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Leland's Approach to Option Pricing: The Evolution of a Discontinuity

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  • Peter Grandits
  • Werner Schachinger

Abstract

A claim of Leland (1985) states that in the presence of transaction costs a call option on a stock S, described by geometric Brownian motion, can be perfectly hedged using Black–Scholes delta hedging with a modified volatility. Recently Kabanov and Safarian (1997) disproved this claim, giving an explicit (up to an integral) expression of the limiting hedging error, which appears to be strictly negative and depends on the path of the stock price only via the stock price at expiry ST. We prove in this paper that the limiting hedging error, considered as a function of ST, exhibits a removable discontinuity at the exercise price. Furthermore, we provide a quantitative result describing the evolution of the discontinuity: Hedging errors, plotted over the price at expiry, show a peak near the exercise price. We determine the rate at which that peak becomes narrower (producing the discontinuity in the limit) as the lengths of the revision intervals shrink.

Suggested Citation

  • Peter Grandits & Werner Schachinger, 2001. "Leland's Approach to Option Pricing: The Evolution of a Discontinuity," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 347-355, July.
    • Handle: RePEc:bla:mathfi:v:11:y:2001:i:3:p:347-355
    DOI: 10.1111/1467-9965.00119
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    Cited by:

    1. Thai Huu Nguyen & Serguei Pergamenschchikov, 2015. "Approximate hedging with proportional transaction costs in stochastic volatility models with jumps," Papers 1505.02627, arXiv.org, revised Sep 2019.
    2. Thai Huu Nguyen & Serguei Pergamenshchikov, 2015. "Approximate hedging problem with transaction costs in stochastic volatility markets," Papers 1505.02546, arXiv.org.
    3. Fathi Abid & Wafa Abdelmalek & Sana Ben Hamida, 2020. "Dynamic Hedging using Generated Genetic Programming Implied Volatility Models," Papers 2006.16407, arXiv.org.
    4. Huu Thai Nguyen & Serguei Pergamenchtchikov, 2012. "Approximate hedging problem with transaction costs in stochastic volatility markets," Working Papers hal-00747689, HAL.
    5. J. S. Kennedy & P. A. Forsyth & K. R. Vetzal, 2009. "Dynamic Hedging Under Jump Diffusion with Transaction Costs," Operations Research, INFORMS, vol. 57(3), pages 541-559, June.
    6. Daniel Sevcovic & Magdalena Zitnanska, 2016. "Analysis of the nonlinear option pricing model under variable transaction costs," Papers 1603.03874, arXiv.org.
    7. Huu Thai Nguyen & Serguei Pergamenchtchikov, 2014. "Approximate hedging with proportional transaction costs in stochastic volatility models with jumps," Working Papers hal-00979199, HAL.

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