投資組合的 Black-Scholes 模型

給定 Black 和 Scholes 模型，考慮投資組合 $ a_t $ = 1/2, $ b_t $ = $ 1/2 $ $ S_t $ $ exp(-rt) $ .

1. 證明這個投資組合複製了一股股票。
2. Show if it is self-financing.
3. Find another portfolio which is self financing and replicates one share of stock.

My Attempt:

I’m fairly sure that for Q1, I need to show that this is a arbitrage free portfolio by showing $ C_t $ = $ V_t $ , and not $ C_t $ > $ V_t $ or $ C_t $ < $ V_t $ with $ V_t $ = $ a_t $ $ S_t $ + $ b_t $ $ β_t $ . However I’m not entirely sure how to find out $ C_t $ .

For Q2. I believe I need to show that $ dV_t $ = $ a_tdS_t+b_tdβ_t $ but am not sure how exactly to do that.

I have no idea how to attempt Q3.

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.

To find another self-financing portfolio that replicates one share of the stock, we can simply set $ a_t=1 $ and $ b_t=0 $ .

引用自：https://quant.stackexchange.com/questions/26161