# How to Simulate Future Price Developments

In addition to the fundamental analysis, which aims to find the real value of an asset independently of its market value, and the technical analysis, the aim of which is to determine future market sentiment, there is a third position for valuing market prices. The efficient market hypothesis is based on the assumption that the current market price of an investment is always the correct one, i.e. that the fundamental value or any market sentiment has already been priced in. A radical interpretation of the efficient market hypothesis therefore takes the view that the market is unbeatable.

## Brownian movement

Even though this third position has been criticized by various sides, it is a good basis for the estimation of long-term developments – here it is true that the market cannot really be estimated on such a long-term basis. This is why statistical methods are used. The basis of the analysis is a so-called random walk model. Here, a random course of the price is calculated. Such approaches have already been tested for Bitcoin and other Altcoins. It also served as an inspiration aid for the specially developed application mentioned above.

For a meaningful estimation it is considered how high the volatility was in the past and where the market has moved overall. The so-called geometric Brownian movement, which can be described in the following sentence, is often used for this purpose: In a Brownian movement, a next step in a random process depends only on the most recent previous values.

Formula friends would express the proposition as follows:

ΔP = P* (µΔt + σϵ(Δt)^0.5)

P is the price, µ is an expected increase (the so-called return), σ is its volatility (which corresponds to the standard deviation of the return), ϵ is a pure random variable and Δt is time. A price change ΔP is thus dependent on time, the old price, a drift and a volatility.

That’s enough theory for now, let’s start applying it! We consider the price development of Bitcoin as a classic example:

The chart shows the Bitcoin price development between 2012 and 2018. In order to be able to study as much of the development as possible, a logarithmic application was chosen. Some important variables can be determined from this price movement:

Return = log (price(t)/price(t-1))
Close = Course(today)
σ(day) = standard deviation(return)
Daily drift = average (Return)

## Return is the profit made every day

Close is Bitcoin’s price today, which will also be the basis for further simulations. The standard deviation is the dispersion of the profit and a measure of the volatility of this investment. Finally, the drift is the mean value of the daily profits made. With the help of these variables and the formula given above, the price movement of Bitoin can be simulated over a period of one year. Since ϵ is a random variable (where the term Random Walk comes from), each simulation calculates a different possible future. To illustrate this, three simulated price curves are shown:

We see that the same formula can generate extremely different price curves using the random element ϵ . Running through a simulation will apparently say little about future price trends. Is it possible to say how a large part of the simulations will behave?

Monte Carlo – it doesn’t have to be just a simulation
We are a bit insane in favour of good statistics and carry out a million simulations. We orientate ourselves on Monte Carlo processes. Admittedly, these are not as complete as the above price trends, since we are only concerned with the question of how Bitcoin will develop in one year. To calculate this, we will upscale the parameters drift and volatility to 365 days:

Drift per year = Daily drift * 365
Volatility per year = σ(day)*365^0.5

With this annual drift and volatility, the following formula can now be applied:

P(year) = Close*exp((drift per year-0.5*volatility per year^2)+volatility per year*ϵ)

If this is now calculated several times, a histogram can give us information about how often which prices were calculated at the end of a year. A logarithmic representation of the histogram was chosen so that the wide range of calculated final prices can be viewed:

We see that most simulations calculate a price of 17,078 US dollars after one year. We also see that the distribution is normally distributed in a logarithmic representation. This lets us say something more: we can look at the standard deviation around this mean. This is quite an interesting measure, since it is known that 68% of the results described by a normal distribution are within this standard deviation.

Thus