The origin of fractality of stock market time series

Krivosheev O.I.

ICP RAS, Profsoyuznaya 65, Moscow, RSXX0063, Russia, 8(926)14-777-36,

We consider several strictly proved facts on smooth continuous functions presenting asset price. In order to prevent some mathematical difficulties we consider time functions instead of discrete time series. We show that no smooth function can present market prices at least if there is unlimited and cheap credit for riskless borrowers and no transaction costs.

We show that under these conditions are possible strategies that provide infinite return at low or even at zero risk. So smooth price function in continuous model is impossible as it violates the no-arbitrage condition.

We show that the value (more o less depending on set if row dimension is not 0.5)

- (1)

one can consider as a fractal resistivity of time series.

where is a price value at some point & is it’s increment .

We prove a some kind of Ohm’s low for market


where is credit leverage – ratio of invested capital to own one,

- final to initial price ratio at the sample.

is proved to be (nearly) optimal leverage and can be calculated for expected values of and .

We show that nearly smooth continuous time functions (with low fractal resistivity) are impossible too, if they provide too high return rate – more than equilibrium one for the market. Then we use these facts as a kind of heuristics to understand real market behavior especially to get ½ (or1 ½ ) price time series dimension.

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