Revision booklet 6 question 3

[NS206]

Hi,

I'm not quite following the following part of the answer, please can someone point me in the right direction?

Sept 2008 Qu9 part (ii)

The answers in the revision booklet give a formula for the pdf of d1 over the pdf of d2. I'm not sure I understand this.

Thanks!

 

[Lolgabby]

Please see the attached PDF

 

[Anna Bishop]

Hello

The derivation of the Greeks from the BS option pricing formulae used to be a syllabus objective. It is no longer, so I would say it is highly unlikely to come up. However, not impossible, if the examiners feel they want to ask a difficult application question and test your differentiation.

I attach some tutorial notes that I produced several years ago that show where the φ(d1)/φ(d2) result comes from and then take you step by step through the differentiation of the call option formula to derive the Greeks.

I wouldn't spend ages on this. It is more important that you can quote the formulae for delta for a call and delta for a put in relation to the BS option pricing formulae, ie:

Delta(call) = exp{-q(T-t)}Φ(d1)
Delta(put) = -exp{-q(T-t)}Φ(-d1)

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In case it's useful for helping to understand some of the differentiation:

Big Φ(x) is the CDF of the N(0,1) distribution.
Little φ(x) is the PDF of the N(0,1) distribution.

dΦ(d1)/dSt
= [dΦ(d1)/dd1] * [dd1/dSt] by the chain rule
= φ(d1) * [dd1/dSt] as when we differentiate the CDF we get the PDF.

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Anna