Why Fat-Tailed Return Models Require More Than Kurtosis Adjustment
Summary
The discussion considers why financial models often use normally distributed returns despite the observed frequency of unusually large moves. It focuses on the trade-off between tractable pricing formulas and representing return tails more realistically. The response notes that models such as Black–Scholes rely on Gaussian dynamics, which support closed-form solutions for standard option pricing problems.
Matching historical kurtosis alone is not enough to specify a usable model. The underlying stochastic process must be defined, and one approach is to allow jumps through a Lévy process. This can better represent excess kurtosis, but pricing may then require simulation or numerical methods such as Fourier techniques. The computational burden can become substantial for exotic products. The answer offers a general modeling explanation rather than empirical tests or a specific calibrated process, and it does not claim that one jump model is suitable for every asset or application.
Key ideas
- Normal return assumptions make some pricing problems analytically tractable.
- Black–Scholes uses Gaussian return dynamics.
- A kurtosis value alone does not define the stochastic process needed for pricing.
- Jump processes can represent excess kurtosis but often require numerical valuation.
- Exotic products can make computation under jump models especially demanding.
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Full text
# Why assume stock returns are normally distributed instead of just adjusting the kurtosis? # Why assume stock returns are normally distributed instead of just adjusting the kurtosis? Most standard models assume stock returns are normally distributed even though everyone agrees that real-world returns have fat tails. We've all heard stories of hedge funds that went bankrupt cause something happened that their model called a "10 sigma event that should only happen once every billion years" and that's obviously a flawed model. My textbooks point this out but hand-wave it away like it's an unavoidable simplification. But why do we have to simplify? We know the real-world historical kurtosis of stock returns, and it's easy to define a distribution that matches that kurtosis. Why can't we simply use that fat tail distribution in all standard models instead of the normal distribution? Is it simply that the PDF of the normal distribution has analytical solutions that are easier to work with? ## Answer by alexbougias (score 6, accepted) https://quant.stackexchange.com/a/46936 If we are talking about risk management (Hence, the risk neutral world), normality allows us to get closed form solutions. For instance, the Black and Scholes equation assumes Gaussian returns (Equivalently, the stock follows a geometric Brownian motion). Your thought is correct, although you can not simply adjust for kurtosis. You need to define properly the stochastic process of the underlying. To allow for excess kurtosis, you need to allow the process to make jumps. This type of process is called Levy process. However, you either have to resort to simulation or other approximation (e.g Fourier). In case you have an exotic product, then computational difficulty is very high. Note that more recent research emphasized on the pricing of options under jump diffusion process, so when these textbooks were written, it was commonly accepted the assumption of normality.
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