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

## Author Info

• Emmanuel Denis

()

• Yuri Kabanov

()

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## Abstract

Lelandâ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 Blackâ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 âÎ± , where Î±â[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 Î±=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 â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]â[0,1] and its inverse f are continuous with their first and second derivatives on the whole interval, or g(t)=1â(1ât) Î² , Î²â¥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

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 14 (2010)
Issue (Month): 4 (December)
Pages: 625-667

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Handle: RePEc:spr:finsto:v:14:y:2010:i:4:p:625-667

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## Related research

Keywords: Black–Scholes formula; European option; Transaction costs; Leland–Lott strategy; Approximate hedging; Martingale limit theorem; Diffusion approximation; 60G44; G11; G13;

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Find related papers by JEL classification:
• G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
• G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

## References

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1. Hayne E. Leland., 1984. "Option Pricing and Replication with Transactions Costs," Research Program in Finance Working Papers 144, University of California at Berkeley.
2. E. R. Grannan & G. H. Swindle, 1996. "Minimizing Transaction Costs Of Option Hedging Strategies," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 341-364.
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. 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.
5. 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.
6. Emmanuel Temam & Emmanuel Gobet, 2001. "Discrete time hedging errors for options with irregular payoffs," Finance and Stochastics, Springer, vol. 5(3), pages 357-367.
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## Citations

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Cited by:
1. Masaaki Fukasawa, 2012. "Efficient Discretization of Stochastic Integrals," Papers 1204.0637, arXiv.org.
2. Masaaki Fukasawa, 2014. "Efficient discretization of stochastic integrals," Finance and Stochastics, Springer, vol. 18(1), pages 175-208, January.
3. Huu Thai Nguyen & Serguei Pergamenchtchikov, 2012. "Approximate hedging problem with transaction costs in stochastic volatility markets," Working Papers hal-00808608, HAL.

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