Very good question.

The most important reason is that for most commonly used distributions, almost none of them need to be determined by higher moments such as kurtosis or skewness. Even though for some asymmetric distribution or fat tail distribution such as exponential or t. They don't need these moments. So you don't need to model them. Of course, there is some exception such Cornish Fisher method for VaR calculation directly uses the 3 and 4 th moment.

The second reason is that most of the models are build on Brownian motion or normal density. Normal distribution can be totally determined by the first two moments. To model the skewness and kurtosis, people do is to introduce the following: time varying volatility, jumping, CEV, Stochastic Volatility, correlation between volatility and asset prices...

In a bullish or bearish market, the distribution of the return may be skew. But perhaps this is only a special sample period of a normal distribution. In the long run, the return is normal.

best,

Thank u and grateful for ur answer.
I wonder if I make it right ——the trick for the first issue is that there is simply few technically-feasible model to incorporate 3 or 4th moment in pricing financial assets?
And for the second problem, I am actually quite into that guess and I hope it is worth researching.

1. Yes, you can think in that way. There is a tradition in finance that new model should more or less have a connection with the old model such as black-scholes.  Otherwise, the model will not be very popular if you introduce a totally different one. The reason is because the people has studied this area for so many years and find the normal framework has many good properties and the appearance of these models are all very beautiful. If you put 3 or 4 moments in, you can imaging how ugly the model will be. Ugly model will never get its place.

The "correct" way is that: when you find the normal distribution not work, try to make some little improvement but not abandon normal and find other distributions. Normal is very good in most cases, you don't need to find a complicated model for a special time period or a special asset class. Actually, stochastic volatility models are very very flexible I think it is more flexible than just include more moments.


2. Actually I have not consider about this topic before, may be you can check some papers. I think most of the study focuses on the empirical part, such as use different type of volatiltiy models (e.g. GARCH) to fit the data to test the some skewness effects such as leverge effect (NGARCH). Personally，I do not recommend you to study this topic. The problem does not need to be modelled in that way. Of course, you can study it for personal interest.

Just some suggestions, good luck

best,

Thank u for your answer. And I benefit a lot from your advice.