Weighted BMO and discrete time hedging within the Black-Scholes model
In stochastic finance the approximation of stochastic integrals, driven by the geometric Brownian motion, by integrals over piece-wise constant integrands has a natural interpretation: the pay-off of a continuously re-balanced portfolio is replaced by the pay-off of a portfolio, rebalanced at finitely many trading dates only. And the approximation error between the first and the second integrals can be interpreted as risk. These two ``small'' observations may be considered as a motivation to approach the following mathematical problems investigated in the paper. Usually, the approximation error is measured in a distributional way by limit distributions or with respect to \(L_2.\) But the \(L_2\)-approach has some drawbacks: the resulting distributional tail-estimates are rather weak and secondly, it might not be clear what sequence of time-nets, realizing the asymptotically optimal \(L_2\)-approximation rate, one should really take. The present paper approaches both problems. In order to replace the \(L_2\)-criterion by a stronger one, one would use \(L_p\)-spaces with \(2<p<\infty.\) However, spaces of stochastic processes having weighted bounded mean oscillation (weighted BMO) provide much more information, while getting the same upper bound for the asymptotics as in the \(L_2\)-case and are of advantage because of several reasons. That's why the paper is divided into two rather different parts. Section 2 deals with the weighted BMO-spaces and in Section 3 the author applies the obtained results to the approximation problem for stochastic integrals.
scientific article; zbMATH DE number 2164753
scientific article