Pricing American options with stochastic volatility and small nonlinear price impact: A PDE approach

Yan, Dong; Lin, Sha; Hu, Zhihao; Yang, Bohan

doi:10.1016/j.chaos.2022.112581

Cited by 6 publications

(2 citation statements)

References 39 publications

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“…The discrete equation (10a) is a nonlinear system with unknown solution V. The computation of V would be a serious problem when h is small (using a small h is necessary if we want to get a more accurate solution, cf. (7).…”

Section: Theorem 1 ([9]) For Nonlinear Dirichlet Boundary Value Probl...mentioning

confidence: 99%

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Numerical simulation of a nonlinear model in finance by Broyden’s method

Zeng,

Liu,

Song

2024

ScienceAsia

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In this paper, we study the stationary Black-Scholes model arising in finance with transaction costs. This model becomes interesting when the time does not play a role such as, for instance, in perpetual options. The equation describing this model is a nonlinear second-order boundary value problem and there is no analytic solutions in closed form for such a nonlinear equation. After discretization via the centered finite difference formula we have to solve a nonlinear algebraic system which would be a serious problem when we use a small discretization mesh. We solve this nonlinear system by the residual-based Broyden's method, which is an efficient quasi-Newton method and is convenient to implement by a desk computer. We give a convergence analysis of the Broyden's method by assuming a lower and upper bound of the converged solution of the Black-Scholes model. Numerical results are given to show that the convergence rate of the method is robust with respect to the discretization mesh and the problem parameters.

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Section: Theorem 1 ([9]) For Nonlinear Dirichlet Boundary Value Probl...mentioning

confidence: 99%

“…Otherwise, a continuous adjustment would imply that those costs, such as taxes or fees, would become infinitely large [4][5][6]. Hence, considering transaction costs in the model is an important issue which has motivated the work of several authors and has led to the study of new Black-Scholes model [7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%