The Föllmer–Schweizer decomposition: Comparison and description

Abstract

This paper proposes two main contributions concerning the Föllmer-Schweizer decomposition (called hereafter the FS-decomposition). First we completely elaborate the relationship between this decomposition and the Galtchouk-Kunita-Watanabe decomposition under the minimal martingale measure. The difference between these two decompositions is highlighted in a very practical example, and the martingale tools that enhance this difference are illustrated in the semimartingale framework as well. The second main contribution focuses on the description of the FS-decomposition using the predictable characteristics.

Robustness of quadratic hedging strategies in finance via Fourier transforms

“…The existence of the FS-decomposition has been studied by many authors, see, e.g., [17] and [18]. In particular, it was shown that the decomposition exists in the case of exponential Lévy models.…”

“…In particular, it was shown that the decomposition exists in the case of exponential Lévy models. From the general formulas in [17] it is easy to derive that the LRM hedging number can be expressed as 12) in case the setting is observed under the historical measure P and where the prices P and the delta ∆ are determined w.r.t. the minimal martingale measure (MMM).…”

“…Now we observe the approximating process under the historical measure P. We compute the LRM strategy as follows. Either we use the results obtained in [17] in which the LRM hedging number is computed based on the option price and its delta computed under the MMM for the approximating model…”

“…Combining the MVT on the function θ → e θz and the property (3.15) leads to 17) where γ(z) = e (1+|θ0|+C(θ0)s…”

Métodos de simulação Monte Carlo para aproximação de estratégias de hedging ideais

“…Para maiores referência neste sentido, citamos os trabalhos de Pham (2000) [79], Heath et al (2001) [49] e Schweizer (2008) [90]. Mais recentemente, temos os trabalhos e Choulli et al (2010) [18] e Choulli et al (2011) [19]. Veja também o trabalho de Delong (2013) [29], em que decomposições de Föllmer-Schweizer são obtidas a partir de soluções de equações diferenciais estocásticas retrógradas (BSDEs).…”