Abstract
We analyze a model equation arising in option pricing. This model equation takes the form of a nonlinear, nonlocal diffusion equation. We prove the well posedness of the Cauchy problem for this equation. Furthermore, we introduce a semidiscrete difference scheme and show its rate of convergence.
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Abergel, F., Tachet, R.: A nonlinear partial integro-differential equation from mathematical finance. Discrete Contin. Dyn. Syst. 27(3), 907–917 (2010)
Black, F., Scholes, M.: The effects of dividend yield and dividend policy on common stock prices and returns. J. Financ. Econ. 1(1), 1–22 (1974)
Coclite, G.M., Reichmann, O., Risebro, N.H.: A convergent difference scheme for a class of partial integro-differential equations modeling pricing under uncertainty. SIAM J. Numer. Anal. 54(2), 588–605 (2016)
Guyon, J., Henry-Labordere, P.: The smile calibration problem solved. Available at SSRN. 1885032 (2011)
Henry-Labordere, P.: Calibration of local stochastic volatility models to market smiles: a Monte-Carlo approach, Risk Mag. (2009)
Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)
Hull, J., White, A.: The general Hull–White model and supercalibration. Financ. Anal. J. 57(6), 34–43 (2001)
Lieb, E.H., Loss, M.: Analysis (Graduate Studies in Mathematics), vol. 14, 2nd edn. American Mathematical Society, Providence (2001)
Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. (4) 146, 65–96 (1987)
Sznitman, A.S.: Topics in propagation of chaos. In: Hennequin P.L. (ed.) Ecole d’Eté de Probabilités de Saint-Flour XIX — 1989. Lecture Notes in Mathematics, vol 1464. Springer, Berlin, Heidelberg (1991)
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G. M. Coclite is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 642768.
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Coclite, G.M., Risebro, N.H. A difference method for the McKean–Vlasov equation. Z. Angew. Math. Phys. 70, 149 (2019). https://doi.org/10.1007/s00033-019-1196-x
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DOI: https://doi.org/10.1007/s00033-019-1196-x