This dissertation studies a couple of variance-dependent instruments in the financial market. Firstly, a number of aspects of the variance swap in connection to the Barndorff-Nielsen and Shephard model are studied. A partial integro-differential equation that describes the dynamics of the arbitrage-free price of the variance swap is formulated. Under appropriate assumptions for the first four cumulants of the driving subordinator, a Ve\v{c}e\v{r}-type theorem is proved. The bounds of the arbitrage-free variance swap price are also found. Finally, a price-weighted index modulated by market variance is introduced. The large-basket limit dynamics of the price index and the ``error term" are derived. Empirical data driven numerical examples are provided in support of the proposed price index. We implemented Feynman path integral method for the analysis of option pricing for certain L\'evy process-driven financial markets. For such markets, we find closed form solutions of transition probability density functions of option pricing in terms of various special functions. Asymptotic analysis of transition probability density functions is provided. We also find expressions for transition probability density functions in terms of various special functions for certain L\'evy process-driven markets where the interest rate is stochastic.