PhD Oral Exam - Renjie Wang, 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.
We first associate the bond price with an optimal measure transformation problem which is closely related to decoupled nonlinear forward-backward stochastic differential equation (FBSDE).1 In the default-free case we prove the equivalence of the optimal measure transformation problem and an optimal stochastic control problem of Gombani and Runggaldier (Math. Financ. 23(4):659-686, 2013) for bond price in the framework of quadratic term structure models. The measure which solves the optimal measure transformation problem is the forward measure. These connections explain why the forward measure transformation employed in the FBSDE approach of Hyndman (Math. Financ. Econ. 2(2):107-128, 2009) is effective. We obtain explicit solutions to FBSDEs with jumps in affine term structure models and quadratic term structure models, which extend Hyndman (Math. Financ. Econ. 2(2):107-128, 2009). From the optimal measure transformation problem for defaultable bonds, we derive FBSDEs with random terminal condition to which we give a partially explicit solution. The futures price and the forward price of a risky asset are also considered in the framework of optimal measure transformation problems.
In the second part we consider trading against a hedge fund or large trader that must liquidate a large position in a risky asset if the market price of the asset crosses a certain threshold.2 Liquidation occurs in a disorderly manner and negatively impacts the market price of the asset. We consider the perspective of small investors whose trades do not induce market impact and who possess different levels of information about the liquidation trigger mechanism and the market impact. We classify these market participants into three types: fully informed, partially informed and uninformed investors. We consider the portfolio optimization problems and compare the optimal trading and wealth processes for the three classes of investors theoretically and by numerical illustrations. Finally we study the portfolio optimization problems with risk constraints and make comparison with the results without risk constraints.
1. Based on the paper with Cody Hyndman.
2. Based on the paper with Caroline Hillairet, Cody Hyndman and Ying Jiao.