使用 Delta 對沖推導 BS PDE 問題
我一直對 Delta 對沖感到困惑。眾所周知，對於（足夠平滑的）函式 $ (S,t) $ 由於伊藤引理，我們有：
$$ \begin{eqnarray*} dC = \left(\frac{\partial C}{\partial t} (S,t) + \mu S \frac{\partial C}{\partial S} (S,t) + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}(S,t)\right)dt + \sigma S \frac{\partial C}{\partial S} (S,t) dX \end{eqnarray*} $$ （來源在這裡，在哪裡 $ X $ 是標準的布朗運動）。然後作者繼續表明，如果我們選擇 $ \Delta = -\frac{\partial C}{\partial S} (S,t) $ ，那麼我們將有 $$ \begin{align*} d(C+\Delta S) &= \left(\frac{\partial C}{\partial t} (S,t) + \mu S \frac{\partial C}{\partial S} (S,t) + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}(S,t) + \Delta \mu S\right) dt\ &+ \sigma S \left(\frac{\partial C}{\partial S}+\Delta\right) dX\ &=\left(\frac{\partial C}{\partial t}(S,t) + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}(S,t)\right)dt \end{align*} $$ （我已經更正了作者的錯字並替換了術語 $ \Delta S (\frac{\partial C}{\partial S}+\Delta) dX $ 在上面的公式中用正確的 $ \sigma S (\frac{\partial C}{\partial S}+\Delta) dX $ ) 我了解通常的程序 $ \Delta $ ，但我從來不明白如何 $ \Delta $ 可以定義為 $ -\frac{\partial C}{\partial S} (S,t) $ . 根據定義，對沖策略必須是一個可預測的過程，是否有任何理由證明 $ \Delta $ 上述給出的是可預測的，而不僅僅是適應於庫存過程產生的過濾？
即使我們接受以下定義 $ \Delta $ ，作者也沒有說明如何計算 $ d(C+\Delta S) $ . 但是，據我所知，如果 $ \Delta $ 真的是像徵 $ -\frac{\partial C}{\partial S} (S,t) $ ，那麼我們將有
$$ d(\Delta S)=Sd(\Delta)+\Delta dS+(dS)\cdot(d\Delta). $$ 至於 $ d\Delta $ ，我認為它只是替換 $ C $ 經過 $ \Delta $ 在表達 $ dC $ ，以及由此產生的 $ d(\Delta S) $ 將超級複雜 $ \partial^3 C/\partial S^3 $ 當下。我最後得到的是一個可怕的爛攤子，危險的術語 $ dX $ 沒有被淘汰和非隨機項 $ dt $ 與 BS PDE 中的不匹配。 問題出在哪裡？
這個問題已經被問過很多次了，似乎需要一些澄清。
- 正如對這個問題的回答所指出的那樣，投資組合
$$ \begin{align*} \Delta_t^1 S_t + \Delta^2_t C, \end{align*} $$ 在哪裡 $ \Delta_t^1 = -\frac{\partial C}{\partial S} $ 和 $ \Delta_t^2 =1 $ ，一般來說，既不是自籌資金，也不是本地無風險的。
- 為了推導出 Black-Scholes 的 PDE，我們尋求一個投資組合 $ \Delta_t^1 S_t + \Delta^2_t C $ 因此，它是自籌資金和本地無風險的。如對上述問題的回答所示，我們推導出了 PDE
$$ \begin{align*} \frac{\partial C}{\partial t} + r S_t \frac{\partial C}{\partial S} + \frac{1}{2}\frac{\partial^2 C}{\partial S^2} \sigma^2S_t^2 -rC = 0. \tag{1} \end{align*} $$ 和投資組合權重 $$ \begin{align*} \Delta_t^1 = -\frac{\frac{\partial C}{\partial S} B_t}{C_t - \frac{\partial C} {\partial S}S}, \quad \Delta_t^2 =\frac{B_t}{C_t - \frac{\partial C}{\partial S}S}, \tag{2} \end{align*} $$ 在哪裡 $ B_t=e^{rt} $ 是貨幣市場賬戶價值。注意 $$ \begin{align*} \Delta_t^1 S_t + \Delta^2_t C = B_t. \end{align*} $$
- 基於偏微分方程 $ (1) $ 和投資組合權重 $ (2) $ , 很容易驗證
$$ \begin{align*} \Delta_t^1 dS_t + \Delta^2_t dC = dB_t. \end{align*} $$ 也就是說，該投資組合確實是自籌資金和本地無風險的。
- 雖然雜亂無章，但基於 PDE $ (1) $ 和投資組合權重 $ (2) $ , 可以直接證明
$$ \begin{align*} d\left(\Delta_t^1 S_t + \Delta^2_t C \right) = \Delta_t^1 dS_t + \Delta^2_t dC \end{align*} $$ 使用伊藤引理。在這裡，我們假設各個偏導數，例如 $ \frac{\partial^2 C}{\partial t\partial S} $ 和 $ \frac{\partial^3 C}{\partial S^3} $ ， 存在。
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下面，我們使用伊藤引理展示自籌資金性質，即
$$ \begin{align*} Sd\Delta_t^1 + C d\Delta_t^2 + d\langle \Delta_t^1,, S \rangle_t +d\langle \Delta_t^2,, C \rangle_t =0. \end{align*} $$
重量 $ \Delta_t^1 $ ,
