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Closed Form Formulas For Exotic Options And Their Lifetime Distribution

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  • RAPHAËL DOUADY

    (C.M.L.A., Ecole Normale Supérieure, 61 av. du Pdt. Wilson, 94235 Cachan Cedex, France)

Abstract

We first recall the well-known expression of the price of barrier options, and compute double barrier options by the mean of the iterated mirror principle. The formula for double barriers provides an intraday volatility estimator from the information of high-low-close prices. Then we give explicit formulas for the probability distribution function and the expectation of the exit time of single and double barrier options. These formulas allow to price time independent and time dependent rebates. They are also helpful to hedge barrier and double barrier options, when taking into account variations of the term structure of interest rates and of volatility. We also compute the price of rebates of double knock-out options that depend on which barrier is hit first, and of the BOOST, an option which pays the time spent in a corridor. All these formulas are either in closed form or double infinite series which converge likee-α n2.

Suggested Citation

  • Raphaël Douady, 1999. "Closed Form Formulas For Exotic Options And Their Lifetime Distribution," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 2(01), pages 17-42.
    • Handle: RePEc:wsi:ijtafx:v:02:y:1999:i:01:n:s0219024999000030
    DOI: 10.1142/S0219024999000030
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    Cited by:

    1. Dell'Era Mario, M.D., 2008. "Pricing of Double Barrier Options by Spectral Theory," MPRA Paper 17502, University Library of Munich, Germany.
    2. Franck Moraux, 2009. "On perpetual American strangles," Post-Print halshs-00393811, HAL.
    3. Sbuelz, A., 2000. "Hedging Double Barriers with Singles," Discussion Paper 2000-112, Tilburg University, Center for Economic Research.
    4. Claude Bardos & Raphaël Douady & Andrei Fursikov, 2002. "Static Hedging Of Barrier Options With A Smile: An Inverse Problem," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01477102, HAL.
    5. Sbuelz, A., 2000. "Hedging Double Barriers with Singles," Other publications TiSEM e810e3ab-1936-457e-a3ae-7, Tilburg University, School of Economics and Management.
    6. Dell'Era Mario, M.D., 2008. "Pricing of the European Options by Spectral Theory," MPRA Paper 17429, University Library of Munich, Germany.
    7. Zhang, Kun & Liu, Jing & Wang, Erkang & Wang, Jin, 2017. "Quantifying risks with exact analytical solutions of derivative pricing distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 757-766.
    8. Pauline Barrieu & Nadine Bellamy & Bernard Sinclair-Desgagné, 2017. "Assessing contaminated land cleanup costs and strategies," Post-Print halshs-02292808, HAL.
    9. Vaibhav Srivastava & Samuel F. Feng & Jonathan D. Cohen & Naomi Ehrich Leonard & Amitai Shenhav, 2015. "A martingale analysis of first passage times of time-dependent Wiener diffusion models," Papers 1508.03373, arXiv.org, revised Sep 2016.

    More about this item

    Keywords

    Double barrier options; Closed forms; First hitting time; JEL classification G130;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G31 - Financial Economics - - Corporate Finance and Governance - - - Capital Budgeting; Fixed Investment and Inventory Studies

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