15 research outputs found
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
Moving boundary transformation for American call options with transaction cost: Finite difference methods and computing
The pricing of American call option with transaction cost is a free boundary problem. Using a new transformation method the boundary is made to follow a certain known trajectory in time. The new transformed problem is solved by various finite difference methods, such as explicit and implicit schemes. Broyden’s and Schubert’s methods are applied as a modification to Newton’s method in the case of nonlinearity in the equation. An alternating direction explicit method with second-order accuracy in time is used as an example in this paper to demonstrate the technique. Numerical results demonstrate the efficiency and the rate of convergence of the methods
XVA for American options with two stochastic factors: modelling, mathematical analysis and numerical methods
En actas del XXVI Congreso de Ecuaciones Diferenciales y Aplicaciones. XVI Congreso de Matemática Aplicada. Gijón, 14-18 junio 2021[Abstract]: In this work, we derive new linear and nonlinear partial differential equations (PDEs) models for pricing American options and total value adjustment in the presence of counterparty risk. Moreover, stochastic spreads are considered, which increases the dimension of the problem