The stock market prices are conceptually driven by numerous market trades. Contrary to Heat diffusion process, the stock market does not diffuse into infinite entropy. It is a self-organizing system with regulations and methods in place to maintain trading orders. The finite entropy is often reflected in long-dated options market (stock, swaption, caps/floors market, etc) in a form of mean-reversion process, even with much fatter tails than normal distribution predicted by Heat diffusion process.
In a smaller time interval, we should observe a Jump process between bid/ask (Dirac Delta density function). And as measured time interval gets larger and larger, it would shape more like Laplace distribution. The evidence is in the graph below for S&P price change distribution for 15/30/60 seconds:
It is clear that Brownian process can not explain such pattern. Below is the daily distributions for S&P, Crude Oil, 10Y Treasury Futures as examples. It shows that they all resembles Laplace distribution much more than Normal distribution:
When time interval becomes larger, the path-decencies started to fade away. Hence, it looks more like the convolution of two Laplace distribution. Below weekly density function for S&P, Dow Jones 30, and Nikkie 225 since 1970:
For participants in option markets, they often observe one side skew distribution. The proposed stochastic process also explain the one side tail, similar to that of comet tail discussed previously. Below is the PDE illustration. When particles is moving to the right, the tail is on the left. When the particles ball is moving to the left, the tail is on the right.
It is harder to show the tail in graphic form, below is the table to show the tail for probability smaller than or greater than 1~5 standard deviation:
(Click to enlarge table)
It is clear to see that in 2007-8 crisis, the tail is on the upside, while the market is moving lower, and in 2010-2012 when market is going up, the tail is thicker on the downside, similar to that of comet tails.
It is also commonly accepted that economic cycles existed. Prevailing diffusion process does not explains waves. The 2nd order PDE of proposed framework have wave-particle duality that allows us to model economic cycles in wave form as well. Each economic policies or events serves as as force to disturb the market and the impact is propagated throughout the system in the form of waves.
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Disclaimer: All contents on this website have not been peer-reviewed, and it may contain errors. It is strictly the author’s personal ideas that do not reflect any organization’s view. Furthermore, it is in disagreement with generally accepted practice of Brownian process. The proposal is original to the best of author’s knowledge. The author also reserves all the rights to the contents, including but not limited to, any copy rights, rights to modify and remove any statements, and the right to distribution. Any application of the model is used at the readers own risks.
In a smaller time interval, we should observe a Jump process between bid/ask (Dirac Delta density function). And as measured time interval gets larger and larger, it would shape more like Laplace distribution. The evidence is in the graph below for S&P price change distribution for 15/30/60 seconds:
It is clear that Brownian process can not explain such pattern. Below is the daily distributions for S&P, Crude Oil, 10Y Treasury Futures as examples. It shows that they all resembles Laplace distribution much more than Normal distribution:
When time interval becomes larger, the path-decencies started to fade away. Hence, it looks more like the convolution of two Laplace distribution. Below weekly density function for S&P, Dow Jones 30, and Nikkie 225 since 1970:
For participants in option markets, they often observe one side skew distribution. The proposed stochastic process also explain the one side tail, similar to that of comet tail discussed previously. Below is the PDE illustration. When particles is moving to the right, the tail is on the left. When the particles ball is moving to the left, the tail is on the right.
It is harder to show the tail in graphic form, below is the table to show the tail for probability smaller than or greater than 1~5 standard deviation:
It is clear to see that in 2007-8 crisis, the tail is on the upside, while the market is moving lower, and in 2010-2012 when market is going up, the tail is thicker on the downside, similar to that of comet tails.
It is also commonly accepted that economic cycles existed. Prevailing diffusion process does not explains waves. The 2nd order PDE of proposed framework have wave-particle duality that allows us to model economic cycles in wave form as well. Each economic policies or events serves as as force to disturb the market and the impact is propagated throughout the system in the form of waves.
Disclaimer: All contents on this website have not been peer-reviewed, and it may contain errors. It is strictly the author’s personal ideas that do not reflect any organization’s view. Furthermore, it is in disagreement with generally accepted practice of Brownian process. The proposal is original to the best of author’s knowledge. The author also reserves all the rights to the contents, including but not limited to, any copy rights, rights to modify and remove any statements, and the right to distribution. Any application of the model is used at the readers own risks.
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