A case study on pricing foreign exchange options using the modified Craig–Sneyd ADI scheme

References

• L.B. Andersen and V.V. Pieterbarg, Interest Rate Modeling Volume I: Foundations and Vanilla Models, 1st ed., Atlantic Financial Press, London, 2010.
   Google Scholar
• Z. Bučková, M. Ehrhardt and M. Günther, Fichera Theory and its Application in Finance. Proceedings of the 18th European Conference on Mathematics for Industry, June 9–13, 2014, Taormina, Italy, Springer Verlag, Berlin, 2015.
   Google Scholar
• I.J. Clark, Foreign Exchange Option Pricing – A Practitioner's Guide, Wiley, West Sussex, 2011.
   Google Scholar
• I.J. Craig and A.D. Sneyd, An alternating-direction implicit scheme for parabolic equations with mixed derivatives, Compt. Math. Appl. 16 (1988), pp. 341–350. doi: 10.1016/0898-1221(88)90150-2
   Web of Science ®Google Scholar
• D-M. Dang, C. Christara, K.R. Jackson and A. Lakhani, An efficient numerical PDE approach for pricing foreign exchange interest rate hybrid derivatives, J. Comput. Financ. 18 (2015), pp. 39–93. doi: 10.21314/JCF.2015.303
   Web of Science ®Google Scholar
• J. Douglas Jr., Alternating direction methods for three space variables, Numer. Math 4 (1962), pp. 41–63. doi: 10.1007/BF01386295
   Google Scholar
• D. Duffy, Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, Wiley Finance, West Sussex, 2006.
   Google Scholar
• W. Feller, Two singular diffusion problems, Ann. Math. 54(1) (1951), pp. 173–182. doi: 10.2307/1969318
   Web of Science ®Google Scholar
• G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order, in boundary value problems, R.E. Langer, ed., University of Wisconsin Press, Madison (1960), pp. 97–120.
   Google Scholar
• T. Haentjens and K.J. in'tHout, Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation, J. Comp. Finan. 16 (2012), pp. 83–110. doi: 10.21314/JCF.2012.244
   Web of Science ®Google Scholar
• T. Haentjens and K.J. in 't Hout, ADI schemes for pricing American options under the Heston model, Appl. Math. Finan 22 (2015), pp. 207–237. doi: 10.1080/1350486X.2015.1009129
   Google Scholar
• C. Hendricks, C. Heuer, M. Ehrhardt and M. Günther, High-order ADI finite difference schemes for parabolic equations in the combination technique with application in finance, J. Comput. Appl. Math 316 (2017), pp. 175–194. doi: 10.1016/j.cam.2016.08.044
   Web of Science ®Google Scholar
• S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Fin. Stud. 6(2) (1993), pp. 327–343. doi: 10.1093/rfs/6.2.327
   Web of Science ®Google Scholar
• W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer-Verlag, Berlin, 2003.
   Google Scholar
• S. Ikonen and J. Toivanen, Componentwise splitting methods for pricing American options under stochastic volatility, Int. J. Theor. Appl. Finan. 10 (2007), pp. 331–361. doi: 10.1142/S0219024907004202
   Google Scholar
• K.J. in 't Hout and S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation, Int. J. Num. Anal. Mod 7 (2010), pp. 303–320.
   Web of Science ®Google Scholar
• K.J. in 't Hout and C. Mishra, Stability of the modified Craig–Sneyd scheme for two-dimensional convection-diffusion equations with mixed derivative term, Math. Comp. Simul. 81 (2011), pp. 2540–2548. doi: 10.1016/j.matcom.2011.04.004
   Web of Science ®Google Scholar
• K.J. in 't Hout and C. Mishra, Stability of ADI schemes for multidimensional diffusion equations with mixed derivative terms, Appl. Numer. Math. 74 (2013), pp. 83–94. doi: 10.1016/j.apnum.2013.07.003
   Web of Science ®Google Scholar
• K.J. in 't Hout and J. Toivanen, ADI schemes for valuing European options under the bates model, Appl. Numer. Math. 130 (2018), pp. 143–156. doi: 10.1016/j.apnum.2018.04.003
   Web of Science ®Google Scholar
• K.J. in 't Hout and B.D. Welfert, Stability of ADI schemes applied to convection-diffusion equations with mixed derivative terms, Appl. Num. Math. 57 (2007), pp. 19–35. doi: 10.1016/j.apnum.2005.11.011
   Web of Science ®Google Scholar
• K.J. in 't Hout and B.D. Welfert, Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms, Appl. Num. Math. 59 (2009), pp. 677–692. doi: 10.1016/j.apnum.2008.03.016
   Web of Science ®Google Scholar
• A. Itkin and P. Carr, Jumps without tears: A new splitting technology for barrier options., Int. J. Numer. Anal. Mod., 8 (2011), pp. 667–704.
   Web of Science ®Google Scholar
• A. Lipton, Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach, World Scientific, Singapore, 2001.
   Google Scholar
• C. Mishra, A new stability result for the modified Craig–Sneyd scheme applied to two-dimensional convection-diffusion equations with mixed derivatives, Appl. Math. Comput., 285 (2016), pp. 41–50.
   Web of Science ®Google Scholar
• K.W. Morton and D. Mayers, Numerical Solution of Partial Differential Equations, 2nd ed., Cambridge University Press, Cambridge, 2005.
   Google Scholar
• R. Rannacher, Finite element solution of diffusion problems with irregular data, Numer. Math. 43 (1984), pp. 309–327. doi: 10.1007/BF01390130
   Web of Science ®Google Scholar
• D. Tavella and C. Randall, Pricing Financial Instruments, Wiley, New York, 2000.
   Google Scholar
• G. Winkler, T. Apel and U. Wystup, Valuation of options in Heston stochastic volatility model using finite element method, in Foreign Exchange Risk, J. Hakala & U.P. Wystup, eds., Risk Books, London (2002), pp. 283–303.
   Google Scholar
• S-P. Zhu and W-T. Chen, A predictor-corrector scheme based on the ADI method for pricing American puts with stochastic volatility, Comput. Math. Appl., 62 (2011), pp. 1–26. doi: 10.1016/j.camwa.2011.03.101
   Web of Science ®Google Scholar