Abstract
This article offers a general unifying treatment of barrier options. The unifying treatment is based on a general representation of the risk-neutral density of the absorbed return process of the underlying asset: the density. On the basis of the convolution density, the article establishes relationships between plain and barrier options as well as knock-outs and knock-ins: the parities. The plain/knock parities provide new static hedging strategies for the replication of double barrier options; a double barrier option is a portfolio of single barrier options. The article then derives new representations for the analytical solution of option prices in the double barrier setting. For the first time, the analytical solution of the price of the contract with a single knock-in triggering a single knock-out is offered, and new representations of the analytical solution of the price of double knock-ins and knock-outs are also offered. The form of these analytical solutions is a series which absolutely converges at a very high rate.
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