We give a study result to analyze a rather different, semi-analytical numerical algorithms based on splitting-step methods with their applications to mathematical finance. As certain subsistent numerical schemes may fail due to producing negative values for financial variables which require non-negativity preserving. These algorithms which we are analyzing preserve not only the non-negativity, but also the character of boundaries (natural, reflecting, absorbing, etc.). The derivatives of the CIR process and the Heston model are being extensively studied. Beyond plain vanilla European options, we creatively apply our splitting-step methods to a path-dependent option valuation. We compare our algorithms to a class of numerical schemes based on Euler discretization which are prevalent currently. The comparisons are given with respect to both accuracy and computational time for the European call option under the CIR model whereas with respect to convergence rate for the path-dependent option under the CIR model and the European call option under the Heston model.
Kloeden, P.E. and Platen, E. (1998) Numerical Solution of Stochastic Differential Equations. Springer, New York.
Kloeden, P.E., Platen, E. and Schurz, H. (1994) Numerical Solution of SDEs through Computer Experiments. Springer, New York. https://doi.org/10.1007/978-3-642-57913-4
Schurz, H. (1997) Stability, Stationarity, and Boundedness of Some Implicit Numerical Methods for Stochastic Differential Equations and Applications. Logos-Verlag, Berlin.
Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985) A Theory of Term Structure of Interest Rates. Econometrica, 53, 385-407.
Heston, S. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Reviews of Financial Studies, 6, 327-343. https://doi.org/10.1093/rfs/6.2.327
Feller, W. (1951) Two Singular Diffusion Problems. Annals of Mathematics, 54, 173-182. https://doi.org/10.2307/1969318
Karlin, S. and Taylor, H.M. (1981) A Second Course in Stochastic Processes. Academic Press, New York.
Moro, E. and Schurz, H. (2007) Boundary Preserving Semi-Analytic Numerical Algorithms for Stochastic Differential Equations. SIAM Journal on Scientific Computing, 29, 1525-1549. https://doi.org/10.1137/05063725X
Glasserman, P. (2003) Monte Carlo Methods in Financial Engineering. Springer Verlag, New York. https://doi.org/10.1007/978-0-387-21617-1
Broadie, M. and Kaya, O. (2004) Exact Simulation of Option Greeks under Stochastic Volatility and Jump Diffusion Models. In: Ingalls, R.G., Rossetti, M.D., Smith, J.S. and Peters, B.A., Eds., Proceedings of the 2004 Winter Simulation Conference, INFORMS, Washington, 1607-1615. https://doi.org/10.1109/WSC.2004.1371506
Broadie, M. and Kaya, O. (2006) Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes. Operations Research, 54, 217-231. https://doi.org/10.1287/opre.1050.0247
Alfonsi, A. (2005) On the Discretization Schemes for the CIR (and Bessel Squared) Processes. Monte Carlo Methods and Applications, 11, 355-384. https://doi.org/10.1515/156939605777438569
Kahl, C. and Schurz, H. (2006) Balanced Milstein Methods for Ordinary SDEs. Monte Carlo Methods and Applications, 12, 143-170. https://doi.org/10.1515/156939606777488842
Dereich, S., Neuenkirch, A. and Szpruch, L. (2012) An Euler-Type Method for the Strong Approximation of the Cox-Ingersoll-Ross Process. Proceedings of the Royal Society of London A, 468, 1105-1115. https://doi.org/10.1098/rspa.2011.0505
Alfonsi, A. (2010) High Order Discretization Schemes for the CIR Process: Application to Affine Term Structure and Heston Models. Mathematics of Computation, American Mathematical Society, 79, 209-237. https://doi.org/10.1090/S0025-5718-09-02252-2
Deelstra, G. and Delbaen, F. (1998) Convergence of Discretized Stochastic (Interest Rate) Processes with Stochastic Drift Term. Applied Stochastic Models and Data Analysis, 14, 77-84. https://doi.org/10.1002/(SICI)1099-0747(199803)14:1 3.0.CO;2-2
Bossy, M. and Diop, A. (2004) An Efficient Discretization Scheme for One Dimensional SDEs with a Diffusion Coefficient Function of the Form. INRIA Working Paper No. 5396.
Berkaoui, A., Bossy, M. and Diop, A. (2008) Euler Scheme for SDEs with Non-Lipchitz Diffusion Coefficient: Strong Convergence. ESAIM Probability and Statistics, 12, 1-11.
Higham, D.J. and Mao, X. (2005) Convergence of Monte Carlo Simulations Involving the Mean-Reverting Square Root Process. Journal of Computational Finance, 8, 35-62.
Lord, R., Koekkoek, R. and van Dijk, D. (2008) A Comparison of Biased Simulation Schemes for Stochastic Volatility Models. Journal of Quantitative Finance, 10, 177-194.
Kahl, C. and Jackel, P. (2006) Fast Strong Approximation Monte-Carlo Schemes for Stochastic Volatility Models. Quantitative Finance, 6, 513-536. https://doi.org/10.1080/14697680600841108
Andersen, L. (2008) Simple and Efficient Simulation of the Heston Stochastic Volatility Model. Journal of Computational Finance, 11, 1-42. https://doi.org/10.21314/JCF.2008.189
Carr, P. and Madan, D. (1999) Option Valuation using the Fast Fourier Transform. Journal of Computational Finance, 2, 61-73. https://doi.org/10.21314/JCF.1999.043
Lewis, A.L. (2001) A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes. http://www.optioncity.net/pubs/ExpLevy.pdf
Lee, R.W. (2004) Option Pricing by Transform Methods: Extensions, Unification and Error Control. Journal of Computational Finance, 7, 51-86. https://doi.org/10.21314/JCF.2004.121
Kahl, C. and Jackel, P. (2005) Not-So-Complex Logarithms in the Heston Model. Wilmott Magazine, 19, 94-103.
Lord, R. and Kahl, C. (2007) Optimal Fourier Inversion in Semi-Analytical Option Pricing. http://papers.tinbergen.nl/06066.pdf
Fang, F. and Oosterlee, C.W. (2008) A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions. SIAM Journal on Scientific Computing, 31, 826-848. https://doi.org/10.1137/080718061
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions. Wiley, New York, 2.
Siegel, A.F. (1979) The Noncentral Chi-Squared Distribution with Zero Degrees of Freedom and Testing for Uniformity. Biometrika, 66, 381-386. https://doi.org/10.1093/biomet/66.2.381
Feller, W. (1971) Probability Theory and Its Applications. 2nd Edition, Wiley, New York.
Gatheral, J. (2006) The Volatility Surface: A Practitioners Guide. John Wiley and Sons, New York.
Makarov, R.N. and Glew, D. (2010) Exact Simulation of Bessel Discussions. Monte Carlo Methods and Applications, 16, 283-306. https://doi.org/10.1515/mcma.2010.010
Shreve, S.E. (1996) Stochastic Calculus and Finance. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.137.6951&rep=rep1&type=pdf
Wilmott, P., Howison, S. and Dewynne, J. (1995) The Mathematics of Financial Derivatives. Cambridge University Press, New York. https://doi.org/10.1017/CBO9780511812545
Merton, R.C. (1969) Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case. Review of Economics and Statistics, 51, 247-257. https://doi.org/10.2307/1926560
Korn, R. and Kraft, H. (2002) A Stochastic Control Approach to Portfolio Problems with Stochastic Interest Rates. SIAM Journal on Control and Optimization, 40, 1250-1269. https://doi.org/10.1137/S0363012900377791