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A note on the numerical resolution of Heston PDEs. (English) Zbl 1466.65060

This paper considered the numerical solution of a PDE for Heston model’s dynamics. Two numerical methods: the alternating direction implicit method and the radial basis functions method are applied the solve the PDE. Comparisons of the two schemes in the case of a vanilla are discussed.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65D05 Numerical interpolation
65D12 Numerical radial basis function approximation
91G20 Derivative securities (option pricing, hedging, etc.)
91G60 Numerical methods (including Monte Carlo methods)

Software:

Matlab
Full Text: DOI

References:

[1] Hull, JC, Options Futures and Other Derivatives (2003), Bengaluru: Pearson Education India, Bengaluru · Zbl 1087.91025
[2] Dewynne, J.; Wilmott, P., Partial to the exotic, Risk, 6, 3, 38-46 (1993)
[3] Hull, J.; White, A., Efficient procedures for valuing European and American path-dependent options, J. Deriv., 1, 1, 21-31 (1993) · doi:10.3905/jod.1993.407869
[4] Kat, HM; Verdonk, LT, Tree surgery. Risk Mag., 8, 2, 53-56 (1995)
[5] Douglas, J., Alternating direction methods for three space variables, Numerische Mathematik, 4, 1, 41-63 (1962) · Zbl 0104.35001 · doi:10.1007/BF01386295
[6] Ballestra, LV; Pacelli, G., Pricing European and American options with two stochastic factors: a highly efficient radial basis function approach, J. Econ. Dyn. Control, 37, 6, 1142-1167 (2013) · Zbl 1402.91887 · doi:10.1016/j.jedc.2013.01.013
[7] Heston, SL, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6, 2, 327-343 (1993) · Zbl 1384.35131 · doi:10.1093/rfs/6.2.327
[8] De Marchi, S., Perracchione, E.: Lectures on radial basis functions (preprint)
[9] Campagna, R.; Cuomo, S.; De Marchi, S.; Perracchione, E.; Severino, G., A stable meshfree pde solver for source-type flows in porous media, Appl. Numer. Math., 149, 30-42 (2020) · Zbl 1440.65242 · doi:10.1016/j.apnum.2019.08.015
[10] Black, F.; Scholes, M., The pricing of options and corporate liabilities, J. Polit. Econ., 81, 637-654 (1973) · Zbl 1092.91524 · doi:10.1086/260062
[11] Fasshauer, G., Meshfree Approximation Methods with Matlab. Interdisciplinary Mathematical Sciences (2007), Singapore: World Scientific Publishers, Singapore · Zbl 1123.65001
[12] Kansa, E., Multiquadrics: a scattered data approximation scheme with applications to computational fluid-dynamics, part i and part ii, Comput. Math. Appl., 19, 127-145, 147-161 (1990) · Zbl 0850.76048 · doi:10.1016/0898-1221(90)90271-K
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