Abstract
Static hedging of complicated payoff structures by standard instruments becomes increasingly popular in finance. The classical approach is developed for quite regular functions, while for less regular cases, generalized functions and approximation arguments are used. In this note, we discuss the regularity conditions in the classical decomposition formula due to P. Carr and D. Madan (in Jarrow ed, Volatility, pp. 417–427, Risk Publ., London, 1998) if the integrals in this formula are interpreted as Lebesgue integrals with respect to the Lebesgue measure. Furthermore, we show that if we replace these integrals by Lebesgue–Stieltjes integrals, the family of representable functions can be extended considerably with a direct approach.
A large part of the research was carried out while the second author was a postdoctoral researcher at the Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland.
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Notes
- 1.
Recall that following e.g. [16, Def. 7.1.4], a finite function f defined on a closed interval [a,b] is absolutely continuous on [a,b] (notation f∈AC[a,b]) if, for every ε>0, there exists a δ>0 such that \(\sum_{k=1}^{n}|f(b_{k})-f(a_{k})|<\varepsilon\) for any a≤a 1<b 1≤a 2<b 2≤…≤a n <b n ≤b for which \(\sum_{k=1}^{n}(b_{k}-a_{k})<\delta\).
- 2.
For the following, see Definition 6.1.2 in [16]. Let \(f\colon[a,b]\to\mathbb{R}\). Assume that P={x 0,x 1,…,x n(P)} is a partition of the interval [a,b]. If
$$ T_f[a,b]=\sup_P\sum _{k=1}^{n\! (P )\!}\bigl\lvert f\! (x_k )\!-f\! (x_{k-1} )\bigr\rvert <\infty, $$where the supremum is taken over all partitions P of [a,b], then f is said to be of bounded variation on [a,b], for short f∈BV[a,b]. If \(f\colon \mathbb{R}\to \mathbb{R}\) or \(f\colon \mathbb{R}_{+}\to \mathbb{R}\) is such that the restriction of f to [a,b] is in BV[a,b] for all a<b, then f is said to be locally of bounded variation (f∈BVloc).
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Acknowledgements
The authors are grateful to Katrin Fässler, Ilya Molchanov, Jean-Francois Renaud, and Thorsten Rheinländer for helpful hints and discussions. This work was supported by the Swiss National Science Foundation Grant Nr. 200021-126503 and PBBEP3_130157.
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Schmutz, M., Zürcher, T. (2014). A Stieltjes Approach to Static Hedges. In: Kabanov, Y., Rutkowski, M., Zariphopoulou, T. (eds) Inspired by Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-02069-3_24
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