Hedging Stock Options Using Futures Contracts on the Stock

Mihai Grigore Bunea Domsa

doi:doi:10.11648/j.acm.20150403.24

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Hedging Stock Options Using Futures Contracts on the Stock

Mihai Grigore Bunea Domsa

Published in Applied and Computational Mathematics (Volume 4, Issue 3)

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), 2024. 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.

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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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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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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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TY  - JOUR
T1  - Hedging Stock Options Using Futures Contracts on the Stock
AU  - Mihai Grigore Bunea Domsa
Y1  - 2015/06/16
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150403.24
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
SN  - 2328-5613
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
IS  - 3
ER  -

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Author Information

Mihai Grigore Bunea Domsa

Department of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania

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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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Hedging Stock Options Using Futures Contracts on the Stock
AU  - Mihai Grigore Bunea Domsa
Y1  - 2015/06/16
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150403.24
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
SN  - 2328-5613
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
IS  - 3
ER  -

Copy | Download