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The discontinuous Galerkin method for discretely observed Asian options
Author(s) -
Hozman Jiří,
Tichý Tomáš
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6160
Subject(s) - mathematics , jump , piecewise , asian option , valuation of options , monte carlo methods for option pricing , stochastic game , valuation (finance) , constraint (computer aided design) , mathematical optimization , finite difference methods for option pricing , discontinuous galerkin method , put option , black–scholes model , econometrics , mathematical economics , mathematical analysis , finance , economics , volatility (finance) , physics , geometry , quantum mechanics , finite element method , thermodynamics
Asian options represent an important subclass of the path‐dependent contracts that are identified by payoff depending on the average of the underlying asset prices over the prespecified period of option lifetime. Commonly, this average is observed at discrete dates, and also, early exercise features can be admitted. As a result, analytical pricing formulae are not always available. Therefore, some form of a numerical approximation is essential for efficient option valuation. In this paper, we study a PDE model for pricing discretely observed arithmetic Asian options with fixed as well as floating strike for both European and American exercise features. The pricing equation for such options is similar to the Black‐Scholes equation with 1 underlying asset, and the corresponding average appears only in the jump conditions across the sampling dates. The objective of the paper is to present the comprehensive methodological concept that forms and improves the valuation process. We employ a robust numerical procedure based on the discontinuous Galerkin approach arising from the piecewise polynomial generally discontinuous approximations. This technique enables a simple treatment of discrete sampling by incorporation of jump conditions at each monitoring date. Moreover, an American early exercise constraint is directly handled as an additional nonlinear source term in the pricing equation. The proposed solving procedure is accompanied by an empirical study with practical results compared to reference values.