$$ \begin{align*} \frac{\partial \Delta_t^1}{\partial t} &= -\frac{\left(\frac{\partial^2 C}{\partial t \partial S} B_t + r\frac{\partial C}{\partial S}B_t\right) \left(C_t - \frac{\partial C}{\partial S}S \right) - \left(\frac{\partial C}{\partial t} - \frac{\partial^2 C}{\partial t \partial S}S \right)\frac{\partial C}{\partial S} B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\ &=-\frac{\frac{\partial^2 C}{\partial t \partial S} B_t C_t + r\frac{\partial C}{\partial S}B_t C_t - r \left(\frac{\partial C}{\partial S} \right)^2B_t S_t - \frac{\partial C}{\partial t}\frac{\partial C}{\partial S} B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\ &=-\frac{\frac{\partial^2 C}{\partial t \partial S} B_t C_t + \frac{1}{2}\frac{\partial^2 C}{\partial S^2} \frac{\partial C}{\partial S}\sigma^2S_t^2 B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2} \qquad\qquad\qquad\qquad\qquad\qquad \mbox{(From Eqn $(1)$)}\ &=-\frac{\frac{\partial^2 C}{\partial t \partial S} B_t C_t^2 + \frac{1}{2}\frac{\partial^2 C}{\partial S^2} \frac{\partial C}{\partial S}\sigma^2S_t^2 B_tC_t - \frac{\partial^2 C}{\partial t \partial S}\frac{\partial C}{\partial S} B_t C_t S - \frac{1}{2}\frac{\partial^2 C}{\partial S^2} \left(\frac{\partial C}{\partial S}\right)^2\sigma^2S_t^3 B_t}{\left(C_t - \frac{\partial C}{\partial S}S\right)^3},\ \frac{\partial \Delta_t^1}{\partial S} &=-\frac{\frac{\partial^2 C}{\partial S^2} B_t \left(C_t - \frac{\partial C}{\partial S}S \right) - \left(\frac{\partial C}{\partial S} - \frac{\partial^2 C}{\partial S^2}S - \frac{\partial C}{\partial S}\right)\frac{\partial C}{\partial S} B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\ &=-\frac{\frac{\partial^2 C}{\partial S^2} B_t C_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2},\ \frac{\partial^2 \Delta_t^1}{\partial S^2} &=-\frac{\left(\frac{\partial^3 C}{\partial S^3} B_tC_t + \frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S} B_t \right) \left(C_t - \frac{\partial C}{\partial S}S \right)^2 +2\left(C_t - \frac{\partial C}{\partial S}S\right) \left(\frac{\partial^2 C}{\partial S^2}\right)^2S B_t C_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^4}\ &=-\frac{\frac{\partial^3 C}{\partial S^3} B_tC_t^2 - \frac{\partial^3 C}{\partial S^3}\frac{\partial C}{\partial S} B_tC_t S + \frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S} B_t C_t - \frac{\partial^2 C}{\partial S^2}\left(\frac{\partial C}{\partial S}\right)^2 B_t S + 2\left(\frac{\partial^2 C}{\partial S^2}\right)^2S B_t C_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^3} \end{align*} $$ 重量 $ \Delta_t^2 $ ,
$$ \begin{align*} \frac{\partial \Delta_t^2}{\partial t} &= \frac{rB_t \left(C_t - \frac{\partial C}{\partial S}S \right) - \left(\frac{\partial C}{\partial t} - \frac{\partial^2 C}{\partial t \partial S}S \right) B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\ &=\frac{ rB_t C_t - r\frac{\partial C}{\partial S} B_t S - B_t \frac{\partial C}{\partial t} +\frac{\partial^2 C}{\partial t \partial S}S B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\ &=\frac{\frac{1}{2}\frac{\partial^2 C}{\partial S^2} \sigma^2S_t^2 B_t +\frac{\partial^2 C}{\partial t \partial S}S B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2} \qquad\qquad\qquad\qquad\qquad\qquad \mbox{(From Eqn $(1)$)}\ &=\frac{\frac{1}{2}\frac{\partial^2 