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edited Sep 23, 2015 at 19:49

I think there are a few conflating ideas here.

With respect to the sum of logs idea, I think you're thinking about infinitely divisible distributions (https://en.wikipedia.org/wiki/Infinite_divisibility_(probability)). These ideas are indeed used to build more complicated models (i.e. Levy processes) for asset returns.
With regards to the Efficient Market Hypothesis, I think most people think of them as asset prices --- under an appropriate measure change --- are martingales (i.e. past history information cannot be used to predict tomorrow's asset price), and not returns.
I'm not why you'd think the central limit theorem here would necessary hold. While there are tons of CLT's out there, having a random variable of the form $S_n = \sum_{i=1}^n X_i$$S_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - E X_i)$ does not necessarily mean that as $n \to \infty$, $S_n$ will converge (in distribution, or some other strong sense of convergence) to a Gaussian random variable. Next, the absence of kurtosis in the Black-Scholes is not surprising, especially since the asset prices there follow a geometric Brownian motion (so effectively, each small time interval is Gaussian). But if one replaces the GBM asset price assumption to something else (i.e. Levy models and/or the current literature favorite flavor of Ito semimartingales), then one will definitely generate kurtosis and other higher moment properties. The downside, of course, is that more sophisticated asset pricing models inevitably implies it'll be a challenge to model derivative prices, and/or econometric inference on the underlying parameters.

But going back to the fundamental question of --- "why is there kurtosis in asset prices / returns" --- this I believe, is at heart a question that's too broad to answer. From a statistical perspective, the response is simply: "it is what it is" and people just build models that build kurtosis into it. From an economics and finance perspective, there are simply too many different theories out there; perhaps the representative agent cares about disaster risks in consumption, or that he has a preference that's state dependent and such macro state variable has some higher order effects. Really, who knows.

I think there are a few conflating ideas here.

With respect to the sum of logs idea, I think you're thinking about infinitely divisible distributions (https://en.wikipedia.org/wiki/Infinite_divisibility_(probability)). These ideas are indeed used to build more complicated models (i.e. Levy processes) for asset returns.
With regards to the Efficient Market Hypothesis, I think most people think of them as asset prices --- under an appropriate measure change --- are martingales (i.e. past history information cannot be used to predict tomorrow's asset price), and not returns.
I'm not why you'd think the central limit theorem here would necessary hold. While there are tons of CLT's out there, having a random variable of the form $S_n = \sum_{i=1}^n X_i$ does not necessarily mean that as $n \to \infty$, $S_n$ will converge (in distribution, or some other strong sense of convergence) to a Gaussian random variable. Next, the absence of kurtosis in the Black-Scholes is not surprising, especially since the asset prices there follow a geometric Brownian motion (so effectively, each small time interval is Gaussian). But if one replaces the GBM asset price assumption to something else (i.e. Levy models and/or the current literature favorite flavor of Ito semimartingales), then one will definitely generate kurtosis and other higher moment properties. The downside, of course, is that more sophisticated asset pricing models inevitably implies it'll be a challenge to model derivative prices, and/or econometric inference on the underlying parameters.

I think there are a few conflating ideas here.

With respect to the sum of logs idea, I think you're thinking about infinitely divisible distributions (https://en.wikipedia.org/wiki/Infinite_divisibility_(probability)). These ideas are indeed used to build more complicated models (i.e. Levy processes) for asset returns.
With regards to the Efficient Market Hypothesis, I think most people think of them as asset prices --- under an appropriate measure change --- are martingales (i.e. past history information cannot be used to predict tomorrow's asset price), and not returns.
I'm not why you'd think the central limit theorem here would necessary hold. While there are tons of CLT's out there, having a random variable of the form $S_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - E X_i)$ does not necessarily mean that as $n \to \infty$, $S_n$ will converge (in distribution, or some other strong sense of convergence) to a Gaussian random variable. Next, the absence of kurtosis in the Black-Scholes is not surprising, especially since the asset prices there follow a geometric Brownian motion (so effectively, each small time interval is Gaussian). But if one replaces the GBM asset price assumption to something else (i.e. Levy models and/or the current literature favorite flavor of Ito semimartingales), then one will definitely generate kurtosis and other higher moment properties. The downside, of course, is that more sophisticated asset pricing models inevitably implies it'll be a challenge to model derivative prices, and/or econometric inference on the underlying parameters.

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created Sep 23, 2015 at 19:29

user32416

I think there are a few conflating ideas here.

With respect to the sum of logs idea, I think you're thinking about infinitely divisible distributions (https://en.wikipedia.org/wiki/Infinite_divisibility_(probability)). These ideas are indeed used to build more complicated models (i.e. Levy processes) for asset returns.
With regards to the Efficient Market Hypothesis, I think most people think of them as asset prices --- under an appropriate measure change --- are martingales (i.e. past history information cannot be used to predict tomorrow's asset price), and not returns.
I'm not why you'd think the central limit theorem here would necessary hold. While there are tons of CLT's out there, having a random variable of the form $S_n = \sum_{i=1}^n X_i$ does not necessarily mean that as $n \to \infty$, $S_n$ will converge (in distribution, or some other strong sense of convergence) to a Gaussian random variable. Next, the absence of kurtosis in the Black-Scholes is not surprising, especially since the asset prices there follow a geometric Brownian motion (so effectively, each small time interval is Gaussian). But if one replaces the GBM asset price assumption to something else (i.e. Levy models and/or the current literature favorite flavor of Ito semimartingales), then one will definitely generate kurtosis and other higher moment properties. The downside, of course, is that more sophisticated asset pricing models inevitably implies it'll be a challenge to model derivative prices, and/or econometric inference on the underlying parameters.