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 \[C - P = {S_0} - K{e^{ - rT}} \] =================== 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}
If you can’t solve a problem, then there is an easier problem you can solve: find it. -George Polya