Optimal Variance Swaps Portfolios and Estimating Greeks for Variance-Gamma

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

Modeling VIX And VIX Derivatives With Mean Reverting Models And Parameter Estimation Using Filter Methods

In this thesis, we study the mean reverting property of the VIX time series, and use the VIX process as the underlying. We employ various mean reverting processes, including the Ornstein-Uhlenbeck (OU) process, the Cox-Ingersoll-Ross (CIR) process and the OU processes driven by Levy processes (Levy OU) to fit historical data of VIX, and calibrate the VIX option prices. The first contribution of this thesis is to use the Levy OU process to model the VIX process, in order to explain the observed high kurtosis. To price the option using the Levy OU process, we develop a FFT method. The second contribution is to build a joint framework to consistently model the VIX and VIX derivatives together on the entire time series of market data. We choose multi-factor mean-reverting models, in which we model the VIX process as a linear combination of latent factors. To estimate the models, we use Euler approximation to find a discrete approximation for the VIX process. Based on this approximate, we consider various filter methods, namely, the Unscented Kalman Filter (UKF), constrained UKF, mixed Gaussian UKF and Particle Filter (PF) for estimation. The performances of these models are compared and discussed. Radon Nikodym derivatives of the risk-neutral measure are discussed with respect to the physical measure for the jumps. A simple dynamic trading strategy was tested on these models