C}{\partial S^2} \sigma^2S_t^2 B_tC_t +\frac{\partial^2 C}{\partial t \partial S}S B_tC_t - \frac{1}{2}\frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S} \sigma^2S_t^3 B_t -\frac{\partial^2 C}{\partial t \partial S}\frac{\partial C}{\partial S}S^2 B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2},\ \frac{\partial \Delta_t^2}{\partial S} &=\frac{ - \left(\frac{\partial C}{\partial S} - \frac{\partial^2 C}{\partial S^2}S - \frac{\partial C}{\partial S}\right) B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\ &=\frac{\frac{\partial^2 C}{\partial S^2} B_t S }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2},\ \frac{\partial^2 \Delta_t^2}{\partial S^2} &=\frac{\left(\frac{\partial^3 C}{\partial S^3} B_t S + \frac{\partial^2 C}{\partial S^2} B_t \right) \left(C_t - \frac{\partial C}{\partial S}S \right)^2 +2\left(C_t - \frac{\partial C}{\partial S}S\right) \left(\frac{\partial^2 C}{\partial S^2}\right)^2S^2 B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^4}\ &=\frac{\frac{\partial^3 C}{\partial S^3} B_tC_t S - \frac{\partial^3 C}{\partial S^3}\frac{\partial C}{\partial S} B_t S^2 + \frac{\partial^2 C}{\partial S^2} B_t C_t - \frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S} B_t S + 2\left(\frac{\partial^2 C}{\partial S^2}\right)^2S^2 B_t }{\left(C_t - \frac{\partial C}{\partial S}S\right)^3}. \end{align*} $$ 然後
$$ \begin{align*} &\ d\langle \Delta_t^1,, S \rangle_t +d\langle \Delta_t^2,, C \rangle_t \ =& \left(-\frac{\frac{\partial^2 C}{\partial S^2} B_t C_t \sigma^2 S^2}{\left(C_t - \frac{\partial C}{\partial S}S\right)^2} +\frac{\frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S}\sigma^2 B_t S^3 }{\left(C_t - \frac{\partial C}{\partial S}S\right)^2}\right)dt\ =& \left(\frac{-\frac{\partial^2 C}{\partial S^2} B_t C_t^2 \sigma^2 S^2 + \frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S} B_t C_t \sigma^2 S^3 + \frac{\partial^2 C}{\partial S^2}\frac{\partial C}{\partial S}\sigma^2 B_t S^3 C_t - \frac{\partial^2 C}{\partial S^2}\left(\frac{\partial C}{\partial S}\right)^2\sigma^2 B_t S^4}{\left(C_t - \frac{\partial C}{\partial S}S\right)^3} \right)dt. \end{align*} $$ 而且， $$ \begin{align*} Sd\Delta_t^1 + C d\Delta_t^2 &= S\left(\frac{\partial \Delta_t^1}{\partial t}dt + \frac{1}{2}\frac{\partial^2 \Delta_t^1}{\partial S^2}\sigma^2 S^2 dt + \frac{\partial \Delta_t^1}{\partial S}dS \right) \ &\qquad + C\left(\frac{\partial \Delta_t^2}{\partial t}dt + \frac{1}{2}\frac{\partial^2 \Delta_t^2}{\partial S^2}\sigma^2 S^2 dt + \frac{\partial \Delta_t^2}{\partial S}dS \right)\ &=S\left(\frac{\partial \Delta_t^1}{\partial t} + \frac{1}{2}\frac{\partial^2 \Delta_t^1}{\partial S^2}\sigma^2 S^2\right)dt + C\left(\frac{\partial \Delta_t^2}{\partial t} + \frac{1}{2}\frac{\partial^2 \Delta_t^2}{\partial S^2}\sigma^2 S^2 \right)dt. \end{align*} $$ 通過將所有術語組合在一起，我們可以證明 $$ \begin{align*} Sd\Delta_t^1 + C d\Delta_t^2 + d\langle \Delta_t^1,, S \rangle_t +d\langle \Delta_t^2,, C \rangle_t =0. \end{align*} $$ 然後 $$ \begin{align*} d\left(\Delta_t^1 S_t + \Delta_t^2 C_t \right) = \Delta_t^1 dS_t + \Delta_t^2 dC_t, \end{align*} $$ 那是， $ \left(\Delta_t^1, , \Delta_t^2\right) $ 構成自籌資金戰略。