Moments, time-depending rebates and the put-call parity for barrier options

This paper derives put-call parity relations for barrier options via a probabilistic approach. As in the case of standard options, the difference between call and put prices depends on the stock price and the discounted exercise price. However, these terms now become functions of the moments of the relevant defective distributions and as such correct for the loss of probability that the barrier provisions bring. Moments are also useful building blocks for the valuation of a wide variety of time-depending rebate specifications that are interesting both from a theoretical as well as a practitioner’s point of view. For instance, rebate payments that are larger when the option is knocked out later recognize the higher opportunity costs caused by the option having immobilized funds over a longer period of time.