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?

  • $\begingroup$ The reality is that the assumtion of a single volatility is not used. Simple derivatives are valued using their prices from the market, or some interpolation/extrapoaltion, and more complex derivatives are priced using PDEs/MonteCarlo, such that the market implied expected distribution of the underlying (including tails as fat as you like) is followed. $\endgroup$ – will Aug 4 '19 at 12:47

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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