Nonlinear Parabolic Equations Arising in Mathematical Finance

Abstract

Summary. 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.

Utility indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation

“…Following Leland's approach, a model involving variable transaction costs has been introduced and analyzed in [21]. For an overview of nonlinear option pricing models of the Leland type under transaction costs we refer to [20].…”

Multigrid Method for a Two Dimensional Fully Nonlinear Black-Scholes Equation with a Nonlinear Volatility Function

“…There is a considerable literature that reviews numerical simulations of the option price when markets are subject to transaction costs using PDEs, we quote [4,19,22], where several nonlinear Black-Scholes models in the presence of transaction costs have been studied for European and American options, [30] presented a positivity preserving scheme for the Barles' and Soner's model and they studied its convergence. [21] used a grid stretching technique on non-uniform meshes for the computation and [24,25] reviewed nonlinear parabolic equations arising from markets under transaction costs. However, there are only few works for the computation of nonlinear Black-Scholes equations depending on more than one underlying asset, among them we cite [29], they gave a numerical analysis for two dimensional nonlinear Black-Scholes equation for Spread option in incomplete markets and [3] where the authors gave a theoretical and numerical study of multi-asset nonlinear Black-Scholes equation arising from models with general transaction costs.…”