Pricing of geometric Asian power barrier options

We study the pricing of fixed strike geometric Asian power barrier options in the Black–Scholes model. Specifically, we derive the joint transforms for the prices of geometric Asian power barrier options. These transforms are expressed in terms of the joint transform of the first passage time and the area covered by a drifted Brownian motion until its first passage time. Additionally, we derive in-out parity and put-call parity relations for barrier options. The prices of geometric Asian power barrier options can be computed by numerical inversion of the transforms for the prices, along with in-out parity and put-call parity relations. To illustrate our theoretical findings, we provide numerical results for the prices of all eight types of single barrier options.