# Black-Scholes投資組合模型

## 最佳答案

To show whether it is self-financing, we need to show whether the equation \begin{align*} dV_t = a_t dS_t+b_t d\beta_t \end{align*} holds. Note that \begin{align*} V_t &= a_t S_t + b_t \beta_t\\ &=\frac{1}{2} S_t + \frac{1}{2} S_t e^{-rt} e^{rt}\\ &=S_t. \end{align*} Then \begin{align*} dV_t = dS_t. \end{align*} On the other hand, \begin{align*} a_t dS_t + b_t d\beta_t &=\frac{1}{2}dS_t + \frac{1}{2}S_t e^{-rt} \big(re^{rt}\big)dt\\ &=\frac{1}{2}dS_t + \frac{1}{2}rS_t dt\\ &\neq dS_t. \end{align*} Therefore, this is not a self-financing portfolio.