The Return Barrier and Return Timer Option with Pricing under Lévy Processes

This work introduces two new financial derivatives into the finance literature. The first is the Return Barrier Option, which has emerged recently as a popular contract in the OTC markets. This contract is similar to a barrier option, but the knock-out event depends on an asset's returns, rather than its level. Its direct exposure to jump risk offers protection from a key source of market incompleteness, with great potential value to hedgers and investors. The second contract is the Return Timer Option, which is proposed in this work for its natural relation to the Return Barrier Option, and by analogy to Timer Options written on realized volatility. This contract offers similar upside potential to a European option, but can be purchased at a substantial discount, a trade-off for bearing jump risk. Due to the dependence of these contracts on single (daily) return events, capturing jump risk is essential to their pricing, as diffusion-based models are unable to generate sufficiently heavy tails at short time horizons. Thus we provide analytical pricing formulas for the Return Barrier Option, along with efficient and accurate numerical methods under Exponential L\'evy dynamics, to capture the heavy-tailed returns.We then prove that the Return Timer Option can be conveniently priced as a linear combination of Return Barrier Options, providing a consistent pricing framework for the combined family of derivatives. Given the recent trend in Return Barrier Options in OTC markets, as well as the interesting mathematics required to price them, this work offers both practical and theoretical contributions to the literature