可违约金融市场中动态VaR约束的最优投资策略

Optimal Investment Strategy with Dynamic VaR Constrain in a Defaultable Financial Market

摘要: 本文研究了可违约金融市场中具有动态风险价值(VaR)约束的最优投资问题。假定投资者将资产投资于由一种无风险资产、股票和可违约债券组成的金融市场中。由于债券的违约可能导致向下跳跃的发生，这样的设定使总财富过程成为跳跃–扩散过程，而不是纯粹的扩散过程。对财富过程进行动态VaR约束达到对风险实时监控的目的，根据随机控制的原理和Karush-Kuhn-Tucker (KKT)条件将最优投资问题转化为求解非线性方程组问题，得到了幂效用函数下具有动态风险约束的最优投资策略，并且给出了相应的验证定理。最后，通过数值分析说明了模型参数以及违约对投资策略的影响。

Abstract: This paper considers the optimal investment problem with dynamic Value-at-Risk (VaR) constraint in a defaultable financial market. The wealth is assumed to be invested in a risk-free asset and two risk assets: a stock and a defaultable bond, which is a discontinuous process since there is a possibility of downside jump caused by the default of bond. Such a setting makes the total wealth process a jump diffusion process, rather than a pure diffusion process. This paper applies dynamic VaR constraint to the wealth process to achieve real-time risk monitoring. Based on the principles of stochastic control and the Karush Kuhn Tucker (KKT) condition, the optimal investment problem is transformed into a problem of solving a nonlinear system of equations. The optimal investment portfolio with dynamic risk constraint on the wealth process in a defaultable financial market is obtained when the utility function is a power function, and corresponding verification theorems are proven. Finally, numerical analysis is conducted to demonstrate the impact of model parameters and default on investment strategies.