Statistical Modeling of Diffusion Processes with Financial Applications

Abstract: This thesis consists of five papers (Paper A-E) on statistical modeling of diffusion processes. Two papers (Paper A & D) consider Maximum Likelihood estimators for non-linear diffusion processes. An offline Maximum Likelihood estimator is derived in Paper A, and it is shown that this estimator is computationally more efficient than other Maximum Likelihood estimators. The offline algorithm is modified into an online algortihm in Paper D, where it is shown that the statistical properites are preserved while the computational cost is reduced. Model validation is discussed in Paper B and C. A general and numerically robust definition of Gaussian residuals for diffusion processes is presented in Paper B, where it is shown that these residuals are independent and identically distributed under the null hypothesis. Paper C adds suggestions on how these residuals can be tested to detect non-linear dependence. Finally, paper E introduces a simple bias correction framework to the Black & Scholes option pricing formula by correcting for parameter uncertainty. This correction improves the predictive performance of the Black & Scholes formula significantly. Furthermore, it is argued that the bias correction in paper E can be extended to more advanced option pricing models.