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Home Tutorials Lognormal Distribution for Asset Pricing (Mathematical Details)


Lognormal Distribution for Asset Pricing (Mathematical Details)

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In order to understand why Lognormal distribution is suitable for stock price returns, let’s first think about stock prices, the following graph is a stock price chart over time:T.

Sample Stock Price


Section A:

Section B:

Why is z(H) normally distributed?
We have to recognize some assumptions in order to realize that z(H) is normally distributed. We know that z(H) is made up of T sub intervals and for each interval we can say that these sub intervals are:

  • Independently distributed
    a. This is because the stock price change (and hence return) over a sub interval is not dependent on the previous price change (return).
  • Identically distributed
    a. With the interval sufficiently small stock return appear to have the same distribution.
    The first two assumptions tell that stock price follows a random walk (see: http://en.wikipedia.org/wiki/Random_walk) and random walk is associated with “efficient market” theory.
  • The expected continuously compounded return is: E[z(j)] = \mu \times h
    a. Here, h is the size of the sub interval and \mu is the return per unit time and is independent of the interval length of the subinterval h. The expected return will increase/decrease based on the size of h.
  • The variance of the continuously compounded return is: var[z(t) ] = \sigma^2 \times h
    a. Here, h is the size of the sub-interval and \sigma^2 is the variance per unit time and is independent of the length of the interval length of the subinterval h.


Based on these 4 assumptions we can say that the returns z(j) are normally distributed. We can also say that z(H) is normally distributed. Why? Because of central limit theorem (see: Central Limit Theorem). The central limit theorem says that the summation of a large number of independent distributions i.e. \sum_{j=1}^T\z(j) results in a normally distributed variable i.e. z(H).
Now, that we have understood that z(H) is normally distributed, please read "Section A" again and to understand why stock return is lognormally distributed.

 

Comments  

 
0 #6 profile 2016-12-19 11:50
 
 
0 #5 luigi4235 2015-03-03 11:06
 
 
0 #4 luigi4235 2015-02-21 13:10
( ͡° ͜ʖ ͡°)
+++thatsafunnypic.com+++
 
 
0 #3 Supat 2013-04-06 01:25
Well explained and easy to follow.
 
 
+2 #2 Matt-uk 2012-12-31 16:24
:lol: one of best simple explanations of assumptions underlying use of lognormal disturb for stock prices i have seen year. keep it up.
 
 
0 #1 ghfh 2012-12-15 21:27
:-*