### Abstract

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^{-}_{B}) or max(1 - ρ τ^{-}_{B}, 0), where τ^{-}_{B} 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.

Original language | English (US) |
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Pages (from-to) | 55-96 |

Number of pages | 42 |

Journal | Mathematical Finance |

Volume | 9 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1999 |

### Keywords

- Barrier options
- Feynman-Kac formula
- Laplace transform
- Occupation time
- Path-dependent options

### ASJC Scopus subject areas

- Accounting
- Finance
- Social Sciences (miscellaneous)
- Economics and Econometrics
- Applied Mathematics

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## Cite this

*Mathematical Finance*,

*9*(1), 55-96. https://doi.org/10.1111/1467-9965.00063