Partial Differentiation $ \frac {log( \frac {S_0}{k}) + (r - \frac {\sigma^2}{2})T} {\sigma \sqrt{T}} $

Question

I'm having difficulty in finding the $ \frac{\partial d1}{\partial \sigma}$ for the equation below.

I tried using quotient rule but I'm getting $ -\sqrt{T} -\frac {log( \frac {S_0}{k}) + (r - \frac {\sigma^2}{2})T} {\sigma^2 \sqrt{T}} $ for $ \frac{\partial d1}{\partial \sigma}$

$ d1 = \frac {log( \frac {S_0}{k}) + (r - \frac {\sigma^2}{2})T} {\sigma \sqrt{T}} $

Answer 1

You can use the quotient rule as you tried already yourself.

So $g(x) = {log( \frac {S_0}{k}) + (r - \frac {\sigma^2}{2})T}$

and $h(x) = \sigma\sqrt{T}$

and $g'(x) = \frac{\partial}{\partial \sigma} log( \frac {S_0}{k}) + \frac{\partial}{\partial \sigma}(r - \frac {\sigma^2}{2})T = -T\sigma $

and $h'(x) = \frac{\partial}{\partial \sigma} \sigma \times \sqrt{T} = \sqrt{T}$

and then it is just applying the rule:

$$ \frac{-T\sigma \times \sigma\sqrt(T) - \sqrt(T)\times[log( \frac {S_0}{k}) + (r - \frac {\sigma^2}{2})T]}{(\sigma\sqrt{T})^2} $$