Abstract

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). 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. 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. 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.