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The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic and Time Value

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  • Carr, Peter P
  • Jarrow, Robert A

Abstract

The downside risk in a leveraged stock position can be eliminated by using stop-loss orders. The upside potential of such a position can be captured using contingent buy orders. The terminal payoff to this stop-loss start-gain strategy is identical to that of a call option, but the strategy costs less initially. This article resolves this paradox by showing that the strategy is not self-financing for continuous stock- price processes of unbounded variation. The resolution of the paradox leads to a new decomposition of an option's price into its intrinsic and time value. When stock price follows geometric Brownian motion, this decomposition is proven to be mathematically equivalent to the Black-Scholes (1973) formula. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

Suggested Citation

  • Carr, Peter P & Jarrow, Robert A, 1990. "The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic and Time Value," The Review of Financial Studies, Society for Financial Studies, vol. 3(3), pages 469-492.
    • Handle: RePEc:oup:rfinst:v:3:y:1990:i:3:p:469-92
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    Cited by:

    1. Zhi Jun Guo & Eckhard Platen, 2012. "The Small And Large Time Implied Volatilities In The Minimal Market Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(08), pages 1-23.
    2. Guillermo Angeris & Alex Evans & Tarun Chitra, 2021. "Replicating Market Makers," Papers 2103.14769, arXiv.org.
    3. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    4. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    5. Peter P. Carr & Zura Kakushadze, 2017. "FX options in target zones," Quantitative Finance, Taylor & Francis Journals, vol. 17(10), pages 1477-1486, October.
    6. Jarrow, Robert & Protter, Philip, 2005. "Large traders, hidden arbitrage, and complete markets," Journal of Banking & Finance, Elsevier, vol. 29(11), pages 2803-2820, November.
    7. Jarrow, Robert & Protter, Philip, 2012. "Discrete versus continuous time models: Local martingales and singular processes in asset pricing theory," Finance Research Letters, Elsevier, vol. 9(2), pages 58-62.
    8. Zura Kakushadze, 2015. "Coping with Negative Short-Rates," Papers 1502.06074, arXiv.org, revised Aug 2015.
    9. A. M. G. Cox & David Hobson & Jan Ob{l}'oj, 2007. "Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping," Papers math/0702173, arXiv.org, revised Nov 2008.
    10. Kraft, Holger, 2007. "Pitfalls in static superhedging of barrier options," Finance Research Letters, Elsevier, vol. 4(1), pages 2-9, March.
    11. Joon Y. Park, 2004. "The Spatial Analysis of Time Series," Econometric Society 2004 North American Winter Meetings 595, Econometric Society.
    12. Cayetano, Gea, 2007. "Studying the Properties of the Correlation Trades," MPRA Paper 22318, University Library of Munich, Germany.
    13. Michele Bonollo & Luca Di Persio & Luca Mammi & Immacolata Oliva, 2017. "Estimating the Counterparty Risk Exposure by using the Brownian Motion Local Time," Papers 1704.03244, arXiv.org.
    14. Mark Broadie & Jérôme Detemple, 1996. "American Options on Dividend-Paying Assets," CIRANO Working Papers 96s-16, CIRANO.
    15. Aleksandar Mijatović, 2010. "Local time and the pricing of time-dependent barrier options," Finance and Stochastics, Springer, vol. 14(1), pages 13-48, January.
    16. J. Scheinkman & W. Xiong, 2002. "Overconfidence, Short-Sale Constraints and Bubbles," Princeton Economic Theory Working Papers 98734966f1c1a57373801367f, David K. Levine.
    17. Xavier Warin, 2016. "The Asset Liability Management problem of a nuclear operator : a numerical stochastic optimization approach," Papers 1611.04877, arXiv.org.
    18. Jason Milionis & Ciamac C. Moallemi & Tim Roughgarden & Anthony Lee Zhang, 2022. "Automated Market Making and Loss-Versus-Rebalancing," Papers 2208.06046, arXiv.org, revised Nov 2023.
    19. Salopek, D. M., 2002. "A new class of nearly self-financing strategies," Statistics & Probability Letters, Elsevier, vol. 56(1), pages 69-75, January.
    20. Guillermo Angeris & Alex Evans & Tarun Chitra, 2023. "Replicating market makers," Digital Finance, Springer, vol. 5(2), pages 367-387, June.
    21. A. Bellier-Delienne, 2005. "Synthèse sur les Options de Livraison dans les Contrats à Terme," THEMA Working Papers 2005-09, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    22. Nicolas Perkowski & David J. Promel, 2014. "Local times for typical price paths and pathwise Tanaka formulas," Papers 1405.4421, arXiv.org, revised Apr 2015.
    23. Simon Ellersgaard & Martin Jönsson & Rolf Poulsen, 2017. "The Fundamental Theorem of Derivative Trading - exposition, extensions and experiments," Quantitative Finance, Taylor & Francis Journals, vol. 17(4), pages 515-529, April.

    More about this item

    JEL classification:

    • B26 - Schools of Economic Thought and Methodology - - History of Economic Thought since 1925 - - - Financial Economics
    • O16 - Economic Development, Innovation, Technological Change, and Growth - - Economic Development - - - Financial Markets; Saving and Capital Investment; Corporate Finance and Governance
    • E44 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Financial Markets and the Macroeconomy

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