Geodesic
Approximation of the solution and its derivative for the singularly perturbed Black–Scholes equation with nonsmooth initial data
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 3, pp. 460-480 
S. Li; G. I. Shishkin; L. P. Shishkina. Approximation of the solution and its derivative for the singularly perturbed Black–Scholes equation with nonsmooth initial data. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 3, pp. 460-480. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_3_a8/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A problem for the Black–Scholes equation that arises in financial mathematics is reduced, by a transformation of variables, to the Cauchy problem for a singularly perturbed parabolic equation with the variables $x$, $t$ and a perturbation parameter $\varepsilon$, $\varepsilon\in(0,1]$. This problem has several singularities such as the unbounded domain, the piecewise smooth initial function (its first-order derivative in $x$ has a discontinuity of the first kind at the point $x=0$), an interior (moving in time) layer generated by the piecewise smooth initial function for small values of the parameter $\varepsilon$, etc. In this paper, a grid approximation of the solution and its first-order derivative is studied in a finite domain including the interior layer. On a uniform mesh, using the method of additive splitting of a singularity of the interior layer type, a special difference scheme is constructed that allows us to $\varepsilon$-uniformly approximate both the solution to the boundary value problem and its first-order derivative in $x$ with convergence orders close to 1 and 0.5, respectively. The efficiency of the constructed scheme is illustrated by numerical experiments.

[1] P. Wilmott, S. Howison, and J. Dewynne, The mathematics of financial derivatives, Cambridge Univ. Press, Cambridge, 2002 | MR

[2] G. I. Shishkin, “A difference scheme for a singularly perturbed equation of parabolic type with a discontinuous initial condition”, Soviet Math. Dokl., 37:3 (1988), 792–796 | MR | Zbl

[3] G. I. Shishkin, “A difference scheme for a singularly perturbed equation of parabolic type with a discontinuous boundary condition”, Comput. Math. and Math. Phys., 28:6 (1988), 32–41 | DOI | MR | Zbl

[4] P. W. Hemker and G. I. Shishkin, “Discrete approximation of singularly perturbed parabolic PDEs with a discontinuous initial condition”, Comput. Fluid Dynamics J., 2:4 (1994), 375–392

[5] G. I. Shishkin, “Approximation of solutions and diffusion flows in the case of singularly perturbed boundary value problems with discontinuous initial condition”, Comput. Math. and Math. Phys., 36:9 (1996), 1233–1250 | MR | Zbl

[6] G. I. Shishkin, “Singularly perturbed boundary value problems with concentrated sources and discontinuous initial conditions”, Comput. Math. and Math. Phys., 37:4 (1997), 417–434 | MR | Zbl

[7] V. L. Kolmogorov and G. I. Shishkin, “Numerical methods for singularly perturbed boundary value problems modelling diffusion processes”, Singular Perturbation Problems in Chemical Physics, Advances in Chemical Physics Series, XCVII, ed. J. J. H. Miller, J. Wiley Sons, 1997, 181–362

[8] G. I. Shishkin, “Grid approximations of parabolic convection-diffusion equations with piecewise smooth initial conditions”, Dokl. Math., 72:3 (2005), 850–853 | MR | Zbl

[9] G. I. Shishkin, “Grid approximation of singularly perturbed parabolic convection-diffusion equations subject to a piecewise smooth initial condition”, Comput. Math. and Math. Phys., 46:1 (2006), 49–72 | DOI | MR | Zbl

[10] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and quasilinear equations of parabolic type, Nauka, Moscow, 1967

[11] P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan, and G. I. Shishkin, Robust computational techniques for boundary layers, Chapman and Hall/CRC, Boca Raton, 2000 | MR | Zbl

[12] S. Li, D. E. Creamer, and G. I. Shishkin, “On numerical methods for a singularly perturbed Black–Scholes equation with nonsmooth initial data”, Proceedings of the International Conference on Computational Mathematics ICCM'2004 (Novosibirsk, June 2004), ICM MG Publisher, Novosibirsk, 2004; Part II, 896–900

[13] S. Li, D. E. Creamer, and G. I. Shishkin, “Discrete approximation of a singularly perturbed Black–Scholes equation with nonsmooth initial data”, An International Conference on Boundary and Interior Layers – Computational and Asymptotic Methods BAIL 2004 (ONERA, Toulouse, 5th–9th July, 2004), 441–446

[14] A. A. Samarskii, Theory of difference schemes, Marcel Dekker, Inc., New York, 2001 | MR | Zbl

[15] G. I. Shishkin, Grid approximations of singularly perturbed elliptic and parabolic equations, Ural Branch of Russian Acad. Sci., Ekaterinburg, 1992 | Zbl

[16] J. J. H. Miller, E. O'Riordan, and G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific, Singapore, 1996 | MR