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Even though a consensus has been reached a long time ago that the interest rate is a stochastic process, most of the existing works on dynamic mean-downside risk portfolio selection still focus on a deterministic interest rate. This work studies a dynamic mean-downside risk portfolio selection problem with a stochastic interest rate in a continuous-time financial market. Specifically, we choose the lower-partial moments(LPM), value-at-risk (VaR), and conditional value-at-risk(CVaR) to model our downside risk measures in the Vasicek interest rate model. By using the martingale method and the inverse Fourier Transformation, we successfully derive the semi-analytical optimal portfolio policies and the optimal wealth processes for the mean-downside risk measures with a stochastic interest rate. The results suggest that the higher the degree of negative correlation, the better the performance of the portfolio valued by mean-downside risk framework when the other market opportunity set is the same.

Instead of using the classical block-shaped market depth to build the optimal execution model, this work studies the constrained optimal execution problem in a limit order book (LOB) market with a power-shaped market depth. Different from the linear price impact derived from the framework of block-shaped market depth, the price impact generated from the framework of our power-shaped market depth becomes a nonlinear function, which is consistent with the empirical studies. We also consider a class of state-dependent upper and lower bound constraints on trading strategies, which includes non-negative constraint (or non-short selling constraint) as its special case. Even though both the power-shaped market depth and trading strategy constraints make it hardly to solve such an optimal execution problem analytically, we still develop some significant properties on the optimal execution policy and optimal execution cost of our model. From some illustrative examples, we find that the optimal execution policy derived from our model is quite different from the one generated from the model with the block-shaped market depth when the market exhibits finite resiliency. However, when the market resilience is infinite, the obtained result becomes different, i.e., the optimal execution policies derived from these two kind of models are equivalent. For a special model with the stochastic block-shaped market depth and infinite market resilience, we successfully derive the analytical solution for such an optimal execution problem by utilizing the state separation property induced from its structure. The revealed optimal execution strategy is a piece-wise affine function with respect to the current remaining position, which can be computed off-line efficiently by solving two coupled equations. Finally, due to its explicit solution, we utilize this optimal execution model to demonstrate that the model admits no price magnification opportunity for the two-sided trading strategy.

This paper studies the dynamic mean-risk portfolio optimization problem with variance and Value-at-Risk(VaR) as the risk measures in recognizing the importance of incorporating different risk measures in the portfolio management model. Using the martingale approach and combining it with the quantile optimization technique, we provide the solution framework for this problem and show that the optimal terminal wealth may have different patterns under a general market setting. When the market parameters are deterministic, we develop the closed-form solution for this problem. Examples are provided to illustrate the solution procedure of our method and demonstrate the beneft of our dynamic portfolio model comparing with its static counterpart.

The housing markets have played an important role in macroeconomic fluctuations, especially during the recent financial crisis. We investigate the correlation between housing dynamics and the business cycle for a variety of countries. Our empirical results exhibit the two daunting facts faced by lots of macroeconomic modelers: (i) house prices are highly volatile and closely correlated with the business cycle, which is at odds with the evidence that rental prices are relatively stable and almost uncorrelated with the business cycle; and (ii) residential investment leads the business cycle while nonresidential investment moves contemporaneously with the business cycle.