M-P逆与一般B-S模型中的等价鞅测度

在一般B-S模型中,为了得到等价鞅测度,首先必须解出风险溢价方程.为此,经典方法总是假定扩散矩阵几乎处处是满秩的.文章利用代数中的M-P逆理论,证明了即使扩散矩阵不满足这个条件,M-P逆方法是一种求解风险溢价方程的有效方法.根据最小化风险溢价过程模的标准,文章利用M-P逆找到了一个唯一的等价鞅测度,并证明了在一定条件下,B-S模型中的Esscher测度、最小熵鞅测度和逆相对熵鞅测度鞅测度实际上就是这个等价鞅测度.

Moore-Penrose Pseudo-Inverse and Equivalent Martingale Measures in General Black-Scholes Model

To find an equivalent martingale measure in General Black-Scholes model,we need to solve the market price of risk equations firstly.However,classical methods always assume the dispersion matrix is full rank almost surely when deriving a specific market price of risk.By Moore-Penrose pseudo-inverse theory in algebra,this paper proofs that Moore-Penrose pseudo-inverse method is an efficient technique to solve the market price of risk even if the dispersion matrix is not always full rank.According to the criterion that minimizes the Frobenius norm of market price of risk,we can find a unique equivalent martingale measure by virtue of the Moore-Penrose pseudo-inverse of dispersion matrix.It goes to prove that our equivalent martingale measure is the same as the Esscher transformed martingale measure,the minimal entropy martingale measure and the minimal reverse entropy martingale measure under the certain conditions in General Black-Scholes model.