Partial Barrier Options

In this paper we study the pricing of barrier options where the period during which the underlying price is monitored for barrier hits is restricted to only part of the options' lifetime. We derive closed-form formulas for the prices of a number of partial barrier options, including partial asset- or-nothing and cash-or-nothing barrier options. Our results clearly show the importance of the length and the location of the monitoring period for the option price. It is also shown that for options where the monitoring period starts after the options' initial starting date, an alternative interpretation of the barrier condition may lead to a very substantial change in the option price