How to Derive the Standard Normal CDF in Option Pricing Analysis?

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yamdizzle
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I was wondering how I can find the derivative of a normal cdf with respect to a boundary parameter?
I can get an answer with Mathematica or something but I have no idea how to actually do this. I don't know how fundamental theorem of calculus can be applied. (if it can be)


[itex]\Phi[/itex]((log(S/K)+(r+σ^2)T)/(σ sqrt(T)))

where phi is the std normal cdf.

I'm doing option pricing analysis but I can't really figure out how to derive this with respect to r or sigma or any other parameter. I would appreciate any help.
 
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standard normal has a mean 0 and variance 1.
 
So basically I need to derive
∫[itex]\frac{1}{\sqrt{2\pi}}[/itex]exp([itex]\frac{-x^2}{2}[/itex]) dx

with upper bound
(log(S/K)+(r+σ^2)T)/(σ sqrt(T))
lower bound -inf

with respect to r, Sigma or any parameter so I can learn how to do this.
 
How do you think those values were found?
The derivative with respect to r is this... according to Mathematica. But I don't know how.


[itex]\frac{\sqrt{T}}{\sqrt{2\pi}σ}[/itex]exp(-([itex]\frac{(T (r + σ^2) + Log[S/K] )}{\sqrt{2 T σ}}[/itex])^2)

with changing a few things ==

T * Normal PDF(-log(S/K),Tσ^2) at point T*(r+^2)

Technically this is suppose to be 0 as normal pdf is 0 at any point since it is continuous but something different can be acquired with deriving with respect to something else.
 
Just use the chain rule. e.g., for r

[tex]\frac{d\Phi(f(r))}{dr} = \frac{d\Phi(y)}{dy} \frac{dy(r)}{dr},[/tex]

where y = f(r) is the argument of your cdf. Since it's a standard normal distribution, [itex]d\Phi(y)/dy = \exp(-y^2/2)/\sqrt{2\pi}[/itex], and then plug in y = f(r), of course.

You can do a similar thing treating the argument of the cdf as a function of [itex]\sigma[/itex].
 
Thank you. That was of great help!