[衍生商品] 淺談 Black-Scholes Model 的性質 (1) - Equivalence of Put Call Parity and Black-Scholes Formula.

延續上篇的 Black-Scholes Model 的性質，這次要介紹 Put-Call parity 與 Black-Scholes Formula 兩者等價。

回憶 Black-Scholes Formula
\[\left\{ \begin{array}{l}
C = {S_0}N\left( {{d_1}} \right) - K{e^{ - rT}}N\left( {{d_2}} \right)\\
P = K{e^{ - rT}}N\left( { - {d_2}} \right) - {S_0}N\left( { - {d_1}} \right)
\end{array} \right.
\] 其中 $C$ 為 Call option 價格，$P$ 為 Put Option 價格，$r$ 為無風險利率，$\sigma$ 為股價波動度，$K$ 為執行價格，$T$ 為到期時間， $N(\cdot)$ 為 Standard Normal Cumulative distribution function (CDF)，且\[\left\{ \begin{array}{l}
{d_1} = \frac{{\ln (S/K) + (r - q + \frac{1}{2}{\sigma ^2})(T)}}{{\sigma \sqrt T }}\\
{d_2} = {d_1} - \sigma \sqrt T
\end{array} \right.\]

與 Put-Call parity
\[C - P = {S_0} - K{e^{ - rT}}
\]
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Claim: 
Black-Scholes Formula 滿足 Put-Call parity。
===================

Proof
計算 $C-P$：
將 Black-Scholes Formula 的結果帶入上式，可得
\[C - P = \left[ {{S_0}N\left( {{d_1}} \right) - K{e^{ - rT}}N\left( {{d_2}} \right)} \right] - \left[ {K{e^{ - rT}}N\left( { - {d_2}} \right) - {S_0}N\left( { - {d_1}} \right)} \right] \ \ \ \ (*)
\]現在利用 Standard Normal Cumulative distribution function 的特性
\[\left\{ \begin{array}{l}
N\left( { - {d_2}} \right) = 1 - N\left( {{d_2}} \right)\\
N\left( { - {d_1}} \right) = 1 - N\left( {{d_1}} \right)
\end{array} \right.
\]我們可改寫 $(*)$ 如下
\[\begin{array}{l}
 \Rightarrow C - P = {S_0}N\left( {{d_1}} \right) - K{e^{ - rT}}N\left( {{d_2}} \right)\\
\begin{array}{*{20}{c}}
{}&{}
\end{array} - \left[ {K{e^{ - rT}} - K{e^{ - rT}}N\left( {{d_2}} \right) - {S_0} + {S_0}N\left( {{d_1}} \right)} \right]\\
 \Rightarrow C - P =  - K{e^{ - rT}} + {S_0}
\end{array}
\] 上式即為 Put-Call parity。 $\square$