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Black-Scholes投資組合模型

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

  1. 顯示此投資組合復制了一股股票。
  2. 顯示它是否自籌資金。
  3. 找到另一個自籌資金投資組合並復制一股股票。

我的嘗試:

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 $C_t$.

對於Q2。我相信我需要證明$ dV_t $ = $ a_tdS_t +b_tdβ_t$但我不確定究竟是怎麽做到的。

我不知道如何嘗試Q3。

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.

要找到另一個自我融資的投資組合,復制一股股票,我們可以設置$ a_t = 1 $和$ b_t = 0 $。

轉載註明原文: Black-Scholes投資組合模型