In this dissertation, we investigate two problems: constructing optimal variance swaps portfolios and estimating Greeks for options with underlying assets following a Variance Gamma process. By modeling the dependent non-Gaussian residual in a linear regression model through a L'evy Mixture (LM) model and a Variance Gamma Correlated (VGC) model, and running some optimizations, we construct an optimal variance swap portfolio. By implementing gradient estimation techniques, we estimate the Greeks for a series of basket options called Mountain Range options.

Constructing an optimal variance swap portfolio consists of two steps: evaluations and optimization. Each variance swap has two legs: a fixed leg (also called the variance strike) and a floating leg (also called the realized variance). The value of a variance swap is the discounted difference between the realized variance and the variance strike. For the latter, one can use an option surface calibration to evaluate. For the former, the procedure is complicated due to the non-negligible residuals from a linear regression model. Through LM and VGC, we can estimate the realized variance on different sample paths and obtain the payoff of a variance swap numerically. Based on these numerical results, we can apply the optimization method to construct an optimal portfolio.

In the second part of this dissertation, we consider gradient estimation for Mountain Range options including Everest options, Atlas options, Altiplano/Annapurna options and Himalayan options. Assuming the underlying assets follow a Variance-Gamma (VG) process, we derive estimators for sensitivities such as Greeks through Monte Carlo simulation. We implement and compare using numerical experiments several gradient estimation approaches: finite difference methods (forward difference), infinitesimal perturbation analysis (IPA), and likelihood ratio (LR) method using either the density function or the characteristic function.