Mean square error for the LelandLott hedging strategy: convex payoffs
Lépinette, Emmanuel; Kabanov, Yuri (2010), Mean square error for the LelandLott hedging strategy: convex payoffs, Finance and Stochastics, 14, 4, p. 625667. http://dx.doi.org/10.1007/s007800100130z
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Article accepté pour publication ou publiéDate
2010Journal name
Finance and StochasticsVolume
14Number
4Publisher
Springer
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625667
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Show full item recordAbstract (EN)
Leland’s approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the Europeantype 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 payoff V T . In this paper, we consider only the Lott case α=1/2. We prove first, for an arbitrary payoff 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 nonuniform 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 twodimensional 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; LelandLott strategy; BlackScholes formulaRelated items
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