Abstract: Value at Risk is defined as the maximum loss of a portfolio given a future time horizon within high confidence (or probability, typical values used are 95% or 99%). In our work we devise novel techniques to minimize the non-convex Value-at-Risk function. VaR has the following properties: 1. VaR is a non-coherent measure of risk; in particular, it is not sub-additive. 2. VaR also happens to be a non-convex (multiple local solutions). The above properties make search for a global minimum of VaR a very diffcult problem, in fact an NP-hard problem. CVaR is a coherent and convex measure of risk and we use homotopy methods to project CVaR optimal solutions to VaR optimum. The results show that optimal VaR is within 1% of global minimum if found and as efficient as finding a solution to a convex conditional-VaR minimization problem.

Abstract: We propose a pairs trading model that incorporates a time-varying volatility of the Constant Elasticity of Variance type. Our approach is based on stochastic control techniques; given a fixed time horizon and a portfolio of two cointegrated assets, we define the trading strategies as the portfolio weights maximizing the expected power utility from terminal wealth. We compute the optimal pairs strategies by using a Finite Difference method. We then show some empirical tests on data of stocks that are dual listed in Shanghai and Hong Kong of China, with low frequency and high frequency.

Abstract: Consider a hedging strategy g(s) for using short-term futures contracts to hedge a long-term exposure. Here the underline commodity $S_t$ follows the stochastic differential equation $d S_t = \mu dt + \sigma dW_t$. It is known that the full hedging is not a good choice in terms of the risk. We establish a numerical approach for searching a strategy g(s) which reduces the running risk of the hedging. The approach also leads to the numerical solution of the optimal strategy for such a hedging problem.

Abstract: Research in finance and lots of other areas often encounter large-scale complex optimization problems that are hard to find solutions. Classic heuristic algorithms often have limitations from the objectives that they are trying to mimic, leading to drawbacks such as lacking memory-efficiency, trapping in local optimal solutions, unstable performances, etc. This work considers imitating market competition behavior (MCB) and develops a novel heuristic algorithm accordingly, which combines characteristics of searching-efficiency, memory-efficiency, conflict avoidance, recombination, mutation and elimination mechanism. In searching space, the MCB algorithm updates solution dots according to the inertia and gravity rule, avoids falling into local optimal solution by introducing new enterprises while ruling out of the old enterprises at each iteration, and recombines velocity vector to speed up solution searching efficiency. This algorithm is capable of solving large-scale complex optimization model of large input dimension, including Over Lapping Generation Models, and can be easily applied to solve for other complex financial models. As a sample case, MCB algorithm is applied to a hybrid investment optimization model on R&D, riskless and risky assets over a continuous time period.