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Hedging Stock Options Using Futures Contracts on the Stock

Received: 8 May 2015     Accepted: 20 May 2015     Published: 16 June 2015
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Abstract

The aim of this paper is to present the price and replicating strategy for an European option on spot (or cash) underlier with continuous dividend yield, when the instrument used in the dynamic hedging of the option is a futures contract on the respective underlier. It formalizes the heuristic practice among option traders to replicate options on a stock index using futures on the respective stock index and investigates weather the obtained results differ significantly from what they would get using the actual stock index, as required by Black-Scholes pricing. Heuristically, the substitution is supported by index and futures prices being close, at least for small dividends and time to maturity. Our method is to express this practice in accounting terms, derive the self-financing portfolio dynamics and then the closed form option price and delta. Finally, run numerical simulations and compare results obtained by Black-Scholes versus our approach. Results show both the price and delta formulas differ from Black-Scholes, however numeric simulation doesn’t yield high enough differences to warrant obvious arbitrage, meaning that while not rigorously exact, the approximation is good enough for most practical use cases.

Published in Applied and Computational Mathematics (Volume 4, Issue 3)
DOI 10.11648/j.acm.20150403.24
Page(s) 214-219
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Options Pricing, Options Replication Using Futures, Arbitrage

References
[1] Antonie Kotze, Delta Hedging: Futures Versus Underlying Spot, quantonline.co.za.
[2] Emanuel Derman, When You Cannot Hedge Continuously, Goldman Sachs Quantitative Strategies Research Note.
[3] Emanuel Derman, Nassim Nicholas Taleb, The Illusions of Dynamic Replication, Quantitative Finance, Vol. 5, No. 4, August 2005, 323–326.
[4] Espen Gaarder Haug, The Complete Guide To Option Pricing Formulas, McGraw-Hill, 1998.
[5] Espen Gaarder Haug, Nassim Nicholas Taleb, Option Traders Use (very) Sophisticated Heuristics, Never the Black–Scholes–Merton Formula, Journal of Economic Behavior and Organization, Vol. 77, No. 2, 2011.
[6] Fabrice Douglas Rouah, Four Derivations of the Black-Scholes Formula, www.frouah.com.
[7] Hayne Leeland, Option Pricing and Replication with Transaction Costs, Journal of The Journal of Finance, Vol 40, No. 5 (Dec 1985), 1283-1301.
[8] John Hull, Options, Futures and Other Derivatives, Prentice Hall, 6nd edition, 2005.
[9] Ralf Korn, Leonard Rogers, Stocks Paying Discrete Dividends: Modelling and Option Pricing, The Journal of Derivatives, Winter 2005, Vol. 13, No. 2: pp. 44-48.
[10] Salih Neftci, An Introduction to the Mathematics of Financial Derivatives, Academic Press, 2nd edition, 2000.
[11] Thomas Björk, Arbitrage Theory in Continuous Time, Oxford Finance, 2nd edition, 2004.
Cite This Article
  • APA Style

    Mihai Grigore Bunea Domsa. (2015). Hedging Stock Options Using Futures Contracts on the Stock. Applied and Computational Mathematics, 4(3), 214-219. https://doi.org/10.11648/j.acm.20150403.24

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    ACS Style

    Mihai Grigore Bunea Domsa. Hedging Stock Options Using Futures Contracts on the Stock. Appl. Comput. Math. 2015, 4(3), 214-219. doi: 10.11648/j.acm.20150403.24

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    AMA Style

    Mihai Grigore Bunea Domsa. Hedging Stock Options Using Futures Contracts on the Stock. Appl Comput Math. 2015;4(3):214-219. doi: 10.11648/j.acm.20150403.24

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  • @article{10.11648/j.acm.20150403.24,
      author = {Mihai Grigore Bunea Domsa},
      title = {Hedging Stock Options Using Futures Contracts on the Stock},
      journal = {Applied and Computational Mathematics},
      volume = {4},
      number = {3},
      pages = {214-219},
      doi = {10.11648/j.acm.20150403.24},
      url = {https://doi.org/10.11648/j.acm.20150403.24},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150403.24},
      abstract = {The aim of this paper is to present the price and replicating strategy for an European option on spot (or cash) underlier with continuous dividend yield, when the instrument used in the dynamic hedging of the option is a futures contract on the respective underlier. It formalizes the heuristic practice among option traders to replicate options on a stock index using futures on the respective stock index and investigates weather the obtained results differ significantly from what they would get using the actual stock index, as required by Black-Scholes pricing. Heuristically, the substitution is supported by index and futures prices being close, at least for small dividends and time to maturity. Our method is to express this practice in accounting terms, derive the self-financing portfolio dynamics and then the closed form option price and delta. Finally, run numerical simulations and compare results obtained by Black-Scholes versus our approach. Results show both the price and delta formulas differ from Black-Scholes, however numeric simulation doesn’t yield high enough differences to warrant obvious arbitrage, meaning that while not rigorously exact, the approximation is good enough for most practical use cases.},
     year = {2015}
    }
    

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    T1  - Hedging Stock Options Using Futures Contracts on the Stock
    AU  - Mihai Grigore Bunea Domsa
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    DO  - 10.11648/j.acm.20150403.24
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 214
    EP  - 219
    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.acm.20150403.24
    AB  - The aim of this paper is to present the price and replicating strategy for an European option on spot (or cash) underlier with continuous dividend yield, when the instrument used in the dynamic hedging of the option is a futures contract on the respective underlier. It formalizes the heuristic practice among option traders to replicate options on a stock index using futures on the respective stock index and investigates weather the obtained results differ significantly from what they would get using the actual stock index, as required by Black-Scholes pricing. Heuristically, the substitution is supported by index and futures prices being close, at least for small dividends and time to maturity. Our method is to express this practice in accounting terms, derive the self-financing portfolio dynamics and then the closed form option price and delta. Finally, run numerical simulations and compare results obtained by Black-Scholes versus our approach. Results show both the price and delta formulas differ from Black-Scholes, however numeric simulation doesn’t yield high enough differences to warrant obvious arbitrage, meaning that while not rigorously exact, the approximation is good enough for most practical use cases.
    VL  - 4
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Author Information
  • Department of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania

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