10.1137/S003614299834736X. [14] 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,, Proceedings of the Royal Society A, 467 (2011), 1563. [15] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients,, The Annals of Applied Probability, 22 (2012), 1611. doi: 10.1214/11-AAP803. [16] A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients,, Numerische Mathematik, 112 (2009), 41. doi: 10.1007/s00211-008-0200-8. [17] C. Kahl, M. Gunther and T. Rosberg, Structure preserving stochastic integration schemes in interest rate derivative modeling,, Applied Numerical Mathematics, 58 (2008), 284. doi: 10.1016/j.apnum.2006.11.013. [18] C. Kahl and H. Schurz, Balanced Milstein methods for ordinary SDEs,, Monte Carlo Methods and Applications, 12 (2006), 143. doi: 10.1515/156939606777488842. [19] I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus,", Springer, (1991). doi: 10.1007/978-1-4612-0949-2. [20] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,", Springer, (1992). [21] A. L. Lewis, "Option Valuation Under Stochastic Volatility,", Finance Press, (2000). [22] R. S. Liptser and A. N. Shiryayev, "Theory of Martingales,", Kluwer Academic Publishers, (1989). doi: 10.1007/978-94-009-2438-3. [23] R. Lord, R. Koekkoek and D. J. C. Van Dijk, A comparison of biased simulation schemes for stochastic volatility models,, Quantitative Finance, 10 (2010), 177. doi: 10.1080/14697680802392496. [24] X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales,", Longman Scientific & Technical, (1991). [25] X. Mao, "Stochastic Differential Equations and Their Applications,", Horwood Pub Ltd, (1997). [26] X. Mao and L. Szpruch, Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients,, Stochastics, 85 (2012), 144. doi: 10.1080/17442508.2011.651213. [27] X. Mao and L. Szpruch, Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients,, J. Comput. Appl. Math., 238 (2013), 14. doi: 10.1016/j.cam.2012.08.015. [28] G. N. Milstein, E. Platen and H. Schurz, Balanced implicit methods for stiff stochastic systems,, SIAM Journal on Numerical Analysis, 35 (1998), 1010. doi: 10.1137/S0036142994273525. [29] G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics. Scientific Computation,", Springer-Verlag, (2004). [30] Tetsuya Misawa, A lie algebraic approach to numerical integration of stochastic differential equations,, SIAM Journal on Scientific Computing, 23 (2001), 866. doi: 10.1137/S106482750037024X. [31] A. Neuenkirch and L. Szpruch, First order strong approximations of scalar sdes with values in a domain,, Preprint. , (). [32] H. Schurz, Convergence and stability of balanced implicit methods for systems of SDEs,, Int. J. Numer. Anal. Model, 2 (2005), 197. [33] Y. Shen, Q. Luo and X. Mao, The improved LaSalle-type theorems for stochastic functional differential equations,, Journal of Mathematical Analysis and Applications, 318 (2006), 134. doi: 10.1016/j.jmaa.2005.05.026. [34] A. N. Shiryaev, "Probability,", Springer-Verlag, (1996). [35] L. Szpruch, X. Mao, D. J. Higham and J. Pan, Numerical simulation of a strongly nonlinear Ait-Sahalia type interest rate model,, BIT Numerical Mathematics, 51 (2011), 405. doi: 10.1007/s10543-010-0288-y. [36] X. Wang, S. Gan and D. Wang, A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise,, BIT, 52 (2012), 741. doi: 10.1007/s10543-012-0370-8. [37] E. Zeidler, "Nonlinear Functional Analysis and Its Applications,", Springer Verlag, (1990).
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