John Hull Chapter 12

[matiz]

Hi,

Equation 12.3 is shown in the book as below

ln ST ~ φ [ln S0 + (μ - σ^2/2)T, σ T^1/2]

I do not understand how (ln So) is now part of the 1st parameter (ie the mean) for the normal distribution shown in the above. (ln So) is shown as below in the previous equation in the book

ln ST - ln So ~ φ [(μ - σ^2/2)T, σ T^1/2]

I would appreciate any help on this.
Thanks in Advance for your help

 

[Lyosha]

sure,
\(ln(\frac{S_T}{S_0})=ln(S_T)-ln(S_0)\)

This is a property of logs. If you would like I can prove it for you.

From there you add the \(ln(S_0)\) to the mean and you're done.

 

[matiz]

Hi,

Thanks for the answer. I understand the property of the logs. However I do not understand why ln(So) is added only to the mean and not to the standard deviation part of the normal distribution in the above. Sorry, I might be missing something basic here but your help is greatly appreciated.

 

[enthusiast]

Hi,

Thanks for the answer. I understand the property of the logs. However I do not understand why ln(So) is added only to the mean and not to the standard deviation part of the normal distribution in the above. Sorry, I might be missing something basic here but your help is greatly appreciated.

Consider ln ST - ln So ~ φ [(μ - σ^2/2)T, σ T^1/2]

Ln (S0) is a constant since S0 is known. When you add (or subtract) a constant from a normal distribution only the mean will shift. There is no impact on the SD of the normal distribution.

A constant A1, has normal distribution with mean A1 and SD = 0. ie A1 ~ φ [A1,0]. So adding this to φ [(μ - σ^2/2)T, σ T^1/2] would give ~ φ [A1+(μ - σ^2/2)T, σ T^1/2]. Hope this helps.

 

[matiz]

got it. Thanks a lot