Numerical solutions of Black-Scholes integro-differential equations with convergence analysis
Numerical solutions of Black-Scholes integro-differential equations with convergence analysis
Stochastic integro-differential equations are obtained when we consider prices jump in financial modelling.In this paper, these equations are solved numerically by applying the two-dimensional Tau method with ordinary bases.Next, the numerical solutions of the equations above are investigated by the ordinary bases to the Hermitian one.Moreover, we provide an error analysis for the Tau method with ordinary bases. Also, we will prove that the errorsof the approximate solutions decay exponentially in weighted $L^2$-norm. At the end, we will provide some numericalexamples which show the efficiency and accuracy of the method
___
- [1] Hull J, White A. The pricing of options on assets with stochastic volatilities. The Journal of Finance. 1987; 42 (2):
281-300.
- [2] Ackerer D, Filipović D, Pulido S. The Jacobi stochastic volatility model. Finance and Stochastics. 2017; 22 (3):
1-34.
- [3] Barles G, Soner HM. Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance and
Stochastics. 1998; 2 (4): 369-397.
- [4] Davis MH, Panas VG, Zariphopoulou T. European option pricing with transaction costs. SIAM Journal on Control
and Optimization. 1993; 31 (2): 470-493.
- [5] Merton RC. On the pricing of corporate debt: the risk structure of interest rates. The Journal of Finance. 1974;
29 (2): 449-470.
- [6] Liang JR, Wang J, Zhang WJ, Qiu WY, Ren FY. Option pricing of a bi-fractional Black–Merton–Scholes model
with the Hurst exponent H in [12, 1]. Applied Mathematics Letters. 2010; 23 (8): 859-863.
- [7] Björk T, Hult H. A note on Wick products and the fractional Black-Scholes model. Finance and Stochastics. 2005;
9 (2): 197-209.
- [8] Duan JS, Lu L, Chen L, An YL. Fractional model and solution for the Black-Scholes equation. Mathematical
Methods in the Applied Sciences. 2018; 41 (2): 697-704.
- [9] Merton RC. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics. 1976;
3 (1-2): 125-144.
- [10] Briani M, La Chioma C, Natalini R. Convergence of numerical schemes for viscosity solutions to integro-differential
degenerate parabolic problems arising in financial theory. Numerische Mathematik. 2004; 98 (4): 607-646.
- [11] Cont R, Voltchkova E. A finite difference scheme for option pricing in jump diffusion and exponential Lévy models.
SIAM Journal on Numerical Analysis. 2005; 43 (4): 1596-1626.
- [12] Matache AM, Schwab C, Wihler TP. Fast numerical solution of parabolic integrodifferential equations with
applications in finance. SIAM Journal on Scientific Computing. 2005; 27 (2): 369-393.
- [13] Matache AM, Von Petersdorff T, Schwab C. Fast deterministic pricing of options on Lévy driven assets. ESAIM:
Mathematical Modelling and Numerical Analysis. 2004; 38 (1): 37-71.
- [14] Sulaiman T, Yokus A, Gulluoglu N, Baskonus H, Bulut H. Regarding the Numerical and Stability Analysis of the
Sharma-Tosso-Olver Equation. In: 3rd International Conference on Computational Mathematics and Engineering
Sciences-(CMES2018), Girne/Cyprus; 2018.
- [15] Patel KS, Mehra M. Compact finite difference method for pricing European and American options under jumpdiffusion
models. arXiv preprint arXiv:180409043. 2018;.
- [16] Rambeerich N, Pantelous AA. A high order finite element scheme for pricing options under regime switching jump
diffusion processes. Journal of Computational and Applied Mathematics. 2016;300:83-96.
- [17] Gencoglu MT, Baskonus HM, Bulut H. Numerical simulations to the nonlinear model of interpersonal relationships
with time fractional derivative. In: AIP Conference Proceedings. vol. 1798. AIP Publishing; 2017. p. 020103.
- [18] Yokus A, Baskonus HM, Sulaiman TA, Bulut H. Numerical simulation and solutions of the two-component second
order KdV evolutionarysystem. Numerical Methods for Partial Differential Equations. 2018; 34 (1): 211-227.
- [19] Veeresha P, Prakasha D, Baskonus HM. Novel simulations to the time-fractional Fisher’s equation. Mathematical
Sciences. 2019; 13: 1-10.
- [20] Ortiz EL, Samara H. An operational approach to the Tau method for the numerical solution of non-linear differential
equations. Computing. 1981; 27 (1): 15-25.
- [21] Tari A. Modified homotopy perturbation method for solving two-dimensional Fredholm integral equations. Int J
Comput Appl Math. 2010; 5 (5): 585-593.
- [22] Xiu D, Karniadakis GE. The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM Journal
on Scientific Computing. 2002; 24 (2): 619-644.
- [23] Adams R, Fournier J. Sobolev Spaces. New York, NY, USA: Academic Press, 1975.
- [24] Guo By. Error estimation of Hermite spectral method for nonlinear partial differential equations. Mathematics of
Computation of the American Mathematical Society. 1999; 68 (227): 1067-1078.
- [25] Shen TT, Zhang ZQ, Ma HP. Optimal error estimates of the Legendre tau method for second-order differential
equations. Journal of Scientific Computing. 2010; 42 (2): 198-215.