Statistical inferences and computing for diffusion models in finance
Stochastic models play a crucial role in modern finance. With uncertainty built in the models, the stochastic models make a complex problem mathematically tractable. For example, diffusion models represented by stochastic differential equations are widely used in asset pricing and financial econometrics. Cutting-edge statistical methods are being used in inferences and computations for these stochastic models. With the advent of modern computer technologies and widely available computing resources, complex stochastic models are becoming more and more popular in the research as well as financial practice. This thesis focuses on estimation and computations for financial problems with diffusion based models. It consists of three parts. Part one investigates efficient Quasi-Monte Carlo (QMC) method for simulating interest rates from a stochastic string model and computing option prices based on the interest rate model. The stochastic string model is governed by a stochastic differential equation driven by a Brownian sheet. Compared with ordinary Monte Carlo(MC) method, QMC method uses low discrepancy sequences which allocates sample points more evenly and achieves a much faster convergence rate. As an alternative simulation scheme, we propose the QMC method to simulate Brownian sheet and then to simulate the interest rate process from the stochastic string model. The QMC method is much more efficient than MC based methods. A real application on pricing interest rate based options is implemented. Part two deals with a nonparametric kernel density estimation of spot volatility based on high frequency financial data. With data from a continuous-time stochastic volatility (SV) model, where the spot volatility is governed by a stochastic differential equation, we propose to estimate the spot volatility based on the data from the SV model by the kernel method. We established asymptotic distributions for the kernel estimator at a fixed time point as well as for the maximum difference between the kernel estimator and the spot volatility over entire time range. Simulations based on a bivariate SV model show that the asymptotic distributions performs well for finite data with reasonably large sample size. Part three studies a wavelet-based analysis of high frequency financial data. We model the true logarithm of price process by jump-diffusion model with stochastic volatility. The observed high frequency data are assumed to have additive microstructure noise. We locate jumps from the noisy high frequency data and then estimate jump variation and integrated volatility separately. Large scale simulation studies are conducted for common SV models to check the performance of the proposed methods. The simulations show that our methods are comparable with existing methods when data are assumed to have no jumps or no noise and are superior to existing methods when data have both jumps and noise.
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