体制转换下等价鞅测度的构造研究

Construction of Equivalent Martingale Measure in the Regime-switching Model

体制转换资产价格模型可以描述宏观经济的影响,但在金融衍生产品定价所涉及的等价鞅测度的构造问题中,利用传统的Esscher变换方法得到的等价鞅测度实质上只考虑了微观市场风险,而没有考虑体制转换所表示的宏观经济风险.此外,经典的几何布朗运动不能刻画资产收益率的尖峰厚尾现象.本文首先利用马尔科夫过程和最大化非广延熵分布建立了一个新的资产价格模型.该模型可以同时描述体制转换和尖峰厚尾现象.然后利用鞅理论,借助微观市场的资产价格过程和宏观经济的马尔科夫过程的乘积给出了一种新的等价鞅测度构造方法,通过该方法构造的等价鞅测度包含了微观市场和宏观经济两种风险.最后,在该等价鞅测度下,给出了资产价格折现过程为鞅的充要条件,为进一步研究金融衍生产品的定价及风险控制提供了理论基础.

Price models with regime switching can describe the impact of macroeconomics. However, when constructing an equivalent martingale measure to price financial derivatives, the equivalent martingale measure obtained by the traditional Esscher transform considers only micro-marketing risks but not macroeeonomic risks represented by the regime switching. In addition, the classical geometric Brownian motion can not characterize higher peak and fat tail phenomena of asset returns. In this paper~ firstly~ by using a Markov process and a non-extensive maximum entropy distribution, a new price model which can describe both the phenomena of higher peak, fat tail and regime-switching is constructed. Then, by utilizing the martingale theory and the product of the price process of micro market and the Markov process of macroeconomics, a new method for constructing equivalent martingale measure is provided. The equivalent martingale measure obtained by this approach includes two kinds of risks： micro-marketing risks and macroeconomic risks. Finally, under the resulted equivalent martingale measure, the necessary and sufficient conditions are given with which the discounted asset price process can become a martingale. The results provide a theoretical basis for further studying derivative pricing and risk control.