4,961 research outputs found
Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations
We analyze and calculate the early exercise boundary for a class of
stationary generalized Black-Scholes equations in which the volatility function
depends on the second derivative of the option price itself. A motivation for
studying the nonlinear Black Scholes equation with a nonlinear volatility
arises from option pricing models including, e.g., non-zero transaction costs,
investors preferences, feedback and illiquid markets effects and risk from
unprotected portfolio. We present a method how to transform the problem of
American style of perpetual put options into a solution of an ordinary
differential equation and implicit equation for the free boundary position. We
finally present results of numerical approximation of the early exercise
boundary, option price and their dependence on model parameters
Adaptive Wavelet Precise Integration Method for Nonlinear Black-Scholes Model Based on Variational Iteration Method
An adaptive wavelet precise integration method (WPIM) based on the variational iteration method (VIM) for Black-Scholes model is proposed. Black-Scholes model is a very useful tool on pricing options. First, an adaptive wavelet interpolation operator is constructed which can transform the nonlinear partial differential equations into a matrix ordinary differential equations. Next, VIM is developed to solve the nonlinear matrix differential equation, which is a new asymptotic analytical method for the nonlinear differential equations. Third, an adaptive precise integration method (PIM) for the system of ordinary differential equations is constructed, with which the almost exact numerical solution can be obtained. At last, the famous Black-Scholes model is taken as an example to test this new method. The numerical result shows the method's higher numerical stability and precision
On the numerical solution of nonlinear Black-Scholes equations
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 be concerned with 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 Ševčovič’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
Nonlinear Parabolic Equations arising in Mathematical Finance
This survey paper is focused on qualitative and numerical analyses of fully
nonlinear partial differential equations of parabolic type arising in financial
mathematics. The main purpose is to review various non-linear extensions of the
classical Black-Scholes theory for pricing financial instruments, as well as
models of stochastic dynamic portfolio optimization leading to the
Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both
problems can be represented by solutions to nonlinear parabolic equations.
Qualitative analysis will be focused on issues concerning the existence and
uniqueness of solutions. In the numerical part we discuss a stable
finite-volume and finite difference schemes for solving fully nonlinear
parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387
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