Numerical treatment of the Black-Scholes variational inequality in computational finance

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Among the central concerns in mathematical finance is the evaluation of American options. An American option gives the holder the right but not the obligation to buy or sell a certain financial asset within a certain time-frame, for a certain strike price. The valuation of American options is formulated as an optimal stopping problem. If the stock price is modelled by a geometric Brownian motion, the value of an American option is given by a deterministic parabolic free boundary value problem (FBVP) or equivalently a non-symmetric variational inequality on weighted Sobolev spaces on the entire real line R. To apply standard numerical methods, the unbounded domain is truncated to a bounded one. Applying the Fourier transform to the FBVP yields an integral representation of the solution including the free boundary explicitely. This integral representation allows to prove explicit truncation errors. Since the variational inequality is formulated within the framework of weighted Sobolev spaces, we establish a weighted Poincaré inequality with explicit determined constants. The truncation error estimate and the weighted Poncaré inequality enable a reliable a posteriori error estimate between the exact solution of the variational inequality and the semi-discrete solution of the penalised problem on R. A sufficient regular solution provides the convergence of the solution of the penalised problem to the solution of the variational inequality. An a priori error estimate for the error between the exact solution of the variational inequality and the semi-discrete solution of the penalised problem concludes the numerical analysis. The established a posteriori error estimates motivates an algorithm for adaptive mesh refinement. Numerical experiments show the improved convergence of the adaptive algorithm compared to uniform mesh refinement. The choice of different truncation points reveal the influence of the truncation error estimate on the total error estimator. This thesis provides a semi-discrete reliable a posteriori error estimates for a variational inequality on an unbounded domain including explicit truncation errors. This allows to determine a truncation point such that the total error (discretisation and truncation error) is below a given error tolerance.