延續上篇 [隨機分析] Black-Scholes PDE for European Call option (0) ,這次要來證明 Black-Scholes Formula \[ {\scriptsize f(t,S_t) = S \Phi \left( \frac{\ln(S_t/K) + (r+\frac{1}{2} \sigma^2) (T-t) }{\sigma \sqrt{T-t}}\right) - K e^{-r (T-t)} \Phi \left( \frac{\ln(S_t/K) + (r - \frac{1}{2}\sigma^2) (T-t) }{\sigma \sqrt{T-t}}\right)} \]其中 $\Phi (\cdot)$ 為 Standard Normal Cumulative distribution function (CDF) 現在令 $x= S_t$,上述的 Black-Scholes Formula 確實為 Black-Scholes PDE 的解,亦即上式為下列PDE的解: \[ rf(t,x) = {f_t}(t,x) + rx{f_x}(t,x) + \frac{1}{2}{f_{xx}}(t,x){\sigma ^2}{x^2} \]且滿足終端邊界條件 $f(T,x) = h(x), \ \forall x \in \mathbb{R}$ 我們將分成下列幾個小步驟逐步完成此證明: 步驟1 :首先證明下列等式成立: Claim 1: $K e^{-r(T-t)} \Phi ' (d_2(T-t,x)) = x \Phi' (d_1(T-t,x))$ 其中 \[ d_1 (T-t, x) = \frac{\ln(x/K) + (r+\frac{1}{2} \sigma^2) (T-t) }{\sigma \sqrt{T-t}} \\ d_2 (T-t,x) = \frac{\ln(x/K) + (r - \frac{1}{2}\sigma^2) (T-t) }{\sigma \sqrt{T-t}} \\ \] Proof 我們證明 \[ K e^{-r(T-t)} \Phi ' (d_2(T-t,x))- x \Phi'
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