Stochastic Modeling with Temporally Dependent Gaussian Processes Applications to Financial Engineering, Pricing and Risk Management.

摘要

In this dissertation we focused on developing modeling techniques to handle the complex and fast paced world of finance. In 2009, trading algorithms had enabled execution of trades in under 128 microseconds. Applications in finance to modeling and decision making requires quick, efficient methods that are accurate on small sample sizes. In this motivation we developed two new estimators for the parameters of a fractional Wiener process. We showed that our Ratio method estimator of the Hurst index is significantly more accurate than Whittle's approximate MLE on small sample sizes (n=128 data points or less) for 0.4H0.8. Additionally, analysis showed that the Ratio method estimator shows no significant difference in accuracy for large sample sizes. To address the need for robust estimation of parameters, our second estimator of the Hurst index, which we call the Quadrant method, ignores the magnitude of movements, but measures the correlation structure of series of two dimensional Gaussian random variables. This method, while not as accurate as Whittle's MLE or the Ratio method, outperforms the second best estimator (according to Taqqu [Taqqu et al. 1995]), known as the Variance of Residuals method. Both of our methods are of the order of 104 times faster than Whittle's approximate MLE, and are approximately 500 times faster than the Variance of Residuals method. Our new methods address the disproportionate trade-off in accuracy and computation time that is present in current methods.