Electronic Resource
Boston, USA and Oxford, UK
:
Blackwell Publishers Inc
Mathematical finance
9 (1999), S. 0
ISSN:
1467-9965
Source:
Blackwell Publishing Journal Backfiles 1879-2005
Topics:
Mathematics
,
Economics
Notes:
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(-ρτB-) or max(1-ρτ-B,0), where &\tau;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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1111/1467-9965.00063
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