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

## Author Info

• Lépinette-Denis, Emmanuel
• Kabanov, Yuri
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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 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.

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## Bibliographic Info

Paper provided by Paris Dauphine University in its series Economics Papers from University Paris Dauphine with number 123456789/4654.

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 Length: Date of creation: 2010 Date of revision: Publication status: Published in Finance and Stochastics, 2010, Vol. 14, no. 4. pp. 625-667.Length: 42 pages Handle: RePEc:dau:papers:123456789/4654 Contact details of provider: Web page: http://www.dauphine.fr/en/welcome.htmlMore information through EDIRC

## References

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1. Emmanuel Temam & Emmanuel Gobet, 2001. "Discrete time hedging errors for options with irregular payoffs," Finance and Stochastics, Springer, vol. 5(3), pages 357-367.
2. Zhao, Yonggan & Ziemba, William T., 2007. "Hedging errors with Leland's option model in the presence of transaction costs," Finance Research Letters, Elsevier, vol. 4(1), pages 49-58, March.
3. Y. M. Kabanov & M. Safarian, 1995. "On Leland's Strategy of Option Pricing with Transaction Costs," SFB 373 Discussion Papers 1995,65, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
4. Leland, Hayne E., 2007. "Comments on "Hedging errors with Leland's option model in the presence of transactions costs"," Finance Research Letters, Elsevier, vol. 4(3), pages 200-202, September.
5. E. R. Grannan & G. H. Swindle, 1996. "Minimizing Transaction Costs Of Option Hedging Strategies," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 341-364.
6. Leland, Hayne E, 1985. " Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
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