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The Value of Timing Risk

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  • Jiro Akahori
  • Flavia Barsotti
  • Yuri Imamura

Abstract

The aim of this paper is to provide a mathematical contribution on the semi-static hedge of timing risk associated to positions in American-style options under a multi-dimensional market model. Barrier options are considered in the paper and semi-static hedges are studied and discussed for a fairly large class of underlying price dynamics. Timing risk is identified with the uncertainty associated to the time at which the payoff payment of the barrier option is due. Starting from the work by Carr and Picron (1999), where the authors show that the timing risk can be hedged via static positions in plain vanilla options, the present paper extends the static hedge formula proposed in Carr and Picron (1999) by giving sufficient conditions to decompose a generalized timing risk into an integral of knock-in options in a multi-dimensional market model. A dedicated study of the semi-static hedge is then conducted by defining the corresponding strategy based on positions in barrier options. The proposed methodology allows to construct not only first order hedges but also higher order semi-static hedges, that can be interpreted as asymptotic expansions of the hedging error. The convergence of these higher order semi-static hedges to an exact hedge is shown. An illustration of the main theoretical results is provided for i) a symmetric case, ii) a one dimensional case, where the first order and second order hedging errors are derived in analytic closed form. The materiality of the hedging benefit gain of going from order one to order two by re-iterating the timing risk hedging strategy is discussed through numerical evidences by showing that order two can bring to more than 90% reduction of the hedging 'cost' w.r.t. order one (depending on the specific barrier option characteristics).

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  • Jiro Akahori & Flavia Barsotti & Yuri Imamura, 2017. "The Value of Timing Risk," Papers 1701.05695, arXiv.org.
    • Handle: RePEc:arx:papers:1701.05695
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    References listed on IDEAS

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    1. Takashi Kato & Akihiko Takahashi & Toshihiro Yamada, 2013. "An Asymptotic Expansion Formula for Up-and-Out Barrier Option Price under Stochastic Volatility Model," CIRJE F-Series CIRJE-F-873, CIRJE, Faculty of Economics, University of Tokyo.
    2. Takashi Kato & Akihiko Takahashi & Toshihiro Yamada, 2014. "A Semi-group Expansion for Pricing Barrier Options," CARF F-Series CARF-F-349, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Yuri Imamura, 2011. "A remark on static hedging of options written on the last exit time," Review of Derivatives Research, Springer, vol. 14(3), pages 333-347, October.
    4. Takashi Kato & Akihiko Takahashi & Toshihiro Yamada, 2013. "An Asymptotic Expansion Formula for Up-and-Out Barrier Option Price under Stochastic Volatility Model," Papers 1302.3306, arXiv.org.
    5. Jirô Akahori & Yuri Imamura, 2014. "On a symmetrization of diffusion processes," Quantitative Finance, Taylor & Francis Journals, vol. 14(7), pages 1211-1216, July.
    6. Yuri Imamura & Katsuya Takagi, 2013. "Semi-Static Hedging Based on a Generalized Reflection Principle on a Multi Dimensional Brownian Motion," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 20(1), pages 71-81, March.
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    Cited by:

    1. Jiro Akahori & Flavia Barsotti & Yuri Imamura, 2018. "Asymptotic Static Hedge via Symmetrization," Papers 1801.04045, arXiv.org.

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