# Black-Scholes方程 - 無風險投資組合推導

$dB_t = B_t r dt$

$dS_t = S_t（\ mu dt + \ sigma dW_t）$

$X_t = V（t，S_t） - \ frac {\ partial V} {\ partial S} S_t$（財富過程的定義）

$dX_t = dV（t，S_t） - \ frac {\ partial V} {\ partial S} dS_t$（假設投資組合為自籌資金）

$dX_t = dV（t，S_t） - d（\ frac {\ partial V} {\ partial S} S_t）$（簡單來說就是差異的定義）

$dV（t，S_t） - \ frac {\ partial V} {\ partial S} dS_t = dV（t，S_t） - d（\ frac {\ partial V} {\ partial S} S_t）$

Using Ito's lemma on the RHS gives: $\frac{\partial V}{\partial S}dS_t = d(\frac{\partial V}{\partial S})S_t + \frac{\partial V}{\partial S}dS_t + d<\frac{\partial V}{\partial S},S>_t$.

And so $d(\frac{\partial V}{\partial S})S_t + d<\frac{\partial V}{\partial S},S>_t = 0$. (*)

Now, $d(\frac{\partial V}{\partial S}) = \frac{\partial^2 V}{\partial S \partial t} dt + \frac{\partial^2 V}{\partial S^2} dS_t + \frac{1}{2}\frac{\partial^3 V}{\partial S^3}d_t$.

Therefore $d<\frac{\partial V}{\partial S},S>_t = \frac{\partial^2 V}{\partial S^2}S_t^2 \sigma^2 dt$.

$\frac{\partial^2 V}{\partial S \partial t} dt + \frac{\partial^2 V}{\partial S^2} dS_t + \frac{1}{2}\frac{\partial^3 V}{\partial S^3}d_t + \frac{\partial^2 V}{\partial S^2}S_t^2 \sigma^2 dt = 0$.

$dS_t$的cofficient必須為零，所以$\ frac {\ partial ^ 2 V} {\ partial S ^ 2} = 0$，所以$V（t，S）= f（t）+ Sg（t）$。我們可以在此時停止，因為我們知道我們不能滿足邊界條件$v（T，S）= \ max（0，SK）$，因此我們假設我們可以用自我來對沖期權融資組合包括1個期權和$- \ frac {\ partial V} {\ partial S}$ shares是錯誤的。

## 最佳答案

$$V_t = \ alpha_tS_t + \ beta_t B_t$$

$$dV_t = \ alpha_tdS_t + \ beta_tdB_t$$

$$DC = \ partial_tCdt + \ partial_sCdS + \壓裂{1} {2} \西格瑪^ 2S_t ^ 2 \ partial_ {SS} CDT = \ partial_SCdS_t +（\ partial_tC + \壓裂{1} {2} \西格瑪^ 2S_t ^ 2 \ partial_ {SS} C）dt的$$

$$\ partial_tC + \ frac {1} {2} \ sigma ^ 2S_t ^ 2 \ partial_ {SS} C = rC-rS_t \ partial_S C$$

$$DC = \ partial_SCdS_t +（C-S_T \ partial_SC）RDT$$

$$dC = \ partial_SC \ cdot dS_t + \ left（\ frac {C_t} {B_t} - \ frac {S_t} {B_t} \ partial_SC \ right）\ cdot dB_t$$

$$\左（\ alpha_t = \ partial_SC，\，\ beta_t = \ dfrac {C_T} {B_T} - \ dfrac {S_T} {B_T} \ partial_SC \右）$$