We propose and study two approaches for the pricing problem of three financial derivatives viz. Asian option, variance swap, and VIX option. In Chapter 1, we review recent research on these three types of financial products. In Chapter 2, we develop a Markov chain-based approximation method to price arithmetic Asian options for short maturities under the case of geometric Brownian motion. We demonstrate that this method achieves faster convergence and exhibits stability properties in hedging parameters. We also consider the pricing and hedging of floating-strike Asian options and fixed-strike in-progress Asian options and present that our method is as good as and sometimes better than existing approximation methods in the literature. In Chapter 3, we utilize Ito-Taylor expansion to solve the variance swap, which is based on discretely sampled variance formula under multi-dimensional stochastic volatility processes. We present numerical results to show that this approach is accurate with short maturities. In Chapter 4, we propose a novel analytical method to valuate VIX derivatives under the general class of stochastic volatility models, within which the current literature only considers a few special cases. The approach is based on a closed-form approximation of the VIX through the Ito-Taylor expansion and the continuous-time Markov chain (CTMC) approximation. The formula is in explicit closed-form and does not involve numerical inversions, in contrast to the existing literature. We test our method under several stochastic volatility models and demonstrate that it is accurate and efficient by comparing it with benchmarks in the literature and Monte Carlo simulations.
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