Valuation of multi-period barrier options and extensions

Barrier options are widely used financial instruments with a long history. Because their payoff depends on whether or not the underlying asset has reached a predetermined barrier level, these options are cheaper than the corresponding standard options. This distinctive feature makes them a popular investment tool traded on various markets, including over-the-counter and foreign exchange. In addition, barrier options are used in pricing and modeling problems in both finance and insurance, including construction of more accurate pricing and hedging models of variable annuity products and pricing of popular auto-callable, or auto-trigger, derivatives.

Motivated by these applications, this thesis focuses on pricing extensions of barrier options that are characterized by having piecewise constant horizontal barriers as well as the vertical ones, which are referred to as icicles. To perform valuation of these options, we develop an instrument called the static hedging theorem that does not rely on traditional tools, such as integration or solving the Black-Scholes differential equation, and instead makes use of the technique of Esscher transform and the reflection principle. By applying this theorem, the pricing formulas for a large variety of barrier options are obtained, including compound, sequential, and double barrier ones, with quite a few formulas not having been derived in the existing literature, to the best of the author’s knowledge. Lastly, we develop variations of the static hedging theorem that allow pricing options with exponential barriers and barrier options written on multiple assets, and demonstrate its use by deriving closed-form expressions for certain options.