Step Options
Motivated by risk management problems with barrier options, we propose a flexible modification of the standard knock‐out and knock‐in provisions and introduce a family of path‐dependent options: " step options". They are parametrized by a "finite knock‐out (knock‐in) rate", ρ. For a down‐and‐out step option, its payoff at expiration is defined as the payoff of an otherwise identical vanilla option discounted by the " knock‐out factor" exp(-ρτ<sub>B</sub>-super-‐) or max(1‐ρτ-super--B,0), where &\tau;<sub>B</sub>-super-‐ is the total time during the contract life that the underlying price was lower than a prespecified barrier level (" occupation time"). We derive closed‐form pricing formulas for step options with any knock‐out rate in the range $[0,∞). For any finite knock‐out rate both the step option's value and delta are continuous functions of the underlying price at the barrier. As a result, they can be continuously hedged by trading the underlying asset and borrowing. Their risk management properties make step options attractive "no‐regrets" alternatives to standard barrier options. As a by‐product, we derive a dynamic almost‐replicating trading strategy for standard barrier options by considering a replicating strategy for a step option with high but finite knock‐out rate. Finally, a general class of derivatives contingent on occupation times is considered and closed‐form pricing formulas are derived. Copyright Blackwell Publishers Inc 1999.
Year of publication: |
1999
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Authors: | Linetsky, Vadim |
Published in: |
Mathematical Finance. - Wiley Blackwell, ISSN 0960-1627. - Vol. 9.1999, 1, p. 55-96
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Publisher: |
Wiley Blackwell |
Saved in:
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