Finite difference schemes and partial differential equations
Finite difference schemes and partial differential equations
European option pricing with transaction costs
SIAM Journal on Control and Optimization
High order difference schemes for unsteady one-dimensional diffusion-convection problems
Journal of Computational Physics
Superreplication Under Gamma Constraints
SIAM Journal on Control and Optimization
Accurate and efficient pricing of vanilla stock options via the Crandall--Douglas scheme
Applied Mathematics and Computation
Tools for Computational Finance (Universitext)
Tools for Computational Finance (Universitext)
Compact finite difference method for American option pricing
Journal of Computational and Applied Mathematics
Computing option pricing models under transaction costs
Computers & Mathematics with Applications
Numerical analysis and simulation of option pricing problems modeling illiquid markets
Computers & Mathematics with Applications
A kernel-based algorithm for numerical solution of nonlinear PDEs in finance
LSSC'11 Proceedings of the 8th international conference on Large-Scale Scientific Computing
A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets
Mathematics and Computers in Simulation
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Nonlinear Black-Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor's preferences or illiquid markets (which may have an impact on the stock price), the volatility, the drift and the option price itself. In this paper we will focus on several models from the most relevant class of nonlinear Black-Scholes equations for European and American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives due to transaction costs. We will analytically approach the option price by transforming the problem for a European Call option into a convection-diffusion equation with a nonlinear term and the free boundary problem for an American Call option into a fully nonlinear nonlocal parabolic equation defined on a fixed domain following Sevcovic's idea. Finally, we will present the results of different numerical discretization schemes for European options for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.