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What is Lognormal Distribution in Option Pricing

March 17th, 2019, 8:12 pm

Hello,

Can someone explain this to me like I am 5.  I always see this term thrown around but what exactly is going on behind the scenes?

Let's say you calculate the historical returns of stock XYZ.   Then you find the standard deviation of the returns.   You plot the returns where x axis is the return amount and y axis is % of times that return occurs.  Let's say it happens to be a normal distribution with mean of 0 and sd of 10%.  To convert your normal distribution to a lognormal distribution, all you do is take each x value and plug it into exp(x) - 1?  So:

[New x value] = exp([old x value]) - 1
What used to be a +10% return is now a exp(0.1)-1 = 10.5% move and what used to be a -10% move is now a -9.5% down move.

Y, which are the probabilities, is still the same?  So this means there is a 68% chance that the continuously compounded stock return falls between -9.5% and +10.5%?

Is this correct?

Thanks
 
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Alan
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Re: What is Lognormal Distribution in Option Pricing

March 18th, 2019, 8:55 pm

Yes. 

For clarity, I suggest distinguishing the random variable names  by 'X' and 'Y' or some such, rather than the -- IMO -- somewhat confusing 'new x value' and 'old x value'. 

The general situation is that, if you have a continuous random variable X with density [$]p_X(x)[$], the probability of finding X between 'a' and 'b' is [$]\int_a^b p_X(x) \, dx[$]. If you decide to change variables to [$]Y = f(X)[$], where the mapping is 1-1, then 'a' maps to [$]A = f(a)[$], etc. As you noted, the probability of finding [$]Y[$] between 'A' and 'B' must be exactly the same as the probability of finding X between 'a' and 'b'.