Closed-form solutions for options with random initiation under asset price monitoring

•This paper considers derivatives to prepare for financial risk that may arise randomly.•Options initiated by an unexpected event and connected with a barrier of knock-in or knock-out type are studied.•Closed-form solutions for pricing options with random initiation are obtained.

This paper studies derivatives to prepare for financial risk from unexpected events. It is difficult for firms and financial institutions to hedge losses triggered by natural catastrophes such as earthquakes, by using derivative securities with fixed initiation and maturities. In this context, we consider an option that is initiated at random by an unexpected event, and moreover, is connected with a barrier of knock-in or knock-out type for asset price monitoring until the time of event. We derive closed-form valuation formulas for these options.