Stochastic approaches are used in modern financial analysis to explore the underlying dynamics of securities like stocks and options. Statistical modeling and inferences within this aspect is an important concern because pricing errors could lead to serious economic losses. In this thesis, statistical estimation motivated by real applications are developed for inferences under stochastic diffusion processes using tensor method and kernel smooth method.

We consider in Chapter 2 parameter estimation for multi–factor stochastic processes defined by stochastic differential equations. The class of processes considered are multivariate diffusion which are popular processes in modeling the dynamics of financial assets. We quantify the bias and variance by developing theoretical expansions for a large class of estimators which includes as special cases estimators based on the maximum likelihood, approximate likelihood and discretizations. We apply the proposed methods to evaluate bias in estimated contingent claims. We also provide simulation results for a set of popular multi-factor processes to confirm our theory.

Our Chapter 3 is dedicated to improve the estimation of the market volatility, specifically the VIX index introduced by Chicago Board Option Exchange (CBOE). This index provides a way to measure the 30–day expected volatility of the S & P 500 index. Among a few ways to estimate it, the CBOE and the Goldman Saches had developed an estimator based on the concept of fair value of future variance. In realizing the discretization error, truncation error, and the approximation error in their estimator, as well as the possible option pricing errors involved, we propose a new method that combines the CBOE method and the kernel smoothing method. We derive the weak convergence property of our estimator. Simulation is run to justify the improvement.