PhD Oral Exam - Xiang Gao, Mathematics & Statistics

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

We consider three problems motivated by mathematical and computational finance which utilize forward-backward stochastic differential equations (FBSDEs) and other techniques from stochastic control. Firstly, we review the case of post-retirement annuitization with labor income as stochastic control and optimal stopping problems. We apply the martingale approach to consider a Cobb–Douglas type objective function. We have proved the theoretical existence and uniqueness of an optimal solution. Several analysis are made based on the simulations for the optimal stopping choice and strategies. Secondly, we review the application of backward stochastic differential equations (BSDEs) and the option pricing problem with Heston model. We obtain the conditional density function for a Heston stochastic volatility model and examined the preliminary result of fast Fourier transform (FFT). Thirdly, we review the yield curve forecasting problem in bond markets. Our data include both U.S. Treasuries and coupon bonds from twelve corporate issuers. Our approach to predicting bond prices is divided into two steps. First, we estimate the zero coupon bond yield from coupon bonds. Second, we forecast an arbitrage-free yield curve with the help of deep neural networks and machine learning techniques. Our main study is focused on predicting the arbitrage-free yield curve using a dynamic Nelson-Siegel model and a Va\v{s}\'{i}\v{c}ek factor model. We obtain the arbitrage penalty by combing the HJM forward rate model with the dynamic Nelson-Siegel term structure. Prediction is conducted by the Kalman filter and the particle filters, with factors selection through sequential method by recurrent neural network (RNN). Our result shows that the predicted yield curve has very small arbitrage opportunity and small errors in both training set and testing set. The predicted bond prices shows the prediction errors follows non-Gaussian distribution with excess kurtosis and fat tails. Future works will be from two aspects, refine the importance sampling by non-parametric distribution and refine the term structure model with jump process and credit risk.