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Does anyone have links to information about the empirical and theoretical justifications for modeling asset prices with Levy distributions?
In my head, the justification for log-normal returns goes roughly like this:
1. Stock returns over a period of time are just the product of returns over smaller periods.
2. EMH says these returns are independent.
3. So I've got the product of a bunch of independent random variables. So what?
4. Well, stock prices can't be smaller than 0 and can't actually reach infinity, so we should have a finite variance.
5. Which means the log should have a finite variance too!
6. Central limit theorem.
7. So the sum of the logs of the returns is normally distributed.
8. Thus, the returns are log-normally distributed.
Yay! So how can anyone argue against this?
Oh, wait, (5) isn't actually true, I realize. Over any given time period a share has a positive probility of becoming permanently worthless. OK, that gets us infinite variance of the log, fine, now we can play with the other stable distributions.
Is it just the possibility of bankruptcy that breaks log-normal returns? Or is there something else at work?
Aside:
(2) also gives me pause. I can imagine a system where the distribution of expected returns for a given asset was a function of its price, and therefore the returns would only be conditionally independent. It's not quite clear to me if this breaks the CLT.
In my head, the justification for log-normal returns goes roughly like this:
1. Stock returns over a period of time are just the product of returns over smaller periods.
2. EMH says these returns are independent.
3. So I've got the product of a bunch of independent random variables. So what?
4. Well, stock prices can't be smaller than 0 and can't actually reach infinity, so we should have a finite variance.
5. Which means the log should have a finite variance too!
6. Central limit theorem.
7. So the sum of the logs of the returns is normally distributed.
8. Thus, the returns are log-normally distributed.
Yay! So how can anyone argue against this?
Oh, wait, (5) isn't actually true, I realize. Over any given time period a share has a positive probility of becoming permanently worthless. OK, that gets us infinite variance of the log, fine, now we can play with the other stable distributions.
Is it just the possibility of bankruptcy that breaks log-normal returns? Or is there something else at work?
Aside:
(2) also gives me pause. I can imagine a system where the distribution of expected returns for a given asset was a function of its price, and therefore the returns would only be conditionally independent. It's not quite clear to me if this breaks the CLT.
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