Fitted strong stability-preserving schemes for the Black-Scholes-Barenblatt equation

References

• L. Angermann and S. Wang, Convergence of a fitted finite volume method for the penalized Black-Sholes equation governing European and American option pricing, Numer. Math. 106 (2007), pp. 1–40.
   Web of Science ®Google Scholar
• M. Avellaneda, A. Levy and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities, Appl. Math. Finance 2 (1995), pp. 73–88.
   Google Scholar
• G. Barles and E.R. Jakobsen, Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations, Math. Comp. 76(260) (2007), pp. 1861–1894.
   Web of Science ®Google Scholar
• G. Barles, C. Daher and M. Romano, Convergence of numerical schemes for parabolic equations arising in finance theory, Math. Methods Appl. Sci. 5(1) (1995), pp. 125–143.
   Web of Science ®Google Scholar
• J.Fr. Bonnans and X. Tan, Monotonicity condition for the θ-scheme for diffusion equations, INRIA Raport de Recherche, No. 7778 (2011) October.
   Google Scholar
• J.H. Bramble, B.E. Hubbard and V. Thomée, Convergence estimates for essentially positive type discrete Dirichlet problems, Math. Comp. 23 (1969), pp. 695–709.
   Web of Science ®Google Scholar
• N. Dokuchaev and A. Savkin, The pricing of options in a financial market model with transaction costs and uncertain volatility, J. Mult. Fin. Man. 8 (1998), pp. 353–364.
   Google Scholar
• M. Ehrhardt, (ed.), Non-linear Models in Mathematical Finance: New Research Trends in Option Pricing, Nova Publishers, Berlin, 2008.
   Google Scholar
• R. Eymard, J. Führmann and K. Gärtner, A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems, Numer. Math. 102 (2006), pp. 463–495.
   Web of Science ®Google Scholar
• X. Feng, R. Glowinski and M. Neilan, Recent developments in numerical methods for fully nonlinear second order partial differential equations, SIAM Review 55(2) (2013), pp. 205–267.
   Web of Science ®Google Scholar
• P.A. Forsyth and G. Labahn, Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance, J. Comp. Fin. 11(2) (2008), pp. 1–43.
   Google Scholar
• R. Frey and U. Polte, Nonlinear Black-Scholes in finance: associated control problems and properties of solutions, SIAM J. Control Optim. 49 (2011), pp. 185–204.
   Web of Science ®Google Scholar
• S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review 43(1) (2001), pp. 89–112.
   Web of Science ®Google Scholar
• S. Gottlieb, D.I. Ketcheson and C.-W. Shu, High order strong stability preserving time discretizations, J. Sci. Comp. 38 (2009), pp. 251–289.
   Web of Science ®Google Scholar
• T. Haentjens and K.J. in 't Hout, Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation, J. Comp. Fin. 16(1) (2012), pp. 83–110.
   Web of Science ®Google Scholar
• W. Hundsdorfer and J. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer-Verlag, Berlin Heidelberg, 2003.
   Google Scholar
• W. Hundsdorfer, A. Mozartova and M.N. Spijker, Stepsize conditions for boundedness in numerical initial value problems, SIAM J. Numer. Anal. 47(5) (2009), pp. 3797–3819.
   Web of Science ®Google Scholar
• N. Ishimura, Remarks on the nonlinear Black-Scholes equations with the effect of transaction costs, Asia-Pacific Finan. Mar. 17 (2010), pp. 241–259.
   Google Scholar
• J.F.B.M. Kraaijevanger, B-convergence of the implicit midpoint rule and the trapezoidal rule, BIT Num. Math. 25(4) (1985), pp. 652–666.
   Web of Science ®Google Scholar
• A.M. Oberman, Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems, SIAM J. Numer. Anal. 44(2) (2006), pp. 879–895.
   Web of Science ®Google Scholar
• J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, London, 1970.
   Google Scholar
• D.M. Pooley, P. Forsyth and K. Vetzal, Numerical convergence properties of option pricing PDEs with uncertain volatility, IMA J. Numer. Anal. 23 (2003), pp. 241–267.
   Web of Science ®Google Scholar
• M. Ruijter and C. Oosterlee, The COS method for pricing options under uncertain volatility, Top. Numer. Meth. Finan., Springer Proceedings in Mathematics & Statistics (M. Cummins, F. Murphy, and J.J.H. Miller, eds.) 19 (2012), pp. 95–113.
   Google Scholar
• P. Wilmott, The Theory and Practice of Financial Engineering, Wiley & Sons, Chichester, UK, 1998.
   Google Scholar
• Y. Zhan, Viscosity solutions of nonlinear degenerate parabolic equations and several applications, Ph.D. thesis, University of Toronto, 1999.
   Google Scholar
• K. Zhang and S. Wang, A computational scheme for uncertain volatility model in option pricing, Appl. Num. Math. 59 (2009), pp. 1754–1767.
   Web of Science ®Google Scholar