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

Lépinette, Emmanuel; Kabanov, Yuri (2010), Mean square error for the Leland-Lott hedging strategy: convex pay-offs, Finance and Stochastics, 14, 4, p. 625-667. http://dx.doi.org/10.1007/s00780-010-0130-z

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Type
Article accepté pour publication ou publié
Date
2010
Journal name
Finance and Stochastics
Volume
14
Number
4
Publisher
Springer
Pages
625-667
Publication identifier
http://dx.doi.org/10.1007/s00780-010-0130-z
Metadata
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Author(s)
Lépinette, Emmanuel

Kabanov, Yuri
Abstract (EN)
Leland’s approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim V T using the classical Black–Scholes formula with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e., a sequence of models with transaction cost coefficients k n =k 0 n −α , where α∈[0,1/2] and n is the number of portfolio revision dates. The enlarged volatility $\widehat{\sigma}_{n}$ in general depends on n except for the case which was investigated in detail by Lott, to whom belongs the first rigorous result on convergence of the approximating portfolio value $V^{n}_{T}$ to the pay-off V T . In this paper, we consider only the Lott case α=1/2. We prove first, for an arbitrary pay-off V T =G(S T ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n −1/2 in L 2 and find the first order term of the asymptotics. We are working in a setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are $t_{i}^{n}=g(i/n)$, 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, or g(t)=1−(1−t) β , β≥1. We show that the sequence $n^{1/2}(V_{T}^{n}-V_{T})$ converges in law to a random variable which is the terminal value of a component of a two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.
Subjects / Keywords
Diffusion approximation; Martingale limit theorem; European option; approximate hedging; transaction costs; Leland-Lott strategy; Black-Scholes formula
JEL
G13 - Contingent Pricing; Futures Pricing
G11 - Portfolio Choice; Investment Decisions

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