Numerical methods for two-dimensional G-heat equation

Abstract

The G-expectation is a sublinear expectation. It is an important tool for pricing financial products and managing risk owing to its ability to deal with model uncertainty. The problem is how to efficiently quantify it since the commonly used Monte Carlo method does not work. Fortunately, the expectation of a G-normal random variable can be linked to the viscosity solution of a fully nonlinear G-heat equation. In this paper, we first identify the limits of the uncertainty in the covariance of a two-dimensional G-normal random variable and determine the corresponding G-heat equation. Then, we propose a novel numerical scheme for solving the two-dimensional G-heat equation and pay more attention to the case where there exists uncertainty on the correlation, especially in the case that the correlation ranges from negative to positive. The scheme is monotone, stable, and convergent. The numerical tests show that the scheme is highly efficient.

Keywords

G-expectation / G-heat equation / model uncertainty / inner iteration / convergence / viscosity solution

Acknowledgements

We are grateful to Professors Shige Peng and Lihe Wang for useful discussions. This work of Yue was partially supported by the NSFC (Grant No. 12371401).

References

[1] Artzner, P., Delbaen, F., Eber, J. M. and Heath, D., Coherent measures of risk, Math. Finance, 1999, 9(3): 203-228.

[2] Barles, G., Daher, C. and Romano, M., Convergence of numerical schemes for parabolic equations arising in finance theory, Math. Mod. Meth. Appl. Sci., 1995, 5(01): 125-143.

[3] Barles, G. and Souganidis, P. E., Convergence of approximation schemes for fully non-linear second order equations, Asymptotic Anal., 1991, 4(3): 271-283.

[4] Bonnans, J. F. and Zidani, H., Consistency of generalized finite difference schemes for the stochastic HJB equation, SIAM J. Numer. Anal., 2003, 41(3): 1008-1021.

[5] Clift, S. S. and Forsyth, P. A., Numerical solution of two asset jump diffusion models for option valuation, Appl. Numer. Math., 2008, 58(6): 743-782.

[6] Debrabant, K. and Jakobsen, E., Semi-Lagrangian schemes for linear and fully non-linear diffusion equations, Math. Comput., 2013, 82(283): 1433-1462.

[7] Forsyth, P. A. and Labahn, G., Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance, J. Comput. Financ., 2007, 11(2): 1-44.

[8] Ma, K. and Forsyth, P. A., An unconditionally monotone numerical scheme for the two factor uncertain volatility model, IMA J. Numer. Anal., 2016, 37(2): 905-944.

[9] Øksendal, B. K. and Sulem, A., Applied stochastic control of jump diffusions, Springer, Berlin, 2005.

[10] Peng, S., G-expectation, G-Brownian motion and related stochastic calculus of Ito’s type, Stochastic Analysis and Applications, The Abel Symposium 2005, Abel Symposia 2, New York: Springer-Verlag, 2006: 541-567.

[11] Peng, S., Nonlinear expectations and stochastic calculus under uncertainty with robust CLT and G-Brownian motion, Springer, Berlin, 2019.

[12] Pooley, D. M., Forsyth, P. A. and Vetzal, K. R., Numerical convergence properties of option pricing pdes with uncertain volatility, IMA J. Numer. Anal., 2003, 23(2): 241-267.

[13] Pooley, D. M., Forsyth, P. A. and Vetzal, K. R., Two factor option pricing with uncertain volatility, in International Conference on Computational Science and Its Applications, Springer, Berlin, Heidelberg, 2003: 158-167.

[14] Krylov, N. V., Controlled diffusion processes, Springer, New York, 1980.

[15] Krylov, N. V., Nonlinear parabolic and elliptic equations of the second order, Reidel, Original Russian version by Nauka, Moscow ( 1985, 1987), 1987.