Focused Enhancements in Exponential Lévy Models: The Bilateral Gamma Motion

In the realm of mathematical finance, the calibration of Lévy models to market options data stands as a pivotal task. This thesis offers a comprehensive exploration of the Bilateral Gamma Motion process, an evolved variant of the Bilateral Gamma process, newly introduced this year. With an incorporation of a diffusion component, this novel development addresses and rectifies the limitations previously identified in the classic process. Alongside this, the thesis compares the Bilateral Gamma Motion model with six other Lévy models in the context of calibration and option pricing, detailing each model's parameters and their effects on exotic options valuation. Options explored include Asian options, Barrier options, and occupation time derivatives such as Step and Fader options.