Radial basis function partition of unity operator splitting method for pricing multi-asset American options
Tóm tắt
The operator splitting method in combination with finite differences has been shown to be an efficient approach for pricing American options numerically. Here, the operator splitting formulation is extended to the radial basis function partition of unity method. An approach that has previously often been used together with radial basis function methods to deal with the free boundary arising in American option pricing is to solve a penalised version of the Black–Scholes equation. It is shown that the operator splitting technique outperforms the penalty approach when used with the radial basis function partition of unity method. Numerical experiments are performed for one, two and three underlying assets. The advantage of the operator splitting technique grows with the number of dimensions.
Từ khóa
Tài liệu tham khảo
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973). doi:10.1086/260062
Cuthill, E., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. In: Proceedings of the 24th ACM National Conference, pp. 157–172. ACM, New York (1969). doi:10.1145/800195.805928
Fasshauer, G.E., Khaliq, A.Q.M., Voss, D.A.: Using mesh free approximation for multi-asset American option problems. J. Chin. Inst. Eng. 27(4), 563–571 (2004)
Fichera, G.: Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine. Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I. VIII, Ser. 5, 3–30 (1956)
Golub, G.H., Van Loan, C.F.: Matrix computations. In: Johns Hopkins Studies in the Mathematical Sciences, 4th edn. Johns Hopkins University Press, Baltimore (2013)
Hairer, E., Nørsett, S., Wanner, G.: Solving ordinary differential equations I. In: Nonstiff Problems, 2nd edn. Springer, Berlin (2000). doi:10.1007/978-3-540-78862-1
Hon, Y.C.: A quasi-radial basis functions method for American options pricing. Comput. Math. Appl. 43(3–5), 513–524 (2002). doi:10.1016/S0898-1221(01)00302-9
Hull, J.: Options. In: Futures and Other Derivatives. Pearson Prentice Hall, Upper Saddle River (2009)
Ikonen, S., Toivanen, J.: Operator splitting methods for American option pricing. Appl. Math. Lett. 17(7), 809–814 (2004). doi:10.1016/j.aml.2004.06.010
Ikonen, S., Toivanen, J.: Operator splitting methods for pricing American options under stochastic volatility. Numer. Math. 113(2), 299–324 (2009). doi:10.1007/s00211-009-0227-5
Larsson, E., Åhlander, K., Hall, A.: Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform. J. Comput. Appl. Math. 222(1), 175–192 (2008). doi:10.1016/j.cam.2007.10.039
Lindström, E., Ströjby, J., Brodén, M., Wiktorsson, M., Holst, J.: Sequential calibration of options. Comput. Stat. Data Anal. 52(6), 2877–2891 (2008). doi:10.1016/j.csda.2007.08.009
Nielsen, B.F., Skavhaug, O., Tveito, A.: Penalty and front-fixing methods for the numerical solution of American option problems. J. Comput. Finan. 5(4), 69–97 (2002)
Nielsen, B.F., Skavhaug, O., Tveito, A.: Penalty methods for the numerical solution of American multi-asset option problems. J. Comput. Appl. Math. 222(1), 3–16 (2008). doi:10.1016/j.cam.2007.10.041
Rieger, C., Zwicknagl, B.: Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning. Adv. Comput. Math. 32(1), 103–129 (2010). doi:10.1007/s10444-008-9089-0
Ruijter, M.J., Oosterlee, C.W.: Two-dimensional Fourier cosine series expansion method for pricing financial options. SIAM J. Sci. Comput. 34(5), B642–B671 (2012). doi:10.1137/120862053
Safdari-Vaighani, A., Heryudono, A., Larsson, E.: A radial basis function partition of unity collocation method for convection-diffusion equations. J. Sci. Comput. 64(2), 341–367 (2015). doi:10.1007/s10915-014-9935-9
Shcherbakov, V., Larsson, E.: Radial basis function partition of unity methods for pricing vanilla basket options. Comput. Math. Appl. 71(1), 185–200 (2016). doi:10.1016/j.camwa.2015.11.007
Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 23rd ACM National Conference, pp. 517–524. ACM, New York (1968). doi:10.1145/800186.810616
von Sydow, L., Höök, L.J., Lindström, E., Milovanović, S., Persson, J., Shcherbakov, V., Shpolyanskiy, Y., Samuel, S., Toivanen, J., Waldén, J., Wiktorsson, M., Levesley, J., Li, J., Oosterlee, C., Ruijter, M., Toropov, A., Zhao, Y.: BENCHOP—the BENCHmarking project in option pricing. Int. J. Comput. Math. 92(12), 2361–2379 (2015). doi:10.1080/00207160.2015.1072172
Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4(4), 389–396 (1995). doi:10.1007/BF02123482