The question is as follows:

Let $v = S_0 \phi(d_1)\sqrt{T}$. Solve the following equation for $K$. $$ \frac{\partial v}{\partial K} = 0 $$

By finding $\frac{\partial v}{\partial d_1}$ and $\frac{\partial d_1}{\partial K}$ I have used the following rule. $ \frac{\partial v}{\partial K} = \frac{\partial v}{\partial d_1}\cdot \frac{\partial d_1}{\partial K}$ and achieved the following result:

$$ \frac{\partial v}{\partial K} = \frac{S_0 d_1 \phi(d_1) \sqrt{T}}{\sigma K \sqrt{T-t} } $$

Now setting this equal to 0 I have been unable to solve for K. The furthest I've got is $$ \frac{S_0 \sqrt{T}}{\sigma^2 K} \Big(\frac{\ln(S)-\ln(K)}{T-t} + r+\frac{\sigma^2}{2}\Big) = 0$$

Any advice? Thanks in advance.

  • 1
    $\begingroup$ You are looking for the numerator of dv/dK to equal zero. It cannot be S, T or phi(d1). Where does d1 achieve zero? $\endgroup$ – James Spencer-Lavan Mar 17 '19 at 7:16

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