Mean square error for the Leland-Lott hedging strategy: convex pay-offs

Emmanuel Denis and Yuri Kabanov ()
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Emmanuel Denis: CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique
Yuri Kabanov: LMB - Laboratoire de Mathématiques de Besançon (UMR 6623) - CNRS - Centre National de la Recherche Scientifique - UFC - Université de Franche-Comté - UBFC - Université Bourgogne Franche-Comté [COMUE]
Authors registered in the RePEc Author Service: Emmanuel Lépinette and Юрий Михайлович Кабанов

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Abstract: Leland's approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim VT using the classical Black Scholes formulae with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e. a sequence of models with the transaction costs coefficients kn and n is the number of the portfolio revision dates. The enlarged volatility, in general, depends on n except the case which was investigated in details by Lott to whom belongs the first rigorous result on convergence of the approximating portfolio value to the pay-off. In this paper we consider only the Lott case alpha= 1/2. We prove first, for an arbitrary pay-off VT = G(ST ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n^1/2 in L2 and find the first order term of asymptotics. We are working in the setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are t_ni=g(i_n) where the strictly increasing scale function g : [0; 1] -> [0; 1] and its inverse f are continuous with their first and second derivatives on the whole interval. We show that the sequence of approximate error converges in law to a random variable which is the terminal value of a component of two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.

Keywords: Black-Scholes formula; Leland-Lott strategy; transaction costs; approximate hedging (search for similar items in EconPapers)
Date: 2010
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Citations: View citations in EconPapers (8)

Published in Finance and Stochastics, 2010, 14 (4), pp.625-667. ⟨10.1007/s00780-010-0130-z⟩

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Journal Article: Mean square error for the Leland–Lott hedging strategy: convex pay-offs (2010)
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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-00488278

DOI: 10.1007/s00780-010-0130-z

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