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Probabilistic methods for semilinear partial differential equations. Applications to finance

Dan Crisan, Konstantinos Manolarakis (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

With the pioneering work of [Pardoux and Peng, Syst. Contr. Lett.14 (1990) 55–61; Pardoux and Peng, Lecture Notes in Control and Information Sciences176 (1992) 200–217]. We have at our disposal stochastic processes which solve the so-called backward stochastic differential equations. These processes provide us with a Feynman-Kac representation for the solutions of a class of nonlinear partial differential equations (PDEs) which appear in many applications in the field of Mathematical Finance....

Solution of option pricing equations using orthogonal polynomial expansion

Falko Baustian, Kateřina Filipová, Jan Pospíšil (2021)

Applications of Mathematics

We study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial differential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre polynomials. We compare the obtained solutions to existing semi-closed pricing formulas. Special attention is paid to the solution of the Heston model at the boundary with vanishing volatility.

Superconvergence estimates of finite element methods for American options

Qun Lin, Tang Liu, Shu Hua Zhang (2009)

Applications of Mathematics

In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. 39 (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation...

Thoughts about Selected Models for the Valuation of Real Options

Mikael Collan (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

This paper discusses option valuation logic and four selected methods for the valuation of real options in the light of their modeling choices. Two of the selected methods the Datar–Mathews method and the Fuzzy Pay-off Method represent later developments in real option valuation and the Black & Scholes formula and the Binomial model for option pricing the more established methods used in real option valuation. The goal of this paper is to understand the big picture of real option valuation models...

Valuing barrier options using the adaptive discontinuous Galerkin method

Hozman, Jiří (2013)

Programs and Algorithms of Numerical Mathematics

This paper is devoted to barrier options and the main objective is to develop a sufficiently robust, accurate and efficient method for computation of their values driven according to the well-known Black-Scholes equation. The main idea is based on the discontinuous Galerkin method together with a spatial adaptive approach. This combination seems to be a promising technique for the solving of such problems with discontinuous solutions as well as for consequent optimization of the number of degrees...

Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

Nils Reich (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

For a class of anisotropic integrodifferential operators ℬ arising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations ℬ u = f on [0,1]n with possibly large n. Under certain conditions on ℬ , the scheme is of essentially optimal and dimension independent complexity 𝒪 (h-1| log h |2(n-1)) without corrupting the convergence or smoothness requirements...

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