Article 3

of time the underlying asset price spends outside of a pre-specified price range (occupation time). Proportional and simple double-barrier step options are gradual knockout options with the principal amortized based on the occupation time outside of the range. Delayed double-barrier options are extinguished when the occupation time outside of the range exceeds a prespecified knock-out window (delayed knockout). These contract designs are proposed as alternatives to the currently traded double-barrier options. They alleviate discrete “barrier event” risk and are easier to hedge.