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可以使用希臘人(delta,gamma,theta)來證明Black-Scholes呼叫公式滿足Black-Scholes PDE嗎?

如果是這樣,是否有任何推導顯示這個?我被告知這可以在課堂上完成,但我不知道它是如何可能的。

最佳答案

當您將希臘語的公式插入PDE時,這非常簡單。

預賽:

$ \ Delta = \ frac {\ partial c_t} {\ partial S_t} = \ Phi(d_1)$

$ \ Gamma = \ frac {\ partial ^ 2 c_t} {\ partial S_t ^ 2} = \ frac {\ phi(d1)} {S_t \ sigma \ sqrt {u}} $

$ \ Theta = \ frac {\ partial c_t} {\ partial t} = - rKe ^ {ru} \ Phi(d_2)-S_t \ phi(d_1)\ frac {\ sigma} {2 \ sqrt {u}} $

\begin{eqnarray} rc_t&=&\Theta+rS_t\Delta + \frac{1}{2}S_t^2\sigma^2\Gamma\\ RHS&=&-rKe^{ru}\Phi(d_2)-S_t\phi(d_1)\frac{\sigma}{2\sqrt{u}}+rS_t\Phi(d_1)+\frac{1}{2}S_t^2\sigma^2 \frac{\phi(d1)}{S_t\sigma\sqrt{u}}\\ &=&-rKe^{ru}\Phi(d_2)-\frac{S_t\phi(d_1)\sigma}{2\sqrt{u}}+rS_t\Phi(d_1)+ \frac{S_t\sigma\phi(d1)}{2\sqrt{u}}\\ &=&rS_t\Phi(d_1)-rKe^{ru}\Phi(d_2)\\ &=&rc_t\\ &=&LHS \end{eqnarray}

轉載註明原文: 可以使用希臘人(delta,gamma,theta)來證明Black-Scholes呼叫公式滿足Black-Scholes PDE嗎?