*Just for a toy model, to start us off, I used excel to insist the log data is strictly linear, implying the price follows a perfect exponential curve. We observe bitcoin hits $1000 on precisely….ok lets not get carried away. The Price cannot be a pure exponential function of time – that seems to violate some economic conservation laws. But these are still fun plots. The source is an excel sheet which you can find here https://drive.google.com/folderview?id=0B2HU2oGcAN_SbDRHZmFsWldSX0E&usp=sharing*

## A somewhat sophistocated model for the price of bitcoin based on diffusion

In a previous post I suggested that the log linear behavior of bitcoin was a trend that will continue and proposed a model by which this would operate. Here is a gogle drive containing the mathematica code I wrote to investigate the matter. These equations do support the long term log linear behavior and produce price curves which seem to match the behavior we have seen from bitcoin.

https://drive.google.com/folderview?id=0B2HU2oGcAN_SSllnYkd0VThTcW8&usp=sharing

You will be able to manipulate a parameter controlling the effective bitcoin supply.

The rules I used to derive the formulas were:

1) the Bitcoin demand D(t) is proportional to the probability that somebody on earth has learned of Bitcoins existence. This function is an exponential in t for small times

D(t) ~ exp[t / to]

However the probability question has vastly distinct long time behavior. The function that really represents the long term demand of bitcoin over time is closely related to the hyperbolic tangent

This family of functions has the form

D(t) = 1 / ( 1 + exp[- t / t0] )

2) The Bitcoin supply is known in the short term to be proportional to t, but since the number of bitcoins generated per unit time halves every fixed amount of years, dictated by the protocol, I said we should model the supply as

S(t) = t * 2^(-t / t’)

Where t’ is in general a parameter that can be adjusted to dictate the effective supply of bitcoin as a function of time.

This incredibly simplistic model was in fact successful at generating the family of curves that the bitcoin price and its log seem to belong to.

Perhaps in the future more sophisticated models for the supply curve can be employed, since it seems clear the supply curve is responsible for the volatility. See tables beneath:

Anyone who has mathematica ready to can run this code. An animation can be generated as well as a dynamic plot you may manipulate. A simple table of outputs for the price and log of the price is given in the above google drive as the supply parameter is swept. You can observe the supply in the model has a strong influence on the price – many orders of magnitude in fact. Below are some excerpts from the PDF and interactive mathematica program provided in the google drive, as well as the generating mathematica code.

Demand[t_] := 1/(1 + Exp[-t/tp]) Supply[t_] := t*2^(-t/th) Slider[Dynamic[tp], {0, 29, .1}, ImageSize -> 1300], Dynamic[tp], "= propogation time tp" , Slider[Dynamic[th], {0, 4, .1}], ImageSize -> 1300, Dynamic[th], "= effective bitcoin supply"} Dynamic[ Plot[Demand[t]/Supply[t], {t, .0000001, 5}, ImageSize -> 500]] Dynamic[ Plot[Log[Demand[t]/Supply[t]], {t, .0000001, 5}, ImageSize -> 500]] {Slider[Dynamic[tp], {0, 29, .1}, ImageSize -> 1300], Dynamic[tp], "= propogation time tp", Slider[Dynamic[th], {0, 2, .001}], ImageSize -> 1300, Dynamic[th], "= effective bitcoin supply"} {Dynamic[ Plot[Demand[t]/Supply[t], {t, .0000001, 5}, ImageSize -> 500]], Dynamic[Plot[Log[Demand[t]/Supply[t]], {t, .0000001, 5}, ImageSize -> 500]]}*)

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[…] I have constructed a log plot and fitted to two parameters, as I have done before, just to demonstrate that since bitcoin grows in powers of 10, (seemingly large) contemporary fluctuations may look alarming. This rests on the observation I and perhaps a few others have made, that the bitcoin price follows a log linear growth model, on average, according to what we may expect from the diffusion equation. […]