Stochastic Modeling of Bitcoin prices using Monte Carlo prediction

Approach

To predict today's closing price st, we take the previous day's closing price sy and raise multiply it by e^r where r is a long term drift plus a random offset.

st = sy * e^(drift + offset)

To calculate the long term drift, we use the historical prices. In this demo, we're using the prices of Bitcoin, which are readily available from several sources.

We use daily data downloaded from: http://api.bitcoincharts.com/v1/csv/

PDR - Periodic Daily Return

ln(Today's closing price / Yesterday's closing price)

We use three other figures based on this PDR

• Mean
• Variance
• Standard Deviation

Drift

This is calculated like so: mean - (variance/2) and represents the thrust of where the price is heading

Stochastic offset

We're adding an offset to the drift. This is selected at random from within the normal distribution. The ppf() (percent point function) from scipy.stats allows to us match an percentage of the area under the Gaussian distribution graph to a standard deviation. stats.ppf(0.95) will return 1.645, which means that 1.645 standard deviations from the mean will contain 95% of the space

Sample output

Graph showing how the price could fluctuate across time periods (here in 10 second intervals) based on historical trends and stochastic fluctuations (as in Brownian motion)

TODO

At first glance, this doesn't seem to model Bitcoin's very volatile price fluctuations. Use an ML-style approach to testing and tuning; i.e. take 60% of the data as a 'training set', and use the other 40% (the more recent prices) as the test set.

Challenges

The historical data available for free is not awesome. The chaps at bitcoincharts have helpfully provided this stuff for free, but the sample is 'every few seconds', which ranges from sub-second to 15 seconds. This means that our predications are going to be a bit out of goose.

Also, the data is not necessarily up to date, and this implementation is pretty naive, so don't use the forcasted prices to plan your whole investment strategy :)

Further reading

https://www.youtube.com/watch?v=3gcLRU24-w0 - this short video explains the maths behind what we're doing here with basic Brownian montecarlo

http://www.turingfinance.com/random-walks-down-wall-street-stochastic-processes-in-python/ - This excellent article goes into a lot more depth Brownian and why more sophisticated algorithms like Merton Jump and Heston are better models for financial modelling