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Log normal distribution

Where do you meet this distribution?

  • Finance, Economics : Change of stock price

Shape of Distribution

Basic Properties

  • Two parameters M, S are required (How can you get these).
    S>0
  • Continuous distribution defined on semi-bounded range x>0
  • This distribution is always asymmetric.

Probability

Quantile

  • Inverse function of cumulative distribution function
    F^{-1}(P)=\exp\left[S\Phi^{-1}(P)+M\right]

    where
    \Phi(\cdot)is cumulative distribution function of standard normal distribution.

  • How to compute this on Excel.

     
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    A B
    Data Description
    0.7 Probability associated with the distribution
    0.1 Value of parameter M
    2 Value of parameter S
    Formula Description (Result)
    =NTLOGNORMINV(A2,A3,A4) Inverse of the cumulative distribution function for the terms above

Characteristics

Mean – Where is the “center” of the distribution? (Definition)

  • Mean of the distribution is given as
    m\sqrt{\omega}

    where

    m=\exp(M),\;\omega=\exp(S^2)
  • How to compute this on Excel

     
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    A B
    Data Description
    0.1 Value of parameter M
    2 Value of parameter S
    Formula Description (Result)
    =NTLOGNORMMEAN(A2,A3) Mean of the distribution for the terms above
  • Function reference : NTLOGNORMMEAN

Standard Deviation – How wide does the distribution spread? (Definition)

  • Variance of the distribution is given as
    m^2\omega(\omega-1)

    where

    m=\exp(M),\;\omega=\exp(S^2)

    Standard Deviation is a positive square root of Variance.

  • How to compute this on Excel

     
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    A B
    Data Description
    0.1 Value of parameter M
    2 Value of parameter S
    Formula Description (Result)
    =NTLOGNORMSTDEV(A2,A3) Standard deviation of the distribution for the terms above
  • Function reference : NTLOGNORMSTDEV

Skewness – Which side is the distribution distorted into? (Definition)

  • Skewness of the distribution is given as
    (\omega+2)\sqrt{\omega-1}

    where

    \omega=\exp(S^2)
  • How to compute this on Excel

     
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    A B
    Data Description
    0.1 Value of parameter M
    2 Value of parameter S
    Formula Description (Result)
    =NTLOGNORMSKEW(A2,A3) Skewness of the distribution for the terms above
  • Function reference : NTLOGNORMSKEW

Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

  • Kurtosis of the distribution is given as
    \omega^4+2\omega^3+3\omega^2-6

    where

    \omega=\exp(S^2)
  • How to compute this on Excel

     
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    A B
    Data Description
    0.1 Value of parameter M
    2 Value of parameter S
    Formula Description (Result)
    =NTLOGNORMKURT(A2,A3) Kurtosis of the distribution for the terms above
  • Function reference : NTLOGNORMKURT

Random Numbers

  • Random number x is generated by inverse function method, which is for uniform random U,
    x=\exp\left[S\Phi^{-1}(U)+M\right]

    where
    \Phi(\cdot)is cumulative distribution function of standard normal distribution.

  • How to generate random numbers on Excel.
     
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    A B
    Data Description
    0.1 Value of parameter M
    2 Value of parameter S
    Formula Description (Result)
    =NTRANDLOGNORM(100,A2,A3,0) 100 log normal deviates based on Mersenne-Twister algorithm for which the parameters above

    Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A7:A106 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

  • Function reference : NTRANDLOGNORM

NtRand Functions

Reference

 

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